Iterative algorithms for attitude estimation using global positioning system phase measurements

Algorithms are sought for attitude determination using global positioning system (GPS) differential phase measurements, assuming that the cycle integer ambiguities are known. The problem of attitude determination is posed as a constrained parameter optimization problem, where a quaternion-based quartic cost function is used. A new general minimization scheme is developed. The new scheme is a continuous version of the well-known Newton-Raphson algorithm and is based on the solution of an ordinary differential equation. The new continuous algorithm converges exponentially from any initial condition to the closest local minimum located on the gradient direction in regions where the associated Hessian matrix is positive definite. Three new algorithms are developed for solving the attitude estimation problem, a discrete Newton-Raphson-based algorithm, a continuous Newton-Raphson algorithm, and an algorithm that is based on the eigenproblem structure of the nonlinear equations, which are related to the minimization of the quartic cost function. The performance of the new algorithms is evaluated via numerical examples and compared with each other and against the well-known QUEST algorithm. The continuous Newton-Raphson algorithm and the eigenproblem algorithm have similar accuracy. The discrete Newton-Raphson algorithm is less efficient than the continuous Newton-Raphson algorithm in the examined minimization because its search may wander and may even reach a nonrelevant extreme. When the GPS satellites are at low elevation, the accuracy of the new algorithms is better than that of QUEST, when the latter is applied to vectorized phase measurements.