Optimization of a Four-Bar Mechanism for Knee Prosthesis Using a Genetic Algorithm Based on Freudenstein’s Equation

Background: The natural motion of the human knee involves a combination of rotation and translation, resulting in a variable Instantaneous Center of Rotation (ICR) throughout the gait cycle. Traditional prosthetic knee designs often fail to reproduce this complex kinematic behavior. Objectives: This study aims to propose a customized, biomimetic knee mechanism through the synthesis of a four-bar linkage capable of approximating the physiological ICR trajectory with high precision. Methods: A Genetic Algorithm (GA) was implemented to optimize the geometric parameters of the four-bar mechanism, specifically its link lengths and inter-link angles. The optimization process is based on Freudenstein’s equation, which analytically relates the input and output angles of the linkage to the lengths of its links. The desired ICR trajectory was derived from experimental data, and the objective function minimized the Euclidean error between the generated and target trajectories. Results: The proposed method yielded customized mechanisms that closely approximate the target ICR curves, achieving an overall mean Euclidean tracking error of 1.95% (±1.68%) across diverse patient profiles, with a best-case optimization error as low as 0.355%. Furthermore, the GA demonstrated high computational efficiency, converging on optimal geometric configurations in an average execution time of just 3.98 min. Conclusions: These numerical results validate the robustness of the GA in navigating the design space while strictly adhering to kinematic constraints, Grashof’s condition, and anatomical motion limits. The integration of Freudenstein’s equation with GA-based optimization techniques enables the customized synthesis of four-bar linkages with a high capacity to reproduce the physiological kinematics of the knee. This computational approach could be highly beneficial for the design of polycentric knee prostheses, as it reduces design and manufacturing time by providing the initial parameters for the development of the four-bar mechanism, ultimately ensuring a better biomechanical fit between prosthetic and natural human movement.