Planning rigid body motions using elastic curves

James Biggs, William Holderbaum

Research output: Contribution to journalArticle

14 Citations (Scopus)
58 Downloads (Pure)

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.
Original languageEnglish
Pages (from-to)351-367
Number of pages17
JournalMathematics of Control, Signals, and Systems
Volume20
Issue number4
DOIs
Publication statusPublished - Dec 2008

Keywords

  • rigid body motion
  • optimal control
  • elastic curves
  • helical motions
  • control systems

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