Approximate End-Effector Tracking Control of Flexible Multibody Systems Using Singular Perturbations

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Thomas Gorius ◽  
Robert Seifried ◽  
Peter Eberhard
Keyword(s):  
Tracking Control ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Multibody System ◽  
Feedforward Control ◽  
Flexible Multibody Systems ◽  
Flexible Manipulator ◽  
End Effector ◽  
Nonminimum Phase ◽  
Flexible Multibody

In many cases, the design of a tracking controller can be significantly simplified by the use of a 2-degrees of freedom (DOF) control structure, including a feedforward control (i.e., the inversion of the nominal system dynamics). Unfortunately, the computation of this feedforward control is not easy if the system is nonminimum-phase. Important examples of such systems are flexible multibody systems, such as lightweight manipulators. There are several approaches to the numerical computation of the exact inversion of a flexible multibody system. In this paper, the singularly perturbed form of such mechanical systems is used to give a semianalytic solution to the tracking control design. The control makes the end-effector to even though not exactly, but approximately track a certain trajectory. Thereby, the control signal is computed as a series expansion in terms of an overall flexibility of the bodies of the multibody system. Due to the use of symbolic computations, the main calculations are independent of given parameters (e.g., the desired trajectories), such that the feedforward control can be calculated online. The effectiveness of this approach is shown by the simulation of a two-link flexible manipulator.

System-Level Modal Representation of Flexible Multibody Systems

2009 ◽  
Author(s):  
Gert H. K. Heirman ◽  
Wim Desmet
Keyword(s):  
Model Reduction ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Flexible Multibody Systems ◽  
System Level ◽  
Algebraic Equations ◽  
Flexible Multibody ◽  
Simulation Results ◽  
Model Equations ◽  
Model Reduction Technique

The presence of both differential and algebraic equations in the model equations, as well as the number of degrees of freedom needed to accurately represent flexibility, prohibit fast simulation of flexible multibody systems (e.g. real-time). In this research, Global Modal Parametrization, a model reduction technique for flexible multibody systems is further developed to speed up simulation of flexible multibody systems. The reduction of the model is achieved by projection on a curvilinear subspace instead of a fixed vector space, requiring significantly less degrees of freedom to represent the system dynamics with the same level of accuracy. The complexity of simulation of the reduced model equations is estimated. In a numerical experiment, simulation results for the original model equations are compared with simulation results for the model equations obtained after model reduction, showing a good match. The dominant sources of error of the proposed methodology are illustrated and explained.

Modeling of Revolute Joints in Topology Optimization of Flexible Multibody Systems

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Ali Moghadasi ◽  
Alexander Held ◽  
Robert Seifried
Keyword(s):  
Topology Optimization ◽  
Multibody Systems ◽  
Multibody System ◽  
Order Reduction ◽  
Flexible Multibody Systems ◽  
Hertzian Contact ◽  
Revolute Joints ◽  
Nonlinear Finite Elements ◽  
Flexible Multibody ◽  
Hertzian Contact Law

In recent years, topology optimization has been used for optimizing members of flexible multibody systems to enhance their performance. Here, an extension to existing topology optimization schemes for flexible multibody systems is presented in which a more accurate model of revolute joints and bearing domains is included. This extension is of special interest since a connection between flexible members in a multibody system using revolute joints is seen in many applications. Moreover, the modeling accuracy of the bearing area is shown to be influential on the shape of the optimized structure. In this work, the flexible bodies are incorporated in the multibody simulation using the floating frame of reference formulation, and their elastic deformation is approximated using global shape functions calculated in the model order reduction analysis. The modeling of revolute joints using Hertzian contact law is incorporated in this framework by introducing a corrector load in the bearing model. Furthermore, an application example of a flexible multibody system with revolute joints is optimized for minimum value of compliance, and a comparative study of the optimization result is performed with an equivalent system which is modeled with nonlinear finite elements.

