Iterative algorithms for attitude estimation using global positioning system phase measurements

Algorithms are sought for attitude determination using global positioning s ystem (GPS) differential phase measurements, assuming that the cycle intege r ambiguities are known. The problem of attitude determination is posed as a constrained parameter optimization problem, where a quaternion-based quar tic cost function is used. A new general minimization scheme is developed. The new scheme is a continuous version of the well-known Newton-Raphson alg orithm and is based on the solution of an ordinary differential equation. T he new continuous algorithm converges exponentially from any initial condit ion to the closest local minimum located on the gradient direction in regio ns where the associated Hessian matrix is positive definite. Three new algo rithms are developed for solving the attitude estimation problem, a discret e Newton-Raphson-based algorithm, a continuous Newton-Raphson algorithm, an d an algorithm that is based on the eigenproblem structure of the nonlinear equations, which are related to the minimization of the quartic cost funct ion. The performance of the new algorithms is evaluated via numerical examp les and compared with each other and against the well-known QUEST algorithm . The continuous Newton-Raphson algorithm and the eigenproblem algorithm ha ve similar accuracy. The discrete Newton-Raphson algorithm is less efficien t than the continuous Newton-Raphson algorithm in the examined minimization because its search may wander and may even reach a nonrelevant extreme. Wh en the GPS satellites are at low elevation, the accuracy of the new algorit hms is better than that of QUEST, when the latter is applied to vectorized phase measurements.