51 Algorithmic Motion Planning

Motion planning is a fundamental problem in robotics. It comes in a variety of forms, but the simplest version is as follows. We are given a robot system B, which may consist of several rigid objects attached to each other through various joints, hinges, and links, or moving independently, and a 2D or 3D environment V cluttered with obstacles. We assume that the shape and location of the obstacles and the shape of B are known to the planning system. Given an initial placement Z1 and a final placement Z2 of B, we wish to determine whether there exists a collisionavoiding motion of B from Z1 to Z2, and, if so, to plan such a motion. In this simplified and purely geometric setup, we ignore issues such as incomplete information, nonholonomic constraints, control issues related to inaccuracies in sensing and motion, nonstationary obstacles, optimality of the planned motion, and so on. Since the early 1980s, motion planning has been an intensive area of study in robotics and computational geometry. In this chapter we will focus on algorithmic motion planning, emphasizing theoretical algorithmic analysis of the problem and seeking worst-case asymptotic bounds, and only mention briefly practical heuristic approaches to the problem. The majority of this chapter is devoted to the simplified version of motion planning, as stated above. Section 51.1 presents general techniques and lower bounds. Section 51.2 considers efficient solutions to a variety of specific moving systems with a small number of degrees of freedom. These efficient solutions exploit various sophisticated methods in computational and combinatorial geometry related to arrangements of curves and surfaces (Chapter 30). Section 51.3 then briefly discusses various extensions of the motion planning problem such as computing optimal paths with respect to various quality measures, computing the path of a tethered robot, incorporating uncertainty, moving obstacles, and more.