##

7.4.1 Adaptation of Motion Planning Algorithms

All of the components from the general motion planning problem of
Formulation 4.1 are included: , ,
, ,
, , , and . It is assumed that the robot is
a collection of links that are possibly attached in loops.

It is assumed in this section that
. If this is not
satisfactory, there are two ways to overcome the assumption. The
first is to represent and as
and
,
respectively, and include the circle or sphere equation as part of the
constraints considered here. This avoids the topology problems. The
other option is to abandon the restriction of using
and
instead use a parameterization of that is of the appropriate
dimension. To perform calculus on such manifolds, a *smooth
structure* is required, which is introduced in Section
8.3.2. In the presentation here, however, vector calculus
on
is sufficient, which intentionally avoids these extra
technicalities.

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