Abstract:

The generalized nonlinear Gauss-Helmert model is a unified expression of explicit and implicit nonlinear function adjustment models that consider the errors of the dependent variable or the whole variable. Aiming at the problem of non-convergence of its Gauss-Newton iterative solution algorithm when the difference between the initial value and the true value is large, the parameter estimation method of the generalized nonlinear Gauss-Helmert model that integrates the homotopy method and nonlinear least squares is proposed. Starting from the nonlinear least-squares adjustment criterion that introduces the homotopy parameter, the system of differential equations for solving the generalized model parameters and the fixed-step prediction formula for tracking the homotopy curve with the Newton's correction formula are derived, and the approximation formula for calculating the residual vector of the implicit function model is given. The complexity of computing the system of differential equations is reduced by introducing the Kronecker product and the matrix straightening operation into the derivation process in order to avoid computing the cubic matrix. The feasibility of the method is verified through three experiments, including distance positioning that only considers the error of the independent variable, pseudo-distance positioning that considers the satellite coordinate error and ranging error, trilateration network that considered the errors in the known coordinates, and circular curve fitting that considers the error of plane coordinates. The experimental results show that the new method converges to a larger range of initial values.

Key words: nonlinear Gauss-Helmert, homotopy method, distance positioning, pseudo-distance positioning, circular curve fitting, trilateration network