Generalized Wahba Problems for Spinning Spacecraft Attitude and Rate Determination

Abstract

Two generalized versions of Wahba’s attitude determination problem have been developed for a spinning spacecraft, and a restricted version of one problem has been solved in closed-form. These problems seek to estimate both attitude and rate based solely on a time series of vector attitude observations along with a spacecraft dynamic model. Algorithms that solve these problems will be useful for spin-stabilized spacecraft that reduce complexity by omitting rate gyros. The first generalized Wahba problem presumes that the spin axis is known and that the spin rate is constant but unknown, as for a spinning spacecraft that has a nutation damper. The second generalized problem includes full rigid-body Euler dynamics, which allow for nutations, and seeks to estimate the unknown initial attitude rate vector. Both problems are recast into the K-matrix form of Wahba’s problem with K matrices that depend on the unknown rates. Restricted problems are developed that use the minimum number of vector measurements, two for the first problem and three for the second problem. The restricted first problem is solved in closed-form. The restricted second problem is shown to be observable, and it is reduced to a small system of nonlinear equations in the axially symmetric case. The possibility of deriving global solutions for these problems makes them attractive to assist or replace an extended Kaiman filter because a global solution cannot suffer from nonlinear divergence.