基于可达集的航天器多对一轨道博弈几何求解

doi:10.7527/S1000-6893.2024.30803

Abstract

Abstract:

The orbital game between spacecraft has significant importance for space safety. Compared to the classical “one-to-one” orbital game problem， the “many-to-one” orbital game faces challenges such as high-dimensional state parameters， undefined roles of pursuers， and variable terminal conditions， making traditional differential strategy methods difficult to solve. To overcome these challenges， this paper proposes a geometric method for the “many-to-one” orbital game based on the reachable set of spacecraft. Firstly， the reachable set equivalence representation of Nash equilibrium points in the orbital game is established based on the theory of reachable sets of spacecraft. Then， the envelopment of the reachable set of spacecraft at any time is accurately solved using a grid point search method. Subsequently， the relative geometric relationship between the reachable sets of spacecraft involved in the game is determined by computing the solid angle of the triangulated closed surface. Finally， the binary search method is used to determine the terminal time of the game， that satisfies the geometric conditions of the reachable set of Nash equilibrium points， thereby determining the spatial position coordinates of the spacecraft at the end of the game and completing the solution to the “many-to-one” orbital game problem. Simulation results demonstrate that for a typical “three-to-one” orbital game scenario， the traditional differential strategy method takes over 2 h on an ordinary personal computer. In contrast， the proposed geometric method can provide the same solution within 12 min.

Key words: orbital game, multiple-pursuer/one-evader, reachable set, Nash equilibrium, geometrical solution

CLC Number:

V412.41

Zhaohang LI, Changxuan WEN, Dong QIAO, Bo PANG. Geometrical solution of multi-pursuer/one-evader orbital pursuit-evasion game based on reachable set theory[J]. Acta Aeronautica et Astronautica Sinica, 2024, 45(S1): 730803.

Figures/Tables 8

Fig.1

Fig.2

Fig.3

Table 1

Initial orbit states of spacecrafts

参数	$p 1$	$p 2$	$p 3$	$e$
$r 0 / 103 k m$	$- 6.20 9.07 0$	$- 34.12 - 16.83 0$	$- 24.05 - 2.74 0$	$- 26.38 - 0.22 0$
$v 0 / k m · s - 1$	$- 6.45 - 3.12 0$	$1.95 - 1.98 0$	$- 0.23 - 3.92 0$	$0.03 - 3.89 0$

Table 1

Fig.4

Table 2

Nash equilibrium point calculated by two methods

方法	$t f / s$	$r N / 103 k m$
几何法	8 976.6	均衡点1： $- 6.369 1 - 25.542 5 0.685 9$
几何法	8 976.6	均衡点2： $- 6.369 1 - 25.542 5 - 0.685 9$
微分博弈^［20］	9 022.8	$- 5.866 5, - 25.494 4,0.688 6 T$

Table 2

Fig.5

Fig.6

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Geometrical solution of multi-pursuer/one-evader orbital pursuit-evasion game based on reachable set theory

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