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## A Dimension-reduction method for the finite-horizon spacecraft pursuit-evasion game

 State Key Laboratory of Ocean Engineering, Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200240, China

*Corresponding author: Guo-ping Cai

Received  June 2021 Revised  December 2021 Early access March 2022

The finite-horizon two-person zero-sum differential game is a significant technology to solve the finite-horizon spacecraft pursuit-evasion game (SPE game). Considering that the saddle point solution of the differential game usually results in solving a high-dimensional (24 dimensional in this paper) two-point boundary value problem (TPBVP) that is challengeable, a dimension-reduction method is proposed in this paper to simplify the solution of the 24-dimensional TPBVP related to the finite-horizon SPE game and to improve the efficiency of the saddle point solution. In this method, firstly, the 24-dimensional TPBVP can be simplified to a 12-dimensional TPBVP by using the linearization of the spacecraft dynamics; then the adjoint variables associated with the relative state variables between the pursuer and evader can be expressed in the form of state transition; after that, based on the necessary conditions for the saddle point solution and the adjoint variables in the form of state transition, the 12-dimensional TPBVP can be transformed into the solving of 6-dimensional nonlinear equations; finally, a hybrid numerical algorithm is developed to solve the nonlinear equations so as to obtain the saddle point solution. The simulation results show that the proposed method can effectively obtain the saddle point solution and is robust to the interception time, the orbital altitude and the initial relative states between the pursuer and evader.

