Abstract
Transversality is a mathematical concept, widely used in singularity theory, which generalises the idea of regularity of a function. One of its uses is in providing a method for certifying that a property of a given set of functions or mappings is generic, which is to say that it holds for almost all members of that set. Its application in kinematics is illustrated via a prototype for showing that in a given class of parallel mechanisms, for almost all choices of design parameter the configuration space is non-singular and hence a manifold.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Amirinezhad, S.V., Donelan, P.: Kinematic constraint maps, C-space singularities and generalised Grashof conditions. In: 2017 IDETC/CIE Conference. ASME (2017)
Brockett, R.W.: Robotic manipulators and the product of exponentials formula. In: Mathematical Theory of Networks and Systems, pp. 120–129. Springer (1984)
Donelan, P.: Genericity conditions for serial manipulators. In: Lenarčič, J., Wenger, P. (eds.) Advances in Robot Kinematics: Analysis and Design, pp. 185–192. Springer, Dordrecht (2008)
Gibson, C.G., Newstead, P.E.: On the geometry of the planar 4-bar mechanism. Acta Applicandae Mathematicae 7(2), 113–135 (1986)
Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics. Springer, New York (2012)
Grashof, F.: Theoretische maschinenlehre, vol. 3. L. Voss, Hamburg (1890)
Guillemin, V., Pollack, A.: Differential Topology, vol. 370. American Mathematical Soc, Providence (2010)
Müller, A.: Generic mobility of rigid body mechanisms. Mech. Mach. Theor. 44(6), 1240–1255 (2009)
Pai, D.K., Leu, M.C.: Generic singularities of robot manipulators. In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 738–744. IEEE, May 1989
Tchoń, K., Muszynski, R.: Singular inverse kinematic problem for robotic manipulators: a normal form approach. IEEE Trans. Robot. Autom. 14(1), 93–104 (1998)
Ting, K.L., Liu, Y.W.: Rotatability laws for N-bar kinematic chains and their proof. J. Mech. Des. 113(1), 32–39 (1991)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Vahid Amirinezhad, S., Donelan, P., Müller, A. (2019). Transversality and Its Applications to Kinematics. In: Lenarcic, J., Parenti-Castelli, V. (eds) Advances in Robot Kinematics 2018. ARK 2018. Springer Proceedings in Advanced Robotics, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-93188-3_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-93188-3_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93187-6
Online ISBN: 978-3-319-93188-3
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)