Uncertainty Singularity Analysis of Parallel Manipulators Based on the Instability Analysis of Structures

When all the inputs of a parallel manipulator (PM) are locked, the manipulator is usually turned into a structure. When an uncertainty singularity occurs for a PM, the latter structure is unstable or, in other words, it may undergo infinitesimal or finite motion. Hence, the investigation of the uncertainty singularities of a PM can be reduced to the instability analysis of its corresponding structure. PMs with a 3-XS structure cover a broad class of PMs. A 3-XS structure is composed of two platforms connected by three XS serial chains in parallel. Here, X and S denote a generalized joint with one degree of freedom (DOF) and a spherical joint, respectively. An X joint can take the form of any kinematic joint with one DOF, such as a revolute joint or a prismatic joint, or the form of any closed kinematic chain with one DOF, such as a parallelogram. In this paper, the instability condition of the 3-XS structure is derived by simply differentiating its constraint equations. The geometric interpretation of the instability condition is revealed using a method based on linear algebra. The uncertainty singularity analysis of the 6-3 Gough-Stewart PM is performed to illustrate the application and efficiency of the proposed approach. Several specific cases of the 6-3 Gough-Stewart PM with singularity surfaces of reduced degree are proposed. The geometric interpretation of the singularity conditions is also given for some of the specific cases. KEY WORDS—singularity, parallel manipulators, instability analysis, singularity locus