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Research Papers

Motion Planning and Tracking for Tip Displacement and Deflection Angle for Flexible Beams

[+] Author and Article Information
Antranik A. Siranosian1

Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093asiranosian@ucsd.edu

Miroslav Krstic, Andrey Smyshlyaev

Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093

Matt Bement

 Los Alamos National Laboratory, Los Alamos, NM 87545

1

Corresponding author.

J. Dyn. Sys., Meas., Control 131(3), 031009 (Mar 20, 2009) (10 pages) doi:10.1115/1.3072152 History: Received December 06, 2007; Revised October 22, 2008; Published March 20, 2009

Explicit motion-planning reference solutions are presented for flexible beams with Kelvin–Voigt (KV) damping. The goal is to generate periodic reference signals for the displacement and deflection angle at the free-end of the beam using only actuation at the base. The explicit deflection angle reference solution is found as a result of writing the shear beam model in a strict-feedback form. Special “partial differential equation (PDE) backstepping” transformations relate the strict-feedback model to a “target system,” governed by an exponentially stable wave equation with KV damping, whose displacement reference solution is relatively easy to find. The explicit beam displacement reference solution is found using the target system solution and an inverse backstepping transformation. The explicit reference solutions for the wave equation and shear beam with KV damping are novel results. State-feedback tracking boundary controllers are found by extending previous PDE backstepping stabilization results. Application of the shear beam results to the more complicated Timoshenko beam is discussed.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Diagram depicting a string/beam with transverse displacement u(x,t): The goal is to generate and track a reference trajectory at x=0. The arrows at x=1 represent actuation, and the circle at x=0 represents the desired reference trajectory.

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Figure 2

Diagram representing the target system

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Figure 3

A differential element of length dx in the Timoshenko beam: The diagram shows the relationship between the beam displacement u(x,t), the slope ux(x,t), and the deflection angle α(x,t). This diagram has been adapted from a figure in Ref. 28.

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Figure 4

Pictorial representation of the structure of the input-output relationship {ur(1,t),αr(1,t)}↦{ur(0,t),αr(0,t)}, and a description of the types of problems involved in solving the simultaneous motion-planning problem: Finding αr(1,t) involves solving a two-point boundary-value problem (TPBVP) for α(x,t), then modifying the resulting boundary input α(1,t) to satisfy both spatial causality of the shear beam and motion planning. Finding ur(1,t) requires solving a PDE for the auxiliary system r(x,t), then employing a direct transformation from wr(x,t) to ur(x,t).

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Figure 5

The string of invertible transformations involved in solving the shear beam motion-planning problem: The functions above and below the arrows represent the appropriate transformation gains

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Figure 6

String simulation showing the state as snapshots in time

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Figure 7

String simulation results comparing the (a) tip displacement u(0,t) and reference trajectory ur(0,t), (b) base displacement u(1,t) and reference displacement ur(1,t), and (c) boundary control input ux(1,t) and reference input uxr(1,t)

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Figure 8

Timoshenko beam simulation results showing snapshots of the beam state u(x,t)

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Figure 9

Timoshenko beam simulation results showing snapshots of the beam state α(x,t)

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Figure 10

Timoshenko beam simulation results showing (a) the tip displacement tracking error u(0,t)−ur(0,t), (b) the base displacement u(1,t) and the reference displacement ur(1,t), and (c) the boundary control ux(1,t) and the reference control uxr(1,t)

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Figure 11

Timoshenko beam simulation results showing (a) the tip deflection angle tracking error α(0,t)−αr(0,t), and (b) the boundary control α(1,t) and reference control αr(1,t)

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Figure 12

Timoshenko beam gains (a) k(1,y) and (b) kx(1,y)

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