1 Copyright © 2003 by ASME Proceedings of DETC’03 ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference Chicago, Illinois USA, September 2-6, 2003 DETC2003/VIB-48314 CONSTRAINT GRADIENT PROJECTIVE METHOD FOR STABILIZED DYNAMIC SIMULATION OF CONSTRAINED MULTIBODY SYSTEMS Zdravko Terze University of Zagreb F. Mech. Eng. Naval Arch. Department of Aerospace Engineering Joris Naudet Vrije Universiteit Brussel Department of Mechanical Engineering Dirk Lefeber Vrije Universiteit Brussel Department of Mechanical Engineering ABSTRACT Constraint gradient projective method for stabilization of constraint violation during integration of constrained multibody systems is in the focus of the paper. Different mathematical models for constrained MBS dynamic simulation on manifolds are surveyed and violation of kinematical constraints is discussed. As an extension of the previous work focused on the integration procedures of the holonomic systems, the constraint gradient projective method for generally constrained mechanical systems is discussed. By adopting differential- geometric point of view, the geometric and stabilization issues of the method are addressed. It is shown that the method can be applied for stabilization of holonomic and non-holonomic constraints in Pfaffian and general form. 1. INTRODUCTION During dynamical simulation of constrained multibody systems, a violation of system kinematical constraints is the basic source of time-integration errors and frequent difficulty that analyst have to cope with. As will be surveyed in the following chapters, if the governing equations are not turned into so called minimal form, but dynamic simulation is based on the mathematical models expressed via redundant coordinates, a constraint violation stabilization method have to be applied during integration procedure. Baumgarte stabilization method that minimizes violations can be applied for this purpose, but this algorithm is dependent on empirical feedback gains and has some limitations [1]. Different methods that provide full stabilization of system constraints are discussed in [2, 3, 4]. The stabilized integration procedure, whose stabilization step is based on projection of the integration results to the underlying constraint manifold via post-integration correction of selected coordinates, is proposed and compared with similar integration schemes in [5]. The integration procedure is compatible with many ODE integrators and provides full stabilization of system constraint violation, but its utilization is confined to the holonomic systems only. As an extension of the previous work, a further elaboration of the projective stabilization step described in [5] is reported in this paper. Based on the gained insight, the geometric and stabilization properties of the projection algorithm are addressed when routine is applied for stabilization of holonomic and non- holonomic constraints in Pfaffian and general form. In the case of holonomic systems it is shown that, once the subvector is optimally partitioned at the position level, it can be used automatically for stabilization at the velocity level as well. The next question is: would it be possible to apply the proposed algorithm in the framework of simulation procedures of non- holonomic systems ? It is shown that in the case of non- holonomic systems, the optimally partitioned subvectors can generally have a different structure for ‘positions’ and velocities. 2. UNCONSTRAINED MBS ON MANIFOLDS Unconstrained multibody system (MBS) is an autonomous Lagrangian system. If n DOF is assumed, the system evolution in configuration space R is described (by definition) by Lagrangian equations [6] * d d * = ∂ ∂ -       ∂ ∂ [ [ / / W  , ( ) ( ) W , , * [ [ 4 [ [    = 0 . (1)