Optimal Attitude Matrix from Two Vector Measurements

A SIMPLE counting argument shows that a minimum of two unit-vector measurements, with two independent scalar pieces of information each, are required to determine the 3 degrees of freedom needed to specify a spacecraft’s attitude. The earliest published algorithm for attitude determination from two vector measurements was Black’s TRIAD algorithm [1,2], also known as the algebraic method [3]. TRIAD has been applied to both groundbased and onboard attitude determination, using either unit vectors to the sun and along the Earth’s magnetic field for a coarse “sun-mag” attitude determination or unit vectors to two stars for a precise attitude determination. TRIAD is suboptimal because it ignores one piece of information from one of the unit vectors. Modern star trackers can track 5, 6, or even 50 stars at a time, creating a need for methods that can use all of the information from two or more unit vectors, weighted in some “optimal” fashion. This need had been anticipated in 1965, when Wahba proposed her famous problem to find the proper orthogonal matrix A (i.e., with determinant 1) that minimizes the loss function [4]