Research Article Optimization of Allocation and Launch Conditions of Multiple Missiles for Three-Dimensional Collaborative Interception of Ballistic Targets Burak Yuksek and N. Kemal Ure Aerospace Research Center, Department of Aeronautical Engineering, Istanbul Technical University, 34469 Istanbul, Turkey Correspondence should be addressed to N. Kemal Ure; [email protected] Received 12 April 2016; Accepted 13 July 2016 Academic Editor: Mahmut Reyhanoglu Copyright © 2016 B. Yuksek and N. K. Ure. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the integrated problem of allocation and control of surface-to-air-missiles for interception of ballistic targets. Previous work shows that using multiple missile and utilizing collaborative estimation and control laws for target interception can significantly decrease the mean miss distance. However, most of these methods are highly sensitive to initial launch conditions, such as the initial pitch and heading angles. In this work we develop a methodology for optimizing selection of multiple missiles to launch among a collection of missiles with prespecified launch coordinates, along with their launch conditions. For the interception we use 3-DoF models for missiles and the ballistic target. e trajectory of the missiles is controlled using three-dimensional extensions of existing algorithms for planar collaborative control and estimation laws. Because the dynamics of the missiles and nature of the allocation problem is highly nonlinear and involves both discrete and continuous variables, the optimization problem is cast as a mixed integer nonlinear programming problem (MINP). e main contribution of this work is the development of a novel probabilistic search algorithm for efficiently solving the missile allocation problem. We verify the algorithm by performing extensive Monte-Carlo simulations on different interception scenarios and show that the developed approach yields significantly less average miss distance and more efficient use of resources compared to alternative methods. 1. Introduction Missile systems form an integral part of a nation’s defensive capability. In particular, high precision control of surface- to-air-missiles (SAMs) is a key technology for intercepting maneuvering targets, such as hostile aircraſt and ballistic missiles [1]. Successful mid-air interception of such targets might prevent significant losses and enable preparing for counter-attacks. Hence, there has been a significant amount of previous work that focuses on development of guidance, control, and estimation laws for interception of maneuvering targets by a single missile in the past years [2]. Intercepting ballistic targets is generally more challenging compared to interception of static targets or targets executing simple maneuvers (such as an aircraſt in level flight), due to lack of precise dynamic models and high reentry speed of the target. Hence, even the smallest perturbations in the guidance errors might translate into large miss distances for the interception of ballistic targets. In order to reduce the miss distance, there has been a growing interest in using multiple collaborating missiles for the interception of a single target. ere are two main advantages for using multiple missiles; (1) probability to hit increases due to increased number of missiles targeted at the hostile; (2) missiles can fuse their measurements to improve the estimation of the target’s state, which results in increased guidance performance and hence reduced miss distance. at being said, the initial conditions of the launched missiles still have a significant impact on their overall per- formance. Ballistic target’s total speed is usually greater than the interceptors speed; hence the launching coordinates of the SAMs directly affect the miss distance. In addition, launching pitch and heading angles of the SAMs are also critical for the guidance performance. Even though errors in these angles are Hindawi Publishing Corporation International Journal of Aerospace Engineering Volume 2016, Article ID 9582816, 14 pages http://dx.doi.org/10.1155/2016/9582816
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Research ArticleOptimization of Allocation and Launch Conditions ofMultiple Missiles for Three-Dimensional CollaborativeInterception of Ballistic Targets
Burak Yuksek and N Kemal Ure
Aerospace Research Center Department of Aeronautical Engineering Istanbul Technical University 34469 Istanbul Turkey
Correspondence should be addressed to N Kemal Ure ureituedutr
Received 12 April 2016 Accepted 13 July 2016
Academic Editor Mahmut Reyhanoglu
Copyright copy 2016 B Yuksek and N K Ure This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We consider the integrated problem of allocation and control of surface-to-air-missiles for interception of ballistic targetsPrevious work shows that using multiple missile and utilizing collaborative estimation and control laws for target interception cansignificantly decrease themeanmiss distance However most of thesemethods are highly sensitive to initial launch conditions suchas the initial pitch and heading angles In this workwe develop amethodology for optimizing selection ofmultiplemissiles to launchamong a collection of missiles with prespecified launch coordinates along with their launch conditions For the interception we use3-DoF models for missiles and the ballistic target The trajectory of the missiles is controlled using three-dimensional extensionsof existing algorithms for planar collaborative control and estimation laws Because the dynamics of the missiles and nature ofthe allocation problem is highly nonlinear and involves both discrete and continuous variables the optimization problem is castas a mixed integer nonlinear programming problem (MINP) The main contribution of this work is the development of a novelprobabilistic search algorithm for efficiently solving themissile allocation problemWe verify the algorithmby performing extensiveMonte-Carlo simulations on different interception scenarios and show that the developed approach yields significantly less averagemiss distance and more efficient use of resources compared to alternative methods
1 Introduction
Missile systems form an integral part of a nationrsquos defensivecapability In particular high precision control of surface-to-air-missiles (SAMs) is a key technology for interceptingmaneuvering targets such as hostile aircraft and ballisticmissiles [1] Successful mid-air interception of such targetsmight prevent significant losses and enable preparing forcounter-attacks Hence there has been a significant amountof previous work that focuses on development of guidancecontrol and estimation laws for interception of maneuveringtargets by a single missile in the past years [2]
Intercepting ballistic targets is generallymore challengingcompared to interception of static targets or targets executingsimple maneuvers (such as an aircraft in level flight) due tolack of precise dynamic models and high reentry speed of thetarget Hence even the smallest perturbations in the guidance
errors might translate into large miss distances for theinterception of ballistic targets In order to reduce the missdistance there has been a growing interest in using multiplecollaborating missiles for the interception of a single targetThere are two main advantages for using multiple missiles(1) probability to hit increases due to increased number ofmissiles targeted at the hostile (2) missiles can fuse theirmeasurements to improve the estimation of the targetrsquos statewhich results in increased guidance performance and hencereduced miss distance
That being said the initial conditions of the launchedmissiles still have a significant impact on their overall per-formance Ballistic targetrsquos total speed is usually greater thanthe interceptors speed hence the launching coordinates of theSAMs directly affect themiss distance In addition launchingpitch and heading angles of the SAMs are also critical for theguidance performance Even though errors in these angles are
Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2016 Article ID 9582816 14 pageshttpdxdoiorg10115520169582816
2 International Journal of Aerospace Engineering
compensated by the missile autopilots in later phases of theflight due to high kinetic energy of the ballistic target having1-2 degrees off in the initial pitch and heading angles of theSAMs might lead to significant deviations from the desiredperformance even in the presence of restoring actions of theautopilotHence the selection of the launching coordinates ofthe SAMs along with their initial launching angles should beadjusted carefully to gain optimal collaborative missile inter-ception performance The main contribution of this paper isthe development of an optimization algorithm that drives theinitial conditions of collaborative guidance and estimationlaws In particular the missile launching parameters and thenumber of launchedmissiles are selected using a probabilisticsearch algorithm which attempts to optimize an objectivefunction that favors minimum miss distance and maximumefficiency in use of resources
11 Previous Work on Collaborative Estimation and Controlfor Missiles There has been a significant amount of previouswork on control and estimation of multiple missiles Chenand Speyery [3] formulated the multiple missile coordina-tion problem as a Linear Exponential Gaussian Differentialgame and applied their algorithm to interception of ballisticmissiles in terminal and boost phase Wang and Fu [4]formulated the multiple missile interception problem asa multiplayer pursuit and evasion game and studied theinterception of a ballistic target in three dimensions Jeon etal [5] developed a cooperative proportional navigation (PN)law that enables multiple missiles to close simultaneouslyon a stationary target Similarly Daughtery and Qu [6]also developed an algorithm for multiple missiles attackingsimultaneously a target and they also showed that theiralgorithm is robust to communication losses between themissiles
Shaferman and Oshman [7] developed an extendedKalman filter (EKF) algorithm that fuses information gath-ered from multiple interceptors They were able to showthat using cooperative estimation algorithms yield improvedguidance and control performance Shaferman and Shima [8]combined adaptive control laws and multiple model filteringalgorithms for collaborative interception Liu et al [9] alsoconsidered using multiple EKFs to improve guidance per-formance in particular they discovered that the estimationperformance improves as the relative line of sight (LOS)between two intercepting missiles gets larger and designeda control law that enforces separation between two missilesRecently Shaferman and Shima [10] also developed guidancelaws for enforcing relative intercept angles Another recentdevelopment was provided by Wang et al [11] where theauthors used a probabilistic framework to maximize hit-to-kill probability for two-missile cooperative interception
These previous works showed that fusing the estimationprocess between multiple missiles almost certainly leadsto improved control performance and hence reduced missdistance Most of these works only study the scenario wheremissiles are about to close on the target the launch conditionsare largely ignored However as explained in the beginningof this section estimation and control performance mightvary significantly under different launch conditions such as
different initial heading and elevation angles of the missilesMoreover launch conditions could also require adjustmentsbased on relative heading and velocity of the ballistic target Inparticular the sensitivity of miss distance based on differentconditions would be higher for targets with high speeds suchas ballistic missiles
12 Previous Work on Missile Allocation The existing algo-rithms for multiple missile interception assume that a fixednumber of missiles have been assigned for the target inter-ception In real scenarios usually a larger set of missilesare available on the ground and a certain subset of thesemissiles should be allocated and launched for interceptionDeciding on which missiles to allocate depends on severalfactors such as the altitude and the velocity of the targetSince increasing the number of allocated missiles improvesthe estimation performance and hence the probability ofsuccessful interception it might be desirable to launch manymissiles as possible However launching more missiles thannecessary would result in inefficient use of resources hence atrade-off exists between the kill probability and the numberof launched missiles
To the best of our knowledge there are no previous worksthat directly address the allocation problem for multiplemissiles that utilize collaborative estimation and controlalgorithms However previous researches on weapon andtask allocationmethods are certainly relevant to this problemand they are reviewed in this subsection
Weapon and task allocation algorithms generally addressthe problem of assigning a finite number of weapons to mul-tiple targets The search space for such problems are usuallyhuge and the objective functions lack useful properties suchas linearity or convexity hence evolutionary algorithms andprobabilistic search approaches are very popular for theseproblems Lee et al [12] used a genetic algorithm based ongreedy eugenics to solve the assignment problem whereasTeng et al [13] utilized particle swarm optimization approachfor the coordinated air combat missile-target assignmentAnt colony optimization [14] and simulated annealing [15]have also been applied to the target assignment problemBesides genetic algorithms game theoretic [16] and rule-based [17] approaches have also been applied to missileallocation problems
These existing works either completely ignore the vehicledynamics or use simplified missile and target models forsolving the allocation problem However utilization of col-laborative estimation and control algorithms is integral to theinterception of ballistic target problem studied in this paperhence the dynamics of estimation and control algorithmscannot be ignored by the allocation algorithm Existingworksusually remedy this problem by assigning a probability ofsuccess to each possible assignment however overestimatingor underestimating these probabilities might lead to seriousperformance problems
13 Overview and Contribution of This Work The mainidea behind the developed algorithm is to cast the missileallocation problem as a mixed integer nonlinear program(MINP) where the discrete decision variables correspond to
International Journal of Aerospace Engineering 3
selection of which missiles to launch and the continuousdecision variables correspond to launch conditions such aselevation and heading angles for each missile In order tosolve this optimization problem a novel probabilistic searchmethod is developed which extends the well-known contin-uous optimization algorithm Covariance Matrix Adaptation(CMA) [18] to solve problems that contain both continuousand discrete variablesTheoptimization problem also embedsthe full dynamics of the missiles where we have extended anexisting collaborative EKF algorithm [7 10] to work in three-dimensional settings
With respect to the state of the art our algorithm offersthe following contributions
(i) The previous work on missile control assumes fixedinitial launch conditions for collaborative targets Ouralgorithm automates the process of selecting missileheading and elevation angles which impacts theestimation and control performance in later stagesand hence reduces the average miss distance
(ii) Besides optimizing the launch conditions the devel-oped algorithm also handles the selection of thesubset ofmissiles for interception among the availablelarger set of missiles by examining the trade-offbetween the missile launch cost and achievable missdistance This feature is captured in MINP formula-tion and the results show that the developed algo-rithm yields lesser miss distances using lesser numberof missiles compared to alternative approaches
(iii) The developed optimization algorithm works bypropagating missile and target dynamics equippedwith estimation and control laws Hence unlike exist-ing missile allocation algorithms there are no sim-plifications or approximations in dynamical modelswhich translates into better prediction of overallsystem performance
(iv) As a minor contribution we would also like tonote that most existing collaborative missile controlpapers consider two-dimensional (planar) missileand target dynamics In this work we have extendedthe collaborative EKF to work in three dimensionsAlthough this is a straightforward extension havingthe results in three dimensions leads to more realisticmiss distances and hence enables a better assessmentof the fitness of the developed algorithms on realbattlefield operations
The paper is structured as follows In Section 2 theballistic target interception problem is formulated and themissilemodes for both the interceptors and the ballistic targetare provided Section 3 explains the collaborative estimationand control algorithms used in this work and Section 4gives the details of the novel optimization algorithm usedfor solving the allocation problem Finally Section 5 studiesthe performance of the algorithm by examining the results ofMonte-Carlo simulations that involve various different targetinitial conditions
2 Problem Formulation and Missile Models
In this section we provide the complete formulation of theoptimization problem for the multiple missile ballistic targetinterception Since the problem formulation involves thedynamics of themissiles themodel of the interceptormissilesand the ballistic target are also given in this section
21 Multiple Missile Ballistic Target Interception Problem Letx
119879= [p
119879 p
119879] isin R6 denote the state vector of the ballistic
target and let x119894
119872= [p119894
119872 p119894
119872] isin R6 119894 = 1 2 119873SAM denote
the state vectors of the SAMs where p denotes the positionp denotes the velocity and 119873SAM denotes the number ofmissiles It is assumed that SAMs initial positions are fixed atthe launch site and the ballistic target is detected by a radar inits terminal phase We assume that closed loop guidance andcontrol algorithms are implemented on SAMs (see Section 3)and the decision maker only needs to decide on the initiallaunching conditions
Let Γ0= 120574
119894(0) 119894 = 1 119873SAM and Ψ
0= 120595
119894(0) 119894 =
1 119873SAM denote the initial launching pitch and headingangles for each SAM Let 119905
119891be the terminal time for the
scenario Define the miss distance as
miss (Γ0 Ψ
0) = min
119894=1119873SAM119905isin[0119905119891]
10038171003817100381710038171003817p
119879 (119905) minus p119894
119872(119905)10038171003817100381710038171003817 (1)
In other words for a given set of launch conditions missdistance is computed by propagating the dynamics of theSAMs and ballistic target over the horizon [0 119905
119891] and finding
the minimum distance achieved between the ballistic targetand all launched SAMs
The objective of the multiple missile ballistic targetinterception problem is to minimize the miss distance byoptimizing over the launch conditions Γ0
Ψ0
minΓ0Ψ0
miss (Γ0 Ψ
0)
subject to x119879= 119891
119879(x
119879)
x119894
119872= 119891
119872(x119894
119879) 119894 = 1 119873SAM
(2)
where 119891119879and 119891
119872are the dynamical models of the ballistic
target and interceptor missiles (see Section 22) For the sakeof simplicity we assume that all SAMs share the same model
Although the optimization problem above reflects themain objective of the mission it assumes that all of the SAMswill be used for interception However in many scenariosonly a strict subset of the available SAMs might be sufficientfor successful interception Hence we should modify theobjective function to reflect the cost of launching a missile inorder to yield solutions that minimize themiss distance whileavoiding inefficient use of resources
Let z = 119911119894isin 0 1 119894 = 1 119873SAM denote the binary
decision vector where 119911119894= 1 means that SAM 119894 is launched
for intercepting the ballistic target and 119911119894= 0 means that
SAM 119894 is not launched and will stay in its initial position
4 International Journal of Aerospace Engineering
throughout the missionThe modified optimization problemis formulated as
minzΓ0Ψ0
miss (Γ0 Ψ
0) +
119873SAM
sum
119894=1
119911119894119903
subject to x119879= 119891
119879(x
119879)
x119894
119872= 119891
119872(x119894
119879) for 119911
119894= 1
x119894
119872= 0 for 119911
119894= 0
(3)
where 119903 gt 0 is a user defined parameter that reflects the costfor launching a missile Larger values of 119903 favor launchinglesser number of missiles Since the optimization problemin (3) contains both continuous variables Γ0
Ψ0 and discrete
variables z and since the dynamics119891119872and119891
119879are nonlinear it
can be classified as a nonlinear mixed integer programming(MINP) problem MINP are known to be very challengingto solve and the major contribution of the paper is thedevelopment of a probabilistic search method (explained inSection 4) for solving the optimization problem in (3)
Note that we are only interested in deciding the allocationof the missiles and their launch conditions and no furthertrajectory optimization is needed since it is assumed thatSAMs have closed loop estimation and guidance laws embed-ded in their dynamics These estimation and control lawsare further detailed in Section 3 However before examiningthe control laws we describe the open loop dynamics of theSAMs and ballistic target in the next subsection
22 Missile Models
221 Interceptor SAM Model The SAM interceptor usedin this study is a 3-degree-of-freedom (DoF) point massmodel of amissile which is controlled via aerodynamic forcesgenerated by the finsThree-dimensional equations ofmotionof the interceptor can be stated as shown in [2]
= 119881119898cos 120574 cos120595
= 119881119898cos 120574 sin120595
= 119881119898sin 120574
= (119886pitch minus cos 120574) (119892
119881119898
)
= (
119886yaw
cos 120574)(
119892
119881119898
)
(4)
where 119909 is the downrange displacement of the SAM 119910 isthe cross-range displacement of the SAM 119911 is the altitudeof the SAM 119892 is the gravity acceleration 120574 is the flight pathangle 119881
119898is SAM velocity and 119886pitch 119886yaw are the vertical and
horizontal load factors
222 Ballistic Target Model In the 3D interception scenarioSCUD-B short range ballistic missile (SRBM) is used asthe incoming ballistic threat Simulation starts from a given
Ze
Fd
T
g
Ye
O Xe
120595T
120574T
VT
Figure 1 Forces acting on the ballistic target model
apogee altitude velocity reentry angle and heading Aero-dynamic data of the SCUD-B is obtained by using MissileDATCOM software [19]
The drag force and the gravity acceleration componentsthat act on the ballistic target during reentry phase are shownin Figure 1 Drag force acts in the opposite direction of thevelocity vector Hence if the effect of the drag force is greaterthan that of gravity the ballistic target decelerates duringits flight The perpendicular component of the decelerationvector to the line of sight is shown as a target maneuver tothe pursuing interceptor Thus the interceptor guidance lawmust take this deceleration maneuver of the ballistic threatinto account [20]
