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Research ArticleInitial Alignment Error On-Line Identification
Based onAdaptive Particle Swarm Optimization Algorithm
Weilin Guo , Yong Xian, Bing Li, and Leliang Ren
Xi’an Research Institute of High Technology, Xi’an, Shaanxi
710025, China
Correspondence should be addressed to Weilin Guo;
[email protected]
Received 23 August 2018; Accepted 13 December 2018; Published 31
December 2018
Academic Editor: Waldemar T. Wójcik
Copyright © 2018 Weilin Guo et al. This 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.
To solve the problem of high accuracy initial alignment of
strap-down inertial navigation system (SINS) for ballistic missile,
anon-line identification method of initial alignment error based on
adaptive particle swarm optimization (PSO) is proposed. Firstly,a
complete navigation model of SINS is established to provide the
accurate model basis for subsequent numerical
optimizationcalculation.Then setting the initial alignment error as
the optimization parameter and regarding the minimum deviation
betweenSINS andGPS output as the objective function, the error
parameter optimizationmodel is designed. At the same time,
themutationidea of genetic algorithm (GA) is introduced into the
PSO; thus the adaptive PSO is adopted to identify the initial
alignmenterror on-line. The simulation results show that it is
feasible to solve the initial alignment error identification
problem of SINS byintelligent optimization algorithm. Compared with
the standard PSO algorithm and the GA, the adaptive PSO algorithm
has thefastest convergence speed and the highest convergence
precision, and the initial pitch error and the initial yaw error
precision arewithin 10 and the initial azimuth error precision is
within 25. The navigation accuracy of SINS is improved effectively.
Finally,the feasibility of the adaptive PSO algorithm to identify
the initial alignment error is further validated based on the test
data.
1. Introduction
Initial alignment for strap-down inertial navigation
system(SINS) plays an important role in the navigation operation
ofthe ballistic missile. The main purpose of initial alignment isto
establish the initial attitudematrix, and the quality of
initialalignment will affect the navigation accuracy of SINS
directly[1] and thus affect the missile firing accuracy
ultimately.Therefore, improving the initial alignment accuracy of
SINSis of great significance to improve the performance of
ballisticmissile weapon.
The propagation process of the initial alignment error ofSINS is
a complex nonlinear problem. The previous solutionis to linearize
the nonlinear problem, and the filtering algo-rithms based on
Kalman filter are widely adopted [2–6]. Afast SINS initial
alignment scheme based on the disturbanceobserver and Kalman filter
is proposed to estimate themisalignment angles in [4], and an
adaptive extendedKalmanfilter algorithm combined with
innovation-based adaptiveestimation is proposed in [7], while these
filtering algorithmsoften have some disadvantages, such as the
difficulty of
model establishing, poor observability of parameters, andlong
alignment time [8]. Consequently, this paper rejects thetraditional
research method based on analytic simplification,linearization and
filtering, attempting to convert the initialalignment problem of
SINS into parameter optimizationidentification problem.The complete
nonlinear optimizationmodel is established, and the intelligent
optimization algo-rithm is used to realize the on-line
identification of the initialalignment error of SINS.
To solve the initial alignment problem of inertial system,the
application of genetic algorithm (GA) in the initialalignment of
SINS on the static base is studied based onthe intelligent
optimization algorithm [9, 10]. The precisionof the initial
alignment error is about 2 in [9], and thealignment accuracy needs
to be improved. At the same time,the GAhas the disadvantages of
large computational capacity,low efficiency, and complicated
coding, while the particleswarm optimization (PSO) is simple in
structure, fast inconvergence, and easy to implement and has the
advantageof dealing with complex systems. The transfer
alignmentbetween the master inertial sensor and the slave
inertial
HindawiMathematical Problems in EngineeringVolume 2018, Article
ID 3486492, 10 pageshttps://doi.org/10.1155/2018/3486492
http://orcid.org/0000-0003-0405-2986https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/3486492
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2 Mathematical Problems in Engineering
sensor is realized by PSO, and the influence of the maneuveron
alignment accuracy is analyzed in [11]. The PSO is appliedinto the
parameter optimization of compass alignment circuitin SINS, and the
performance of strap-down gyrocompassinitial alignment is improved
in [12]. In view of the aboveanalysis and research results, the PSO
algorithm is consideredto solve the initial alignment problem of
the ballistic missileSINS.
In this paper, a complete SINS navigation model isestablished,
and then the error parameter optimizationmodelis constructed based
on the minimum deviation betweenthe position parameters outputted
by SINS and the positionparameters measured by GPS. The mutation
idea of GA isintroduced into PSO, and the inertial weight and
learningfactors are improved to obtain adaptive PSO. Finally,
thestandard PSO, GA, and adaptive PSO are adopted to identifythe
initial alignment error with the flight software of a certaintype
of ballistic missile. At the same time, the test dataare used to
inspect the identification effect of intelligentoptimization
algorithm.
