Quasi-velocities definition in Lagrangian multibody dynamics

2021 ◽  
pp. 095440622199585
Author(s):  
S Mohammad Mirtaheri ◽  
Hassan Zohoor
Keyword(s):  
Computational Efficiency ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Matrix Inversion ◽  
Lagrangian Mechanics ◽  
Holonomic Constraints ◽  
Velocity Constraints ◽  
Analytical Relations ◽  
Explicit Equations Of Motion

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.

Explicit Equations of Motion for Mechanical Systems With Nonideal Constraints

2000 ◽  
Vol 68 (3) ◽  
pp. 462-467 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Lagrangian Mechanics ◽  
Constrained Motion ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Lagrangian Formulations ◽  
Explicit Equations Of Motion ◽  
D'alembert's Principle

Since its inception about 200 years ago, Lagrangian mechanics has been based upon the Principle of D’Alembert. There are, however, many physical situations where this confining principle is not suitable, and the constraint forces do work. To date, such situations are excluded from general Lagrangian formulations. This paper releases Lagrangian mechanics from this confinement, by generalizing D’Alembert’s principle, and presents the explicit equations of motion for constrained mechanical systems in which the constraints are nonideal. These equations lead to a simple and new fundamental view of Lagrangian mechanics. They provide a geometrical understanding of constrained motion, and they highlight the simplicity with which Nature seems to operate.

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

1999 ◽  
Vol 66 (4) ◽  
pp. 986-996 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Kinematic Chain ◽  
Rigid Bodies ◽  
Dynamic Equations ◽  
Orthogonal Complement ◽  
Forward Dynamics ◽  
Velocity Constraints

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics

2006 ◽  
Vol 462 (2071) ◽  
pp. 2097-2117 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Explicit Form ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Constrained Motion ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
Mass Matrices ◽  
Body Dynamics ◽  
Multi Body ◽  
Explicit Equations Of Motion

We present the new, general, explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices. The systems may have holonomic and/or non-holonomic constraints, which may or may not satisfy D'Alembert's principle at each instant of time. The equation provides new insights into the behaviour of constrained motion and opens up new ways of modelling complex multi-body systems. Examples are provided and applications of the equation to such systems are illustrated.

An Ordinary Differential Equation Formulation for Multibody Dynamics: Nonholonomic Constraints

2016 ◽  
Vol 17 (1) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Nonholonomic Constraints ◽  
State Variables ◽  
Well Conditioning ◽  
Fifth Order

A method is presented for formulating and numerically integrating ordinary differential equations of motion for nonholonomically constrained multibody systems. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation, yielding ordinary differential equations of motion. Orthogonal-dependent coordinates and velocities are used to enforce constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for updating local coordinates on configuration and velocity constraint manifolds, transparent to the user and at minimal computational cost. The formulation is developed for multibody systems with nonlinear holonomic constraints and nonholonomic constraints that are linear in velocity coordinates and nonlinear in configuration coordinates. A computational algorithm for implementing the approach is presented and used in the solution of three examples: one planar and two spatial. Numerical results using a fifth-order Runge–Kutta–Fehlberg explicit integrator verify that accurate results are obtained, satisfying all the three forms of kinematic constraint, to within error tolerances that are embedded in the formulation.

Selective Generalized Coordinates Partitioning Method for Multibody Systems With Non-Holonomic Constraints

2017 ◽  
Author(s):  
Ayman A. Nada ◽  
Abdullateef H. Bashiri
Keyword(s):  
Multibody Systems ◽  
Large Scale ◽  
Research Work ◽  
Nonholonomic Constraints ◽  
Integration Step ◽  
Large Scale Systems ◽  
Holonomic Constraints ◽  
Generalized Coordinates ◽  
Holonomic And Nonholonomic Constraints ◽  
Multi Body

The goal of this research work is to extend the method of generalized coordinates partitioning to include both holonomic and nonholonomic constraints. Furthermore, the paper proposes a method for selective coordinates for integration instead of identifying a set of independent coordinates at each integration step. The effectiveness of the proposed method is presented and compared with full-coordinates integration as well as generalized co-ordinates partitioning method. The proposed method can treat large-scale systems as one of the main advantages of multi-body systems.

What is the General Form of the Explicit Equations of Motion for Constrained Mechanical Systems?

2002 ◽  
Vol 69 (3) ◽  
pp. 335-339 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Analytical Mechanics ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Explicit Equations Of Motion ◽  
D'alembert's Principle

This paper presents the general form of the explicit equations of motion for mechanical systems. The systems may have holonomic and/or nonholonomic constraints, and the constraint forces may or may not satisfy D’Alembert’s principle at each instant of time. The explicit equations lead to new fundamental principles of analytical mechanics.

