Workspace and Singularity Analysis of 4-DOF 3R1T Parallel Mechanism with a Circular Rail

Limited workspace and singularities are major challenges for parallel mechanisms. This article addresses these issues for a 4-DOF 1-SPS/3-RRRRR parallel mechanism with a circular rail, proposed in our prior work. The mechanism has a 3R1T motion type with a movable center of spherical motion. The paper begins with a detailed description of the mechanism design. A closed-form solution of the inverse kinematics follows next, which computes the active joint coordinates and determines the spatial positions of all joints and links. Based on this solution, an iterative approach is applied to analyze the workspace for three different heights of the spherical motion center. The analysis reveals the regions of a full twist about the platform symmetry axis, bounded by maximum tilt angles of 51°, 38°, and 23°, respectively. Introducing joint constraints significantly reduces the workspace, limiting the tilt angles to 21°, 26°, and 0° at the same heights. Subsequently, screw theory is applied to identify serial, parallel, and constraint singularities, and an iterative approach is used to find the boundary of the singularity-free workspace. The analysis shows that the full-twist tilt angles are limited to 33°, a value determined solely on the platform geometry and independent of the spherical motion center height. These results establish a foundation for the design optimization and prototyping of the mechanism.