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14.6.3 Path-Constrained Trajectory Planning

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.

**Subsections**
Steven M LaValle
2012-04-20