GB2577371A - Neural network-based method for pursuit-evasion game of spacecrafts - Google Patents

Neural network-based method for pursuit-evasion game of spacecrafts Download PDF

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GB2577371A
GB2577371A GB1910670.7A GB201910670A GB2577371A GB 2577371 A GB2577371 A GB 2577371A GB 201910670 A GB201910670 A GB 201910670A GB 2577371 A GB2577371 A GB 2577371A
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pursuit
evasion
spacecraft
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Yuan Yuan
Li Xuelong
Zhang Peng
Sun Chong
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Northwestern Polytechnical University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • G—PHYSICS
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    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/0202—Control of position or course in two dimensions specially adapted to aircraft
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/0265—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion
    • G05B13/027—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion using neural networks only
    • G—PHYSICS
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
    • B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
    • B64G1/242—Orbits and trajectories
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
    • B64G1/64—Systems for coupling or separating cosmonautic vehicles or parts thereof, e.g. docking arrangements
    • B64G1/646—Docking or rendezvous systems

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Abstract

A method for a pursuit-evasion game of spacecrafts based on a neural network, comprising: establishing a discrete system model for the pursuit-evasion game of the spacecraft, designing an iterative control strategy with synchronous convergence of zero-sum game based on an adaptive dynamic programming, and deriving an approximate optimal control strategy using the neural network.

