After Chapter 13, it seems that differential constraints arise nearly everywhere. For example, they may arise when wheels roll, aircraft fly, and when the dynamics of virtually any mechanical system is considered. This makes the basic model used for motion planning in Part II invalid for many applications because differential constraints were neglected. Formulation 4.1, for example, was concerned only with obstacles in the C-space.

This chapter incorporates the differential models of Chapter 13 into sampling-based motion planning. The detailed modeling (e.g., Lagrangian mechanics) of Chapter 13 is not important here. This chapter works directly with a given system, expressed as ẋ = f(x, u). The focus is limited to sampling-based approaches because very little can be done with combinatorial methods if differential constraints exist. However, if there are no obstacles, then powerful analytical techniques may apply. This subject is complementary to motion planning with obstacles and is the focus of Chapter 15.

Section 14.1 provides basic definitions and concepts for motion planning under differential constraints. It is particularly important to explain the distinctions made in literature between nonholonomic planning, kinodynamic planning, and trajectory planning, all of which are cases of planning under differential constraints. Another important point is that obstacles may be somewhat more complicated in phase spaces, which were introduced in Section 13.2. Section 14.2 introduces sampling over the space of action trajectories, which is an essential part of later planning algorithms.