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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Polygerinos, P. , Wang, Z. , Galloway, K. , Wood, R. , and Walsh, C. , 2015, “ Soft Robotic Glove for Combined Assistance and At-Home Rehabilitation,” Rob. Auton. Syst., 73, pp. 135–143. [CrossRef]
Degani, A. , Choset, H. , Zubiate, B. , Ota, T. , and Zenati, M. , 2008, “ Highly Articulated Robotic Probe for Minimally Invasive Surgery,” IEEE International Conference on Robotics and Automation (ICRA), Orlando, FL, May 15–19, pp. 3273–3276.
Antman, S. , 2006, Nonlinear Problems of Elasticity (Applied Mathematical Sciences), Springer, New York.
Bisshopp, K. , and Drucker, D. , 1945, “ Large Deflection of Cantilever Beams,” Q. Appl. Math., 3(3), pp. 272–275. [CrossRef]
Frisch-Fay, R. , 1962, Flexible Bars, Butterworths, London.
Navaee, S. , and Elling, R. , 1992, “ Equilibrium Configurations of Cantilever Beams Subjected to Inclined End Loads,” ASME J. Appl. Mech., 59(3), pp. 572–579. [CrossRef]
Zhang, A. , and Chen, G. , 2013, “ A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms,” ASME J. Mech. Rob., 5(2), p. 021006. [CrossRef]
Batista, M. , 2014, “ Analytical Treatment of Equilibrium Configurations of Cantilever Under Terminal Loads Using Jacobi Elliptical Functions,” Int. J. Solids Struct., 51(13), pp. 2308–2326. [CrossRef]
Wang, C. , 1981, “ Large Deflections of an Inclined Cantilever With an End Load,” Int. J. Nonlinear Mech., 16(2), pp. 155–164. [CrossRef]
Wang, C. , and Kitipornchai, S. , 1992, “ Shooting-Optimization Technique for Large Deflection Analysis of Structural Members,” Eng. Struct., 14(4), pp. 231–240. [CrossRef]
Wang, C. , Lam, K. , He, X. , and Chucheepsakul, S. , 1997, “ Large Deflections of an End Supported Beam Subjected to a Point Load,” Int. J. Nonlinear Mech., 32(1), pp. 63–72. [CrossRef]
Shvartsman, B. , 2013, “ Analysis of Large Deflections of a Curved Cantilever Subjected to a Tip-Concentrated Follower Force,” Int. J. Nonlinear Mech., 50, pp. 75–80. [CrossRef]
Watson, L. , and Wang, C. , 1981, “ A Homotopy Method Applied to Elastica Problems,” Int. J. Solids. Struct., 17(1), pp. 29–37. [CrossRef]
Zhang, X. , and Yang, J. , 2005, “ Inverse Problem of Elastica of a Variable-Arc-Length Beam Subjected to a Concentrated Load,” Acta Mech. Sin., 21(5), pp. 444–450. [CrossRef]
Hinze, M. , Pinnau, R. , Ulbrich, M. , and Ulbrich, S. , 2008, Optimization With PDE Constraints. Mathematical Modelling: Theory and Applications, Springer, Dordrecht, The Netherlands.
Haslinger, J. , 2003, Introduction to Shape Optimization: Theory, Approximation, and Computation, Vol. 7, SIAM, Philadelphia, PA.