The rest of this paper is organized as follows. A nav-igation
model of SINS, including initial alignment errormodel and error
compensation model, is established in thesecond section. In the
third section, an error parameteroptimization model is constructed
and the adaptive PSOis designed for the ballistic missile SINS. In
the fourthsection, the simulation for identification of initial
alignmenterror is given to demonstrate the feasibility of the
intelligentoptimization algorithm. Finally, we conclude in the
fifthsection.
2. Establishment of SINS Navigation Model
Themain coordinate frames used in this paper different fromother
references are defined as follows: the body coordinateof the
ballistic missile is the orthogonal reference framealigned with the
inertial measurement unit (IMU) axes, andthe origin locates the
mass of the ballistic missile, the x-axis along the longitudinal
direction forward, opposite thedirection of gravity, the y-axis is
perpendicular to the lon-gitudinal direction upward, the z-axis
along the transversaldirection right, completing a right-handed
system. Launchinertial coordinate (inertial coordinate) is a
coordinate whoseorigin is the launch point, the x-axis points to
target in thelocal level of launch point, and the y–axis is
perpendicularto the launch point’s local level (upward) and
constitutesthe right-handed Cartesian coordinates with the axes
ofx, z. The inertial coordinate is used as the navigationframe.
2.1. Initial Alignment Error Model. The initial alignmenterrors
of SINS, including initial pitch angle error Δ𝜑0, initialyaw angle
errorΔ𝜓0, and initial azimuth errorΔ𝛾0, are causedby vertical
degree, installation error, and aiming error of theballistic
missile. The initial attitude matrix 𝐴 between thebody coordinate
and the inertial coordinate can be describedas follows by using the
quaternion (𝑞0, 𝑞1, 𝑞2, 𝑞3):
𝐴
= [[[[𝑞21 + 𝑞20 − 𝑞22 − 𝑞23 2 (𝑞1𝑞2 − 𝑞0𝑞3) 2 (𝑞0𝑞2 + 𝑞1𝑞3)2
(𝑞1𝑞2 + 𝑞0𝑞3) 𝑞20 + 𝑞22 − 𝑞21 − 𝑞23 2 (𝑞2𝑞3 − 𝑞0𝑞1)2 (𝑞1𝑞3 − 𝑞0𝑞2)
2 (𝑞0𝑞1 + 𝑞2𝑞3) 𝑞20 + 𝑞23 − 𝑞21 − 𝑞22
]]]](1)
The initial values of the quaternion (𝑞0, 𝑞1, 𝑞2, 𝑞3) are𝑞0 =
𝑞00 − 𝛾02 𝑞20𝑞1 = 𝑞10 + 𝛾02 𝑞30𝑞2 = 𝑞20 + 𝛾02 𝑞00𝑞3 = 𝑞30 − 𝛾02
𝑞10
(2)
where
𝑞00 = √22 (1 − Δ𝜑02 )𝑞10 = −𝜓02 𝑞00𝑞20 = 𝜓02 𝑞30𝑞30 = √22 (1 +
Δ𝜑02 )
(3)
From (2) and (3), we can see that the initial alignmenterror
will affect the initial values of the quaternion and thusaffect the
calculation precision of the initial attitude matrix𝐴.
2.2. Error Compensation Model of SINS. During the flightcourse
of ballistic missile, the inertial measurement unit(IMU) of SINS,
including gyroscope and accelerometer, canmeasure apparent
acceleration and angular velocity in realtime and output the data
in pulse form. The pulse outputsof the accelerometer and the
gyroscope under a navigationcycle are (Δ𝑁𝑤𝑥1, Δ𝑁wy1, Δ𝑁𝑤𝑧1) and
(Δ𝑁𝑏𝑥1, Δ𝑁𝑏𝑦1, Δ𝑁𝑏𝑧1),respectively, in which the pulse number
remains the same asthe actual missile, and the outputs are
integers.
After the IMU sends the pulse signals to the onboardcomputer,
the error compensation calculation is completedby the onboard
computer in real time. The equations ofthe error compensation
calculation of the apparent velocityincrement and the angular
increment under a navigationcycle in body coordinate are shown in
(4) and (5), respectively[13]:
[[[Δ𝑊𝑥𝑏0Δ𝑊𝑦𝑏0Δ𝑊𝑧𝑏0
]]] =[[[[[[[[[
(Δ𝑁𝑤𝑥1 − 𝐾0𝑥)𝐾1𝑥(Δ𝑁𝑤𝑦1 − 𝐾0𝑦)𝐾1𝑦(Δ𝑁𝑤𝑧1 − 𝐾0𝑧)𝐾1𝑧
]]]]]]]]]
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Mathematical Problems in Engineering 3
[[[Δ𝑊𝑥𝑏Δ𝑊𝑦𝑏Δ𝑊𝑧𝑏
]]] =[[[Δ𝑊𝑥𝑏0Δ𝑊𝑦𝑏0Δ𝑊𝑧𝑏0
]]] −[[[
0𝐸𝑥𝑦Δ𝑊𝑥10𝐸𝑥𝑧Δ𝑊𝑥10]]](4)
[[[Δ𝜃𝑥𝑏0Δ𝜃𝑦𝑏0Δ𝜃𝑧𝑏0
]]] =[[[[[[[[[[[
Δ𝑁𝑏𝑥1𝐾𝑥Δ𝑁𝑏𝑦1𝐾𝑦Δ𝑁𝑏𝑧1𝐾𝑧
]]]]]]]]]]]− [[[
𝐷0𝑥𝐷0𝑦𝐷0𝑧]]]
[[[Δ𝜃𝑥𝑏Δ𝜃𝑦𝑏Δ𝜃𝑧𝑏
]]] = [[[Δ𝜃𝑥𝑏0Δ𝜃𝑦𝑏0Δ𝜃𝑧𝑏0
]]] −[[[[
0 𝐸𝑦𝑥 𝐸𝑧𝑥𝐸𝑥𝑦 0 𝐸𝑧𝑦𝐸𝑥𝑧 𝐸𝑦𝑧 0]]]][[[Δ𝜃𝑥𝑏0Δ𝜃𝑦𝑏0Δ𝜃𝑧𝑏0]]]
− [[[𝐷1𝑥 𝐷2𝑥 𝐷3𝑥𝐷1𝑦 𝐷2𝑦 𝐷3𝑦𝐷1𝑧 𝐷2𝑧 𝐷3𝑧
]]][[[Δ𝑊𝑥1Δ𝑊𝑦1Δ𝑊𝑧1]]]
(5)
where𝐾𝑖, 𝐾0𝑖, 𝐾1𝑖, 𝐷0𝑖, 𝐷1𝑖, 𝐷2𝑖, 𝐷3𝑖, and 𝐸𝑖𝑗 are the tool
errorcoefficient of IMU which are calibrated in the missile
techni-cal site.Δ𝑊𝑖𝑏0 andΔ𝜃𝑖𝑏0 are the calculated intermediate
valuesof the apparent velocity and angle increment,
respectively.Δ𝑊𝑖𝑏 and Δ𝜃𝑖𝑏 are the apparent velocity and the
angleincrement in body coordinate after the error
compensation,respectively, where 𝑖, 𝑗 = 𝑥, 𝑦, 𝑧 represents the
three direc-tions of the x, y, and z axes.
2.3. Calculation of Velocity and Position in Inertial
Coordinate.According to the error compensation model of SINS,
theapparent velocity increment in body coordinate can be
cal-culated. The apparent velocity increment in body coordinateis
converted to inertial coordinate, and it can be presented as
[[[Δ𝑊𝑥𝑎Δ𝑊𝑦𝑎Δ𝑊𝑧𝑎]]] = 𝐴[[[
Δ𝑊𝑥𝑏Δ𝑊𝑦𝑏Δ𝑊𝑧𝑏]]] (6)whereΔ𝑊𝑥𝑎,Δ𝑊𝑦𝑎, and Δ𝑊𝑧𝑎 are the projections
of apparentvelocity increment in inertial coordinate and the matrix
𝐴is calculated by (1), where the calculation equation of
thequaternion is as follows:
[[[[[𝑞0𝑞1𝑞2𝑞3]]]]]𝑗= [[[[[[
𝑞0 −𝑞1 −𝑞2 −𝑞3𝑞1 𝑞0 −𝑞3 𝑞2𝑞2 𝑞3 𝑞0 −𝑞1𝑞3 −𝑞2 𝑞1 𝑞0]]]]]]𝑗−1
[[[[[[[[[[
1 − 18Δ𝜃2𝑗(12 − 148Δ𝜃2𝑗)Δ𝜃𝑥𝑏(12 − 148Δ𝜃2𝑗)Δ𝜃𝑦𝑏(12 −
148Δ𝜃2𝑗)Δ𝜃𝑧𝑏
]]]]]]]]]]𝑗
(7)
where Δ𝜃𝑗 = √Δ𝜃2𝑥𝑏 + Δ𝜃2𝑦𝑏 + Δ𝜃2𝑧𝑏. According to (7),
thequaternion is calculated by the recursion method whichbased on
the value of the previous moment, recursion to getthe quaternion of
the current moment.The initial value of thequaternion can be
calculated by (2).
By integrating the apparent velocity increment in
inertialcoordinate, the recursive value of the velocity and
positionin inertial coordinate at any moment of the missile can
beobtained, and the calculation equation is as follows:
[[[V𝑥𝑎V𝑦𝑎V𝑧𝑎
]]]𝑗 =[[[V𝑥𝑎V𝑦𝑎V𝑧𝑎
]]]𝑗−1 +[[[Δ𝑊𝑥𝑎Δ𝑊𝑦𝑎Δ𝑊𝑧𝑎
]]]𝑗 +[[[𝑔𝑥𝑎𝑔𝑦𝑎𝑔𝑧𝑎
]]]𝑗Δ𝑇2
+ [[[𝑔𝑥𝑎𝑔𝑦𝑎𝑔𝑧𝑎
]]]𝑗−1Δ𝑇2
(8)
[[[𝑥𝑎𝑦𝑎𝑧𝑎]]]𝑗 =
[[[𝑥𝑎𝑦𝑎𝑧𝑎]]]𝑗−1 +
[[[V𝑥𝑎V𝑦𝑎V𝑧𝑎
]]]𝑗−1 Δ𝑇+([[[
Δ𝑊𝑥𝑎Δ𝑊𝑦𝑎Δ𝑊𝑧𝑎]]]𝑗 +
[[[𝑔𝑥𝑎𝑔𝑦𝑎𝑔𝑧𝑎
]]]𝑗−1 Δ𝑇)Δ𝑇2
(9)
where V𝑥𝑎, V𝑦𝑎, V𝑧𝑎 are the projections of velocity in
inertialcoordinate;𝑥𝑎, 𝑦𝑎, 𝑧𝑎 are the projections of position in
inertialcoordinate; 𝑔𝑥𝑎, 𝑔𝑦𝑎, 𝑔𝑧𝑎 are the projections of gravity
accel-eration in inertial coordinate, which can be computed bythe
ellipsoid gravity acceleration model; Δ𝑇 is the
navigationcycle.
3. Error Parameter Optimization Modeland Algorithm Design
3.1. Establishment of Error Parameter Optimization Model
3.1.1. Select the Optimization Variable. Set the initial
align-ment error of SINS as the optimization parameter; that is,𝑋 =
(𝑥1, 𝑥2, 𝑥3)T = (Δ𝜑0, 𝜓0, 𝛾0)T (10)3.1.2. Determine the Objective
Function. Regard the mini-mum deviation between the position
parameters outputtedby SINS and the position parameters measured by
GPS as theobjective function, namely,
𝐽 (𝑋) = min𝑁𝑢𝑚∑𝑖=1
√𝛿𝑥 (𝑖)2 + 𝛿𝑦 (𝑖)2 + 𝛿𝑧 (𝑖)2𝛿𝑥 (𝑖) = 𝑥𝐼𝑁𝑆 (𝑖) − 𝑥𝐺𝑃𝑆 (𝑖)𝛿𝑦 (𝑖) =
𝑦𝐼𝑁𝑆 (𝑖) − 𝑦𝐺𝑃𝑆 (𝑖)𝛿𝑧 (𝑖) = 𝑧𝐼𝑁𝑆 (𝑖) − 𝑧𝐺𝑃𝑆 (𝑖)
(11)
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4 Mathematical Problems in Engineering
where𝑥𝐼𝑁𝑆(𝑖),𝑦𝐼𝑁𝑆(𝑖), and 𝑧𝐼𝑁𝑆(𝑖) are the position
parametersoutputted by SINS at the navigation cycle 𝑖, which can
becalculated by (9); 𝑥𝐺𝑃𝑆(𝑖),𝑦𝐺𝑃𝑆(𝑖), and 𝑧𝐺𝑃𝑆(𝑖) are the
positionparameters measured by GPS at the navigation cycle i;
𝛿𝑥(𝑖),𝛿𝑦(𝑖), and 𝛿𝑧(𝑖) are the position deviations. 𝑁𝑢𝑚 is
thenumber of navigation cycle used to optimize alignment, and𝑁𝑢𝑚 ⋅
Δ𝑇 ≤ 𝑇𝑠, 𝑇𝑠 is the total simulation test time.3.2. Design of
Adaptive Particle Swarm Optimization Algo-rithm. PSO is an
intelligent optimization algorithm for find-ing optimal region of
search spaces through the interactionof individuals in a swarm
[14], and it has been widelyapplied in the fields of aeronautics
and astronautics becauseof its advantages such as fast convergence,
simple structure,and strong versatility [15–17]. However, the
standard PSOalgorithm has the disadvantage of premature
convergenceand low efficiency in optimization iteration.
Consequently,the standard PSO is improved to the adaptive PSO, and
theimproved strategy is as follows:(1) The mutation idea is
introduced into PSO algorithmbased on GA algorithm. Mutation
operation is an importantmeans to increase population diversity in
GA algorithm,which can expand search space and avoid falling into
localoptimization. Therefore, the mutation idea is introducedinto
PSO algorithm, and through the mutation operation ofparticles, the
population can jump out of the current localoptimal position in
foraging process and search for a largerspace range. Thus, the
global search ability is enhanced toovercome the shortcoming of
premature convergence of PSO.(2) The learning factors and inertia
weight of PSOare designed as dynamic adjustment form to improve
theconvergence speed and overcome the disadvantage of lowefficiency
of PSO in the late optimization. By dynamicallyadjusting the values
of the learning factors and inertia weightof PSO algorithm, it is
ensured that PSO has strong globalsearch ability at the initial
stage of optimization and a fastsearch speed at the later stage of
optimization.
The basic optimization flow of initial alignment errorparameter
by adaptive PSO is shown in Figure 1, and thespecific steps for
optimization calculations are as follows.
Step 1 (initialize the population). Set particle population
size𝑁, maximum iterations 𝑀, maximum position 𝑥max, andminimum
position 𝑥min = −𝑥max. The initial position andthe initial velocity
of the particles are randomly generated,and each particles fitness
value of initialization population iscalculated to determine the
individual best and the global bestof the particles.
Step 2 (update particle swarm velocity and position).
Theequation for calculating the velocity and position of
particlesis as follows:
V𝑖𝑗 (𝑘 + 1) = 𝑤 (𝑘) V𝑖𝑗 (𝑘) + 𝑟1𝑐1 (𝑘) (𝑃𝑖𝑗 (𝑘) − 𝑥𝑖𝑗 (𝑘))+ 𝑟2𝑐2
(𝑘) (𝑃𝑔𝑗 (𝑘) − 𝑥𝑖𝑗 (𝑘))𝑥𝑖𝑗 (𝑘 + 1) = 𝑥𝑖𝑗 (𝑘) + V𝑖𝑗 (𝑘 + 1)(12)
where 𝑥𝑖𝑗(𝑘) and V𝑖𝑗(𝑘) (𝑗 = 1, 2, 3) are the position
andvelocity of the jth dimension of particle i at iteration k;
𝑤(𝑘)is the inertia weight; 𝑟1 and 𝑟2 are the random
numbersdistributed in the range [0, 1]; 𝑐1(𝑘) and 𝑐2(𝑘) are the
learningfactors;𝑃𝑖𝑗(𝑘) and𝑃𝑔𝑗(𝑘) are the individual best and the
globalbest at iteration k, respectively.
In order to enhance the global exploration ability in theearly
stage of optimization and improve the convergencespeed at the later
stage of optimization, the inertia weight isdesigned to be the
dynamic adjustment mode. The inertialweight has a large value at
the beginning, and the weightdecreases with the increase of the
number of iterations.Consequently, the inertia weight 𝑤(𝑘) can be
designed as
𝑤 (𝑘) = 𝑤max − ( 𝑘𝑀)2 (𝑤max − 𝑤min) (13)Learning factors 𝑐1(𝑘)
and 𝑐2(𝑘) are the factors which
can control degree of self-learning and group learning
ofparticles, respectively. The particle swarm needs a
largeself-learning ability to enhance the global search effect
inthe early stage of optimization, and it needs a large
grouplearning ability to speed up the convergence speed in the
lateoptimization. Therefore, with the increase of the number
ofiterations, learning factors 𝑐1(𝑘) continues to decrease
andlearning factors 𝑐2(𝑘) increases gradually and the
learningfactors can be described as
𝑐1 (𝑘) = 𝑐1max − ( 𝑘𝑀)2 (𝑐1max − 𝑐1min)𝑐2 (𝑘) = 𝑐2min + ( 𝑘𝑀)2
(𝑐2max − 𝑐2min)
(14)
Step 3. Calculate fitness value of population, and
updateindividual best and global best. According to (11), the
fitnessvalue 𝑓(𝑥𝑖(𝑘 + 1)) of particles is calculated at iteration 𝑘
+ 1and compared with the previous fitness value 𝑓(𝑃𝑖(𝑘))
ofindividual best. If 𝑓(𝑥𝑖(𝑘 + 1)) < 𝑓(𝑃𝑖(𝑘)), the
individualbest is updated, namely, 𝑃𝑖(𝑘 + 1) = 𝑥𝑖(𝑘 + 1).
Similarly,comparing the fitness value of the individual best𝑓(𝑃𝑖(𝑘
+ 1))at iteration 𝑘 + 1 with the previous fitness value of the
globalbest 𝑓(𝑃𝑔(𝑘)), if 𝑓(𝑃𝑖(𝑘 + 1)) < 𝑓(𝑃𝑔(𝑘)), the global best
ofparticles is updated, namely, 𝑃𝑔(𝑘 + 1) = 𝑃𝑖(𝑘 + 1).Step 4.
