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Research ArticleImpacts of Deflection Nose on Ballistic
Trajectory Control Law
Bo Zhang,1,2 Shushan Wang,1 Mengyu Cao,3 and Yuxin Xu1
1 State Key Laboratory of Explosion Science and Technology,
Beijing Institute of Technology, Beijing 100081, China2 College of
Engineering, Bohai University, Liaoning 121013, China3 AVIC Keeven,
Beijing 100081, China
Correspondence should be addressed to Bo Zhang; [email protected]
and Yuxin Xu; [email protected]
Received 9 January 2014; Accepted 27 January 2014; Published 2
March 2014
Academic Editor: Shen Yin
Copyright © 2014 Bo Zhang et al.This is an open access article
distributed under theCreativeCommonsAttribution License,
whichpermits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The deflection of projectile nose is aimed at changing the
motion of the projectile in flight with the theory of motion
control andchanging the exterior ballistics so as to change its
range and increase its accuracy. The law of external ballistics
with the deflectablenose is considered as the basis of the design
of a flight control system and an important part in the process of
projectile development.Based on the existing rigid external
ballistic model, this paper establishes an external ballistic
calculationmodel for deflectable noseprojectile and further
establishes the solving programs accordingly. Different angle of
attack, velocity, coefficients of lift, resistance,and moment under
the deflection can be obtained in this paper based on the previous
experiments and emulation researches. Inthe end, the author pointed
out the laws on the impaction of external ballistic trajectory by
the deflection of nose of the missile.
1. Introduction
Targets in the modern war are changing constantly. Theincreasing
targets with high mobility, extensive depth, andmultiple levels put
a higher demand on the renovation ofthe missile technology.
Developing the long distance, highaccuracy, and powerful missile is
regarded as the ultimatedestination for this technology. Human
beings are dedicatedto developing the lighter and more intelligent
missile nowand even for quite far future. Research on the
creativeintelligent control technology has very important
significanceand practical value, where the external ballistics
plays akey role in this modern missile control technology. Thebomb
integrates sensor with actuator, which was developedbased on
intelligent material and structure. The sensordetects environment
information continuously during flightand passes it to a
missile-borne computer; the actuatorthen deforms the nose or wing
of the missile accordingto the decision made by computer. The local
structuraldeformations give the missile additional aerodynamic
forceand moment, which result in velocity, direction, and centerof
mass coordinates changes; the center of mass motion isalso
affected. The trajectory is controlled by changing the
missile’s aerodynamic characteristics and ballistic
trajectory.Trajectory control can improve the missile’s target
accuracyand range capability.With the increasing demands
ofmodernwar and advancement of scientific technology, the
externalballistics has enjoyed rapid development in the past 20
years.The external ballistics is a science of mechanics based
onmotion stability, vibration theory, and aerodynamics, relyingon
modern control theory and computer technology. Onthe other hand, it
is closely related to the measurementtechnology. The external
ballistics deals with the flight,especially bullets, gravity bombs,
aviation bomb, rocket,missile, and the like. Traditionally, the
external ballisticsonly studies the behavior of the projectile and
providesthe simple trajectory design and firing tables. Some
controlmethods such as data driven fault tolerant control
[1–5],robust control [6–9], and so forth have been applied inthis
field. Recently, the external ballistics has been widelystudied in
the research fields like trajectory calculation, flightstability,
initial disturbance analysis, dissemination, control-lable
trajectory, integral optimization design, and experimenttechnology
and parameter identification [10–13]. Experts andprofessors put
forward a number of new projects regardingthis subject.
Hindawi Publishing CorporationMathematical Problems in
EngineeringVolume 2014, Article ID 984840, 6
pageshttp://dx.doi.org/10.1155/2014/984840
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2 Mathematical Problems in Engineering
2. Rigid-Body Dynamic Modeling forDeflectable Nose
Projectile
2.1. Analysis of External Ballistic Trajectory Features andBasic
Assumption. Compared with the external trajectory ofnormal
projectile, projectile with deflectable trajectory hasthe following
features.
(1) In the flight of a normal projectile, the aerodynamicforce
imposed on it is not controllable, while, forthe projectile with
deflectable nose, its trajectory ispartially controllable because
there is a control unitcontrolling the deflection of nose so that
its externaltrajectory is no longer a free trajectory.
