Projection Method With Minimal Correction Procedure for Numerical Simulation of Constrained Dynamics

2017 ◽  
Author(s):  
Patrick S. Heaney ◽  
Gene Hou
Keyword(s):  
Projection Method ◽  
Numerical Technique ◽  
Correction Method ◽  
Differential Algebraic Equations ◽  
Constrained Systems ◽  
Correction Procedure ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Index Reduction ◽  
Algebraic Constraints

This paper describes a numerical technique for simulating the dynamics of constrained systems, which are described generally by differential-algebraic equations. The Projection Method for index reduction of a differential-algebraic equation and a minimal correction procedure are described. This procedure ensures algebraic constraints are satisfied during the numerical integration of the reduced index system of differential equations. Two examples illustrate how the method can be utilized to solve constrained multibody and rotational dynamics problems. The efficiency and accuracy of the proposed index-reduction and minimal correction method are then evaluated.

Control of Constrained Systems Described by Lagrangian DAEs

2011 ◽  
Author(s):  
Jason P. Frye ◽  
Brian C. Fabien
Keyword(s):  
Motion Tracking ◽  
Controller Design ◽  
Numerical Technique ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Constrained Systems ◽  
Algebraic Equations ◽  
Nonlinear Controller ◽  
Constrained System ◽  
Constraint Equations

In this paper, a nonlinear controller design for constrained systems described by Lagrangian differential algebraic equations (DAEs) is presented. The controller design utilizes the structure introduced by the coordinate splitting formulation, a numerical technique used for integration of DAEs. In this structure, the Lagrange multipliers associated with the constraint equations are eliminated, and the equations of motion are transformed into implicit differential equations. Making use of this, a feedback linearizing controller can be chosen for successful motion tracking of the constrained system. Numerical examples demonstrate the controller design can be successfully applied to fully actuated and underactuated systems.

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.

On QZ steps with perfect shifts and computing the index of a differential-algebraic equation

2020 ◽  
Author(s):  
Nicola Mastronardi ◽  
Paul Van Dooren
Keyword(s):  
Differential Algebraic Equations ◽  
Important Application ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Triangular Form ◽  
Finite Precision ◽  
Exact Arithmetic ◽  
Exact Eigenvalue ◽  
Algebraic Constraints ◽  
The Given

Abstract In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

Index reduction of differential algebraic equations by differential Dixon resultant

2018 ◽  
Vol 328 ◽  
pp. 189-202 ◽  
Author(s):  
Xiaolin Qin ◽  
Lu Yang ◽  
Yong Feng ◽  
Bernhard Bachmann ◽  
Peter Fritzson
Keyword(s):  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Index Reduction

scholarly journals The Pantelides algorithm for delay differential-algebraic equations

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Ines Ahrens ◽  
Benjamin Unger
Keyword(s):  
Differential Algebraic Equations ◽  
Equivalence Classes ◽  
Point Of View ◽  
Algebraic Equations ◽  
Sufficient Criterion ◽  
Differential Algebraic Equation ◽  
Graph Theoretical Approach ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Structural Point

Abstract We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

scholarly journals Modeling and PDC fuzzy control of planar parallel robot

2017 ◽  
Vol 14 (1) ◽  
pp. 172988141668711
Author(s):  
Benyamine Allouche ◽  
Antoine Dequidt ◽  
Laurent Vermeiren ◽  
Michel Dambrine
Keyword(s):  
Structural Characteristics ◽  
Differential Algebraic Equations ◽  
Parallel Robots ◽  
Linear Matrix ◽  
Parallel Mechanisms ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Trajectory Tracking Control ◽  
Model Based ◽  
Takagi Sugeno

Many works in the literature have studied the kinematical and dynamical issues of parallel robots. But it is still difficult to extend the vast control strategies to parallel mechanisms due to the complexity of the model-based control. This complexity is mainly caused by the presence of multiple closed kinematic chains, making the system naturally described by a set of differential–algebraic equations. The aim of this work is to control a two-degree-of-freedom parallel manipulator. A mechanical model based on differential–algebraic equations is given. The goal is to use the structural characteristics of the mechanical system to reduce the complexity of the nonlinear model. Therefore, a trajectory tracking control is achieved using the Takagi-Sugeno fuzzy model derived from the differential–algebraic equation forms and its linear matrix inequality constraints formulation. Simulation results show that the proposed approach based on differential–algebraic equations and Takagi-Sugeno fuzzy modeling leads to a better robustness against the structural uncertainties.

LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1

2012 ◽  
Vol 12 (04) ◽  
pp. 1250002 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
NGUYEN THI THE
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lyapunov Exponents ◽  
Algebraic Equation ◽  
Differential Algebraic Equations ◽  
Lyapunov Spectrum ◽  
Stochastic Flow ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Two Parameter

We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

Modular Modeling Using Lagrangian DAEs

2000 ◽  
Author(s):  
Robert Piché ◽  
Mikko Palmroth
Keyword(s):  
Software Design ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Simulation Software ◽  
Algebraic Equations ◽  
Feedback Control System ◽  
Physical Systems ◽  
Modular Modeling ◽  
Index Reduction ◽  
Analytical System

Abstract Layton’s Analytical System Dynamics theory for modeling multidisciplinary physical systems with Lagrangian differential algebraic equations (DAEs) is extended by introducing a technique for using hierarchical reusable modules. Connections between submodels are represented in a systematic manner using kinematic constraints and Lagrange multipliers. Simulation software design issues are discussed: data structures, consistent initial conditions, index reduction, and DAE solvers. An example of an electromechanical feedback control system is presented in detail.

scholarly journals A New Block Structural Index Reduction Approach for Large-Scale Differential Algebraic Equations

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2057
Author(s):  
Juan Tang ◽  
Yongsheng Rao
Keyword(s):  
Large Scale ◽  
Differential Algebraic Equations ◽  
Coupled Systems ◽  
Algebraic Equations ◽  
Triangular Form ◽  
Large Scale Systems ◽  
Index Reduction ◽  
Dae Systems ◽  
New Generation ◽  
Block Structures

A new generation of universal tools and languages for modeling and simulation multi-physical domain applications has emerged and became widely accepted; they generate large-scale systems of differential algebraic equations (DAEs) automatically. Motivated by the characteristics of DAE systems with large dimensions, high index or block structures, we first propose a modified Pantelides’ algorithm (MPA) for any high order DAEs based on the Σ matrix, which is similar to Pryce’s Σ method. By introducing a vital parameter vector, a modified Pantelides’ algorithm with parameters has been presented. It leads to a block Pantelides’ algorithm (BPA) naturally which can immediately compute the crucial canonical offsets for whole (coupled) systems with block-triangular form. We illustrate these algorithms by some examples, and preliminary numerical experiments show that the time complexity of BPA can be reduced by at least O(ℓ) compared to the MPA, which is mainly consistent with the results of our analysis.

scholarly journals Index Reduction for Differential-algebraic Equations with Mixed Matrices

2019 ◽  
Vol 66 (5) ◽  
pp. 1-34 ◽  
Author(s):  
Satoru Iwata ◽  
Taihei Oki ◽  
Mizuyo Takamatsu
Keyword(s):  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Index Reduction
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