МОДЕЛИРОВАНИЕ ОПТИМАЛЬНОЙ ТРАЕКТОРИИ ПРОТИВОТАНКОВОЙ УПРАВЛЯЕМОЙ РАКЕТЫ С УЧЕТОМ УГЛА ПАДЕНИЯ

Чан Ван Хай
Tran Van Hai
Магистр
MS
Нгуен Нгок Диен
Nguyen Ngoc Dien
Кандидат технических наук
PhD
Нгуен Тхань Тунг
Nguyen Thanh Tung
Магистр
MS
Факультет Техники управления
Вьетнамский государственный технический университет
имени Ле Куй Дона
Faculty of Control engineering
Le Quy Don Technical University
МОДЕЛИРОВАНИЕ ОПТИМАЛЬНОЙ ТРАЕКТОРИИ
ПРОТИВОТАНКОВОЙ УПРАВЛЯЕМОЙ РАКЕТЫ С УЧЕТОМ УГЛА
ПАДЕНИЯ
MODELING THE OPTIMAL TRAJECTORY OF AN ANTI-TANK
GUIDED MISSILE TAKING INTO ACCOUNT THE IMPACT ANGLE
CONSTRAINT
Аннотация на русском языке: В настоящей работе произведена попытка
моделирования оптимальной траектории противотанковой управляемой ракеты (ПТУР) с
учетом угла падения. В качестве критерия оптимальности выбирается интеграл (по
расстоянию ракеты-цели) отношения квадрата ускорения ракеты на расстояние ракетыцели. В качестве оптимального сигнала управления выбирается ускорение ракеты.
Предлагая, что скорость ракеты неизменна, с помощью теории оптимального управления
позволяется определить функцию оптимального сигнала управления. Для проверки
результатов расчета, автором выбрана конкретная модель ПТУР типа Javelin и сделано
моделирование оптимальной траектории ПТУР. Результаты моделирования показаны, что
цель уничтожена ПТУР-ом с совсем маленьким промахом в случае разных значений угла
падения.
The summary in English: In this paper, an attempt is made to simulate the optimal
trajectory of an anti-tank guided missile (ATGM) taking into account the impact angle (IA). The
cost function weighted by a power of range-to-go (by the integral according to relative distance
between the missile and target). The selected control signal is the missile's LA, using an optimal
control theory with the assumption that the missile’s velocity is unchanged and the angle
between the missile velocity and line-of-sight (LOS) vector is small enough to linearize the
engagement kinematics and allow the optimum control signal function to be found. To verify the
calculation results, the author selected a specific Javelin type anti-tank missile and simulated the
optimal trajectory of the anti-tank missile. The simulation results show that the target is
destroyed by ATGM satisfying the zero miss distance and impact angle constraints (IAC).
Ключевые слова: Угол падения, наведения ракеты, оптимальное наведение, метод
пропорционального сближения, расстояние цели-ракеты.
Международный научный журнал «Синергия наук»
Keywords: impact angle, Anti-tank Guided Missile, Optimal Guidance Law (OGL),
Proportional Guidance, Range-to-Go
Introduction
Requirements of increasing angle of approach to the target for ATGMs are
playing a crucial role in the future warfare. For ATGM weapons, many guidance
algorithms with IAC are mentioned in order to increase the probability of
destroying target and attack tank’s weak areas.
Many published papers in this field that using optimal control theory mainly
have been solved the quadratic linear optimization problem for the missile's motion
equations system. For the predecessor, Kim and Grider [1] was one of the earliest
pioneers where an OGL has been presented for re-entry vehicles with an IAC to
attack stationary targets. Many scholars modified traditional proportional
navigation guidance (PNG) law to solve the IACs to satisfy constraints such as
zero miss distance, terminal IA and zero terminal accelerations [2, 3, 4, 5, 6, 7].
Numerical simulations are carried out for missile models with constant speed or
considering the change in the velocity of missile, but not examining the actual
parameters that affect the velocity change such as thrust, mass of the ATGM
engine and the gravitational force [8, 9].
Although the authors have proposed OGLs for solving IAC problems, they have
not yet applied these proposed OGLs into a specific ATGM model.
In this paper, the author has mentioned the application of an OGL to control the
IA of a third generation fire-and-forget ATGM with variable speed to strike
stationary tanks or armored devices on the ground. The guidance algorithm is
