A Non-Redundant Formulation for the Dynamics Simulation of Multibody Systems in Terms of Unit Dual Quaternions

2016 ◽  
Author(s):  
Andreas Müller ◽  
Zdravko Terze ◽  
Viktor Pandza
Keyword(s):  
Rigid Body ◽  
Time Derivative ◽  
Natural Extension ◽  
Differential Algebraic Equations ◽  
Dynamics Simulation ◽  
Integration Scheme ◽  
Integration Step ◽  
Algebraic Equations ◽  
Dual Quaternions ◽  
Rigid Body Motions

Quaternions are favorable parameters to describe spatial rotations of rigid bodies since they give rise to simple equations governing the kinematics and attitude dynamics in terms of simple algebraic equations. Dual quaternions are the natural extension to rigid body motions. They provide a singularity-free purely algebraic parameterization of rigid body motions, and thus serve as global parameters within the so-called absolute coordinate formulation of MBS. This attractive feature is owed to the inherent redundancy of these parameters since they must satisfy two quadratic conditions (unit condition and Plcker condition). Formulating the MBS kinematics in terms of dual quaternions leads to a system of differential-algebraic equations (DAE) with index 3. This is commonly transformed to an index 1 DAE system by replacing the algebraic constraints with their time derivative. This leads to the well-known problem of constraint violation. A brute force method, enforcing the unit constraint of quaternions, is to normalize them after each integration step. Clearly this correction affects the overall solution and the dynamic consistency. Moreover, for unit dual quaternions the two conditions cannot simply be enforced in such a way. In this paper a non-redundant formulation of the motion equations in terms of dual quaternions is presented. The dual quaternion constraints are avoided by introducing a local canonical parameterization. The key to this formulation is to treat dual unit quaternions as Lie group. The formulation can be solved with any standard integration scheme. Examples are reported displaying the excellent performance of this formulation regarding the constraint satisfaction as well as the solution accuracy.

scholarly journals Estimation and Global Stability Analysis of PDAEs with Singular Time Derivative Matrix and Wetland Conservation Application

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Yushan Jiang ◽  
Qingling Zhang
Keyword(s):  
Global Stability ◽  
Singular Systems ◽  
Time Derivative ◽  
Estimation Method ◽  
Differential Algebraic Equations ◽  
Wetland Conservation ◽  
Descriptor System ◽  
Algebraic Equations ◽  
Energy Estimation ◽  
Singular Time

Some partial differential algebraic equations (PDAEs) system with singular time derivative matrices is analyzed. First, by PDE spectrum theory, this system is formulated as infinite-dimensional singular systems. Second, the state space and its properties of the system are built according to descriptor system theory. Third, the admissible property of the PDAEs is given via LMIs. Finally, the developed energy estimation method is proposed to investigate the global stability of PDAEs. The proposed approach is evaluated by an application in numerical simulations on some wetland conservation system with social behavior.

Synthesis of Six-Bar Timed Curve Generators of Stephenson-Type Using Random Monodromy Loops

2020 ◽  
Vol 13 (1) ◽  
Author(s):  
Aravind Baskar ◽  
Mark Plecnik
Keyword(s):  
Rigid Body ◽  
Complete Solution ◽  
Numerical Algebraic Geometry ◽  
Algebraic Equations ◽  
Solution Sets ◽  
Root Finding ◽  
Rigid Body Motions ◽  
Joint Variable ◽  
Four Bar Linkage ◽  
Parameter Continuation

Abstract Synthesis of rigid-body mechanisms has traditionally been motivated by the design for kinematic requirements such as rigid-body motions, paths, or functions. A blend of the latter two leads to timed curve synthesis, the goal of which is to produce a path coordinated to the input of a joint variable. This approach has utility for altering the transmission of forces and velocities from an input joint onto an output point path. The design of timed curve generators can be accomplished by setting up a square system of algebraic equations and obtaining all isolated solutions. For a four-bar linkage, obtaining these solutions is routine. The situation becomes much more complicated for the six-bar linkages, but the range of possible output motions is more diverse. The computation of nearly complete solution sets for these six-bar design equations has been facilitated by recent root finding techniques belonging to the field of numerical algebraic geometry. In particular, we implement a method that uses random monodromy loops. In this work, we report these solution sets to all relevant six-bars of the Stephenson topology. The computed solution sets to these generic problems represent a design library, which can be used in a parameter continuation step to design linkages for different subsequent requirements.

