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This paper presents the singularity analysis of four 3-DOF symmetric zero-torsion parallel mechanisms. These mechanisms are composed of three identical legs ending with a spherical joint that is constrained to move in one of three equally spaced planes intersecting at one line. The computation of the singularity loci is based on the degeneracy of the system of screws applied on the platform by the legs. The whole study is based on the use of a special orientation representation, previously introduced under the name of Tilt-and-Torsion angles. This representation is briefly introduced. Then the interdependence between the Cartesian coordinates of the general class of parallel mechanisms is derived. Finally, the singularity loci are derived and the size of the workspace taking into account all singular configurations is shown.