On-line Symbolic Constraint Embedding for Simulation of
Hybrid Dynamical Systems
R. B. Gillespie, V. Patoglu, E. Westervelt, and I. Hussein
Journal of Multibody System Dynamics, accepted for publication
In this paper we present a simulator designed to handle multibody systems with
changing constraints, wherein the equations of motion for each of its constraint
configurations are formulated in minimal ODE form with constraints embedded
before they are passed to an ODE solver. The constraint-embedded equations are
formulated symbolically according to a recombination of terms of the unconstrained
equations, and this symbolic process is undertaken on-line by the simulator.
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 all constraint conditions would be unwieldy
because of their complexity or number. Issues of drift associated with DAE
solvers that usually require stabilization are sidestepped with the
constraint-embedding approach. We apply nomenclature developed for hybrid
dynamical systems to describe the system with changing constraints and to
distinguish the roles of the forward dynamics solver, a collision detector, and
an impact resolver. We have developed a MATLAB toolbox for
symbolically generating equations of motion using Kane's method and prototyped a
simulator with on-line constraint embedding and demonstrated the design in three
representative examples.