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
configuration space,
or
. Let
denote a part configuration. The transformed part model
is denoted as
.
Figure 7.15:
Examples of several important subsets of
for manipulation planning.
|
The combined configuration space, , is defined as the
Cartesian product
|
(7.14) |
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
|
(7.15) |
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
|
(7.16) |
denote the open set for which the interior of the part intersects
. Certainly, if the part penetrates , then the configuration
should be avoided.
Consider
. 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
|
(7.17) |
Removing all of these bad configurations yields
|
(7.18) |
which is called the set of admissible configurations.
Steven M LaValle
2020-08-14