Even Order Explicit Symplectic Geometric Algorithms for Solving Quaternions in Guidance Navigation and Control via Diagonal Pad\'{e} Approximation and Cayley Transform

TL;DR

This paper introduces high-order explicit symplectic algorithms for solving quaternion kinematic differential equations, enhancing accuracy and efficiency for real-time navigation and control applications.

Contribution

The paper develops a unified framework for constructing high-order symplectic difference schemes using Padé approximation and Cayley transform, with rigorous proofs of symplectic properties.

Findings

Maximum absolute error scales with rac{1}{2 ext{ell}}

Algorithms have linear time complexity and constant space complexity

Validated through mathematical analysis and numerical simulations

Abstract

Quaternion kinematical differential equation (QKDE) plays a key role in navigation, control and guidance systems. Although explicit symplectic geometric algorithms (ESGA) for this problem are available, there is a lack of a unified way for constructing high order symplectic difference schemes with configurable order parameter and the fractional interval sampling problem should be treated carefully. We present even order explicit symplectic geometric algorithms to solve the QKDE with diagonal Pad\'{e} approximation via a four-step strategy. Firstly, the Pad\'{e}-Cayley lemma is proved and used to simplify the symplectic Pad\'{e} approximation for the linear Hamiltonian system with infinitesimal symplectic structure. Secondly, both parallel and alternative iterative methods are proposed to construct the symplectic difference schemes with even order accuracy. Thirdly, the symplecity, orthogonality and invertibility of the single-step transition matrices are proved rigorously. Finally, the explicit symplectic geometric algorithms are designed for both the linear time-invariant and linear time-varying QKDE. The maximum absolute error for solving the QKDE is $\mathcal{O}((t_f-t_0)\tau^{2\ell})$ where $\tau$ is the time step, $\ell$ is the order parameter and $[t_0,t_f]$ is the time span. The linear time complexity and constant space complexity of computation as well as the simple algorithmic structure show that our algorithms are appropriate for real-time applications in aeronautics, astronautics, robotics and so on. The performance of the proposed algorithms are verified and validated by mathematical analysis and numerical simulation.