Comparison of Different Methods to Control Constraints Violation in Forward Multibody Dynamics

2013 ◽  
Author(s):  
Paulo Flores ◽  
Parviz E. Nikravesh
Keyword(s):  
Multibody Dynamics ◽  
Multibody Systems ◽  
Generalized Inverse ◽  
Jacobian Matrix ◽  
Equations Of Motion ◽  
Control Constraints ◽  
Algebraic Equations ◽  
Kinematic Constraint ◽  
Constraint Equations ◽  
Velocity Constraint

The dynamic equations of motion for constrained multibody systems are frequently formulated using the Newton-Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. It is known that the standard resolution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general review of the main methods commonly used to control or eliminate the violation of the constraint equations in the context of multibody dynamics formulation is presented and discussed. Furthermore, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is also presented. The basic idea of this approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as function of the Moore-Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations.

Numerical Computation of Differential-Algebraic Equations for Non-Linear Dynamics of Multibody Systems Involving Contact Forces

1999 ◽  
Vol 123 (2) ◽  
pp. 272-281 ◽  
Author(s):  
B. Fox ◽  
L. S. Jennings ◽  
A. Y. Zomaya
Keyword(s):  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Differential Algebraic Equations ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Lagrange Equations ◽  
Constraint Equations

The well known Euler-Lagrange equations of motion for constrained variational problems are derived using the principle of virtual work. These equations are used in the modelling of multibody systems and result in differential-algebraic equations of high index. Here they concern an N-link pendulum, a heavy aircraft towing truck and a heavy off-highway track vehicle. The differential-algebraic equation is cast as an ordinary differential equation through differentiation of the constraint equations. The resulting system is computed using the integration routine LSODAR, the Euler and fourth order Runge-Kutta methods. The difficulty to integrate this system is revealed to be the result of many highly oscillatory forces of large magnitude acting on many bodies simultaneously. Constraint compliance is analyzed for the three different integration methods and the drift of the constraint equations for the three different systems is shown to be influenced by nonlinear contact forces.

Enforcing Constraints in Multibody Systems: A Review

2007 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Andre´ Laulusa
Keyword(s):  
Multibody Dynamics ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Constrained Systems ◽  
Algebraic Equations ◽  
Constraint Violation ◽  
Index Reduction ◽  
Practical Algorithms ◽  
Theoretical Foundations

A hallmark of multibody dynamics is that most formulations involve a number of constraints. Typically, when redundant generalized coordinates are used, equations of motion are simpler to derive but constraint equations are present. While the dynamic behavior of constrained systems is well understood, the numerical solution of the resulting equations, potentially of differential-algebraic nature, remains problematic. Many different approaches have been proposed over the years, all presenting advantages and drawbacks: the sheer number and variety of methods that have been proposed indicate the difficulty of the problem. A cursory survey of the literature reveals that the various methods fall within broad categories sharing common theoretical foundations. This paper summarizes the theoretical foundations to the enforcement in constraints in multibody dynamics problems. Next, methods based on the use of Lagrange’s equation of the first kind, which are index-3 differential algebraic equations are reviewed. Methods leading to a minimum set of equations are discussed; in view of the numerical difficulties associated with index-3 approaches, reduction to a minimum set is often performed, leading to a number of practical algorithms using methods developed for ordinary differential equations. Finally, alternative approaches to dealing with high index differential algebraic equations, based on index reduction techniques, are reviewed and discussed. Constraint violation stabilization techniques that have been developed to control constraint drift are also reviewed. These techniques are used in conjunction with algorithms that do not exactly enforce the constraints. Control theory forms the basis for a number of these methods. Penalty based techniques have also been developed, but the augmented Lagrangian formulation presents a more solid theoretical foundation. In contrast to constraint violation stabilization techniques, constraint violation elimination techniques enforce exact satisfaction of the constraints, at least to machine accuracy. Finally, as the finite element method has gained popularity for the solution of multibody systems, new techniques for the enforcement of constraints has been developed in that framework. The goal of this paper is to review the features of these methods, assess their accuracy and efficiency, underline the relationship among the methods, and recommend approaches that seem to perform better than others.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

