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An Optimal Design Scheme of Missile Trajectory
To cite this article: Shilin Zhang 2022 J. Phys.: Conf. Ser. 2220 012012
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2022 International Conference on Automation and Space Science & Technology
Journal of Physics: Conference Series 2220 (2022) 012012
IOP Publishing
doi:10.1088/1742-6596/2220/1/012012
1
An Optimal Design Scheme of Missile Trajectory
Shilin Zhang
1College of Aerospace Science and Engineering, National University of Defense
Technology, Changsha, Hunan, 410073
2Hunan Provincial Key Laboratory of Image Measurement and Vision Navigation,
Changsha, Hunan, 410073
zhang_sl@nudt.edu.cn
Abstract—The optimization of missile trajectory is an important part of missile
design. Its difficulty lies in the complex force of the missile and the different laws of
the trajectory parameters in each stage. The existing trajectory simulation models have
problems such as complex models, low accuracy, simplifying the initial motion state
of the missile, and failing to fully reflect the characteristics of the active and passive
motions. In order to solve these problems, this paper extracts important model
variables on the basis of analyzing the force of the missile at different stages, and
establishes a ballistic model with time-varying parameters. At the same time,
according to the engineering design method of the missile flight program, the change
rule of the pitch angle of the active phase of the missile with time is drawn up. Finally,
an intercontinental ballistic missile is used as the object to simulate and analyze the
proposed model. The results show that the method proposed in this paper can design
and optimize missile trajectory well.
1. Introduction
The optimization of missile trajectory is a classic problem in aircraft design and simulation. Based on
the given parameters, designing the optimal missile trajectory (usually the trajectory with the farthest
range) through computer simulation has important guiding significance for the actual engineering
realization[1]. In order to describe the flight process of the missile, previous scholars have established
various ballistic models. However, the existing ballistic models are often too complex and redundant
in parameters, which are disadvantageous to the efficiency and accuracy of the solution[2]. In this
regard, this paper proposes a simplified model for missile trajectory optimization, which greatly
improves the calculation efficiency without affecting the accuracy, providing a new idea for the
optimization of missile trajectory design.
2. Model establishment and simplification
2.1. mathematical model
During the movement of the missile, the instantaneous equilibrium assumption is used to ignore the
movement of the missile around the center of mass and only consider the movement of the center of
mass[3]. The factors that affect the acceleration of the center of mass are: thrust P, aerodynamic force
2022 International Conference on Automation and Space Science & Technology
Journal of Physics: Conference Series 2220 (2022) 012012
IOP Publishing
doi:10.1088/1742-6596/2220/1/012012
2
R, gravitational force mg, additional Coriolis force F
and centrifugal inertial force F
. Consider in
the launch coordinate system:
Thrust P:
𝑃
𝑃
𝑃
𝐺
𝑃
0
0
(1)
Aerodynamic force R:
𝑅
𝑥
𝑅
𝑦
𝑅
𝑧
𝐺
𝐵
𝐶
𝑥
𝑞𝑆
𝑚
𝐶
𝑦
𝛼
𝑞𝑆
𝑚
𝛼
𝐶
𝑧𝛽
𝑞𝑆
𝑚
𝛽
(2)
Gravitational force mg:
𝑚𝑔
𝑚𝑔
𝑟
𝑟
0
𝑚𝑔
𝜔
𝑒
𝜔
𝑒0
(3)
𝑔
𝑟
𝑓𝑀
𝑟
2
1𝐽
𝑎
𝑒
𝑟
2
15sin
2
𝜙
(4)
𝑔
2
𝐽
𝑠𝑖𝑛𝜙
(5)
Additional Coriolis force F
:
m
𝑎
𝑘
2𝑚𝜔
𝑒
𝑣
(6)
Centrifugal inertial force F
:
m
𝑎
𝑒
𝑚𝜔
𝑒
𝜔
𝑒
𝑟
(7)
Therefore the acceleration is:
𝑎
𝑔𝑎
𝑎
(8)
The overall quality at a certain moment is:
𝑚𝑚
𝑑𝑚𝑡
(9)
The above formulas are the equations of the center of mass motion of the missile during flight.
Among them, the atmospheric density refers to the US1976 table.
2.2. Basic assumptions
2.2.1. Earth's gravitational field and shape model. The accurate calculation of the Earth's gravitational
field is very complicated. In this model, in addition to the basic spherical gravity, only the J2 term is
considered for non-spherical gravity. Assume that the shape of the earth is a standard ellipsoid.
2.2.2. Thrust model. The engine thrust and specific impulse are both at nominal values. After turning
on, the thrust immediately reaches the rated value; after turning off, the thrust instantly drops to 0. The
mass of the missile is calculated according to the consumption per second, and the mass of the missile
is the mass of the warhead after shutting down.
2.2.3. Earth’s atmosphere parameters. The Earth’s atmosphere model adopts the standard atmosphere
table, and the aerodynamic force is only considered when the altitude is below 90km. The influence of
aerodynamic force is not considered in the reentry section.
2022 International Conference on Automation and Space Science & Technology
Journal of Physics: Conference Series 2220 (2022) 012012
IOP Publishing
doi:10.1088/1742-6596/2220/1/012012
3
2.2.4. Pitch angle model. Assuming that the yaw angle and the roll angle are constant at 0, only the
change in the pitch angle is considered. The pitch program angle of the active section is assumed to be
in the form of a function shown in Figure 1.