The approximate mathematical model of the target in thereentry phase is given in [20]
119879119909
=minus119865
119889
119898cos (120595
119879) cos (120574
119879)
119879119910
=119865
119889
119898sin (120595
119879) cos (120574
119879)
119879119911
=119865
119889
119898cos (120595
119879) sin (120574
119879) minus 119892
(5)
where 119865119889is the drag force 119881
119879119909 119881
119879119910 and 119881
119879119911are the velocity
components119898 is the targetmissilemass in the reentry phase119892 is the gravity acceleration 120574
119879is the reentry angle and 120595
119879is
the heading angle
3 Control and Estimation Algorithms
31 RadarMeasurement for Initialization For initialization ofEoMs of the missiles and the estimation process a groundbased radar is used The radar is fixed in a prespecifiedposition Relative positions of the interceptors and the targetaccording to the radar in the 3D space are shown in Figure 2
International Journal of Aerospace Engineering 5
Ze
Xe
Ye
120588ti120588tR
VtT120574tT
120595tT
antT
120582ti
120582tR
120595tR
120595ti
mi(xti yti zti)
T(xtT ytT ztT)
R(xtR ytR ztR)
Figure 2 Radar geometry
Ze
Xe
Ye
120588t1 120588t2120588tn
VtT120574tT
120595tT
antT
T(xtT ytT ztT)
xt2 yt2 zt2)
120582t1 120582t2
120582tn
120595t1
120595t2
120595tn
m1(xt1 yt1 zt1) mn(xtn ytn ztn)
m2(
Figure 3 3D engagement geometry
We assume that the radar estimates the initial state vector
0119877without any error and send the initialization data to the
interceptors without any delay
x0119877
= [1205880119877120582
0119877120574
0119879119886
0119879] (6)
The geometric relation between the 119894th interceptor andthe target is given in (7) In this work it is assumed that 120574
0119879
and 1198860119879
are estimated perfectly by the radar
1205880119894= radic(119909
0119879minus 119909
0119894)
2+ (119910
0119879minus 119910
0119894)
2+ (119911
0119879minus 119911
0119894)
2
1205820119894= arctan[[
[
1199110119879
minus 1199110119894
radic(1199090119879
minus 1199090119894)
2+ (119910
0119879minus 119910
0119894)
2
]]
]
(7)
32 Relative Kinematics 3D engagement geometry betweenthe target and interceptors is shown in Figure 3 It is assumedthat each interceptormissile canmeasure its own inertial statevector as shown in (8) and measurements are exact
x119868
119905119894= [119909119905119894
119910119905119894
119911119905119894
120574119905119894
120595119905119894]
119879 (8)
The state vector of the 119894thmissile according to the target attime 119905 is given in (9) where 119894 = 1 2 119899 In the informationsharing mode each missile could transmit and receive theestimated target data without any loss and delay
x119877
119905119894= [120588119905119894
120582119905119894
120574119905119879
119886119905119879]
119879 (9)
Relative kinematics between the interceptors and theballisticmissile in 119878
119909119911 119878
119909119910 and 119878
119910119911planes are defined by using
119909119911119894
= 119881120588119909119911
119909119911119894
=
119881120582119909119911119894
120588119909119911119894
119879119909119911
= minus
119886119879119909119911
119881119879119909119911
119879119909119911
= 119892
119886119879119909119911
119881119879119909119911
sin (120574119879119909119911)
119909119910119894
= 119881120588119909119910
119909119910119894
=
119881120582119909119910119894
120588119909119910119894
119879119909119910
= 0
119879119909119910
= 0
119910119911119894
= 119881120588119910119911
119910119911119894
=
119881120582119910119911119894
120588119910119911119894
119879119910119911
= minus
119886119879119910119911
119881119879119910119911
119879119910119911
= 119892
119886119879119910119911
119881119879119910119911
sin (120574119879119910119911)
(10)
where
119881120588119909119911
= minus119881119898119909119911
cos (120574119898119909119911
minus 120582119909119911) + 119881
119879119909119911cos (120582
119909119911minus 120574
119879119909119911)
119881120582119909119911
= minus119881119898119909119911
sin (120574119898119909119911
minus 120582119909119911) minus 119881
119879119909119911sin (120582
119909119911minus 120574
119879119909119911)
119881120588119909119910
= minus119881119898119909119910
cos (120574119898119909119910
minus 120582119909119910)
+ 119881119879119909119910
cos (120582119909119910minus 120574
119879119909119910)
119881120582119909119910
= minus119881119898119909119910
sin (120574119898119909119910
minus 120582119909119910)
minus 119881119879119909119910
sin (120582119909119910minus 120574
119879119909119910)
119881120588119910119911
= minus119881119898119910119911
cos (120574119898119910119911
minus 120582119910119911)
+ 119881119879119910119911
cos (120582119910119911minus 120574
119879119910119911)
6 International Journal of Aerospace Engineering
119881120582119910119911
= minus119881119898119910119911
sin (120574119898119910119911
minus 120582119910119911)
minus 119881119879119910119911
sin (120582119910119911minus 120574
119879119910119911)
(11)
The relative kinematics equations can be described in thediscrete-time by using
x119896= 119891
119896minus1(119909
119896minus1) (12)
where x119896is state vector of the 119894th missile at time 119905
119896and 119891
119896minus1is
obtained by integrating the relative kinematics EoMs in (10)[7]
33 Measurement Model Each interceptor is equipped withthe infrared seeker that measures the line of sight (LOS)angle 120582
119896119894Thismeasurement has a zero-meanGaussian noise
with standard deviation 120590120582119894 LOS angle measurements are
performed by each interceptor missile separately therefore119864(V
119896119894 V
119896119895) = 0 forall119894 = 119895 The 119894th missile LOS measurement is
119911119896119894= ℎ
119896119894(119909
119896) + V
119896119894 (13)
where
V119896119894sim alefsym (0 120590
2
120582119894) (14)
The interceptor missiles can operate in two modes The firstmode is information nonsharing mode In this mode eachinterceptor measures only its own LOS angle and uses it inthe estimation process of the relative statesThemeasurementvector in the information nonsharingmode is given as shownin (13)
The second mode is information sharing mode in whicheach interceptor not only measures its own LOS angle butalso calculates the LOS angle of the other interceptors asshown in (15) [7] Also to improve the estimation qualityeach interceptor calculates the range-to-go distance by usingposition and LOS angle data as shown in (16)
119911120582119894119895
=
[[[[[[[
[
1199111205821198941
1199111205821198942
119911120582119894119899
]]]]]]]
]
(15)
119911120588119894119895
=
[[[[[[[
[
1199111205881198941
1199111205881198942
119911120588119894119899
]]]]]]]
]
(16)
Here 119894 is the missile which performs measurement and 119899is the missile observed by the missile 119894
The LOS angle of the missile 119895 measured by the missile 119894is calculated by using
119911120582119894119895
= arctan120588
119894sin 120582
119894minus (119911
119895minus 119911
119894)
120588119894cos 120582
119894+ (119909
119894minus 119909
119895)
(17)
In addition the range-to-go of the 119895th missile measuredby the 119894th missile is obtained by using
119911120588119894119895
= radic(120588119894sin 120582
119894minus (119911
119895minus 119911
119894))
2
+ (120588119894cos 120582
119894+ (119909
119894minus 119909
119895))
2
(18)
In the information sharing mode the measurementmodel is given by
z119894119895= h (119909
119894119895) + k
119894119895 (19)
where
z119894119895= [119911120588(119894119895)
119911120582(119894119895)]
119879
k119894119895= [V120588(119894119895)
V120582(119894119895)]
119879
(20)
34 ExtendedKalmanFilter Because of the nonlinear relativekinematics between the interceptors and the ballistic targetan extended Kalman filter (EKF) is used for estimatingthe unmeasured data and to filter the noisy LOS anglemeasurements [7] The prediction error covariance matrix isgiven in
P119896|119896minus1
= 120601119896|119896minus1
P119896minus1|119896minus1120601
119879
119896|119896minus1+Q
119896minus1 (21)
where
120601119896|119896minus1
= 119890F119896minus1119879
cong I + F119896minus1
119879 (22)
is the transition matrix associated with the relative kinemat-ics 119879 is sampling time and I is the identity matrix withappropriate dimensionsQ
119896minus1is the covariance matrix of the
equivalent discrete process noise and it is calculated as shownin
Q119896minus1
= int
119879
0
120601119896|119896minus1
Q120601119879119896|119896minus1
119889119879 (23)
F119896minus1
is the Jacobian matrix associated with the nonlinearrelative kinematics
F119896minus1
=120597119891 (119909)
120597119909
10038161003816100381610038161003816100381610038161003816119909=119896minus1
(24)
The state estimation is performed by using
x119896|119896
= x119896|119896minus1
+ K119896[z
119896minus h
119896x
119896|119896minus1] (25)
where x119896|119896minus1
is the predicted state vector andK is the Kalmangain as shown in
x119896|119896minus1
= 120601119896|119896minus1
x119896minus1|119896minus1
K = P119896|119896minus1
H119879
119896[H
119896P
119896|119896minus1H119879
119896+ R
119896]
minus1
(26)
Here H119896is the measurement Jacobian matrix and R
119896is the
measurement noise covariancematrixThe covariancematrixis updated as shown in
P119896|119896
= [I minus K119896H
119896]P
119896|119896minus1[I minus K
119896H
119896]119879+ K
119896R
119896K119879
119896 (27)
International Journal of Aerospace Engineering 7
Z
X
nc119909119911
Vm119909119911
120588xz
VT119909119911
120574T119909119911
T
120574m119909119911120582xz
M
Figure 4 Relative kinematics on 119878119909119911plane
Y
X
nc119909y
Vm119909y
120588xy
VT119909y
120574T119909y
T
120574m119909y
120582xy
M
Figure 5 Relative kinematics on 119878119909119910
plane
35 3D True Proportional Navigation Algorithm The pro-portional algorithm is one of the most common and effectiveguidance techniques because of its simple structure andimplementation The true proportional navigation (TPN)system generates the acceleration command perpendicular tothe LOS As shown in (29) the acceleration command is afunction of closing velocity 119881
119888and LOS rate
119899119888= 119873119881
119888 (28)
where 119899119888is the acceleration command perpendicular to the
LOS 119881119888is closing velocity and 119873 is navigation ratio which
is generally between 3 and 5 [20] In this 3D interceptionstudy TPN algorithm is applied for 119878
119909119911 119878
119909119910 and 119878
119910119911planes
separately [21] Geometry of relative kinematics for eachdifferent plane is displayed in Figures 4 5 and 6
Acceleration commands in 119878119909119911 119878
119909119910 and 119878
119910119911are obtained
as shown in
119899119888119909119911
= 119873119881119888119909119911
119909119911
119899119888119909119910
= 119873119881119888119909119910
119909119910
119899119888119910119911
= 119873119881119888119910119911
119910119911
(29)
Z
Y
ncyz
Vmyz
120588yz
VTyz
120574Tyz
T
120574myz120582yz
M
Figure 6 Relative kinematics on 119878119910119911
plane
The acceleration components of the interceptor in the 119909- 119910-and 119911-axis (119886
119898119909 119886
119898119910 119886
119898119911) can be obtained from (30) by using
the trigonometric relations
119886119898119909
= minus119899119888119909119910
sin (120582119909119910) minus 119899
119888119909119911sin (120582
119909119911)
119886119898119910
= minus119899119888119909119910
cos (120582119909119910) minus 119899
119888119910119911sin (120582
119910119911)
119886119898119911
= 119899119888119909119911
cos (120582119909119911) + 119899
119888119910119911cos (120582
119910119911)
(30)
Before applying the control commands to the interceptorvertical and horizontal components 119886pitch and 119886yaw should becalculated Here 119886pitch is in the pitch plane and perpendicularto the velocity vector of the interceptor and 119886yaw is perpendic-ular to both velocity vector and vertical acceleration vectorFor TPN these acceleration components are calculated using
119886pitch = 119886119898119911
cos (120574119898) minus 119886
119898119909sin (120574
119898) minus 119892 cos (120574
119898)
119886yaw = 119886119898119910
cos (120595119898) minus 119886
119898119909sin (120595
119898)
(31)
4 Optimization Algorithm
41 CMA Algorithm Consider the general form uncon-strained optimization problem
minimize 119891 (x) 119891 R119899997888rarr R (32)
It is well known that when 119891 possess a certain structure(such as being continuous linear or convex) there arevariety of local search algorithms that can be applied tosolve this optimization problem efficiently However when119891 does not possess these desirable properties local searchmethods either fail to find an answer or get stuck in localminima Global search methods [22] remedy this problem bygeneralizing the search over the entire state space Althoughglobal methods can also exploit the structure of 119891 manyglobal methods treat 119891 as a black box function and hencethe solution is found entirely by examining the input-outputpairs (x 119891(x))
8 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899rarr R Number of Samples per Iteration 120582 Number of Iterations 119899iter Weights 119908
1 119908
120583
Output Approximate optimal solution xlowast
Initialization(1) m
0 120590
0 C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895 119888
1larr 2119899
2 119888120583larr 120583
119908119899
2 119888minus1
120590larr 1198993 119888minus1
119888larr 1198994 120572 larr 15 120583 larr 1205822
(2) while 119896 lt 119899iter do(3) for 119894 in 1 120582 do
Sample Candidate Solutions
(4) x119894sim N(m 120590
2
119896C
119896) 119891
119894larr 119891(x
119894)
Sort the Candidate Solutions Based on Their Cost
(5) x1120582
larr x119905(1) x
119905(120582) such that 119891
119905(1)le sdot sdot sdot le 119891
119905(120582)
Move the mean to low cost solutions
(6) m119896+1
larr m119896+ sum
120583
119894=1119908
119894(x
119894minusm
119896)
Update Evolution Path Variables
(7) p120590larr (1 minus 119888
120590)p
120590+ radic1 minus (1 minus 119888
120590)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(8) 120590119896+1
larr 120590119896times exp (119888
120590(p
120590EN(0 I) minus 1))
Update The Covariance Matrix
(9) if p120590 lt 120572radic(119899) then
(10) 119889119896larr 1
(11) else(12) 119889
119896larr 0
(13) 119888119904larr (1 minus 119889
2
119896)119888
1119888119888(2 minus 119888
119888)
p119888larr (1 minus 119888
119888)p
119888+ 119889
119896radic(1 minus 119888
119888)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(14) C119896+1
larr (1 minus 1198881minus 119888
120583+ 119888
119904)C
119896+ 119888
1p⊤
119888p
119888+ 119888
120583sum
120583
119894=1119908
119894((x
119894minusm
119896)120590
119896)((x
119894minusm
119896)120590
119896)⊤
(15) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(16) xlowastlarr x
119905(1)
Algorithm 1 Covariance Matrix Adaptation (CMA)
Covariance Matrix Adaptation (CMA) [18] is a popularglobal search method that usually ranks among the bestsolvers in global search benchmarks [23] The basic ideabehind CMA is to place a multivariate normal distributionover the search space R119899 and sample candidate solutions(x
119894 119891(x
119894)) from this distributionThemean vector and covari-
ance matrix of the distribution are incrementally updatedat each step based on the values of the sampled solutionsThe objective is to eventually steer the mean vector to theoptimal solution xlowast and shrink the covariance matrix toidentity matrix hence in the limit the distribution will yieldthe optimal solution when it is sampled
For completeness we provide the pseudocode for theCMA algorithm in Algorithm 1 taken from [24] The algo-rithm starts by initializing its internal parameters (Line (1))At the 119896th iteration the algorithm samples 120582 number ofsamples from amultivariate Gaussian distribution withmeanm
119896and covariance C
119896(Line (4)) Next the samples x
119894are
sorted according to their costs119891119894Theweighted average of top
120583 number of solutions is computed to find the mean vectorm
119896+1for the next iteration (Line (6)) which moves the mean
of the distribution towards samples with lower costs Nextalgorithm updates the covariance matrix with the help of theevolution path variables which are p
120590(Line (7)) and p
119888(Line
(13)) which ensures that the adaptation steps are conjugatedirectionsThe interested reader is referred to [24] for the fullderivation of the algorithm and the intuition for updating thepath parameters
42 CMA-MV Algorithm Unfortunately CMA algorithm isonly applicable to continuous optimization problems hencewe cannot use it to solve the missile launch condition settingproblem given in (3) since the allocation of missiles isdetermined by the integer variables
To overcome this issue we develop a novel algorithmnamed Covariance Matrix Adaptation with Mixed Variables(CMA-MV) which extends the classical CMA algorithm towork on nonlinearmixed integer optimization problemsThegeneric nonlinear mixed integer programming problem is ofthe form
minimize 119891 (x z) 119891 R119899times Z
119889997888rarr R (33)
where Z is the set of integers The special case we areinterested in is the problem where the discrete variable z isa binary vector that is z isin 0 1
119889 This is also the case forthe missile allocation problem in (3) where 119911
119894= 1 refers to
missile 119894 being launchedThe main idea behind the CMA-MV algorithm is to
define and update two probability distributions for samplingcontinuous and discrete variables For the continuous vari-ables x we use a multivariate normal distribution and we usethe exact same procedure followed in the CMA algorithm(Algorithm 1) to update the mean and covariance of thedistribution For the discrete variables we use a multivariateBernoulli distribution and update the mean and covarianceof this distribution based on the costs of sampled variables
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
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2 International Journal of Aerospace Engineering
compensated by the missile autopilots in later phases of theflight due to high kinetic energy of the ballistic target having1-2 degrees off in the initial pitch and heading angles of theSAMs might lead to significant deviations from the desiredperformance even in the presence of restoring actions of theautopilotHence the selection of the launching coordinates ofthe SAMs along with their initial launching angles should beadjusted carefully to gain optimal collaborative missile inter-ception performance The main contribution of this paper isthe development of an optimization algorithm that drives theinitial conditions of collaborative guidance and estimationlaws In particular the missile launching parameters and thenumber of launchedmissiles are selected using a probabilisticsearch algorithm which attempts to optimize an objectivefunction that favors minimum miss distance and maximumefficiency in use of resources
11 Previous Work on Collaborative Estimation and Controlfor Missiles There has been a significant amount of previouswork on control and estimation of multiple missiles Chenand Speyery [3] formulated the multiple missile coordina-tion problem as a Linear Exponential Gaussian Differentialgame and applied their algorithm to interception of ballisticmissiles in terminal and boost phase Wang and Fu [4]formulated the multiple missile interception problem asa multiplayer pursuit and evasion game and studied theinterception of a ballistic target in three dimensions Jeon etal [5] developed a cooperative proportional navigation (PN)law that enables multiple missiles to close simultaneouslyon a stationary target Similarly Daughtery and Qu [6]also developed an algorithm for multiple missiles attackingsimultaneously a target and they also showed that theiralgorithm is robust to communication losses between themissiles
Shaferman and Oshman [7] developed an extendedKalman filter (EKF) algorithm that fuses information gath-ered from multiple interceptors They were able to showthat using cooperative estimation algorithms yield improvedguidance and control performance Shaferman and Shima [8]combined adaptive control laws and multiple model filteringalgorithms for collaborative interception Liu et al [9] alsoconsidered using multiple EKFs to improve guidance per-formance in particular they discovered that the estimationperformance improves as the relative line of sight (LOS)between two intercepting missiles gets larger and designeda control law that enforces separation between two missilesRecently Shaferman and Shima [10] also developed guidancelaws for enforcing relative intercept angles Another recentdevelopment was provided by Wang et al [11] where theauthors used a probabilistic framework to maximize hit-to-kill probability for two-missile cooperative interception
These previous works showed that fusing the estimationprocess between multiple missiles almost certainly leadsto improved control performance and hence reduced missdistance Most of these works only study the scenario wheremissiles are about to close on the target the launch conditionsare largely ignored However as explained in the beginningof this section estimation and control performance mightvary significantly under different launch conditions such as
different initial heading and elevation angles of the missilesMoreover launch conditions could also require adjustmentsbased on relative heading and velocity of the ballistic target Inparticular the sensitivity of miss distance based on differentconditions would be higher for targets with high speeds suchas ballistic missiles
12 Previous Work on Missile Allocation The existing algo-rithms for multiple missile interception assume that a fixednumber of missiles have been assigned for the target inter-ception In real scenarios usually a larger set of missilesare available on the ground and a certain subset of thesemissiles should be allocated and launched for interceptionDeciding on which missiles to allocate depends on severalfactors such as the altitude and the velocity of the targetSince increasing the number of allocated missiles improvesthe estimation performance and hence the probability ofsuccessful interception it might be desirable to launch manymissiles as possible However launching more missiles thannecessary would result in inefficient use of resources hence atrade-off exists between the kill probability and the numberof launched missiles
To the best of our knowledge there are no previous worksthat directly address the allocation problem for multiplemissiles that utilize collaborative estimation and controlalgorithms However previous researches on weapon andtask allocationmethods are certainly relevant to this problemand they are reviewed in this subsection
Weapon and task allocation algorithms generally addressthe problem of assigning a finite number of weapons to mul-tiple targets The search space for such problems are usuallyhuge and the objective functions lack useful properties suchas linearity or convexity hence evolutionary algorithms andprobabilistic search approaches are very popular for theseproblems Lee et al [12] used a genetic algorithm based ongreedy eugenics to solve the assignment problem whereasTeng et al [13] utilized particle swarm optimization approachfor the coordinated air combat missile-target assignmentAnt colony optimization [14] and simulated annealing [15]have also been applied to the target assignment problemBesides genetic algorithms game theoretic [16] and rule-based [17] approaches have also been applied to missileallocation problems
