15. System Theory and Analytical Techniques

This chapter is complementary to Chapter 14 in that it provides tools and concepts that can be used to develop better local planning methods (LPMs). Most of the material was developed in the field of control theory, which focuses mainly on characterizing the behavior of particular classes of systems, and controlling them in the absence of obstacles. The two-point boundary value problem (BVP), which was a frequent nuisance in Chapter 14, can be better understood and solved for many systems by using the ideas of this chapter. Keep in mind that throughout this chapter there are no obstacles. Although planning for this case was trivial in Part II, the presence of differential constraints brings many challenges.

The style in this chapter is to provide a brief survey of concepts and techniques, with the hope of inspiring further study in other textbooks and research literature. Modern control theory is a vast and fascinating subject, of which only the surface can be scratched in one chapter. Section 15.1 introduces stability and controllability concepts, both of which characterize possible arrivals in a goal state. Stability characterizes how the integral curves of a vector field behave around a goal point, and controllability indicates whether an action trajectory exists that arrives at a specified goal.

Section 15.2 revisits dynamic programming one last time. Here it becomes a partial differential equation expressed in terms of the optimal cost-to-go function. In some cases, it actually has a closed-form solution, as opposed to its main use in computer science, which is to obtain algorithm constraints. The powerful Pontryagin's minimum principle, which can be derived from dynamic programming, is also covered.

The remainder of the chapter is devoted to nonholonomic systems, which often arise from underactuated mechanical systems. Section 15.3 expresses the shortest paths between any pair of points for the Dubins car, the Reeds-Shepp car, and a differential drive, all of which were introduced in Section 13.1.2. The paths are a beautiful solution to the BVP and are particularly valuable as an LPM; for example, some have been used in the plan-and-transform method of Section 14.6.2. Section 15.4 addresses some basic properties of nonholonomic systems. The most important issues are determining whether nonholonomic constraints are actually integrable (which removes all ${\dot x}_i$ variables) and characterizing reachable sets that arise due to nonholonomic constraints. Section 15.5 attempts to do the same as Section 15.3, but for more challenging nonholonomic systems. In these cases, the BVP problem may not be solved optimally, and some methods may not even reach the goal point precisely. Nevertheless, when applicable, they can be used to build powerful LPMs in a sampling-based motion planning algorithm.