Automatic determination of possible velocity and applicable force of frictionless objects in contact from a geometric model

Investigates motions of a polyhedron in contact with a fixed polyhedron under the frictionless assumption. The authors propose a complete algorithm for determining possible velocity of the moving polyhedron and force applicable from the moving one to the fixed one automatically from their geometric models. The algorithm consists of two parts. The first part derives the constraints for the velocity of the moving polyhedron from their shape descriptions. The constraints are represented by linear inequalities of the velocity. The algorithm is complete in the sense that it can be applied to any case in which polyhedra with arbitrary shapes are in contact with arbitrary state. It includes the degenerate case in which a vertex contacts another vertex or edge. The second part solves the inequalities and obtains the set of possible velocity vectors of the moving polyhedron. The authors prove that this part is equivalent to the algorithm for enumerating all vertices of a compact polytope in higher dimensional space. The solution is the direct sum of a nonnegative linear combination of the vectors that break the contact state and a linear combination of the vectors that maintain the state. The minimum set of force vectors that are applicable from the moving polyhedron to the fixed one can be obtained from the set of possible velocity vectors of the moving one. The algorithm is fully implemented in an object-oriented lisp with a solid modeler and in C. The possible applications of the algorithm are also presented.