Keywords: elastic multibody systems, shape optimization, particle swarm optimization, control, manipulators, vibrations, dynamics.

Modern lightweight construction of machine tools or manipulators provide many advantages such as better mass to payload ratios and higher working speeds. However with the lower mass the stiffness decreases as well causing unintentional vibrations and elastic deformations in the machines. The method of elastic multibody systems with a floating frame of reference approach is an efficient way to model both the large nonlinear motions and the small but undesired elastic deformations of lightweight manipulators. Thereby a reference frame is used to describe the global nonlinear motions while the elastic deformations are described with respect to the reference frame. The elastic properties of the bodies can be derived from finite element models. They are stored in SID files to reduce the calculation time for the simulation. Due to the complexity of the model generation and dynamical load cases shape optimization of elastic bodies in multibody systems is not easy to perform. This paper presents a procedure to establish a workflow to optimize the shape of elastic bodies in controlled multibody systems. By means of a two arm manipulator the optimization goal is to achieve a lightweight design whose elasticity does not detract from the performance in the end effect or trajectory tracking.

At first two approaches are stated which show how to create SID files with respect to the design parameters. In this example the design variables are the cross section of the manipulator arms, which can change along the arm axis. Firstly a combination of ANSYS and Matlab is used for that task. Thereby a parameterized ANSYS script creates the system matrices and all necessary calculations to obtain the elastic data are then completed using Matlab. Secondly it is shown that for simple beam structures Matlab only can be used to derive the SID files.

In the next step it is shown how to set up all necessary equations to simulate a controlled elastic multibody system. The task can be divided into deriving the equations of motion and then setting up a simulation of a controlled elastic multibody system. The equations of motion are derived with Neweul-M² which is a multibody system tool based on Matlab's Symbolic Math Toolbox. It derives the equations of motion in a symbolical way and enables the user to export them to Matlab/Simulink. Latter it is used to create a simulation model. In the end the Simulink model combines the equations of motion, with an inverse dynamic controller to ensure that the manipulator follows a desired trajectory. This simulation returns the value of a criteria function to the optimizer which closes the optimization loop.

The last part deals with the optimization problem itself. For many mechanical optimization tasks gradient based algorithms are characterized on the one hand by fast convergence and low computational effort but on the other hand these algorithms tend to converge to local minima. Particularly if the starting point is unknown stochastic optimization approaches such as evolutionary computing, genetic algorithms or particle swarm optimization might be the better choice despite calculation times which are much higher. In this contribution a particle swarm optimization algorithm is applied to find the best global solution for the problem. As optimization criterion the sum or the maximum of the end effect or tracking error in trajectory tracking is used. It is shown that the shape of the bodies determine significantly the dynamical behavior of the overall controlled system.

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