Research On Dual-Quaternion Navigation Algorithm And Nonlinear Gaussian Filtering

The thesis has investigated two theoretical problems involved in navigation systems. One is the mathematical representation and computation of navigation information; the other is the optimal real-time fusion strategy, i.e., filtering, of navigation information. Inertial navigation is essentially to solve the kinematic problem of a three-dimensional rigid body. As a subset of geometry algebra, dual quaternion is the most concise and efficient mathematical tool to represent the general rigid motion. It can be used to address all rigid kinematic (and dynamic) problems, including inertial navigation of course. The first half of this thesis has done some pilot works on the strapdown inertial navigation theory founded on dual quaternion.1. Through reinterpreting the rationale of the strapdown inertial navigation in terms of dual quaternion algebra, we obtain three dual quaternion kinematic equations that take the same forms as the conventional attitude quaternion rate equation. Borrowing the traditional two-speed approach originally developed in conventional attitude integration, we design one new numerical integration algorithm to solve the three kinematic equations. The new navigation algorithm based on dual quaternion is thus constituted. It integrates the coning, sculling and scrolling corrections all together, simplifying the algorithm structure and implementation complexity.2. The new navigation algorithm is analyzed and compared with the conventional one from various aspects. The duality between the coning and sculling corrections, raised in the recent literature, is fundamentally explained. The superiority of the new algorithm in accuracy is analytically derived. A variety of simulations are carried out to support the analytic conclusions, including those with ideal inertial sensors and those with non-ideal inertial sensors. The numerical results agree well with the analyses. The new algorithm turns out to be a better choice than the conventional algorithm for high-precision navigation systems and high-maneuver applications. Several guidelines in choosing a suitable navigation algorithm are also provided based on the inertial sensors configuration and the turning frequency of the conventional algorithm.3. Error characteristics of the strapdown inertial navigation are investigated using dual quaternion. Two new error models in terms of quaternion algebra are developed: the additive dual quaternion error model and multiplicative dual quaternion error model. Both are expected to facilitate the future dual quaternion-based integrated navigation filter.4. As a sub-problem involved in the new navigation algorithm, the transformation from Earth-centered Earth-fixed coordinates to geodetic coordinates is addressed. We come up with an iterative approach using the Newton-Raphson method that has good efficiency and accuracy and is free from singularity and non-convergence except in a small region near the center of the Earth. Comparisons with other well-known approximate methods indicate the new transformation algorithm has higher accuracy and lower arithmetic complexity.On the other hand, inertial navigation must be integrated with extraneous information feedback to form a stable and closed-form integrated navigation system, so as to limit the error accumulation. The integrated navigation system reckons on an optimal on-line information fusing strategy, i.e., filtering, to integrate various information sources. Therefore, to some extent, nonlinear filtering is the cornerstone of an integrated...