14. Sampling-Based Planning Under Differential Constraints

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 ${\dot x}= f(x,u)$. 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.

Section 14.3 revisits the incremental sampling and searching framework of Section 5.4 and extends it to handle differential constraints. This leads to several sampling-based planning approaches, which are covered in Section 14.4. Familiar choices such as dynamic programming or the RDTs of Section 5.5 appear once again. The resulting planning methods can be used for a wide variety of problems that involve differential constraints on C-spaces or phase spaces.

Section 14.5 briefly covers feedback motion planning under differential constraints. Approximate, optimal plans can be obtained by a simple adaptation of value iteration from Section 8.5.2. Section 14.6 describes decoupled methods, which start with a collision-free path that ignores differential constraints, and then perform refinements to obtain the desired trajectory. Such approaches often lose completeness and optimality, but they offer substantial computational savings in many settings. Section 14.7 briefly surveys numerical techniques for optimizing a trajectory subjected to differential constraints; the techniques can be used to improve solutions computed by planning algorithms.