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
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'' ,
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