designed based on the application of research results of author Bong Gyun Park [4]
by solving the problem of minimizing the energy cost function weighted by a
power of range-to-go with linear engagement models. The aim of solving this
problem is to find out the expression of the missile lateral acceleration that can
adjust the ATGM’s flight trajectory.
Международный научный журнал «Синергия наук»
Problem Formulation
As illustrated in Figure 1, consider the trajection of an ATGM in the polar
coordinate plane, it hits a stationary tank target on the ground. At a time t during
flight, M (x, y) is the missile 2-D coordinates, V is the missile velocity. The r
variable is the relative distance between the ATGM and the tank. s , θ, and λ
represent the angle between the missile velocity and line-of-sight (LOS) vectors,
the angle between the velocity (flight path) vector and the local horizon, and the
LOS angle, respectively. The desired coordinates of the target impact point T ( x f ,
y f ), and the desired IAC is  f . The ATGM trajection at launch is also
characterized with the initial condition 0   0 . Where σ can be indicated as the
missile’s heading angle in the assumption that the angle of attack (  ) of the
ATGM is insignificant. The ATGM’s velocity depends on thrust, the gravitational
force and atmospheric resistances.
Figure 1. 2-D engagement geometry
The nonlinear equations of motion in a polar coordinate system are given by
.
r = -Vcosσ
(1)
. . . a
Vsinσ
σ = θ- λ = M +
V
r
(2)
Международный научный журнал «Синергия наук»
.
H = Vsin
(3)
. a
θ= M
V
(4)
. -Rcos
 g sin   12m Cx V 2S
V=
m
(5)
   ( H ) ; R  R(t ) ; m  m(t ) ;
(6)
Where, Cx , 𝜌, 𝑆, 𝑚, g denote the aerodynamic coefficients, the atmospheric
density, the characteristic area of the missile, mass of the missile, and the
gravitational acceleration, respectively. a M is the lateral acceleration (LA)
perpendicular to the velocity vector. The subscript 0 denote the initial state and f is
final state. The boundary conditions for the flight path of missile are given as
r(t 0 ) = r0 ,σ(t 0 ) = σ0 ,θ(t 0 ) = θ0
(7)
r(t f ) = rf ,σ(t f ) = σ f ,θ(t f ) = θ f
(8)
It assumes that the missile velocity V is constant and  is small enough to
linearize the missile-target engagement kinematics, by dividing Eqs. (2) and (3) for
Eq. (1), we will get formulas for the linearized system which are given below:
d  1
= -  2 aM
r V
dr
(9)
d
1
=  2 aM
dr
V
(10)
Optimal IA control guidance law
In this section, target IA control guidance laws using linear optimal control
theory are derived minimizing the energy cost function weighted by a power of
range-to-go subject to the terminal conditions which are chosen as the interception
with IAC.
Международный научный журнал «Синергия наук»
The cost function weighted a power of range-to-go can be written as shown
below:
J=
1
rf
aM 2
2 тr0 (r - rf ) N
dr,
(11)
Nі 0
Where N denotes a guidance gain. In case of N = 0, the problem becomes
optimized for pure energy minimization. Also, in case of N > 0, the control energy
are distributed by the weight function of (r - rf ) N in Eq. (11), so the guidance law
will create the guidance commands very large at the initial phase and at the end of
the guidance phase, the acceleration becomes zero.
We are given a Hamiltonian function of the guidance problem is
H=
a
a σ
a
- λ σ ( M2 + ) - λ θ M2
2(r - rf )
V
r
V
2
M
N
(12)
Where λ σ , λ θ are called co-states. Their differential equations follow variable r
are given by :
dλ σ
dr
dλ γ
dr
=-
=-
H
σ
H
θ
= λσ
1
(13)
r
(14)
=0
Where the terminal conditions of the co-states are  f   and  f   .
Integrating Eqs. (13) and (14) with the terminal boundary conditions.
λ σ = σ
r
; λθ = θ
rf
(15)
Necessary optimal conditions are derived of the problem ∂H/∂r=0, we have
the following optimal LA:
aM =
σ
2
V rf
r(r - r f ) N +
θ
V
2
(r - r f ) N
Международный научный журнал «Синергия наук»
(16)
According to equation (16), the acceleration commands for N >0 will become
zero at the end of the guidance phase. However, when N =0, the LA generated will
change linearly and aM (rf )  0 because of the constant  .
Substituting Eq. (16) into Eqs. (9) and (10) then integrating these equations,
we have

 r r
f
 = - 4 
V rf r
4



  2r  r  r   r  r  r   -    r  r   r  r  r    C
(17)
  r  r  r   -   r  r   C
(18)

  r  rf
 =- 4
V rf  3

4
3
2
f
f
f
3
3
3
f
2
2

 V 4r 
f


2
2
f
f
3
2



r
2
f
f
2
f
f

 V4

2

where C and C are integration constants to be determined from the initial
conditions.
Set the shortened variables as follows :
3
2
r  r 4
r  rf 
r  rf  


f
2

 2 rf
 rf
A(r) = 
 4

3
2


(20)
2
  r  r 3
r  rf  

f

 rf
B(r) = 


3
2


(21)
r  r   r r  r 
C(r) =
3
f
f
3
r  r 
D(r)=
2
f
(22)
2
2
f
(23)
2
Therefore, applying the initial conditions of Eq. (7), Eqs. (17) and (18) can be
rewritten as
=


r
 A(r0 )  A(r )   4  B(r0 )  B(r )    0 0
V rr
V
r
4
f
Международный научный журнал «Синергия наук»
(24)
=


C (r0 )  C (r )   4  D(r0 )  D(r )   0
V
V r
4
(25)
f
Also, from the terminal conditions of Eq. (8), we have




    B(r )   r   r D(r ) 
0
0
0 0
0 
f f
 f

 = V 4rf 
C (r0 ) B(r0 )  D(r0 ) A(r0 ) 






