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.