This article develops two new algorithms of three-term conjugate gradient (TTCG) coefficients to handle nonlinear least-squares (NLS) problems using the structured secant equation. Motivated by improved conjugacy and sufficient descent conditions, we developed two new formulations of CG coefficients. Furthermore, with these parameters, two search directions are proposed that satisfy the sufficient descent condition, which further enhances the efficiency of the proposed strategies. One key advantage of the proposed techniques is their low memory requirements, rendering them suitable for large-scale nonlinear least squares problems. Moreover, some mild suppositions and a non-monotone line search are used to establish the global convergence properties of the methods. More so, we investigate the robustness and effectiveness of the proposed methods numerically by performing experiments on benchmark test problems, and their performance is compared against existing methods. The outcomes of these experiments indicate that the proposed methods outperform the other techniques regarding the metrics of comparison adopted. Finally, the algorithms are applied to an extension of the model of robotic motion control of four degrees of freedom (4DOF), resulting in positive outcomes for the robot�s motion traits. ©2024 The Authors.