The speed graph method: Time optimal navigation among obstacles subject to safe braking constraint

This paper describes a method for computing the global time optimal path of a mobile robot navigating among obstacles subject to safe braking constraints. The paper first generalizes the classical Brachistochrone problem into a time optimal navigation problem, where the mobile robot navigates under a braking safety constraint near a point obstacle or a wall segment. The time optimal navigation problem is then formulated for general polygonal environments. Based on this formulation, the paper constructs a speed graph for the environment which consists of time optimal arcs that connect critical via points. The speed graph is then used to identify the path homotopy class which most likely contains the global time optimal path. Once a candidate homotopy class is selected, the exact time optimal path subject to safe braking constraints is computed within the homotopy class based on convexity properties of these paths. The results are illustrated with examples, described as readily implementable procedures, and demonstrated with experiments.