Numerically stable covariance intersection for spacecraft formation flying
Banas, William D.
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The goal of this dissertation is to achieve more numerically stable attitude estimates for distributed systems of spacecraft. The method used is the Covariance Intersection (CI) algorithm, and this research continues and expands upon the foundation laid in the seminal Julier & Uhlman paper entitled A Non-divergent Estimation Algorithm in the Presence of Unknown Correlations and the body of related work that is now scarcely 15 years old. An optimization problem is developed using the quaternion to parameterize the attitude of each spacecraft. The norm constraint on the quaternion makes it necessary to augment the cost function using the method of Lagrange multipliers. Closed-form solutions for two or more quaternions have yet to be found (as far as the author or his collaborators are aware), and numerical solutions have been unstable and difficult, or they have not been demonstrated at all. Several numerically stable solutions are presented here. For two quaternions, a solution is found using a multi-variate Newton-Raphson algorithm and using built-in Matlab functions (employing only standard double precision variables), but these are not guaranteed to find the optimal solution. For two and three quaternions, homotopy continuation methods are used, which guarantee optimal solutions. For three quaternions, a square-root algorithm is developed for use with a homotopy continuation method. In the process, the design space of the Lagrange multipliers is explored, simplified forms of the quaternion constraints cost function are presented, an analogous optimization problem in two-dimensions is developed and analyzed, and some insight into the fundamental nature of the problem is gained.