There are two main philosophies for addressing the motion planning problem, in Formulation 4.1 from Section 4.3.1. This chapter presents one of the philosophies, sampling-based motion planning, which is outlined in Figure 5.1. The main idea is to avoid the explicit construction of Cobs as described in Section 4.3, and instead conduct a search that probes the C-space with a sampling scheme. This probingis enabled by a collision detection module, which the motion planning algorithm considers as a “black box.” This enables the development of planning algorithms that are independent of the particular geometric models. The collision detection module handles concerns such as whether the models are semi-algebraic sets, 3D triangles, nonconvex polyhedra, and so on. This general philosophy has been very successful in recent years for solving problems from robotics, manufacturing, and biological applications that involve thousands and even millions of geometric primitives. Such problems would be practically impossible to solve using techniques that explicitly represent Cobs.

Notions of completeness

It is useful to define several notions of completeness for sampling-based algorithms. These algorithms have the drawback that they result in weaker guarantees that the problem will be solved. An algorithm is considered complete if for any input it correctly reports whether there is a solution in a finite amount of time. If a solution exists, it must return one in finite time. The combinatorial motion planning methods of Chapter 6 will achieve this. Unfortunately, such completeness is not achieved with sampling-based planning.