Revisiting the two-side optimization problem in satellite pursuit-evasion

. In the present paper two-side optimization problem where the intercepting satellite pursuits the target satellite was studied. In the pursuit-evasion game the continuous thruster and the simple central gravity with J2 perturbation were involved for two satellites. With applying the functional extremum condition, the optimal control outputs of both side satellites were obtained by building the systems Hamilton function. Further, the two side optimal problem was transformed to the TPBVP, which contributed the solution with the mixed numerical method. Finally, the proposed method was validated by the numerical simulation, where the optimal interception and escaping trajectories were illustrated.


Introduction
In previous studies, for the satellite interception question, the targets were assumed generally to be static without any motion ability and moved in the pre-assigned orbits. Thus the solver only needs to design a maneuvering trajectory for the attacking satellite in solving the interception which could be regarded as single side optimization question. The design and optimization method was different as the propulsive force exerting on the interceptor. Luo, etc. studied the multiple impulsive fuel optimal orbit rendezvous problem under the linear [1] and nonlinear [2] dynamics, they directly applied the simplex, quadratic, genetic [3] and simulated annealing algorithms [4] to optimize the impulse vectors. Simulation results show that combining with the meta optimization method and the SQP method can effectively solve the problem [5]. Lawden, etc. proposed the principal vector theory to determine the trajectory optimal property [6] and added the coast stage or middle impulse to achieve the fuel optimal trajectory [7]. For the continuous thruster, the indirect method was applied based on the calculus of variation and maximum principle, causing that the interception question was transformed to the two point boundary value problem (TPBVP). Greenwood solved the transformed problem by the shooting method and applied it to study the small thrust fuel optimal transferring problem in the co-plane orbit [8]. Unfortunately, the initial value of the indirect method is difficult to estimate and is sensitive for the result achievement. The direct method scatters the total transferring procedure to small steps and transfers the dynamical optimization to the constrained minimum solving problem. Hargraves, etc. put forward the collocation method to solve the time optimal interception climbs problem [9]. Enright [10]

Pursuit and Evasion Dynamics with JPerturbation
This intercepting event ends when the pursuer catches the target, namely both satellites arrives at the same location in the space. During the catching and escaping process, the pursuer adopts the control strategy to catch the target as soon as possible while the later tries its best to postpone the intercept terminal time. In order to make the question simple, it was assumed that both players can completely acquire the counter dynamics information.
Firstly, aiming to complete the interception, it is required that UP (the specific thrusts of the interceptor) is bigger than UE (the specific thrusts of the escaper). Mauro [16] used a group of new state variables to describe the spacecraft's motion to make the simulation more real. Thus the same variables were adopted in the paper. The state variables of the pursuer and evader can be defined as , , , , , with representing respectively the satellite's geocentric distance, geometrical longitude, latitude, path angle, speed and azimuth angle, and , i P E stand for the two sides in the interception, respectively.
The simple two-body gravity exerts a centripetal force on the satellite which produces an acceleration vectors can been represent in the geodetic coordinate system as following (1) where P represents the gravitational constant of the earth. The acceleration arose by the J2 non-spherical perturbation can been decomposed in the same coordinate as below   (4) Obviously the pursuer and evader optimize the angle of the thruster to compete for the interception ending time, thus the control variables of satellites are given as below Further, the dynamics of both sides can be rewritten as below , , , , (6) Interception ending requires that the location coordinate of both satellites is equal, namely the following conditions are satisfied at the terminal time f 0 Rewritten them to a compact vector form as following , 0 f t t t N x (8) Finally, the orbital elements can be transformed to the state variables according to the relation equation in reference [13].

Saddel Solution of the Pursuit-evasion Game
In terms of the pursuer, the interception ending time is expected to become short as little as possible, while the counterpart escapes the catch to the best of its ability. Thus the ending time can be chosen as the objective namely f J t (9) This interception is a traditional zero sum differential game. When the control strategies variables pair , P E U U satisfies the condition below, it is defined as the saddle solution (10) which means that if only the purser adopts the optimal control strategy, the objective will be better than that they both make use the optimal control strategies, and the interception ending time will shorten. Otherwise, the time will increase.
The open loop solution of the differential game is obtained based on the initial states 0 x of both sides and the current time, whilst the close loop solution which is more flexible to the real situation can be solved according to the initial states, current time and the current states. Conclusion has been made that if the value function is continuous in the admissible domain of control strategies, the open loop solution and close loop solution are identical [18]. And for the system dynamics with high complexity and nonlinearity, it is almost impossible to analytically solve the saddle solution. Thus obtaining the close loop solution is unpractical currently, the optional way is to solve the open loop control strategy and optimal trajectory relevant. For the two-side dynamical optimization problem with system (5), Maya type objective (8) and terminal condition collection (7), the Hamiltonian function can be defined as the optimal control theory (11) where P λ and E λ stand for the conjugates of the pursuer and evader's state variables, respectively. P H is given as equation (12) ^` ^1  (13) , (13) Substituting the correlative variables and equations, f t λ is given below 0 And collecting the equations, and the conjugates satisfy the following conditions Then applying the function extremum theory to system Hamiltonian function P H , the co-state equations are derived as the following And using s and c with replacing sin and cos respectively and substituting P f as equations (6), then the co-state equations above can be written as equation (17).  5  4  6  5  4  6  2  1  2  3  4  4  3  3  5  6  2  2  5  1  3  1  1  5   5  4  4  4  5  4  3  3  5  6  3  2  5  3  1 5  1  1  1   3  6  5  4  6  3  6  2  5  1  3  1  5  4   2   2   c  c  c  s  3  s  3c 2  2  2s 2  c  s  c   c  3  2 c  2 s  s  3c 2  2  2s 2  c  s   6  s The Hamilton function can be divided to two parts P H and E H , so applying the min-max condition to H is equal to deriving the minimum of P H and maximum of E H . Then the optimal control strategies of the pursuer and evader satisfy the equations as below > @ > @ 10 12 , 11 arg max arg max sin cos sin

Simulations and Two Side Optimal Trajectories
The optimization method described in section 4 was applied to solve the pursuit-evasion game of satellites with J2 non-spherical perturbation. Dimensional normalization was implemented as the reference [18] where  Substituting the initials of co-states and system states to the integration of the conjugate and dynamic equation, the states variations of the pursuer and evader were obtained, as shown in figure 1.

Conclusion
In the present paper the typical satellite interception problem was revised, where both the pursuer and the evader can apply the continuous thrust to perform orbital maneuver in the earth central force with the J2 non-spherical perturbation. The two-side optimization was involved since the interceptor and the escaper will try their best to compete in the attacking-defense scenario. The satellite dynamic with J2 perturbation was derived. The function extremum theory was used to obtain the necessary condition of the Hamilton system and the optimal control strategies of both sides when the pursuer and evader game have got its saddle solution. Then the pursuit-evasion game was turned into a two point boundary value problem and a further constrained optimization problem. Genetic algorithm was employed to search the approximation of the global optimal solution. The accurate solution was achieved by application of the SQP method to re-optimize the objective with the approximating solution as starting initials. Lastly, the two side optimal control angles and the optimal trajectories of the pursuer and evader were simulated and calculated to validate the efficiency of the method.