A Gauge-Invariant Formulation for Constrained Mechanical Systems Using Square-Root Factorization and Unitary Transformation

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Farhad Aghili
Keyword(s):  
Motion Control ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Control Action ◽  
Gauge Invariant ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
Weighting Matrix ◽  
Rotational Components ◽  
Invariant Formulation

A gauge-invariant formulation for deriving the dynamic equations of constrained multibody systems in terms of (reduced) quasivelocities is presented. This formulation does not require any weighting matrix to deal with the gauge-invariance problem when both translational and rotational components are involved in the generalized coordinates or in the constraint equations. Moreover, in this formulation the equations of motion are decoupled from those of constrained force, and each system has its own independent input. This allows the possibility to develop a simple force control action that is totally independent from the motion control action facilitating a hybrid force/motion control. Tracking force/motion control of constrained multibody systems based on a combination of feedbacks on the vectors of the quasivelocities and the configuration-variables are presented.

Invariant Modeling of Mechanical Systems in Terms of Quasi-Velocities and Quasi-Forces

2009 ◽  
Author(s):  
Farhad Aghili
Keyword(s):  
Motion Control ◽  
Equations Of Motion ◽  
Control Action ◽  
Constraint Equations ◽  
Weighting Matrix ◽  
Rotational Components ◽  
Tracking Force ◽  
Invariant Formulation ◽  
Invariance Problem ◽  
Multi Body

A gauge-invariant formulation for deriving the dynamic equations of constrained multi-body systems (MBS) in terms of (reduced) quasi–velocities is presented. This formulation does not require any weighting matrix to deal with the gauge-invariance problem when both translational and rotational components are involved in the generalized coordinates or in the constraint equations. Moreover, in this formulation the equations of motion are decoupled from those of constrained force and each system has its own independent input. This allows the possibility to develop a simple force control action that is totally independent from the motion control action facilitating a hybrid force/motion control. Tracking force/motion control of constrained multi-body systems based on a combination of feedbacks on the vectors of the quasi–velocities and the configuration variables are presented.

On the Use of the Canonical Equations of Motion for the Dynamic Analysis of Constrained Multibody Systems

1992 ◽  
Author(s):  
E. Bayo ◽  
J. M. Jimenez
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
First Order ◽  
Canonical Variables ◽  
Lagrange’S Multipliers

Abstract We investigate in this paper the different approaches that can be derived from the use of the Hamiltonian or canonical equations of motion for constrained mechanical systems with the intention of responding to the question of whether the use of these equations leads to more efficient and stable numerical algorithms than those coming from acceleration based formalisms. In this process, we propose a new penalty based canonical description of the equations of motion of constrained mechanical systems. This technique leads to a reduced set of first order ordinary differential equations in terms of the canonical variables with no Lagrange’s multipliers involved in the equations. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency. In addition, we examine the use of the canonical equations based on independent coordinates, and conclude that in this second case the use of the acceleration based formulation is more advantageous than the canonical counterpart.

scholarly journals Constraint Gradient Projective Method for Stabilized Dynamic Simulation of Constrained Multibody Systems

2003 ◽  
Author(s):  
Zdravko Terze ◽  
Joris Naudet ◽  
Dirk Lefeber
Keyword(s):  
Mathematical Models ◽  
Dynamic Simulation ◽  
Multibody Systems ◽  
Point Of View ◽  
Constraint Violation ◽  
Projective Method ◽  
Holonomic Constraints ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
Holonomic Systems

Constraint gradient projective method for stabilization of constraint violation during integration of constrained multibody systems is in the focus of the paper. Different mathematical models for constrained MBS dynamic simulation on manifolds are surveyed and violation of kinematical constraints is discussed. As an extension of the previous work focused on the integration procedures of the holonomic systems, the constraint gradient projective method for generally constrained mechanical systems is discussed. By adopting differentialgeometric point of view, the geometric and stabilization issues of the method are addressed. It is shown that the method can be applied for stabilization of holonomic and non-holonomic constraints in Pfaffian and general form.

