| 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. A 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.