Time-optimal Dubins trajectory for moving obstacle avoidance

In this paper, the problem of computing the global time-optimal Dubins trajectory between two configurations (two points with prescribed heading angles) while avoiding a moving obstacle by a specified safe distance is discussed. The obstacle is allowed to move on any arbitrary trajectory as long as its trajectory satisfies some regularity constraints. By Pontryagin’s maximum principle, the fundamental segments of the time-optimal trajectory are proposed. Along each segment, the control input is either constant or is expressed in state-feedback form. Thereafter, using the necessary conditions for state-inequality constraints, the underlying geometric and analytical properties that govern the concatenation of these fundamental segments are revealed. All these properties exhibit solely state-dependent relations. This has two-fold applications: (1) such relations are essential in ruling out spurious extremals obtained by numerical optimal control methods, and (2) they enable the formulation of a set of nonlinear equations whose solution gives rise to the exact time-optimal trajectory. Numerical examples are presented, validating and highlighting the application of these properties.