Singularity analysis and representation of the general Gough-Stewart platform

In this paper; the singularity loci of the Gough-Stewart platform are studied and a graphical representation of these loci in the manipulator's workspace is obtained. The algorithm presented is based on analytical expressions of the determinant of the Jacobian matrix, using two different approaches, namely, linear decomposition and cofactor expansion. The first approach is used to assess the effect of the architecture parameters on the nature of the singularity loci, while the second approach leads to a significant reduction of the computational complexity of the determinant. It is shown that, for a given orientation of the platform, the singularity locus in the Cartesian space is represented by a polynomial of degree three. Moreover; this polynomial equation is applied to several simplified Gough-Stewart architectures and it is shown that the expression is reduced when the base of the mechanism is coplanar and for other special geometries. A comparison with the results obtained using Grassmann geometry is then presented, which illustrates the advantages of using one single compact equation for the singularity loci. The generalization of Fichter's singular configuration is also developed, and several observations are then made. Finally, a brief discussion on the architecture singularities is presented and a graphical representation of the singularity loci in the Cartesian workspace of the manipulator is obtained. Two examples clearly illustrate the use fulness of the results for the analysis and design of parallel manipulators.