Algorithms for attitude determination using the global positioning system

Techniques are discussed for attitude determination using the global positioning system (GPS) differential phase measurements, assuming that the cycle integer ambiguities are known. The problem of attitude determination is posed as a parameter optimization problem. One proposed set of optimal solutions, which includes solutions of Wahba's problem, is based on least-squares fit of some attitude parameters to a set of vector measurements. The use of these algorithms requires the conversion of the basic GPS scalar phase measurements into unit vectors. It is shown that when the GPS antennas constitute tao axes of a Cartesian coordinate system, the conversion is immediate. When this is not the case, a more elaborate transformation is required. The necessary conversion formulae for both cases are developed and demonstrated in an example. Another possible approach is based on a least squares fit of the attitude quaternion to the GPS phase measurements themselves. The cost function of the fit is given in the literature in the most straightforward formulation as a function of the attitude matrix. Conversion is presented of the matrix-based cost function to a quaternion-based cost function that corresponds to the cost function minimized by QUEST. However, unlike the QUEST cost function, the converted cost function is not a simple quadratic form; therefore, the simple QUEST solution is not applicable in this case. An iterative solution for finding the optimal quaternion is derived and demonstrated through numerical examples. The algorithms can handle cases of planar antenna arrays and, thus, cover a deficiency in earlier algorithms.