Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations

2011 ◽  
Author(s):  
Olivier Bru¨ls ◽  
Martin Arnold ◽  
Alberto Cardona
Keyword(s):  
Numerical Solution ◽  
Lie Group ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Computational Cost ◽  
Differential Algebraic Equations ◽  
Rigid Body Dynamics ◽  
Time Step ◽  
Iteration Matrix

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations can then be solved using a Lie group time integration method proposed in a previous work. The general structure of the equations of motion are derived from Hamilton principle in a general and unifying framework. Then, in the case of rigid body dynamics, two particular formulations are developed and compared from the viewpoint of the structure of the equations of motion, of the accuracy of the numerical solution obtained by time integration, and of the computational cost of the iteration matrix involved in the Newton iterations at each time step. In the first formulation, the equations of motion are described on a Lie group defined as the Cartesian product of the group of translations R3 (the Euclidean space) and the group of rotations SO(3) (the special group of 3 by 3 proper orthogonal transformations). In the second formulation, the equations of motion are described on the group of Euclidean transformations SE(3) (the group of 4 by 4 homogeneous transformations). Both formulations lead to a second-order accurate numerical solution. For an academic example, we show that the formulation on SE(3) offers the advantage of an almost constant iteration matrix.

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Alexander Epple ◽  
Carlo L. Bottasso
Keyword(s):  
Physical Properties ◽  
Augmented Lagrangian ◽  
Equations Of Motion ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Integration Schemes

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

Explicit Time Integration of Multibody Systems Modelled With Three Rotation Parameters

2020 ◽  
Author(s):  
Stefan Holzinger ◽  
Johannes Gerstmayr
Keyword(s):  
Lie Group ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Time Integration ◽  
Rigid Bodies ◽  
Rotation Vector ◽  
Body Motion ◽  
Euler Parameters ◽  
Time Step ◽  
Rotation Parameters

Abstract Rigid bodies are an essential part of multibody systems. As there are six degrees of freedom in rigid bodies, it is natural but also precarious to use three parameters for the displacement and three parameters for the rotation parameters — since there is no singularity-free description of spatial rotations based on three rotation parameters. Standard formulations based on three rotation parameters avoid singularities, e.g. by applying reparameterization strategies during the time integration of the rotational kinematic equations. Alternatively, Euler parameters are commonly used to avoid singularities. State of the art approaches use Lie group methods, specifically integrators, to model rigid body motion without the need for the above mentioned solutions. However, the methods so far have been based on additional information, e.g., the rotation matrix, which has to been computed in each step. The latter procedure is thus difficult to be implemented in existing codes that are based on three rotation parameters. In this paper, we use the rotation vector to model large rotations. Whereby Lie group integration methods are used to compute consistent updates for the rotation vector in every time step. The resulting rotation vector update is finite, while the derivative of the rotation vector in the singularity becomes unbounded. The advantages of this method are shown in an example of a gyro. Additionally, the method is applied to a multibody system and the effects of crossing singularities are presented.

Adaptive Time Integration Method for DAEs of Multibody Systems

2012 ◽  
Author(s):  
Jieyu Ding ◽  
Zhenkuan Pan
Keyword(s):  
Multibody Systems ◽  
Integration Method ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Richardson Extrapolation ◽  
Time Interval ◽  
Time Step ◽  
Time Integration Method ◽  
Adaptive Time Integration ◽  
Adaptive Time

An adaptive time integration method is developed for the index-3 differential-algebraic equations (DAEs) of multibody systems to improve the computational efficiency as well as the accuracy of the results. Based on the modified general-α method, the adaptive time integration is presented. At each discrete time interval, the time step size is changed through Richardson extrapolation with definable computation accuracy. A rotary rod slider system is used to validate the presented adaptive time integration. The accuracy and efficiency are determined by the expected order of the accuracy in Richardson extrapolation.

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.

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.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

A Parametric Study on the Baumgarte Stabilization Method for Forward Dynamics of Constrained Multibody Systems

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Paulo Flores ◽  
Margarida Machado ◽  
Eurico Seabra ◽  
Miguel Tavares da Silva
Keyword(s):  
Parametric Study ◽  
Multibody Systems ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Forward Dynamics ◽  
Stabilization Method ◽  
Time Step ◽  
Constrained Multibody Systems ◽  
Laplace Transform Technique ◽  
Baumgarte Stabilization

This paper presents and discusses the results obtained from a parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. The main purpose of this work is to analyze the influence of the variables that affect the violation of constraints, chiefly the values of the Baumgarte parameters, the integration method, the time step, and the quality of the initial conditions for the positions. In the sequel of this process, the formulation of the rigid multibody systems is reviewed. The generalized Cartesian coordinates are selected as the variables to describe the bodies’ degrees of freedom. The formulation of the equations of motion uses the Newton–Euler approach, augmented with the constraint equations that lead to a set of differential algebraic equations. Furthermore, the main issues related to the stabilization of the violation of constraints based on the Baumgarte approach are revised. Special attention is also given to some techniques that help in the selection process of the values of the Baumgarte parameters, namely, those based on the Taylor’s series and the Laplace transform technique. Finally, a slider-crank mechanism with eccentricity is considered as an example of application in order to illustrate how the violation of constraints can be affected by different factors.

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