Analytical Strategies and Winning Conditions for Elliptic-Orbit Target-Attacker-Defender Game

TL;DR

This work addresses the orbital Target-Attacker-Defender game with a non-maneuvering target on elliptic orbits. It delivers an analytical $LQ$ solution by solving the matrix Riccati equation using transversality and Tschauner–Hempel state-transition dynamics, yielding explicit Nash-equilibrium strategies for the attacker and defender. It further derives analytical sufficient and necessary winning conditions for the attacker, expressed via geometric ellipsoids that constrain the defender's initial position within the true anomaly window $[f_0,f_f]$ and depend on eccentricity $e$. Numerical simulations validate the analytical results with near-perfect accuracy (cost relative error around $0.004\%$) while achieving CPU-time reductions over $99.9\%$, underscoring real-time applicability; eccentricity mainly shifts the capture epoch without changing the winner in tested scenarios.

Abstract

This paper proposes an analytical framework for the orbital Target-Attacker-Defender game with a non-maneuvering target along elliptic orbits. Focusing on the linear quadratic game, we derive an analytical solution to the matrix Riccati equation, which yields analytical Nash-equilibrium strategies for the game. Based on the analytical strategies, we derive the analytical form of the necessary and sufficient winning conditions for the attacker. The simulation results show good consistency between the analytical and numerical methods, exhibiting 0.004$\%$ relative error in the cost function. The analytical method achieves over 99.9$\%$ reduction in CPU time compared to the conventional numerical method, strengthening the advantage of developing the analytical strategies. Furthermore, we verify the proposed winning conditions and investigate the effects of eccentricity on the game outcomes. Our analysis reveals that for games with hovering initial states, the initial position of the defender should be constrained inside a mathematically definable set to ensure that the attacker wins the game. This constrained set further permits geometric interpretation through our proposed method. This work establishes the analytical framework for orbital Target-Attacker-Defender games, providing fundamental insights into the solution analysis of the game.

Figures (9)

• Figure 1: The LVLH frame.
• Figure 2: The schematic of the TAD game in the LVLH frame.
• Figure 3: The trajectories of the attacker and defender in the LVLH frame. (a) Trajectories obtained from the analytical method; (b) Trajectories obtained from the numerical method.
• Figure 4: The variation in the position and control inputs of attacker and defender during the game. (a) The position of the attacker; (b) The position of the defender; (c) The control inputs of the attacker; (d) The control inputs of the defender.
• Figure 5: The distance of the attacker and target, the distance between attacker and defender, $g_1\left(f\right)$, and $g_2\left(f\right)$ during the game.