This section assumes that a path has been given. It may be computed by a motion planning algorithm from Part II or given by hand. The remaining task is to determine the speed along the path in a way that satisfies differential constraints on the phase space . Assume that each state represents both a configuration and its time derivative, to obtain . Let denote the dimension of ; hence, the dimension of is . Once a path is given, there are only two remaining degrees of freedom in : 1) the position along the domain of , and 2) the speed at each . The full state, , can be recovered from these two parameters. As the state changes, it must satisfy a given system, . It will be seen that a 2D planning problem arises, which can be solved efficiently using many alternative techniques. Similar concepts appeared for decoupled versions of time-varying motion planning in Section 7.1. The presentation in the current section is inspired by work in time-scaling paths for robot manipulators [456,876,879], which was developed a couple of decades ago. At that time, computers were much slower, which motivated the development of strongly decoupled approaches.