These existing works either completely ignore the vehicledynamics or use simplified missile and target models forsolving the allocation problem However utilization of col-laborative estimation and control algorithms is integral to theinterception of ballistic target problem studied in this paperhence the dynamics of estimation and control algorithmscannot be ignored by the allocation algorithm Existingworksusually remedy this problem by assigning a probability ofsuccess to each possible assignment however overestimatingor underestimating these probabilities might lead to seriousperformance problems
13 Overview and Contribution of This Work The mainidea behind the developed algorithm is to cast the missileallocation problem as a mixed integer nonlinear program(MINP) where the discrete decision variables correspond to
International Journal of Aerospace Engineering 3
selection of which missiles to launch and the continuousdecision variables correspond to launch conditions such aselevation and heading angles for each missile In order tosolve this optimization problem a novel probabilistic searchmethod is developed which extends the well-known contin-uous optimization algorithm Covariance Matrix Adaptation(CMA) [18] to solve problems that contain both continuousand discrete variablesTheoptimization problem also embedsthe full dynamics of the missiles where we have extended anexisting collaborative EKF algorithm [7 10] to work in three-dimensional settings
With respect to the state of the art our algorithm offersthe following contributions
(i) The previous work on missile control assumes fixedinitial launch conditions for collaborative targets Ouralgorithm automates the process of selecting missileheading and elevation angles which impacts theestimation and control performance in later stagesand hence reduces the average miss distance
(ii) Besides optimizing the launch conditions the devel-oped algorithm also handles the selection of thesubset ofmissiles for interception among the availablelarger set of missiles by examining the trade-offbetween the missile launch cost and achievable missdistance This feature is captured in MINP formula-tion and the results show that the developed algo-rithm yields lesser miss distances using lesser numberof missiles compared to alternative approaches
(iii) The developed optimization algorithm works bypropagating missile and target dynamics equippedwith estimation and control laws Hence unlike exist-ing missile allocation algorithms there are no sim-plifications or approximations in dynamical modelswhich translates into better prediction of overallsystem performance
(iv) As a minor contribution we would also like tonote that most existing collaborative missile controlpapers consider two-dimensional (planar) missileand target dynamics In this work we have extendedthe collaborative EKF to work in three dimensionsAlthough this is a straightforward extension havingthe results in three dimensions leads to more realisticmiss distances and hence enables a better assessmentof the fitness of the developed algorithms on realbattlefield operations
The paper is structured as follows In Section 2 theballistic target interception problem is formulated and themissilemodes for both the interceptors and the ballistic targetare provided Section 3 explains the collaborative estimationand control algorithms used in this work and Section 4gives the details of the novel optimization algorithm usedfor solving the allocation problem Finally Section 5 studiesthe performance of the algorithm by examining the results ofMonte-Carlo simulations that involve various different targetinitial conditions
2 Problem Formulation and Missile Models
In this section we provide the complete formulation of theoptimization problem for the multiple missile ballistic targetinterception Since the problem formulation involves thedynamics of themissiles themodel of the interceptormissilesand the ballistic target are also given in this section
21 Multiple Missile Ballistic Target Interception Problem Letx
119879= [p
119879 p
119879] isin R6 denote the state vector of the ballistic
target and let x119894
119872= [p119894
119872 p119894
119872] isin R6 119894 = 1 2 119873SAM denote
the state vectors of the SAMs where p denotes the positionp denotes the velocity and 119873SAM denotes the number ofmissiles It is assumed that SAMs initial positions are fixed atthe launch site and the ballistic target is detected by a radar inits terminal phase We assume that closed loop guidance andcontrol algorithms are implemented on SAMs (see Section 3)and the decision maker only needs to decide on the initiallaunching conditions
Let Γ0= 120574
119894(0) 119894 = 1 119873SAM and Ψ
0= 120595
119894(0) 119894 =
1 119873SAM denote the initial launching pitch and headingangles for each SAM Let 119905
119891be the terminal time for the
scenario Define the miss distance as
miss (Γ0 Ψ
0) = min
119894=1119873SAM119905isin[0119905119891]
10038171003817100381710038171003817p
119879 (119905) minus p119894
119872(119905)10038171003817100381710038171003817 (1)
In other words for a given set of launch conditions missdistance is computed by propagating the dynamics of theSAMs and ballistic target over the horizon [0 119905
119891] and finding
the minimum distance achieved between the ballistic targetand all launched SAMs
The objective of the multiple missile ballistic targetinterception problem is to minimize the miss distance byoptimizing over the launch conditions Γ0
Ψ0
minΓ0Ψ0
miss (Γ0 Ψ
0)
subject to x119879= 119891
119879(x
119879)
x119894
119872= 119891
119872(x119894
119879) 119894 = 1 119873SAM
(2)
where 119891119879and 119891
119872are the dynamical models of the ballistic
target and interceptor missiles (see Section 22) For the sakeof simplicity we assume that all SAMs share the same model
Although the optimization problem above reflects themain objective of the mission it assumes that all of the SAMswill be used for interception However in many scenariosonly a strict subset of the available SAMs might be sufficientfor successful interception Hence we should modify theobjective function to reflect the cost of launching a missile inorder to yield solutions that minimize themiss distance whileavoiding inefficient use of resources
Let z = 119911119894isin 0 1 119894 = 1 119873SAM denote the binary
decision vector where 119911119894= 1 means that SAM 119894 is launched
for intercepting the ballistic target and 119911119894= 0 means that
SAM 119894 is not launched and will stay in its initial position
4 International Journal of Aerospace Engineering
throughout the missionThe modified optimization problemis formulated as
minzΓ0Ψ0
miss (Γ0 Ψ
0) +
119873SAM
sum
119894=1
119911119894119903
subject to x119879= 119891
119879(x
119879)
x119894
119872= 119891
119872(x119894
119879) for 119911
119894= 1
x119894
119872= 0 for 119911
119894= 0
(3)
where 119903 gt 0 is a user defined parameter that reflects the costfor launching a missile Larger values of 119903 favor launchinglesser number of missiles Since the optimization problemin (3) contains both continuous variables Γ0
Ψ0 and discrete
variables z and since the dynamics119891119872and119891
119879are nonlinear it
can be classified as a nonlinear mixed integer programming(MINP) problem MINP are known to be very challengingto solve and the major contribution of the paper is thedevelopment of a probabilistic search method (explained inSection 4) for solving the optimization problem in (3)
Note that we are only interested in deciding the allocationof the missiles and their launch conditions and no furthertrajectory optimization is needed since it is assumed thatSAMs have closed loop estimation and guidance laws embed-ded in their dynamics These estimation and control lawsare further detailed in Section 3 However before examiningthe control laws we describe the open loop dynamics of theSAMs and ballistic target in the next subsection
22 Missile Models
221 Interceptor SAM Model The SAM interceptor usedin this study is a 3-degree-of-freedom (DoF) point massmodel of amissile which is controlled via aerodynamic forcesgenerated by the finsThree-dimensional equations ofmotionof the interceptor can be stated as shown in [2]
= 119881119898cos 120574 cos120595
= 119881119898cos 120574 sin120595
= 119881119898sin 120574
= (119886pitch minus cos 120574) (119892
119881119898
)
= (
119886yaw
cos 120574)(
119892
119881119898
)
(4)
where 119909 is the downrange displacement of the SAM 119910 isthe cross-range displacement of the SAM 119911 is the altitudeof the SAM 119892 is the gravity acceleration 120574 is the flight pathangle 119881
119898is SAM velocity and 119886pitch 119886yaw are the vertical and
horizontal load factors
222 Ballistic Target Model In the 3D interception scenarioSCUD-B short range ballistic missile (SRBM) is used asthe incoming ballistic threat Simulation starts from a given
Ze
Fd
T
g
Ye
O Xe
120595T
120574T
VT
Figure 1 Forces acting on the ballistic target model
apogee altitude velocity reentry angle and heading Aero-dynamic data of the SCUD-B is obtained by using MissileDATCOM software [19]
The drag force and the gravity acceleration componentsthat act on the ballistic target during reentry phase are shownin Figure 1 Drag force acts in the opposite direction of thevelocity vector Hence if the effect of the drag force is greaterthan that of gravity the ballistic target decelerates duringits flight The perpendicular component of the decelerationvector to the line of sight is shown as a target maneuver tothe pursuing interceptor Thus the interceptor guidance lawmust take this deceleration maneuver of the ballistic threatinto account [20]
The approximate mathematical model of the target in thereentry phase is given in [20]
119879119909
=minus119865
119889
119898cos (120595
119879) cos (120574
119879)
119879119910
=119865
119889
119898sin (120595
119879) cos (120574
119879)
119879119911
=119865
119889
119898cos (120595
119879) sin (120574
119879) minus 119892
(5)
where 119865119889is the drag force 119881
119879119909 119881
119879119910 and 119881
119879119911are the velocity
components119898 is the targetmissilemass in the reentry phase119892 is the gravity acceleration 120574
119879is the reentry angle and 120595
119879is
the heading angle
3 Control and Estimation Algorithms
31 RadarMeasurement for Initialization For initialization ofEoMs of the missiles and the estimation process a groundbased radar is used The radar is fixed in a prespecifiedposition Relative positions of the interceptors and the targetaccording to the radar in the 3D space are shown in Figure 2
International Journal of Aerospace Engineering 5
Ze
Xe
Ye
120588ti120588tR
VtT120574tT
120595tT
antT
120582ti
120582tR
120595tR
120595ti
mi(xti yti zti)
T(xtT ytT ztT)
R(xtR ytR ztR)
Figure 2 Radar geometry
Ze
Xe
Ye
120588t1 120588t2120588tn
VtT120574tT
120595tT
antT
T(xtT ytT ztT)
xt2 yt2 zt2)
120582t1 120582t2
120582tn
120595t1
120595t2
120595tn
m1(xt1 yt1 zt1) mn(xtn ytn ztn)
m2(
Figure 3 3D engagement geometry
We assume that the radar estimates the initial state vector
0119877without any error and send the initialization data to the
interceptors without any delay
x0119877
= [1205880119877120582
0119877120574
0119879119886
0119879] (6)
The geometric relation between the 119894th interceptor andthe target is given in (7) In this work it is assumed that 120574
0119879
and 1198860119879
are estimated perfectly by the radar
1205880119894= radic(119909
0119879minus 119909
0119894)
2+ (119910
0119879minus 119910
0119894)
2+ (119911
0119879minus 119911
0119894)
2
1205820119894= arctan[[
[
1199110119879
minus 1199110119894
radic(1199090119879
minus 1199090119894)
2+ (119910
0119879minus 119910
0119894)
2
]]
]
(7)
32 Relative Kinematics 3D engagement geometry betweenthe target and interceptors is shown in Figure 3 It is assumedthat each interceptormissile canmeasure its own inertial statevector as shown in (8) and measurements are exact
x119868
119905119894= [119909119905119894
119910119905119894
119911119905119894
120574119905119894
120595119905119894]
119879 (8)
The state vector of the 119894thmissile according to the target attime 119905 is given in (9) where 119894 = 1 2 119899 In the informationsharing mode each missile could transmit and receive theestimated target data without any loss and delay
x119877
119905119894= [120588119905119894
120582119905119894
120574119905119879
119886119905119879]
119879 (9)
Relative kinematics between the interceptors and theballisticmissile in 119878
119909119911 119878
119909119910 and 119878
119910119911planes are defined by using
119909119911119894
= 119881120588119909119911
119909119911119894
=
119881120582119909119911119894
120588119909119911119894
119879119909119911
= minus
119886119879119909119911
119881119879119909119911
119879119909119911
= 119892
119886119879119909119911
119881119879119909119911
sin (120574119879119909119911)
119909119910119894
= 119881120588119909119910
119909119910119894
=
119881120582119909119910119894
120588119909119910119894
119879119909119910
= 0
119879119909119910
= 0
119910119911119894
= 119881120588119910119911
119910119911119894
=
119881120582119910119911119894
120588119910119911119894
119879119910119911
= minus
119886119879119910119911
119881119879119910119911
119879119910119911
= 119892
119886119879119910119911
119881119879119910119911
sin (120574119879119910119911)
(10)
where
119881120588119909119911
= minus119881119898119909119911
cos (120574119898119909119911
minus 120582119909119911) + 119881
119879119909119911cos (120582
119909119911minus 120574
119879119909119911)
119881120582119909119911
= minus119881119898119909119911
sin (120574119898119909119911
minus 120582119909119911) minus 119881
119879119909119911sin (120582
119909119911minus 120574
119879119909119911)
119881120588119909119910
= minus119881119898119909119910
cos (120574119898119909119910
minus 120582119909119910)
+ 119881119879119909119910
cos (120582119909119910minus 120574
119879119909119910)
119881120582119909119910
= minus119881119898119909119910
sin (120574119898119909119910
minus 120582119909119910)
minus 119881119879119909119910
sin (120582119909119910minus 120574
119879119909119910)
119881120588119910119911
= minus119881119898119910119911
cos (120574119898119910119911
minus 120582119910119911)
+ 119881119879119910119911
cos (120582119910119911minus 120574
119879119910119911)
6 International Journal of Aerospace Engineering
119881120582119910119911
= minus119881119898119910119911
sin (120574119898119910119911
minus 120582119910119911)
minus 119881119879119910119911
sin (120582119910119911minus 120574
119879119910119911)
(11)
The relative kinematics equations can be described in thediscrete-time by using
x119896= 119891
119896minus1(119909
119896minus1) (12)
where x119896is state vector of the 119894th missile at time 119905
119896and 119891
119896minus1is
obtained by integrating the relative kinematics EoMs in (10)[7]
33 Measurement Model Each interceptor is equipped withthe infrared seeker that measures the line of sight (LOS)angle 120582
119896119894Thismeasurement has a zero-meanGaussian noise
with standard deviation 120590120582119894 LOS angle measurements are
performed by each interceptor missile separately therefore119864(V
119896119894 V
119896119895) = 0 forall119894 = 119895 The 119894th missile LOS measurement is
119911119896119894= ℎ
119896119894(119909
119896) + V
119896119894 (13)
where
V119896119894sim alefsym (0 120590
2
120582119894) (14)
The interceptor missiles can operate in two modes The firstmode is information nonsharing mode In this mode eachinterceptor measures only its own LOS angle and uses it inthe estimation process of the relative statesThemeasurementvector in the information nonsharingmode is given as shownin (13)
The second mode is information sharing mode in whicheach interceptor not only measures its own LOS angle butalso calculates the LOS angle of the other interceptors asshown in (15) [7] Also to improve the estimation qualityeach interceptor calculates the range-to-go distance by usingposition and LOS angle data as shown in (16)
119911120582119894119895
=
[[[[[[[
[
1199111205821198941
1199111205821198942
119911120582119894119899
]]]]]]]
]
(15)
119911120588119894119895
=
[[[[[[[
[
1199111205881198941
1199111205881198942
119911120588119894119899
]]]]]]]
]
(16)
Here 119894 is the missile which performs measurement and 119899is the missile observed by the missile 119894
The LOS angle of the missile 119895 measured by the missile 119894is calculated by using
119911120582119894119895
= arctan120588
119894sin 120582
119894minus (119911
119895minus 119911
119894)
120588119894cos 120582
119894+ (119909
119894minus 119909
119895)
(17)
In addition the range-to-go of the 119895th missile measuredby the 119894th missile is obtained by using
119911120588119894119895
= radic(120588119894sin 120582
119894minus (119911
119895minus 119911
119894))
2
+ (120588119894cos 120582
119894+ (119909
119894minus 119909
119895))
2
(18)
In the information sharing mode the measurementmodel is given by
z119894119895= h (119909
119894119895) + k
119894119895 (19)
where
z119894119895= [119911120588(119894119895)
119911120582(119894119895)]
119879
k119894119895= [V120588(119894119895)
V120582(119894119895)]
119879
(20)
34 ExtendedKalmanFilter Because of the nonlinear relativekinematics between the interceptors and the ballistic targetan extended Kalman filter (EKF) is used for estimatingthe unmeasured data and to filter the noisy LOS anglemeasurements [7] The prediction error covariance matrix isgiven in
P119896|119896minus1
= 120601119896|119896minus1
P119896minus1|119896minus1120601
119879
119896|119896minus1+Q
119896minus1 (21)
where
120601119896|119896minus1
= 119890F119896minus1119879
cong I + F119896minus1
119879 (22)
is the transition matrix associated with the relative kinemat-ics 119879 is sampling time and I is the identity matrix withappropriate dimensionsQ
119896minus1is the covariance matrix of the
equivalent discrete process noise and it is calculated as shownin
Q119896minus1
= int
119879
0
120601119896|119896minus1
Q120601119879119896|119896minus1
119889119879 (23)
F119896minus1
is the Jacobian matrix associated with the nonlinearrelative kinematics
F119896minus1
=120597119891 (119909)
120597119909
10038161003816100381610038161003816100381610038161003816119909=119896minus1
(24)
The state estimation is performed by using
x119896|119896
= x119896|119896minus1
+ K119896[z
119896minus h
119896x
119896|119896minus1] (25)
where x119896|119896minus1
is the predicted state vector andK is the Kalmangain as shown in
x119896|119896minus1
= 120601119896|119896minus1
x119896minus1|119896minus1
K = P119896|119896minus1
H119879
119896[H
119896P
119896|119896minus1H119879
119896+ R
119896]
minus1
(26)
Here H119896is the measurement Jacobian matrix and R
119896is the
measurement noise covariancematrixThe covariancematrixis updated as shown in
P119896|119896
= [I minus K119896H
119896]P
119896|119896minus1[I minus K
119896H
119896]119879+ K
119896R
119896K119879
119896 (27)
International Journal of Aerospace Engineering 7
Z
X
nc119909119911
Vm119909119911
120588xz
VT119909119911
120574T119909119911
T
120574m119909119911120582xz
M
Figure 4 Relative kinematics on 119878119909119911plane
Y
X
nc119909y
Vm119909y
120588xy
VT119909y
120574T119909y
T
120574m119909y
120582xy
M
Figure 5 Relative kinematics on 119878119909119910
plane
35 3D True Proportional Navigation Algorithm The pro-portional algorithm is one of the most common and effectiveguidance techniques because of its simple structure andimplementation The true proportional navigation (TPN)system generates the acceleration command perpendicular tothe LOS As shown in (29) the acceleration command is afunction of closing velocity 119881
119888and LOS rate
119899119888= 119873119881
119888 (28)
where 119899119888is the acceleration command perpendicular to the
LOS 119881119888is closing velocity and 119873 is navigation ratio which
is generally between 3 and 5 [20] In this 3D interceptionstudy TPN algorithm is applied for 119878
119909119911 119878
119909119910 and 119878
119910119911planes
separately [21] Geometry of relative kinematics for eachdifferent plane is displayed in Figures 4 5 and 6
Acceleration commands in 119878119909119911 119878
119909119910 and 119878
119910119911are obtained
as shown in
119899119888119909119911
= 119873119881119888119909119911
119909119911
119899119888119909119910
= 119873119881119888119909119910
119909119910
119899119888119910119911
= 119873119881119888119910119911
119910119911
(29)
Z
Y
ncyz
Vmyz
120588yz
VTyz
120574Tyz
T
120574myz120582yz
M
Figure 6 Relative kinematics on 119878119910119911
plane
The acceleration components of the interceptor in the 119909- 119910-and 119911-axis (119886
119898119909 119886
119898119910 119886
119898119911) can be obtained from (30) by using
the trigonometric relations
119886119898119909
= minus119899119888119909119910
sin (120582119909119910) minus 119899
119888119909119911sin (120582
119909119911)
119886119898119910
= minus119899119888119909119910
cos (120582119909119910) minus 119899
119888119910119911sin (120582
119910119911)
119886119898119911
= 119899119888119909119911
cos (120582119909119911) + 119899
119888119910119911cos (120582
119910119911)
(30)
Before applying the control commands to the interceptorvertical and horizontal components 119886pitch and 119886yaw should becalculated Here 119886pitch is in the pitch plane and perpendicularto the velocity vector of the interceptor and 119886yaw is perpendic-ular to both velocity vector and vertical acceleration vectorFor TPN these acceleration components are calculated using
119886pitch = 119886119898119911
cos (120574119898) minus 119886
119898119909sin (120574
119898) minus 119892 cos (120574
119898)
119886yaw = 119886119898119910
cos (120595119898) minus 119886
119898119909sin (120595
119898)
(31)
4 Optimization Algorithm
41 CMA Algorithm Consider the general form uncon-strained optimization problem
minimize 119891 (x) 119891 R119899997888rarr R (32)
It is well known that when 119891 possess a certain structure(such as being continuous linear or convex) there arevariety of local search algorithms that can be applied tosolve this optimization problem efficiently However when119891 does not possess these desirable properties local searchmethods either fail to find an answer or get stuck in localminima Global search methods [22] remedy this problem bygeneralizing the search over the entire state space Althoughglobal methods can also exploit the structure of 119891 manyglobal methods treat 119891 as a black box function and hencethe solution is found entirely by examining the input-outputpairs (x 119891(x))
8 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899rarr R Number of Samples per Iteration 120582 Number of Iterations 119899iter Weights 119908
1 119908
120583
Output Approximate optimal solution xlowast
Initialization(1) m
0 120590
