A New Newton-Type Iterative Formula for Over-Determined Distance Equations

Newton-type methods were ordinarily developed by using approximate Hessian matrices to solve the nonlinear equations. For dealing with ill-posed ranging resection problems, we propose a new Newton-type iterative formula based on the precise Hessian matrix, called closed-form Newton iterative formula. An orthogonal condition is introduced to the nonlinear least squares solution of an over-determined distance equation system. It is revealed that the solution is the barycentre of a particle system composed of unit masses at the endpoints of the ranging vectors. Then a closed-form of the Newton method is given by compactly expressing the Hessian matrix. The simulation result shows that the closed-form Newton method can improve the efficiency and the stability of the convergence, especially in the case of the ill-posed positioning configurations.