Picture of a sphere with binary code

Making Strathclyde research discoverable to the world...

The Strathprints institutional repository is a digital archive of University of Strathclyde research outputs. It exposes Strathclyde's world leading Open Access research to many of the world's leading resource discovery tools, and from there onto the screens of researchers around the world.

Explore Strathclyde Open Access research content

Planning rigid body motions using elastic curves

Biggs, James and Holderbaum, William (2008) Planning rigid body motions using elastic curves. Mathematics of Control, Signals, and Systems, 20 (4). pp. 351-367. ISSN 0932-4194

[img]
Preview
PDF (strathprints007309.pdf)
strathprints007309.pdf

Download (200kB) | Preview

Abstract

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.