0 C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895 119888
1larr 2119899
2 119888120583larr 120583
119908119899
2 119888minus1
120590larr 1198993 119888minus1
119888larr 1198994 120572 larr 15 120583 larr 1205822
(2) while 119896 lt 119899iter do(3) for 119894 in 1 120582 do
Sample Candidate Solutions
(4) x119894sim N(m 120590
2
119896C
119896) 119891
119894larr 119891(x
119894)
Sort the Candidate Solutions Based on Their Cost
(5) x1120582
larr x119905(1) x
119905(120582) such that 119891
119905(1)le sdot sdot sdot le 119891
119905(120582)
Move the mean to low cost solutions
(6) m119896+1
larr m119896+ sum
120583
119894=1119908
119894(x
119894minusm
119896)
Update Evolution Path Variables
(7) p120590larr (1 minus 119888
120590)p
120590+ radic1 minus (1 minus 119888
120590)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(8) 120590119896+1
larr 120590119896times exp (119888
120590(p
120590EN(0 I) minus 1))
Update The Covariance Matrix
(9) if p120590 lt 120572radic(119899) then
(10) 119889119896larr 1
(11) else(12) 119889
119896larr 0
(13) 119888119904larr (1 minus 119889
2
119896)119888
1119888119888(2 minus 119888
119888)
p119888larr (1 minus 119888
119888)p
119888+ 119889
119896radic(1 minus 119888
119888)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(14) C119896+1
larr (1 minus 1198881minus 119888
120583+ 119888
119904)C
119896+ 119888
1p⊤
119888p
119888+ 119888
120583sum
120583
119894=1119908
119894((x
119894minusm
119896)120590
119896)((x
119894minusm
119896)120590
119896)⊤
(15) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(16) xlowastlarr x
119905(1)
Algorithm 1 Covariance Matrix Adaptation (CMA)
Covariance Matrix Adaptation (CMA) [18] is a popularglobal search method that usually ranks among the bestsolvers in global search benchmarks [23] The basic ideabehind CMA is to place a multivariate normal distributionover the search space R119899 and sample candidate solutions(x
119894 119891(x
119894)) from this distributionThemean vector and covari-
ance matrix of the distribution are incrementally updatedat each step based on the values of the sampled solutionsThe objective is to eventually steer the mean vector to theoptimal solution xlowast and shrink the covariance matrix toidentity matrix hence in the limit the distribution will yieldthe optimal solution when it is sampled
For completeness we provide the pseudocode for theCMA algorithm in Algorithm 1 taken from [24] The algo-rithm starts by initializing its internal parameters (Line (1))At the 119896th iteration the algorithm samples 120582 number ofsamples from amultivariate Gaussian distribution withmeanm
119896and covariance C
119896(Line (4)) Next the samples x
119894are
sorted according to their costs119891119894Theweighted average of top
120583 number of solutions is computed to find the mean vectorm
119896+1for the next iteration (Line (6)) which moves the mean
of the distribution towards samples with lower costs Nextalgorithm updates the covariance matrix with the help of theevolution path variables which are p
120590(Line (7)) and p
119888(Line
(13)) which ensures that the adaptation steps are conjugatedirectionsThe interested reader is referred to [24] for the fullderivation of the algorithm and the intuition for updating thepath parameters
42 CMA-MV Algorithm Unfortunately CMA algorithm isonly applicable to continuous optimization problems hencewe cannot use it to solve the missile launch condition settingproblem given in (3) since the allocation of missiles isdetermined by the integer variables
To overcome this issue we develop a novel algorithmnamed Covariance Matrix Adaptation with Mixed Variables(CMA-MV) which extends the classical CMA algorithm towork on nonlinearmixed integer optimization problemsThegeneric nonlinear mixed integer programming problem is ofthe form
minimize 119891 (x z) 119891 R119899times Z
119889997888rarr R (33)
where Z is the set of integers The special case we areinterested in is the problem where the discrete variable z isa binary vector that is z isin 0 1
119889 This is also the case forthe missile allocation problem in (3) where 119911
119894= 1 refers to
missile 119894 being launchedThe main idea behind the CMA-MV algorithm is to
define and update two probability distributions for samplingcontinuous and discrete variables For the continuous vari-ables x we use a multivariate normal distribution and we usethe exact same procedure followed in the CMA algorithm(Algorithm 1) to update the mean and covariance of thedistribution For the discrete variables we use a multivariateBernoulli distribution and update the mean and covarianceof this distribution based on the costs of sampled variables
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 3
selection of which missiles to launch and the continuousdecision variables correspond to launch conditions such aselevation and heading angles for each missile In order tosolve this optimization problem a novel probabilistic searchmethod is developed which extends the well-known contin-uous optimization algorithm Covariance Matrix Adaptation(CMA) [18] to solve problems that contain both continuousand discrete variablesTheoptimization problem also embedsthe full dynamics of the missiles where we have extended anexisting collaborative EKF algorithm [7 10] to work in three-dimensional settings
With respect to the state of the art our algorithm offersthe following contributions
(i) The previous work on missile control assumes fixedinitial launch conditions for collaborative targets Ouralgorithm automates the process of selecting missileheading and elevation angles which impacts theestimation and control performance in later stagesand hence reduces the average miss distance
(ii) Besides optimizing the launch conditions the devel-oped algorithm also handles the selection of thesubset ofmissiles for interception among the availablelarger set of missiles by examining the trade-offbetween the missile launch cost and achievable missdistance This feature is captured in MINP formula-tion and the results show that the developed algo-rithm yields lesser miss distances using lesser numberof missiles compared to alternative approaches
(iii) The developed optimization algorithm works bypropagating missile and target dynamics equippedwith estimation and control laws Hence unlike exist-ing missile allocation algorithms there are no sim-plifications or approximations in dynamical modelswhich translates into better prediction of overallsystem performance
(iv) As a minor contribution we would also like tonote that most existing collaborative missile controlpapers consider two-dimensional (planar) missileand target dynamics In this work we have extendedthe collaborative EKF to work in three dimensionsAlthough this is a straightforward extension havingthe results in three dimensions leads to more realisticmiss distances and hence enables a better assessmentof the fitness of the developed algorithms on realbattlefield operations
The paper is structured as follows In Section 2 theballistic target interception problem is formulated and themissilemodes for both the interceptors and the ballistic targetare provided Section 3 explains the collaborative estimationand control algorithms used in this work and Section 4gives the details of the novel optimization algorithm usedfor solving the allocation problem Finally Section 5 studiesthe performance of the algorithm by examining the results ofMonte-Carlo simulations that involve various different targetinitial conditions
2 Problem Formulation and Missile Models
In this section we provide the complete formulation of theoptimization problem for the multiple missile ballistic targetinterception Since the problem formulation involves thedynamics of themissiles themodel of the interceptormissilesand the ballistic target are also given in this section
21 Multiple Missile Ballistic Target Interception Problem Letx
119879= [p
119879 p
119879] isin R6 denote the state vector of the ballistic
target and let x119894
119872= [p119894
119872 p119894
119872] isin R6 119894 = 1 2 119873SAM denote
the state vectors of the SAMs where p denotes the positionp denotes the velocity and 119873SAM denotes the number ofmissiles It is assumed that SAMs initial positions are fixed atthe launch site and the ballistic target is detected by a radar inits terminal phase We assume that closed loop guidance andcontrol algorithms are implemented on SAMs (see Section 3)and the decision maker only needs to decide on the initiallaunching conditions
Let Γ0= 120574
119894(0) 119894 = 1 119873SAM and Ψ
0= 120595
119894(0) 119894 =
1 119873SAM denote the initial launching pitch and headingangles for each SAM Let 119905
119891be the terminal time for the
scenario Define the miss distance as
miss (Γ0 Ψ
0) = min
119894=1119873SAM119905isin[0119905119891]
10038171003817100381710038171003817p
119879 (119905) minus p119894
119872(119905)10038171003817100381710038171003817 (1)
In other words for a given set of launch conditions missdistance is computed by propagating the dynamics of theSAMs and ballistic target over the horizon [0 119905
119891] and finding
the minimum distance achieved between the ballistic targetand all launched SAMs
The objective of the multiple missile ballistic targetinterception problem is to minimize the miss distance byoptimizing over the launch conditions Γ0
Ψ0
minΓ0Ψ0
miss (Γ0 Ψ
0)
subject to x119879= 119891
119879(x
119879)
x119894
119872= 119891
119872(x119894
119879) 119894 = 1 119873SAM
(2)
where 119891119879and 119891
119872are the dynamical models of the ballistic
target and interceptor missiles (see Section 22) For the sakeof simplicity we assume that all SAMs share the same model
Although the optimization problem above reflects themain objective of the mission it assumes that all of the SAMswill be used for interception However in many scenariosonly a strict subset of the available SAMs might be sufficientfor successful interception Hence we should modify theobjective function to reflect the cost of launching a missile inorder to yield solutions that minimize themiss distance whileavoiding inefficient use of resources
Let z = 119911119894isin 0 1 119894 = 1 119873SAM denote the binary
decision vector where 119911119894= 1 means that SAM 119894 is launched
for intercepting the ballistic target and 119911119894= 0 means that
SAM 119894 is not launched and will stay in its initial position
4 International Journal of Aerospace Engineering
throughout the missionThe modified optimization problemis formulated as
minzΓ0Ψ0
miss (Γ0 Ψ
0) +
119873SAM
sum
119894=1
119911119894119903
subject to x119879= 119891
119879(x
119879)
x119894
119872= 119891
119872(x119894
119879) for 119911
119894= 1
x119894
119872= 0 for 119911
119894= 0
(3)
where 119903 gt 0 is a user defined parameter that reflects the costfor launching a missile Larger values of 119903 favor launchinglesser number of missiles Since the optimization problemin (3) contains both continuous variables Γ0
Ψ0 and discrete
variables z and since the dynamics119891119872and119891
119879are nonlinear it
can be classified as a nonlinear mixed integer programming(MINP) problem MINP are known to be very challengingto solve and the major contribution of the paper is thedevelopment of a probabilistic search method (explained inSection 4) for solving the optimization problem in (3)
Note that we are only interested in deciding the allocationof the missiles and their launch conditions and no furthertrajectory optimization is needed since it is assumed thatSAMs have closed loop estimation and guidance laws embed-ded in their dynamics These estimation and control lawsare further detailed in Section 3 However before examiningthe control laws we describe the open loop dynamics of theSAMs and ballistic target in the next subsection
22 Missile Models
221 Interceptor SAM Model The SAM interceptor usedin this study is a 3-degree-of-freedom (DoF) point massmodel of amissile which is controlled via aerodynamic forcesgenerated by the finsThree-dimensional equations ofmotionof the interceptor can be stated as shown in [2]
= 119881119898cos 120574 cos120595
= 119881119898cos 120574 sin120595
= 119881119898sin 120574
= (119886pitch minus cos 120574) (119892
119881119898
)
= (
119886yaw
cos 120574)(
119892
119881119898
)
(4)
where 119909 is the downrange displacement of the SAM 119910 isthe cross-range displacement of the SAM 119911 is the altitudeof the SAM 119892 is the gravity acceleration 120574 is the flight pathangle 119881
119898is SAM velocity and 119886pitch 119886yaw are the vertical and
horizontal load factors
222 Ballistic Target Model In the 3D interception scenarioSCUD-B short range ballistic missile (SRBM) is used asthe incoming ballistic threat Simulation starts from a given
Ze
Fd
T
g
Ye
O Xe
120595T
120574T
VT
Figure 1 Forces acting on the ballistic target model
apogee altitude velocity reentry angle and heading Aero-dynamic data of the SCUD-B is obtained by using MissileDATCOM software [19]
The drag force and the gravity acceleration componentsthat act on the ballistic target during reentry phase are shownin Figure 1 Drag force acts in the opposite direction of thevelocity vector Hence if the effect of the drag force is greaterthan that of gravity the ballistic target decelerates duringits flight The perpendicular component of the decelerationvector to the line of sight is shown as a target maneuver tothe pursuing interceptor Thus the interceptor guidance lawmust take this deceleration maneuver of the ballistic threatinto account [20]
The approximate mathematical model of the target in thereentry phase is given in [20]
119879119909
=minus119865
119889
119898cos (120595
119879) cos (120574
119879)
119879119910
=119865
119889
119898sin (120595
119879) cos (120574
119879)
119879119911
=119865
119889
119898cos (120595
119879) sin (120574
119879) minus 119892
(5)
where 119865119889is the drag force 119881
119879119909 119881
119879119910 and 119881
119879119911are the velocity
components119898 is the targetmissilemass in the reentry phase119892 is the gravity acceleration 120574
119879is the reentry angle and 120595
119879is
the heading angle
3 Control and Estimation Algorithms
31 RadarMeasurement for Initialization For initialization ofEoMs of the missiles and the estimation process a groundbased radar is used The radar is fixed in a prespecifiedposition Relative positions of the interceptors and the targetaccording to the radar in the 3D space are shown in Figure 2
International Journal of Aerospace Engineering 5
Ze
Xe
Ye
120588ti120588tR
VtT120574tT
120595tT
antT
120582ti
120582tR
120595tR
120595ti
mi(xti yti zti)
T(xtT ytT ztT)
R(xtR ytR ztR)
Figure 2 Radar geometry
Ze
Xe
Ye
120588t1 120588t2120588tn
VtT120574tT
120595tT
antT
T(xtT ytT ztT)
xt2 yt2 zt2)
120582t1 120582t2
120582tn
120595t1
120595t2
120595tn
m1(xt1 yt1 zt1) mn(xtn ytn ztn)
m2(
Figure 3 3D engagement geometry
We assume that the radar estimates the initial state vector
0119877without any error and send the initialization data to the
interceptors without any delay
x0119877
= [1205880119877120582
0119877120574
0119879119886
0119879] (6)
The geometric relation between the 119894th interceptor andthe target is given in (7) In this work it is assumed that 120574
0119879
and 1198860119879
are estimated perfectly by the radar
1205880119894= radic(119909
0119879minus 119909
0119894)
2+ (119910
0119879minus 119910
0119894)
2+ (119911
0119879minus 119911
0119894)
2
1205820119894= arctan[[
[
1199110119879
minus 1199110119894
radic(1199090119879
minus 1199090119894)
2+ (119910
0119879minus 119910
0119894)
2
]]
]
(7)
32 Relative Kinematics 3D engagement geometry betweenthe target and interceptors is shown in Figure 3 It is assumedthat each interceptormissile canmeasure its own inertial statevector as shown in (8) and measurements are exact
x119868
119905119894= [119909119905119894
119910119905119894
119911119905119894
120574119905119894
120595119905119894]
119879 (8)
The state vector of the 119894thmissile according to the target attime 119905 is given in (9) where 119894 = 1 2 119899 In the informationsharing mode each missile could transmit and receive theestimated target data without any loss and delay
x119877
119905119894= [120588119905119894
120582119905119894
120574119905119879
119886119905119879]
119879 (9)
Relative kinematics between the interceptors and theballisticmissile in 119878
119909119911 119878
119909119910 and 119878
119910119911planes are defined by using
119909119911119894
= 119881120588119909119911
119909119911119894
=
119881120582119909119911119894
120588119909119911119894
119879119909119911
= minus
119886119879119909119911
119881119879119909119911
119879119909119911
= 119892
119886119879119909119911
119881119879119909119911
sin (120574119879119909119911)
119909119910119894
= 119881120588119909119910
119909119910119894
=
119881120582119909119910119894
120588119909119910119894
119879119909119910
= 0
119879119909119910
= 0
119910119911119894
= 119881120588119910119911
119910119911119894
=
119881120582119910119911119894
120588119910119911119894
119879119910119911
= minus
119886119879119910119911
119881119879119910119911
119879119910119911
= 119892
119886119879119910119911
119881119879119910119911
sin (120574119879119910119911)
(10)
where
119881120588119909119911
= minus119881119898119909119911
cos (120574119898119909119911
minus 120582119909119911) + 119881
119879119909119911cos (120582
119909119911minus 120574
119879119909119911)
119881120582119909119911
= minus119881119898119909119911
sin (120574119898119909119911
minus 120582119909119911) minus 119881
119879119909119911sin (120582
119909119911minus 120574
119879119909119911)
119881120588119909119910
= minus119881119898119909119910
cos (120574119898119909119910
minus 120582119909119910)
+ 119881119879119909119910
cos (120582119909119910minus 120574
119879119909119910)
119881120582119909119910
= minus119881119898119909119910
sin (120574119898119909119910
minus 120582119909119910)
minus 119881119879119909119910
sin (120582119909119910minus 120574
119879119909119910)
119881120588119910119911
= minus119881119898119910119911
cos (120574119898119910119911
minus 120582119910119911)
+ 119881119879119910119911
cos (120582119910119911minus 120574
119879119910119911)
6 International Journal of Aerospace Engineering
119881120582119910119911
= minus119881119898119910119911
sin (120574119898119910119911
minus 120582119910119911)
minus 119881119879119910119911
sin (120582119910119911minus 120574
119879119910119911)
(11)
The relative kinematics equations can be described in thediscrete-time by using
x119896= 119891
119896minus1(119909
119896minus1) (12)
where x119896is state vector of the 119894th missile at time 119905
119896and 119891
119896minus1is
obtained by integrating the relative kinematics EoMs in (10)[7]
33 Measurement Model Each interceptor is equipped withthe infrared seeker that measures the line of sight (LOS)angle 120582
119896119894Thismeasurement has a zero-meanGaussian noise
with standard deviation 120590120582119894 LOS angle measurements are
performed by each interceptor missile separately therefore119864(V
119896119894 V
119896119895) = 0 forall119894 = 119895 The 119894th missile LOS measurement is
119911119896119894= ℎ
119896119894(119909
119896) + V
119896119894 (13)
where
V119896119894sim alefsym (0 120590
2
120582119894) (14)
The interceptor missiles can operate in two modes The firstmode is information nonsharing mode In this mode eachinterceptor measures only its own LOS angle and uses it inthe estimation process of the relative statesThemeasurementvector in the information nonsharingmode is given as shownin (13)
The second mode is information sharing mode in whicheach interceptor not only measures its own LOS angle butalso calculates the LOS angle of the other interceptors asshown in (15) [7] Also to improve the estimation qualityeach interceptor calculates the range-to-go distance by usingposition and LOS angle data as shown in (16)
119911120582119894119895
=
[[[[[[[
[
1199111205821198941
1199111205821198942
119911120582119894119899
]]]]]]]
]
(15)
119911120588119894119895
=
[[[[[[[
[
1199111205881198941
1199111205881198942
119911120588119894119899
]]]]]]]
]
(16)
Here 119894 is the missile which performs measurement and 119899is the missile observed by the missile 119894
The LOS angle of the missile 119895 measured by the missile 119894is calculated by using
119911120582119894119895
= arctan120588
119894sin 120582
119894minus (119911
119895minus 119911
119894)
120588119894cos 120582
119894+ (119909
119894minus 119909
119895)
(17)
In addition the range-to-go of the 119895th missile measuredby the 119894th missile is obtained by using
119911120588119894119895
= radic(120588119894sin 120582
119894minus (119911
119895minus 119911
119894))
2
+ (120588119894cos 120582
119894+ (119909
119894minus 119909
119895))
2
(18)
In the information sharing mode the measurementmodel is given by
z119894119895= h (119909
119894119895) + k
119894119895 (19)
where
z119894119895= [119911120588(119894119895)
119911120582(119894119895)]
119879
k119894119895= [V120588(119894119895)
V120582(119894119895)]
119879
(20)
34 ExtendedKalmanFilter Because of the nonlinear relativekinematics between the interceptors and the ballistic targetan extended Kalman filter (EKF) is used for estimatingthe unmeasured data and to filter the noisy LOS anglemeasurements [7] The prediction error covariance matrix isgiven in
P119896|119896minus1
= 120601119896|119896minus1
P119896minus1|119896minus1120601
119879
119896|119896minus1+Q
119896minus1 (21)
where
120601119896|119896minus1
= 119890F119896minus1119879
cong I + F119896minus1
119879 (22)
is the transition matrix associated with the relative kinemat-ics 119879 is sampling time and I is the identity matrix withappropriate dimensionsQ
119896minus1is the covariance matrix of the
equivalent discrete process noise and it is calculated as shownin
Q119896minus1
= int
119879
0
120601119896|119896minus1
Q120601119879119896|119896minus1
119889119879 (23)
F119896minus1
is the Jacobian matrix associated with the nonlinearrelative kinematics
F119896minus1
=120597119891 (119909)
120597119909
10038161003816100381610038161003816100381610038161003816119909=119896minus1
(24)
The state estimation is performed by using
x119896|119896
= x119896|119896minus1
+ K119896[z
119896minus h
119896x
119896|119896minus1] (25)
where x119896|119896minus1
is the predicted state vector andK is the Kalmangain as shown in
x119896|119896minus1
= 120601119896|119896minus1
x119896minus1|119896minus1
K = P119896|119896minus1
H119879
119896[H
119896P
119896|119896minus1H119879
119896+ R
119896]
minus1
(26)
Here H119896is the measurement Jacobian matrix and R
119896is the
measurement noise covariancematrixThe covariancematrixis updated as shown in
P119896|119896
= [I minus K119896H
119896]P
119896|119896minus1[I minus K
119896H
119896]119879+ K
119896R
119896K119879
119896 (27)
International Journal of Aerospace Engineering 7
Z
X
nc119909119911
Vm119909119911
120588xz
VT119909119911
120574T119909119911
T
120574m119909119911120582xz
M
Figure 4 Relative kinematics on 119878119909119911plane
Y
X
nc119909y
Vm119909y
120588xy
VT119909y
120574T119909y
T
120574m119909y
120582xy
M
Figure 5 Relative kinematics on 119878119909119910
plane
