Nonlinear analysis of position estimate in global navigation satellite systems

The paper presents a nonlinear analysis of position estimation based on a global navigation satellite system. A classical problem formulation and iterative solution that results in the weighted least squares estimate of the receiver state are assumed. The analysis employs the Taylor's theorem to express the nonlinear measurement model using the first order Taylor polynomial at the state estimate and the Lagrange form of the remainder. A sensitivity analysis of the Jacobian matrix pseudoinverse is performed, and an upper bound on the size of the Lagrange remainder is derived using eigenstructure of the Hessian matrix. The results obtained show that both the sensitivity of the pseudoinverse and the size of the quadratic term are not significant, and thus the linear approximation commonly used to derive stochastic properties of the state estimate is reasonable. Although this result has been experimentally confirmed by numerous successful applications, this analysis can serve as a more rigorous basis when the design procedures for a safety critical system have to be satisfied