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Volume 11 Issue 2
Feb.  2024

IEEE/CAA Journal of Automatica Sinica

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N. Chen, L. Li, and  W.  Mao,  “Equilibrium strategy of the pursuit-evasion game in three-dimensional space,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 2, pp. 446–458, Feb. 2024. doi: 10.1109/JAS.2023.123996
Citation: N. Chen, L. Li, and  W.  Mao,  “Equilibrium strategy of the pursuit-evasion game in three-dimensional space,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 2, pp. 446–458, Feb. 2024. doi: 10.1109/JAS.2023.123996

Equilibrium Strategy of the Pursuit-Evasion Game in Three-Dimensional Space

doi: 10.1109/JAS.2023.123996
Funds:  This work was supported in part by the Strategic Priority Research Program of Chinese Academy of Sciences (XDA27030100), and National Natural Science Foundation of China (72293575, 11832001)
More Information
  • The pursuit-evasion game models the strategic interaction among players, attracting attention in many realistic scenarios, such as missile guidance, unmanned aerial vehicles, and target defense. Existing studies mainly concentrate on the cooperative pursuit of multiple players in two-dimensional pursuit-evasion games. However, these approaches can hardly be applied to practical situations where players usually move in three-dimensional space with a three-degree-of-freedom control. In this paper, we make the first attempt to investigate the equilibrium strategy of the realistic pursuit-evasion game, in which the pursuer follows a three-degree-of-freedom control, and the evader moves freely. First, we describe the pursuer’s three-degree-of-freedom control and the evader’s relative coordinate. We then rigorously derive the equilibrium strategy by solving the retrogressive path equation according to the Hamilton-Jacobi-Bellman-Isaacs (HJBI) method, which divides the pursuit-evasion process into the navigation and acceleration phases. Besides, we analyze the maximum allowable speed for the pursuer to capture the evader successfully and provide the strategy with which the evader can escape when the pursuer’s speed exceeds the threshold. We further conduct comparison tests with various unilateral deviations to verify that the proposed strategy forms a Nash equilibrium.


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    • We make the first attempt to theoretically investigate the equilibrium strategy for the players conducting realistic motions in three-dimensional space
    • We derive the equilibrium strategy to tackle the realistic pursuit-evasion game by modeling the three-degree-of-freedom kinematics of the pursuer, which is typical in many real-world applications
    • We provide the theoretical derivation of the equilibrium strategy based on the HJBI equation to ensure the minimax property of the equilibrium strategy
    • We analyze the velocity threshold for a successful capture and then derive the optimal acceleration scheme for the pursuer and the escape strategy for the evader whenever the pursuer's speed exceeds the threshold
    • Our proposed solution supports real-time decisions for the pursuit-evasion games in the three-dimensional space


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