Combinatorial approaches to motion planning find paths through the continuous configuration space without resorting to approximations. Due to this property, they are alternatively referred to as exact algorithms. This is in contrast to the sampling-based motion planning algorithms from Chapter 5.

Introduction

All of the algorithms presented in this chapter are complete, which means that for any problem instance (over the space of problems for which the algorithm is designed), the algorithm will either find a solution or will correctly report that no solution exists. By contrast, in the case of sampling-based planning algorithms, weaker notions of completeness were tolerated: resolution completeness and probabilistic completeness.

Representation is important

When studying combinatorial motion planning algorithms, it is important to carefully consider the definition of the input. What is the representation used for the robot and obstacles? What set of transformations may be applied to the robot? What is the dimension of the world? Are the robot and obstacles convex? Are they piecewise linear? The specification of possible inputs defines a set of problem instances on which the algorithm will operate. If the instances have certain convenient properties (e.g., low dimensionality, convex models), then a combinatorial algorithm may provide an elegant, practical solution. If the set of instances is too broad, then a requirement of both completeness and practical solutions may be unreasonable. Many general formulations of general motion planning problems are PSPACE-hard; therefore, such a hope appears unattainable. Nevertheless, there exist general, complete motion planning algorithms.