Simulation and feed-forward control of a flexible parallel manipulator

The importance of lightweight constructions are steady increasing since they promise a low energy consumption together with higher movement speeds. However these demand modern, model-based feed-forward control designs. Especially the undesired vibrations due to the reduced overall stiffness of such manipulators have to be taken into account. A convenient way to model the dynamical behavior of systems that perform large, nonlinear motions superposed with small, elastic deformations is the floating frame of reference approach in a flexible multibody system. The application of the Newton-Euler-Formalism together with D'Alembert's principle to parallel manipulators results in a set of differential-algebraic equations. Therefore, the consideration of the trajectory tracking problem with so-called servo constraints appears to be promising. In case of a non-flat system, the arising set of differential-algebraic equations, which consists of the system dynamics, the holonomic loop closing constraint equations and the servo constraints embodies nontrivial dynamics. With an oblique projection, the embedded set of ordinary differential equations describing the internal dynamics can be obtained. The stability properties of these dynamics determines the complexity of the feed-forward control design, as two-point boundary value problems might have to be solved. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparing exact inversion and singular perturbation approaches for a serial flexible manipulator

This paper reviews the singular perturbation control theory in the context of flexible multibody systems. The theory is motivated and explained by a simple, but sufficiently complex model. It is explained that based on the singular perturbed model an end-effector trajectory tracking can be achieved by an integral manifold controller. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of bounded feed-forward control for underactuated multibody systems using nonlinear optimization

Underactuation occurs, when only some generalized coordinates have a control input. For end-effector trajectory tracking a combined feed-forward and feedback control is often a suitable approach. Feed-forward control design based on an inverse model for underactuated multibody systems is presented. The starting point is the transformation of the multibody system into a nonlinear input-output normal-form. The inverse model follows from this and consists of chains of differentiators, driven internal dynamics and an algebraic part. Especially when using the end-effector as system output the internal dynamics is often unbounded. In order to obtain a viable feed-forward control, a bounded solution must be determined. For this task the internal dynamics is solved as a nonlinear optimization problem. Thereby, the coordinates of the internal dynamics define the objective function which is minimized. The equation of the internal dynamics must be fulfilled at each point of a discrete time grid. In addition continuity of the solution is achieved by adding as equality constraint an integration formula, e.g. trapezoidal rule. The optimization problem is then solved by a SQP-method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact Modeling in Multibody Systems with Elastic Bodies in High-Frequency Applications

This paper discusses the refinement of multibody models by integration of flexible bodies and by considering nonlinearities from contacts. It presents common approaches for contact modeling in multibody simulations and strategies to include flexible bodies. A contact model is implemented in the elastic multibody model. Experimental results show that significant effects of system dynamics can be modeled by use of a multibody model including elastic bodies and contacts. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamical Modeling and Flatness Based Control of a Belt Drive System

Belt driven systems are part of many industrial applications, like computerized numerical control (CNC) machines in particular cutting machines and 3D-printers. In this paper the dynamical modeling and a flatness based controller design for belt driven systems are proposed. Due to the special kinematics, the stiffness of the belt is nonlinear, leading to nonlinear equations of motion. By neglecting some minor dynamical effects, the resulting system simplifies to a differentially flat one. This allows to calculate nominal feed-forward control torques by using the flat output of the system. To stabilize the error dynamics, an additional PD control law is introduced. The proposed method is compared with a controller, where elastic deflections for the feed forward part are neglected and elastic deformations are compensated by modifying the desired trajectories in a model-based manner. The tracking performance of both methods is evaluated in certain simulations and experiments. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Rotationless formulation for large deformations in flexible multibody dynamics

Especially for specific applications, such as contact problems, computer methods for flexible multibody dynamics that are able to treat large deformation phenomena are important. Classical formalisms for multibody dynamics are based on rigid bodies. Their extension to flexible multibody systems is typically restricted to linear elastic material behavior whereas large deformation phenomena are formulated in the framework of the nonlinear finite element method. In the talk we address computer methods that can handle large deformations in the context of multibody systems. In particular, the link between nonlinear continuum mechanics and multibody systems is facilitated by a specific formulation of rigid body dynamics [1]. It makes possible the incorporation of state-of-the-art computer methods for large deformation problems. In the talk we focus on the treatment of large deformation contact whithin flexible multibody dynamics [2]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of a running gear with IRWs on curved tracks for a robust control development

