Spacecraft Pursuit-Evasion Game

Author: Tianjiang Says
This article is based on Zhang Chengming (2021) Research on Guidance Strategies for Spacecraft Pursuit-Evasion Games.

Definition

The spacecraft pursuit-evasion game is a hot research topic in the field of space security. It studies the adversarial maneuvering problem between maneuverable on-orbit spacecraft. The pursuing spacecraft aims to approach the target spacecraft at minimal cost and complete a capture mission, while the evading spacecraft adopts corresponding strategies to avoid capture. The problem is essentially a bilateral control continuous dynamic game, which differs from cooperative rendezvous and docking in that it requires consideration of the strategic game between both pursuing and evading parties.

Problem Classification

Spacecraft pursuit-evasion problems can be classified along different dimensions:

Classification Criterion	Type
Orbit type	Near-circular orbit, elliptical orbit
Thrust form	Continuous low-thrust, impulsive thrust
Number of players	Two-player, multi-spacecraft
Objective function	Zero-sum, non-zero-sum
Game duration	Fixed-time, free-time
Relative distance	Long-range, short-range

Dynamic Modeling Methods

CW Equation Description

Short-range pursuit-evasion problems typically use the CW equation (Clohessy-Wiltshire equation) to describe the relative motion of two spacecraft. The CW equation is applicable to near-circular orbits with relative distances much smaller than the semi-major axis.

State Equations

In the LVLH coordinate frame, the spacecraft state vector is $\boldsymbol{X} = [x, y, z, \dot{x}, \dot{y}, \dot{z}]^T$ , and the state equation is:

\dot{\boldsymbol{X}}_i = \boldsymbol{A}\boldsymbol{X}_i + T_i\boldsymbol{B}\boldsymbol{U}_i

where $\boldsymbol{U}_i = [\cos\alpha_i\cos\beta_i, \sin\alpha_i\cos\beta_i, \sin\beta_i]^T$ is the control variable, and $\alpha_i, \beta_i$ are thrust direction angles.

Objective Function

A typical objective function for the free-time pursuit-evasion problem is:

J = J_p = -J_e = \int_0^{t_f} 1 \, dt

i.e., minimizing the pursuit-evasion time (for the pursuer) or maximizing it (for the evader).

Solution Methods

Differential Game Approach

Based on differential game theory, the pursuit-evasion problem is formulated as a zero-sum game. By deriving the necessary conditions for the saddle-point strategy, a two-point boundary value problem is obtained. Typical solution methods include:

Costate normalization: Eliminates non-uniqueness of the saddle-point solution
Dimensionality reduction: Converts the high-dimensional two-point boundary value problem into a lower-dimensional optimization problem
Semi-direct method: Uses the analytical necessary conditions of one party to reduce problem dimensionality

Deep Learning Methods

In recent years, deep neural networks have shown promising applications in solving pursuit-evasion problems:

DRD algorithm: A dimensionality reduction-DNN-based approach for near-circular orbit pursuit-evasion problems
DNN-pseudospectral method: A Radau pseudospectral method based on deep neural networks
Combined optimization methods: Integrating traditional optimization algorithms with neural networks

Numerical Solution Methods

Shooting method and its improved variants
Pseudospectral method
Global optimization algorithms such as genetic algorithms and particle swarm optimization
Hybrid methods combining multiple shooting with sequential quadratic programming

Typical Application Scenarios

Space situational awareness: Approach operations on non-cooperative targets (e.g., defunct satellites, space debris)
Space confrontation: Strategy design in orbital offense-defense games
On-orbit servicing: Relative maneuvering in satellite inspection, repair, and refueling missions
Rendezvous and docking: Although primarily cooperative, game analysis can enhance mission safety

Research Frontiers

Multi-spacecraft pursuit-evasion problems (cooperation or competition within the pursuing or evading side)
Pursuit-evasion problems considering orbital perturbations ( $J_2$ perturbation, atmospheric drag, etc.)
Applications of deep reinforcement learning in pursuit-evasion games
Pursuit-evasion-defense three-body problems (complex game scenarios introducing a defending party)

References

Zhang Chengming. Research on Guidance Strategies for Spacecraft Pursuit-Evasion Games[D]. National University of Defense Technology, 2021. [in Chinese]
Isaacs R. Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Optimization and Control[M]. John Wiley & Sons, 1965.
Lachner R, et al. Air Combat Analysis Using Differential Games[C]. AIAA Guidance, Navigation, and Control Conference, 1999.