Collision Cones for Quadric Surfaces in $n$ -Dimensions

Abstract

In this letter, we present analytical expressions for collision cones associated with a class of hyperquadric surfaces moving in $n$ -dimensional configuration space. Using a relative velocity paradigm, a geometric analysis of the distance, time, and point of closest approach between moving objects in $n$ -dimensional space, is carried out to obtain a characterization of the collision cone between a point and a hyperspheroid as well as a constrained hyperboloid, which represent an interesting and useful class of objects in configuration space. It is shown that these $n$ -dimensional collision cones can be integrated with sampling-based motion planners, avoiding the need to evaluate waypoints that lie inside the collision cone. The cones can also consist of the heading angles toward desirable regions in the configuration space, in which case planners may evaluate more waypoints inside the cone. Finally, analytical expressions of the collision cones are used, in conjunction with the concept of level sets, and incorporated into a Lyapunov-based design approach, to determine analytical expressions of nonlinear guidance laws that can manipulate the velocity vector of an object in $n$ -dimensional space.