(2) Although the trajectory of deflectable nose is
control-lable, it is different from the controllable trajectoryof
other projectiles. The control unit can only controlthe ballistic
trajectory to some degree or within somecertain section of
trajectory, other than completecontrol.
We can see that the external trajectory is equippedwith the
features of both normal projectile trajectory andcontrollable
trajectory.
The equation of trajectory of the deflectable nose projec-tile
is set under the following basic assumptions.
Assumption 1. Standard meteorological condition, calmwind, and
no rain.
Assumption 2. Nomass eccentricity in the projectile; centroidof
the whole projectile is kept in the same point after the
nosedeflection, and the plane is exactly symmetric.
Assumption 3. Omitting the changes to Coriolis inertial forceand
gravity acceleration along with the changes in latitude.
Assumption 4. Omitting the changes to earth curvature andgravity
acceleration along with the changes in height; gravityacceleration
𝑔 ≈ 9.8m/s2 and its direction are vertical toground.
Assumption 5. No spinning to the projectile (i.e., to neglectthe
Magnus force, moment, damping moment, and angularmoment in the
empennage); the projectile flies within thefore-and-aft plane.
2.2. Force Analysis. According to the aerodynamic simula-tion,
due to the existence of an upward relative angle in thenose and the
projectile body, the air around the upper surfaceand lower surface
areas is not equal. Pressure in the lowersurface is larger than the
upper; thus the nose will suffer fromlift force [14].
In the flight of the projectile, regardless of the spinning,in
order to measure the effects from each force and joinforces, forces
and moments are simplified to the centroid ofprojectile. For
convenient for illustrating, is given in Figure 1.
Rx
Ry
V
𝛿
Mz
O
Figure 1: Diagram for simplifying aerodynamic forces.
(1) 𝑅𝑥is drag and expressed as
𝑅𝑥= (𝜌V2
2) 𝑆𝑀𝐶𝑥, (1)
where 𝐶𝑥is drag coefficient and 𝑆
𝑀is reference area
(m2).(2) 𝑅𝑦is lift and expressed as
𝑅𝑦= (𝜌V2
2) 𝑆𝑀𝐶𝑦, (2)
where 𝐶𝑦is lift coefficient.
(3) 𝑀𝑧is static moment and expressed as
𝑀𝑧= (𝜌V2
2) 𝑆𝑀𝑙𝑚𝑧, (3)
where𝑀𝑧is moment coefficient.
3. Establishing Rigid-Body Dynamic Modeling
The motion equation system of the projectile deals withthe
relationship of forces, moments, and motion parameters.It is
composed of equations of dynamics, kinematics, andgeometrical
relationship. The motion of the projectile in thespace is strictly
divided into 6 free degrees, where 4 freedegrees can indicate its
motion laws through the equation, ifyaw angle and roll are
neglected. By shifting the coordinatesystems and referring the
motion equation, equations ofrigid-body external trajectory can be
obtained [14–16].
3.1. The Coordinate System and the Conversion of the Coor-dinate
System. The movement rules of the projectile do notchange with the
selection of the coordinate system. Thecoordinate system affects
the difficulties of establishing andsolving the motion equations or
whether it is easy to readthe motion equations. References [15, 16]
are a commoncoordinate system.
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Mathematical Problems in Engineering 3
(1) Ground coordinate system𝐴𝑥𝑦𝑧 is used to determinespatial
coordinate of the projectile’s centroid. Theearth can be regarded
as holding still; that is, theground coordinate system can be
regarded as theinertial coordinate system.
(2) Body coordinate system 𝑂𝑥1𝑦1𝑧1: body coordinate
system is a moving coordinate system and bodycoordinate system
𝑂𝑥
1𝑦1𝑧1describes the change in
the spatial attitude of the projectile, which is relativeto the
pitch, yaw, and roll motion of the groundcoordinate system
𝐴𝑥𝑦𝑧.
(3) Trajectory coordinate system 𝑂𝑥2𝑦2𝑧2is formed by
the reference coordinate system’s twice rotations.Trajectory
coordinate system is fixed with the velocityvector 𝑉 of the bullet
centroid, so it is also a movingcoordinate system. It is used to
establish the kineticsscalar equation of the projectile centroid’s
movementand study the trajectory’s characteristics.
