**Feedback Stabilized Minimum Distance Maintenance for Convex
Parametric Surfaces**

Volkan Patoglu and R. Brent Gillespie

*to appear, IEEE Transactions on Robotics*

A new minimum distance tracking algorithm is presented for moving convex bodies represented using tiled together parametric surface patches. The algorithm is formulated by differentiating the geometric minimization problem with respect to time. This produces a hybrid dynamical system that incorporates dependence on rigid body motion, surface shape, and surface boundary interconnectedness. The minimum distance between a pair of previously identified closest features is found by feedback stabilizing the dynamical equations and numerically solving the resulting closed loop system equations. Maintenance of the minimum distance and the associated closest points during motion is achieved through the action of a feedforward controller and a switching algorithm. The feedforward controller simultaneously accounts for surface shape and motion while the switching controller triggers updates to the extremal feature pair when extremal points on one body cross between Voronoi regions of the other body. The algorithm may be implemented within the same framework used for multibody simulation since the minimum distance tracking algorithm itself follows as the simulation of a hybrid dynamical system. In contrast to previously available minimum distance determination algorithms, attractive properties of the new algorithm include a means of determining the highest gain K that maintains stability under a given discretization scheme and a large and easily characterized basin of attraction of the stabilized closest points. These properties may be used to achieve higher computational efficiency. Simulation results are presented for various planar and spatial systems composed of a body and point or composed of two bodies.