This research presents Pure Condition approach, which has used in analyzing\nsimultaneously the singularity configuration and the rigidity of mechanism.\nThe study cases analysis is implemented on variable joints orientation of 6R\n(Revolute) Serial Manipulators (SMs) and variable actuated joints position of\n3-PRS (Prismatic-Revolute-Spherical) Parallel Manipulators (PMs) using\nGrassmann-Cayley Algebra (GCA). In this work we require in Projective\nSpace both Twist System (TS) and Global Wrench System (GWS) respectively\nfor serial and parallel manipulators which represent the Jacobian Matrix (J) in\nsymbolic approach to Plücker coordinate vector of lines and unify framework\non static and kinematics respectively. This paper, works, is designed to determine\ngeometrically at symbolic level the vanished points of inverse form of\nthis Jacobian Matrix (J) which called superbracket in GCA. The investigation\nvary to those reported early by introducing GCA approach on the singularity\nof serial robot, variable joints orientation and actuated positions on robot\nmanipulators (RMs) to analyze rigidity frame work and singularity configuration\nwhich involve simultaneously Pure Condition. And the results also revealed\na single singularity condition which contains all particulars cases and\nthree general cases such as the shoulder, elbow and wrist singularity for\nSMs while double, single and undermined singularities respectively for\n3-PRS, 3-PRS and 3-PRS PMs which contain all generals and particulars\ncases.
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