A New Iterative Method for Solving Nonlinear Equation

Abstract

Nonlinear equations are known to be difficult to solve, and numerical methods are used to solve systems of nonlinear equations. The objective of this research was to evaluate the potential application of a hybrid mathematical approach to improve the stability of nonlinear equations, with a focus on inverse kinematics. This hybrid approach combines the Newton’s method, an existing iterative technique, with the Vector Epsilon Algorithm, a type of convergence accelerator. However, problems arise when the method diverges. In this research, four numerical methods (all based on the classical Newton’s method) were used to solve 3 cases studies: 1) a sinusoidal function, which is fundamental to kinematic analyses where position vectors are defined by trigonometric functions; 2) a robot arm with two links pivoted about a pin; and 3) a robot arm with two links on a changeable pole. For single degree-of-freedom problem, the Newton’s method was not able to converge to the closest root when the initial guess was close to a critical point. However, other numerical methods, such as the hybrid method, were able to converge. Throughout the research, inverse kinematics problems were solved, and results are presented for both existing and new methods.