Analysis of a Nonlinear Least Squares Procedure Used in Global Positioning Systems

Iterated least squares (ILS) is a widely used parameter estimation algorithm for nonlinear least squares problems. The ILS estimation error covariance is usually written
(
GTR
^-1
G)
^-1
, where G is the Jacobian matrix at the solution and
R
is the noise covariance. Using a first-order expansion of the “gain matrix” in ILS, we provide a rigorous justification for the covariance formula. The analysis includes uncertainty in the initial estimate and is capable of modeling the transient performance of the algorithm. Given convergence, the usual ILS covariance is obtained asymptotically. The analysis makes use of matrix differential calculus to obtain the first differential of the ILS gain matrix, and includes, as a special case, the R=I case, where the gain matrix is the pseudoinverse of the Jacobian matrix. The results are harnessed to obtain a sensitivity analysis of the ILS algorithm to additional random parametric variations. The analysis is then applied to a Global Positioning System problem to characterize the effect of ephemeris errors on the ILS position estimates. Results from a comparative Monte Carlo simulation demonstrate the approach's effectiveness.