Under the term parallel manipulators of the Stewart-Gough type, we summarize mechanisms, where the moving platform is connected to the base by a certain number of prismatic (P) legs according to the robots degree of freedom (dof). For planar devices, the legs are anchored by passive revolute (R) joints, and for spatial ones by passive spherical (S) joints.
In so-called singular (also known as shaky) configurations these manipulators gain at least one instantaneous DoF. Therefore, minor variations in the manipulator geometry (e.g. backlash in passive joints or uncertainties in the actuation of the P joints) can significantly affect the realized configuration. Another phenomenon that also appears close to these configurations is that the prismatic actuator forces can become very large, resulting in a breakdown of the manipulator. Therefore, singular configurations and their vicinity should be avoided. In this context we consider 3-RPR manipulators and present a comparison of singular distances with respect to extrinsic and intrinsic metrics along a 1-parametric motion. Note that different metrics can be used depending on the chosen interpretations of the platform/base; e.g. as triangular plate or as pin-jointed triangular bar structure.
There also exist so-called architecture singularities referring to robot designs, which are shaky in every configuration. Clearly, these designs have to be avoided but also their vicinity, as every anchor point can be associated with a space of uncertainties (e.g. tolerances in the passive joints or deviations of the platform/base from the geometric model). In this context we consider linear pentapods (5-SPS manipulators with linear platform) and present an approach to measure the distance of a given design from being architectural singular.
For both kinds of singularities the distances are computed as the global minima of constrained optimization problems. Their critical points are found through a generic computational pipeline that relies on algorithms from symbolic and numerical algebraic geometry implemented in Maple, Bertini and Paramotopy. Note that we do not only obtain the singularity distance but also the corresponding closest singular configuration and architecture singularity, respectively. All presented approaches are demonstrated on the basis of illustrative examples.