(4) Velocity coordinate system 𝑂𝑥3𝑦3𝑧3: velocity coordi-
nate system is also a moving coordinate system, andit is usually
used to study the aerodynamic of theprojectile.
To establish the projectile motion equation often needsto
convert the force or torque in a coordinate system intoanother
coordinate system, so it needs to establish theconversion
relationship between the coordinate systems, andthese relationships
can be obtained by projection or matrixoperation.
3.2. Kinematics Equation. Kinematics equation, namely,
therelationship equations between the bullet’s center of
gravityposition and the velocity, mainly includes the
kinematicsequation of the centroid relative to the ground
coordinatesystem and relative centroid of missile body
kinematicsequation.
3.2.1. Centroid Movement Kinematics Equation. When westudy the
motion law of missile body, it is generally relativeto the ground,
therefore, based on the ground coordinatesystem, the relationship
between velocity coordinate systemand the ground coordinate system,
to get the missile body’scentroid kinematics equation:
𝑑𝑥
𝑑𝑡= 𝑉 cos 𝜃,
𝑑𝑦
𝑑𝑡= 𝑉 sin 𝜃,
(4)
where 𝜃 is trajectory angle.
3.2.2. Kinematics Equation around the Centroid.
Kinematicsequation is the relation between the angular
displacementand the angular velocity. It can be obtained by
taking
advantage of the relationship between the missile body
coor-dinate system and the ground coordinate system. Describethe
kinematics equation of missile body relative to centroid’srotation
4.2:
𝑑𝜗
𝑑𝑡= 𝜔𝑧, (5)
where 𝜗 is pitch angle and𝜔𝑧is angular velocity of the
missile
relative to ground coordinates.
3.3. Dynamic Equation. Missile body’smotion can be decom-posed
into centroid in vertical plane of translation and themissile body
rotation around the centroid. The role on themissile body’s force
and moment, through the relationshipbetween the two coordinate
systems, and projection to ballis-tic coordinates system according
to the Newtonian dynamicsrelations are as follows.
3.3.1. The Missile Body’s Movement around the CentroidDynamic
Equation. Suppose that the missile body’s rota-tional velocity
relative to the ground coordinates is 𝜔, andthe missile body swings
only in the longitudinal plane, so therotation dynamics equation
can be obtained:
𝑑𝜔𝑧
𝑑𝑡=𝑀𝑧
𝐽𝑧
, (6)
where 𝑑𝜔𝑧/𝑑𝑡 is the rotation angle acceleration of missile
body, 𝑀𝑧is the torque of all the external force in the
missile body to centroid, and 𝐽𝑧is the missile body
equatorial
rotation inertia.
3.3.2. The Missile Body Centroid Dynamic Equation
𝑚(𝑑𝑉
𝑑𝑡) = −𝑅
𝑋− 𝑚𝑔 sin 𝜃,
𝑚𝑉(𝑑𝜃
𝑑𝑡) = 𝑅
𝑌− 𝑚𝑔 cos 𝜃,
(7)
where 𝑑𝑉/𝑑𝑡 is the tangential acceleration; 𝑉(𝑑𝜃/𝑑𝑡) isnormal
acceleration, 𝜃 is trajectory angle, and𝛼 is flight
attackangle.
There are six equations,𝑉, 𝜃, 𝛼, 𝜗, 𝑥, 𝑦, 𝜔 seven
unknownnumbers. And the certain relationship exists between
angles𝜃, 𝛼, and 𝜗. Therefore, another angle relation needs to
besupplied:
𝜃 = 𝜗 − 𝛼. (8)
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4 Mathematical Problems in Engineering
Table 1: Range of different shooting angles.
Shooting angle (∘) 3 10 20 30 40 50Ranges (m, 𝑉 = 1020m/s) 2,040
2,275 2,369 2,324 2,149 1,835Ranges (m, 𝑉 = 300m/s) 597 1,152 1,407
1,500 1,483 1,377
10 20 30 40 501100
1150
1200
1250
1300
1350
1400
1450
1500
1550
Rang
e (m
)
Angle of elevation (∘)
Initial velocity 300m/s
(a) Range changing with different shooting angles in 𝑉 =
300m/s
1800
1900
2000
2100
2200
2300
2400
Rang
e (m
)10 20 30 40 50
Angle of elevation (∘)
Initial velocity 1020m/s
(b) Range changing with different shooting angles in 𝑉 =
1,020m/s
Figure 2: Comparison of different ranges with different shooting
angles in 2 velocities.