Citation: Qi-shuai Wang, Pei Li, Ting Lei, Xiao-feng Liu, Guo-ping Cai. A Dimension-reduction method for the finite-horizon spacecraft pursuit-evasion game. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022028
##### References:
 [1] A. Flores-Abad, O. Ma, K. Pham and S. Ulrich, A review of space robotics technologies for on-orbit servicing, Progress in Aerospace Sciencesl, 68 (2014), 1-26. [2] H.Z. Liang, J.Y. Wang, J.Q. Liu and P. Liu, Guidance strategies for interceptor against active defense spacecraft in two-on-two engagement, Aerospace Science and Technologyl, 96 (2020), 105529. [3] D. Ye, M.M. Shi and Z.W. Sun, Satellite proximate pursuit-evasion game with different thrust configurations, Aerospace Science and Technologyl, 99 (2020), 105715. [4] A. Jagat and A.J. Sinclair, Nonlinear control for spacecraft pursuit-evasion game using the state-dependent Riccati equation method, IEEE Transactions on Aerospace and Electronic Systems, 53 (2017), 3032-3042. [5] T. Woodbury, J.E. Hurtado, Adaptive play via estimation in uncertain nonzero-sum orbital pursuit-evasion games, in AIAA SPACE and Astronautics Forum and Exposition, Orlando, Florida, 2017. [6] W.T. Hafer, H.L. Reed, Orbital pursuit-evasion hybrid spacecraft controllers, in AIAA Guidance, Navigation, and Control Conference, Kissimmee, Florida, 2015. [7] R. Isaacs, Games of Pursuit, Rand Corporation, Santa Monica, California, 1951. [8] R. Isaacs, Differential Games, Wiley, New York, 1965. [9] K. Horie and B.A. Conway, Optimal fighter pursuit-evasion maneuvers found via two-sided optimization, Journal of Guidance Control and Dynamics, 29 (2006), 105-112. [10] M.H. Breitner, H.J. Pesch and W. Grimm, Complex differential games of pursuit-evasion type with state constraints, Part 1: necessary conditions for optimal open-loop strategies, Journal of Optimization Theory and Applications, 78 (1993), 419-441.  doi: 10.1007/BF00939876. [11] M. Pachter and Y. Yavin, A stochastic homicidal chauffeur pursuit-evasion differential game, Journal of Optimization Theory and Applications, 34 (1981), 405-424.  doi: 10.1007/BF00934680. [12] V. Macias, I. Becerra, R. Murrieta-Cid, H.M. Becerra and S. Hutchinson, Image feedback based optimal control and the value of information in a differential game, Automatica, 90 (2018), 271-285.  doi: 10.1016/j.automatica.2017.12.045. [13] W. Lin, Distributed UAV formation control using differential game approach, Aerospace Science and Technology, 35 (2014), 54-62. [14] F. Liu, X.W. Dong, Q.D. Li and Z. Ren, Robust multi-agent differential games with application to cooperative guidance, Aerospace Science and Technologyl, 111 (2021), 106568. [15] Y. Chai, J.J. Luo, N. Han and J. Sun, Robust event-triggered game-based attitude control for on-orbit assembly, Aerospace Science and Technologyl, 103 (2020), 105894. [16] C. Venigalla, D.J. Scheeres, Spacecraft rendezvous and pursuit/evasion analysis using reachable sets, in AIAA/AAS Space Flight Mechanics Meeting, Kissimmee, Florida, USA, 2018. [17] M. Pontani and B.A. Conway, Numerical solution of the three-dimensional orbital pursuit-evasion game, Journal of Guidance Control and Dynamics, 32 (2009), 474-487. [18] H.X. Shen and L. Casalino, Revisit of the three-dimensional orbital pursuit-evasion game, Journal of Guidance Control and Dynamics, 41 (2018), 1823-1831. [19] G.M. Anderson and V.W. Grazier, Barrier in pursuit-evasion problems between two low-thrust orbital spacecraft, AIAA Journal, 14 (1976), 158-163.  doi: 10.2514/3.61350. [20] Z.Y. Li, H. Zhu, Z. Yang and Y.Z. Luo, Saddle point of orbital pursuit-evasion game under $J_2$-perturbed dynamics, Journal of Guidance Control and Dynamics, 43 (2020), 1733-1739. [21] R.W. Carr, R.G. Cobb, M. Pachter and S. Pierce, Solution of a pursuit-evasion game using a near-optimal strategy, Journal of Guidance Control and Dynamics, 41 (2018), 841-850. [22] Z.Y. Li, H. Zhu, Z. Yang and Y.Z. Luo, A dimension-reduction solution of free-time differential games for spacecraft pursuit-evasion, Acta Astronautica, 163 (2019), 201-210. [23] S. Sun, Q. Zhang, R. Loxton and B. Li, Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit, Journal of Industrial and Management Optimization, 11 (2017), 1127-1147.  doi: 10.3934/jimo.2015.11.1127. [24] A. Colorni, M. Dorigo, V. Maniezzo, Distributed optimization by ant colonies, in Ecal91-European Conference on Artificial Life, York, USA, 2015. [25] M.J.D. Powell, Numerical Methods for Nonlinear Algebraic Equations, Gordon & Breach, London, 1970. [26] S. Lee and S.Y. Park, Approximate analytical solutions to optimal reconfiguration problems in perturbed satellite relative motion, Journal of Guidance Control and Dynamics, 34 (2011), 1097-1111. [27] T. Basar, G.J. Olsder, Dynamic Noncooperative Game Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1999. [28] S.Y. Balaman, Decision-Making for Biomass-Based Production Chains, Academic Press, Pittsburgh, 2018. [29] J.J. More, B.S. Garbow, K.E. Hillstorm, User Guide for Minpack-1, Argonne National Laboratory, USA, 1980.