As a major task of the DLR-internal project “Next Generation Train”, robust state feedback control with gain scheduling was sucessfully applied to guide the experimental running gear with independently rotating wheels (IRWs) at a 1:5 scaled roller rig, see [1]. However, the adaptation of the control structure to the 1:1 multibody model requires to additionally consider the properties of curved tracks. For that reason, an analytical model of a running gear with IRWs is deduced using Euler-Lagrange-equations and taking superelevation and track curvature into account. Furthermore, the complexity of the system is reduced to allow for a robust feedback control synthesis and feed-forward control including model inversion. Finally, the model is discussed and a first approach for a feed-forward control in transition curves is shown. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled differential algebraic equations in the simulation of flexible multibody systems with hydrodynamic force elements

The mathematical modelling of elastohydrodynamic bearings in combustion engines leads to a coupled system of (partial) differential algebraic equations, which is represented by a flexible multibody system model of the engine including crankshaft and bearing and by the Reynolds equation that describes the non-linear effects in the fluid film. The hydrodynamic forces depend strongly on the position and the elastic deformation of crankshaft and bearing shell. We discuss the influence of the spatial discretization on accuracy and numerical effort. Since a fine spatial discretization substantially slows down the numerical solution, we propose a semi-analytical method based on singular perturbation theory to speed-up time integration. Numerical tests for a simplified benchmark problem illustrate the advantages of this approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlinear finite element framework for flexible multibody dynamics

We present a uniform treatment of rigid body dynamics and nonlinear structural dynamics. The advocated approach is based on a rotationless formulation of rigid bodies, nonlinear beams and shells. In this connection, the specific kinematic assumptions are taken into account by the explicit incorporation of holonomic constraints. This approach facilitates the straightforward extension to flexible multibody dynamics by including additional constraints due to the interconnection of rigid and flexible bodies. We further address the design of energy-momentum schemes for the stable numerical integration of the underlying finite-dimensional Hamiltonian systems. To demonstrate the superior numerical performance of the proposed methodology, the numerical examples deals with a multibody system containing both rigid and flexible bodies undergoing large deformations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear control of a variable displacement axial piston pump

In this contribution a mathematical model of a variable displacement axial piston pump controlled by a solenoid valve is derived. For the purpose of a controller design the mathematical model is simplified using singular perturbation arguments. The goal of the controller design is to track prescribed trajectories in the load pressure for arbitrary unknown load conditions. The control concept being proposed comprises a feedforward controller and a nonlinear backstepping controller combined with a load estimator for the trajectory error system. The feasibility of this control concept is shown by means of measurements on a test-stand. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification and Self Tuning Control for Active Magnetic Bearing Systems: Levitation of Unknown Rotors

The present paper deals with the problem of levitating rotors with unknown characteristics by means of active magnetic bearings whose properties are known. This problem is of interest in a technical setting to shorten the development time of AMB systems, in particular for controller design. Theoretical interest arises from the fact that several issues in the area of identification and self tuning control are addressed for an unstable system. Our aim is to identify the flexible rotor including gyroscopic effects and to automatically design a robustly stabilizing controller for this system that can be used for running the system under regular operating conditions. To this end, a rigid body model of the rotor is identified based on measured step responses from the plant. Then, the bearings are adjusted to have very low stiffness, and a controller with steep roll‐off is designed in order to avoid excitation of the unknown flexible modes of the system. Once the rotor is floating, the identification algorithm from [1] is applied to obtain information on the flexible modes of the system. Based on this extended model, a robust controller allowing for slow rotation of the rotor is designed. With the rotor rotating at a moderate speed, the frequency response functions are measured, and based on these measurements, the gyroscopic matrixof the system is identified, completing the system model and allowing for design of the desired controller for normal operation. The present contribution focusses on identification of the rigid body model of the flexible rotor.     

