Extending the Modified Inertia Representation to Constrained Rigid Multibody Systems

2019 ◽  
Vol 87 (1) ◽  
Author(s):  
X. M. Xu ◽  
J. H. Luo ◽  
Z. G. Wu
Keyword(s):  
Kinetic Energy ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Constrained System ◽  
Energy Error ◽  
Numerical Performance ◽  
Rigid Multibody Systems ◽  
Rigid Multibody

Abstract The inertia representation of a constrained system includes the formulation of the kinetic energy and its corresponding mass matrix, given the coordinates of the system. The way to find a proper inertia representation achieving better numerical performance is largely unexplored. This paper extends the modified inertia representation (MIR) to the constrained rigid multibody systems. By using the orthogonal projection, we show the possibility to derive the MIR for many types of non-minimal coordinates. We present examples of the derivation of the MIR for both planar and spatial rigid body systems. Error estimation shows that the MIR is different from the traditional inertia representation (TIR) in that its parameter γ can be used to reduce the kinetic energy error. With preconditioned γ, numerical results show that the MIR consistently presents significantly higher numerical accuracy and faster convergence speed than the TIR for the given variational integrator. The idea of using different inertia representations to improve the numerical performance may go beyond constrained rigid multibody systems to other systems governed by differential algebraic equations.

Natural Coordinates in the Optimal Control of Multibody Systems

2011 ◽  
Author(s):  
Peter Betsch ◽  
Ralf Siebert ◽  
Nicolas Sa¨nger
Keyword(s):  
Optimal Control ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Open Loop ◽  
Algebraic Equations ◽  
Natural Coordinates ◽  
Joint Forces ◽  
New Energy

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Secondly, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

Natural Coordinates in the Optimal Control of Multibody Systems

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
Peter Betsch ◽  
Ralf Siebert ◽  
Nicolas Sänger
Keyword(s):  
Optimal Control ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Open Loop ◽  
Algebraic Equations ◽  
Natural Coordinates ◽  
Joint Forces ◽  
New Energy

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

Improved Differential Nullspace Matrix Method for the Dynamic Analysis of Multibody Systems

2018 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Computational Time ◽  
Algebraic Equations ◽  
Accurate Analysis ◽  
Dynamics Of Multibody Systems ◽  
General Dynamics ◽  
Novel Method ◽  
Rigid Multibody Systems

This paper presents a novel method for motion analysis of rigid multibody systems. In general, dynamics of multibody systems is described by differential algebraic equations with Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called nullspace method and Maggi’s method eliminate the Lagrange multipliers by using the nullspace matrix for the constraint Jacobian. In a previous report, the author presented a method in which the nullspace matrix is obtained by solving a differential equation together with the equation of motion of the system. In that method QR decomposition is used. In this report, reduction in computational time with the LU decomposition is attempted. In addition, treatment of singular configurations for accurate analysis is presented. Validity of the presented method is confirmed via numerical 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.

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