35 3D True Proportional Navigation Algorithm The pro-portional algorithm is one of the most common and effectiveguidance techniques because of its simple structure andimplementation The true proportional navigation (TPN)system generates the acceleration command perpendicular tothe LOS As shown in (29) the acceleration command is afunction of closing velocity 119881
119888and LOS rate
119899119888= 119873119881
119888 (28)
where 119899119888is the acceleration command perpendicular to the
LOS 119881119888is closing velocity and 119873 is navigation ratio which
is generally between 3 and 5 [20] In this 3D interceptionstudy TPN algorithm is applied for 119878
119909119911 119878
119909119910 and 119878
119910119911planes
separately [21] Geometry of relative kinematics for eachdifferent plane is displayed in Figures 4 5 and 6
Acceleration commands in 119878119909119911 119878
119909119910 and 119878
119910119911are obtained
as shown in
119899119888119909119911
= 119873119881119888119909119911
119909119911
119899119888119909119910
= 119873119881119888119909119910
119909119910
119899119888119910119911
= 119873119881119888119910119911
119910119911
(29)
Z
Y
ncyz
Vmyz
120588yz
VTyz
120574Tyz
T
120574myz120582yz
M
Figure 6 Relative kinematics on 119878119910119911
plane
The acceleration components of the interceptor in the 119909- 119910-and 119911-axis (119886
119898119909 119886
119898119910 119886
119898119911) can be obtained from (30) by using
the trigonometric relations
119886119898119909
= minus119899119888119909119910
sin (120582119909119910) minus 119899
119888119909119911sin (120582
119909119911)
119886119898119910
= minus119899119888119909119910
cos (120582119909119910) minus 119899
119888119910119911sin (120582
119910119911)
119886119898119911
= 119899119888119909119911
cos (120582119909119911) + 119899
119888119910119911cos (120582
119910119911)
(30)
Before applying the control commands to the interceptorvertical and horizontal components 119886pitch and 119886yaw should becalculated Here 119886pitch is in the pitch plane and perpendicularto the velocity vector of the interceptor and 119886yaw is perpendic-ular to both velocity vector and vertical acceleration vectorFor TPN these acceleration components are calculated using
119886pitch = 119886119898119911
cos (120574119898) minus 119886
119898119909sin (120574
119898) minus 119892 cos (120574
119898)
119886yaw = 119886119898119910
cos (120595119898) minus 119886
119898119909sin (120595
119898)
(31)
4 Optimization Algorithm
41 CMA Algorithm Consider the general form uncon-strained optimization problem
minimize 119891 (x) 119891 R119899997888rarr R (32)
It is well known that when 119891 possess a certain structure(such as being continuous linear or convex) there arevariety of local search algorithms that can be applied tosolve this optimization problem efficiently However when119891 does not possess these desirable properties local searchmethods either fail to find an answer or get stuck in localminima Global search methods [22] remedy this problem bygeneralizing the search over the entire state space Althoughglobal methods can also exploit the structure of 119891 manyglobal methods treat 119891 as a black box function and hencethe solution is found entirely by examining the input-outputpairs (x 119891(x))
8 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899rarr R Number of Samples per Iteration 120582 Number of Iterations 119899iter Weights 119908
1 119908
120583
Output Approximate optimal solution xlowast
Initialization(1) m
0 120590
0 C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895 119888
1larr 2119899
2 119888120583larr 120583
119908119899
2 119888minus1
120590larr 1198993 119888minus1
119888larr 1198994 120572 larr 15 120583 larr 1205822
(2) while 119896 lt 119899iter do(3) for 119894 in 1 120582 do
Sample Candidate Solutions
(4) x119894sim N(m 120590
2
119896C
119896) 119891
119894larr 119891(x
119894)
Sort the Candidate Solutions Based on Their Cost
(5) x1120582
larr x119905(1) x
119905(120582) such that 119891
119905(1)le sdot sdot sdot le 119891
119905(120582)
Move the mean to low cost solutions
(6) m119896+1
larr m119896+ sum
120583
119894=1119908
119894(x
119894minusm
119896)
Update Evolution Path Variables
(7) p120590larr (1 minus 119888
120590)p
120590+ radic1 minus (1 minus 119888
120590)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(8) 120590119896+1
larr 120590119896times exp (119888
120590(p
120590EN(0 I) minus 1))
Update The Covariance Matrix
(9) if p120590 lt 120572radic(119899) then
(10) 119889119896larr 1
(11) else(12) 119889
119896larr 0
(13) 119888119904larr (1 minus 119889
2
119896)119888
1119888119888(2 minus 119888
119888)
p119888larr (1 minus 119888
119888)p
119888+ 119889
119896radic(1 minus 119888
119888)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(14) C119896+1
larr (1 minus 1198881minus 119888
120583+ 119888
119904)C
119896+ 119888
1p⊤
119888p
119888+ 119888
120583sum
120583
119894=1119908
119894((x
119894minusm
119896)120590
119896)((x
119894minusm
119896)120590
119896)⊤
(15) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(16) xlowastlarr x
119905(1)
Algorithm 1 Covariance Matrix Adaptation (CMA)
Covariance Matrix Adaptation (CMA) [18] is a popularglobal search method that usually ranks among the bestsolvers in global search benchmarks [23] The basic ideabehind CMA is to place a multivariate normal distributionover the search space R119899 and sample candidate solutions(x
119894 119891(x
119894)) from this distributionThemean vector and covari-
ance matrix of the distribution are incrementally updatedat each step based on the values of the sampled solutionsThe objective is to eventually steer the mean vector to theoptimal solution xlowast and shrink the covariance matrix toidentity matrix hence in the limit the distribution will yieldthe optimal solution when it is sampled
For completeness we provide the pseudocode for theCMA algorithm in Algorithm 1 taken from [24] The algo-rithm starts by initializing its internal parameters (Line (1))At the 119896th iteration the algorithm samples 120582 number ofsamples from amultivariate Gaussian distribution withmeanm
119896and covariance C
119896(Line (4)) Next the samples x
119894are
sorted according to their costs119891119894Theweighted average of top
120583 number of solutions is computed to find the mean vectorm
119896+1for the next iteration (Line (6)) which moves the mean
of the distribution towards samples with lower costs Nextalgorithm updates the covariance matrix with the help of theevolution path variables which are p
120590(Line (7)) and p
119888(Line
(13)) which ensures that the adaptation steps are conjugatedirectionsThe interested reader is referred to [24] for the fullderivation of the algorithm and the intuition for updating thepath parameters
42 CMA-MV Algorithm Unfortunately CMA algorithm isonly applicable to continuous optimization problems hencewe cannot use it to solve the missile launch condition settingproblem given in (3) since the allocation of missiles isdetermined by the integer variables
To overcome this issue we develop a novel algorithmnamed Covariance Matrix Adaptation with Mixed Variables(CMA-MV) which extends the classical CMA algorithm towork on nonlinearmixed integer optimization problemsThegeneric nonlinear mixed integer programming problem is ofthe form
minimize 119891 (x z) 119891 R119899times Z
119889997888rarr R (33)
where Z is the set of integers The special case we areinterested in is the problem where the discrete variable z isa binary vector that is z isin 0 1
119889 This is also the case forthe missile allocation problem in (3) where 119911
119894= 1 refers to
missile 119894 being launchedThe main idea behind the CMA-MV algorithm is to
define and update two probability distributions for samplingcontinuous and discrete variables For the continuous vari-ables x we use a multivariate normal distribution and we usethe exact same procedure followed in the CMA algorithm(Algorithm 1) to update the mean and covariance of thedistribution For the discrete variables we use a multivariateBernoulli distribution and update the mean and covarianceof this distribution based on the costs of sampled variables
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
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Active and Passive Electronic Components
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International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
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Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
4 International Journal of Aerospace Engineering
throughout the missionThe modified optimization problemis formulated as
minzΓ0Ψ0
miss (Γ0 Ψ
0) +
119873SAM
sum
119894=1
119911119894119903
subject to x119879= 119891
119879(x
119879)
x119894
119872= 119891
119872(x119894
119879) for 119911
119894= 1
x119894
119872= 0 for 119911
119894= 0
(3)
where 119903 gt 0 is a user defined parameter that reflects the costfor launching a missile Larger values of 119903 favor launchinglesser number of missiles Since the optimization problemin (3) contains both continuous variables Γ0
Ψ0 and discrete
variables z and since the dynamics119891119872and119891
119879are nonlinear it
can be classified as a nonlinear mixed integer programming(MINP) problem MINP are known to be very challengingto solve and the major contribution of the paper is thedevelopment of a probabilistic search method (explained inSection 4) for solving the optimization problem in (3)
Note that we are only interested in deciding the allocationof the missiles and their launch conditions and no furthertrajectory optimization is needed since it is assumed thatSAMs have closed loop estimation and guidance laws embed-ded in their dynamics These estimation and control lawsare further detailed in Section 3 However before examiningthe control laws we describe the open loop dynamics of theSAMs and ballistic target in the next subsection
22 Missile Models
221 Interceptor SAM Model The SAM interceptor usedin this study is a 3-degree-of-freedom (DoF) point massmodel of amissile which is controlled via aerodynamic forcesgenerated by the finsThree-dimensional equations ofmotionof the interceptor can be stated as shown in [2]
= 119881119898cos 120574 cos120595
= 119881119898cos 120574 sin120595
= 119881119898sin 120574
= (119886pitch minus cos 120574) (119892
119881119898
)
= (
119886yaw
cos 120574)(
119892
119881119898
)
(4)
where 119909 is the downrange displacement of the SAM 119910 isthe cross-range displacement of the SAM 119911 is the altitudeof the SAM 119892 is the gravity acceleration 120574 is the flight pathangle 119881
119898is SAM velocity and 119886pitch 119886yaw are the vertical and
horizontal load factors
222 Ballistic Target Model In the 3D interception scenarioSCUD-B short range ballistic missile (SRBM) is used asthe incoming ballistic threat Simulation starts from a given
Ze
Fd
T
g
Ye
O Xe
120595T
120574T
VT
Figure 1 Forces acting on the ballistic target model
apogee altitude velocity reentry angle and heading Aero-dynamic data of the SCUD-B is obtained by using MissileDATCOM software [19]
The drag force and the gravity acceleration componentsthat act on the ballistic target during reentry phase are shownin Figure 1 Drag force acts in the opposite direction of thevelocity vector Hence if the effect of the drag force is greaterthan that of gravity the ballistic target decelerates duringits flight The perpendicular component of the decelerationvector to the line of sight is shown as a target maneuver tothe pursuing interceptor Thus the interceptor guidance lawmust take this deceleration maneuver of the ballistic threatinto account [20]
The approximate mathematical model of the target in thereentry phase is given in [20]
119879119909
=minus119865
119889
119898cos (120595
119879) cos (120574
119879)
119879119910
=119865
119889
119898sin (120595
119879) cos (120574
119879)
119879119911
=119865
119889
119898cos (120595
119879) sin (120574
119879) minus 119892
(5)
where 119865119889is the drag force 119881
119879119909 119881
119879119910 and 119881
119879119911are the velocity
components119898 is the targetmissilemass in the reentry phase119892 is the gravity acceleration 120574
119879is the reentry angle and 120595
119879is
the heading angle
3 Control and Estimation Algorithms
31 RadarMeasurement for Initialization For initialization ofEoMs of the missiles and the estimation process a groundbased radar is used The radar is fixed in a prespecifiedposition Relative positions of the interceptors and the targetaccording to the radar in the 3D space are shown in Figure 2
International Journal of Aerospace Engineering 5
Ze
Xe
Ye
120588ti120588tR
VtT120574tT
120595tT
antT
120582ti
120582tR
120595tR
120595ti
mi(xti yti zti)
T(xtT ytT ztT)
R(xtR ytR ztR)
Figure 2 Radar geometry
Ze
Xe
Ye
120588t1 120588t2120588tn
VtT120574tT
120595tT
antT
T(xtT ytT ztT)
xt2 yt2 zt2)
120582t1 120582t2
120582tn
120595t1
120595t2
120595tn
m1(xt1 yt1 zt1) mn(xtn ytn ztn)
m2(
Figure 3 3D engagement geometry
We assume that the radar estimates the initial state vector
0119877without any error and send the initialization data to the
interceptors without any delay
x0119877
= [1205880119877120582
0119877120574
0119879119886
0119879] (6)
The geometric relation between the 119894th interceptor andthe target is given in (7) In this work it is assumed that 120574
0119879
and 1198860119879
are estimated perfectly by the radar
1205880119894= radic(119909
0119879minus 119909
0119894)
2+ (119910
0119879minus 119910
0119894)
2+ (119911
0119879minus 119911
0119894)
2
1205820119894= arctan[[
[
1199110119879
minus 1199110119894
radic(1199090119879
minus 1199090119894)
2+ (119910
0119879minus 119910
0119894)
2
]]
]
(7)
32 Relative Kinematics 3D engagement geometry betweenthe target and interceptors is shown in Figure 3 It is assumedthat each interceptormissile canmeasure its own inertial statevector as shown in (8) and measurements are exact
x119868
119905119894= [119909119905119894
119910119905119894
119911119905119894
120574119905119894
120595119905119894]
119879 (8)
The state vector of the 119894thmissile according to the target attime 119905 is given in (9) where 119894 = 1 2 119899 In the informationsharing mode each missile could transmit and receive theestimated target data without any loss and delay
x119877
119905119894= [120588119905119894
120582119905119894
120574119905119879
119886119905119879]
119879 (9)
Relative kinematics between the interceptors and theballisticmissile in 119878
119909119911 119878
119909119910 and 119878
119910119911planes are defined by using
119909119911119894
= 119881120588119909119911
119909119911119894
=
119881120582119909119911119894
120588119909119911119894
119879119909119911
= minus
119886119879119909119911
119881119879119909119911
119879119909119911
= 119892
119886119879119909119911
119881119879119909119911
sin (120574119879119909119911)
119909119910119894
= 119881120588119909119910
119909119910119894
=
119881120582119909119910119894
120588119909119910119894
119879119909119910
= 0
119879119909119910
= 0
119910119911119894
= 119881120588119910119911
119910119911119894
=
119881120582119910119911119894
120588119910119911119894
119879119910119911
= minus
119886119879119910119911
119881119879119910119911
119879119910119911
= 119892
119886119879119910119911
119881119879119910119911
sin (120574119879119910119911)
(10)
where
119881120588119909119911
= minus119881119898119909119911
cos (120574119898119909119911
minus 120582119909119911) + 119881
119879119909119911cos (120582
119909119911minus 120574
119879119909119911)
119881120582119909119911
= minus119881119898119909119911
sin (120574119898119909119911
minus 120582119909119911) minus 119881
119879119909119911sin (120582
119909119911minus 120574
119879119909119911)
119881120588119909119910
= minus119881119898119909119910
cos (120574119898119909119910
minus 120582119909119910)
+ 119881119879119909119910
cos (120582119909119910minus 120574
119879119909119910)
119881120582119909119910
= minus119881119898119909119910
sin (120574119898119909119910
minus 120582119909119910)
minus 119881119879119909119910
sin (120582119909119910minus 120574
119879119909119910)
119881120588119910119911
= minus119881119898119910119911
cos (120574119898119910119911
minus 120582119910119911)
+ 119881119879119910119911
cos (120582119910119911minus 120574
119879119910119911)
6 International Journal of Aerospace Engineering
119881120582119910119911
= minus119881119898119910119911
sin (120574119898119910119911
minus 120582119910119911)
minus 119881119879119910119911
sin (120582119910119911minus 120574
119879119910119911)
(11)
The relative kinematics equations can be described in thediscrete-time by using
x119896= 119891
119896minus1(119909
119896minus1) (12)
where x119896is state vector of the 119894th missile at time 119905
119896and 119891
119896minus1is
obtained by integrating the relative kinematics EoMs in (10)[7]
33 Measurement Model Each interceptor is equipped withthe infrared seeker that measures the line of sight (LOS)angle 120582
119896119894Thismeasurement has a zero-meanGaussian noise
with standard deviation 120590120582119894 LOS angle measurements are
performed by each interceptor missile separately therefore119864(V
119896119894 V
119896119895) = 0 forall119894 = 119895 The 119894th missile LOS measurement is
119911119896119894= ℎ
119896119894(119909
119896) + V
119896119894 (13)
where
V119896119894sim alefsym (0 120590
2
120582119894) (14)
The interceptor missiles can operate in two modes The firstmode is information nonsharing mode In this mode eachinterceptor measures only its own LOS angle and uses it inthe estimation process of the relative statesThemeasurementvector in the information nonsharingmode is given as shownin (13)
The second mode is information sharing mode in whicheach interceptor not only measures its own LOS angle butalso calculates the LOS angle of the other interceptors asshown in (15) [7] Also to improve the estimation qualityeach interceptor calculates the range-to-go distance by usingposition and LOS angle data as shown in (16)
119911120582119894119895
=
[[[[[[[
[
1199111205821198941
1199111205821198942
119911120582119894119899
]]]]]]]
]
(15)
119911120588119894119895
=
[[[[[[[
[
1199111205881198941
1199111205881198942
119911120588119894119899
]]]]]]]
]
(16)
Here 119894 is the missile which performs measurement and 119899is the missile observed by the missile 119894
The LOS angle of the missile 119895 measured by the missile 119894is calculated by using
119911120582119894119895
= arctan120588
119894sin 120582
119894minus (119911
119895minus 119911
119894)
120588119894cos 120582
119894+ (119909
119894minus 119909
119895)
(17)
In addition the range-to-go of the 119895th missile measuredby the 119894th missile is obtained by using
119911120588119894119895
= radic(120588119894sin 120582
119894minus (119911
119895minus 119911
119894))
2
+ (120588119894cos 120582
119894+ (119909
119894minus 119909
119895))
2
(18)
In the information sharing mode the measurementmodel is given by
z119894119895= h (119909
119894119895) + k
119894119895 (19)
where
z119894119895= [119911120588(119894119895)
119911120582(119894119895)]
119879
k119894119895= [V120588(119894119895)
V120582(119894119895)]
119879
(20)
34 ExtendedKalmanFilter Because of the nonlinear relativekinematics between the interceptors and the ballistic targetan extended Kalman filter (EKF) is used for estimatingthe unmeasured data and to filter the noisy LOS anglemeasurements [7] The prediction error covariance matrix isgiven in
P119896|119896minus1
= 120601119896|119896minus1
P119896minus1|119896minus1120601
119879
119896|119896minus1+Q
119896minus1 (21)
where
120601119896|119896minus1
= 119890F119896minus1119879
cong I + F119896minus1
119879 (22)
is the transition matrix associated with the relative kinemat-ics 119879 is sampling time and I is the identity matrix withappropriate dimensionsQ
119896minus1is the covariance matrix of the
equivalent discrete process noise and it is calculated as shownin
Q119896minus1
= int
119879
0
120601119896|119896minus1
Q120601119879119896|119896minus1
119889119879 (23)
F119896minus1
is the Jacobian matrix associated with the nonlinearrelative kinematics
F119896minus1
=120597119891 (119909)
120597119909
10038161003816100381610038161003816100381610038161003816119909=119896minus1
(24)
The state estimation is performed by using
x119896|119896
= x119896|119896minus1
+ K119896[z
119896minus h
119896x
119896|119896minus1] (25)
where x119896|119896minus1
is the predicted state vector andK is the Kalmangain as shown in
x119896|119896minus1
= 120601119896|119896minus1
x119896minus1|119896minus1
K = P119896|119896minus1
H119879
119896[H
119896P
119896|119896minus1H119879
119896+ R
119896]
minus1
(26)
Here H119896is the measurement Jacobian matrix and R
119896is the
measurement noise covariancematrixThe covariancematrixis updated as shown in
P119896|119896
= [I minus K119896H
119896]P
119896|119896minus1[I minus K
119896H
119896]119879+ K
119896R
119896K119879
119896 (27)
International Journal of Aerospace Engineering 7
Z
X
nc119909119911
Vm119909119911
120588xz
VT119909119911
120574T119909119911
T
120574m119909119911120582xz
M
Figure 4 Relative kinematics on 119878119909119911plane
Y
X
nc119909y
Vm119909y
120588xy
VT119909y
120574T119909y
T
120574m119909y
120582xy
M
Figure 5 Relative kinematics on 119878119909119910
plane
35 3D True Proportional Navigation Algorithm The pro-portional algorithm is one of the most common and effectiveguidance techniques because of its simple structure andimplementation The true proportional navigation (TPN)system generates the acceleration command perpendicular tothe LOS As shown in (29) the acceleration command is afunction of closing velocity 119881
119888and LOS rate
119899119888= 119873119881
119888 (28)
where 119899119888is the acceleration command perpendicular to the
LOS 119881119888is closing velocity and 119873 is navigation ratio which
is generally between 3 and 5 [20] In this 3D interceptionstudy TPN algorithm is applied for 119878
119909119911 119878
119909119910 and 119878
119910119911planes
separately [21] Geometry of relative kinematics for eachdifferent plane is displayed in Figures 4 5 and 6
Acceleration commands in 119878119909119911 119878
119909119910 and 119878
119910119911are obtained
as shown in
119899119888119909119911
= 119873119881119888119909119911
119909119911
119899119888119909119910
= 119873119881119888119909119910
119909119910
119899119888119910119911
= 119873119881119888119910119911
119910119911
(29)
Z
Y
ncyz
Vmyz
120588yz
VTyz
120574Tyz
T
120574myz120582yz
M
Figure 6 Relative kinematics on 119878119910119911
plane
The acceleration components of the interceptor in the 119909- 119910-and 119911-axis (119886
119898119909 119886
119898119910 119886
119898119911) can be obtained from (30) by using
the trigonometric relations
119886119898119909
= minus119899119888119909119910
sin (120582119909119910) minus 119899
119888119909119911sin (120582
119909119911)
119886119898119910
= minus119899119888119909119910
cos (120582119909119910) minus 119899
119888119910119911sin (120582
119910119911)
119886119898119911
= 119899119888119909119911
cos (120582119909119911) + 119899
119888119910119911cos (120582
119910119911)
(30)
