DEVELOPMENT OF AN ALGORITHM FOR DETERMINING THE POSITION OF A SPHERICAL PARALLEL MECHANISM PLATFORM IN 3D SPACE AND ITS SOFTWARE IMPLEMENTATION

Abstract

The paper addresses the problem of mathematical modeling and software implementation of algorithms for spherical parallel mechanisms (SPM), which are widely used in modern engineering due to their high rigidity and accuracy. Potential applications of SPM include training simulators, robotic manipulators, medical equipment, and stabilization systems. It is shown that the key task is not only to reproduce the spatial orientation of the moving platform but also to develop a mathematical framework for determining relative angular displacements and direct control signals for servo drives. The method is proposed for calculating the coordinates of the moving platform based on rotation matrices, which allows precise consideration of rotations relative to the X, Y, and Z axes. Special attention is given to determining the coordinates of the intermediate reference of the mechanism located at the circle in the XY-plane. The system of geometric and kinematic equations is formulated, the solution of which yields multiple possible coordinates of the mechanism links positions. To select the only physically feasible solution, a multi-step filtering algorithm has been developed, including circle-membership verification, minimum distance checks, link intersection analysis, and the choice of the coordinates configuration closest to the previous state. The study introduces a method for determining the relative angular displacements of the intermediate points, enabling the estimation of both direction and magnitude of their shifts. This provides a transition from the platform’s conditional rotations to the computation of servo drive rotation angles, which act as direct control signals. Based on the developed algorithm, a software module was implemented in Python using NumPy and Matplotlib libraries, enabling both numerical calculations and two- and three-dimensional visualization of the mechanism’s operation. The results obtained confirm the adequacy of the mathematical model and its suitability for integration into automated control systems. The proposed approach ensures accurate tracking of platform position, determination of relative angles, and calculation of control signals for servo drive. This makes the method applicable in robotic systems, stabilization systems, training simulators, and medical devices, where high precision, reliability, and continuous control are crucial.