Quasi-velocities definition in Lagrangian multibody dynamics

Based on Lagrangian mechanics, use of velocity constraints as a special set of quasi-velocities helps derive explicit equations of motion. The equations are applicable to holonomic and nonholonomic constrained multibody systems. It is proved that in proposed quasi-spaces, the Lagrange multipliers are eliminated from equations of motion; however, it is possible to compute these multipliers once the equations of motion have been solved. The novelty of this research is employing block matrix inversion to find the analytical relations between the parameters of quasi-velocities and equations of motion. In other words, this research identifies arbitrary submatrices and their effects on equations of motion. Also, the present study aimed to provide appropriate criteria to select arbitrary parameters to avoid singularity, reduce constraints violations, and improve computational efficiency. In order to illustrate the advantage of this approach, the simulation results of a 3-link snake-like robot with nonholonomic constraints and a four-bar mechanism with holonomic constraints are presented. The effectiveness of the proposed approach is demonstrated by comparing the constraints violation at the position and velocity levels, conservation of the total energy, and computational efficiency with those obtained via the traditional methods.

[1]  Elzbieta Jarzebowska.  Model-Based Tracking Control of Nonlinear Systems , 2012 .

[2]  J. Baumgarte.  Stabilization of constraints and integrals of motion in dynamical systems , 1972 .

[3]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[4]  S. S. Kim,et al.  QR Decomposition for State Space Representation of Constrained Mechanical Dynamic Systems , 1986 .

[5]  W. Blajer.  Methods for constraint violation suppression in the numerical simulation of constrained multibody systems – A comparative study , 2011 .

[6]  P. Nikravesh,et al.  Comparison of Different Methods to Control Constraints Violation in Forward Multibody Dynamics , 2013 .

[7]  Parviz E. Nikravesh,et al.  Some Methods for Dynamic Analysis of Constrained Mechanical Systems: A Survey , 1984 .

[8]  Jerry H. Ginsberg,et al.  Advanced Engineering Dynamics , 1988 .

[9]  A. G. Greenhill.  Analytical Mechanics , 1890, Nature.

[10]  R. Ledesma,et al.  Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics , 1996 .

[11]  T. Kane.  Dynamics of Nonholonomic Systems , 1961 .

[12]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[13]  E. Haug.  Extension of Maggi and Kane Equations to Holonomic Dynamic Systems , 2018, Journal of Computational and Nonlinear Dynamics.

[14]  Q. Yu,et al.  A direct violation correction method in numerical simulation of constrained multibody systems , 2000 .

[15]  O. Bauchau,et al.  Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems , 2008 .

[16]  Hossein Nejat Pishkenari,et al.  Comparison of Kane’s and Lagrange’s Methods in Analysis of Constrained Dynamical Systems , 2020, Robotica.

[17]  Aravindh Krishnamoorthy,et al.  Matrix inversion using Cholesky decomposition , 2011, 2013 Signal Processing: Algorithms, Architectures, Arrangements, and Applications (SPA).

[18]  A. M. Shafei,et al.  A General Formulation for Managing Trajectory Tracking in Non-holonomic Moving Manipulators with Rotary-Sliding Joints , 2020, J. Intell. Robotic Syst..

[19]  Elżbieta Jarzębowska,et al.  MODELING AND CONTROL DESIGN USING THE BOLTZMANN-HAMEL EQUATIONS: A ROLLER-RACER EXAMPLE , 2006 .

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

[21]  J. G. Papastavridis,et al.  Maggi's equations of motion and the determination of constraint reactions , 1990 .

[22]  THE EXPLICIT GIBBS-APPELL EQUATIONS of MOTION for RIGID-BODY CONSTRAINED MECHANICAL SYSTEM , 2018, 2018 6th RSI International Conference on Robotics and Mechatronics (IcRoM).

[23]  George M. Lance,et al.  A Differentiable Null Space Method for Constrained Dynamic Analysis , 1987 .

[24]  M. A. Serna,et al.  A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems , 1988 .

[25]  B. Leimkuhler,et al.  Numerical solution of differential-algebraic equations for constrained mechanical motion , 1991 .

[26]  C. W. Gear,et al.  Automatic integration of Euler-Lagrange equations with constraints , 1985 .

[27]  Hossein Nejat Pishkenari,et al.  A novel computer-oriented dynamical approach with efficient formulations for multibody systems including ignorable coordinates , 2018, Applied Mathematical Modelling.

[28]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

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

[30]  E. A. Desloge.  Relationship between Kane's equations and the Gibbs-Appell equations , 1987 .

[31]  Domenico Guida,et al.  On the Computational Methods for Solving the Differential-Algebraic Equations of Motion of Multibody Systems , 2018 .

[32]  P. Flores,et al.  On the constraints violation in forward dynamics of multibody systems , 2017 .

[33]  Domenico Guida,et al.  A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems , 2018, Archive of Applied Mechanics.

[34]  Andrés F. Hidalgo,et al.  Efficient Solution of Maggi’s Equations , 2011 .

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

[36]  R. Kalaba,et al.  A new perspective on constrained motion , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[37]  O. Bauchau,et al.  Review of Classical Approaches for Constraint Enforcement in Multibody Systems , 2008 .

[38]  Gian Antonio Maggi,et al.  Principii della teoria matematica del movimento dei corpi : corso di meccanica razionale , 1896 .

[39]  J. Yen,et al.  Generalized Coordinate Partitioning Methods for Numerical Integration of Differential-Algebraic Equations of Dynamics , 1990 .

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

[41]  A. Kurdila,et al.  Role of Maggi's Equations in Computational Methods for Constrained Multibody Systems , 1990 .

[42]  Satoru Iwata,et al.  Index Reduction for Differential-algebraic Equations with Mixed Matrices , 2017, J. ACM.