Mutation operation and update the individual bestand the global
best again.
(A) Select Mutation Particles. The fitness values of
particleswarm at iteration 𝑘+1were sorted by descending order,
andthe numbers𝑁𝑚 of particles arranged in front and with
largefitness value were selected as the mutation objects.
(B) Mutation Operation. First, generate the random number𝑟, and
then compare the sizes of the random number 𝑟 and𝑟𝑚, where 𝑟𝑚 is
the mutation probability of particle. If thegenerated random number
𝑟 is less than 𝑟𝑚, the mutationoperation is performed as
follows:𝑥𝑖𝑗 (𝑘) = (2𝑟 − 1) (𝑥max (𝑗) − 𝑥min (𝑗)) (15)
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Mathematical Problems in Engineering 5
Initialize population
Update inertia weight and learning factors
Calculate the fitness of particle swarm
Update individual best value and global best value
Update particle position and velocity
Output optimal solution
End
no
yes
Mutation operation
Reach maximum iterations or meet precision requirements
Start
no
yes
The generated random
Select mutation particles and generate the random number r
Update individual best value and global best value
number r is less than
Figure 1: Calculation flow chart of adaptive PSO.
where 𝑟 is the random number distributed in the range [0, 1]and
𝜆 is the mutation coefficient. Otherwise, go to Step 5 forthe next
calculation.
(C) Update Particle’s Best. For the new particles generatedby
mutation, update the individual best and the global bestagain.
Step 5 (end condition). If reaching maximum iterations,namely, 𝑘
> 𝑀, or the search results satisfy the accuracyrequirement, the
calculation is stopped and the optimalparameter of initial
alignment error is output. Otherwise,return Step 2 for the next
generation calculation.
4. Simulation Experiment and Result Analysis
The simulation study is presented to confirm the feasibility
ofinitial alignment error on-line identification based on adap-tive
PSO in this section. Firstly, the simulation conditionsare set up,
then the simulation experiment and results areanalyzed. At last,
the test data is introduced.
4.1. SimulationCondition Settings. Thesimulation conditionsof
the initial alignment error optimization model for SINS areset as
follows:(1) Setting initial pitch angle error is 90, initial yaw
angleerror is -90 and initial azimuth error is -150 .The
navigationcycleΔ𝑇 is 0.1 s, the number of navigation cycles𝑁𝑢𝑚 is
500,and the total simulation test time 𝑇𝑠 is 50 s.(2) Tool error of
SINS.The constant and random drifts ofgyro are chosen as 0.02∘/h
and 0.01∘/h/Hz, respectively. Theconstant and random drifts of
accelerators are chosen as 100ug and 50 ug/Hz, respectively.(3)GPS
navigation error. The position error is 5.0 m, andthe velocity
error is 0.1 m/s.(4) Simulation conditions of PSO algorithm. The
maxi-mum iterations 𝑀 is 100, the population size 𝑁 is 40, andthe
maximum position 𝑥max = (300 300 500). The inertiaweights 𝑤max and
𝑤min are 0.8 and 0.4, respectively. Thelearning factors 𝑐1max and
𝑐2max are all 3.0, and the learningfactors 𝑐1min and 𝑐2min are all
0.5. The mutation numbers ofparticles 𝑁𝑚 are 10, the mutation
probability 𝑟𝑚 is 0.2, andthe mutation coefficient 𝜆 is 0.5.
-
6 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 100Evolution time
GA optimizationStandard PSOAdaptive PSO
70
75
80
85
90
95
100
105
Initi
al p
itch
angl
e err
or()
Figure 2: Convergence process of initial pitch angle error.
4.2. Simulation Results and Analyses. Based on the aboveerror
parameter optimization model and taking the flightsoftware of a
certain ballistic missile as the simulationexperiment environment,
the standard PSO algorithm, theGA, and the adaptive PSO algorithm
are adopted to optimizethe initial alignment error parameter of
SINS, and theconvergence process of the initial pitch angle error,
the initialyaw angle error, and the initial azimuth angle error are
shownin Figures 2–4, and the fitness convergence diagrams of
thethree algorithms are shown in Figure 5. The optimizationresults
of initial alignment error are shown in Table 1. Itshows the
accuracy and convergence speed of three differentalgorithms.
It is shown from Table 1 that the GA, the standard PSO,and the
adaptive PSO can be used to optimize the initialalignment error
parameter; the maximum residual error ofthe initial alignment error
of three algorithms is not morethan 30, the optimization
calculation time is less than 4.0s, and the convergence speed is
fast. It shows that it is feasibleand effective in identifying the
initial alignment error of SINSby using intelligent optimization
algorithm.
