Application of the epsilon technique to a realistic optimal pursuit-evasion problem

Abstract

The Balakrishnan epsilon technique is applied to a pursuit—evasion problem in which both the pursuer and evader act optimally. The problem consists of determining the control functions for both the pursuer and the evader that result in the time of interception being minimized and maximized, respectively. The pursuer is considered to be a point mass confined to a horizontal plane and subjected to realistic aerodynamic and propulsive forces which are nonlinear functions of the state and the control. The evader has a constant speed and has a limit on the rate with which it can change direction. The Balakrishnan epsilon technique involves the insertion of a penalty term for not satisfying the dynamic equations directly. A modified Newton-Raphson method is used to solve the coefficients of a functional expansion of the state variables which minimizes the cost; a simple search is used to find the control at each time point which minimizes the integrand of the cost function. Repeated application of both the Newton-Raphson method and the search results in rapid convergence to the minimum-time solution to a fixed point. Six to eight iterations are typically required.

A complete family of minimum-time flight paths to points in several directions are computed and subsequently used to determine the intercept point for virtually any evader flight path. Interpolation between solutions yields both the optimal path and corresponding control for the interception. A similar set of solutions are generated for the evader. The general solution of the evader is superimposed on that of the pursuer in a way that is consistent with the problem's initial conditions. The intercept point for the max-min solution corresponds to the maximum of the minimum time required for both the pursuer and evader to reach that point. Once the intercept point is determined, the corresponding trajectories and control functions for both the pursuer and evader can be obtained by interpolating between adjacent solutions. The approach used is particularly efficient when a large number of optimal pursuit-evasion solutions are needed.