A generalization of the roider method to solve the robot collision problem in 3D

Abstract

The Roider Method is a method to test by means of computational geometry whether two convex, compact objects, say A and B, in two dimensions intersect. Roughly, this iterative method constructs a witness to disjointness (a wedge formed by a pair of touching-lines from some P(∈ A) to B that separates A and B) if the objects are disjoint. If the objects intersect then a witness to intersection, i.e. a point in common to both objects, is constructed. We generalize the Roider Method in two aspects: Firstly, we generalize the algorithm such that it is also applicable to convex, compact objects in three dimensions. In 3D, a witness to disjointness is a cone formed by the touching-lines from some P ∈ A to B that separates the objects. A witness to intersection is again a point in common to both objects. The consideration of all touching-lines from P to B is necessary to find the point P for the next iteration step. It is not possible to take the intersection-point of an arbitrary touching-line as the P for the next iteration step. However, one may replace the cone by a three-faced pyramid using some heuristics in the choice of the three touching-planes forming the pyramid. Secondly, we generalize the method such that it can be used to test whether a non-moving object A collides with a moving object B. This is done by testing at finitely many positions of B along its path whether it collides with A. By two theorems one may guarantee that the path between these two positions is also collision-free. This conversion of a static collision check into a dynamic collision check has not been pursued in the literature so far. We believe that our theorems are the first contributions is such a direction. In order to obtain maximal generality, as in 2D, the Roider Method is formulated w.r.t. required subprocedures rather than w.r.t. a concrete representation of the objects and their positions. An implementation of the algorithm and a comparison with an algorithm that wraps each object by unions of spheres and tests whether two spheres (belonging to different objects) intersect showed that the Roider Method is a very efficient method to solve the Collision Detection Problem.