A PID Type Constraint Stabilization Method for Numerical Integration of Multibody Systems

2011 ◽  
Vol 6 (4) ◽  
Author(s):  
Shih-Tin Lin ◽  
Ming-Wen Chen
Keyword(s):  
Numerical Integration ◽  
Pid Controller ◽  
Controller Design ◽  
Equations Of Motion ◽  
Constraint Equation ◽  
Correct Choice ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Stabilization Method ◽  
Constraint Stabilization

The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAEs). The numerical solution of the DAE systems solved using ordinary-differential equation (ODE) solvers may suffer from constraint drift phenomenon. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. Baumgarte’s method is a proportional-derivative (PD) type controller design. In this paper, an Iintegrator controller is included to form a proportional-integral-derivative (PID) controller so that the steady state error of the numerical integration can be reduced. Stability analysis methods in the digital control theory are used to find out the correct choice of the coefficients for the PID controller.

Stabilization Method for Numerical Integration of Multibody Mechanical Systems

1998 ◽  
Vol 120 (4) ◽  
pp. 565-572 ◽  
Author(s):  
Shih-Tin Lin ◽  
Ming-Chong Hong
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Integration Step ◽  
Algebraic Equations ◽  
Stabilization Method ◽  
Step Size ◽  
Integration Methods

The object of this study is to solve the stability problem for the numerical integration of constrained multibody mechanical systems. The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAE). In applying numerical integration methods to this equation, constrained equations and their first and second derivatives must be satisfied simultaneously. That is, the generalized coordinates and their derivatives are dependent. Direct integration methods do not consider this dependency and constraint violation occurs. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. The disadvantage of this method is that there is no reliable method for selecting the coefficients of the position and velocity terms. Improper selection of these coefficients can lead to erroneous results. In this study, stability analysis methods in digital control theory are used to solve this problem. Correct choice of the coefficients for the Adams method are found for both fixed and variable integration step size.

Stabilization of Baumgarte’s Method Using the Runge-Kutta Approach

2002 ◽  
Vol 124 (4) ◽  
pp. 633-641 ◽  
Author(s):  
Shih-Tin Lin ◽  
Jiann-Nan Huang
Keyword(s):  
Equations Of Motion ◽  
Constraint Equation ◽  
Correct Choice ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Runge Kutta Method ◽  
Dae Systems ◽  
Numerical Integration Methods ◽  
Runge Kutta ◽  
Selection Of

The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAE). The DAE systems cannot be solved using numerical integration methods that are commonly used for solving ordinary differential equations. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. The disadvantage of this method is that there is no reliable method for selecting the coefficients of the position and velocity terms. Improper selection of these coefficients can lead to erroneous results. In this study, stability analysis methods in digital control theory are used to solve this problem. Correct choice of the coefficients for the Runge-Kutta method is found.

A Geometric Derivation of the Equations of Motion for Mechanical Systems With Scleronomic Constraints

2015 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Elias Paraskevopoulos
Keyword(s):  
Numerical Methods ◽  
Lagrange Multipliers ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Main Idea ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Motion Constraints ◽  
Stiffness Properties

A new set of equations of motion is presented for a class of mechanical systems subjected to equality motion constraints. Specifically, the systems examined satisfy a set of holonomic and/or nonholonomic scleronomic constraints. The main idea is to consider the equations describing the action of the constraints as an integral part of the overall process leading to the equations of motion. The constraints are incorporated one by one, in a process analogous to that used for setting up the equations of motion. This proves to be equivalent to assigning appropriate inertia, damping and stiffness properties to each constraint equation and leads to a system of second order ordinary differential equations for both the coordinates and the Lagrange multipliers associated to the motion constraints automatically. This brings considerable advantages, avoiding problems related to systems of differential-algebraic equations or penalty formulations. Apart from its theoretical value, this set of equations is well-suited for developing new robust and accurate numerical methods.

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.

Analytical Modelling of the Dynamics of the Four-Bar Mechanism: A Comparison of Some Methods

2018 ◽  
Author(s):  
J. P. Meijaard ◽  
V. van der Wijk
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Force Balance ◽  
Dynamic Force ◽  
Algebraic Equations ◽  
Principle Of Virtual Work ◽  
Principal Vector ◽  
Reaction Forces ◽  
Principal Points

Some thoughts about different ways of formulating the equations of motion of a four-bar mechanism are communicated. Four analytic methods to derive the equations of motion are compared. In the first method, Lagrange’s equations in the traditional form are used, and in a second method, the principle of virtual work is used, which leads to equivalent equations. In the third method, the loop is opened, principal points and a principal vector linkage are introduced, and the equations are formulated in terms of these principal vectors, which leads, with the introduced reaction forces, to a system of differential-algebraic equations. In the fourth method, equivalent masses are introduced, which leads to a simpler system of principal points and principal vectors. By considering the links as pseudorigid bodies that can have a uniform planar dilatation, a compact form of the equations of motion is obtained. The conditions for dynamic force balance become almost trivial. Also the equations for the resulting reaction moment are considered for all four methods.

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.

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.

Time-Differentiable Null Space Method for Constrained Mechanical Systems (Application to a Rheonomic System)

2013 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Null Space ◽  
Constraint Equation ◽  
Coefficient Matrix ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Accurate Analysis ◽  
Constrained Mechanical Systems ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the coefficient matrix which appears in the constraint equation in velocity level. In a previous report, the author presented a method to obtain a time differentiable null space matrix for scleronomic systems, whose constraint does not depend on time explicitly. In this report, the method is generalized to rheonomic systems, whose constraint depends on time explicitly. Finally, the presented method is applied to four-bar linkages.

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