Description

NEURAL NETWORK-BASED METHOD FOR PURSUIT-EVASION GAME OF SPACECRAFTS
TECHNICAL FIELD
[0001] The invention belongs to a field of spacecraft pursuit-evasion control, *especially 5 relating to an adaptive dynamic programming-based zero-sum game optimal control algorithm.
BACKGROUND OF THE INVENTION
[0002] In recent years, relative motion problem for spacecrafts has been widely concerned by scholars and scientific and research institutions at home and abroad. In particular, a pursuit-evasion game of spacecrafts is a hot research topic in recent years, At present most of pursuit-evasion game algorithms for spacecrafts are mainly based on a linear spacecraft dynamics model for spacecraft *where a linear feedback controller is designed in virtue of a theory of linear quadratic differential game. However, in practical systems, the dynamic model for spacecraft has a characteristics of strong non-linearity. if a general linear controller is used, a control performance of the system will be greatly reduced. Therefore, it is very important to design an adaptive nonlinear controller for nonlinear spacecraft model [0003] Recent years various game-based control algorithms have been proposed for the pursuit-evasion game of spacecrafts. In the field of spacecraft, the common control algorithms are a linear L4 control algorithm, a linear optimal control algorithm and so on However, the practical spacecraft model has strong nonlinearifies. If the above control algorithms are used for designing the controller, the control performance will be reduced inevitably Therefore, it is urgent to design a nonlinear controller for the nonlinear system of spacecraft. For an optimal controller for the nonlinear system, an adaptive dynamic programming (ADP) algorithm is mainly used at present. For the discrete zero-sum game problems, there has been no ADP algorithm to guarantee that the control strategies meet the condition of converging simultaneously.
SUMMARY OF THE INVENTION
[0004] The purpose of the invention is to provide a method for a pursuit-evasion game of spacecrafts, so as to overcome the shortcomings of prior technologies. The invention adopts the ADP-based control strategy with synchronous convergence or zero sum game, which can ensure the system obtains an optimal performance.
[0005] To achieve the above purpose, the invention adopts the following technical so I Lit on.
[0006] A method for a pursuit-evasion game of spacecrafts includes the following steps; [0007] Step 1: establishing a discrete system model for the Pursuit-evasion game of the spacecrafts; and 00081 Step 2: designing an iterative control strategy with synchronous convergence of zero-sum game based on an adaptive dynamic programming, and deriving approximatively an optimal control strategy by using a neural network [0000] Specifically, Step I is given as follows: [0010] In the Euler-Hill reference frame, equations for a nonlinear relative motion of the spacecrafts are given as 1 = Vy 2c-y+ .x)+u, Ti.
y = -itA ± fly-21)i [0011] =-=-z+tig, (1) P3 --\Ai* + x)2 V2 Z2 -2r F; fd, rc- [0012] Where y, 2 denote information of a position of the spacecraft in the Euler-Hill reference frame; u. u1, and is, are control inputs; it is a gravitational constant; r and ra, respectively, denote orbit radii of a pursuit spacecraft and an evasion spacecraft v denotes a true anomaly of the orbit of the pursuit spacecraft, ,x denotes a first-order derivative of x, and denotes a second-order derivative of x.
[0013] By denoting ri =4' [x y z i±17, U [if, u.. it,], the equations for the nonlinear relative motion can be reformulated into a state-space form as poi 4] F./ -ft.i.7YE flu [0015] where x [0016] f(7-7)=4 c ra [r) rf p - + 1", y -2f).+"( v
--Cl
[0017] where 77 denotes a state set. if denotes an input set B denotes a coefficient matrix of the input, and I denotes a unit matrix.
[0018] Hence, the respective nonlinear dynamics models for the pursuer and the 10 evader are described, by Expressions (2), (3).
pal 9] =1(17 (2) [0020] , p)± Btl° (3) [0021] th = PP-) ± -thee respectively, a state vector and a control input of the where and uPare pursuer: ':and ti-are, respectively, a state vector and a control input of the evader. 15 Subtracting Expression (3) from Expression (2), a model for the nonlinear pursuit-evasion game is derived as: [0022] 11= -1177,0-.I07e)+Bpup-FBA (4) [0023] where 4 qv -qr. B2,4 B, a difference between 72 and, Bp is a coefficient matrix of the inpu t for the pursuit spacecraft, and Beis a coefficien t matrix of the input for the evasion spacecraft [0024] Because the function iv rs differentiable with arbitrary order at a point Ve, 1(77P) is expanded using Taylor series at the pointii,as Expression (5) [0025] ) = @id +vt (11,)17+ 0(11) (5): [0026] A further Expression (6) is derived as [0027] f (1-7,)=vr Jr' (77,)1j+0(17),4 F (6) [0028] whereot7)) is an infinitesimal of higher order than 7). \ is a symbol for gradient, and VT fr)18 defined as vr fr (71,) tn) o7), [0029] 7i=767.
[0030] Substituting Expression (6) into Expression (4). Expression (7) is derived as FrTh+ Bpi/Jr, + B"u" [0031] (7) [0032] Them by using Euler discrete method, system (7) can be discretizerl with a sampling period 1as L.0033] 01,7,) 4, Epup [0034] where +r-2F(113. L3P*1±.L. B TB Y7g. is a value of 0 at time k 11 Pic is the control input of the pursuit spacecraft at time k and e'k is the control input of the evasion spacecraft at time k [0035] Furthermore, e, Step 2 further includes [0036] Step 2.1: Initialize an error threshold,9, taking control strategies {PP' ih and a weight matrix letting s e° [0037] where Rq is a positive, number: 77 C7'7 ianditi (rj)are initial control strategies; V4 is an initial value of the weight matrix; s is a number of iterative times.
[0038] Sept 2. 2 Computing a residual error of Hamilton equation: [0039] es-s-fri (Eh)11u 0)4)-s- (i),)+ fr1f, 17 7._ +13,14,7 [0040] where Q, E are positive definite matrices; is a predetermined positive number; and/4, )are the control strategies at step s; denotes a value o* tne weight matrix at step s: rff denotes theiesidual error; and ka is expressed a [0041] V (ft, )(fit),)+Ep 14 Kfik)± ttitie,s(fid [0042] where C (11) is a neural network -based function. The weight matrix ift, is updated according to an expression below Otu-, 147 = 1± ,$ E r t.k's [0043] [0044] where 0 is a real number satisfying 0< 0 <1 [0045] Step 2.3: setting s ±-6±11 computing a value function and the control strategies: 1 o [0o46] [0047] [0048] Step 2.4: computing and determining whether It ilk)-, Cid < is true If it is not true, turn back to Step 2,2; and if it is true, iterating is end ed, outputting the bhtrol strategies tic, -I ire, trip Ir [0049] Compared with the existing technology, the invention has the foilowing beneficial technical effects: [0050] The discrete ADP-based pursuit-evasion game controller designed according to the present invention is convenient to be implemented in engineering. The iterative algorithm with synchronous convergence based on an adaptive dynamic programming can effectively ensure that the control strategies can be converged synchronously to an optimal value by using the algorithm, the strong nonlinearities of the system can be effectively solved and an approxitnative optimal control strategy can be derived, which can ensure the system obtains an optimal performance.
