Multiple Flight Vehicles Trajectory Global Optimization Based on Improved Natural Running Process Algorithm Haoxiang Chen1, Ying Nan1 1. College of Astronautics, Nanjing University of Aeronautics and Astronautics Nanjing, 210006 E-mail:
[email protected] Abstract: Natural running process algorithm (NRPA) is a novel algorithm for multiple flight vehicles trajectories optimization problem. In this paper we provide an improved NRPA to optimize cooperation and confrontation trajectories for multiple flight vehicles. A new concept of rational value is proposed to make combined substances to evolve or eliminate depending on their states current and subsequent. This paper also introduces the calculation steps of the improved NRPA, and confirms the accuracy in solving optimization problem of multi-phase, multi-target, multilateral confrontations. Regarding on the cooperation and confrontation among multiply flight vehicles, several numerical simulation results have been provided to prove the feasibility of the improved NRPA, as well as superiority for the multilateral confrontations problem. Key Words: Multiple flight vehicles, Cooperation, Confrontation, Trajectory Optimization, Improved NRPA, Rational Value
It should be concerned not only the current states, but also the subsequent states, to develop the control strategy of flight vehicles. In view of this aspect, Nan put forward natural running process algorithm (NRPA) firstly [18], which is applied in multiple flight vehicle trajectory designing of global integration. Although NRPA is adapted to cooperative and confrontation trajectories optimization problems with general applicability, when it comes to objects with large amount of data, the calculating time increases substantially as the decision-making phase multiplies, which is so called ‘dimension disaster’. This paper puts forward a novel elimination rate to endow each material with ration based on the original NPRA, hence, materials can choose decomposition or evolution depending on their own ration, which contributes to enhance computing speed and reduce calculation time.
1 Introduction In the modern warfare with systematization of high-techs, the conventional mode of single vehicle has been unable to meet the requirements of actual combat [1], such as large amounts of units, high speed, globally optimal cooperation, high accuracy, high reliability, and so on. As aviation technologies have developed massively these years, multivehicles cooperation is becoming a popular form of warfare. Thus, the method for flight vehicles to cooperate or confront is a problem that must be concerned [2, 3]. Currently, scholars have done massive work on flight trajectory optimization, most of them focus on the algorithms [4], which are mainly divided into two categories: optimization and heuristic [5]; Optimization algorithms provide better result for trajectory optimization, nevertheless, they are sensitive to system uncertainty, enemy operations and environment changes; On the other hand, heuristic algorithms are unable to ensure the optimal solutions, but they have stronger robustness. For multicoupled NLP of multi-vehicles cooperation [5, 6], from the aspects of trajectory optimization, those algorithms include the following: conjugate gradient method, projection restoration gradient method, variation method, differential dynamic method, nonlinear programming method [7, 8], multiple shooting method, bargaining differential game theory, parallel iterative dynamic programming [9], and many intelligent algorithms such as genetic algorithm [10, 11], particle swarm algorithm [12, 13], ant colony algorithm [14, 15], neural network algorithm [16, 17], etc. However, little of them could solve the multiple flight vehicles trajectory optimization problem, as well as cooperation and confrontation under the complex and unknown situation.
2 Problem Formulation In the certain flight combat area, assuming the groups of attacking flight vehicle are distributed in a region on the surface of the earth, while the groups of intercepting flight vehicle are distributed over a certain number of different zones (dynamic or static). The mission of the attacking flight vehicle groups is to destroy specified targets in different areas with high precision, or fly past some certain waypoints within a limited time. Intercepting flight vehicle groups, which are widely distributed over the ground and air, are responsible for resisting continuous saturation attacks by all possible means. Regarding on this aspect of combat, the concept of offensive and defensive confrontation is proposed for describing the entire operation process. The motion of various flight vehicles can be described as following differential equations:
This work has been funded by the Jiangsu Province ordinary university graduate student research and innovation programˈGrand No. KYLX15_0319.
