Third-Order Differential-Algebraic Equations for Improved Integration of Multibody Dynamics

2017 ◽  
Author(s):  
H. J. Sommer
Keyword(s):  
Multibody Systems ◽  
Computation Time ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Planar Mechanisms ◽  
Third Order ◽  
Simulation Accuracy ◽  
Constrained Multibody Systems ◽  
Predictor Corrector ◽  
Reduce Computation Time

This paper introduces the concept of third-order differential-algebraic equations (DAE) for dynamics of constrained multibody systems. Third-order DAE provide jerk of components which can be integrated simultaneously with acceleration to provide improved simulation accuracy. A new Obreshkov predictor-corrector multistep integrator was developed to test this concept. Results from simulations of two planar mechanisms indicate that third-order DAE can reduce computation time by a factor of ten with equivalent accuracy compared to classical methods.

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.

Null Space Method of Differential Equation Type for Motion Analysis of Multibody Systems

2017 ◽  
Author(s):  
Keisuke Kamiya ◽  
Yusaku Yamashita
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Dependent Variables ◽  
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 constraint Jacobian. In previous reports, one of the authors presented methods which use the null space matrix. In the procedure to obtain the null space matrix, the inverse of a matrix whose regularity may not be always guaranteed. In this report, a new method is proposed in which the null space matrix is obtained by solving differential equations that can be always defined by using the QR decomposition, even if the constraints are redundant. Examples of numerical analysis are shown to validate the proposed method.

Control Constraint Realization for Multibody Systems

2009 ◽  
Author(s):  
Alessandro Fumagalli ◽  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Paolo Mantegazza
Keyword(s):  
Linear Systems ◽  
Multibody Systems ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Control Constraint ◽  
Practical Observation ◽  
Alternative Condition ◽  
Deformable Bodies

This paper discusses the problem of control constraint realization applied to generic under-actuated multibody systems. The conditions for the realization are presented. Focus is placed on the tangent realization of the control constraint. An alternative condition is formulated, based on the practical observation that Differential-Algebraic Equations (DAE) need to be integrated using implicit algorithms, thus naturally leading to the solution of the problem in form of matrix pencil. The analogy with the representation of linear systems in Laplace’s domain is also discussed. The formulation is applied to the solution of simple, yet illustrative problems, related to rigid and deformable bodies. Some implications of considering deformable continua are addressed.

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.

The Dynamic Analysis of Multibody Systems with Uncertain Parameters Using Interval Method

2012 ◽  
Vol 152-154 ◽  
pp. 1555-1561 ◽  
Author(s):  
Jing Lai Wu ◽  
Yun Qing Zhang
Keyword(s):  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Uncertain Parameters ◽  
Algebraic Equations ◽  
Scanning Method ◽  
Inclusion Function ◽  
Bounded Interval ◽  
Interval Variables ◽  
Multibody Dynamic ◽  
Computational Aspects

The theoretical and computational aspects of interval methodology based on Chebyshev polynomials for modeling multibody dynamic systems in the presence of parametric uncertainties are proposed, where the uncertain parameters are modeled by uncertain-but-bounded interval variables which only need the bounds of uncertain parameters, not necessarily knowing the probabilistic distribution. The Chebyshev inclusion function which employs the truncated Chevbyshev series expansion to approximate the original function is proposed. Based on Chebyshev inclusion function, the algorithm for solving the nonlinear equations with interval parameters is proposed. Combining the HHT-I3 method, this algorithm is used to calculate the multibody systems dynamic response which is governed by differential algebraic equations (DAEs). A numerical example that is a slider-crank with uncertain parameters is presented, which shows that the novel methodology can control the overestimation effectively and is computationally faster than the scanning method.

