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

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.

💡 Research Summary

The paper addresses a three‑player differential game—Target‑Attacker‑Defender (TAD)—in which the target follows a non‑maneuvering Keplerian trajectory on an elliptic orbit. By modeling the relative motion of the attacker, defender, and target with the linearized Tschaüner‑Hempel (TH) equations, the authors obtain a state‑space representation ˜X′ = A˜X + B u where the true anomaly f is used as the independent variable and the scaling factor ρ = 1 + e cos f accounts for orbital eccentricity.

The game is cast in a linear‑quadratic (LQ) framework. The cost functional J combines quadratic penalties on the attacker’s and defender’s states (weighted by positive‑semi‑definite matrices Sₐ and S_d) and on their control efforts (weighted by positive‑definite matrices Rₐ and R_d). The target is fixed at the origin of the LVLH frame, which simplifies the relative dynamics to those of the attacker and the relative attacker‑defender separation.

Applying Pontryagin’s minimum‑maximum principle yields the Hamiltonian and the optimality conditions ∂H/∂uₐ = 0 and ∂H/∂u_d = 0. The resulting Nash‑equilibrium feedback laws are
uₐ* = –Rₐ⁻¹ Bᵀ(λ – ν), u_d* = R_d⁻¹ Bᵀ ν,
where λ and ν are the costate vectors associated with the attacker and defender, respectively. Their dynamics are linear: λ′ = –Aᵀλ and ν′ = –Aᵀν.

The crucial coupling between λ and ν is expressed through a matrix Riccati differential equation (RDE) for P(f) = ν λ⁻¹:
P′ = W₂₂ P – P W₁₁ – P W₁₂ P,
with W₁₁, W₁₂, and W₂₂ constructed from A, B, Rₐ, and R_d. Traditional approaches solve this RDE numerically via backward integration, which is computationally intensive and hampers real‑time decision making.

The authors exploit the analytical state‑transition matrix Ω₁₁(f,f₀) of the uncontrolled TH dynamics and the corresponding transition matrix Ω₂₂(f,f₀) for the costates. By integrating the linear costate equations, they derive a closed‑form expression for P(f):
P(f) = Ω₂₂(f,f₀) P(f₀) Ω₂₂⁻¹(f,f₀).
The terminal transversality condition P(f_f) = diag(Sₐ, –S_d) provides the initial value P(f₀) analytically, eliminating any need for numerical integration. Consequently, the entire Nash strategy (both control laws and the associated costates) can be computed directly from the initial relative states.

With the analytical strategies in hand, the paper proceeds to derive necessary and sufficient winning conditions for the attacker. By evaluating the difference Jₐ – J_d under the optimal policies, the authors show that the attacker succeeds if and only if the defender’s initial position lies inside a specific set 𝔇 defined by the inequality
˜x_dᵀ M ˜x_d ≤ 0,
where M is a symmetric matrix built from P(f₀) and the STM Ω₁₁. Geometrically, 𝔇 is a six‑dimensional ellipsoidal region that, when projected onto physical space, appears as an ellipsoidal cone surrounding the line from the attacker to the target. Thus, the defender must start within this cone for the attacker to be able to capture the target before being intercepted.

The influence of orbital eccentricity e is examined analytically. As e increases, the scaling factor ρ varies more strongly with f, making Ω₁₁ and Ω₂₂ more nonlinear. This contraction of the ellipsoidal region 𝔇 implies that high‑eccentricity orbits tighten the defender’s admissible initial region, making attacker victory more difficult. Numerical examples for e ranging from 0 to 0.8 confirm the theoretical trend.

Simulation results compare the analytical solution with a conventional numerical backward‑integration approach. The cost‑function discrepancy is less than 0.004 %, confirming the analytical solution’s accuracy. More strikingly, the analytical method reduces CPU time by over 99.9 % (the analytical computation consumes only about 0.1 % of the time required for the numerical method), demonstrating its suitability for real‑time guidance and space‑security applications where rapid response is essential.

In conclusion, the paper delivers a complete analytical framework for elliptic‑orbit TAD games: it provides closed‑form Nash strategies, explicit attacker‑winning conditions with clear geometric interpretation, and quantifies the effect of orbital eccentricity. The dramatic computational savings open the door to on‑board implementation in autonomous spacecraft and missile‑defense scenarios. Future work may extend the methodology to impulsive thrust models, fuel‑constrained games, and multi‑defender configurations, further enriching the toolbox for orbital differential games.