Shoup, T. , and McLarnan, C. , 1971, “ On the Use of the Undulating Elastica for the Analysis of Flexible Link Mechanisms,” J. Eng. Ind., 93(1), pp. 263–267. [CrossRef]
Wilson, J. , and Snyder, J. , 1988, “ The Elastica With End-Load Flip-Over,” ASME J. Appl. Mech., 55(4), pp. 845–848. [CrossRef]
Gravagne, I. A. , and Walker, I. D. , 2002, “ Manipulability, Force, and Compliance Analysis for Planar Continuum Manipulators,” IEEE Trans. Rob. Autom., 18(3), pp. 263–273. [CrossRef] [PubMed]
Li, C. , and Rahn, C. , 2002, “ Design of Continuous Backbone, Cable-Driven Robots,” ASME J. Mech. Des., 124(2), pp. 265–271. [CrossRef]
Trivedi, D. , Lotfi, A. , and Rahn, C. , 2008, “ Geometrically Exact Models for Soft Robotic Manipulators,” IEEE Trans. Rob., 24(4), pp. 773–780. [CrossRef]
Rucker, C. , and Webster, R. , 2011, “ Statics and Dynamics of Continuum Robots With General Tendon Routing and External Loading,” IEEE Trans. Rob., 27(6), pp. 1033–1044. [CrossRef]
Camarillo, D. , Milne, C. , Carlson, C. , Zinn, M. , and Salisbury, K. , 2008, “ Mechanics Modeling of Tendon-Driven Continuum Manipulators,” IEEE Trans. Rob., 24(6), pp. 1262–1273. [CrossRef]
Webster , R. J., III. , and Jones, B. A. , 2010, “ Design and Kinematic Modeling of Constant Curvature Continuum Robots: A Review,” Int. J. Rob. Res., 29(13), pp. 1661–1683. [CrossRef]
Chirikjian, G. , 1994, “ Hyper-Redundant Manipulator Dynamics: A Continuum Approximation,” Adv. Rob., 9(3), pp. 217–243. [CrossRef]
Challamel, N. , Kocsis, A. , and Wang, C. , 2015, “ Discrete and Non-Local Elastica,” Int. J. Nonlinear Mech., 77, pp. 128–140. [CrossRef]
Wang, C. , 2015, “ Longest Reach of a Cantilever With a Tip Load,” Eur. J. Phys., 37(1), p. 012001. [CrossRef]
Mochiyama, H. , Watari, M. , and Fujimoto, H. , 2007, “ A Robotic Catapult Based on the Closed Elastica and Its Application to Robotic Tasks,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Diego, CA, Oct. 29–Nov. 2, pp. 1508–1513.
Patricio, P. , Adda-Bedia, M. , and Ben Amar, M. , 1999, “ An Elastica Problem: Instabilities of an Elastic Arch,” Phys. Sect. D, 124(1–3), pp. 285–295.
Plaut, R. , and Virgin, L. , 2009, “ Vibration and Snap-Through of Bent Elastica Strips Subjected to End Rotations,” ASME J. Appl. Mech., 76(4), p. 041011. [CrossRef]
Armanini, C. , Dal Corso, F. , Misseroni, D. , and Bigoni, D. , 2017, “ From the Elastica Compass to the Elastica Catapult: An Essay on the Mechanics of Soft Robot Arm,” Proc. R. Soc. A, 473(2198), p. 20160870. [CrossRef]
Griner, G. , 1984, “ A Parametric Solution to the Elastic Pole-Vaulting Pole Problem,” ASME J. Appl. Mech., 51(2), pp. 409–414. [CrossRef]
Zhang, Y. , Wang, Y. , Li, Z. , Huang, Y. , and Li, D. , 2007, “ Snap-Through and Pull-in Instabilities of an Arch-Shaped Beam Under an Electrostatic Loading,” J. Microelectromech. Syst., 16(3), pp. 684–693. [CrossRef]
Rus, D. , and Tolley, M. T. , 2015, “ Design, Fabrication and Control of Soft Robots,” Nature, 521(7553), pp. 467–475. [CrossRef] [PubMed]
Mosadegh, B. , Polygerinos, P. , Keplinger, C. , Wennstedt, S. , Shepherd, R. , Gupta, U. , Shim, J. , Bertoldi, K. , Walsh, C. , and Whitesides, G. , 2014, “ Pneumatic Networks for Soft Robotics That Actuate Rapidly,” Adv. Funct. Mater., 24(15), pp. 2163–2170. [CrossRef]
Connolly, F. , Walsh, C. , and Bertoldi, K. , 2017, “ Automatic Design of Fiber-Reinforced Soft Actuators for Trajectory Matching,” Proc. Natl. Acad. Sci., 114(1), pp. 51–56. [CrossRef]
Yang, D. , Mosadegh, B. , Ainla, A. , Lee, B. , Khashai, F. , Suo, Z. , Bertoldi, K. , and Whitesides, G. , 2015, “ Buckling of Elastomeric Beams Enables Actuation of Soft Machines,” Adv. Mater., 27(41), pp. 6323–6327. [CrossRef] [PubMed]
Huang, W. , 2002, “ On the Selection of Shape Memory Alloys for Actuators,” Mater. Des., 23(1), pp. 11–19. [CrossRef]
Bar-Cohen, Y. , 2004, “ Electroactive Polymer (EAP) Actuators as Artificial Muscles: Reality, Potential, and Challenges,” SPIE Press Monographs, SPIE, Bellingham, WA.
