Symplectic Geometric Algorithm for Quaternion Kinematical Differential Equation

TL;DR

This paper introduces symplectic geometric algorithms for solving quaternion kinematical differential equations, ensuring norm preservation and reducing long-term errors in applications like aerospace and navigation.

Contribution

The paper develops novel symplectic algorithms for both autonomous and non-autonomous quaternion equations, with rigorous proofs of their properties and practical implementation advantages.

Findings

Algorithms preserve quaternion norm over long-term simulations.

Proved symplecticity, orthogonality, and invertibility of transition operators.

Validated efficiency and accuracy through numerical simulations.

Abstract

Solving quaternion kinematical differential equations 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 symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modeling its time-invariant and time-varying versions with Hamiltonian systems by adopting a three-step strategy. Firstly, a generalized Euler's formula for the autonomous quaternion kinematical differential equation are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecitiy, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the main results obtained are generalized to design symplectic geometric algorithm for the time-varying quaternion kinematical differential equation which is a non-autonomous and nonlinear Hamiltonian system essentially. Our novel algorithms have simple algorithmic structures and low time complexity of computation, which are easy to be implemented with real-time techniques. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.