From Figures 2–5, we can see that the adaptive PSO has afastest
convergence speed than the GA and the standard PSO.It is shown
fromTable 1 that the residual errors of initial pitchangle and the
initial yaw angle calculated by the adaptive PSOare less than 10,
and the residual error of the initial azimuthis less than 25, which
shows that the adaptive PSO canimprove the convergence accuracy of
the initial alignmenterror.
The initial alignment error parameter calculated by theadaptive
PSO is compensated, and then the navigationparameters of the SINS
are recalculated. The deviationbetween the navigation parameters
obtained by error com-pensation and the actual navigation
parameters of the missileis called the optimization residual. By
simulation, the SINSposition error, GPS position error and
optimization residual
of position are shown in Figure 6, and the SINS velocity
error,GPS velocity error, and optimization residual of velocity
areshown in Figure 7. The root mean square (RMS) statisticresults
of navigation parameter error are listed in Table 2.
From Figures 6 and 7, we can find out that the optimiza-tion
residuals of position and velocity are not only far lessthan SINS,
but also significantly less than GPS navigationsystem. It is shown
from Table 2 that the RMS position errorsof SINS are within 10 m,
and the RMS velocity errors of SINSare within 1 m/s while the RMS
position errors are less than1 m, and the RMS velocity errors are
less than 0.1 m/s, afterthe initial alignment error is identified
and compensated bythe adaptive PSO algorithm. Obviously, compared
with SINS,the RMS errors of the navigation parameters compensated
bythe adaptive PSO are reduced by 10 times, and the
navigationaccuracy is greatly improved. The simulation results
indicatethat the adaptive PSO can effectively identify the
initialalignment error and improve the navigation accuracy
ofSINS.
4.3. Test Data Validation. In order to verify the
effectivenessof adaptive PSO algorithm to identify initial
alignment error,the data collected from the test are analyzed.
Among them,the gyroscope constant drift and the accelerometer
constantbias are about 0.01∘/h and 100𝜇g, respectively; SINS
dataupdate frequency is 10.0HZ; GPS data update frequency is1.0Hz.
The actual measured output values of SINS and GPSwere collected and
recorded during the test. The deviationbetween the position and the
speed of SINS andGPS output iscalled the position error and the
velocity error of SINS, thenthe position error and the velocity
error curve of SINS areshown in Figures 8 and 9, respectively.
Based on the above test data, the adaptive PSO algorithmis used
to optimize the initial alignment error parameters,and the results
of the optimization parameters are shown inTable 3.
-
Mathematical Problems in Engineering 7
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
10 20 30 40 50 60 70 80 90 100Evolution time
GA optimization Standard PSOAdaptive PSO
Initi
al y
aw an
gle e
rror
()
Figure 3: Convergence process of initial yaw angle error.
−500
−400
−300
−200
−100
0
100
200
300
10 20 30 40 50 60 70 80 90 100Evolution time
GA optimizationStandard PSOAdaptive PSO
Initi
al az
imut
h an
gle e
rror
()
Figure 4: Convergence process of initial azimuth angle
error.
Table 1: Optimization results of initial alignment error.
Optimization algorithm Parameter Initial alignment error
Calculation time/(s)Δ𝜑0/() Δ𝜓0/() Δ𝛾0/()GA optimization
Optimization value 92.42 -108.67 -122.62 3.85
Residual error 2.42 -18.67 27.38
Standard PSO Optimization value 91.03 -102.35 -124.53
3.46Residual error 1.03 -12.35 25.47
Adaptive PSO Optimization value 90.51 -99.29 -127.56
3.48Residual error 0.51 -9.29 22.44
-
8 Mathematical Problems in Engineering
Table 2: RMS statistical results of navigation parameter
errors.
Error type Navigation parameter Pure SINS error Compensation
error
Position error (m)𝑥𝑎 6.9748 0.1730𝑦𝑎 2.4885 0.3683𝑧𝑎 10.7109
0.0960
Velocity error (m/s)V𝑎𝑥 0.3748 0.0147V𝑎𝑦 0.1790 0.0306V𝑎𝑧 0.6336
0.0044
GA optimization Standard PSOAdaptive PSO
60
70
80
90
100
110
120
130
Fitn
ess v
alue
10 20 30 40 50 60 70 80 90 1000Evolution time
Figure 5: Fitness convergence diagram.
Table 3: Results of optimization parameters.
Optimizationparameter
Lowerbound
Upperbound
OptimizationvalueΔ𝜑0/() -90.00 90.00 90.00Δ𝜓0/() -90.00 90.00
73.46Δ𝛾0/() -180.00 180.00 -104.04
Table 4: RMS statistical results of errors.