scholarly journals L-Stable Method for Differential-Algebraic Equations of Multibody System Dynamics

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Bowen Li ◽  
Jieyu Ding ◽  
Yanan Li
Keyword(s):  
System Dynamics ◽  
Multibody System ◽  
Differential Algebraic Equations ◽  
Multibody System Dynamics ◽  
Dynamics Simulation ◽  
Time Interval ◽  
Algebraic Equations ◽  
Stable Method ◽  
Long Term Conditions ◽  
Equidistant Nodes

An L-stable method over time intervals for differential-algebraic equations (DAEs) of multibody system dynamics is presented in this paper. The solution format is established based on equidistant nodes and nonequidistant nodes such as Chebyshev nodes and Legendre nodes. Based on Ehle’s theorem and conjecture, the unknown matrix and vector in the L-stable solution formula are obtained by comparison with Pade approximation. Newton iteration method is used during the solution process. Taking the planar two-link manipulator system as an example, the results of L-stable method presented are compared for different number of nodes in the time interval and the step size in the simulation, and also compared with the classic Runge-Kutta method, A-stable method, Radau IA, Radau IIA, and Lobatto IIIC methods. The results show that the method has the advantages of good stability and high precision and is suitable for multibody system dynamics simulation under long-term conditions.

On the Numerical Scheme to Solve a Realistic Chemical Vapor Infiltration Reactor Model

2007 ◽  
Author(s):  
John K. Kamel ◽  
Samuel Paolucci
Keyword(s):  
Mathematical Model ◽  
Numerical Integration ◽  
Chemical Vapor ◽  
Differential Algebraic Equations ◽  
Chemical Vapor Infiltration ◽  
Integration Scheme ◽  
Algebraic Equations ◽  
Splitting Algorithm ◽  
Reactor Model ◽  
Vapor Infiltration

We describe the general mathematical model as well as the numerical integration procedure arising in modeling a realistic chemical vapor infiltration process. The numerical solution of the model ultimately leads to the solution of a large system of stiff differential algebraic equations. An operator splitting algorithm is employed to overcome the stiffness associated with chemical reactions, whereas a projection method is employed to overcome the difficulty arising from having to solve a large coupled system for the velocity and pressure fields. The resulting mathematical model and the numerical integration scheme are used to explore temperature, velocity, and concentration fields inside a chemical vapor infiltration reactor used in the manufacturing of aircraft brakes.

Implicit Integration of the Equations of Multibody Dynamics in Descriptor Form

1997 ◽  
Author(s):  
Edward J. Haug ◽  
Mirela Iancu ◽  
Dan Negrut
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Kinetic Equations ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Integration Formulas ◽  
Multibody Dynamic ◽  
Constrained Equations

Abstract An implicit numerical integration approach, based on generalized coordinate partitioning of the descriptor form of the differential-algebraic equations of motion of multibody dynamics, is presented. This approach is illustrated for simulation of stiff mechanical systems using the well known Newmark integration method from structural dynamics. Second order Newmark integration formulas are used to define independent generalized coordinates and their first time derivative as functions of independent accelerations. The latter are determined as the solution of discretized equations obtained using the descriptor form of the equations of motion. Dependent variables in the formulation, including Lagrange multipliers, are determined to satisfy all the kinematic and kinetic equations of multibody dynamics. The approach is illustrated by solving the constrained equations of motion for mechanical systems that exhibit stiff behavior. Results show that the approach is robust and has the capability to integrate differential-algebraic equations of motion for stiff multibody dynamic systems.