Before applying the control commands to the interceptorvertical and horizontal components 119886pitch and 119886yaw should becalculated Here 119886pitch is in the pitch plane and perpendicularto the velocity vector of the interceptor and 119886yaw is perpendic-ular to both velocity vector and vertical acceleration vectorFor TPN these acceleration components are calculated using
119886pitch = 119886119898119911
cos (120574119898) minus 119886
119898119909sin (120574
119898) minus 119892 cos (120574
119898)
119886yaw = 119886119898119910
cos (120595119898) minus 119886
119898119909sin (120595
119898)
(31)
4 Optimization Algorithm
41 CMA Algorithm Consider the general form uncon-strained optimization problem
minimize 119891 (x) 119891 R119899997888rarr R (32)
It is well known that when 119891 possess a certain structure(such as being continuous linear or convex) there arevariety of local search algorithms that can be applied tosolve this optimization problem efficiently However when119891 does not possess these desirable properties local searchmethods either fail to find an answer or get stuck in localminima Global search methods [22] remedy this problem bygeneralizing the search over the entire state space Althoughglobal methods can also exploit the structure of 119891 manyglobal methods treat 119891 as a black box function and hencethe solution is found entirely by examining the input-outputpairs (x 119891(x))
8 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899rarr R Number of Samples per Iteration 120582 Number of Iterations 119899iter Weights 119908
1 119908
120583
Output Approximate optimal solution xlowast
Initialization(1) m
0 120590
0 C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895 119888
1larr 2119899
2 119888120583larr 120583
119908119899
2 119888minus1
120590larr 1198993 119888minus1
119888larr 1198994 120572 larr 15 120583 larr 1205822
(2) while 119896 lt 119899iter do(3) for 119894 in 1 120582 do
Sample Candidate Solutions
(4) x119894sim N(m 120590
2
119896C
119896) 119891
119894larr 119891(x
119894)
Sort the Candidate Solutions Based on Their Cost
(5) x1120582
larr x119905(1) x
119905(120582) such that 119891
119905(1)le sdot sdot sdot le 119891
119905(120582)
Move the mean to low cost solutions
(6) m119896+1
larr m119896+ sum
120583
119894=1119908
119894(x
119894minusm
119896)
Update Evolution Path Variables
(7) p120590larr (1 minus 119888
120590)p
120590+ radic1 minus (1 minus 119888
120590)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(8) 120590119896+1
larr 120590119896times exp (119888
120590(p
120590EN(0 I) minus 1))
Update The Covariance Matrix
(9) if p120590 lt 120572radic(119899) then
(10) 119889119896larr 1
(11) else(12) 119889
119896larr 0
(13) 119888119904larr (1 minus 119889
2
119896)119888
1119888119888(2 minus 119888
119888)
p119888larr (1 minus 119888
119888)p
119888+ 119889
119896radic(1 minus 119888
119888)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(14) C119896+1
larr (1 minus 1198881minus 119888
120583+ 119888
119904)C
119896+ 119888
1p⊤
119888p
119888+ 119888
120583sum
120583
119894=1119908
119894((x
119894minusm
119896)120590
119896)((x
119894minusm
119896)120590
119896)⊤
(15) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(16) xlowastlarr x
119905(1)
Algorithm 1 Covariance Matrix Adaptation (CMA)
Covariance Matrix Adaptation (CMA) [18] is a popularglobal search method that usually ranks among the bestsolvers in global search benchmarks [23] The basic ideabehind CMA is to place a multivariate normal distributionover the search space R119899 and sample candidate solutions(x
119894 119891(x
119894)) from this distributionThemean vector and covari-
ance matrix of the distribution are incrementally updatedat each step based on the values of the sampled solutionsThe objective is to eventually steer the mean vector to theoptimal solution xlowast and shrink the covariance matrix toidentity matrix hence in the limit the distribution will yieldthe optimal solution when it is sampled
For completeness we provide the pseudocode for theCMA algorithm in Algorithm 1 taken from [24] The algo-rithm starts by initializing its internal parameters (Line (1))At the 119896th iteration the algorithm samples 120582 number ofsamples from amultivariate Gaussian distribution withmeanm
119896and covariance C
119896(Line (4)) Next the samples x
119894are
sorted according to their costs119891119894Theweighted average of top
120583 number of solutions is computed to find the mean vectorm
119896+1for the next iteration (Line (6)) which moves the mean
of the distribution towards samples with lower costs Nextalgorithm updates the covariance matrix with the help of theevolution path variables which are p
120590(Line (7)) and p
119888(Line
(13)) which ensures that the adaptation steps are conjugatedirectionsThe interested reader is referred to [24] for the fullderivation of the algorithm and the intuition for updating thepath parameters
42 CMA-MV Algorithm Unfortunately CMA algorithm isonly applicable to continuous optimization problems hencewe cannot use it to solve the missile launch condition settingproblem given in (3) since the allocation of missiles isdetermined by the integer variables
To overcome this issue we develop a novel algorithmnamed Covariance Matrix Adaptation with Mixed Variables(CMA-MV) which extends the classical CMA algorithm towork on nonlinearmixed integer optimization problemsThegeneric nonlinear mixed integer programming problem is ofthe form
minimize 119891 (x z) 119891 R119899times Z
119889997888rarr R (33)
where Z is the set of integers The special case we areinterested in is the problem where the discrete variable z isa binary vector that is z isin 0 1
119889 This is also the case forthe missile allocation problem in (3) where 119911
119894= 1 refers to
missile 119894 being launchedThe main idea behind the CMA-MV algorithm is to
define and update two probability distributions for samplingcontinuous and discrete variables For the continuous vari-ables x we use a multivariate normal distribution and we usethe exact same procedure followed in the CMA algorithm(Algorithm 1) to update the mean and covariance of thedistribution For the discrete variables we use a multivariateBernoulli distribution and update the mean and covarianceof this distribution based on the costs of sampled variables
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 5
Ze
Xe
Ye
120588ti120588tR
VtT120574tT
120595tT
antT
120582ti
120582tR
120595tR
120595ti
mi(xti yti zti)
T(xtT ytT ztT)
R(xtR ytR ztR)
Figure 2 Radar geometry
Ze
Xe
Ye
120588t1 120588t2120588tn
VtT120574tT
120595tT
antT
T(xtT ytT ztT)
xt2 yt2 zt2)
120582t1 120582t2
120582tn
120595t1
120595t2
120595tn
m1(xt1 yt1 zt1) mn(xtn ytn ztn)
m2(
Figure 3 3D engagement geometry
We assume that the radar estimates the initial state vector
0119877without any error and send the initialization data to the
interceptors without any delay
x0119877
= [1205880119877120582
0119877120574
0119879119886
0119879] (6)
The geometric relation between the 119894th interceptor andthe target is given in (7) In this work it is assumed that 120574
0119879
and 1198860119879
are estimated perfectly by the radar
1205880119894= radic(119909
0119879minus 119909
0119894)
2+ (119910
0119879minus 119910
0119894)
2+ (119911
0119879minus 119911
0119894)
2
1205820119894= arctan[[
[
1199110119879
minus 1199110119894
radic(1199090119879
minus 1199090119894)
2+ (119910
0119879minus 119910
0119894)
2
]]
]
(7)
32 Relative Kinematics 3D engagement geometry betweenthe target and interceptors is shown in Figure 3 It is assumedthat each interceptormissile canmeasure its own inertial statevector as shown in (8) and measurements are exact
x119868
119905119894= [119909119905119894
119910119905119894
119911119905119894
120574119905119894
120595119905119894]
119879 (8)
The state vector of the 119894thmissile according to the target attime 119905 is given in (9) where 119894 = 1 2 119899 In the informationsharing mode each missile could transmit and receive theestimated target data without any loss and delay
x119877
119905119894= [120588119905119894
120582119905119894
120574119905119879
119886119905119879]
119879 (9)
Relative kinematics between the interceptors and theballisticmissile in 119878
119909119911 119878
119909119910 and 119878
119910119911planes are defined by using
119909119911119894
= 119881120588119909119911
119909119911119894
=
119881120582119909119911119894
120588119909119911119894
119879119909119911
= minus
119886119879119909119911
119881119879119909119911
119879119909119911
= 119892
119886119879119909119911
119881119879119909119911
sin (120574119879119909119911)
119909119910119894
= 119881120588119909119910
119909119910119894
=
119881120582119909119910119894
120588119909119910119894
119879119909119910
= 0
119879119909119910
= 0
119910119911119894
= 119881120588119910119911
119910119911119894
=
119881120582119910119911119894
120588119910119911119894
119879119910119911
= minus
119886119879119910119911
119881119879119910119911
119879119910119911
= 119892
119886119879119910119911
119881119879119910119911
sin (120574119879119910119911)
(10)
where
119881120588119909119911
= minus119881119898119909119911
cos (120574119898119909119911
minus 120582119909119911) + 119881
119879119909119911cos (120582
119909119911minus 120574
119879119909119911)
119881120582119909119911
= minus119881119898119909119911
sin (120574119898119909119911
minus 120582119909119911) minus 119881
119879119909119911sin (120582
119909119911minus 120574
119879119909119911)
119881120588119909119910
= minus119881119898119909119910
cos (120574119898119909119910
minus 120582119909119910)
+ 119881119879119909119910
cos (120582119909119910minus 120574
119879119909119910)
119881120582119909119910
= minus119881119898119909119910
sin (120574119898119909119910
minus 120582119909119910)
minus 119881119879119909119910
sin (120582119909119910minus 120574
119879119909119910)
119881120588119910119911
= minus119881119898119910119911
cos (120574119898119910119911
minus 120582119910119911)
+ 119881119879119910119911
cos (120582119910119911minus 120574
119879119910119911)
6 International Journal of Aerospace Engineering
119881120582119910119911
= minus119881119898119910119911
sin (120574119898119910119911
minus 120582119910119911)
minus 119881119879119910119911
sin (120582119910119911minus 120574
119879119910119911)
(11)
The relative kinematics equations can be described in thediscrete-time by using
x119896= 119891
119896minus1(119909
119896minus1) (12)
where x119896is state vector of the 119894th missile at time 119905
119896and 119891
119896minus1is
obtained by integrating the relative kinematics EoMs in (10)[7]
33 Measurement Model Each interceptor is equipped withthe infrared seeker that measures the line of sight (LOS)angle 120582
119896119894Thismeasurement has a zero-meanGaussian noise
with standard deviation 120590120582119894 LOS angle measurements are
performed by each interceptor missile separately therefore119864(V
119896119894 V
119896119895) = 0 forall119894 = 119895 The 119894th missile LOS measurement is
119911119896119894= ℎ
119896119894(119909
119896) + V
119896119894 (13)
where
V119896119894sim alefsym (0 120590
2
120582119894) (14)
The interceptor missiles can operate in two modes The firstmode is information nonsharing mode In this mode eachinterceptor measures only its own LOS angle and uses it inthe estimation process of the relative statesThemeasurementvector in the information nonsharingmode is given as shownin (13)
The second mode is information sharing mode in whicheach interceptor not only measures its own LOS angle butalso calculates the LOS angle of the other interceptors asshown in (15) [7] Also to improve the estimation qualityeach interceptor calculates the range-to-go distance by usingposition and LOS angle data as shown in (16)
119911120582119894119895
=
[[[[[[[
[
1199111205821198941
1199111205821198942
119911120582119894119899
]]]]]]]
]
(15)
119911120588119894119895
=
[[[[[[[
[
1199111205881198941
1199111205881198942
119911120588119894119899
]]]]]]]
]
(16)
Here 119894 is the missile which performs measurement and 119899is the missile observed by the missile 119894
The LOS angle of the missile 119895 measured by the missile 119894is calculated by using
119911120582119894119895
= arctan120588
119894sin 120582
119894minus (119911
119895minus 119911
119894)
120588119894cos 120582
119894+ (119909
119894minus 119909
119895)
(17)
In addition the range-to-go of the 119895th missile measuredby the 119894th missile is obtained by using
119911120588119894119895
= radic(120588119894sin 120582
119894minus (119911
119895minus 119911
119894))
2
+ (120588119894cos 120582
119894+ (119909
119894minus 119909
119895))
2
(18)
In the information sharing mode the measurementmodel is given by
z119894119895= h (119909
119894119895) + k
119894119895 (19)
where
z119894119895= [119911120588(119894119895)
119911120582(119894119895)]
119879
k119894119895= [V120588(119894119895)
V120582(119894119895)]
119879
(20)
34 ExtendedKalmanFilter Because of the nonlinear relativekinematics between the interceptors and the ballistic targetan extended Kalman filter (EKF) is used for estimatingthe unmeasured data and to filter the noisy LOS anglemeasurements [7] The prediction error covariance matrix isgiven in
P119896|119896minus1
= 120601119896|119896minus1
P119896minus1|119896minus1120601
119879
119896|119896minus1+Q
119896minus1 (21)
where
120601119896|119896minus1
= 119890F119896minus1119879
cong I + F119896minus1
119879 (22)
is the transition matrix associated with the relative kinemat-ics 119879 is sampling time and I is the identity matrix withappropriate dimensionsQ
119896minus1is the covariance matrix of the
equivalent discrete process noise and it is calculated as shownin
Q119896minus1
= int
119879
0
120601119896|119896minus1
Q120601119879119896|119896minus1
119889119879 (23)
F119896minus1
is the Jacobian matrix associated with the nonlinearrelative kinematics
F119896minus1
=120597119891 (119909)
120597119909
10038161003816100381610038161003816100381610038161003816119909=119896minus1
(24)
The state estimation is performed by using
x119896|119896
= x119896|119896minus1
+ K119896[z
119896minus h
119896x
119896|119896minus1] (25)
where x119896|119896minus1
is the predicted state vector andK is the Kalmangain as shown in
x119896|119896minus1
= 120601119896|119896minus1
x119896minus1|119896minus1
K = P119896|119896minus1
H119879
119896[H
119896P
119896|119896minus1H119879
119896+ R
119896]
minus1
(26)
Here H119896is the measurement Jacobian matrix and R
119896is the
measurement noise covariancematrixThe covariancematrixis updated as shown in
P119896|119896
= [I minus K119896H
119896]P
119896|119896minus1[I minus K
119896H
119896]119879+ K
119896R
119896K119879
119896 (27)
International Journal of Aerospace Engineering 7
Z
X
nc119909119911
Vm119909119911
120588xz
VT119909119911
120574T119909119911
T
120574m119909119911120582xz
M
Figure 4 Relative kinematics on 119878119909119911plane
Y
X
nc119909y
Vm119909y
120588xy
VT119909y
120574T119909y
T
120574m119909y
120582xy
M
Figure 5 Relative kinematics on 119878119909119910
plane
35 3D True Proportional Navigation Algorithm The pro-portional algorithm is one of the most common and effectiveguidance techniques because of its simple structure andimplementation The true proportional navigation (TPN)system generates the acceleration command perpendicular tothe LOS As shown in (29) the acceleration command is afunction of closing velocity 119881
119888and LOS rate
119899119888= 119873119881
119888 (28)
where 119899119888is the acceleration command perpendicular to the
LOS 119881119888is closing velocity and 119873 is navigation ratio which
is generally between 3 and 5 [20] In this 3D interceptionstudy TPN algorithm is applied for 119878
119909119911 119878
119909119910 and 119878
119910119911planes
separately [21] Geometry of relative kinematics for eachdifferent plane is displayed in Figures 4 5 and 6
Acceleration commands in 119878119909119911 119878
119909119910 and 119878
119910119911are obtained
as shown in
119899119888119909119911
= 119873119881119888119909119911
119909119911
119899119888119909119910
= 119873119881119888119909119910
119909119910
119899119888119910119911
= 119873119881119888119910119911
119910119911
(29)
Z
Y
ncyz
Vmyz
120588yz
VTyz
120574Tyz
T
120574myz120582yz
M
Figure 6 Relative kinematics on 119878119910119911
plane
The acceleration components of the interceptor in the 119909- 119910-and 119911-axis (119886
119898119909 119886
119898119910 119886
119898119911) can be obtained from (30) by using
the trigonometric relations
119886119898119909
= minus119899119888119909119910
sin (120582119909119910) minus 119899
119888119909119911sin (120582
119909119911)
119886119898119910
= minus119899119888119909119910
cos (120582119909119910) minus 119899
119888119910119911sin (120582
119910119911)
119886119898119911
= 119899119888119909119911
cos (120582119909119911) + 119899
119888119910119911cos (120582
119910119911)
(30)
Before applying the control commands to the interceptorvertical and horizontal components 119886pitch and 119886yaw should becalculated Here 119886pitch is in the pitch plane and perpendicularto the velocity vector of the interceptor and 119886yaw is perpendic-ular to both velocity vector and vertical acceleration vectorFor TPN these acceleration components are calculated using
119886pitch = 119886119898119911
cos (120574119898) minus 119886
119898119909sin (120574
119898) minus 119892 cos (120574
119898)
119886yaw = 119886119898119910
cos (120595119898) minus 119886
119898119909sin (120595
119898)
(31)
4 Optimization Algorithm
41 CMA Algorithm Consider the general form uncon-strained optimization problem
minimize 119891 (x) 119891 R119899997888rarr R (32)
It is well known that when 119891 possess a certain structure(such as being continuous linear or convex) there arevariety of local search algorithms that can be applied tosolve this optimization problem efficiently However when119891 does not possess these desirable properties local searchmethods either fail to find an answer or get stuck in localminima Global search methods [22] remedy this problem bygeneralizing the search over the entire state space Althoughglobal methods can also exploit the structure of 119891 manyglobal methods treat 119891 as a black box function and hencethe solution is found entirely by examining the input-outputpairs (x 119891(x))
8 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899rarr R Number of Samples per Iteration 120582 Number of Iterations 119899iter Weights 119908
1 119908
120583
Output Approximate optimal solution xlowast
Initialization(1) m
0 120590
0 C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895 119888
1larr 2119899
2 119888120583larr 120583
119908119899
2 119888minus1
120590larr 1198993 119888minus1
119888larr 1198994 120572 larr 15 120583 larr 1205822
(2) while 119896 lt 119899iter do(3) for 119894 in 1 120582 do
Sample Candidate Solutions
(4) x119894sim N(m 120590
2
119896C
119896) 119891
119894larr 119891(x
119894)
Sort the Candidate Solutions Based on Their Cost
(5) x1120582
larr x119905(1) x
119905(120582) such that 119891
119905(1)le sdot sdot sdot le 119891
119905(120582)
Move the mean to low cost solutions
(6) m119896+1
larr m119896+ sum
120583
119894=1119908
119894(x
119894minusm
119896)
Update Evolution Path Variables
(7) p120590larr (1 minus 119888
120590)p
120590+ radic1 minus (1 minus 119888
120590)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(8) 120590119896+1
larr 120590119896times exp (119888
120590(p
120590EN(0 I) minus 1))
Update The Covariance Matrix
(9) if p120590 lt 120572radic(119899) then
(10) 119889119896larr 1
(11) else(12) 119889
119896larr 0
(13) 119888119904larr (1 minus 119889
2
119896)119888
1119888119888(2 minus 119888
119888)
p119888larr (1 minus 119888
119888)p
119888+ 119889
119896radic(1 minus 119888
119888)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(14) C119896+1
larr (1 minus 1198881minus 119888
120583+ 119888
119904)C
119896+ 119888
1p⊤
119888p
119888+ 119888
120583sum
120583
119894=1119908
119894((x
119894minusm
119896)120590
119896)((x
119894minusm
119896)120590
119896)⊤
(15) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(16) xlowastlarr x
119905(1)
Algorithm 1 Covariance Matrix Adaptation (CMA)
Covariance Matrix Adaptation (CMA) [18] is a popularglobal search method that usually ranks among the bestsolvers in global search benchmarks [23] The basic ideabehind CMA is to place a multivariate normal distributionover the search space R119899 and sample candidate solutions(x
119894 119891(x
119894)) from this distributionThemean vector and covari-
ance matrix of the distribution are incrementally updatedat each step based on the values of the sampled solutionsThe objective is to eventually steer the mean vector to theoptimal solution xlowast and shrink the covariance matrix toidentity matrix hence in the limit the distribution will yieldthe optimal solution when it is sampled
For completeness we provide the pseudocode for theCMA algorithm in Algorithm 1 taken from [24] The algo-rithm starts by initializing its internal parameters (Line (1))At the 119896th iteration the algorithm samples 120582 number ofsamples from amultivariate Gaussian distribution withmeanm
119896and covariance C
119896(Line (4)) Next the samples x
119894are
sorted according to their costs119891119894Theweighted average of top
120583 number of solutions is computed to find the mean vectorm
119896+1for the next iteration (Line (6)) which moves the mean
of the distribution towards samples with lower costs Nextalgorithm updates the covariance matrix with the help of theevolution path variables which are p
120590(Line (7)) and p
119888(Line
(13)) which ensures that the adaptation steps are conjugatedirectionsThe interested reader is referred to [24] for the fullderivation of the algorithm and the intuition for updating thepath parameters
42 CMA-MV Algorithm Unfortunately CMA algorithm isonly applicable to continuous optimization problems hencewe cannot use it to solve the missile launch condition settingproblem given in (3) since the allocation of missiles isdetermined by the integer variables
To overcome this issue we develop a novel algorithmnamed Covariance Matrix Adaptation with Mixed Variables(CMA-MV) which extends the classical CMA algorithm towork on nonlinearmixed integer optimization problemsThegeneric nonlinear mixed integer programming problem is ofthe form
minimize 119891 (x z) 119891 R119899times Z
119889997888rarr R (33)
where Z is the set of integers The special case we areinterested in is the problem where the discrete variable z isa binary vector that is z isin 0 1
119889 This is also the case forthe missile allocation problem in (3) where 119911
119894= 1 refers to
missile 119894 being launchedThe main idea behind the CMA-MV algorithm is to
define and update two probability distributions for samplingcontinuous and discrete variables For the continuous vari-ables x we use a multivariate normal distribution and we usethe exact same procedure followed in the CMA algorithm(Algorithm 1) to update the mean and covariance of thedistribution For the discrete variables we use a multivariateBernoulli distribution and update the mean and covarianceof this distribution based on the costs of sampled variables
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
6 International Journal of Aerospace Engineering
119881120582119910119911
= minus119881119898119910119911
