Another form of equations of motion for constrained multibody systems

2007 ◽  
Vol 51 (3) ◽  
pp. 465-475 ◽  
Author(s):  
C. Q. Liu ◽  
R. L. Huston
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Constrained Multibody Systems

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.

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 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.

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.

Symbolic linearization of equations of motion of constrained multibody systems

2014 ◽  
Vol 33 (2) ◽  
pp. 143-161 ◽  
Author(s):  
Dale L. Peterson ◽  
Gilbert Gede ◽  
Mont Hubbard
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Constrained Multibody Systems

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

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