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dx1 (t ) f1[t , X (t ), u1 (t ), w1 (t ), p1 ] ° dt ° ° dx2 (t ) f [t , X (t ), u (t ), w (t ), p ] 2 2 2 2 ° dt ° °....... ° dxN (t ) A f N A [t , X (t ), u N A (t ), wN A (t ), pN A ] ° ° dt ® dx (t ) ° N A 1 f N A 1[t , X (t ), u N A 1 (t ), wN A 1 (t ), pN A 1 ] ° dt ° dx (t ) ° N A 2 f N A 2 [t , X (t ), u N A 2 (t ), wN A 2 (t ), pN A 2 ] ° dt °...... ° ° dxN A M T (t ) f N A M T [t , X (t ), u N A M T (t ), wN A M T (t ), pN A M T ] ° dt ¯
i 1, 2, } N A , N A 1, N A 2, }, N A M T ; X
{x1 , x2 , x3 ,..., xN A , xN A 1 , xN A 2 , xN A 3 , ..., xN A MT } (1)
where,
{x1 , x2 , x3 ,..., xN A }
attacking
flight
vehicles,
stand for the states of N A and
{xN A 1 , xN A 2 ,
xN A 3 ,..., xN A MT } are the flight states of MT interception flight vehicles.
xi
[V , J ,\ V , hA ,
xA , z A ]i includes
velocity, attitude and coordinates of the ith flight vehicle. ui [ax , a y , az ] is the control command of the ith flight vehicle, and for most flight vehicles in the atmosphere,
[ax , a y , az ] are the accelerations calculated by ui . pi represents for altitude of the ith flight vehicle, and
wi (t ) is
random environment interference variable, regarding on random wind field, wi (t ) [vwxi , vwyi , vwzi , awxi , awyi , awzi ] - velocity and acceleration of wind field. For attacking flight vehicle j, to destroy targets as much as possible and evade the interception from flight vehicle i and find the optimal control ui to optimize the
where, LA is the performance index for attacking flight vehicles, and LM is the one for intercepting flight vehicles;
ri (tif )
is miss distance,
tif
-final flight time of the ith
xTj , yTj , zTj
attacking flight vehicle, xAi , y Ai , z Ai and
represent the height, coordinates of latitude and longitude of the attacking flight vehicles and their targets. For interception flight vehicle i, to seek the following performance index: kMax
LA ri (tif )
min{¦ ri (ti ) |ti ui
tif
}
(3)
i 1
( xAi xTj )2 ( y Ai yTj )2 ( z Ai zTj )2 ri (tif ) d diMin
ti tif
(4)
where, diMin represents for the minimum miss distance allowed. Formula (4) is applied to interception flight vehicles equipped with short-distance guns and missiles. Otherwise, optimal control decision u j of attacking flight vehicle is calculated through the optimization of large system, to complete the penetration, formula (4) below also must be satisfied:
di , j ,Md (t ) ! diMin
(5)
3 Improved NRPA The initial settings of NPRA are presented in [18, 19]. As for flight space of aircraft, space S is contained with time slice [t0, tf] and realistic 3D physical space [xMin, xMax], [yMin, yMax], [zMin, zMax]. Determine the control and strategy vector U (t ) [u(t )P (t )] , to maximize the performance indexes of all substance individuals and their combinations of every race in every level (Fig.1 shows their sequence.):
max J
U ( t ):U
ª max « P ¬u (t )
º E » P (t ) ¼
T
performance indexes which can be described as follows: kMax
LA
max{¦ ri (ti ) |ti ui
ri (tif )
tif
}
i 1
( xAi xTj ) 2 ( y Ai yTj ) 2 ( z Ai zTj ) 2
Figure1. Sequence map of essential ranks ti tif
