Keywords: Euler Bernoulli beam, distributed parameter system, flatness based feedforward control, state controller, residual vibrations, Luenberger observer.

Elasticities in industrial machinery have become more and more important for control design. There are three common ways to tackle residual vibrations as a result of elasticities in the structure of today's standard drives. The first way is to move slow enough avoiding big amplitudes. The second way is to add after the fast positioning process a wait time. Both ways ensure that residual vibrations are smaller than a required limit. The third way is the same as the second way but advanced by an active damping strategy, trying to improve the damping charasteristics of the system, thus, the added wait time can be shortend. The backlash in the drive limits the possibilities of a closed loop control approach by limiting the maximum achievable bandwidth. Therefore the actuator's dynamic possibilities can not be utilised fully for active damping purposes.

This paper presents a controller architecture incorporating analytical feed forward force trajectories assuring minimum residual vibrations. This concept includes a highly dynamic phase where only the feed forward control law is active. Together with a well planed trajectory highly dynamic positioning is also available for a drive with backlash. Moreover the only hard limit is given by the maximum bending moment sustainable by the beam itself and not any longer by the amplitudes of residual vibrations.

In the first part of the paper, the Euler Bernoulli beam dynamic is reduced to a finite dimensional state space using finite differences. At first the time invariant model consisting of only of the perpendicular clamped guided beam is derived. After this the payload is modelled as a point mass, connected to the beam by a spring.

The approach presented is a flatness based trajectory planning approach for distributed parameter dynamics [1,2,3,4,5,6,7]. The main advantage of this mathematical exact method compared to numerical optimization results or the variational calculus results is the realtime capability of the resulting feedforward law.

To take care of the spill-over effects numerical studies have been performed. It is affirmed that a Kalman observer is robust against these differences in dimensionality, while observers with free placed poles mostly resulted in an overall system that was not stable.

1

M. Bachmayer, H. Ulbrich, "Trajectory planning for linearly actuated elastic robots using flatness based control theory", Proceedings of the International Conference of Theoretical and Applied Mechanics 2008, Adelaide, Australia, 2008.

2

M. Bachmayer, J. Rudolph, H. Ulbrich, "Flatness based feed forward control for a horizontally moving beam with a point mass", Fourth European Conference of Structural Control, St. Petersburg, Russia, 2008.

3

M. Bachmayer, J. Rudolph, H. Ulbrich, "Acceleration of linearly actuated elastic robots avoiding residual vibrations", Proceedings of the 9th International Conference on Motion and Vibration Control, Munich, Germany, 2008.

4

J. Rudolph, "Flatness Based Control of Distributed Parameter Systems", Shaker Verlag, Aachen, Germany, 2003.

5

J. Rudolph, "Beiträge zur flachheitsbasierten Folgeregelung linearer und nichtlinearer Systeme endlicher und unendlicher Dimension", Shaker Verlag, Aachen, Germany, 2003.

6

J. Rudolph, F. Woittennek, "Flachheitsbasierte Steuerung eines Timoshenko-Balkens", ZAMM, 83, 119-127, 2003. doi:10.1002/zamm.200310011

7

Y. Aoustin, M. Fliess, H. Mounier, P. Rouchon, J. Rudolph, "Theory and practice in the motion planning and control of a flexible robot arm using Mikusinski operators", Proc. 5th Symposium on Robotics and Control, Nantes, France, 287-293, 1997.

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