Geometric structure-preserving optimal control of a rigid body

A. M. Bloch, I. I. Hussein, M. Leok, A. K. Sanyal

Research output: Contribution to journalArticlepeer-review

35 Scopus citations


In this paper, we study a discrete variational optimal control problem for a rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange-d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use the Lagrange method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.

Original languageEnglish (US)
Pages (from-to)307-330
Number of pages24
JournalJournal of Dynamical and Control Systems
Issue number3
StatePublished - 2009
Externally publishedYes


  • Geometric integrators
  • Lie group integrators
  • Optimal control
  • Rigid body
  • Variational methods

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Algebra and Number Theory
  • Numerical Analysis
  • Control and Optimization


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