A novel approach to the numerical solution of the vector field associated with constrained mechanical systems, called the perturbation approach, is introduced. It is a specialization of a method of solving general vector fields due to Shampine. The perturbation approach has several advantages over other approaches. The performance of the approaches on a constrained robot problem is used to show the efficacy of the new approach.>

The quantification of proximity between a pair of objects whose point descriptions are given is considered. Four problems of proximity between two convex polytopes in R/sup 3/ are considered. The convex polytopes are represented as convex hulls of finite sets of points. The authors discuss the complexity of solving the four problems. They analyze algorithms for the four problems in terms of two complexity types. Let the total number of points in the two finite sets be n. It is shown that three of the proximity problems, checking intersection, checking whether the polytopes are just touching, and finding the distance between them, can be solved in O(n) time for fixed s and in polynomial time for varying s. It is also shown that the fourth proximity problem of finding the intensity of collision for varying s is NP-complete.>

An improved potential-based method for robot path planning is developed by connecting all the local minima and local maxima of the potential function defined in the configuration space of the robot. The authors construct an adjacency graph of the local minima and maxima of the potential function. An edge connecting a local minimum and a local maximum has an associated pair of perturbations which gives a way of moving between them. The method is based on the stability theory of dynamical systems. The usefulness of the method was demonstrated on a two-dimensional piano mover's problem with three degrees of freedom.>