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Characterisation of Frenet–Serret and Bishop motions with applications to needle steering

Published online by Cambridge University Press:  12 April 2013

J. M. Selig
J. M. Selig*
Affiliation:
Faculty of Business, London South Bank University, London, UK
*
*Corresponding author. E-mail: [email protected]
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Summary

Frenet–Serret and Bishop rigid-body motions have many potential applications in robotics, graphics and computer-aided design. In order to study these motions, new characterisations in terms of their velocity twists are derived. This is extended to general motions based on any moving frame to a space curve. Furthermore, it is shown that any such general moving frame motion is the product of a Frenet–Serret motion with a rotation about the tangent vector.

These ideas are applied to a simple model of needle steering. A simple kinematic model of the path of the needle is derived. It is then shown that this leads to Frenet–Serret motions of the needle tip but with constant curvature. Finally, some remarks about curves with constant curvature are made.

Keywords

Rigid-body motionsFrenet–Serret motionsBishop motionsScrew theoryNeedle steering
Type
Articles
Information
Robotica , Volume 31 , Issue 6 , September 2013 , pp. 981 - 992
Copyright
Copyright © Cambridge University Press 2013 

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