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

Steven M LaValle 2020-08-14