Designing good motion primitives

Up to this point, only the discrete-time model has been considered. Although it is the most straightforward and general, there are often many better motion primitives that can be used. For a particular system, it may be possible to design a nice family of trajectories off-line in the absence of obstacles and then use them as motion primitives in the RDT construction. If possible, it is important to carefully analyze the system under consideration to try to exploit any special structure it may have or techniques that might have been developed for it. For motion planning of a vehicle, symmetries can be exploited to apply the primitives from different states. For example, in flying a helicopter, the yaw angle and the particular position (unless it is close to the ground) may not be important. A family of trajectories designed for one yaw angle and position should work well for others.

Using more complicated motion primitives may increase the burden on the LPM. In some cases, a simple control law (e.g., PID [50]) may perform well. Ideally, the LPM should behave like a good steering method, which could be obtained using methods in Chapter 15. It is important to note, though, that the RDT's ability to solve problems does not hinge on this. It will greatly improve performance if there are excellent motion primitives and a good steering method in the LPM. The main reason for this is that the difficulties of the differential constraints have essentially been overcome once this happens (except for the adverse effects of drift). Although having good motion primitives can often improve performance in practice, it can also increase the difficulty of ensuring resolution completeness.