Admissible configurations

Assume that ${\cal W}$, ${\cal O}$, and ${\cal A}$ from Formulation 4.1 are used. For manipulation planning, ${\cal A}$ is called the manipulator, and let ${\cal C}^a$ refer to the manipulator configuration space. Let ${\cal P}$ denote a part, which is a rigid body modeled in terms of geometric primitives, as described in Section 3.1. It is assumed that ${\cal P}$ is allowed to undergo rigid-body transformations and will therefore have its own part configuration space, ${\cal C}^p = SE(2)$ or ${\cal C}^p = SE(3)$. Let ${q}^p \in {\cal C}^p$ denote a part configuration. The transformed part model is denoted as ${\cal P}(q^p)$.

Figure 7.15: Examples of several important subsets of ${\cal C}$ for manipulation planning.

The combined configuration space, ${\cal C}$, is defined as the Cartesian product

$${\cal C}= {\cal C}^a \times {\cal C}^p ,$$ (7.14)

in which each configuration $q \in {\cal C}$ is of the form $q = (q^a,q^p)$. 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

$${{\cal C}^a_{obs}}= \{ (q^a,q^p) \in {\cal C}\;\vert\; {\cal A}(q^a) \cap {\cal O}\not = \emptyset \} .$$ (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

$${{\cal C}^p_{obs}}= \{ (q^a,q^p) \in {\cal C}\;\vert\; \operatorname{int}({\cal P}(q^p)) \cap {\cal O}\not = \emptyset \}$$ (7.16)

denote the open set for which the interior of the part intersects ${\cal O}$. Certainly, if the part penetrates ${\cal O}$, then the configuration should be avoided.

Consider ${\cal C}\setminus ({{\cal C}^a_{obs}}\cup {{\cal C}^p_{obs}})$. The configurations that remain ensure that the robot and part do not inappropriately collide with ${\cal O}$. Next consider the interaction between ${\cal A}$ and ${\cal P}$. The manipulator must be allowed to touch the part, but penetration is once again not allowed. Therefore, let

$${{\cal C}^{ap}_{obs}}= \{ (q^a,q^p) \in {\cal C}\;\vert\; {\cal A}(q^a) \cap \operatorname{int}({\cal P}(q^p)) \not = \emptyset \} .$$ (7.17)

Removing all of these bad configurations yields

$${{\cal C}_{adm}}= {\cal C}\setminus ({{\cal C}^a_{obs}}\cup {{\cal C}^p_{obs}}\cup {{\cal C}^{ap}_{obs}}) ,$$ (7.18)

which is called the set of admissible configurations.