When the missile takes off vertically, the starting program angle is
090
p
. After 1
t time of
fixed program angle flight, it enters the turning phase. At this stage, the program angle decreases
linearly. After 21
tt time, the fixed program angle flight is maintained again. After reaching the
predetermined shutdown time k
t, the engine shuts down.
Figure １. Pitch program angle
Among them, the flight time of the two fixed program angles are both set to 5s, that is,t
5s ,
t
t
5s. The intermediate linear descent process can be expressed by the following formula
()
pp0
tKt
(10)
It can be seen from Figure 1 that
2
()
pf p t
, and the terminal pitch program angle satisfies the
constraint
10 70
pf
.
3. Missile trajectory optimization
According to the initial time conditions, the acceleration can be calculated and iterated. The velocity
vector can be found by performing the first-order integration on the acceleration; similarly, the
acceleration vector can be obtained by performing the second-order integration on the acceleration.
The acceleration at the next moment can be solved by using the fourth-order Runge-Kutta equations[4],
so that the velocity and position at each moment can be calculated.
Given the pitch program angle law shown in Figure 1, the parameters that need to be designed are
mainly the terminal pitch program angle
p
f
. By adjusting this parameter, the missile's flight
trajectory can be changed. When this parameter is determined, the missile's maximum range flight
trajectory is also uniquely determined. With the aid of the simulated annealing algorithm[5], the
terminal pitch program angle is optimized. Set the optimization interval to [10°,70°], and finally
output the maximum range and the program angle corresponding to the maximum range.
4. Optimization results
Taking a certain type of missile as an example, its parameters are shown in the tables.
2022 International Conference on Automation and Space Science & Technology
Journal of Physics: Conference Series 2220 (2022) 012012
IOP Publishing
doi:10.1088/1742-6596/2220/1/012012
4
Table 1. Basic parameters of missile.
Parameter Data Unit
Reference area 2.2 m
Maximum takeoff mass 30000 kg
Warhead mass 1200 kg
Shell quality 1800 kg
Rated specific impulse of engine 320 s
Rated engine thrust 400 kN
Axial force coefficient 0.30 /
Derivative of normal force coefficient to angle of attack 0.05 1/deg
Table 2. Launch mission parameters.
Parameter
Data Unit
longitude 112.6 deg
Geographic latitude 37.5 deg
Launch height 1500 m
Launch azimuth 270 deg
Using computer for simulation, the best optimized missile trajectory parameters are shown in Table
3:
Table 3. Optimization Results.
Maximum ran
g
e an
g
le Maximum ran
g
e The best terminal pro
g
ram an
g
le Simulation time
41.788° 4646.913k
m
24.349° 1267s
Draw the change graph of each time-varying parameter, as shown in Figure 2.
(a)Pitch program angle. (b) Flight altitude.
2022 International Conference on Automation and Space Science & Technology
Journal of Physics: Conference Series 2220 (2022) 012012
IOP Publishing
doi:10.1088/1742-6596/2220/1/012012
5
(c)Velocity. (d)Velocity inclination.
(e)Angle of attack. (f)Mass.
Figure 2. Time-varying parameters.
5. Data analysis
Through computer simulation, it can be found that the optimal fuel loading is not intuitively full load
(27000kg, 4568.882km), but slightly lower than full load (26962.23kg, 4646.913km). This is due to
the impact of iteration accuracy. In the case of little difference in mass (20-40kg), the magnitude of the
difference in shutdown time is only 0.1s. In order to ensure the optimization efficiency, the selected
step size is 1s. This may cause the missile with more fuel to fail to perform effective calculations at the
last shutdown point, making the range smaller than that of the missile with slightly less fuel. Plot the
In order to study the relationship between step length and range, the Figure of range difference versus
step length is drawn.:
Figure 3. Range difference.
2022 International Conference on Automation and Space Science & Technology
Journal of Physics: Conference Series 2220 (2022) 012012
IOP Publishing
doi:10.1088/1742-6596/2220/1/012012
6
It can be seen that as the accuracy increases (the step size is reduced), the range difference
gradually converges, but the calculation efficiency also decreases. Therefore, when the fuel mass is not
much different, due to the limitation of calculation accuracy, there will be cases where the missile with
less fuel mass will fly farther.
In addition, the theoretical landing point of a missile launched in the west direction should remain
unchanged, but the actual landing point is southerly due to inertial centrifugal force. In the same way,
if it is launched in the southern hemisphere, the landing site will be northerly. This is also consistent
with the conclusion of theoretical mechanics[6].
6. Conclusion
In actual engineering applications, the missile trajectory is often optimized to make the corresponding
performance of the missile meet the design requirements. The difficulty is that the optimization design
of missile trajectory is tedious and complicated. The method proposed in this paper can effectively
improve the optimization efficiency by simplifying the missile trajectory model within the allowable
range of error. It has good application value in the initial stage of overall missile design. However, this
algorithm cannot get rid of the limitation of accuracy on efficiency. But with the improvement of
computer performance and the innovation of algorithms, this method will have a larger application
space.
References
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