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Research Papers

Refined Wave-Based Control Applied to Nonlinear, Bending, and Slewing Flexible Systems

[+] Author and Article Information
William J. O’Connor

School of Electrical Electronic and Mechanical Engineering, University College Dublin, Dublin 4, Irelandwilliam.oconnor@ucd.ie

Alessandro Fumagalli

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, 20156 Milan, Italyalessandro.fumagalli@polimi.it

If the strain gauge is not placed exactly at the root of the beam, the moment of interface forces about the actuator axis will have additional contributions from the shear force and even from the axial force, multiplied by their respective moment arms.

J. Appl. Mech 76(4), 041005 (Apr 22, 2009) (9 pages) doi:10.1115/1.3086434 History: Received May 22, 2007; Revised December 30, 2008; Published April 22, 2009

The problem of controlling the position and vibration of open-chain flexible structures undergoing fast maneuvers is of wide interest. In this work, the general flexible structure is actuated by a single actuator at one end, which, depending on the case of interest, is capable of rotating, translating, or simultaneously translating and rotating the root of the flexible system. The goal is to control the motion of the entire flexible system from rest to rest. This needs a simultaneous synthesis of position control and active vibration damping. A new strategy is presented based on further developments of wave-based control. As before it views the actuator motion as simultaneously launching and absorbing mechanical waves into and out of the system. But a new simple method of resolving the actuator motion into two waves is presented. By measuring the elastic forces exchanged at the interface between the actuator and the rest of the system, a returning displacement wave can be resolved. This is then added to a set, launch wave to determine the actuator motion. Typically the launch wave is set to reach half the target displacement, and the addition of the return wave absorbs the vibration while simultaneously moving the system the second half of the target displacement, neatly achieving the two goals in one controlled motion. To date wave-based control has been applied to lumped, second-order, longitudinally vibrating systems. The refined method avoids a difficulty that previously arose in some contexts, thereby making wave-based control even more generic. It can easily control nonlinear elastic systems, laterally bending systems (in the sense of Euler–Bernoulli beams), and slewing systems where lateral translation and system rotation are strongly coupled. Numerical simulation results are presented for controlled, rest-to-rest maneuvers of representative flexible structures, all controlled using the same (linear) algorithm. The first case is control of a string of rigid bodies interconnected by nonlinear springs. The second problem is the rotational control of a very flexible one-link planar manipulator. Finally, in an extension of the previous system, the actuator both translates and rotates, slewing the flexible system to a target lateral displacement and a target rotation angle simultaneously. The strategy is found to be remarkably effective with many advantages. It seamlessly integrates position and vibration control. It is rapid, robust, energy efficient, and computationally light. It requires little sensing, little knowledge of the flexible system dynamics, and copes well with nonideal actuator behavior. It is generic and easily handles a wide variety of flexible systems. It can get the entire system to stop dead exactly at target with little vibration in transit.

Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

A simple flexible system: rigid bodies connected by linear springs

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Figure 2

A simple flexible system modeled as the superposition of two semi-infinite systems using WTFs with Xi=Ai+Bi

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Figure 3

Analog approximation of the wave transfer function of Eq. 7, with Xi−1 as the input, Xi the output: cf. Eq. 3

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Figure 4

Position-position control: only the first part of the system is of interest

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Figure 5

Position-position control scheme

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Figure 6

Wave-based control using cross-over WTFs: only the interface between the actuator and the flexible system is of interest

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Figure 7

Wave-based control using cross-over WTFs: only the interface between the actuator and the flexible system is of interest

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Figure 8

Block scheme implementing Eqs. 26,23, which becomes the xWTFH¯ in Fig. 7

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Figure 9

Tip mass response to a step input using standard and cross-over WTFs

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Figure 10

The one-link planar manipulator with the required strain gauge

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Figure 11

Beam tip rotations for a very fast rest-to-rest maneuver. The reference input is to rotate the system by 1 rad in 0.5 s, following a ramp. Responses are shown for no control and with G(s) (Eq. 8) in the controller tuned to different natural frequencies.

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Figure 12

Actuator rotations for a very fast rest-to-rest maneuver. The total actuator motion is the sum of the launch wave a0 and the measured returning wave b0. The curves refer to cases with no control and with controls tuned at different natural frequencies for G(s) (Eq. 8).

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Figure 13

Tip translation absorbing translational (shear) waves

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Figure 14

Actuator translation absorbing translational (shear) waves

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Figure 15

y movement of the system tip under combined translation and rotation controlled by absorbing both translational and rotational waves simultaneously

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Figure 16

Rotation of the system tip under combined translation and rotation controlled by absorbing both translational and rotational waves simultaneously

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