where, P is the performance index of basic element units
§ ° min ®¦ max ¨ ¦ di , j , Md (t ) |t ui ° i 1 uj © j 1 ¯ NA
LM
di , j ,Md (t )
MT
tf
·½ ° ¸¾ ° ¹¿
( xAi xmj )2 ( yAi ymj )2 ( z Ai zmj )2
(2)
i 1,2, }, N A ; j 1, 2,..., M T
&
(BEUs) while Vector E is the performance index vectors of the combinants (combination of several different BEUs that share the same interest) of every race in every level. So, considering the running state of the whole system, the entire running process is divided into N steps; each step includes the performance indexes of all BEUs and combinants
(2 984
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° JD ,GD ,Ki d JD ,GD ,K j ® ° ¯ D ,GD ,Ki d D ,GD ,K j
Aim to enhance the calculating velocity and reduce the data amount of NPRA, we proposed a novel concept: rational value. 3.1 Potential value and rational choice
3.2 The calculation steps of NRPA
During the process of natural running, assuming that all BEUs and combinants have ration values, if the combinants judge themselves cannot exceed the other combinants, they have rational choice to finish their lives. The aim of rational choice is to reduce the amount of calculation and promote efficiency. In actual situation, abandoned combinants with poor development earlier could be beneficial globally. For the combinant Kof race Gin level Dwhich is denoted as ( D , G , K ), by the end of each time subsection ªtb,; , tb,;1 º , each combinant ¬ ¼ estimates its future potential after a finite time point W D ,G ,K .
W D ,G ,K
The larger
, the higher rational degree of
combinants, which means the clever combinants take the longer-term view. JD ,GD ,K stands for the performance index of the whole system, which includes the performance index of BEU(the ikth BEU of qkth level in kth kind ) pk ,qk ,ik and the performance index of combinant ( D , G , K )- ED ,GD ,K . The potential value
is posed
D ,G ,K
as follows:
WD ,GD ,K ED ,GD ,K ¦
¦ ¦
wk ,qk ,ik p k ,q ,i
JD ,GD ,K
WD ,G ,K jD ,GD ,K ,max
max ¦
¦
jD ,GD ,K ,1
1
Bmax qk ,max lk ,ik ,max
(WD ,GD ,K , jD ,G ,K ,i ED* ,GD ,K , jD ,G ,K ,i ¦ ¦ D
D
ED ,GD ,K , jD GD K *
,
p*k ,q
k ,ik
, ,i
max ED ,GD ,K , jD ,G
D ,K ,i
max pk ,q
k ,ik
¦w
k , qk ,ik
p*k ,q ,i ) k k
k 1 ik 1 qk ,ik 1
where:
(
tb1,Min tb,Max
from
), b
time
tb ,Min
to
tb ,Max
0 means the first motion
(trajectory) state in the space. Eliminating the combinants with poor development, the control and strategy vector
U (1) [u (1), P (1)] of the 1st level is obtained by the following formula: J1 [ X (1),1] min {Lk ,i , j [ X (1), uk , i (1), wk , i (1), 4(1),1] J 0 [ x (0),0]} uk , i (0) ouk , i (1) ° °° x (1) f [ X (0), w (0), 4(0), u (0),0] k ,i k ,i k ,i ® k ,i ° FG , K (1) gG , K [ X (0), 4(0), PG ,K (0),0] ° °¯4(1) h[ X (0), 4(0),0] (8) Remain the fittest combinants on the basis of (6), also the optimal control:
u [ X (1)] and J1 [ X (1),1] Save the states of BEUs and combinants in physical storage Psi [1][iSX ][iSY ][iSZ ][0] , save the performance index interrelation function in storage Psi [1] [iSX ][iSY ][iSZ ][2] , save the optimal control in storage Usi [1][lC , Max ] .