Coordinate Mappings for Rigid Body Motions

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Equations Of Motion ◽  
Time Integration ◽  
Exponential Mapping ◽  
Group Setting ◽  
Dual Quaternions ◽  
Integration Schemes ◽  
Rigid Body Motions ◽  
Lie Group Integration

Geometric methods have become increasingly accepted in computational multibody system (MBS) dynamics. This includes the kinematic and dynamic modeling as well as the time integration of the equations of motion. In particular, the observation that rigid body motions form a Lie group motivated the application of Lie group integration schemes, such as the Munthe-Kaas method. Also established vector space integration schemes tailored for structural and MBS dynamics were adopted to the Lie group setting, such as the generalized α integration method. Common to all is the use of coordinate mappings on the Lie group SE(3) of Euclidean motions. In terms of canonical coordinates (screw coordinates), this is the exponential mapping. Rigid body velocities (twists) are determined by its right-trivialized differential, denoted dexp. These concepts have, however, not yet been discussed in compact and concise form, which is the contribution of this paper with particular focus on the computational aspects. Rigid body motions can also be represented by dual quaternions, that form the Lie group Sp̂(1), and the corresponding dynamics formulations have recently found a renewed attention. The relevant coordinate mappings for dual quaternions are presented and related to the SE(3) representation. This relation gives rise to a novel closed form of the dexp mapping on SE(3). In addition to the canonical parameterization via the exponential mapping, the noncanonical parameterization via the Cayley mapping is presented.

scholarly journals Maximal Regularity for Non-autonomous Evolutionary Equations

2021 ◽  
Vol 93 (3) ◽  
Author(s):  
Sascha Trostorff ◽  
Marcus Waurick
Keyword(s):  
Differential Equations ◽  
Maximal Regularity ◽  
Normal Operator ◽  
Time Derivative ◽  
Parabolic Type ◽  
Regularity Result ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Evolutionary Equations

AbstractWe discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

scholarly journals Modeling Of A Reactive Distillation Column: Methyl Tertiary Butyl Ether (Mtbe) Simulation Studies

1970 ◽  
Vol 4 (2) ◽  
Author(s):  
Ismail Mohd Saaid Abdul Rahman Mohamed and Subhash Bhatia
Keyword(s):  
Distillation Column ◽  
Reactive Distillation ◽  
Relaxation Method ◽  
Differential Algebraic Equations ◽  
Operating Conditions ◽  
Integration Scheme ◽  
Algebraic Equations ◽  
Butyl Ether ◽  
Tertiary Butyl ◽  
Proposed Model

A process simulation stage-wise reactive distillation column model formulated from equilibrium stage theory was developed. The algorithm for solving mathematical model represented by sets of differential-algebraic equations was based on relaxation method. Numerical integration scheme based on backward differentiation formula was selected for solving the stiffness of differential-algebraic equations. Simulations were performed on a personal computer (PC Pentium processor) through a developed computer program using FORTRAN90 programming language. The proposed model was validated by comparing the simulated results with the published simulation results and with the pilot plant data from the literature. The model was capable of predicting high isobutene conversion for heterogeneous system, as desirable in industrial MTBE production process. The comparisons on temperature profiles, liquid composition profile and operating conditions of reactive distillation column also showed promising results. Therefore the proposed model can be used as a tool for the development and simulation of reactive distillation column.Keywords: Modeling, simulation, reactive distillation, relaxation method, equilibrium stage, heterogeneous, MTBE

Dynamic Modeling and Simulation of Flexible Robots With Prismatic Joints

1990 ◽  
Vol 112 (3) ◽  
pp. 307-314 ◽  
Author(s):  
Ye-Chen Pan ◽  
R. A. Scott ◽  
A. Galip Ulsoy
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Numerical Procedure ◽  
Differential Algebraic Equations ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Integration Scheme ◽  
System Response ◽  
Constraint Equations ◽  
Prismatic Joints

A dynamic model for flexible manipulators with prismatic joints is presented in Part I of this study. Floating frames following a nominal rigid body motion are introduced to describe the kinematics of the flexible links. A Lagrangian approach is used in deriving the equations of motion. The work done by the rigid body axial force through the axial shortening of the link due to transverse deformations is included in the Lagrangian function. Kinematic constraint equations are used to describe the compatibility conditions associated with revolute joints and prismatic joints, and incorporated into the equations of motion by Lagrange multipliers. The small displacements due to the flexibility of the links are then discretized by a displacement based finite element method. Equations of motion are derived for the cases of prescribed rigid body motion as well as prescribed joint torques/forces through application of Lagrange’s equations. The equations of motion and the constraint equations result in a set of differential algebraic equations. A numerical procedure combining a constraint stabilization method and a Newmark direct integration scheme is then applied to obtain the system response. An example, previously treated in the literature, is presented to validate the modeling and solution methods used in this study.

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