Journal of Dynamical and Control Systems

, Volume 15, Issue 3, pp 307–330 | Cite as

Geometric structure-preserving optimal control of a rigid body

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


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 \(\mathfrak{s}\mathfrak{o}(3)\). 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.

Key words and phrases

geometric integrators Lie group integrators optimal control variational methods rigid body 

2000 Mathematics Subject Classification

37M15 65K10 49K15 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • A. M. Bloch
    • 1
  • I. I. Hussein
    • 2
  • M. Leok
    • 3
  • A. K. Sanyal
    • 4
  1. 1.University of MichiganAnn ArborUSA
  2. 2.Worcester Polytechnic InstituteWorcesterUSA
  3. 3.Purdue UniversityWest LafayetteUSA
  4. 4.University of HawaiiHonoluluUSA

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