Extension of Maggi and Kane Equations to Holonomic Dynamic Systems

2018 ◽  
Vol 13 (12) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamic Systems ◽  
Tangent Space ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Extended Formulation ◽  
Holonomic Constraints ◽  
Numerical Examples ◽  
Holonomic Systems

The Maggi and Kane equations of motion are valid for systems with only nonholonomic constraints, but may fail when applied to systems with holonomic constraints. A tangent space ordinary differential equation (ODE) extension of the Maggi and Kane formulations that enforces holonomic constraints is presented and shown to be theoretically sound and computationally effective. Numerical examples are presented that demonstrate the extended formulation leads to solutions that satisfy position, velocity, and acceleration constraints for holonomic systems to near computer precision.

A General Approach to the Dynamics of Nonholonomic Mobile Manipulator Systems

2002 ◽  
Vol 124 (4) ◽  
pp. 512-521 ◽  
Author(s):  
Qing Yu ◽  
I-Ming Chen
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Mobile Platform ◽  
Mobile Manipulator ◽  
Dynamic Interactions ◽  
Dynamics Of Multibody Systems ◽  
Minimum Number ◽  
Simulation And Control ◽  
And Control

This paper studies the dynamic modeling of a nonholonomic mobile manipulator that consists of a multi-degree of freedom serial manipulator and an autonomous wheeled mobile platform. The manipulator is rigidly mounted on the mobile platform, and the wheeled mobile platform moves on the ground subjected to nonholonomic constraints. Forward Recursive Formulation for the dynamics of multibody systems is employed to obtain the governing equation of the mobile manipulator system. The approach fully utilizes the existing equations of motion of the manipulator and that of the mobile platform. Furthermore, terms representing the dynamic interactions between the manipulator and the mobile platform can be observed. The resulting dynamic equation of the mobile manipulator has the minimum number of generalized coordinates and can be used for the purpose of dynamic simulation and control design, etc. The implementation issues of the model are discussed.

A Reduced and Linearized High Fidelity Waveboard Multibody Model for Stability Analysis

2022 ◽  
Author(s):  
Alfonso García-Agúndez Blanco ◽  
Daniel García Vallejo ◽  
Emilio Freire ◽  
Aki Mikkola
Keyword(s):  
Multibody Systems ◽  
Jacobian Matrix ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Nonholonomic Constraints ◽  
Design Parameters ◽  
Constrained Multibody Systems ◽  
Multibody Model ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract In this paper, the stability of a waveboard, a human propelled two-wheeled vehicle consisting in two rotatable platforms, joined by a torsion bar and supported on two caster wheels, is analysed. A multibody model with holonomic and nonholonomic constraints is used to describe the system. The nonlinear equations of motion, which constitute a Differential-Algebraic system of equations (DAE system), are linearized along the steady forward motion resorting to a recently validated linearization procedure, which allows the maximum possible reduction of the linearized equations of motion of constrained multibody systems. The approach enables the generation of the Jacobian matrix in terms of the geometric and dynamic parameters of the multibody system, and the eigenvalues of the system are parameterized in terms of the design parameters. The resulting minimum set of linear equations leads to the elimination of spurious null eigenvalues, while retaining all the stability information in spite of the reduction of the Jacobian matrix. The linear stability results of the waveboard obtained in previous work are validated with this approach. The procedure shows an excellent computational efficiency with the waveboard, its utilization being highly advisable to linearize the equations of motion of complex constrained multibody systems.

Numerical Solution of Constrained Systems Using Jet Spaces

2007 ◽  
Author(s):  
Jukka Tuomela ◽  
Teijo Arponen ◽  
Villesamuli Normi
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Constrained Systems ◽  
Lagrangian Formalism ◽  
Jet Space ◽  
Jet Spaces ◽  
Conservation Of Energy ◽  
Holonomic Constraints ◽  
Motion Constraints ◽  
Spurious Oscillations

The major difficulty in simulations of constrained systems is how to avoid drift off and spurious oscillations. We describe results obtained by our method which addresses these issues. Our computational model considers differential equations in jet spaces. In case of multibody systems we use Lagrangian formalism to derive the equations of motion. Constraints and possible invariants (like conservation of energy) are taken into account by restricting the dynamics to an appropriate submanifold of a jet space. We will consider only holonomic constraints in this paper. However, nonholonomic problems can also be formulated and solved with our method.

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