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Advances in Mathematical Physics
Volume 2017, Article ID 6302430, 8 pages
Research Article

Intrinsic Optimal Control for Mechanical Systems on Lie Group

Key Laboratory of Dynamics and Control of Flight Vehicle, Ministry of Education, School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Shengjing Tang; nc.ude.tib@jsgnat

Received 31 March 2017; Revised 10 May 2017; Accepted 30 May 2017; Published 12 July 2017

Academic Editor: Juan C. Marrero

Copyright © 2017 Chao Liu 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 intrinsic infinite horizon optimal control problem of mechanical systems on Lie group is investigated. The geometric optimal control problem is built on the intrinsic coordinate-free model, which is provided with Levi-Civita connection. In order to obtain an analytical solution of the optimal problem in the geometric viewpoint, a simplified nominal system on Lie group with an extra feedback loop is presented. With geodesic distance and Riemann metric on Lie group integrated into the cost function, a dynamic programming approach is employed and an analytical solution of the optimal problem on Lie group is obtained via the Hamilton-Jacobi-Bellman equation. For a special case on , the intrinsic optimal control method is used for a quadrotor rotation control problem and simulation results are provided to show the control performance.