Haptic rendering of parametric surfaces using a feedback stabilized extremal distance tracking algorithm

An extremal distance tracking algorithm is presented for convex parametric curves and surfaces undergoing rigid body motion. The geometric extremization problem is differentiated with respect to time to produce a dynamical system that incorporates dependence on both surface shape and rigid body motion. Extremization then takes place by integrating these dynamical equations, but with a feedback controller in place to stabilize the solution. A controller design using feedback linearization is developed that simultaneously accounts for surface shape and motion while asymptotically achieving (and maintaining) the extremal pair. Collision detection then takes place in a framework fully analogous to that used for multibody simulation. Local stability results are extended to provide global stability for body shapes composed of pieced-together convex parametric surface patches using a switching algorithm.