Analytic Optimal-Based Time Horizon Using Pontryagin's Maximum Principle for Spatial Navigation

Abstract

This paper addresses the issue of agent motion planning for spatial applications using an analytic optimal time horizon solution as basic character of our search. Specifically, we propose the analytic optimal time horizon concept as a leading feature for our local on-line planner for omni-directional robots using Velocity Obstacle (VO). For the first time, we propose a solution to the basic limitation of the VO search and planning method, i.e. when all the dynamic available velocities for the next time step are blocked inside the velocity space and there is no legal node at the next time step of the greedy search using classic VO concept. In this paper, we present unified computation of the minimum time horizon which is formulated as a minimum time problem for omni-directional models. The analytic solution describes minimal and safe VO and allows efficient on-line planning in dynamic and static environments searching safe nodes. At each time step, a local greedy search in velocity space is explored. The analytic solution defines VO shape and by that set the bounded velocity space and the next optimal node outside VO explored in the next time step. We introduce on-line planner for omni-directional robot that generates near-time optimal trajectories to the goal by using optimal time horizon. We demonstrate the solution of our approach in simulations showing the efficiency relative to the traditional VO for on-line motion planning that can be extended for spatial applications in urban scenes