On the Geometry of the Homogeneous Representation for the Group of Proper Rigid-Body Displacements
Keywords:
rigid-body displacements, birational transformations, kinematicsAbstract
This work investigates the geometry of the homogeneous representation of the group of proper rigid-body displacements. In particular it is shown that there is a birational transformation from the Study quadric to the variety determined by the homogeneous representation. This variety is shown to be the join of a Veronese variety with a 2-plane. The rest of the paper looks at sub-varieties, first those which are sub-groups of the displacement group and then some examples defined by geometric constraints. In many cases the varieties are familiar as sub-varieties of the Study quadric, here their transforms to the homogeneous representation is considered. A final section deals with the map which sends each displacements to its inverse. This is shown to be a quadratic birational transformation.
References
BLASCHKE, W., Kinematik und Quaternionen, Mathematische Monographien 4, VEB Deutscher Verlag der Wissenschaften, Berlin, 1960.
BOTTEMA, O., ROTH, B., Theoretical Kinematics, Dover Publications, New York, 1990.
EBERHARTER, J.K., POTTMANN, H., RAVANI, B., Stereographic projection of Study’s quadric, Dresden Symposium Geometry: Constructive and kinematic/konstruktiv und kinematisch. Zum Gedenken an/in commemoration of Rudolf Bereis (1903 1966), Weiß, Gunter (ed.), Dresden, Germany, February 27 – March 1, 2003, Proceedings Technische Universitat Dresden, ¨ 2003, pp. 82–89.
GROH, F., GROH, K., VERL, A., On the inverse kinematics of an a priori unknown general 6R-robot, Robotica, Available on CJO 2012 doi:10.1017/S0263574712000471.
HARRIS, J., Algebraic Geometry a first course, Springer Verlag, New York, 1992.
HUSTY, M. L., An algorithm for solving the direct kinematics of general Stewart-Gough platforms, Mechanism and Machine Theory, 31,4, pp. 365–380, 1996.
JUTTLER, B., ¨ Uber zwangl ¨ aufige rationale Bewegungsvorg ¨ ange ¨ , Sitzungsber. d. Osterr. Akad. d. ¨ Wiss., 202, pp. 117–132, 1993.
MOURRAIN, B., Enumeration problems in Geometry, Robotics and Vision, In Algorithms in Algebraic Geometry and Applications, Prog. in Math., 143, pp. 285–306, eds. L. Gonzalez and T. ´ Recio. Birkhauser, Basel, 1996. ¨
PARK, F.C., MARTIN, B.J., Robot sensor calibration: Solving AX = XB on the Euclidean group, IEEE Trans. Robotics and Automation, 10, 5, pp. 717–721, 1994.
SELIG, J.M., Geometric Fundamentals of Robotics, Springer Verlag, New York, 2005.
SELIG, J.M., On the geometry of point-plane constraints on rigid-body displacements, Acta Applicandae Mathematicae, 116, 2, pp. 133–155, 2011; doi:10.1007/s10440-011-9634-6
SELIG, J.M., Quadratic constraints on rigid-body displacements, ASME Journal of Mechanisms Robotics, 2, 4, 041009 (7 pages), 2010; doi: 10.1115/1.4002344.
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