Explicit Symplectic Geometric Algorithms for Quaternion Kinematical Differential Equation |
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Authors: | Hong-Yan Zhang Zi-Hao Wang Lu-Sha Zhou Qian-Nan Xue Long Ma Yi-Fan Niu |
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Affiliation: | 1.School of Information Science and Technology, Hainan Normal University, Haikou 571158, China2.Department of Engineering, Sino-European Institute of Aviation Engineering, Civil Aviation University of China, Tianjin 300300, China3.Civil Aviation University of China, Tianjin 300300, China |
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Abstract: | 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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Keywords: | Linear time-varying system navigation system quaternion kinematical differential equation (QKDE) real-time computation symplectic method |
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