Abstract—This paper presents a kinematic analysis of a three-link manipulator system, focusing on the calculation and visualization of joint angles (q₁, q₂, q₃) as functions of the end-effector coordinates (x_S, y_S) and the orientation angle (α). The goal is to determine the values of the generalized coordinates (q₁, q₂, q₃) that allow the manipulator's Working Body (WB) to reach a given spatial point S in its workspace. The system's configuration and inverse kinematics problem are addressed through geometric and trigonometric constraints, with the joint angles determined by solving relevant equations involving the manipulator's link lengths and the positioning of the end-effector. We introduce a redundancy elimination technique by minimizing the positioning error, ensuring the accuracy of the manipulator's trajectory. The results are visualized using 3D plots, which depict the relationship between the joint angles and the coordinates of the manipulator, aiding in the understanding of the system's behavior under different configurations. The proposed method demonstrates efficient handling of kinematic redundancy and offers insights into optimizing the manipulator's positioning accuracy. The findings contribute to improving the control and motion planning of robotic systems in precision tasks.

Keywords—end-effector, manipulator's positioning accuracy, visualization, manipulator’s trajectory, robotic systems

Cite: Marzhan Azilkiyasheva, Kuanysh Alipbayev, Algazy Zhauyt, Alisher Aden, and Aray Orazaliyeva, "Solution of the Positioning Problem for a Planar Parallel Manipulator Based on a Third-Class Mechanism," International Journal of Mechanical Engineering and Robotics Research, Vol. 14, No. 4, pp. 454-464, 2025. doi: 10.18178/ijmerr.14.4.454-464

Copyright © 2025 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).