Differential-Geometric Modelling and Dynamic Simulation of Multibody Systems

A formulation for the kinematics of multibody systems is presented, that uses Lie group concepts. With line coordinates the kinematics is parameterized in terms of the screw coordinates of the joints. Thereupon, the Lagrangian motion equations are derived, and explicit expressions are given for the objects therein. It is shown how the kinematics and thus the motion equations can be expressed without the introduction of body-fixed reference frames. This admits the processing of CAD data, which refers to a single (world) frame. For constrained multibody systems, the Lagrangian motion equations are projected to the constraint manifold, which yields the equations of Woronetz. The mathematical models for numerical integration routines of MBS are surveyed and constraint gradient projective method for stabilization of constraint violation is presented.

[1]  Adam Morawiec,et al.  Orientations and Rotations , 2004 .

[2]  W. Blajer.  Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems , 2002 .

[3]  Joel W. Burdick,et al.  Geometric Perspectives on the Mechanics and Control of Robotic Locomotion , 1996 .

[4]  Frank Chongwoo Park,et al.  A Lie Group Formulation of Robot Dynamics , 1995, Int. J. Robotics Res..

[5]  Jaime Gallardo-Alvarado,et al.  Lie Algebra and the Mobility of Kinematic Chains , 2003, J. Field Robotics.

[6]  A. Mu¨ller.  Geometric Characterization of the Configuration Space of Rigid Body Mechanisms in Regular and Singular Points , 2005 .

[7]  F. Park,et al.  Symbolic formulation of closed chain dynamics in independent coordinates , 1999 .

[8]  Romeo Ortega,et al.  Passivity-based Control of Euler-Lagrange Systems , 1998 .

[9]  K. Anderson.  An order n formulation for the motion simulation of general multi-rigid-body constrained systems , 1992 .

[10]  P. Maisser,et al.  A Lie-Group Formulation of Kinematics and Dynamics of Constrained MBS and Its Application to Analytical Mechanics , 2003 .

[11]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[12]  Zdravko Terze,et al.  Geometric properties of projective constraint violation stabilization method for generally constrained multibody systems on manifolds , 2008 .

[13]  Amir Fijany,et al.  Parallel O(log N) algorithms for computation of manipulator forward dynamics , 1994, IEEE Trans. Robotics Autom..

[14]  J. C. Simo,et al.  Conserving algorithms for the dynamics of Hamiltonian systems on lie groups , 1994 .

[15]  P. Maisser,et al.  A differential-geometric approach to the multi body system dynamics , 1991 .

[16]  W. Schiehlen,et al.  A projective criterion to the coordinate partitioning method for multibody dynamics , 1994, Archive of Applied Mechanics.

[17]  Guillermo Rodríguez-Ortiz,et al.  Spatial operator factorization and inversion of the manipulator mass matrix , 1992, IEEE Trans. Robotics Autom..

[18]  J. Dai.  An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist , 2006 .

[19]  Russell H. Taylor,et al.  On homogeneous transforms, quaternions, and computational efficiency , 1990, IEEE Trans. Robotics Autom..

[20]  Werner Schiehlen,et al.  RECURSIVE KINEMATICS AND DYNAMICS FOR PARALLEL STRUCTURED CLOSED-LOOP MULTIBODY SYSTEMS* , 2001 .

[21]  R. Featherstone.  The Calculation of Robot Dynamics Using Articulated-Body Inertias , 1983 .

[22]  J. M. Hervé.  Intrinsic formulation of problems of geometry and kinematics of mechanisms , 1982 .

[23]  Alon Wolf,et al.  Screw theory tools for the synthesis of the geometry of a parallel robot for a given instantaneous task , 2006 .

[24]  E. Haug,et al.  Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems , 1982 .

[25]  J. G. Papastavridis.  Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems , 2014 .

[26]  Jonghoon Park,et al.  Geometric integration on Euclidean group with application to articulated multibody systems , 2005, IEEE Transactions on Robotics.

[27]  Andreas Müller,et al.  A conservative elimination procedure for permanently redundant closure constraints in MBS-models with relative coordinates , 2006 .

[28]  Frank Chongwoo Park,et al.  A geometrical formulation of the dynamical equations describing kinematic chains , 1993, [1993] Proceedings IEEE International Conference on Robotics and Automation.

[29]  McCarthy,et al.  Geometric Design of Linkages , 2000 .

[30]  F. Park.  Distance Metrics on the Rigid-Body Motions with Applications to Mechanism Design , 1995 .

[31]  Frank Chongwoo Park,et al.  Coordinate-invariant algorithms for robot dynamics , 1999, IEEE Trans. Robotics Autom..

[32]  S. Sastry,et al.  Nonholonomic motion planning: steering using sinusoids , 1993, IEEE Trans. Autom. Control..

[33]  José María Rico Martínez,et al.  On Mobility Analysis of Linkages Using Group Theory , 2003 .