Attitude estimation with gps-like measurements

The paper presents a walk-through of different aspects of estimation which occur if we face attitude measurements consisting of projections of precisely known directions in inertial space onto body-line vectors of prescribed direction and length. Emphasis is put on both the geometrical aspects and the estimation optimality with the purpose of identifying and exploiting any mutually independent geometrical constraints and to take any stochastic detail we know of into account. The estimation problem is broken down into two steps. In the first step we determine the body-line vectors in inertial space. In the second step we estimate the rotation from these determined line-vectors onto the line-vectors of the body reference system. We consider three situations. First, we only determine the rotation, provided that the line-vectors in the body system are perfectly known. Second, we tackle the case where the body-line vectors are inaccurately known and we optimally determine a rotation and the mismatch between the observed line-vector geometry and the body-line vectors. Finally, we also look at the ultimate calibration limit of this mismatch, employing perfect attitude knowledge. For all three cases we present numerical examples based on a GPS scenario, in which three antennas are electronically connected to give rise to three baselines, each able to measure the GPS carrier phase differences to a precision of ± one millimeter standard deviation. Similarly, inaccuracies of the body-line vectors in the order of one millimeter standard deviation are added to each Cartesian coordinate at each end of the baselines. All results of Monte-Carlo simulations involving the attitude determination are compared with the results obtained by means of a method based on the Wahba problem approach. Ultimately, we study the attitude estimation accuracies in the context of stochastic variations of the directions of either three or five lines of sight of GPS satellites. Measures are worked out to sample data so as to improve the attitude estimation accuracy.