Then (9) can be derived by (4)–(8):𝑑𝑉
𝑑𝑡=−𝑅𝑋− 𝑚𝑔 sin 𝜃𝑚
,
𝑑𝜃
𝑑𝑡=𝑅𝑌− 𝑚𝑔 cos 𝜃𝑚𝑉
,
𝑑𝜔𝑧
𝑑𝑡=𝑀𝑧
𝐽𝑧
,
𝑑𝑥
𝑑𝑡= 𝑉 cos 𝜃,
𝑑𝑦
𝑑𝑡= 𝑉 sin 𝜃,
𝑑𝜗
𝑑𝑡= 𝜔𝑧,
𝜃 = 𝜗 − 𝛼.
(9)
4. Simulation Calculations andResults Analysis
In this paper, trajectory calculation program is designedbased
on the rigid-body ballistic trajectory modeling toconduct the
stimulation calculation for the external trajectoryof the
deflectable nose projectile. During the ascending pass,the
projectile flies in the uncontrollable trajectory, whileafter the
boosting phase, the smart controlling device will betransformed and
cause the deflection in nose. A deflectionangle relative to the
projectile is formed so that the projectile
can keep flying along this angle. General analytical solutionsof
the trajectory equations are always obtained under someassumptions
of approximation. Only when we need to takean accurate numerical
integration methods for solving thetrajectory equations, some
numerical integral methods areused such as Euler method,
Runge-Kutta method and Adamsmethod. Four-stage Runge-Kutta method
is applied in oursimulation calculations because of that this
methods is ahigher accuracy in calculation and easy for designing
theprogram [17–19].
The nose deflection controlling unit is made of piezo-ceramic
[20]. And its deflection scope is controlled within0 degree to 8
degrees, with initial velocity at 300m/s to1,020m/s. We make the
simulation calculation on externalballistic trajectory in different
shooting angles and nosedeflection angles so as to evaluate
different nose deflectionangles’ impacts on range and further to
achieve themaximumcorrection [21, 22].
4.1. Impacts of Shooting Angle on External Ballistic
Trajectory.When a projectile is launched at a certain initial
velocity,there will be a best angle for maximum range. In order
toevaluate the impacts of shooting angle on external trajectory,we
calculate and summarize the data of different shootingangles and
different shooting range accordingly. We set theinitial velocity of
the projectile at 1,020m/s and 300m/s,respectively; nose deflection
angle at 4∘; shooting angles at 3∘,10∘, 20∘, 30∘, 40∘, and 50∘,
respectively, as shown in Table 1.
Figure 2 shows initial velocity at 300m/s and 1,020m/sand the
comparison of different ranges in different shooting
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Mathematical Problems in Engineering 5
0 2 4 6 8
0
5
10
15
20
25Ra
nge (
m)
Nose declination (∘)
Initial velocity 300m/sInitial velocity 1020m/s
(a) Curve of range changing over nose deflection
0 2 4 6 8
0
5
10
Shoo
ting
heig
ht (m
)
Nose declination (∘)
Initial velocity 300m/sInitial velocity 1020m/s
(b) Curve of shooting height changing over nose deflection
Figure 3: Curve for corrections of range and shooting height
growing over the nose deflection under tow kinds of velocity.
angles and clearly demonstrates that the shooting anglehas
significant impacts on the external ballistic trajectory.Through
the calculation, we work out that velocity at1,020m/s, shooting
angle at 22∘, the range of the projectile is2,371m, which is the
longest range. While the shooting angleincreases from 3∘ to 22∘,
the range is increased along withthe shooting angle. When shooting
angle further increasesfrom 22∘ to 55∘, however, the range is
decreased. In theother case, velocity is 300m/s and the best
shooting angleis 33∘, which makes the longest range to 1507m. Where
theshooting angle increases from 3∘ to 33∘, the range is
increased;while shooting angle increases from 33∘ to 50∘, the range
isdecreased.The best shooting angles with two kinds of velocitywill
be adopted in the later calculation and analysis.