show all references

##### References:
 [1] A. Flores-Abad, O. Ma, K. Pham and S. Ulrich, A review of space robotics technologies for on-orbit servicing, Progress in Aerospace Sciencesl, 68 (2014), 1-26. [2] H.Z. Liang, J.Y. Wang, J.Q. Liu and P. Liu, Guidance strategies for interceptor against active defense spacecraft in two-on-two engagement, Aerospace Science and Technologyl, 96 (2020), 105529. [3] D. Ye, M.M. Shi and Z.W. Sun, Satellite proximate pursuit-evasion game with different thrust configurations, Aerospace Science and Technologyl, 99 (2020), 105715. [4] A. Jagat and A.J. Sinclair, Nonlinear control for spacecraft pursuit-evasion game using the state-dependent Riccati equation method, IEEE Transactions on Aerospace and Electronic Systems, 53 (2017), 3032-3042. [5] T. Woodbury, J.E. Hurtado, Adaptive play via estimation in uncertain nonzero-sum orbital pursuit-evasion games, in AIAA SPACE and Astronautics Forum and Exposition, Orlando, Florida, 2017. [6] W.T. Hafer, H.L. Reed, Orbital pursuit-evasion hybrid spacecraft controllers, in AIAA Guidance, Navigation, and Control Conference, Kissimmee, Florida, 2015. [7] R. Isaacs, Games of Pursuit, Rand Corporation, Santa Monica, California, 1951. [8] R. Isaacs, Differential Games, Wiley, New York, 1965. [9] K. Horie and B.A. Conway, Optimal fighter pursuit-evasion maneuvers found via two-sided optimization, Journal of Guidance Control and Dynamics, 29 (2006), 105-112. [10] M.H. Breitner, H.J. Pesch and W. Grimm, Complex differential games of pursuit-evasion type with state constraints, Part 1: necessary conditions for optimal open-loop strategies, Journal of Optimization Theory and Applications, 78 (1993), 419-441.  doi: 10.1007/BF00939876. [11] M. Pachter and Y. Yavin, A stochastic homicidal chauffeur pursuit-evasion differential game, Journal of Optimization Theory and Applications, 34 (1981), 405-424.  doi: 10.1007/BF00934680. [12] V. Macias, I. Becerra, R. Murrieta-Cid, H.M. Becerra and S. Hutchinson, Image feedback based optimal control and the value of information in a differential game, Automatica, 90 (2018), 271-285.  doi: 10.1016/j.automatica.2017.12.045. [13] W. Lin, Distributed UAV formation control using differential game approach, Aerospace Science and Technology, 35 (2014), 54-62. [14] F. Liu, X.W. Dong, Q.D. Li and Z. Ren, Robust multi-agent differential games with application to cooperative guidance, Aerospace Science and Technologyl, 111 (2021), 106568. [15] Y. Chai, J.J. Luo, N. Han and J. Sun, Robust event-triggered game-based attitude control for on-orbit assembly, Aerospace Science and Technologyl, 103 (2020), 105894. [16] C. Venigalla, D.J. Scheeres, Spacecraft rendezvous and pursuit/evasion analysis using reachable sets, in AIAA/AAS Space Flight Mechanics Meeting, Kissimmee, Florida, USA, 2018. [17] M. Pontani and B.A. Conway, Numerical solution of the three-dimensional orbital pursuit-evasion game, Journal of Guidance Control and Dynamics, 32 (2009), 474-487. [18] H.X. Shen and L. Casalino, Revisit of the three-dimensional orbital pursuit-evasion game, Journal of Guidance Control and Dynamics, 41 (2018), 1823-1831. [19] G.M. Anderson and V.W. Grazier, Barrier in pursuit-evasion problems between two low-thrust orbital spacecraft, AIAA Journal, 14 (1976), 158-163.  doi: 10.2514/3.61350. [20] Z.Y. Li, H. Zhu, Z. Yang and Y.Z. Luo, Saddle point of orbital pursuit-evasion game under $J_2$-perturbed dynamics, Journal of Guidance Control and Dynamics, 43 (2020), 1733-1739. [21] R.W. Carr, R.G. Cobb, M. Pachter and S. Pierce, Solution of a pursuit-evasion game using a near-optimal strategy, Journal of Guidance Control and Dynamics, 41 (2018), 841-850. [22] Z.Y. Li, H. Zhu, Z. Yang and Y.Z. Luo, A dimension-reduction solution of free-time differential games for spacecraft pursuit-evasion, Acta Astronautica, 163 (2019), 201-210. [23] S. Sun, Q. Zhang, R. Loxton and B. Li, Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit, Journal of Industrial and Management Optimization, 11 (2017), 1127-1147.  