A Stable Inversion Method for Feedforward Control of Constrained Flexible Multibody Systems

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Olivier Brüls ◽  
Guaraci Jr. Bastos ◽  
Robert Seifried
Keyword(s):  
Multibody Systems ◽  
Inverse Model ◽  
Flexible Multibody Systems ◽  
Initial Time ◽  
Multiple Shooting ◽  
Inversion Method ◽  
Nonminimum Phase ◽  
Nonminimum Phase Systems ◽  
Flexible Multibody ◽  
Phase Systems

The inverse dynamics of flexible multibody systems is formulated as a two-point boundary value problem for an index-3 differential-algebraic equation (DAE). This DAE represents the equation of motion with kinematic and trajectory constraints. For so-called nonminimum phase systems, the remaining dynamics of the inverse model is unstable. Therefore, boundary conditions are imposed not only at the initial time but also at the final time in order to obtain a bounded solution of the inverse model. The numerical solution strategy is based on a reformulation of the DAE in index-2 form and a multiple shooting algorithm, which is known for its robustness and its ability to solve unstable problems. The paper also describes the time integration and sensitivity analysis methods that are used in each shooting phase. The proposed approach does not require a reformulation of the problem in input-output normal form, which is known from nonlinear control theory. It can deal with serial and parallel kinematic topology, minimum phase and nonminimum phase systems, and rigid and flexible mechanisms.

Flexible Multibody Systems With Large Deformations Using Absolute Nodal Coordinates for Isoparametric Solid Brick Elements

2003 ◽  
Author(s):  
Lars Ku¨bler ◽  
Peter Eberhard ◽  
Johannes Geisler
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Large Deformations ◽  
Flexible Multibody Systems ◽  
Body Motion ◽  
Absolute Nodal Coordinates ◽  
Shell Elements ◽  
Flexible Multibody ◽  
Absolute Coordinates ◽  
Solid Bodies

In this paper a formulation for flexible Multibody Systems (MBS) is proposed where flexible bodies are described using absolute coordinates for isoparametric brick elements. The use of absolute coordinates allows for large deformations and provides an accurate description of rigid body motion and inertia in the case of large rotations without additional considerations. Further, constant mass matrices are obtained for isoparametric elements. Brick elements are important, e. g. if general solid bodies with low stiffness, i. e. not negligible large deformations, are part of the MBS and cannot be modeled using beam, plate, or shell elements. Since only nodal translational degrees of freedom are used for brick elements additional questions arise. For example, imposing joint constraints for relative rotations between two bodies requires a nodal reference frame at connection points. An approach is proposed to define such a reference system utilizing displacement information of three finite element nodes.

Reduced Isogeometric Analysis Models for Impact Simulations

2021 ◽  
Author(s):  
Tobias Rückwald ◽  
Alexander Held ◽  
Robert Seifried
Keyword(s):  
Isogeometric Analysis ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Computation Time ◽  
Flexible Multibody Systems ◽  
Shape Functions ◽  
Elastic Rod ◽  
Flexible Multibody ◽  
Impact Simulations ◽  
The Impact

Abstract Detailed impact simulations in flexible multibody systems are usually based on isoparametric finite element models. For modeling the dynamics of an impact, a precise representation of the geometry is essential. However, isoparametric finite elements involve the discretization of the geometry. This work tests the isogeometric analysis (IGA) as an alternative approach in flexible multibody systems. The IGA enables the exact representation of the geometry by using non-uniform rational basis splines (NURBS) as element shape functions. In the context of an efficient impact simulation a model reduction and a possible inclusion of the floating frame of reference formulation is beneficial. The degrees of freedom of the flexible bodies are reduced using component mode synthesis to save computation time in the multibody simulation. For the precise description of deformations and stresses in the contact area as well as elastodynamic effects, a large number of global shape functions is required. As testing examples, the impact of two elastic spheres and a multibody multicontact problem including wave propagation in a long elastic rod are simulated and compared to reference solutions.