4.2. Impacts of Nose Deflection Angle on External
BallisticTrajectory. In order to explore the impacts of nose
deflectionon the external ballistic trajectory, we calculate
externalballistic trajectory with different nose deflection angle
indifferent initial velocity and best shooting angle. We set
thevelocity at 1,020m/s, shooting angle at 22∘, after the
boostingstage, and the nose deflection angle at 2∘, 4∘, 6∘, and 8∘
andcalculate their external ballistic trajectory. On the other
hand,we set the velocity at 300m/s, shooting angle at 33∘, andthe
nose deflection angle at 2∘, 4∘, 6∘, and 8∘ and calculateagain
their external trajectory. Shooting ranges and shootingheight
changed with the different deflection angles, as shownin Tables 2
and 3.
From the data on its shooting height and range along thetime of
flight, we can know that the correction value growsover the nose
deflection degree. Figure 3 shows the curve ofcorrections of
shooting range and height growing over thenose deflection.
A conclusion can be drawn that initial velocity at 300m/sand
nose deflection angle of 8∘ can provide the maximum
Table 2: Shooting ranges and shooting height with initial
velocity1,020m/s and shooting angle 22∘.
Nose deflection (∘) 0 2 4 6 8Shooting ranges (m) 2,366 2,369
2,375 2,379 2,388Correction values ofshooting ranges (m) 0 3 9 13
22
Correction values ofshooting height (m) 0 5 9 11 12
Table 3: Shooting ranges and shooting height with initial
velocity300m/s and shooting angle 33∘.
Nose deflection (∘) 0 2 4 6 8Shooting ranges (m) 1,498 1,499
1,500 1,502 1,504Correction values ofshooting ranges (m) 0 1 2 4
6
Correction values ofshooting height (m) 0 4 5 7 8
correction value, where maximum correction of shootingrange is
6m and shooting height is 8m.While initial velocityat 1,020m/s and
nose deflection angle of 8∘ can provide themaximum correction
value, where maximum correction ofshooting range is 22m and
shooting height is 12m.
5. Conclusions
From the stimulation calculation and analysis on trajectoryof
deflectable nose projectile, the following conclusions canbe
obtained.
(1) Because the nose deflection is upward, pressurearound the
upper surface of the projectile is differentfrom the lower surface.
The pressure gap will cause
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6 Mathematical Problems in Engineering
the lift force, whichmakes the projectile rotate aroundthe
centroid and finally create a new angle of attack.Under a same
working condition, the larger angle fornose deflection, themore
changes to the aerodynamiccoefficients and the greater correction
obtained.
(2) As to the range extension, when the missile body isset at
the initial velocity of 1,020m/s, its range willbe longer than at
the velocity of 300m/s. When it isset at the deflection angle of 8∘
and shooting speedof 1,020m/s, the shooting range can be extended
with0.93%; when shooting speed is set at 300m/s, 0.40%of range can
be extended. And with the increase ofangle of attack, shooting
range and shooting heightpresent a rising trend. Thus, the
conclusion can beobtained that comparatively more obvious effects
ofnose deflection control to ballistic trajectory at thehigh speed
with large angle of attack conditions canbe seen than those at low
speed with small angle ofattack.
(3) In the future studies, by applying the impacts of angleof
pitch on ballistic trajectory in the yaw angle, wecan build up the
relevant models and programs tomake the calculations on
two-dimensional trajectoryof deflectable nose projectile and
external ballistictrajectory.
In conclusion, the deflection in nose can cause theprojectile to
change its motion in flight. Therefore, it isfeasible and effective
to control the ballistic trajectory bychanging the nose deflection
angle.
Conflict of Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
References
[1] S. Yin, S. X. Ding, A. H. A. Sari, and H. Hao,
“Data-drivenmonitoring for stochastic systems and its application
on batchprocess,” International Journal of Systems Science, vol.
44, no. 7,pp. 1366–1376, 2013.
[2] S. Yin, S. X. Ding, A. Haghani, H. Hao, and P. Zhang,
“Acomparison study of basic data-driven fault diagnosis andprocess
monitoring methods on the benchmark TennesseeEastman process,”
Journal of Process Control, vol. 22, no. 9, pp.1567–1581, 2012.
[3] X. Zhao, L. Zhang, P. Shi, and M. Liu, “Stability of
switchedpositive linear systems with average dwell time
switching,”Automatica, vol. 48, no. 6, pp. 1132–1137, 2012.