sin (120574119898119910119911
minus 120582119910119911)
minus 119881119879119910119911
sin (120582119910119911minus 120574
119879119910119911)
(11)
The relative kinematics equations can be described in thediscrete-time by using
x119896= 119891
119896minus1(119909
119896minus1) (12)
where x119896is state vector of the 119894th missile at time 119905
119896and 119891
119896minus1is
obtained by integrating the relative kinematics EoMs in (10)[7]
33 Measurement Model Each interceptor is equipped withthe infrared seeker that measures the line of sight (LOS)angle 120582
119896119894Thismeasurement has a zero-meanGaussian noise
with standard deviation 120590120582119894 LOS angle measurements are
performed by each interceptor missile separately therefore119864(V
119896119894 V
119896119895) = 0 forall119894 = 119895 The 119894th missile LOS measurement is
119911119896119894= ℎ
119896119894(119909
119896) + V
119896119894 (13)
where
V119896119894sim alefsym (0 120590
2
120582119894) (14)
The interceptor missiles can operate in two modes The firstmode is information nonsharing mode In this mode eachinterceptor measures only its own LOS angle and uses it inthe estimation process of the relative statesThemeasurementvector in the information nonsharingmode is given as shownin (13)
The second mode is information sharing mode in whicheach interceptor not only measures its own LOS angle butalso calculates the LOS angle of the other interceptors asshown in (15) [7] Also to improve the estimation qualityeach interceptor calculates the range-to-go distance by usingposition and LOS angle data as shown in (16)
119911120582119894119895
=
[[[[[[[
[
1199111205821198941
1199111205821198942
119911120582119894119899
]]]]]]]
]
(15)
119911120588119894119895
=
[[[[[[[
[
1199111205881198941
1199111205881198942
119911120588119894119899
]]]]]]]
]
(16)
Here 119894 is the missile which performs measurement and 119899is the missile observed by the missile 119894
The LOS angle of the missile 119895 measured by the missile 119894is calculated by using
119911120582119894119895
= arctan120588
119894sin 120582
119894minus (119911
119895minus 119911
119894)
120588119894cos 120582
119894+ (119909
119894minus 119909
119895)
(17)
In addition the range-to-go of the 119895th missile measuredby the 119894th missile is obtained by using
119911120588119894119895
= radic(120588119894sin 120582
119894minus (119911
119895minus 119911
119894))
2
+ (120588119894cos 120582
119894+ (119909
119894minus 119909
119895))
2
(18)
In the information sharing mode the measurementmodel is given by
z119894119895= h (119909
119894119895) + k
119894119895 (19)
where
z119894119895= [119911120588(119894119895)
119911120582(119894119895)]
119879
k119894119895= [V120588(119894119895)
V120582(119894119895)]
119879
(20)
34 ExtendedKalmanFilter Because of the nonlinear relativekinematics between the interceptors and the ballistic targetan extended Kalman filter (EKF) is used for estimatingthe unmeasured data and to filter the noisy LOS anglemeasurements [7] The prediction error covariance matrix isgiven in
P119896|119896minus1
= 120601119896|119896minus1
P119896minus1|119896minus1120601
119879
119896|119896minus1+Q
119896minus1 (21)
where
120601119896|119896minus1
= 119890F119896minus1119879
cong I + F119896minus1
119879 (22)
is the transition matrix associated with the relative kinemat-ics 119879 is sampling time and I is the identity matrix withappropriate dimensionsQ
119896minus1is the covariance matrix of the
equivalent discrete process noise and it is calculated as shownin
Q119896minus1
= int
119879
0
120601119896|119896minus1
Q120601119879119896|119896minus1
119889119879 (23)
F119896minus1
is the Jacobian matrix associated with the nonlinearrelative kinematics
F119896minus1
=120597119891 (119909)
120597119909
10038161003816100381610038161003816100381610038161003816119909=119896minus1
(24)
The state estimation is performed by using
x119896|119896
= x119896|119896minus1
+ K119896[z
119896minus h
119896x
119896|119896minus1] (25)
where x119896|119896minus1
is the predicted state vector andK is the Kalmangain as shown in
x119896|119896minus1
= 120601119896|119896minus1
x119896minus1|119896minus1
K = P119896|119896minus1
H119879
119896[H
119896P
119896|119896minus1H119879
119896+ R
119896]
minus1
(26)
Here H119896is the measurement Jacobian matrix and R
119896is the
measurement noise covariancematrixThe covariancematrixis updated as shown in
P119896|119896
= [I minus K119896H
119896]P
119896|119896minus1[I minus K
119896H
119896]119879+ K
119896R
119896K119879
119896 (27)
International Journal of Aerospace Engineering 7
Z
X
nc119909119911
Vm119909119911
120588xz
VT119909119911
120574T119909119911
T
120574m119909119911120582xz
M
Figure 4 Relative kinematics on 119878119909119911plane
Y
X
nc119909y
Vm119909y
120588xy
VT119909y
120574T119909y
T
120574m119909y
120582xy
M
Figure 5 Relative kinematics on 119878119909119910
plane
35 3D True Proportional Navigation Algorithm The pro-portional algorithm is one of the most common and effectiveguidance techniques because of its simple structure andimplementation The true proportional navigation (TPN)system generates the acceleration command perpendicular tothe LOS As shown in (29) the acceleration command is afunction of closing velocity 119881
119888and LOS rate
119899119888= 119873119881
119888 (28)
where 119899119888is the acceleration command perpendicular to the
LOS 119881119888is closing velocity and 119873 is navigation ratio which
is generally between 3 and 5 [20] In this 3D interceptionstudy TPN algorithm is applied for 119878
119909119911 119878
119909119910 and 119878
119910119911planes
separately [21] Geometry of relative kinematics for eachdifferent plane is displayed in Figures 4 5 and 6
Acceleration commands in 119878119909119911 119878
119909119910 and 119878
119910119911are obtained
as shown in
119899119888119909119911
= 119873119881119888119909119911
119909119911
119899119888119909119910
= 119873119881119888119909119910
119909119910
119899119888119910119911
= 119873119881119888119910119911
119910119911
(29)
Z
Y
ncyz
Vmyz
120588yz
VTyz
120574Tyz
T
120574myz120582yz
M
Figure 6 Relative kinematics on 119878119910119911
plane
The acceleration components of the interceptor in the 119909- 119910-and 119911-axis (119886
119898119909 119886
119898119910 119886
119898119911) can be obtained from (30) by using
the trigonometric relations
119886119898119909
= minus119899119888119909119910
sin (120582119909119910) minus 119899
119888119909119911sin (120582
119909119911)
119886119898119910
= minus119899119888119909119910
cos (120582119909119910) minus 119899
119888119910119911sin (120582
119910119911)
119886119898119911
= 119899119888119909119911
cos (120582119909119911) + 119899
119888119910119911cos (120582
119910119911)
(30)
Before applying the control commands to the interceptorvertical and horizontal components 119886pitch and 119886yaw should becalculated Here 119886pitch is in the pitch plane and perpendicularto the velocity vector of the interceptor and 119886yaw is perpendic-ular to both velocity vector and vertical acceleration vectorFor TPN these acceleration components are calculated using
119886pitch = 119886119898119911
cos (120574119898) minus 119886
119898119909sin (120574
119898) minus 119892 cos (120574
119898)
119886yaw = 119886119898119910
cos (120595119898) minus 119886
119898119909sin (120595
119898)
(31)
4 Optimization Algorithm
41 CMA Algorithm Consider the general form uncon-strained optimization problem
minimize 119891 (x) 119891 R119899997888rarr R (32)
It is well known that when 119891 possess a certain structure(such as being continuous linear or convex) there arevariety of local search algorithms that can be applied tosolve this optimization problem efficiently However when119891 does not possess these desirable properties local searchmethods either fail to find an answer or get stuck in localminima Global search methods [22] remedy this problem bygeneralizing the search over the entire state space Althoughglobal methods can also exploit the structure of 119891 manyglobal methods treat 119891 as a black box function and hencethe solution is found entirely by examining the input-outputpairs (x 119891(x))
8 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899rarr R Number of Samples per Iteration 120582 Number of Iterations 119899iter Weights 119908
1 119908
120583
Output Approximate optimal solution xlowast
Initialization(1) m
0 120590
0 C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895 119888
1larr 2119899
2 119888120583larr 120583
119908119899
2 119888minus1
120590larr 1198993 119888minus1
119888larr 1198994 120572 larr 15 120583 larr 1205822
(2) while 119896 lt 119899iter do(3) for 119894 in 1 120582 do
Sample Candidate Solutions
(4) x119894sim N(m 120590
2
119896C
119896) 119891
119894larr 119891(x
119894)
Sort the Candidate Solutions Based on Their Cost
(5) x1120582
larr x119905(1) x
119905(120582) such that 119891
119905(1)le sdot sdot sdot le 119891
119905(120582)
Move the mean to low cost solutions
(6) m119896+1
larr m119896+ sum
120583
119894=1119908
119894(x
119894minusm
119896)
Update Evolution Path Variables
(7) p120590larr (1 minus 119888
120590)p
120590+ radic1 minus (1 minus 119888
120590)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(8) 120590119896+1
larr 120590119896times exp (119888
120590(p
120590EN(0 I) minus 1))
Update The Covariance Matrix
(9) if p120590 lt 120572radic(119899) then
(10) 119889119896larr 1
(11) else(12) 119889
119896larr 0
(13) 119888119904larr (1 minus 119889
2
119896)119888
1119888119888(2 minus 119888
119888)
p119888larr (1 minus 119888
119888)p
119888+ 119889
119896radic(1 minus 119888
119888)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(14) C119896+1
larr (1 minus 1198881minus 119888
120583+ 119888
119904)C
119896+ 119888
1p⊤
119888p
119888+ 119888
120583sum
120583
119894=1119908
119894((x
119894minusm
119896)120590
119896)((x
119894minusm
119896)120590
119896)⊤
(15) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(16) xlowastlarr x
119905(1)
Algorithm 1 Covariance Matrix Adaptation (CMA)
Covariance Matrix Adaptation (CMA) [18] is a popularglobal search method that usually ranks among the bestsolvers in global search benchmarks [23] The basic ideabehind CMA is to place a multivariate normal distributionover the search space R119899 and sample candidate solutions(x
119894 119891(x
119894)) from this distributionThemean vector and covari-
ance matrix of the distribution are incrementally updatedat each step based on the values of the sampled solutionsThe objective is to eventually steer the mean vector to theoptimal solution xlowast and shrink the covariance matrix toidentity matrix hence in the limit the distribution will yieldthe optimal solution when it is sampled
For completeness we provide the pseudocode for theCMA algorithm in Algorithm 1 taken from [24] The algo-rithm starts by initializing its internal parameters (Line (1))At the 119896th iteration the algorithm samples 120582 number ofsamples from amultivariate Gaussian distribution withmeanm
119896and covariance C
119896(Line (4)) Next the samples x
119894are
sorted according to their costs119891119894Theweighted average of top
120583 number of solutions is computed to find the mean vectorm
119896+1for the next iteration (Line (6)) which moves the mean
of the distribution towards samples with lower costs Nextalgorithm updates the covariance matrix with the help of theevolution path variables which are p
120590(Line (7)) and p
119888(Line
(13)) which ensures that the adaptation steps are conjugatedirectionsThe interested reader is referred to [24] for the fullderivation of the algorithm and the intuition for updating thepath parameters
42 CMA-MV Algorithm Unfortunately CMA algorithm isonly applicable to continuous optimization problems hencewe cannot use it to solve the missile launch condition settingproblem given in (3) since the allocation of missiles isdetermined by the integer variables
To overcome this issue we develop a novel algorithmnamed Covariance Matrix Adaptation with Mixed Variables(CMA-MV) which extends the classical CMA algorithm towork on nonlinearmixed integer optimization problemsThegeneric nonlinear mixed integer programming problem is ofthe form
minimize 119891 (x z) 119891 R119899times Z
119889997888rarr R (33)
where Z is the set of integers The special case we areinterested in is the problem where the discrete variable z isa binary vector that is z isin 0 1
119889 This is also the case forthe missile allocation problem in (3) where 119911
119894= 1 refers to
missile 119894 being launchedThe main idea behind the CMA-MV algorithm is to
define and update two probability distributions for samplingcontinuous and discrete variables For the continuous vari-ables x we use a multivariate normal distribution and we usethe exact same procedure followed in the CMA algorithm(Algorithm 1) to update the mean and covariance of thedistribution For the discrete variables we use a multivariateBernoulli distribution and update the mean and covarianceof this distribution based on the costs of sampled variables
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
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Navigation and Observation
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DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 7
Z
X
nc119909119911
Vm119909119911
120588xz
VT119909119911
120574T119909119911
T
120574m119909119911120582xz
M
Figure 4 Relative kinematics on 119878119909119911plane
Y
X
nc119909y
Vm119909y
120588xy
VT119909y
120574T119909y
T
120574m119909y
120582xy
M
Figure 5 Relative kinematics on 119878119909119910
plane
35 3D True Proportional Navigation Algorithm The pro-portional algorithm is one of the most common and effectiveguidance techniques because of its simple structure andimplementation The true proportional navigation (TPN)system generates the acceleration command perpendicular tothe LOS As shown in (29) the acceleration command is afunction of closing velocity 119881
119888and LOS rate
119899119888= 119873119881
119888 (28)
where 119899119888is the acceleration command perpendicular to the
LOS 119881119888is closing velocity and 119873 is navigation ratio which
is generally between 3 and 5 [20] In this 3D interceptionstudy TPN algorithm is applied for 119878
119909119911 119878
119909119910 and 119878
119910119911planes
separately [21] Geometry of relative kinematics for eachdifferent plane is displayed in Figures 4 5 and 6
Acceleration commands in 119878119909119911 119878
119909119910 and 119878
119910119911are obtained
as shown in
119899119888119909119911
= 119873119881119888119909119911
119909119911
119899119888119909119910
= 119873119881119888119909119910
119909119910
119899119888119910119911
= 119873119881119888119910119911
119910119911
(29)
Z
Y
ncyz
Vmyz
120588yz
VTyz
120574Tyz
T
120574myz120582yz
M
Figure 6 Relative kinematics on 119878119910119911
plane
The acceleration components of the interceptor in the 119909- 119910-and 119911-axis (119886
119898119909 119886
119898119910 119886
119898119911) can be obtained from (30) by using
the trigonometric relations
119886119898119909
= minus119899119888119909119910
sin (120582119909119910) minus 119899
119888119909119911sin (120582
119909119911)
119886119898119910
= minus119899119888119909119910
cos (120582119909119910) minus 119899
119888119910119911sin (120582
119910119911)
119886119898119911
= 119899119888119909119911
cos (120582119909119911) + 119899
119888119910119911cos (120582
119910119911)
(30)
Before applying the control commands to the interceptorvertical and horizontal components 119886pitch and 119886yaw should becalculated Here 119886pitch is in the pitch plane and perpendicularto the velocity vector of the interceptor and 119886yaw is perpendic-ular to both velocity vector and vertical acceleration vectorFor TPN these acceleration components are calculated using
119886pitch = 119886119898119911
cos (120574119898) minus 119886
119898119909sin (120574
119898) minus 119892 cos (120574
119898)
119886yaw = 119886119898119910
cos (120595119898) minus 119886
119898119909sin (120595
119898)
(31)
4 Optimization Algorithm
41 CMA Algorithm Consider the general form uncon-strained optimization problem
minimize 119891 (x) 119891 R119899997888rarr R (32)
It is well known that when 119891 possess a certain structure(such as being continuous linear or convex) there arevariety of local search algorithms that can be applied tosolve this optimization problem efficiently However when119891 does not possess these desirable properties local searchmethods either fail to find an answer or get stuck in localminima Global search methods [22] remedy this problem bygeneralizing the search over the entire state space Althoughglobal methods can also exploit the structure of 119891 manyglobal methods treat 119891 as a black box function and hencethe solution is found entirely by examining the input-outputpairs (x 119891(x))
8 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899rarr R Number of Samples per Iteration 120582 Number of Iterations 119899iter Weights 119908
1 119908
120583
Output Approximate optimal solution xlowast
Initialization(1) m
0 120590
0 C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895 119888
1larr 2119899
2 119888120583larr 120583
119908119899
2 119888minus1
120590larr 1198993 119888minus1
119888larr 1198994 120572 larr 15 120583 larr 1205822
(2) while 119896 lt 119899iter do(3) for 119894 in 1 120582 do
Sample Candidate Solutions
(4) x119894sim N(m 120590
2
119896C
119896) 119891
119894larr 119891(x
119894)
Sort the Candidate Solutions Based on Their Cost
(5) x1120582
larr x119905(1) x
119905(120582) such that 119891
119905(1)le sdot sdot sdot le 119891
119905(120582)
Move the mean to low cost solutions
(6) m119896+1
larr m119896+ sum
120583
119894=1119908
119894(x
119894minusm
119896)
Update Evolution Path Variables
(7) p120590larr (1 minus 119888
120590)p
120590+ radic1 minus (1 minus 119888
120590)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(8) 120590119896+1
larr 120590119896times exp (119888
120590(p
120590EN(0 I) minus 1))
Update The Covariance Matrix
(9) if p120590 lt 120572radic(119899) then
(10) 119889119896larr 1
(11) else(12) 119889
119896larr 0
(13) 119888119904larr (1 minus 119889
2
119896)119888
1119888119888(2 minus 119888
119888)
p119888larr (1 minus 119888
119888)p
119888+ 119889
119896radic(1 minus 119888
119888)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(14) C119896+1
larr (1 minus 1198881minus 119888
120583+ 119888
119904)C
119896+ 119888
1p⊤
119888p
119888+ 119888
120583sum
120583
119894=1119908
119894((x
119894minusm
119896)120590
119896)((x
119894minusm
119896)120590
119896)⊤
(15) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(16) xlowastlarr x
119905(1)
Algorithm 1 Covariance Matrix Adaptation (CMA)
Covariance Matrix Adaptation (CMA) [18] is a popularglobal search method that usually ranks among the bestsolvers in global search benchmarks [23] The basic ideabehind CMA is to place a multivariate normal distributionover the search space R119899 and sample candidate solutions(x
119894 119891(x
119894)) from this distributionThemean vector and covari-
ance matrix of the distribution are incrementally updatedat each step based on the values of the sampled solutionsThe objective is to eventually steer the mean vector to theoptimal solution xlowast and shrink the covariance matrix toidentity matrix hence in the limit the distribution will yieldthe optimal solution when it is sampled
For completeness we provide the pseudocode for theCMA algorithm in Algorithm 1 taken from [24] The algo-rithm starts by initializing its internal parameters (Line (1))At the 119896th iteration the algorithm samples 120582 number ofsamples from amultivariate Gaussian distribution withmeanm
119896and covariance C
119896(Line (4)) Next the samples x
119894are
sorted according to their costs119891119894Theweighted average of top
120583 number of solutions is computed to find the mean vectorm
119896+1for the next iteration (Line (6)) which moves the mean
of the distribution towards samples with lower costs Nextalgorithm updates the covariance matrix with the help of theevolution path variables which are p
120590(Line (7)) and p
119888(Line
(13)) which ensures that the adaptation steps are conjugatedirectionsThe interested reader is referred to [24] for the fullderivation of the algorithm and the intuition for updating thepath parameters
42 CMA-MV Algorithm Unfortunately CMA algorithm isonly applicable to continuous optimization problems hencewe cannot use it to solve the missile launch condition settingproblem given in (3) since the allocation of missiles isdetermined by the integer variables
To overcome this issue we develop a novel algorithmnamed Covariance Matrix Adaptation with Mixed Variables(CMA-MV) which extends the classical CMA algorithm towork on nonlinearmixed integer optimization problemsThegeneric nonlinear mixed integer programming problem is ofthe form
minimize 119891 (x z) 119891 R119899times Z
119889997888rarr R (33)
where Z is the set of integers The special case we areinterested in is the problem where the discrete variable z isa binary vector that is z isin 0 1
119889 This is also the case forthe missile allocation problem in (3) where 119911
119894= 1 refers to
missile 119894 being launchedThe main idea behind the CMA-MV algorithm is to
define and update two probability distributions for samplingcontinuous and discrete variables For the continuous vari-ables x we use a multivariate normal distribution and we usethe exact same procedure followed in the CMA algorithm(Algorithm 1) to update the mean and covariance of thedistribution For the discrete variables we use a multivariateBernoulli distribution and update the mean and covarianceof this distribution based on the costs of sampled variables
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
8 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899rarr R Number of Samples per Iteration 120582 Number of Iterations 119899iter Weights 119908
1 119908
120583
Output Approximate optimal solution xlowast
Initialization(1) m
0 120590
0 C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895 119888
1larr 2119899
2 119888120583larr 120583
119908119899
2 119888minus1
120590larr 1198993 119888minus1
119888larr 1198994 120572 larr 15 120583 larr 1205822
(2) while 119896 lt 119899iter do(3) for 119894 in 1 120582 do
Sample Candidate Solutions
(4) x119894sim N(m 120590
2
119896C
119896) 119891
119894larr 119891(x
119894)
Sort the Candidate Solutions Based on Their Cost
(5) x1120582
larr x119905(1) x
119905(120582) such that 119891
119905(1)le sdot sdot sdot le 119891
119905(120582)
Move the mean to low cost solutions
(6) m119896+1
larr m119896+ sum
120583
119894=1119908
119894(x
119894minusm
119896)
Update Evolution Path Variables
(7) p120590larr (1 minus 119888
120590)p
120590+ radic1 minus (1 minus 119888
120590)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(8) 120590119896+1
larr 120590119896times exp (119888
120590(p
120590EN(0 I) minus 1))
Update The Covariance Matrix
(9) if p120590 lt 120572radic(119899) then
(10) 119889119896larr 1
(11) else(12) 119889
119896larr 0
(13) 119888119904larr (1 minus 119889
2
119896)119888
1119888119888(2 minus 119888
119888)
p119888larr (1 minus 119888