    A(r )   r   r C (r ) 
0
0
0 0
0 
f
f f
 = -V
C (r0 ) B(r0 )  D(r0 ) A(r0 ) 
4 
(26)
(27)
The closed-loop optimal solution is obtained as
V2
6  r  rf  f   )   12(rf  f  r ) 
aM = 
(r - r f ) 2 
(28)
The conditions to intercept a stationary target at the final time t f are
lim r  0 or
t t f
t t
(29)
t t
(30)
f
rf 
0
f
0
lim   0 or  f 
t t f
By applying the interception conditions of Eqs. (29) and (30) to Eq. (28), we
have
V2
aM = - 6  f     12 
r
(31)
The optimal acceleration command in the formula (31) is shown in the
research [4] by Bong Gyun Park. Replacing aM with the expression (31) found
above into the nonlinear differential equations from (1) to (6). Solving differential
equations analytically by using the ODE45 function in MATLAB software, with
initial and terminal conditions for a type of ATGM. Numerical simulations and
analysis of results are shown in section 4 below.
Международный научный журнал «Синергия наук»
Numerical Simulations and Result analysis
In this section, the effectiveness of the guidance law is investigated through
nonlinear numerical simulations with various terminal IACs and comparison
control energy of different target IA scenarios. Consider five typical desired IAs
include q f = -70°, -50°, - 40°, -30°, and 0°. The proposed guidance law is applied
to the model of an ATGM and examined through the parameters are shown in
Table 1 and Table 2. All simulation cases are terminated when the height of the
missile's location becomes zero.
Table 1. Initial Conditions for Nonlinear Simulations
Parameters
The initial position of the missile, ( x0 ; y0 )
Target Position, ( x f ; y f )
Values
(0; 0) km
(0.8 ; 0) km
The initial velocity of the missile, V0
Launch Angle, s 0 = q0
Terminal IA, q f
250 m/s
30 deg
-70 deg ~ 0 deg
It is assumed that the flight motor is designed in a manner similar to the
launch motor. The performance characteristics of the flight motors are known [10],
(Table 2).
Table 2. Actual Flight Motor Performance
Time (s)
0
0.3
0.6
1.2
1.8
2.4
4.2
5.2
Thrust (N)
0
570
650
750
770
650
50
0
Mass (kg)
11.25
11.16
11.06
10.82
10.58
10.38
10.16
10.15
Международный научный журнал «Синергия наук»
The intercept trajectories of an ATGM with different IA requirements are
shown in Figure 2. Meanwhile, Figure 3 shows the time histories of guidance
acceleration commands for the missile with various desired IAs. The flight
trajectory for the case of q f = -70 deg tends to be more curved than the others, and
thus, the corresponding magnitude of the acceleration command is much larger and
control energy also requires a higher level as shown in Table 4. As shown in
Figure 4, the look angles converge on zero near the final time.
The IA errors and miss distances for this set of simulations are shown in
Table 3, which validates that the zero miss distance and angle constraints are
satisfied as well.
As it shows, the proposed guidance law generates large LA at early part of the
engagement. As the missiles approach to the target, the terminal guidance
commands converge to zero. The acceleration commands have been generated
form of polynomial of range-to-go, therefore, the commands should converge to
zero as the missile approaches the target. In most cases, the missile with larger
desired IA requires more LA. However, in five typical desired IAs are considered,
the case of terminal IA is 30 degrees achieves optimum control energy and has the
smallest command acceleration.
Figure 2. Missile trajectory for various
impact angles
Figure 3. Lateral acceleration for
impact angles
Международный научный журнал «Синергия наук»
Figure 4. Look angles for
various impact angles
Figure 5 - Flight Path angles for
various impact angles
It can be seen from Figure 5 that the desired IACs are satisfied for all cases.
In any case, the proposed OGL through acceleration command adjustment to
achieve the terminal AIs as required.
Table 3. Miss distances and angle errors for different IA cases
Desired IA (°)
q f = 0 deg
Miss distance (m)
0.1072
IA error (°)
0.0996
q f = -30 deg
0.0189
0.1
q f = -40 deg
0.0207
0.1
q f = -50 deg
0.0275
0.1
q f = -70 deg
0.0593
0.1002
Figure 6. Missile velocity for various impact angles
Международный научный журнал «Синергия наук»
Table 4. Comparison of Control Energy for various IAs
f
J
tf
J
1 2
aM dt , m 2 / s 3
2 t
qf =
deg
8622
0 qf =
-30 q f =
deg
6465.5
deg
10490
-40 q f =
-50 q f =-70
deg
16294
deg
32505
The missile velocity depends on thrust, the gravitational force and air
resistance in which thrust value and mass of missile are determined via Table 2
“Actual Flight Motor Performance”. Figure 6 illustrates the time histories of
missile speed for various IAs.
Conclusions
In this paper, a proposed OGL with IAC has been applied for the antitank
guided missile with time-varying velocity. The ATGM model took into account the
factors such as thrust, weight, and velocity change over time.
Furthermore, range-to-go weighted energy costs are compared with different
IAs, allows the selection of an optimal target IA in the actual launch of anti-tank
missiles. Numerical simulations are conducted to verify the performance of
suggested guidance law which successfully meets the terminal constraints for
ATGMs. For the further study, this area could be focus on expanding the proposed
guidance law to deal with moving ground targets.
Использованная литература/ References
1.
M. Kim, K.V. Grider /Terminal guidance for impact attitude angle
constrained flight trajectories IEEE Trans Aerospace Electron Syst, 1973.
2.
B.S. Kim, J.G. Lee, H.S. Han /Biased PNG law for impact with angular
constraint IEEE Trans Aerosp Electron Syst, 1998.
Международный научный журнал «Синергия наук»
3.
Jeong SK, Cho SJ, Kim EG /Angle constraint biased PNG. In: Proceedings
of 5th Asian control conference; Melbourne, Australia. Piscataway (NJ): IEEE
Press, 2004.
4.
Bong Gyun Park. Optimal impact angle constrained guidance with the
seeker’s lock-on condition J. KSIAM. Vol.19, No.3, 2015.
5.
A. Ratnoo, D. Ghose /Impact angle constrained guidance against
nonstationary nonmaneuvering targets. J Guid Control Dyn, 2010.
6.
A. Ratnoo, D. Ghose /Impact angle constrained interception of stationary
targets J Guid Control Dyn, 2008.
7.
C.K. Ryoo, H. Cho, M.J. Tahk /Optimal guidance laws with terminal IAC J
Guid Control Dyn, 2005.
8.
Sang-Wook Shim, Seong-Min Hong, Gun-Hee Moon, and Min-Jea Tahk
/Time-to-go Polynomial Guidance with IAC for Missiles of Time-Varying
Velocity International Journal of Mechanical Engineering and Robotics Research,
2017.
9.
Jun Zhou, Yang Wang, Bin Zhao /Impact-Time-Control Guidance Law for
Missile with Time-Varying Velocity, Institute of Precision Guidance and Control,
Northwestern Polytechnical University, Xi’an 710072, China, 2016.
10.
Harris, J., Slegers, N. /Performance of a fire-and-forget anti-tank missile
with a damaged wing. (Mechanical and Aerospace Engineering Department,
University of Alabama), 298 p.
Международный научный журнал «Синергия наук»