An Indirect Approach to Solve a Pursuit-Evasion War Game between Two Spacecraft

Pursuit-evasion (PE) games define a subclass of zero-sum differential games in which an agent desires to “capture” an adversarial agent. The use of indirect optimization to solve zero-sum differential games is not extensively documented in literature due to their complex dynamics and highly sensitive nature. An indirect approach known as the multi-order shooting scheme (MOSS) is proposed to solve pursuit-evasion war games between two spacecraft in a robust and efficient manner. A traditional shooting scheme employs Newton's method to solve the resulting two-point boundary value problem (TPBVP). Consequently, it is susceptible to divergence when the initial guess is in a region where gradient information alone is insufficient to progress the guess toward the saddle-point equilibrium solution. The MOSS employs Halley's method to develop second and third order root-solving updates, which utilize the higher-order derivatives of the inherent error surface to increase the domain of convergence. Furthermore, a non-product quadrature method known as the Conjugate Unscented Transformation (CUT) is utilized to compute higher-order sensitivities in a derivative-free manner. A key feature of MOSS is that, for each iteration, it automatically selects the update order that ensures the most effective step toward satisfaction of the terminal boundary conditions. An orbital PE game corresponding to the interception of an adversarial spacecraft in minimum time is considered to showcase the efficacy of MOSS in efficiently solving TPBVPs. The numerical results demonstrate that MOSS is able to identify a solution to the PE game when the traditional shooting method struggles to converge due to a poor-quality initial guess.