# Geometric structure-preserving optimal control of a rigid body

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

Research output: Contribution to journalArticle

27 Citations (Scopus)

### Abstract

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 language English (US) 307-330 24 Journal of Dynamical and Control Systems 15 3 https://doi.org/10.1007/s10883-009-9071-2 Published - Aug 14 2009 Yes

### Fingerprint

Geometric Structure
Rigid Body
Equations of motion
Optimal Control
Lie groups
Equations of Motion
Discrete-time
Algebra
Lagrange Method
Kinematics
Torque
Calculus of variations
Discrete Equations
Lagrange
Optimal Control Problem
Initial conditions
Costs
Necessary Conditions
Numerical Examples
Three-dimensional

### Keywords

• 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

### Cite this

Geometric structure-preserving optimal control of a rigid body. / Bloch, A. M.; Hussein, I. I.; Leok, M.; Sanyal, Amit.

In: Journal of Dynamical and Control Systems, Vol. 15, No. 3, 14.08.2009, p. 307-330.

Research output: Contribution to journalArticle

Bloch, A. M. ; Hussein, I. I. ; Leok, M. ; Sanyal, Amit. / Geometric structure-preserving optimal control of a rigid body. In: Journal of Dynamical and Control Systems. 2009 ; Vol. 15, No. 3. pp. 307-330.
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