Numerical Solution of a Generalized Wahba Problem for a Spinning Spacecraft

Abstract

A numerical solution algorithm has been developed to solve a generalization of Wahba's attitude determination problem to the case of unknown attitude and attitude rate. It provides a robust global solution to nonlinear batch attitude and rate estimation problems for spinning spacecraft. The original Wahba problem seeks the attitude that fits a set of unit-normalized direction vector measurements. The generalized problem seeks an initial attitude and rate that, when combined with a torque-free rigid-body attitude dynamics model, fit a time series of direction vector measurements. The new algorithm combines an inner analytic solution for the unknown initial attitude quaternion with an outer numerical solution for the unknown initial rate. The inner problem can be cast in a standard Wahba form, and it is solved in closed form using an adaptation of the q method. The outer problem relies in a trust-region implementation of Newton's method in order to ensure global convergence. The global solution is determined by re-solving starting from a random set of first guesses of the attitude rate that cover the space of non-aliasing rates. Tests on truth-model simulation data for minor-axis and major-axis spinning spacecraft confirm the new algorithm's ability to achieve global convergence and its estimation accuracy's consistency with its computed estimation error covariance matrix.