k k
k 1 ik 1 qk ,ik 1
D ,G ,K
Step1 ˖ Calculate
J1 [ X (1),1] in storage Psi [1][iSX ][iSY ][iSZ ][1] , save the
Bmax qk ,max lk ,ik ,max
JD ,GD ,K
(7)
................ Step k: In the time segment t
tN 1,Min tN ,Min
tN ,Max ), b N . Eliminating the combinants
( tN 1, Min
(6)
with poor development, the control and strategy
(t , X (t ), 4(t ))
vector U (1) [u (N ), P (N )] of the kth level is obtained by the following formula:
˅
(t , X (t ), wk ,i , 4(t ))
WD ,GD ,K and wk ,qk ,ik stand for the inner parameters determined by the essential attribute of combinants and BEUs; X (t ) is the state function, 4(t ) is the impact function between BEUs and combinants. The potential value measures the maximum performance index of combinant ( D , G , K ) in the finite time while the other BEUs and combinants cooperate fully with it. If the potential value conform to formula (7) below, then the combinant ( D , G , K ) would choose to end itself, hence we consider that ( D , G , K ) has little chance to obtain the final victory in the entire space.
JN [ X (N ), N ] min {Lk ,i , j [ X (N ), uk , i (N ), wk , i (N ), 4(N ), N ] uk , i (N 1) ouk , i (N ) ° ° J [ x (N 1), N 1]} ° N 1 ° ® xk ,i (N ) f k ,i [ X (N 1), wk ,i (N 1), 4(N 1), uk ,i (N 1), N 1] ° ° FG , K (N ) gG , K [ X (N 1), 4(N 1), PG ,K (N 1), N 1] ° °¯4(N ) h[ X (N 1), 4(N 1), N 1] (9) Also, remain the fittest combinants on the basis of (6), and get the optimal control:
u [ X (N )] and JN [ X (N ), N ] Save the optimal control, the states of BEUs and combinants in corresponding physical storages, then repeat these operations for step K+1. .................
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Vx (m/sec)~t
Vx (m/sec)
500 0 -500
Vy (m/sec)
J [ X ( N ), N ] min {Lk ,i , j [ X ( N ), uk , i ( N ), wk , i ( N ), 4( N ), N ] uk , i ( N 1) ouk , i ( N ) ° ° J [ x ( N 1), N 1]} ° N 1 ° ® xk ,i ( N ) f k ,i [ X ( N 1), wk ,i ( N 1), 4( N 1), uk ,i ( N 1), N 1] ° ° FG , K ( N ) gG , K [ X ( N 1), 4( N 1), PG ,K ( N 1), N 1] ° °¯4( N ) h[ X ( N 1), 4( N 1), N 1] (10) At the terminal time of the entire process, the states of all BEUs and combinants, as same as histories of theirs states, control variables, performance indexes on each time before, are stored in the physical storage space. Select the optimal combinant with minimum performance index and its history as the optimal control plan.
N
500 0 -500
Vz (m/sec)
control U ( N ) [u ( N ), P ( N )] of the level N is obtained by the following formula:
From the figures presented, simulation 1 proves that for the cooperative problem of multiple flight vehicles, NRPA can plan the paths reasonably and obtain the global optimal solution.
200 0 -200
Vel (m/sec)
Step N: The maximization process in the above formula is as the same as the one in step k, that is ‘k’ becomes ‘ N ’, and satisfy the terminal restriction ) k ,i , j [ ] .The optimal
285 280 275
0
500
1000 1500 time (sec) Vy (m/sec)~t
2000
2500
0
500
1000 1500 time (sec) Vz (m/sec)~t
2000
2500
0
500
1000 1500 time (sec) Vel (m/sec)~t
2000
2500
0
500
1000 1500 time (sec)
2000
2500
Figure 3: Velocity components and velocity module of all 30 vehicles 2
2
ax (m/sec )
a (m/sec )~t
4 Simulation results
0 -10
by
500
1000 1500 time (sec)
2000
2500
2000
2500
2000
2500
a (m/sec )~t y
20 0 -20
0
500
1000 1500 time (sec) 2
a (m/sec )~t 2
az (m/sec )
30 vehicles fly from different start points in the complex terrain environment, in which distributed 13 cylindrical threat zones (represented by larger, hollow and solid cylinders in Fig.1) and 15 task waypoints (represented by smaller solid cylinders).