Dynamics of Constrained Multibody Systems

1984 ◽  
Vol 51 (4) ◽  
pp. 899-903 ◽  
Author(s):  
J. W. Kamman ◽  
R. L. Huston
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Governing Equations ◽  
Dynamical Equations ◽  
Closed Loops ◽  
Constrained Multibody Systems ◽  
Constraint Equations ◽  
Cable Dynamics ◽  
Hanging Chain ◽  
A Chain

A new automated procedure for obtaining and solving the governing equations of motion of constrained multibody systems is presented. The procedure is applicable when the constraints are either (a) geometrical (for example, “closed-loops”) or (b) kinematical (for example, specified motion). The procedure is based on a “zero eigenvalues theorem,” which provides an “orthogonal complement” array which in turn is used to contract the dynamical equations. This contraction, together with the constraint equations, forms a consistent set of governing equations. An advantage of this formulation is that constraining forces are automatically eliminated from the analysis. The method is applied with Kane’s equations—an especially convenient set of dynamical equations for multibody systems. Examples of a constrained hanging chain and a chain whose end has a prescribed motion are presented. Applications in robotics, cable dynamics, and biomechanics are suggested.

A Comparative Study on Some Different Formulations of the Dynamic Equations of Constrained Mechanical Systems

1987 ◽  
Vol 109 (4) ◽  
pp. 466-474 ◽  
Author(s):  
J. Unda ◽  
J. Garci´a de Jalo´n ◽  
F. Losantos ◽  
R. Enparantza
Keyword(s):  
Comparative Study ◽  
Relative Efficiency ◽  
Multibody Systems ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Dynamic Equations ◽  
Natural Coordinates ◽  
Constrained Mechanical Systems ◽  
Computer Codes

This paper presents a comparative theoretical and numerical study on the efficiency of several numerical methods for the dynamic analysis of constrained mechanical systems, also called in the literature multibody systems. This comparative study has been performed between methods based on the use of “reference point” coordinates and those based on the use of “natural” coordinates. This study embraces different possibilities to formulate the differential equations of motion. The relative efficiency of the resulting algorithms has been analyzed theoretically in terms of the number of multiplications needed to evaluate the mechanism accelerations. This efficiency has also been studied implementing the methods into computer codes and testing them with different examples. Conclusions on the relative efficiency of the methods are finally presented.

New Approach to the Modeling of Complex Multibody Dynamical Systems

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Aaron Schutte ◽  
Firdaus Udwadia
Keyword(s):  
Multibody Systems ◽  
Multibody System ◽  
Equations Of Motion ◽  
Multiple Time ◽  
New Approach ◽  
Constrained Mechanical Systems ◽  
Step Procedure ◽  
Highly Nonlinear ◽  
Mass Matrices ◽  
General Complex

In this paper, a general method for modeling complex multibody systems is presented. The method utilizes recent results in analytical dynamics adapted to general complex multibody systems. The term complex is employed to denote those multibody systems whose equations of motion are highly nonlinear, nonautonomous, and possibly yield motions at multiple time and distance scales. These types of problems can easily become difficult to analyze because of the complexity of the equations of motion, which may grow rapidly as the number of component bodies in the multibody system increases. The approach considered herein simplifies the effort required in modeling general multibody systems by explicitly developing closed form expressions in terms of any desirable number of generalized coordinates that may appropriately describe the configuration of the multibody system. Furthermore, the approach is simple in implementation because it poses no restrictions on the total number and nature of modeling constraints used to construct the equations of motion of the multibody system. Conceptually, the method relies on a simple three-step procedure. It utilizes the Udwadia–Phohomsiri equation, which describes the explicit equations of motion for constrained mechanical systems with singular mass matrices. The simplicity of the method and its accuracy is illustrated by modeling a multibody spacecraft system.

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.

Application of the Canonical Equations of Motion in Problems of Constrained Multibody Systems With Intermittent Motion

1988 ◽  
Author(s):  
H. M. Lankarani ◽  
P. E. Nikravesh
Keyword(s):  
Differential Equations ◽  
Canonical Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Momentum Balance ◽  
Constrained Multibody Systems ◽  
Intermittent Motion

Abstract For mechanical systems that undergo intermittent motion, the usual formulation of the equations of motion is not valid over the periods of the discontinuity, and a procedure for balancing the momenta of the system is often performed. A canonical form of the equations of motion is used here as the differential equations of motion. A set of momentum balance-impulse equations are derived in terms of the system total momenta by explicitly integrating the canonical equations. The method shows to be stable while numerically integrating the canonical equations, and efficient while solving the momentum balance-impulse equations. Examples are provided to illustrate the validity of the method.

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