Singularity-Free Dynamic Equations of Open-Chain Mechanisms With General Holonomic and Nonholonomic Joints

Standard methods to model multibody systems are aimed at systems with configuration spaces isomorphic to Ropfn. This limitation leads to singularities and other artifacts in case the configuration space has a different topology, for example, in the case of ball joints or a free-floating mechanism. This paper discusses an extension of classical methods that allows for a more general class of joints, including all joints with a Lie group structure as well as nonholonomic joints. The model equations are derived using the Boltzmann-Hamel equations and have very similar structure and complexity as obtained using classical methods. However, singularities are avoided through the use of global non-Euclidean configuration coordinates, together with mappings describing a local Euclidean structure around each configuration. The resulting equations are explicit (unconstrained) differential equations, both for holonomic and nonholonomic joints, which do not require a coordinate atlas and can be directly implemented in simulation software.

[1]  J. Gibbs.  On the Fundamental Formulae of Dynamics , 1879 .

[2]  T. Kane,et al.  ON THE DERIVATION OF EQUATIONS OF MOTION , 1965 .

[3]  Roy Featherstone,et al.  Robot Dynamics Algorithms , 1987 .

[4]  Edmund Taylor Whittaker,et al.  A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: INDEX OF TERMS EMPLOYED , 1988 .

[5]  M. Lesser.  A geometrical interpretation of Kane’s Equations , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[6]  A. J. van der Schaft,et al.  Port-controlled Hamiltonian Systems:Modelling Origins and System-Theoretic Properties , 1992 .

[7]  van der Arjan Schaft,et al.  On the Hamiltonian Formulation of Nonholonomic Mechanical Systems , 1994 .

[8]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

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

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

[11]  A. D. Lewis.  The geometry of the Gibbs-Appell equations and Gauss' principle of least constraint , 1996 .

[12]  P. Krishnaprasad,et al.  Nonholonomic mechanical systems with symmetry , 1996 .

[13]  J. M. Selig.  Geometric Fundamentals of Robotics , 2004, Monographs in Computer Science.

[14]  Wayne J. Book,et al.  Modeling Mechanisms with Nonholonomic Joints Using the Boltzmann-Hamel Equations , 1997, Int. J. Robotics Res..

[15]  H. Munthe-Kaas.  Runge-Kutta methods on Lie groups , 1998 .

[16]  Amir Fijany,et al.  A technique for analyzing constrained rigid-body systems, and its application to the constraint force algorithm , 1999, IEEE Trans. Robotics Autom..

[17]  W. Blajer.  A geometrical interpretation and uniform matrix formulation of multibody system dynamics , 2001 .

[18]  Stefano Stramigioli,et al.  Modeling and IPC Control of Interactive Mechanical Systems - A Coordinate-Free Approach , 2001 .

[19]  K Yagasaki,et al.  Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; For Engineers, Physicists, and Mathematicians , 2002 .

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

[21]  Hilding Elmqvist,et al.  The New Modelica MultiBody Library , 2003 .

[22]  F. Aghili,et al.  Simulation of Motion of Constrained Multibody Systems Based on Projection Operator , 2003 .

[23]  Stefano Stramigioli,et al.  Port-Based Modeling and Analysis of Snakboard Locomotion , 2004 .

[24]  Jonghoon Park.  Principle of Dynamical Balance for Multibody Systems , 2005 .

[25]  A. D. Lewis,et al.  Geometric control of mechanical systems : modeling, analysis, and design for simple mechanical control systems , 2005 .

[26]  G. Quispel,et al.  Geometric integrators for ODEs , 2006 .

[27]  R. Rosenberg,et al.  System Dynamics: Modeling and Simulation of Mechatronic Systems , 2006 .

[28]  S. Welch,et al.  Parameter-Free Second Order Numerical Scheme for Constrained Multibody Dynamical Systems , 2007 .

[29]  Stefano Stramigioli,et al.  Lagrangian dynamics of open multibody systems with generalized holonomic and nonholonomic joints , 2007, 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems.