Even Order Explicit Symplectic Geometric Algorithms for Solving Quaternions in Guidance Navigation and Control via Diagonal Padé Approximation and Cayley Transform

TL;DR

The Padé-Cayley lemma is proved and used to simplify the symplectic Padé approximation for the linear Hamiltonian system with infinitesimal symplectic structure and both parallel and alternative iterative methods are proposed to construct the symplectic difference schemes with even order accuracy.

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 Padé approximation via a four-step strategy. Firstly, the Padé-Cayley lemma is proved and used to simplify the symplectic Padé 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 $\mathcal{O}((t_f-t_0)τ^{2\ell})$ where $τ$ 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.

Figures (14)

• Figure 1: Flow chart of the EoESGA for solving the LTV-QKDE with accuracy parameter $\ell$.
• Figure 2: Flow chart of computing the key auxiliary parameters $c$ and $\beta$.
• Figure 3: Iterative processes for calculating the coefficients $a^\ell_j$ and $b^\ell_j$ correspond to paths on graph.
• Figure 4: Impact of $\ell$ and $c$ on $\beta(\ell,c)$
• Figure 5: Compute the Parameter $\eta^\ell_k$

Theorems & Definitions (9)

• Theorem 1
• Lemma 2: Generalized Euler's formula
• Lemma 3: Cayley-Euler formula
• Lemma 4: Padé-Cayley
• Lemma 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9