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Research Papers

Shape Control for the Elastica Through Load Optimization

[+] Author and Article Information
Arvind Nayak, Poornakanta Handral

Department of Mechanical Engineering,
Indian Institute of Science,
Bengaluru 560012, India

Ramsharan Rangarajan

Department of Mechanical Engineering,
Indian Institute of Science,
Bengaluru 560012, India
e-mail: rram@iisc.ac.in

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 5, 2018; final manuscript received October 4, 2018; published online November 2, 2018. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 86(1), 011011 (Nov 02, 2018) (10 pages) Paper No: JAM-18-1510; doi: 10.1115/1.4041678 History: Received September 05, 2018; Revised October 04, 2018

Flexible elastic beams can function as dexterous manipulators at multiple length-scales and in various niche applications. As a step toward achieving controlled manipulation with flexible structures, we introduce the problem of approximating desired quasi-static deformations of a flexible beam, modeled as an elastica, by optimizing the loads applied. We presume the loads to be concentrated, with the number and nature of their application prescribed based on design considerations and operational constraints. For each desired deformation, we pose the problem of computing the requisite set of loads to mimic the target shape as one of optimal approximations. In the process, we introduce a novel generalization of the forward problem by considering the inclinations of the loads applied to be functionals of the solution. This turns out to be especially beneficial when analyzing tendon-driven manipulators. We demonstrate the shape control realizable through load optimization using a diverse set of experiments.

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Figures

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Fig. 1

An experimental realization demonstrating the manipulation of a flexible polycarbonate beam using three tendon loads

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Fig. 2

Illustration of the choice of coordinates and the loading configuration for the problem of computing the deformation of an elastica discussed in Sec. 2

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Fig. 3

Plots (a)–(c) compare our finite element approximations of elastica solutions with elliptic integral-based solutions derived in the literature. Plot (d) shows a comparison with an alternative numerical method.

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Fig. 4

With the manipulator shown in (a) designed with one end vertical load, we seek a load P that approximates a given target deformation (θd) as an elastica solution. Plots (b) and (c) show the approximation achieved by computing optimal loads as outlined in Sec. 3.2 to approximate target inclinations of the form θd=βs(s−2). While (b) compares the functions θd(s) and θP(s), (c) compares the corresponding set of deformations. For the specific case of β = 1, details of the convergence of the load optimization scheme to a minimizer of the objective functional are shown in (d) and (e). The algorithm terminates after 8 iterations, when the sensitivity |dJ/dP| becomes smaller than the specified tolerance 10−10.

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Fig. 5

Demonstrating the capability of a manipulator designed with four vertical loads P1−4 to approximate a series of different target deformations by optimizing the set of loads. The target inclinations are indicated alongside each example.

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Fig. 6

Experimental setup for demonstrating shape control in a manipulator actuated using a pair of tendons routed through fixed posts. All coordinates mentioned in the figure are in mm.

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Fig. 7

A comparison of the finite element approximation of an elastica loaded by a tendon shown in (a) with a Jacobi-elliptic integral solution from the literature is shown in (b). The plot in (c) compares experimental measurements of the deformation with numerical solutions while allowing for a 5% uncertainty in the load.

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Fig. 8

Progressive approximation of a one-parameter set of target deformations achieved using the load optimization algorithm for the setup in Fig. 6. Experimental measurements approximate the optimized solution well.

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Fig. 9

More examples and experiments demonstrating shape control for the manipulator in Fig. 6 to approximate varied target solutions

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