  This paper introduces a methodology to analyze geometrically the
singularities of manipulators, of which legs apply both actuation forces and
constraint moments to their moving platform. Lower-mobility parallel
manipulators and parallel manipulators, of which some legs do not have any
spherical joint, are such manipulators. The geometric conditions associated
with the dependency of six Pl\"ucker vectors of finite lines or lines at
infinity constituting the rows of the inverse Jacobian matrix are formulated
using Grassmann-Cayley Algebra. Accordingly, the singularity conditions are
obtained in vector form. This study is illustrated with the singularity
analysis of four manipulators.
