Assume that , , and from Formulation 4.1 are
used. For manipulation planning, is called the manipulator, and let
refer to the manipulator
configuration space. Let denote a part, which is a
rigid body modeled in terms of geometric primitives, as described in
Section 3.1. It is assumed that is allowed to undergo
rigid-body transformations and will therefore have its own part
denote a part configuration. The transformed part model
is denoted as
Examples of several important subsets of
for manipulation planning.
The combined configuration space, , is defined as the
in which each configuration
is of the form
The first step is to remove all configurations that must be avoided.
Parts of Figure 7.15 show examples of these sets.
Configurations for which the manipulator collides with obstacles are
The next logical step is to remove configurations for which the part
collides with obstacles. It will make sense to allow the part to
``touch'' the obstacles. For example, this could model a part sitting
on a table. Therefore, let
denote the open set for which the interior of the part intersects
. Certainly, if the part penetrates , then the configuration
should be avoided.
. The configurations that
remain ensure that the robot and part do not inappropriately collide
with . Next consider the interaction between and . The
manipulator must be allowed to touch the part, but penetration is once
again not allowed. Therefore, let
Removing all of these bad configurations yields
which is called the set of admissible configurations.
Steven M LaValle