多柔体系统动力学符号演算的研究

多柔体系统动力学涉及繁杂的数学运算,对系统动力学方程和系数矩阵进行手工推导是低效而不可靠的．采用数值化方法进行系统动力学分析将包括许多虚运算和重复计算,使得计算效率和精度受到了影响．同时对约束和外力的变化适应性较差．本文在通用计算机符号演算软件ＭＡＴＨＥＭＡＴＩＣＡ环境下,研究了多柔体系统动力学的计算机符号演算方法,提出将多柔体系统动力学建模和数值分析的问题在ＭＡＴＨＥＭＡＴＩＣＡ环境中一体化解决,并通过实践说明这一方法是可行的和有效的．     

A flexible approach to the simulation of hybrid multibody systems

The present works deals with the incorporation of both flexible beam and shell structures into the realm of flexible multibody dynamics. Geometrically exact beam formulations based on classical Simo-Reissner kinematics are suitable for modelling beam-type flexible components in the context of finite-deformation multibody dynamics. So geometrically exact shell formulations are based on Reissner-Mindlin kinematics. In [2], a flexible framework for dealing with flexible structural elements in a multibody context is described. A specific isoparametric finite element discretization of a shell formulation leads to semi-discrete equations of motions assuming the form of differential-algebraic equations (DAEs). A compatible isoparametric formulation of beams has already been developed in [1]. The uniform DAE framework makes possible the incorporation of alternative finite element formulations. In addition to that, various time-stepping schemes such as energy-momentum methods or variational integrators can be applied. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of model reduction techniques on the impact force calculation of two flexible bodies

One important issue for the simulation of flexible multibody systems is the reduction of the flexible body's degrees of freedom. For the reduction process finite element data and user inputs are necessary. The model reduction program for elastic multibody systems MOREMBS, which is developed at the ITM, has an easy-to-use interface and the data can be gained from the programs ABAQUS or ANSYS. In this work, the simulation of a fuel injection process is investigated with MOREMBS. We focus on the interaction between valve and armature. These two bodies impact in every injection circle. The impacting bodies are modeled as flexible and the contact force is calculated by a penalty approach. One essential part of this work is the investigation of the influence of different model reduction techniques on the impact force calculation of the flexible multibody system. The main reduction techniques modal reduction, Krylov-subspace based and Gramian matrix based techniques are compared. The results achieved with modal reduction, the state of the art reduction method, are not acceptable here. Krylov-subspace based techniques are especially well-suited for large sparse systems but are not error controlled. However, by choosing appropriate moment-matching properties the impact force calculation is nearly as good as with a full finite element model. The Gramian matrix based reduction techniques can be fully automated and are error controlled but require high computational effort. Hence, appropriate approximation schemes have to be used for them. With Gramian matrix based methods we can even further reduce the size of the subsystems compared to Krylov-subspace based methods and still have an impact force calculation nearly as good as with finite element results, but we gain a simulation speedup by the factor 4000. In addition, a parameter study of the parameters involved in the model reduction process is presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Real-Time Simulation of Elastic Multibody Systems with Application in Vehicle Dynamics

Real-time simulation models are widely used for vehicle development, usually built up as rigid multibody systems. However, since lightweight structures are commonly used, body deformation is no longer negligible and rigid multibody simulations may be inaccurate. This work presents a real-time capable full vehicle model with a flexible car body, derived from a finite element model, whose performance has been improved by model order reduction. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enhancing the generality of fuzzy relational models for control

A promising area of research in fuzzy control is the model-based fuzzy controller. At the heart of this approach is a fuzzy relational model of the process to be controlled. Since this model is identified directly from process input-output data it is likely that ‘holes’ will be present in the identified relational model. These holes are real problems when the model is incorporated into a model-based controller since the model will be unable to make any predictions whatsoever if the system drifts into an unknown region. The present work deals with the completeness of the fuzzy relational model which forms the core of the controller. This work proposes a scheme of post-processing to ‘fiil in’ the fuzzy relational model once it has been built and thereby improve its applicability for on-line control. A comparative study of the post-processed model and conventional relational model is presented for Box-Jenkins data identification system and a real-time, highly non-linear application of pH control identification.