On-Line Symbolic Constraint Embedding for Simulation of Hybrid Dynamical Systems

Abstract

Abstract. 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 re-combination 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 r 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.

A momentum form of Kane’s equations for scleronomic systems

“…For systems with closed-chain kinematics, however, the loop closure conditions remove the independence of the joint coordinates, making embedding of the closure constraints at the position level inordinately complex. For such systems, constraint embedding can still be applied at the velocity level, by differentiating the closure constraints, and using the differentiated constraints to eliminate linearly dependent generalized speeds [14,15]. Again, this method can be applied without modification to the momentum formulation.…”

Haptic manipulation of virtual linkages with kinematic loops

“…In (11), D, W, w stab and G denote the values of D, W, w stab and G, respectively, computed by the graphics engine and sent to the local model at the update. Furthermore, the directions of joint constraint are approximated through their values in the VE at the update, by computing the SVD of Λ −1 h on the graphics processor: (12) and using the columns of U in the local model to estimate these directions.…”

“…Furthermore, the directions of joint constraint are approximated through their values in the VE at the update, by computing the SVD of Λ −1 h on the graphics processor: (12) and using the columns of U in the local model to estimate these directions. Equations (11) and (12) allow the VT dynamics to be simulated at the frequency of the haptic controller. Operation of VTs with kinematic loops using these approximations is demonstrated experimentally in the following section.…”

“…Computational efficiency has been achieved via application-specific [14] and general purpose algorithms [17], [19]. More recently, a multi-rate architecture has been proposed whereby linkage dynamics are simulated in minimal coordinates at the speed of the haptic controller [10], [11]. The efficient dynamics in minimal coordinates are obtained via symbolic constraint M embedding performed outside the main integration loop.…”

Impact forces in the simulation of simultaneous impacts and contacts in multibody systems with friction

“…This approach eliminates the need to remove generalized coordinates in order to enforce contact constraints, as is done in [9], which can be an arbitrary process [10]. It also allows contact points to easily alternate between sticking and slipping tangentially to the surface.…”