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Volume 5 Issue 2
Mar.  2018

IEEE/CAA Journal of Automatica Sinica

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Hong-Yan Zhang, Zi-Hao Wang, Lu-Sha Zhou, Qian-Nan Xue, Long Ma and Yi-Fan Niu, "Explicit Symplectic Geometric Algorithms for Quaternion Kinematical Differential Equation," IEEE/CAA J. Autom. Sinica, vol. 5, no. 2, pp. 479-488, Mar. 2018. doi: 10.1109/JAS.2017.7510829
Citation: Hong-Yan Zhang, Zi-Hao Wang, Lu-Sha Zhou, Qian-Nan Xue, Long Ma and Yi-Fan Niu, "Explicit Symplectic Geometric Algorithms for Quaternion Kinematical Differential Equation," IEEE/CAA J. Autom. Sinica, vol. 5, no. 2, pp. 479-488, Mar. 2018. doi: 10.1109/JAS.2017.7510829

Explicit Symplectic Geometric Algorithms for Quaternion Kinematical Differential Equation

doi: 10.1109/JAS.2017.7510829
Funds:

the Fundamental Research Funds for the Central Universities of China ZXH2012H005

the National Natural Science Foundation of China 61201085

the National Natural Science Foundation of China 51402356

the National Natural Science Foundation of China 51506216

the Joint Fund of National Natural Science Foundation of China and Civil Aviation Administration of China U1633101

the Joint Fund of the Natural Science Foundation of Tianjin 15JCQNJC42800

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  • Solving quaternion kinematical differential equations (QKDE) is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of quaternions nor avoid errors accumulated in the sense of long term time. We present explicit symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modelling its time-invariant and time-varying versions with Hamiltonian systems and adopting a three-step strategy. Firstly, a generalized Euler's formula and Cayley-Euler formula are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecticity, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the explicit symplectic geometric algorithm for the time-varying quaternion kinematical differential equation, i.e., a non-autonomous and non-linear Hamiltonian system essentially, is designed with the theorems proved. Our novel algorithms have simple structures, linear time complexity and constant space complexity of computation. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.

     

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