Abstract
This paper presents a new method based on geometric algebra for the singularity analysis of 3-degrees of freedom overconstrained 3-RPR planar parallel manipulators. Constraint wrenches acting on the moving platform are obtained using the outer product and dual operations. After the redundant constraint wrenches are identified and removed, a singular polynomial is derived as the coefficient of the outer product of all the non-redundant constraint wrenches. This polynomial provides an overall perspective of the singularity of the 3-RPR parallel manipulators and enables the drawing of 3-dimensional singularity loci, which are important in trajectory planning and workspace design. The main advantage of using geometric algebra is the compact and geometrically intuitive formulation of the singularity polynomial of the 3-RPR parallel manipulators.
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Bayro-Corrochano, E., Daniilidis, K., Sommer, G.: Motor algebra for 3D kinematics: the case of the hand-eye calibration. J. Math. Imaging Vis. 13, 79–100 (2000)
Bayro-Corrochano, E., Reyes-Lozano, L., Zamora-Esquivel, J.: Conformal geometric algebra for robotic vision. J. Math. Imaging Vis. 24, 55–81 (2006)
Bayro-Corrochano, E.: Geometric Computing: For Wavelet Transforms, Robot Vision, Learning. Control and Action. Springer Publishing Company, Incorporated, Berlin (2010)
Ben, H.P., Shoham, M.: Singularity analysis of a class of parallel robots based on Grassmann–Cayley algebra. Mech. Mach. Theory 41(8), 958–970 (2006)
Ben, H.P., Shoham, M.: Singularity condition of six-degree-of-freedom three-legged parallel robots based on Grassmann–Cayley algebra. IEEE Trans. Robot. 22(4), 577–590 (2006)
Bonev, I.A., Zlatanov, D., Gosselin, C.M.M.: Singularity analysis of 3-DOF planar parallel mechanisms via screw theory. J. Mech. Des. 125(3), 573 (2003)
Chen, X., Liu, X.J., Xie, F.: Screw theory based singularity analysis of lower-mobility parallel robots considering the motion/force transmissibility and constrainability. Math. Probl. Eng. (2015) 2015, 487956-1–487956-11 (2015). doi:10.1155/2015/487956
Choi, H.B., Konno, A., Uchiyama, M.: Analytic singularity analysis of a 4-DOF parallel robot based on Jacobian deficiencies. Int. J. Control. Autom. 8(2), 378–384 (2010)
Clifford, P.: Applications of Grassmann’s extensive algebra. Am. J. Math. 1(4), 350–358 (1878)
Collins, C., Mccarthy, J.: The quartic singularity surfaces of planar platforms in the Clifford algebra of the projective plane. Mech. Mach. Theory 33(7), 931–944 (1998)
Daniali, H.M., Zsombor, M.P., Angeles, J.: Singularity analysis of planar parallel manipulators. Mech. Mach. Theory 30(5), 665–678 (1995)
Degani, A., Wolf, A.: Graphical singularity analysis of planar parallel manipulators. In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 751–756 (2006)
Fang, H., Fang, Y., Zhang, K.: Reciprocal screw theory based singularity analysis of a novel 3-DOF parallel manipulator. Chin. J. Mech. Eng. 25(4), 647–653 (2012)
Firmani, F., Podhorodeski, R.P.: Singularity analysis of planar parallel manipulators based on forward kinematic solutions. Mech. Mach. Theory 44(7), 1386–1399 (2009)
Gosselin, C., Angeles, J.: Singularity analysis of closed-loop kinematic chains. IEEE Trans. Robot. Autom. 6(3), 281–290 (1990)
Hestenes, D.: New Foundations for Classical Mechanics, vol. 15. Springer Science & Business Media, New York (2012)
Hildenbrand, D.: Foundations of Geometric Algebra Computing. Springer, Berlin (2013)
Hitzer, E., Nitta, T., Kuroe, Y.: Applications of Clifford’s geometric algebra. Adv. Appl. Clifford Algebras 23(2), 377–404 (2013)
Husty, M., Gosselin, C.: On the singularity surface of planar 3-RPR parallel mechanisms. Mech. Based Des. Struct. Mach. 36(4), 411–425 (2008)
Jones, A.C.: An Introduction to Algebraical Geometry. Clarendon Press, Wotton-under-Edge (1912)
Kanaan, D., Wenger, P., Caro, S.: Singularity analysis of lower mobility parallel manipulators using Grassmann–Cayley algebra. IEEE. Trans. Robot. 25(5), 995–1004 (2009)
Kim, J.S., Jeong, J.H., Park, J.H.: Inverse kinematics and geometric singularity analysis of a 3-SPS/S redundant motion mechanism using conformal geometric algebra. Mech. Mach. Theory 90, 23–36 (2015)
Lasenby, J., Bayro, C.E., Lasenby, A.N.: A new methodology for computing invariants in computer vision. ICPR 393 (1996)
Li, Q., Xiang, J.N., Chai, X.: Singularity analysis of a 3-RPS parallel manipulator using geometric algebra. Chin. J. Mech. Eng. 28(6), 1204–1212 (2015)
Li, Q., Chai, X.: Mobility analysis of limited-DOF parallel mechanisms in the framework of geometric algebra. J. Mech. Robot. 8(4), 041005-1–041005-9 (2016). doi:10.1115/1.4032210
Lipkin, H., Duffy, J.: The elliptic polarity of screws. J. Mech. Trans. Autom. 107(3), 377–386 (1985)
Masouleh, M.T., Gosselin, C.: Singularity analysis of 5-RPUR parallel mechanisms (3T2R). Int. J. Adv. Manuf. Tech. 57(9–12), 1107–1121 (2011)
Merlet, J.P.: Singular configurations of parallel manipulators and Grassmann geometry. Int. J. Rob. Res. 8(5), 45–56 (1989)
Monsarrat, B., Gosselin, C.M.: Singularity analysis of a three-leg six-degree-of-freedom parallel platform mechanism based on Grassmann line geometry. Int. J. Rob. Res. 20(4), 312–328 (2001)
Park, F., Kim, J.W.: Singularity analysis of closed kinematic chains. J. Mech. Des. 121(1), 32–38 (1999)
Perwass, C., Edelsbrunner, H., Kobbelt, L.: Geometric Algebra with Applications in Engineering. Springer, Berlin (2009)
Sefrioui, J., Gosselin, C.M.: On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators. Mech. Mach. Theory. 30(4), 533–551 (1995)
Tanev, T.K.: Singularity analysis of a 4-DOF parallel manipulator using geometric algebra. Advances in Robot Kinematics, pp. 275–284. Springer, The Netherlands (2006)
Tanev, T.K.: Geometric algebra approach to singularity of parallel manipulators with limited mobility. In: Lenarčič, J., Wenger, P. (eds.) Advances in Robot Kinematics: Analysis and Design, pp. 39–48. Springer, The Netherlands (2008)
Wenger, P., Chablat, D.: Kinematic Analysis of a Class of Analytic Planar 3-RPR Parallel Manipulators. Springer, The Netherlands (2009)
Zhu, S.J., Huang, Z., Zhao, M.Y.: Singularity analysis for six practicable 5-DoF fully-symmetrical parallel manipulators. Mech. Mach. Theory 44(4), 710–725 (2009)
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Yao, H., Chen, Q., Chai, X. et al. Singularity Analysis of 3-RPR Parallel Manipulators Using Geometric Algebra. Adv. Appl. Clifford Algebras 27, 2097–2113 (2017). https://doi.org/10.1007/s00006-017-0794-y
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DOI: https://doi.org/10.1007/s00006-017-0794-y