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Optimal plane beams modelling elastic linear objects

Published online by Cambridge University Press:  15 May 2009

Sung K. Koh  and
Guangjun Liu
Sung K. Koh
Affiliation:
Department of Aerospace Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada
Guangjun Liu*
Affiliation:
Department of Aerospace Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada
*
*Corresponding author. E-mail: [email protected]
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Summary

This paper discusses analytical and deterministic models for a plane curve with minimum deformation that may be utilized in planning the motion of elastic linear objects and investigating the inverse kinematics of a hyper-redundant robot. It usually requires intensive computation to determine the configuration of elastic linear objects. In addition, conventional optimization-based numerical techniques that identify the shape of elastic linear objects in equilibrium involve non-deterministic aspects. Several analytical models that produce the configuration of elastic linear objects in an efficient and deterministic manner are presented in this paper. To develop the analytical expressions for elastic linear objects, we consider a cantilever beam where the deflections are determined according to the Euler–Bernoulli beam theory. The deflections of the cantilever beam are determined for prescribed constraints imposed on the deflections at the free end to replicate various elastic linear objects. Deflections of a cantilever beam with roller supports are explored to replicate elastic linear objects in contact with rigid objects. We verify the analytical models by comparing them with exact beam deflections. The analytical model is precisely accurate for beams with small deflections as it is developed on the basis of the Euler–Bernoulli beam theory. Although it is applied to beams undergoing large deflections, it is still reasonably accurate and at least as precise as the conventional pseudo-rigid-body model. The computational demand involved in using the analytical models is negligible. Therefore, efficient motion planning for elastic linear objects can be realized when the proposed analytical models are combined with conventional motion planning algorithms. We also demonstrate that the analytical model solves the inverse kinematics problem in an efficient and robust manner through numerical simulations.

Keywords

Elastic linear objectsEuler–Bernoulli beam theoryElasticity theoryHyper-redundant robotsMotion planningInverse kinematics
Type
Article
Information
Robotica , Volume 28 , Issue 1 , January 2010 , pp. 135 - 148
Copyright
Copyright © Cambridge University Press 2009

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