[4] X. Zhao, L. Zhang, P. Shi, andH.Karimi, “Novel stability
criteriafor T-S fuzzy systems,” IEEE Transactions on Fuzzy
Systems,2013.
[5] S. Yin, X. Yang, and H. R. Karimi, “Data-driven
adaptiveobserver for fault diagnosis,” Mathematical Problems in
Engi-neering, vol. 2012, Article ID 832836, 21 pages, 2012.
[6] X. Zhao, P. Shi, and L. Zhang, “Asynchronously switched
controlof a class of slowly switched linear systems,” Systems &
ControlLetters, vol. 61, no. 12, pp. 1151–1156, 2012.
[7] S. Yin, G. Wang, and H. Karimi, “Data-driven design of
robustfault detection system for wind turbines,”Mechatronics,
2013.
[8] X. Zhao, L. Zhang, P. Shi, and H. Karimi, “Robust control
ofcontinuous time systems with state dependent uncertaintiesand its
application to electronic circuits,” IEEE Transactions onIndustrial
Electronics, vol. 61, no. 8, pp. 4161–4170, 2014.
[9] S. Yin, H. Luo, and S. Ding, “Real-time implementation of
fault-tolerant control systems with performance optimization,”
IEEETransactions on Industrial Electronics, vol. 64, no. 5, pp.
2402–2411, 2014.
[10] W. Zhong-Yuan, “Aerodynamic force and external trajec-tory
optimization design of Armour-Piercing Discard Sabot(APDS),” Acta
Aerodynamica Sinica, vol. 11, no. 3, pp. 270–276,1993.
[11] X. Ming-You, Advanced External Ballistics, Higher
EducationPress, Beijing, China, 2003.
[12] W. Zhong-Yuan and Z. Wei-Ping, Theory and Method ofExternal
Ballistics Design, Science Press, Beijing, China, 2004.
[13] J. Sahu, Time-Accurate Numerial Prediction of Free Flight
Aero-dynamics of Projectiles, IEEE Computer Society, 2006.
[14] H. Zi-Peng, X. Xiao-Zhaong, and Z. Ying-Yi, External
Ballistics,National Defence Industry Press, Beijing, China,
2000.
[15] Q. Xing-Fang, L. Rui-Xiong, and Z. Ya-Nan, Missile
FlightMechanics, Beijing Institute of Technology Press,
Beijing,China, 2006.
[16] W. Fei, W. Zhi-Jun, and W. Guo-Dong, “Numerical
calculationof extended-range artillery rocket base on intelligent
material,”Journal of Projectiles Rockets Missiles and Guidance,
vol. 20, no.4, pp. 325–327, 2004.
[17] F. Wang, G.-D. Wu, Z.-J. Wang, and X.-H. Kang,
“Numericalcalculation of aerodynamic characteristics of shell with
attackangle at the shell head,” Journal of North China Institute
ofTechnology, vol. 26, no. 3, pp. 177–179, 2005.
[18] W. Fang-Hai, W. Zhi-Jun, and W. Guo-Dong, “A new methodof
ballistic correction by controlling the nose angle of
rocket,”Journal of Projectiles Rockets Missiles and Guidance, vol.
26, no.2, pp. 928–930, 2006.
[19] H. Ji-Chuan, L. Zhan-Chen, and X. Zeng-Hui, “Research ofthe
exterior trajectory correction technology based on
piezo-electricity ceramic,” Journal of Projectiles, Rockets,
Missiles andGuidance, vol. 28, no. 6, pp. 201–204, 2008.
[20] L. Shuai, W. Jin-Zhu, Z. Wei-Jun et al., “Trajectory
simulationfor guided rocket,” Foreign Electronic Measurement
Technology,vol. 30, no. 12, pp. 69–71, 2011.
[21] W. Meng-Long, W. Hua, and H. Jing, “Rockets
trajectorycorrection method based on nose cone swinging,” Journal
ofDetection & Control, vol. 33, no. 4, pp. 23–27, 2011.
[22] Z. Ying-Hun, T. Guo-Hui, and D. Ming-Li, “Research
onaerodynamic characteristics and ballistic characteristics of
fin-stabilized rocket at high altitude,” Journal of Projectiles,
Rockets,Missiles and Guidance, vol. 31, no. 2, pp. 142–144,
2011.
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