Pure Condition of Both 6R Serial and 3-PRS Parallel Robots Using Grassmann-Cayley Algebra

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.

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.