Once it has been decided where the samples will be placed, the next problem is to determine whether the configuration is in collision. Thus, collision detection is a critical component of sampling-based planning. Even though it is often treated as a black box, it is important to study its inner workings to understand the information it provides and its associated computational cost. In most motion planning applications, the majority of computation time is spent on collision checking.
A variety of collision detection algorithms exist, ranging from theoretical algorithms that have excellent computational complexity to heuristic, practical algorithms whose performance is tailored to a particular application. The techniques from Section 4.3 can be used to develop a collision detection algorithm by defining a logical predicate using the geometric model of . For the case of a 2D world with a convex robot and obstacle, this leads to an linear-time collision detection algorithm. In general, however, it can be determined whether a configuration is in collision more efficiently by avoiding the full construction of .