To prevent the singularity of serial robot 's

This research presents Pure Condition approach, which has used in analyzing simultaneously the singularity configuration and the rigidity of mechanism. The study cases analysis is implemented on variable joints orientation of 6R (Revolute) Serial Manipulators (SMs) and variable actuated joints position of 3-PRS (Prismatic-Revolute-Spherical) Parallel Manipulators (PMs) using Grassmann-Cayley Algebra (GCA). In this work we require in Projective Space both Twist System (TS) and Global Wrench System (GWS) respectively for serial and parallel manipulators which represent the Jacobian Matrix (J) in symbolic approach to Plücker coordinate vector of lines and unify framework on static and kinematics respectively. This paper, works, is designed to determine geometrically at symbolic level the vanished points of inverse form of this Jacobian Matrix (J) which called superbracket in GCA. The investigation vary to those reported early by introducing GCA approach on the singularity of serial robot, variable joints orientation and actuated positions on robot manipulators (RMs) to analyze rigidity frame work and singularity configuration which involve simultaneously Pure Condition. And the results also revealed a single singularity condition which contains all particulars cases and three general cases such as the shoulder, elbow and wrist singularity for SMs while double, single and undermined singularities respectively for 3-PRS, 3-PRS and 3-PRS PMs which contain all generals and particulars cases.

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