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Research Papers

Analytic Formulation for Kinematics, Statics, and Shape Restoration of Multibackbone Continuum Robots Via Elliptic Integrals

[+] Author and Article Information
Kai Xu

Department of Mechanical Engineering, ARMA—Laboratory for Advanced Robotics and Mechanism Applications, Columbia University, New York, NY 10027kx2102@columbia.edu

Nabil Simaan1

Department of Mechanical Engineering, ARMA—Laboratory for Advanced Robotics and Mechanism Applications, Columbia University, New York, NY 10027ns2236@columbia.edu

1

Corresponding author.

J. Mechanisms Robotics 2(1), 011006 (Nov 24, 2009) (13 pages) doi:10.1115/1.4000519 History: Received November 07, 2008; Revised July 14, 2009; Published November 24, 2009

This paper presents a novel and unified analytic formulation for kinematics, statics, and shape restoration of multiple-backbone continuum robots. These robots achieve actuation redundancy by independently pulling and pushing three backbones to carry out a bending motion of two-degrees-of-freedom (DoF). A solution framework based on constraints of geometric compatibility and static equilibrium is derived using elliptic integrals. This framework allows the investigation of the effects of different external loads and actuation redundancy resolutions on the shape variations in these continuum robots. The simulation and experimental validation results show that these continuum robots bend into an exact circular shape for one particular actuation resolution. This provides a proof to the ubiquitously accepted circular-shape assumption in deriving kinematics for continuum robots. The shape variations due to various actuation redundancy resolutions are also investigated. The simulation results show that these continuum robots have the ability to redistribute loads among their backbones without introducing significant shape variations. A strategy for partially restoring the shape of the externally loaded continuum robots is proposed. The simulation results show that either the tip orientation or the tip position can be successfully restored.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 2

Kinematics nomenclature with the definition of δ for (a) a bent robot, (b) a straight robot, and (c) the distal subsegment

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Figure 3

Gravitational energy over the elastic energy ratio

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Figure 4

Static equilibrium of a spacer disk

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Figure 5

Deformed primary backbone of the tth subsegment as a result of force fp(t) and moment mp(t)

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Figure 6

The actual shape and circular arcs of one subsegment in actuation mode 1

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Figure 7

The calculated shape (solid lines) and circular arc (dashed lines) overlaid over the actual shape of one subsegment in actuation mode 1

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Figure 14

The shooting method is initialized using the local tangents of the subsegments based on the shape of an unloaded robot

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Figure 15

Experimental setup for validating shape restoration

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Figure 16

Shape restoration simulation and experiment: (a) simulated deflected and simulated undeflected shapes overlaid over an actual externally loaded subsegment, (b) simulation of the deflected subsegment in (a) with and without tip orientation restoration, and (c) simulation of the deflected subsegment in (a) with and without tip position restoration

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Figure 1

Continuum robots with actuation redundancy: (a) a ∅7.5 mm one, (b) a ∅4.2 mm one, and (c) a two-segment robot

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Figure 8

Calculated shapes of the last subsegment under different actuation modes: insets (a) and (b) provide enlarged side views

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Figure 9

Diagram for qualitative justification of the simulation results

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Figure 10

A ∅7.5 mm continuum robot with its actual bending shape under configurations of (a) θL=60 deg,δ=0 deg; (b) θL=15 deg,δ=0 deg, and (c) a close-up view of one spacer disk

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Figure 11

Bending shape along the primary backbone of the continuum robot

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Figure 12

Actual θL value versus desired θ¯L value

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Figure 13

The theoretical results from Sec. 4 compared with the experimentally corrected results in the joint space

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