14.6.3 Path-Constrained Trajectory Planning

This section assumes that a path $\tau: [0,1] \rightarrow {\cal C}_{free}$ 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 $X$. Assume that each state $x \in X$ represents both a configuration and its time derivative, to obtain $x = (q,{\dot q})$. Let $n$ denote the dimension of ${\cal C}$; hence, the dimension of $X$ is $2n$. Once a path is given, there are only two remaining degrees of freedom in $X$: 1) the position $s \in [0,1]$ along the domain of $\tau$, and 2) the speed ${\dot s}= ds/dt$ at each $s$. The full state, $x$, can be recovered from these two parameters. As the state changes, it must satisfy a given system, ${\dot x}= f(x,u)$. 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.