doi: 10.3934/jimo.2015.11.1127. [24] A. Colorni, M. Dorigo, V. Maniezzo, Distributed optimization by ant colonies, in Ecal91-European Conference on Artificial Life, York, USA, 2015. [25] M.J.D. Powell, Numerical Methods for Nonlinear Algebraic Equations, Gordon & Breach, London, 1970. [26] S. Lee and S.Y. Park, Approximate analytical solutions to optimal reconfiguration problems in perturbed satellite relative motion, Journal of Guidance Control and Dynamics, 34 (2011), 1097-1111. [27] T. Basar, G.J. Olsder, Dynamic Noncooperative Game Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1999. [28] S.Y. Balaman, Decision-Making for Biomass-Based Production Chains, Academic Press, Pittsburgh, 2018. [29] J.J. More, B.S. Garbow, K.E. Hillstorm, User Guide for Minpack-1, Argonne National Laboratory, USA, 1980.
Illustration of the coordinate system of spacecraft pursuit-evasion game
The optimal motion trajectories of the pursuer and evader in Case 1, Case 2 and Case 3: (a) motion trajectory; (b) x-t curve; (c) y-t curve; (d) z-t curve; (e) relative position; (e) relative velocity
The optimal motion trajectories of the pursuer and evader in Case 4: (a) motion trajectory; (b) x-t curve; (c) y-t curve; (d) z-t curve; (e) relative position; (e) relative velocity
The optimal motion trajectories of the pursuer and evader in Case 5, Case 6 and Case 7: (a) motion trajectory; (b) x-t curve; (c) y-t curve; (d) z-t curve; (e) relative position; (e) relative velocity
Seven cases of the finite-horizon SPE game \\ with different inputs
 Case Orbital Interception Acceleration Initial relative state altitude time variables h ($\rm km$) $t_f ({\rm{s}})$ $[u_P, u_E)] ({\rm{m/s^2}})$ [$\bigtriangleup x$, $\bigtriangleup y$, $\bigtriangleup z$, $\bigtriangleup v_x$, $\bigtriangleup v_y$, $\bigtriangleup v_z$] ($\rm km$) 1 1000 595.5 [0.2, 0.03] [0, -30, -0.1, 0, 0, 0] 2 10000 595.5 [0.2, 0.03] [0, -30, -0.1, 0, 0, 0] 3 35786 595.5 [0.2, 0.03] [0, -30, -0.1, 0, 0, 0] 4 1000 1879.7 [0.2, 0.03] [0, -300, -1, 0, 0, 0] 5 1000 4356.6 [0.2, 0.03] [0, -3000, -10, 0, 0, 0] 6 1000 4600.5 [0.2, 0.03] [0, -3000, -10, 0, 0, 0] 7 1000 4865.5 [0.2, 0.03] [0, -3000, -10, 0, 0, 0]
 Case Orbital Interception Acceleration Initial relative state altitude time variables h ($\rm km$) $t_f ({\rm{s}})$ $[u_P, u_E)] ({\rm{m/s^2}})$ [$\bigtriangleup x$, $\bigtriangleup y$, $\bigtriangleup z$, $\bigtriangleup v_x$, $\bigtriangleup v_y$, $\bigtriangleup v_z$] ($\rm km$) 1 1000 595.5 [0.2, 0.03] [0, -30, -0.1, 0, 0, 0] 2 10000 595.5 [0.2, 0.03] [0, -30, -0.1, 0, 0, 0] 3 35786 595.5 [0.2, 0.03] [0, -30, -0.1, 0, 0, 0] 4 1000 1879.7 [0.2, 0.03] [0, -300, -1, 0, 0, 0] 5 1000 4356.6 [0.2, 0.03] [0, -3000, -10, 0, 0, 0] 6 1000 4600.5 [0.2, 0.03] [0, -3000, -10, 0, 0, 0] 7 1000 4865.5 [0.2, 0.03] [0, -3000, -10, 0, 0, 0]
Setting of the parameters in the hybrid algorithm
 Algorithm Parameters Value ACO Number of ants 50 Number of iterations 100 Importance of heuristic factors 12 Importance of pheromones 0.3 Pheromone evaporation coefficient 0.5 Initial values of $\tau_i (i=1,2,3)$ [-5, 5] Initial values of $x_{r_f}$, $y_{r_f}$ and $z_{r_f}$ [-1400, 1400] The tolerance for integration $1e-5$ Powell algorithm The precision of the terminal distance $1e-8$ The precision of the terminal Hamiltonian $1e-9$ The tolerance for integration $1e-10$
 Algorithm Parameters Value ACO Number of ants 50 Number of iterations 100 Importance of heuristic factors 12 Importance of pheromones 0.3 Pheromone evaporation coefficient 0.5 Initial values of $\tau_i (i=1,2,3)$ [-5, 5] Initial values of $x_{r_f}$, $y_{r_f}$ and $z_{r_f}$ [-1400, 1400] The tolerance for integration $1e-5$ Powell algorithm The precision of the terminal distance $1e-8$ The precision of the terminal Hamiltonian $1e-9$ The tolerance for integration $1e-10$
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