Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves

To identify the parameters of transition curves in as-built alignments by measured points, orthogonal least-squares fitting is studied on the basis of the parameter equation of transition curves. First, eigenvalue analysis has clarified that the Gauss-Newton (GN) algorithm usually fails to converge because of the existence of the ill-condition. Next, a bi-objective optimization model is proposed and the Levenberg-Marquardt (LM) algorithm combining the GN algorithm with the steepest descent method is constructed to fit a transition curve to points orthogonally. The LM parameter is updated dynamically during iterations according to the evaluation of the distance between the current and the optimum locations. Finally, Monte Carlo simulations are employed to test the performances of the GN and LM algorithms with measured points and the same 5 000 initial values. Experimental results show that the GN algorithm diverges while the LM algorithm converges to the same optimum under different initial values. The number of iterations, with an average of 16.8 times and the minimum of 5 times and the maximum of 50 times, is related to the distance between the initial and the optimum locations. The LM algorithm shows a better robustness than the GN algorithm.