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IMA Journal of Mathematical Control and Information Advance Access published online on June 7, 2006
IMA Journal of Mathematical Control and Information, doi:10.1093/imamci/dnl017
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Received September 21, 2005
Accepted April 4, 2006

Article

Curves of stationary acceleration in SE(3)

J. M. Selig 1 *

1 Faculty of Business, Computing and Information Management, London South Bank University, London SE1 0AA, UK

* To whom correspondence should be addressed.
J. M. Selig, E-mail: seligjm{at}lsbu.ac.uk


  Abstract

The concept of curves of minimal acceleration seems to have been introduced by Zefran and Kumar and independently by Noakes, Heinzinger and Paden. In part, the motivation was to extend the notion of spline curves to curves in groups, specifically the groups associated with robotics. A curve in the rigid-body motion group SE(3), e.g. can be thought of as a trajectory of a rigid body. Hence, these ideas have applications to motion planning and interpolation. In this work, the analysis is repeated but using bi-invariant metrics on the group. Since these metrics are not positive definite, the curves specified by the equations derived are only stationary, not minimal. It is possible to solve these non-linear coupled differential equations in some simple cases. However, these simple cases turn out to be highly relevant to robotics and mechanism theory.

Keywords: rigid-body motions; motion interpolation; robotics; differential geometry.
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