Once the CFS equation has been determined, the problem is almost solved. The action trajectory was determined from the given state trajectory and the backward P. Hall coordinate trajectory is determined by (15.143). The only remaining problem is that the action variables from to are fictitious because their associated vector fields are not part of the system. They were instead obtained from Lie bracket operations. When these are applied, they interfere with each other because many of them may try to use the same variables from the original system at the same time.
The CBHD formula is used to determine the solution in terms of the system action variables , , . The differential equation now becomes
When applying vector fields over time, the CBHD formula becomes
Steven M LaValle 2012-04-20