Guaranteed stability for collision detection and simulation of hybrid dynamical systems

Both Collision Detection for models composed of parametric surfaces and Dynamic Simulation for multibody systems subject to intermittent contact can be formulated as hybrid system simulation problems. Collision Detection involves tracking surface parameters across boundaries between surface patches, where the surface parameters locate points that are pairwise closest to each other. Dynamic Simulation involves transitions between various constraint conditions, occasionally with jumps in the state due to impacts. Guaranteed stability is a critical property for algorithms in both areas, especially since constrained system simulation may drift from the constraint manifold or suffer other instabilities due to inexact initializations. This dissertation contributes new algorithms for Collision Detection on parametric models and Dynamic Simulation of hybrid dynamical systems that enjoy guaranteed stability properties.; Part I presents a simulator designed to handle multibody systems with changing constraints, wherein the equations of motion for the system in each of its constraint configurations are formulated in minimal ODE form with constraints embedded before they are passed to an ODE solver. Issues of drift associated with DAE solvers that usually require stabilization are sidestepped with the constraint-embedding approach. The constraint-embedded equations are formulated symbolically on-line according to a re-combination of terms of the unconstrained equations. Constraint-embedding undertaken on-the-fly enables the simulation of systems with an ODE solver for which constraints are not known prior to simulation start or for which the enumeration of constraint conditions would be unwieldy.; In Part II, a novel minimum distance tracking algorithm is presented for parametric models formed by tiling together convex surface patches. The essentially geometric minimization problem is differentiated with respect to time to form a dynamical non-linear control problem. Minimization is then solved with the design of a switching stabilizing controller based on a common control Lyapunov function. Together with a top-level switching algorithm based on Voronoi diagrams, the controller accounts for the combined and interacting effects of object shape and object motion while achieving global uniform asymptotic stability for the pair of closest points. Limits of performance are available, delineating values for control gains needed to suppress motion (and shape) disturbances and preserve convergence under discretization.