Extremal Distance Maintenance for Parametric Curves and Surfaces

A new extremal distance tracking algorithm is presented for parametric curves and surfaces undergoing rigid body motion. The essentially geometric extremization problem is transformed into a dynamical control problem by differentiating with respect to time. Extremization is then solved with the design of a stabilizing controller. We use a feedback linearizing controller. The controller simultaneously accounts for the surface shape and motion while asymptotically achieving (and maintaining) the extremal pair. Thus collision detection takes place in a framework fully analogous to the framework used for the simulation of dynamical response.