#### Obstacle region for a rigid body

Suppose that the world, or , contains an obstacle region, . Assume here that a rigid robot, , is defined; the case of multiple links will be handled shortly. Assume that both and are expressed as semi-algebraic models (which includes polygonal and polyhedral models) from Section 3.1. Let denote the configuration of , in which for and for ( represents the unit quaternion).

The obstacle region, , is defined as

 (4.34)

which is the set of all configurations, , at which , the transformed robot, intersects the obstacle region, . Since and are closed sets in , the obstacle region is a closed set in .

The leftover configurations are called the free space, which is defined and denoted as . Since is a topological space and is closed, must be an open set. This implies that the robot can come arbitrarily close to the obstacles while remaining in . If touches'' ,

 (4.35)

then (recall that means the interior). The condition above indicates that only their boundaries intersect.

The idea of getting arbitrarily close may be nonsense in practical robotics, but it makes a clean formulation of the motion planning problem. Since is open, it becomes impossible to formulate some optimization problems, such as finding the shortest path. In this case, the closure, , should instead be used, as described in Section 7.7.

Steven M LaValle 2012-04-20