Error type NavigationparameterInitialerror
Optimizationerror
Position error(m)
𝑥𝑎 6.2535 5.4109𝑦𝑎 10.3145 8.2356𝑧𝑎 8.6935 7.5713Velocity
error(m/s)
V𝑎𝑥 0.4784 0.2770V𝑎𝑦 0.7956 0.6541V𝑎𝑧 0.2769 0.2540
The initial alignment error calculated by the adaptivePSO is
compensated, and then the navigation parameters ofSINS are
recalculated. After calculation and compensation,the position and
velocity error curves are shown in Figures10 and 11 respectively,
and the error RMS statistic results areshown in Table 4.
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 50Time (s)
−10
0
10
20
30
Posit
ion
erro
r in
z axi
s (m
)
−5
0
5
10
Posit
ion
erro
r in
y ax
is (m
)
−20−15−10
−505
Posit
ion
erro
r in
x ax
is (m
)
SINS errorGPS erroroptimization residual
Figure 6: Position error simulation curve.
From Figures 10 and 11, we can see that the optimizationerrors
of the position and velocity are less than the initialerrors of the
position and velocity of the test data, afterthe initial alignment
error is compensated. It is shown fromTable 4 that the RMS errors
of the position and velocityof optimization calculation are less
than the RMS errors ofthe test data. Therefore, it is proved that
the adaptive PSOalgorithm is effective in identifying the initial
alignment errorand can improve the navigation accuracy of missile
flight.
5. Conclusion
The initial alignment error identification of ballistic
missileSINS is studied in this paper. The real and complete
naviga-tion model of SINS is established, which provides an
accuratemodel basis for the initial alignment error identification.
At
-
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 35 40 45 50−0.8−0.6−0.4−0.2
00.2
Velo
city
erro
r in
x ax
is (m
/s)
SINS errorGPS erroroptimization residual
0 5 10 15 20 25 30 35 40 45 50
5 10 15 20 25 30 35 40 45 500Time (s)
−0.20.10.40.7
11.3
Velo
city
erro
r in
z axi
s (m
/s)
−0.10
0.10.20.30.40.5
Velo
city
erro
r in
y ax
is (m
/s)
Figure 7: Velocity error simulation curve.
0 10 20 30 40 50 60−20−10
010
Δxa
(m)
0 10 20 30 40 50 600
102030
Δya
(m)
0 10 20 30 40 50 60−40−20
020
t (s)
Δza
(m)
Figure 8: Position error curve of SINS.
the same time, the error parameter optimization model
isdesigned, and the initial alignment error is identified on-line
by the intelligent optimization algorithm. What is more,the inertia
weight and learning factors of PSO are designedas dynamic
adjustment form to improve search speed andsearch accuracy, and the
mutation operation of the GA is
0 10 20 30 40 50 60
0 10 20 30 40 50 60−1
012
Δvy
a (m
/s)
0 10 20 30 40 50 60−1−0.5
00.5
t (s)
Δvz
a (m
/s)
−1−0.5
00.5
Δvx
a (m
/s)
Figure 9: Velocity error curve of SINS.
0 10 20 30 40 50 60−20
0
20
Δxa
(m)
0 10 20 30 40 50 600
10
20
30
Δya
(m)
−50
0
50
0 10 20 30 40 50 60t (s)
Δza
(m)
initial erroroptimization error
initial erroroptimization error
initial erroroptimization error
Figure 10: Position optimization error curve.
introduced into the PSO algorithm to jump out local optimalvalue
and enhance global convergence ability.
The simulation results show that the intelligent optimiza-tion
algorithm is efficient to solve the problem of initialalignment
error identification. Of course, the results showthat the adaptive
PSO algorithm has fastest search efficiencyand highest convergence
accuracy than the standard PSOalgorithm and the GA, and the
residuals of the initial pitchangle and the initial yaw angle are
less than 10, and theresidual of the initial azimuth is less than
25. Finally, thevalidity of the adaptive PSO algorithm to identify
the initialalignment error is validated based on the test data.
Therefore,
-
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60−1
−0.5
0
0.5Δ
vxa (
m/s
)
0 10 20 30 40 50 60−1
0
1
2
Δvy
a (m
/s)
−1
−0.5
0
0.5
0 10 20 30 40 50 60t (s)
Δvz
a (m
/s)
initial erroroptimization error
initial erroroptimization error
initial erroroptimization error
Figure 11: Velocity optimization error curve.
the content of this paper has a certain reference value for
theimprovement of the initial alignment accuracy of the
ballisticmissile SINS.
Data Availability
The test data used to support the findings of this study havenot
been made available because the data are currently underembargo,
and requests for data will be considered by thecorresponding author
at the right time.
Conflicts of Interest
The authors declare that there are no conflicts of
interestregarding the publication of this paper.
Acknowledgments
This work was supported in part by the National NaturalScience
Foundation of China (no. 61374054).
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