| Parallel mechanisms are being introduced as platforms for machine tools, where stiffness and vibration supression is vital. A novel parallel architecture, called double triangular (DeltaDelta), was proposed and studied in depth by Daniali in anticipation of applications which require fast and precise motion. However, Daniali failed to find the minimal solution to the spherical DeltaDelta parallel manipulator, which he suspected is quadratic. Here, projective geometry and Grassmannian incidence relationships are used to unify the method of direct kinematic analysis (DKP) of two types of three degree-of-freedom manipulators, viz., the planar and spherical versions of DeltaDeltaPM, while preserving the geometric meaning of the solution. This method was to demonstrate for the first time that SDeltaDeltaPM can have only two real assembly modes; however, the minimum solution is found to be of order eight. An example is included to show that SDeltaDeltaPM can actually possess eight real assembly modes. Only two of the eight real poses lie within the workspace, while the other six arise due to a quadruple triangular tesselation of the sphere. Furthermore, a method to solve the general three-points-on-three-lines problem along with application to statics and spatial parallel manipulators is presented. |
