Conserving Integrators for Parallel Manipulators

Abstract

The present work deals with the development of time stepping schemes for the dynamics of parallel manipulators. In particular, we aim at energy and momentum conserving algorithms for a robust time integration of the differential algebraic equations (DAEs) which govern the motion of closed-loop multibody systems. It is shown that a rotationless formulation of multibody dynamics is especially well-suited for the design of energymomentum schemes. Joint coordinates and associated forces can still be used by applying a specific augmentation technique which retains the advantageous algorithmic conservation properties. It is further shown that the motion of a manipulator can be partially controlled by appending additional servo constraints to the DAEs. Starting with the pioneering works by Simo and co-workers [SW91, STW92, ST92], energymomentum conserving schemes and energy-decaying variants thereof have been developed primarily in the context of nonlinear finite element methods. In this connection, representative works are due to Brank et al. [BBTD98], Bauchau & Bottasso [BB99], Crisfield & Jeleni c [CJ00], Ibrahimbegovic et al. [IMTC00], Romero & Armero [RA02], Betsch & Steinmann [BS01a], Puso [Pus02], Laursen & Love [LL02] and Armero [Arm06], see also the references cited in these works. Problems of nonlinear elastodynamics and nonlinear structural dynamics can be characterized as stiff systems possessing high frequency contents. In the conservative case, the corresponding semi-discrete systems can be classified as finite-dimensional Hamiltonian systems with symmetry. The time integration of the associated nonlinear ODEs by means of energy-momentum schemes has several advantages. In addition to their appealing algorithmic conservation properties energy-momentum schemes are known to possess enhanced numerical stability properties (see Gonzalez & Simo [GS96]). Due to these advantageous properties energy-momentum schemes have even been successfully applied to penalty formulations of multibody dynamics, see Goicolea & Garcia Orden [GGO00]. Indeed, the enforcement of holonomic constraints by means of penalty methods again yields stiff systems possessing high frequency contents. The associated equations of motion are characterized by ODEs containing strong constraining forces. In the limit of infinitely large penalty parameters these ODEs replicate Lagrange’s equations of motion of the first kind (see Rubin & Ungar [RU57]), which can be id entified as index-3 differential-algebraic equations (DAEs). This observation strongly supports the expectation that energy