Abstract
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 Plucker 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.
Highlights
Pure Condition is a scalar value that is needed to identify both singularity configuration and rigidity of robot framework
This paper presented the geometric interpretation of pure condition which means simultaneous singularity and rigidity analysis on both 6R Serial Manipulators (SMs) through a variable orientation joint and 3-PRS Parallel Manipulators (PMs) within variable actuated joints with no coordinate approach based on Grassmann-Cayley Algebra (GCA)
It was deduced that wrist singularity for 6R SMs could be possible if and only if the design of two of the last three axes were parallel according to the orientation of the actuators
Summary
Pure Condition is a scalar value that is needed to identify both singularity configuration and rigidity of robot framework. Caro et al [12] have suggested the geometric method associated with dependency of Plücker vector lines to get the determinant of Matrix J which is formulated in GCA language [10] [13]. The key contribution in this paper is a simultaneous determination of both singularity condition of Robots Manipulators and rigidity framework without algebraic calculus by Grassmann-Cayley Algebra approach. This paper is organized as follows: Section 2 recalls mathematics background of projective space such as Plücker coordinate of vectors line, screw theory, twist system, Global Wrench System with their associate graphs before the concept of brackets which represent the Jacobian matrix used in GCA applied to robot manipulators.
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