Time-Minimal Motion: Basics

Abstract

We have seen in Chap. 2 that the visibility graph plays an important role in planning shortest paths among stationary obstacles in a two-dimensional world. In this chapter, we study the concept of accessibility (Fujimura and Samet 1988), which is a generalization of the concept of visibility. Making use of accessibility, we define a graph called the accessibility graph to represent moving objects for the purpose of planning the motion of a robot. The robot is assumed to be a point that moves in a two-dimensional world in which polygonal obstacles, as well as the destination point, are in motion. The accessibility graph is shown to be a generalization of the visibility graph in the sense that paths to the destination point are found as sequences of edges of the graph. In fact, when all the obstacles have zero velocity, the accessibility graph becomes the visibility graph of these polygonal obstacles. More importantly, if the robot is able to move faster than any of the obstacles, then the graph exhibits a property: a time-minimal motion is represented as a sequence of edges of the accessibility graph. In this chapter, we describe an algorithm for generating a time-minimal motion, prove its timeminimality, and analyze its execution time. The utility of the concept of accessibility is further demonstrated in Chap. 4 and Chap. 5 by solving a number of motion planning problems in dynamic domains.