Research Papers: Design of Mechanisms and Robotic Systems

Six-Dimensional Compliance Analysis and Validation of Orthoplanar Springs

[+] Author and Article Information
Chen Qiu

Centre for Robotics Research,
King's College London,
London WC2R 2LS, UK
e-mail: chen.qiu@kcl.ac.uk

Peng Qi

Centre for Robotics Research,
King's College London,
London WC2R 2LS, UK
e-mail: peng.qi@kcl.ac.uk

Hongbin Liu

Centre for Robotics Research,
King's College London,
London WC2R 2LS, UK
e-mail: hongbin.liu@kcl.ac.uk

Kaspar Althoefer

Centre for Robotics Research,
King's College London,
London WC2R 2LS, UK
e-mail: k.althoefer@kcl.ac.uk

Jian S. Dai

Fellow ASME
Chair of Mechanisms and Robotics
MoE Key Laboratory for Mechanism
Theory and Equipment Design,
Tianjin University,
Tianjin 300072, China;
Centre for Robotics Research,
King's College London,
London WC2R 2LS, UK
e-mail: jian.dai@kcl.ac.uk

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 5, 2015; final manuscript received January 17, 2016; published online February 19, 2016. Assoc. Editor: Oscar Altuzarra.

J. Mech. Des 138(4), 042301 (Feb 19, 2016) (9 pages) Paper No: MD-15-1621; doi: 10.1115/1.4032580 History: Received September 05, 2015; Revised January 17, 2016

This paper for the first time investigates the six-dimensional compliance characteristics of orthoplanar springs using a compliance-matrix based approach, and validates them with both finite element (FEM) simulation and experiments. The compliance matrix is developed by treating an orthoplanar spring as a parallel mechanism and is revealed to be diagonal. As a consequence, corresponding diagonal compliance elements are evaluated and analyzed in forms of their ratios, revealing that an orthoplanar spring not only has a large linear out-of-plane compliance but also has a large rotational bending compliance. Both FEM simulation and experiments were then conducted to validate the developed compliance matrix. In the FEM simulation, a total number of 30 types of planar-spring models were examined, followed by experiments that examined the typical side-type and radial-type planar springs, presenting a good agreement between the experiment results and analytical models. Further a planar-spring based continuum manipulator was developed to demonstrate the large-bending capability of its planar-spring modules.

Copyright © 2016 by ASME
Topics: Springs
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Fig. 2

Isometric view of an orthoplanar spring model with three limbs

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Fig. 1

A limb of the orthoplanar spring

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Fig. 5

Variation of Cb3/Cb4 with variables m/L and n/L

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Fig. 3

Variations of Cb2/Cb1 (ratio of linear out-of-plane compliance to linear in-plane compliance) and Cb3/Cb4 (ratio of rotational bending compliance to rotational torsional compliance) with respect to variables d/L and h/b. (a) α=Cb2/Cb1 with variables d/L and h/b,m/L=−0.5,n/L=0.5 and (b) β=Cb3/Cb4 with variables d/L and h/b, m/L=−0.5, n/L=0.5.

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Fig. 4

Comparisons of Cb3/Cb4 with variables m/L and n/L: (a) γ=Cb3/Cb4 with variable m/L and h/b, d/L=0.2, n/L=0.5 and (b) δ=Cb3/Cb4 with variable n/L and h/b, d/L=0.2, m/L=0.4

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Fig. 8

Experiment samples of both side-type and radial-type planar springs

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Fig. 6

A complete list of planar-spring models used in FEM simulations

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Fig. 7

Discrepancies between FEM simulations and analytical models in accordance with the comparison results shown Figs. 3(a), 3(b), 4(a), and 4(b)

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Fig. 9

(a) Experiment setup to evaluate the linear compliance Cb2, (b) displacement–force comparison results of compliance Cb2, where solid-circle line represents analytical models, dash lines represent experiment results. Error bars represent standard deviations at corresponding tested points.

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Fig. 10

Mathematical model used to evaluate the coupling compliance performance of Cb2 and Cb3

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Fig. 12

Prototype of a continuum manipulator based on the serial integration of orthoplanar springs

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Fig. 11

(a) Deformation of a planar-spring sample during the testing of coupling compliance Cb2 and Cb3 test, (b) displacement–force comparison results of coupling compliance Cb2 and Cb3, where solid lines represent analytical models, dashed lines represent experiment results. Error bars represent standard deviations at corresponding tested points.



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