Integrated Design and Minimum Sensitivity Design of Controlled Multibody Systems

2005 ◽  
Author(s):  
Jason Carrigan ◽  
Atul Kelkar ◽  
Prakash Krishnaswami
Keyword(s):  
Design Process ◽  
Multibody Systems ◽  
System Analysis ◽  
Mechanical Design ◽  
Integrated Design ◽  
Solution Process ◽  
Differential Algebraic Equations ◽  
Element Formulation ◽  
Algebraic Equations ◽  
System Variables

The design methods that have been traditionally used for controlled mechanical systems suffer from three major drawbacks. First, the design process is generally sequential, with the mechanical design being done first and frozen before the control system design is done. Secondly, the design is usually tuned to improve performance only without worrying about the sensitivity of the system’s performance to small variations in the system parameters. Third, there is a lack of systematic guidance for traversing the design space and arriving at a high quality design. In this work, we propose a design approach that addresses all three of these concerns. This approach first extends the constrained multi-element formulation for multibody systems to include a generic controller model. This gives the basic capability to simulate controlled multibody systems in a general way by numerically solving a set of differential-algebraic equations (DAE’s). A direct differentiation technique is then applied to the unified mathematical model to obtain a set of DAE’s in the sensitivities of the system variables. This is then used to compute the sensitivity of any performance function of interest. The system analysis and sensitivity analysis are then treated as inputs to a suitable nonlinear programming problem (NLP). The NLP serves as a vehicle to unify mechanical system and control criteria in the design process, and to incorporate sensitivity considerations along with performance considerations. The NLP also provides the means for automating the solution process through the use of optimization algorithms. Two representative example, including an industrial problem, are solved using this method. The results clearly show that the methodology is feasible and leads to a vast improvement in the quality of the final design, whatever the design considerations may be.

A Novel Approach to Real-Time Flexible Multibody Simulation: Sub-System Global Modal Parameterization

2011 ◽  
Author(s):  
Frank Naets ◽  
Gert H. K. Heirman ◽  
Wim Desmet
Keyword(s):  
Model Reduction ◽  
Real Time ◽  
Relative Motion ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Flexible Multibody Systems ◽  
System Level ◽  
Algebraic Equations ◽  
Global Modal Parameterization ◽  
Flexible Multibody

This paper introduces a novel model reduction technique, namely Sub-System Global Modal Parameterization (SS-GMP), for real-time simulation of flexible multibody systems. In the past, other system-level model reduction techniques have been proposed for this purpose, but these were limited in applicability due to the large storage requirements for systems with many rigid degrees-of-freedom (DOFs). However, in the SS-GMP approach, the motion of a mechanism is split up into a global motion and a relative motion of the (sub-)system. The relative motion is then reduced according to the Global Modal Parameterization, which is a model reduction procedure suitable for closed chain flexible multibody systems. In combination with suitable explicit solvers, the SS-GMP approach enables (hard) real-time simulations due to the strong reduction in the number of DOFs and the conversion of a system of differential-algebraic equations into a system of ordinary differential equations. The proposed approach is validated numerically with a quarter-car model. This fully flexible mechanism is simulated faster than real-time on a regular PC with the SS-GMP approach while providing accurate results.

Globally Independent Coordinates for Real-Time Vehicle Simulation

1998 ◽  
Vol 122 (4) ◽  
pp. 575-582 ◽  
Author(s):  
Radu Serban ◽  
Edward J. Haug
Keyword(s):  
State Space ◽  
Real Time ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Vehicle Simulation ◽  
Generalized Coordinates ◽  
Dynamics Of Multibody Systems ◽  
Improved Performance ◽  
Control Forces

Models of the dynamics of multibody systems generally result in a set of differential-algebraic equations (DAE). State-space methods for solving the DAE of motion are based on reduction of the DAE to ordinary differential equations (ODE), by means of local parameterizations of the constraint manifold that must be often modified during a simulation. In this paper it is shown that, for vehicle multibody systems, generalized coordinates that are dual to suspension and/or control forces in the model are independent for the entire range of motion of the system. Therefore, these additional coordinates, together with Cartesian coordinates describing the position and orientation of the chassis, form a set of globally independent coordinates. In addition to the immediate advantage of avoiding the computationally expensive redefinition of local parameterization in a state-space formulation, the existence of globally independent coordinates leads to efficient algorithms for recovery of dependent generalized coordinates. A topology based approach to identify efficient computational sequences is presented. Numerical examples with realistic vehicle handling models demonstrate the improved performance of the proposed approach, relative to the conventional Cartesian coordinate formulation, yielding real-time for vehicle simulation. [S1050-0472(00)00404-9]

Sign in / Sign up

Export Citation Format

Share Document