Computing the Pursuing Control in Proximate Orbital Pursuit-Evasion Game by Polynomial Approximation

In this paper, we focus on the proximate orbital pursuit-evasion game of two spacecraft with magnitude-bounded continuous controls. Two scenarios are considered depending on whether the pursuer can access the control magnitude of the evader initially. When the pursuer accesses such information, we propose a fast numerical method for computing a sub-optimal control of the pursuer that guarantees the capture of the evader. The key to accelerating the solving is using a polynomial to approximate an important integration in the control computation, whose direct computing involves repeated calculations of matrices' singular values. When the control magnitude of the evader is unavailable, we first propose a simple estimator for the evader's control magnitude, by which the pursuer can estimate the maximal control effort of the evader disclosed over the history based on measured states. Based on the estimate, another fast method for computing the sub-optimal pursuing control is proposed based again on polynomial approximation. Then, considering practical measurements, we analyze how measurement noises influence the estimation of the control magnitude and the pursuing control computation. Finally, we present numerical examples to test the proposed computing methods and discuss the influence of noises.