The minimum principle can be helpful in developing a steering method. Due to the close connection between the Euler-Lagrange equation and Hamilton's equations, as mentioned in Section 13.4.4, this should not be surprising. The Euler-Lagrange equation was used in Section 15.5.2 to determine an optimal steering method for the nonholonomic integrator. A steering methodology based on the minimum principle is described in [846]. The optimal curves of Section 15.3 actually represent steering methods obtained from the minimum principle. Unfortunately, for the vast majority of problems, numerical techniques are needed to solve the resulting differential equations. It is generally expected that techniques developed for specific classes, such as the nilpotent, chained-form, or differentially flat systems, perform much better than general-purpose numerical techniques applied to the Euler-Lagrange equation, Hamilton's equations or Pontryagin's minimum principle.

Steven M LaValle 2020-08-14