Now the formal algebra concepts can be applied to the steering problem. The variables become the system vector fields: for all from to . For the P. Hall basis elements, each becomes . The Lie group becomes the state space , and the Lie algebra is the familiar Lie algebra over the vector fields, which was introduced in Section 15.4.3. Consider how an element of the Lie group must evolve over time. This can be expressed using the differential equation
The solution at every time can be written using the Chen-Fliess series, (15.127):
The next step is to figure out how the backward P. Hall coordinates evolve. Differentiating (15.129) with respect to time yields
Steven M LaValle 2020-08-14