Aspects of Symbolic Formulations in Flexible Multibody Systems

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Markus Burkhardt ◽  
Robert Seifried ◽  
Peter Eberhard
Keyword(s):  
Multibody Systems ◽  
Multibody System ◽  
Dynamic Properties ◽  
Equations Of Motion ◽  
Flexible Multibody Systems ◽  
Fem Model ◽  
Local Shape ◽  
Flexible Bodies ◽  
Elastic Bodies ◽  
Flexible Multibody

The symbolic modeling of flexible multibody systems is a challenging task. This is especially the case for complex-shaped elastic bodies, which are described by a numerical model, e.g., an FEM model. The kinematic and dynamic properties of the flexible body are in this case numerical and the elastic deformations are described with a certain number of local shape functions, which results in a large amount of data that have to be handled. Both attributes do not suggest the usage of symbolic tools to model a flexible multibody system. Nevertheless, there are several symbolic multibody codes that can treat flexible multibody systems in a very efficient way. In this paper, we present some of the modifications of the symbolic research code Neweul-M2 which are needed to support flexible bodies. On the basis of these modifications, the mentioned restrictions due to the numerical flexible bodies can be eliminated. Furthermore, it is possible to re-establish the symbolic character of the created equations of motion even in the presence of these solely numerical flexible bodies.

A Recursive Algorithm for Solving the Generalized Velocities From the Momenta of Flexible Multibody Systems

2008 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Numerical Solution ◽  
Multibody Systems ◽  
Multibody System ◽  
Recursive Algorithm ◽  
Mass Matrix ◽  
Direct Methods ◽  
Flexible Multibody Systems ◽  
Hamiltonian Form ◽  
Generalized Coordinates ◽  
Flexible Multibody

The computation of the generalized velocities from the generalized momenta of a multibody system is a part of the numerical solution of the dynamics equations when they are given in the Hamiltonian form. The states of these equations are the generalized coordinates and momenta, (q, p). The generalized velocity, q˙, is defined by q˙ = J−1p, where J is the system mass matrix. The effort in solving q˙ by direct methods is order(N3) where N is the number of bodies in the system. This paper presents an order(N) recursive algorithm to compute q˙ for flexible multibody systems.

Servo-Constraints for Control of Flexible Multibody Systems With Contact

2013 ◽  
Author(s):  
Robert Seifried ◽  
Markus Burkhardt
Keyword(s):  
Numerical Solution ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Inverse Model ◽  
Flexible Multibody Systems ◽  
Algebraic Equations ◽  
Internal Dynamics ◽  
End Effector ◽  
Flexible Multibody ◽  
Servo Constraints

This paper presents inversion based feedforward control design for flexible multibody systems with kinematic loops and end-effector contact. The inverse model provides for a given desired output trajectories, e.g. end-effector point and contact force, the required control inputs for exact output reproduction. A very appealing and efficient model inversion approach for such multibody systems is the use of so-called servo-constraints. These can be seen as an extension of classical mechanical constraints and yield a set of differential-algebraic equations. This allows an efficient numerical solution without burdensome symbolic manipulations. In addition, the use of servo-constraints allows the straight-forward treatment of flexible multibody systems with various topologies. The arising set of differential-algebraic equations describes the inverse model. The inverse model might be purely algebraic or include a dynamical part, which is called internal dynamics in nonlinear control theory. For its numerical solution it is advisable to transform the set of differential-algebraic equations to its underlying set of ordinary differential equations. The solution method for this internal dynamics depends then on its stability. For systems with unstable internal dynamics, as considered in this paper, a solution can be computed from a boundary-value problem. The efficiency of this approach is demonstrated for a flexible multibody system with a kinematic loop and a closed end-effector contact.

Monte Carlo Simulation of Flexible Multibody System with Uncertainties

2011 ◽  
Vol 55-57 ◽  
pp. 1382-1385
Author(s):  
Ting Pi ◽  
Yun Qing Zhang
Keyword(s):  
Monte Carlo ◽  
Multibody Systems ◽  
Multibody System ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Systems ◽  
Flexible Multibody System ◽  
System Behavior ◽  
Flexible Multibody ◽  
And Performance ◽  
Crank Mechanism

Practical mechanical systems often contain several flexible components and uncertain parameters which makes it hard to predict the system behavior and performance exactly. This research presents the uncertainty analysis of flexible multibody systems with random variables. Absolute nodal coordinate formulation (ANCF), which is different from the traditional finite element method, is employed to model the flexibility here. Monte Carlo method is successfully used to simulate flexible multibody systems of index-3. The method proposed is demonstrated by an example of flexible slider-crank mechanism.

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