7.7 Optimal Motion Planning

This section can be considered transitional in many ways. The main
concern so far with motion planning has been *feasibility* as
opposed to *optimality*. This placed the focus on finding *any* solution, rather than further requiring that a solution be
optimal. In later parts of the book, especially as uncertainty is
introduced, optimality will receive more attention. Even the most
basic forms of decision theory (the topic of Chapter
9) center on making optimal choices. The requirement
of optimality in very general settings usually requires an exhaustive
search over the state space, which amounts to computing continuous
cost-to-go functions. Once such functions are known, a feedback plan
is obtained, which is much more powerful than having only a path.
Thus, optimality also appears frequently in the design of feedback
plans because it sometimes comes at no additional cost. This will
become clearer in Chapter 8. The quest for optimal
solutions also raises interesting issues about how to approximate a
continuous problem as a discrete problem. The interplay between time
discretization and space discretization becomes very important in
relating continuous and discrete planning problems.