The main objective of this paper is to study the Euclidean displacement of a 5-DOF parallel mechanism performing three translation and two independent rotations with identical limb structures-recently revealed by performing the type synthesis-in a higher dimensional projective space, rather than relying on classical recipes, such as Cartesian coordinates and Euler angles. In this paper, Study's kinematic mapping is considered which maps the displacements of three-dimensional Euclidean space to points on a quadric, called Study quadric, in a seven-dimensional projective space, P7. The main focus of this contribution is to lay down the essential features of algebraic geometry for our kinematics purposes, where, as case study, a 5-DOF parallel mechanism with identical limb structures is considered. The forward kinematic problem is reviewed and the kinematic mapping is introduced for both general and first-order kinematics, i.e., velocity, which provides some insight into the better understanding of the kinematic behaviour of the mechanisms under study in some particular configurations for the rotation of the platform and also the constant-position workspace.
D. A. Cox, J. B. Little, and D. O’shea, Using Algebraic Geometry. Springer Verlag, 2005.
E. Study, “ Von den Bewegungen und Umlegungen,” Math. Ann., vol. 39, pp. 441–566, 1891.
M. L. Husty and H.-P. Schrocker,¨ “Algebraic Geometry and Kinematics.” Nonlinear Computational Geometry edited by Emiris, I. Z., Sottile F. and Theobald T., 2007, pp. 85–106.
X. Kong and C. Gosselin, Type Synthesis of Parallel Mechanisms. Springer, Heidelberg, 2007, vol. 33.
F. Gao, B. Peng, H. Zhao, and W. Li, “A Novel 5-DOF Fully Parallel Kinematic Machine Tool,” The International Journal of Advanced Manufacturing Technology, vol. 31, no. 1, pp. 201–207, 2006.
O. Piccin, B. Bayle, B. Maurin, and M. de Mathelin, “Kinematic Modeling of a 5-DOF Parallel Mechanism for Semi-Spherical Workspace,” Mechanism and Machine Theory, vol. 44, no. 8, pp. 1485–1496, 2009.
C. Gosselin, M. Tale Masouleh, V. Duchaine, P. L. Richard, S. Foucault, and X. Kong, “Parallel Mechanisms of the Multipteron Family: Kinematic Architectures and Benchmarking,” in IEEE International Conference on Robotics and Automation, Roma, Italy, 10-14 April 2007, pp. 555–560.
M. Tale Masouleh and C. Gosselin, “Singularity Analysis of 5-RPRRR Parallel Mechanisms via Grassmann Line Geometry,” in Proceedings of the 2009 ASME Design Engineering Technical Conferences, DETC2009-86261.
M. Tale Masouleh, M. Husty, and C. Gosselin, “Forward Kinematic Problem of 5-PRUR Parallel Mechanisms Using Study Parameters,” in Advances in Robot Kinematics: Motion in Man and Machine. Springer, 2010, pp. 211–221.
M. Tale Masouleh, M. H. Saadatzi, C. Gosselin, and H. D. Taghirad, “A Geometric Constructive Approach for the Workspace Analysis of Symmetrical 5-PRUR Parallel Mechanisms (3T2R),” in Proceedings of the 2010 ASME Design Engineering Technical Conferences, DETC2010-28509.
M. Tale Masouleh, M. Husty, and C. Gosselin, “A General Methodology for the Forward Kinematic Problem of Symmetrical Parallel Mechanisms and Application to 5-PRUR parallel mechanisms (3T2R),” in Proceedings of the 2010 ASME Design Engineering Technical Conferences, DETC2010-28222.
M. L. Husty, “An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mechanism and Machine Theory, vol. 31, no. 4, pp. 365–379, 1996.
D. R. Walter, M. Husty, and M. Pfurner, “The SNU-3UPU Parallel Robot from a Theoretical Viewpoint ,” in Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, France, 21–22 September 2008, pp. 151–158.
K. Brunnthaler, “Synthesis of 4R Linkages Using Kinematic Mapping,” Ph.D. dissertation, Institute for Basic Sciences in Engineering, Unit Geometry and CAD, Innsbruck, Austria, December 2006.
M. Tale Masouleh, C. Gosselin, M. H. Saadatzi, and H. D. Taghirad, “Forward Kinematic Problem and Constant Orientation Workspace of 5-RPRRR (3T2R) Parallel Mechanisms,” in 18th Iranian Conference on Electrical Engineering (ICEE). IEEE, 2010, pp. 668–673.
X. Kong and C. Gosselin, “Type Synthesis of 5-DOF Parallel Manipulators Based on Screw Theory,” Journal of Robotic Systems, vol. 22, no. 10, pp. 535–547, 2005.
M. Tale Masouleh and C. Gosselin, “Kinematic Analysis and Singularity Representation of 5-RPRRR Parallel Mechanisms,” in Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, France, 21–22 September 2008, pp. 79–90.
M. Tale Masouleh, C. Gosselin, M. H. Saadatzi, X. Kong, and H. D. Taghirad, “Kinematic Analysis of 5-RPUR (3T2R) Parallel Mechanisms,” Meccanica,vol. 46, no. 1, pp. 131–146, 2011.
I. A. Bonev, “Geometric Analysis of Parallel Mechanisms,” Ph.D. dissertation, Laval University.