Trivedi, D. , Rahn, C. , Kier, W. , and Walker, I. , 2008, “ Soft Robotics: Biological Inspiration, State of the Art, and Future Research,” Appl. Bionics Biomech., 5(3), pp. 99–117. [CrossRef]
Kier, W. , and Smith, K. , 1985, “ Tongues, Tentacles and Trunks: The Biomechanics of Movement in Muscular-Hydrostats,” Zool. J. Linnean Soc., 83(4), pp. 307–324. [CrossRef]
Hirose, S. , 1993, Biologically Inspired Robots: Snake-like Locomotors and Manipulators, Vol. 1093, Oxford University Press, Oxford, UK.
Cicconofri, G. , and DeSimone, A. , 2015, “ A Study of Snake-like Locomotion Through the Analysis of a Flexible Robot Model,” Proc. R. Soc. A, 471(2184), p. 20150054. [CrossRef]
Hannan, M. , and Walker, I. , 2003, “ Kinematics and the Implementation of an Elephant's Trunk Manipulator and Other Continuum Style Robots,” J. Rob. Syst., 20(2), pp. 45–63. [CrossRef]
Laschi, C. , Cianchetti, M. , Mazzolai, B. , Margheri, L. , Follador, M. , and Dario, P. , 2012, “ Soft Robot Arm Inspired by the Octopus,” Adv. Rob., 26(7), pp. 709–727. [CrossRef]
Immega, G. , and Antonelli, K. , 1995, “ The KSI Tentacle Manipulator,” IEEE International Conference on Robotics and Automation, Nagoya, Japan, May 21–27, pp. 3149–3154.
Buckingham, R. , 2002, “ Snake Arm Robots,” Ind. Rob., 29(3), pp. 242–245. [CrossRef]
Yau, J. , 2010, “ Closed-Form Solutions of Large Deflection for a Guyed Cantilever Column Pulled by an Inclination Cable,” J. Mar. Sci. Technol., 18(1), pp. 130–136.
Batista, M. , 2015, “ Large Deflection of Cantilever Rod Pulled by Cable,” Appl. Math. Model., 39(10–11), pp. 3175–3182. [CrossRef]
Brander, D. , Gravesen, J. , and Nørbjerg, T. , 2017, “ Approximation by Planar Elastic Curves,” Adv. Comput. Math., 43(1), pp. 25–43. [CrossRef]
Søndergaard, A. , Feringa, J. , Nørbjerg, T. , Steenstrup, K. , Brander, D. , Graversen, J. , Markvorsen, S. , Bærentzen, A. , Petkov, K. , Hattel, J. , Clausen, K., Jensen, K., Knudsen, L., and Kortbek, J., 2016, “ Robotic Hot-Blade Cutting,” Robotic Fabrication in Architecture, Art and Design, Springer, Cham, Switzerland, pp. 150–164.
Wriggers, P. , 2008, Nonlinear Finite Element Methods, Springer, Berlin.
Rao, N. , and Rao, V. , 1986, “ On the Large Deflection of Cantilever Beams With End Rotational Load,” Z. Angew. Math. Mech., 66(10), pp. 507–509. [CrossRef]
Elishakoff, I. , 2005, “ Controversy Associated With the so-Called Follower Forces: Critical Overview,” Appl. Mech. Rev., 58(2), pp. 117–142. [CrossRef]
Munson, T. , Sarich, J. , Wild, S. , Benson, S. , and McInnes, L. , 2017, “ Tao 3.8 Users Manual,” Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL.
Nocedal, J. , and Wright, S. , 2006, Numerical Optimization (Springer Series in Operations Research and Financial Engineering), Springer, New York.
Simo, J. , 1985, “ A Finite Strain Beam Formulation. the Three-Dimensional Dynamic Problem—Part I,” Comput. Methods Appl. Mech. Eng., 49(1), pp. 55–70. [CrossRef]
Chen, J. , and Li, H. , 2011, “ On an Elastic Rod Inside a Slender Tube Under End Twisting Moment,” ASME J. Appl. Mech, 78(4), p. 041009. [CrossRef]
Mahvash, M. , and Dupont, P. , 2011, “ Stiffness Control of Surgical Continuum Manipulators,” IEEE Trans. Rob., 27(2), pp. 334–345. [CrossRef]
Okubo, S. , and Tortorelli, D. , 2004, “ Control of Nonlinear, Continuous, Dynamic Systems Via Finite Elements, Sensitivity Analysis, and Optimization,” Struct. Multidiscip. Optim., 26(3–4), pp. 183–199. [CrossRef]


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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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