0
2
2
1: Multiple vehicles flight coordination in complex terrain environment.
ay (m/sec )
4.1 Simulation
x
10
z
20 0 -20
0
500
1000 1500 time (sec)
Figure 4: The control variables ax, ay and az of first vehicle
4.2 Simulation 2: Multiple vehicles cooperative interception problem. A hypersonic reentry vehicle penetrates from near space to specified target point on the ground, and four hypersonic interception vehicles maneuver to intercept. The flight environment for all the vehicles is similar, also flight restricted areas are forbidden to get in. In Fig. 5, 6 and 7, lines represent for optimal trajectories of 7 intercepting flight vehicles, while “*” for the penetration trajectory of reentry vehicle.
Figure 2: 3-D optimal trajectories in the complex terrain environment
The entire 30 vehicles aim to avoid all threat zones and ensure that each waypoint must be passed by at least one vehicle. The optimal solution are shown in Figure 2, which shows the 3D optimal trajectories in the complex terrain environment, all waypoint (except the one in the threat zone) have been passed by; Fig. 3 shows the velocity components and velocity modules of all 30 vehicles; Fig. 4 shows the first vehicle's accelerations ax , ay and az as control variables.
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Figure 5: Threat area distribution and optimal trajectories
2017 29th Chinese Control And Decision Conference (CCDC)
Velocity - t 8000 7000 6000
Vel (m/sec)
5000 4000 3000 2000 1000 0
0
100
200
300 t (sec)
400
500
600
Attack angle (deg)
Figure 9: Velocity of each flight vehicle Attack angle ~ Flight time 40 20 0
0
100
0
100
0
100
200
300 400 t (sec) Rolling angle ~ Flight time
500
600
200 300 400 t (sec) Total control acceleration ~ Flight time
500
600
500
600
100 0 -100
2
Total acceleration(m/s ) Rolling angle (deg)
Figure 6: Trajectory of cooperative interception
100 0 -100
200
300 t (sec)
400
Figure 10: Control variables of flight vehicles both sides Figure 7: Velocities of all the 5 flight vehicles
Figures above show that, the penetrating reentry flight vehicle takes the strategy to fly in the valley (TF/TA) and avoid radar detection, threat areas and interception, while the interception flight vehicles follow the optimal planed intercept trajectories and successfully intercepted the penetrating reentry vehicle. 4.3 Simulation 3: Multiple flight vehicles confrontation problem. There are 17 flight vehicles in the free flight space(without any terrain), 3 penetrating vehicles aim to break through the interception and reach the target point on ground, while 14 interceptor vehicles attempt to intercept all penetrating flight vehicles. The target point's coordinate is (-1000, 0, 0), the initial velocities of all vehicles are 7800m/s, supposed that the characteristic (aerodynamic coefficients, mass, specific impulse, maximum available overload) of all vehicles are similar.
From simulation results above, compared with missiles, interception flight vehicles in this case have no priority, such as faster speed and larger maximum available overload, which means the confrontation conditions are “fair” for flight vehicles both sides. Results also prove that in multilateral confrontation problem, if the conditions of both side are quite, as well as both sides adopt the globally optimal cooperation and counter plot based on the control strategy calculated by NRPA, the final solution could be “well-matched” for both sides.
5 Conclusions Improved NRPA can be widely used to get global optimal control and trajectories optimization in unknown stochastic environment for multitudinous natural physical units, such as flight vehicles. A mass of numerical simulation results shown that, based on this optimal control algorithm, the attacking vehicles can also pass all waypoints inside the anfractuous distribution of threat areas with the minimum threat; multitudinous attacking flight vehicles can meet maneuvering flight target vehicles with high accuracy; intercepting flight vehicles are able to cooperatively block penetrating flight vehicles from their targets. Compared with traditional optimization mythologies, NRPA has properties as following: multi-objective optimization, cooperation and/or confrontation among multi systems, global optimization, no sensitivity of initial values, and so on.
Figure 8: 3-D trajectories of flight vehicles both sides
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