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.
References
[1]
Downing, D.M., Samuel, A.E. and Hunt, K.H. (2002) Identification of the Special Configurations of the Octahedral manipulator Using the Pure Condition. The International Journal of Robotics Research, 21, 147.
https://doi.org/10.1177/027836402760475351
[2]
Hayes, M.J.D., Husty, M.L. and Zsombor-Murray, P.J. (2002) Singular Configurations of Wrist-Partitioned 6R Serial Robots: A Geometric Perspective for Users. Transactions of the Canadian Society for Mechanical Engineering, 26, 41-55.
https://doi.org/10.1139/tcsme-2002-0003
[3]
Chen, Y.-C. and Chen, C.L.P. (1994) An Analysis to the Singularity of Serial Manipulator Using Reciprocal Screw Theory. IEEE International Conference on Systems, Man, and Cybernetics, Humans, Information and Technology, 1, 148-153.
[4]
Boudreau, R. and Podhorodeski, R.P. (2010) Singularity Analysis of a Kinematically Simple Class of 7 Jointed Revolute Manipulators. Transactions of the Canadian Society for Mechanical Engineering, 34, 105-117.
[5]
Carretero, J.A., Podhorodeski, R.P., Nahon, M.A. and Gosselin, C.M. (2000) Kinematic Analysis and Optimization of a New Three Degree-of-Freedom Spatial Parallel Manipulator. ASME. Journal of Mechanical Design, 122, 17-24.
[6]
Tchoń, K. and Muszysńki, R. (1998) Singular Inverse Kinematic Problem for Robotic Manipulators: A Normal Form Approach. IEEE Transactions on Robotics and Automation, 14, 93-104. https://doi.org/10.1109/70.660848
[7]
Rocha, D.M. and da Silva, M.M. (2013) Workspace and Singularity Analysis of Redundantly Actuated Planar Parallel Kinematic Machines. Proceedings of the XV International Symposium on Dynamic Problems of Mechanics.
[8]
Husty, M., Ottaviano, E. and Ceccarelli, M. (2008) Advances in Robot Kinematics: Analysis and Design. A Geometrical Characterization of Workspace Singularities in 3R Manipulators. Springer Netherlands, 411-418.
[9]
Selvakumar, A.A., Babu, R.P. and Sivaramakrishnan, R. (2012) Simulation and Singularity Analysis of 3-PRS Parallel Manipulator. Proceedings of IEEE, International Conference on Mechatronics and Automation, 2203-2207.
[10]
Kanaan, D., Wenger, P., Caro, S. and Chablat, D. (2009) Singularity Analysis of Lower Mobility Parallel Manipulators Using Grassmann-Cayley Algebra. IEEE Transactions on Robotics, 25, 995-1004. https://doi.org/10.1109/TRO.2009.2017132
[11]
Amine, S., Tale-Masouleh, M., Caro, S., Wenger, P. and Gosselin, C. (2011) Singularity Analysis of the 4-RUU Parallel Manipulator using Grassmann-Cayley Algebra. CCToMM Symposium on Mechanisms, Machines and Mechatronics, Montreal.
[12]
Amine, S. and Caro, S. Singularity Conditions of Lower-Mobility Parallel Manipulators Based on Grassmann-Cayley Algebra. L’Agence nationale de la recherche.
http://www.irccyn.ec-nantes.fr/~caro/SIROPA/GUIGCASiropa.jar
[13]
Arvin, R. and Mehdi Tale, M. Singularity Configurations Analysis of a 4-DOF Parallel Mechanism Using Grassmann-Cayley Algebra.
[14]
Ben-Horin, P. and Shoham, M. (2009) Application of Grassmann-Cayley Algebra to Geometrical Interpretation of Parallel Robot Singularities. The International Journal of Robotics Research, 28, 127-141. https://doi.org/10.1177/0278364908095918
[15]
Amine, S., Masouleh, M.T., Caro, S., Wenger, P. and Gosselin, C. (2012) Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures Semaan. ASME Journal of Mechanisms and Robotics, 4, 1.
[16]
Wen, K., Won Seo, T., et al. (2015) A Geometric Approach for Singularity Analysis of 3-DOF Planar Parallel Manipulators Using Grassman-Cayley Algebra. Cambridge University Press, Robotica, 1-10.
[17]
Amine, S., Caro, S., Wenger, P. and Kanaan, D. (2012) Singularity Analysis of the H4 Robot Using Grassmann-Cayley Algebra. Robotica, 1-10.
[18]
Staffetti, E. and Thomas, F. (2000) Kinestatic Analysis of Serial and Parallel Robot Manipulators Using Grassman-Cayley Algebra. In: Lenarcic, J. and Stanisic, M.M., Eds., Advances in Robot Kinematics, Springer Netherlands, 17-27.
https://doi.org/10.1007/978-94-011-4120-8_2
[19]
Norton, R.L. (2001) Design of Machinery. An Introduction to the Synthesis and Analysis of Mechanism and Machines. 2 Edition, McGraw-Hill Series in Mechanical Engineering, 32.
