A Structured Liu-Storey-type Conjugate Gradient Approach for Nonlinear Least-Squares with Application to a 3DOF Robot Model

A nonlinear least squares (NLS) problem involves finding the parameters that minimize the sum of the squares of the differences between observed and predicted values in a nonlinear mathematical model. The classical methods for solving NLS problems require calculating the objective function's gradient and the Hessian matrix. However, because it is difficult to acquire inexpensive second derivatives of the residuals function, so handling the complete Hessian matrix takes time. This paper proposes a modification of the Liu-storey type (LS) formula by deriving a structured vector approximation of the Hessian of the objective function using Taylor series approximation. The sufficient descent results of the new formula are established using some assumptions. The research conducts experiment on several benchmark functions to showcase the computing efficiency of the new technique. The outcomes show the effectiveness of the algorithm compared to other available techniques. Finally, we implement the method to solve the inverse kinematics problem for a 3DOF robotic manipulator. The goal is to minimize the error in reaching the desired trajectory configuration accurately.