Geometric characterization and parametric representation of the singularity manifold of a 6–6 Stewart platform manipulator

In this paper, we present a compact closed-form expression for the singularity manifold of a class of 6–6 Stewart platform manipulators most commonly used in research and industry. The singularity manifold is obtained as the hyper-surface in the task-space, SE(3), on which the wrench transformation matrix for the top platform degenerates. This condition leads to an extremely large expression containing algebraic and trigonometric functions of the architecture, position and orientation variables. We present algorithms for efficient symbolic simplification of such large expressions. Using these algorithms, for a given architecture and orientation  , the singularity manifold is obtained as a cubic surface in R3R3. The symbolic computations yield a simple parametric expression for the surface in terms of the architectural and orientation parameters of the manipulator, and allows us to completely characterize and visualize the singularity manifold. We show that, in general, the cubic surface is a one-parameter family of hyperbolas in planes parallel to the base of the manipulator. It is further shown that the hyperbola degenerates to a parabola in a unique plane, and to a pair of straight lines in four other planes. The explicit parameterization allows us to obtain the location of each of these special planes analytically. For a given architecture and position, the singularity manifold is a surface in SO(3), which can be, in general, algebraically described by a 6th degree polynomial in the Rodrigue’s parameters. In this paper, we present explicit expressions for the polynomial defining the orientation singularity manifold in terms architecture and orientation parameters. The theoretical results are illustrated with several numerical examples.