In Formulation 14.1, the specification of the obstacle region in Item 4 was intentionally left ambiguous. Now it will be specified in more detail. If , then , which was defined in (4.34) for a rigid robot and in (4.36) for a robot with multiple links. The more interesting case occurs if is a phase space that includes velocity variables in addition to configuration information.
Any state for which its associated configuration lies in must also be a member of . The velocity is irrelevant if a collision occurs in the world . In most cases that involve a phase space, the obstacle region is therefore defined as