Critical Poses Investigation of Hybrid Parallel Robot with No Identical Legs Based on Double Algebra

In this paper, the critical poses of PPS-RRS-PRS Hybrid Parallel Robot Manipulators (HPRMs) are geometrically investigated in Double Algebra (DA) approach. The screw theory and a reciprocal screw of dyad joints borrowed from the projective space to obtain the geometrically symbolic form of the inverse of the Jacobian matrix (J) which is expressed in the Global Wrench System (GWS) term called superbrackets. These superbrackets mean the symbolic form of the joints screw lines which are the Plücker coordinate finite lines or lines at infinity related to the Hybrid Parallel Robot Manipulators (HPRMs). The critical configurations arise when these Plücker coordinate lines vectors become linearly dependent at the vanished points of the superbrackets. The results of the investigation are the following: the four planes defined by the position of the joints intersected at last at one point which means that the fourth plane passes through the point defined by the other three. Both the base frame and mobile platform lie in a parallel plan. The key contribution in this paper is a determination of singularity condition of Robots Manipulators and rigidity framework without algebraic calculus by Grassmann-Cayley Algebra approach. This paper calculated the determinant of the Jacobian Matrices in a coordinate- free manner by developing and reducing the Superbracket expression. A novelty of this research from other research is that the Hybrid Parallel Robot with no identical leg was cheeked and investigated by Grassmann-Cayley Algebra. Not only fully parallel robot may be studied using Double Algebra, but no-identical legs hybrid parallel robot should be also analysed using Double Algebra.