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Mathematical Problems in Engineering
Volume 2015, Article ID 790409, 14 pages
Research Article

Geometric Collocation Method on SO(3) with Application to Optimal Attitude Control of a 3D Rotating Rigid Body

College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, Hunan 410073, China

Received 22 August 2015; Accepted 13 October 2015

Academic Editor: Naohisa Otsuka

Copyright © 2015 Xiaojia Xiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The collocation method is extended to the special orthogonal group SO(3) with application to optimal attitude control (OAC) of a rigid body. A left-invariant rigid-body attitude dynamical model on SO(3) is established. For the left invariance of the attitude configuration equation in body-fixed frame, a geometrically exact numerical method on SO(3), referred to as the geometric collocation method, is proposed by deriving the equivalent Lie algebra equation in of the left-invariant configuration equation. When compared with the general Gauss pseudo-spectral method, the explicit RKMK, and Lie group variational integrator having the same order and stepsize in numerical tests for evolving a free-floating rigid-body attitude dynamics, the proposed method is higher in accuracy, time performance, and structural conservativeness. In addition, the numerical method is applied to solve a constrained OAC problem on SO(3). The optimal control problem is transcribed into a nonlinear programming problem, in which the equivalent Lie algebra equation is being considered as the defect constraints instead of the configuration equation. The transcription method is coordinate-free and does not need chart switching or special handling of singularities. More importantly, with the numerical advantage of the geometric collocation method, the proposed OAC method may generate satisfying convergence rate.