REEF DESCRIPTION rtr THE DRAWINGS
[0051] Fig. 1 is a flowchart in accordance with the present invention.
[0052] Fig. 2 is a diagram illustrating a simulation result in accordance with the present invention.
DETAILED DESCRIPTION OF THE EMBODIMENTS
[0053] A further description in details is made with reference to the present invention below [0054] For a chat acteristics of strong non-linearity of a spacecraft model, the present invention proposes an optimal control strategy with synchronous convergence of zero-sum game based on an adaptive dynamic programming (ADP) algorithm. Firstly. a discrete system model for a pursuit-evasion game of spacecrafts is established. S'econdly, an ADP iterative control strategy with synchronous convergence of zero-sum game is designed. Finally, the optimal control strategy is derived approximatively by using.a neural network, [0055] As shown by Fig 1, the method includes the to! owing steps: [0056] Step 1: Establish'ng the discrete model for the 'pursuit-evasion game of spacecrafts; [0057] In the. Euler-Hill reference frame, equations for a nonlinear relative motion of the spacecrafts are given as: = e a [0058] it (1) r ' d r el 2 du -21-I/ P --= r -1., -a-e 1 V [0059] where x, y. z denote information of a position of the spacecraft in the Euler-Hill reference frame; lc Ur and 1d, are the control inputs; pis a gravitational constant; and, respectively, denote the orbit radii of a chief spacecraft and a deputy spacecraft; it denotes a true anomaly of the orbit of the chief spacecraft, z denotes a first-order derivative of ', and denotes a second-order derivative of [0060] By denoting 7 e=1,..)c 7.] 4-4-, kg, u u;], the equations for the nonlinear relative motion can be reformulated into a state space. form as [0061] pi) + Bu [0062] where [0063] f t /)= ± ) B=10 1 27(r V X t.V r-fr r3 [0064] where)7 denotes a state set, It denotes an input set, B denotes a coefficient matrix of the input, and ir denotes a unit matrix [006.5] Hence, the respective nonlinear dynamics models for the pursuer and the evader are described by Expressions (2), (3) [0066] p)+ flu p (2) [0067] t +Bic (3) [0068] where TIP and are, respectively, a state vector and a control input of the pursuer; lit and m=are, respectively, a state vector and a control input of the evader Subtracting Expression (3) from Expression (2), a model for the nonlinear pursuit-evasion game is derived as: [0069] = ft). )-(rh)+ B + Belie (4) [0070] where 1,7-= tip - ,k4 -R Fps a differs nce between 7L, and Ths, a coefficient matrix of the input for the pursuit spacecraft, and B, is a coefficient matrix of the input for the evasion spacecraft [0071] Because the function PAis differentiable with arbitrary order at a point 17,, .1(77dis expanded using Taylor series at the pointilras Expression (5) (7 7= 07,)+71 (Viz o(17) [0072] (5): [0073] A further Expression (6) is derived as 10 0 7 41 f) - = f (7 ± ) (6) [0075] whereoui is an infinitesimal of higher order than 7). \ is a symbol for gradient, and VT f ( ) IS defined as (6) into Expression (4), Expression (7) is derived as (7) vj I tn) method, system (7) can be discretized with a [0076] discrete i17= F(0)+Bpfir, [0078] [0079] Them by using B7), Expression + B"u" Euler sampling period11 as [0080] /7,"", = + 1-3,1(.9,7 [0081] where -TClikt-k17,-+ Trtn,). 8,4 -k- fig. is a value of at time 1 5 11 PX is the control input of the pursuit spacecraft at time k, and lie,k is the control input of the evasion spacecraft at little ic [0082] Step 2: Designing an ADP-based pursuit-evasion game algorithm.
[0083] An iterative algorithm of a synchronous ADP pursuit-evasion game based on a neural network is given as follows.
u (77-)} [0084] 1) Initializing an error threshold,9, takin(lmntrol strategies * zo' .c
-
and a weight matrix If.), letting s0 [0085] where 8 is a very small positive number pp 0(1)3 and pe, 0)3 are initial control strategies; fnis an initial value of the weight matrix; s is a number of iterative times In
this example,
0:=101 0t2 0 511. 4); 5 -0.21T 9, = [0.8 0,3 05 0,7 0,2 0,5]T [0086] Computing a residual error of Hamilton equation: j + ..(14.)-R(14,0i9.)+Wfilu, [0087] [0088] where Q B are positive definite matrices, / is a predetermined positive number; p"j and fin (ik) are the control strategies at steps, W+1 denotes a value of, the weight matrix, at step s; and denotes the residual error. In this example. Q = 81ag ([1 1 I 11) cliag([1 1 11) = 2° and ta. is expressed ) R as- [0089] re/ (77) (-1(lik)+ 0')k) [0090] where c(iik) is a neural network-based function. In this example is defined as [0091] cr( = {tailh(r,) thither, tanh(1,) tanli(y,) tanh(x5) [0092] The weight matrix Tic is updated according to an expression below OUT, 1 r S 1+ ft 7.
where 0 is a real number satisfying 3): Setting s computing a value function and the control strategies: 17; (114-= PT:si" c(fik) [0093] [0094] [0095] [0096] L1,1\ at (t)1T2 Q5 R15TV o-(.14)W [0097] [0098] 4y: computing and determining whether )1< 3 is true or not If it is not true, turn back. to Step 2): else, the, iterative algorithm is ended and the control strategiestuf,3(k)..ite".(1), )3 is outputted.
[0099] The simulation results are derived by using the proposed method of the present invention. As shown in Fig. 2, , are the elements of 14 From Fig. 2, it can be seen that a state error converges to 0 finally, indicating that the pursuit spacecraft has tracked down the evasion spacecraft and can maintain stability. Therefore, 5 the method for a pursuit-evasion game of spacecraft based on neural network proposed by the present invention is very effective.
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CN115454122A (en) * 2022-08-15 2022-12-09 北京航空航天大学 Adjacent convex optimization method for high-speed aircraft pursuit escape differential game

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CN113552872B (en) * 2021-01-28 2024-05-14 北京理工大学 Method for controlling pursuit and escape of pursuit robot at different speeds
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