Time-optimal guidance for intercepting moving targets by Dubins vehicles

Abstract

This paper is concerned with a Minimum-Time Intercept Problem (MTIP), for which a Dubins vehicle is guided from a position with a prescribed initial heading angle to intercept a moving target in minimum time. Some geometric properties for the solution of the MTIP are presented, showing that the solution paths must lie in a sufficient family of 4 candidates. Necessary and sufficient conditions for optimality of each candidate are established. By employing the geometric properties, it is found that the 4 candidates are determined by the zeros of some real-valued functions when the target’s velocity is constant. In order to compute all the 4 candidates, the derivatives of these real-valued functions are transformed to polynomials so that the extrema of those real-valued functions can be computed by standard polynomial solvers. This allows employing a simple bisection method to find all the 4 candidates. Since the MTIP with a constant target’s velocity is equivalent to the path planning problem of Dubins vehicle in a constant drift field, developing such an algorithm also enables efficiently finding the shortest Dubins path in a constant drift field. Finally, some numerical examples are presented, demonstrating and verifying the developments of the paper.