119888)p
119888+ 119889
119896radic(1 minus 119888
119888)2
radic120583119908119862
minus12
119896((m
119896+1minusm
119896)120590
119896)
(14) C119896+1
larr (1 minus 1198881minus 119888
120583+ 119888
119904)C
119896+ 119888
1p⊤
119888p
119888+ 119888
120583sum
120583
119894=1119908
119894((x
119894minusm
119896)120590
119896)((x
119894minusm
119896)120590
119896)⊤
(15) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(16) xlowastlarr x
119905(1)
Algorithm 1 Covariance Matrix Adaptation (CMA)
Covariance Matrix Adaptation (CMA) [18] is a popularglobal search method that usually ranks among the bestsolvers in global search benchmarks [23] The basic ideabehind CMA is to place a multivariate normal distributionover the search space R119899 and sample candidate solutions(x
119894 119891(x
119894)) from this distributionThemean vector and covari-
ance matrix of the distribution are incrementally updatedat each step based on the values of the sampled solutionsThe objective is to eventually steer the mean vector to theoptimal solution xlowast and shrink the covariance matrix toidentity matrix hence in the limit the distribution will yieldthe optimal solution when it is sampled
For completeness we provide the pseudocode for theCMA algorithm in Algorithm 1 taken from [24] The algo-rithm starts by initializing its internal parameters (Line (1))At the 119896th iteration the algorithm samples 120582 number ofsamples from amultivariate Gaussian distribution withmeanm
119896and covariance C
119896(Line (4)) Next the samples x
119894are
sorted according to their costs119891119894Theweighted average of top
120583 number of solutions is computed to find the mean vectorm
119896+1for the next iteration (Line (6)) which moves the mean
of the distribution towards samples with lower costs Nextalgorithm updates the covariance matrix with the help of theevolution path variables which are p
120590(Line (7)) and p
119888(Line
(13)) which ensures that the adaptation steps are conjugatedirectionsThe interested reader is referred to [24] for the fullderivation of the algorithm and the intuition for updating thepath parameters
42 CMA-MV Algorithm Unfortunately CMA algorithm isonly applicable to continuous optimization problems hencewe cannot use it to solve the missile launch condition settingproblem given in (3) since the allocation of missiles isdetermined by the integer variables
To overcome this issue we develop a novel algorithmnamed Covariance Matrix Adaptation with Mixed Variables(CMA-MV) which extends the classical CMA algorithm towork on nonlinearmixed integer optimization problemsThegeneric nonlinear mixed integer programming problem is ofthe form
minimize 119891 (x z) 119891 R119899times Z
119889997888rarr R (33)
where Z is the set of integers The special case we areinterested in is the problem where the discrete variable z isa binary vector that is z isin 0 1
119889 This is also the case forthe missile allocation problem in (3) where 119911
119894= 1 refers to
missile 119894 being launchedThe main idea behind the CMA-MV algorithm is to
define and update two probability distributions for samplingcontinuous and discrete variables For the continuous vari-ables x we use a multivariate normal distribution and we usethe exact same procedure followed in the CMA algorithm(Algorithm 1) to update the mean and covariance of thedistribution For the discrete variables we use a multivariateBernoulli distribution and update the mean and covarianceof this distribution based on the costs of sampled variables
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 9
Input Meanm1015840isin [0 1]
119889 Covariance C1015840
Output Sample z isin 0 1119889 Compute the corresponding multivariate Normal distrubtion
(1) for 119894 in 1 119889 do(2) 120574
119894larr Φ
minus1(119898
1015840
119894)
(3) for 119895 in 1 119889 do(4) if 119894 = 119895 then(5) Λ
119894119895larr Solve C1015840
119894119895minus Ψ(120574
119894 120574
119895 Λ
119894119895) = 0
(6) else(7) Λ
119894119895= 1
Sample from the corresponding multivariate Normal distrubtion and transform the results
(8) 119906 sim N(120574Λ)
(9) for 119897 in 1 119889 do(10) if 119906
119897gt 0 then
(11) 119911119897larr 1
(12) else(13) 119911
119897larr 0
Algorithm 2 Sample from a multivariate Bernoulli distribution
However sampling from the multivariate Bernoulli dis-tribution is not as straightforward as sampling from a mul-tivariate normal distribution We use the method describedin [25] for this purpose The pseudocode for the samplingprocess is given in Algorithm 2 The algorithm takes thegiven mean vector m1015840 and the covariance matrix for C1015840
and computes a correspondingmultivariate distribution withmean 120574 and covariance Λ by solving the equations given onLines (2) and (4) In these equations Φ is the cumulativedistribution of a univariate normal variable with zero meanand unit variance Ψ(119909 119910 119911) = Φ
2(119909 119910 119911) minus Φ(119909)Φ(119910)
whereΦ2(119909 119910 119911) is the cumulative distribution of a bivariate
normal variable with mean [119909 119910] and correlation 119911 Aftersolving these equations using numerical techniques we sam-ple the normal variable in Line (8)Then we loop through thecomponents of the sample and set 119911
119897= 1 if the components
are positive and set 119911119897= 0 otherwise It can be shown that the
multivariate sample generated via this fashion comes froma distribution with first and second moments m1015840 and C1015840respectively
The pseudocode for the complete CMA-MV algorithmis given in Algorithm 3 In the 119896th iteration algorithm fixesthe value of the discrete variables and hence recovers thefunction (x) = 119891(x z119896
) Note that is a function of acontinuous variable hence we can apply the CMA algorithm(Algorithm 1) to obtain a solution (Line (3)) Next we fixthe value of the continuous variable to x119896+1 to recover thefunction (z) = 119891(x119896+1
z) Then we sample 1205821015840 solution
candidates from the Bernoulli distribution with mean m1015840
119896
and C1015840
119896using the sampling algorithm given in Algorithm 2
(Line (5)) Then we sort the solution from the lowest cost tohighest cost (Line (6)) Next we use a weighted average ofthe low cost solutions to compute the updated mean m119896+1
of the Bernoulli distribution (Line (7)) Similarly we use theweighted sample covariance estimate of the low cost solutioncandidates to compute the updated covariance matrix C1015840
119896+1
After each update the distribution puts more mass on
low cost solution candidates and hence with each iterationprobability of sampling the optimal solution increases
5 Simulation Results
In this section we fuse our optimization algorithm (Algo-rithm 3) with the control and estimation methods given inSection 3 to create an integrated solution to multiple missileallocation and control for ballistic target interception Wefirst give detailed results for two specific missions in orderto give a better understanding of how the algorithm worksand thenwe demonstrate the effectiveness of the algorithmbycomparing its performance toHeuristic and noncollaborativemethods in Monte-Carlo simulations
In all experiments we use the following parameters forSAM defense system
(i) Thenumber of SAMs119873SAM is 5 and they are arrangedin two parallel lines with the back line containing3 SAMs and the front line containing 2 SAMs Thearrangement can be seen in the upper left corners ofFigures 7 and 9 Radar is placed in front of the frontline
(ii) SAM velocity is set to Mach = 35(iii) Maximum number of iterations for CMA-MV is set
to 50 The number of samples is set to 100 for bothcontinuous and discrete variables The rest of theparameters are tuned manually
51 Results for a High Altitude-Low Velocity Target First weexamine a mission where the ballistic target has relativelylow kinetic energy The ballistic targetrsquos initial conditionsare set to 80000 meters of altitude and speed equivalent toMach number 5 This is a less challenging scenario since theballistic threat has relatively longer time till it hits the groundgiving the enough time for the filters of the SAM defensesystem to converge Resulting trajectory of the target and
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 International Journal of Aerospace Engineering
Input Objective Function 119891 R119899times 0 1
119889rarr R
Number of Continuous Samples per Iteration 120582Number of Discrete Samples per Iteration 120582Number of Iterations 119899iter Weights for continuous samples 119908
1 119908
120583 Weights for
discrete samples 1199081015840
1 119908
1015840
1205831015840 Initial guess x0
z0
Output Approximate optimal solution xlowast zlowast
Initialization(1) m
0 120590
0C
0larr I p
120590 p
119888 119896 larr 0 120583
119908larr 1sum
120583
119895=1119908
2
119895
(2) while 119896 lt 119899iter do Fix the discrete variables to z119896 and use CMA to solve for x
(3) x119896+1larr Call CMA (Algorithm 1) with (x) = 119891(x z119896
)
Fix the continous variables and sample from the multivariate Bernoulli distrubution
(4) for 119894 in 1 1205821015840 do
Sample(5) z
119894sim B(m1015840
119896C1015840
119896) using Algorithm 2
119894larr 119891(x119896+1
z119894)
Sort the candidate Solutions Based on Their Cost
(6) z11205821015840 larr z
119905(1) z
119905(1205821015840) such that
119905(1)le sdot sdot sdot le
119905(1205821015840)
Move the mean to low cost solutions
(7) m1015840
119896+1larr sum
1205831015840
119894=1119908
1015840
119894z
119894
Update The Covariance Matrix
(8) C1015840
119896+1larr
1
1205821015840minus 1
sum1205831015840
119894=1119908
1015840
119894(z
119894minusm1015840
119896+1)(z
119894minusm1015840
119896+1)⊤
(9) z119896+1larr z119896
119905(1)
(10) 119896 larr 119896 + 1
After the algorithm stops output the best sample
(11) xlowastlarr x119896
119905(1) zlowast
larr z119896
119905(1)
Algorithm 3 Covariance Matrix Adaptation with Mixed Variables (CMA-MV)
15
10
5
0minus6minus4
minus20
8
6
2
0
4
Ze
(m)
Xe(m)Y
e (m)
times104
times104
times104
Figure 7 Interception of a high altitude-low velocity ballistic targetRed line depicts the trajectory of the ballistic target Algorithmchooses only 1 missile for interception
the launched missile is shown in Figure 7 The optimizationalgorithm also recognizes that filters have enough time toconverge in this case and launches only a single missile Themissile intercepts the ballistic threat with a miss distance lessthan 1meter
Figures 8(a) 8(b) and 8(c) show the estimation perfor-mance of the filter of themissile for range-to-go line of sightand target acceleration estimations It can be seen that filtersconverged rapidly in the terminal phase of themissionTheseplots justify the decision of the algorithm to launch only asingle missile in this case algorithm recognized that a singlefilter would yield sufficient performance and did not chooseto allocate more missiles in order to keep the cost as close tominimumas possible Also note that no collaborative filteringis performed in this mission since only a single missile islaunched
52 Results for a Low Altitude-High Velocity Target To com-plement the results of the previous subsection now we lookat a mission that corresponds to high kinetic energy targetFor this simulation the ballistic targetrsquos initial conditions areset to 60000meters of altitude and speed equivalent to Machnumber 7 This scenario is much more challenging since thetargetrsquos established time of impact is much shorter Resultingtrajectory of the target and the launched missile is shown inFigure 9 In this case it is seen that the algorithm launches 3collaborative missiles to intercept the targetThe interceptionis achieved with a miss distance of approximately 1meter
Figures 10(a) 10(b) and 10(c) show the estimation per-formance of the filter of the missiles for range-to-go lineof sight and target acceleration estimations averaged overthe 3 launched missiles For comparison performance of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 11
Time (sec)40 45 50 55 60 65 70 75 80
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
5
10
15
20
25
30
(a) Norm of the range-to-go estimation error versus time in terminalphase
Time (sec)40 45 50 55 60 65 70 75 80
Line
of s
ight
estim
atio
n er
ror (
deg)
0
0002
0004
0006
0008
001
0012
(b) Norm of the line of sight estimation error versus time in terminal phase
Time (sec)40 45 50 55 60 65 70 75 80
0
1
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
(c) Norm of the target acceleration estimation error versus time interminal phase
Figure 8 Filter performance versus time in terminal phase for intercepting high altitude-low velocity target
the individual filters is also plotted in these figures Theseindividual filter performances correspond to the case wheremissiles do not cooperate hence no information is sharedbetween them Examining these plots gives us a good insighton algorithmrsquos decision to launch 3 missiles It is seen thatindividual filters did not have enough time to converge forthis high kinetic energy target hence launching a singleor even two missiles would result in high miss distancesAlgorithm made the necessary trade-off analysis and foundout that launching 3 collaborative missiles would generateenough information flow for estimators to converge
53 Monte-Carlo Results for Multiple Scenarios The previoussimulation results demonstrated that the algorithm yieldssound decisions on selected scenarios However in orderto truly assess the performance of the algorithm a widerange of initial conditions that corresponds to different
ballistic threat should be analyzed Also we need to comparethe performance of the algorithm to alternative methodsFor this purpose we conducted a Monte-Carlo test over100 randomly sampled initial conditions for the ballistictarget The initial altitude of the target was sampled in theinterval [40000 80000] meters and the speed was sampledin the Mach number interval [5 8] The following alternativemethodologies are compared
(i) Heuristic Collaborative Interception In this simplealgorithm no launch condition or missile alloca-tion optimization is conducted This method alwayslaunches the same number of missiles that are closestto the ballistic target at the beginning of the simula-tion Launch conditions of the missiles are always setto 0 heading and 90 degrees of pitch angle Missilesuse collaborative filtering for interception
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 International Journal of Aerospace Engineering
8
6
4
2
0minus500005000
6
3
2
0
5
1
4
Ze
(m)
Xe(m)
Ye (m
)
times104
times104
Figure 9 Interception of a low altitude-high velocity ballistic target Red line depicts the trajectory of the ballistic target Algorithm launches3 missiles to intercept the target
Table 1 Average number of missiles and average miss distance obtained by different methods averaged over 100 random initial conditionsof the ballistic target
Average number of missiles launched Average miss distance (m)Heuristic method 1 missile 1 502078 plusmn 60021Heuristic method 2 missiles collaborative 2 49022 plusmn 4108Heuristic method 3 missiles collaborative 3 6055 plusmn 112Optimized noncollaborative 105 plusmn 023 32055 plusmn 5077Optimized collaborative 156 plusmn 034 143 plusmn 022
(ii) Optimized Noncollaborative Interception In thismethodology optimization algorithm CMA-MV isused for optimizing the launching conditions and theallocation of the missiles However missiles do notrun collaborative filtering algorithms on-board
(iii) Optimized Collaborative Interception This is theapproach developed in this paper The CMA-MValgorithm (Algorithm 3) is used for optimizationof launch conditions and missile allocation and themissiles run collaborative filtering algorithms
Table 1 depicts the results of the Monte-Carlo analysisWe see that Heuristic methodrsquos performance gets better asthe number of missiles used by the method increases Thisis expected since increased number of missiles translatesto improved estimation performance However even using3 missiles for all conditions does not reduce the averagemiss distance substantially This is due to fact that Heuristicmethod does not optimize the launch conditions hence themissile autopilots do not have enough time to restore themissiles into the desired trajectories in the terminal phaseOnthe other hand optimized noncollaborative method yieldssubstantially lower miss distances than Heuristic methods
that use 1 or 2 missiles while launching only 105 missileson average This is because the optimized noncollaborativemethod optimizes the launch conditions for the missileswhich leads to improved interception performanceHoweverthis method is outperformed by Heuristic method that uses3missiles because optimized noncollaborative method doesnot utilize collaborative filters hence the algorithm can nottake advantage of improved estimation performance gainedby launching multiple missiles against high kinetic energyballistic targets
Finally we see that the approach developed in thispaper the optimized collaborative method outperforms thecompared approaches in terms of both resourcemanagementefficiency and miss distance This is because unlike thecompared approaches the developed method optimizes thelaunch conditions and missile allocation simultaneously andhence it is able to assess the right trade-off between thenumber of missiles launched and attainable miss distance
6 Conclusions and Future Work
In this work we have developed a novel probabilistic searchalgorithm for allocation and launch condition optimization
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 13
Time (sec)0 10 20 30 40 50 60
Rang
e-to
-go
estim
atio
n er
ror (
m)
0
20
40
60
80
100
120
Individual estimatorCooperative estimator
(a) Average norm of the range-to-go estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
Line
of s
ight
estim
atio
n er
ror (
deg)
0
001
002
003
004
005
006
007
008
Individual estimatorCooperative estimator
(b) Average norm of the line of sight estimation error versus time interminal phase for individual and collaborative estimators
Time (sec)0 10 20 30 40 50 60
2
3
4
5
6
7
8
9
10
Targ
et ac
cele
ratio
n es
timat
ion
erro
r (m
s2)
Individual estimatorCooperative estimator
(c) Average norm of the target acceleration estimation error versus timein terminal phase for individual and collaborative estimators
Figure 10 Filter performance versus time in terminal phase for intercepting low altitude-high velocity target
of multiple missiles intercepting a ballistic target Throughsimulation studies we have verified that the algorithmmakes sound decisions and yields more efficient resourcemanagement and lowermiss distance compared to alternativeapproaches
Future works involve generalization of the problem intointercepting multiple ballistic targets developing alternativecooperative estimationcontrol algorithms and investigationof theoretical properties of the CMA-MV algorithm
Competing Interests
The authors declare that they have no competing interests
References
[1] B-Z Naveh and A Lorber ldquoTheater ballistic missile defenserdquoProgress in Astronautics and Aeronautics vol 192 pp 1ndash3972001
[2] G M Siouris Missile Guidance and Control Systems SpringerNew York NY USA 2004
[3] R H Chen and J L Speyery ldquoTerminal and boost phaseintercept of ballistic missile defenserdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and ExhibitHonolulu Hawaii USA August 2008
[4] T-K Wang and L-C Fu ldquoA guidance strategy for multi-playerpursuit and evasion game in maneuvering target interceptionrdquo
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 International Journal of Aerospace Engineering
in Proceedings of the 9th Asian Control Conference (ASCC rsquo13)June 2013
[5] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010
[6] E Daughtery and Z Qu ldquoOptimal design of cooperativeguidance law for simultaneous strikerdquo in Proceedings of the 53rdIEEE Annual Conference on Decision and Control (CDC rsquo14) pp988ndash993 Los Angeles Calif USA December 2014
[7] V Shaferman and Y Oshman ldquoCooperative interception in amulti-missile engagementrdquo in Proceedings of the AIAA Guid-ance Navigation and Control Conference Chicago Ill USAAugust 2009
[8] V Shaferman and T Shima ldquoCooperative multiple-modeladaptive guidance for an aircraft defending missilerdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1801ndash18132010
[9] Y Liu N Qi and J Shan ldquoCooperative interception withdouble-line-of-sight measuringrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference Boston MassUSA August 2013
[10] V Shaferman and T Shima ldquoCooperative optimal guidancelaws for imposing a relative intercept anglerdquo Journal of Guid-ance Control and Dynamics vol 38 no 8 pp 1395ndash1408 2015
[11] L Wang H Fenghua and Y Yu ldquoGuidance law design fortwo flight vehicles cooperative interceptionrdquo in Proceedings oftheAIAAGuidance Navigation andControl Conference Kissim-mee Fla USA January 2015
[12] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003
[13] P Teng H Lv J Huang and L Sun ldquoImproved particle swarmoptimization algorithm and its application in coordinated aircombat missile-target assignmentrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 2833ndash2837 IEEE Chongqing China June 2008
[14] J Wang X-G Gao Y-W Zhu and H Wang ldquoA solvingalgorithm for target assignment optimization model based onACOrdquo in Proceedings of the 6th International Conference onNatural Computation (ICNC rsquo10) vol 7 pp 3753ndash3757 IEEEYantai China August 2010
[15] J Wang and Y-W Zhu ldquoA solving algorithm for target assign-ment optimization model based on SArdquo in Proceedings of theInternational Conference on Artificial Intelligence and Com-putational Intelligence (AICI rsquo10) pp 489ndash493 Sanya ChinaOctober 2010
[16] M Wei G Chen K Pham E Blasch and Y Wu ldquoGame theo-retic target assignment approach in ballistic missile defenserdquo inProceedings of the SPIE Defense and Security Symposium Inter-national Society for Optics and Photonics 2008
[17] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 41 no 3 pp 598ndash606 2011
[18] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996
[19] Z B TatasModeling and autopilot design for a SCUD Type bal-listicmissile [MS thesis] Department of Electrical EngineeringHacettepe University Ankara Turkey 2006
[20] P Zarchan Tactical and Strategic Missile Guidance vol 199 ofProgress in Aeronautics and Astronautics AIAA 4th edition2002
[21] I Moran and D T Altilar ldquoThree plane approach for 3D trueproportional navigationrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference San Francisco Calif USAAugust 2005
[22] E K P Chong and H Stanislaw An Introduction to Optimiza-tion vol 76 John Wiley amp Sons New York NY USA 2013
[23] N Hansen A Auger R Ros S Finck and P Posık ldquoComparingresults of 31 algorithms from the black-box optimization bench-marking BBOB-2009rdquo in Proceedings of the 12th Annual Con-ference Companion on Genetic and Evolutionary Computation(GECCO rsquo10) ACM July 2010
[24] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[25] J H Macke P Berens A S Ecker A S Tolias and M BethgeldquoGenerating spike trains with specified correlation coefficientsrdquoNeural Computation vol 21 no 2 pp 397ndash423 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Related Documents