Efficient Dynamic Analysis Method for Multibody Systems With Constraint Violation Stabilization Method

1999 ◽  
Author(s):  
Apiwat Reungwetwattana ◽  
Shigeki Toyama
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Analysis Method ◽  
Stabilization Method ◽  
Kinematic Constraint ◽  
Constraint Violation ◽  
Closed Loops ◽  
Constraint Stabilization ◽  
Constraint Equations ◽  
Crank Mechanism

Abstract This paper presents an efficient extension of Rosenthal’s order-n algorithm for multibody systems containing closed loops. Closed topological loops are handled by cut joint technique. Violation of the kinematic constraint equations of cut joints is corrected by Baumgarte’s constraint violation stabilization method. A reliable approach for selecting the parameters used in the constraint stabilization method is proposed. Dynamic analysis of a slider crank mechanism is carried out to demonstrate efficiency of the proposed method.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

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.

Redundancy Resolution of Robotic Manipulators Using Normalized Generalized Inverses

1995 ◽  
Vol 117 (3) ◽  
pp. 454-459 ◽  
Author(s):  
J. T. Wang
Keyword(s):  
Generalized Inverse ◽  
Robotic Manipulators ◽  
Redundancy Resolution ◽  
Objective Functions ◽  
Dynamical Equations ◽  
Kinematic Constraint ◽  
Joint Torques ◽  
Constraint Equations ◽  
Force Method ◽  
Inverse Form

A dynamical formulation based upon the undetermined force method is presented for analyzing redundant robotic manipulators. The equivalence between this dynamical formulation and the general solution of kinematic constraint equations is then obtained through use of a normalized generalized inverse. This leads to a special form of the dynamical equations, called the N-inverse form of the dynamical equations. A class of problems, associated with the optimization of quadratic objective functions, are then studied. We find that the N-inverse form of the dynamical equations is the solution of this class of problems. Examples, including local minimization of joint torques and global minimization of kinetic energy, are presented.

Steady-State Responses of an RSRC-Type Spatial Flexible Linkage

1994 ◽  
Vol 116 (1) ◽  
pp. 264-269 ◽  
Author(s):  
R. Y. Chang ◽  
C. K. Sung
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Body Motion ◽  
Flexible Link ◽  
Predictive Capability ◽  
Kinematic Constraint ◽  
Constraint Equations ◽  
Steady State Responses ◽  
State Responses ◽  
Flexible Linkage

This paper presents an analytical and experimental investigation on the steady-state responses of an RSRC-type spatial flexible linkage. The kinematic analysis of the spatial rigid-body motion is performed using kinematic constraint equations to yield the linear and angular positions, velocities, and accelerations of the linkage. A mixed variational principle is, then, employed to derive the equations of motion governing the longitudinal, transverse, and torsional vibrations of the flexible link and the associated boundary conditions. Based on these equations, the steady-state responses are predicted. Finally, an RSRC-type four-bar spatial flexible linkage is constructed and the experimental study is performed to examine the predictive capability proposed in this investigation. Favorable comparisons between the analytical and experimental results are obtained.

Steady-State Responses of an RSRC-Type Spatial Flexible Linkages

1992 ◽  
Author(s):  
R. Y. Chang ◽  
C. K. Sung
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Body Motion ◽  
Flexible Link ◽  
Predictive Capability ◽  
Kinematic Constraint ◽  
Constraint Equations ◽  
Steady State Responses ◽  
State Responses ◽  
Flexible Linkage

Abstract This paper presents an analytical and experimental investigation on the steady-state responses of an RSRC-type spatial flexible linkage. The kinematic analysis of the spatial rigid-body motion is performed using kinematic constraint equations to yield linear and angular positions, velocities and accelerations of the linkage. A mixed variational principle is, then, employed to derive the equations of motion governing the longitudinal, transversal and torsional vibrations of the flexible link and the associated boundary conditions. Based on these equations, the steady-state responses are predicted. Finally, an RSRC-type four-bar spatial flexible linkage is constructed and the experimental study is performed to examine the predictive capability proposed in this investigation. Favorable comparisons between the analytical and experimental results are obtained.

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