Abstract
This paper investigates a situation pointed out in a recent paper, in which a non-singular change of assembly mode of a planar 2-RPR-PR parallel manipulator was realized by encircling a point of multiplicity 4. It is shown that this situation is, in fact, a non-generic one and gives rise to cusps under a small perturbation. Furthermore, we show that, for a large class of singularities of multiplicity 4, there are only two types of stable singularities occurring in a small perturbation: these two types are given by the complex square mapping and the quarto mapping. Incidentally, this paper confirms the fact that, generically, a local non-singular change of solution must be accomplished by encircling a cusp point.
Authors partially supported by ANR-14-CE34-0008-01 Kapamat.
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
Similar content being viewed by others
References
Innocenti, C., Parenti-Castelli, V.: Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators. J. Mech. Design 120, 73–79 (1998)
McAree, P.R., Daniel, R.W.: An explanation of never-special assembly changing motions for 3–3 parallel manipulators. Int. J. Robot. Res. 18(6), 556–574 (1999)
Zein, M., Wenger, P., Chablat, D.: Singular curves in the joint space and cusp points of 3-RPR parallel manipulators. Robotica 25(6), 717–724 (2007)
Caro, S., Wenger, P., Chablat, D.: Non-singular assembly mode changing trajectories of a 6-DOF parallel robot. In: Proceedings of the ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2012, pp. 1245–1254 (2012)
Husty, M., Schadlbauer, J., Caro, S., Wenger, P.: The 3-RPS manipulator can have nonsingular assembly-mode changes. Computational Kinematics, Mechanisms and Machine Science, pp. 339–348. Springer, Berlin (2014)
Bamberger, H., Wolf, A., Shoham, M.: Assembly mode changing in parallel mechanisms. IEEE Trans. Robot. 24(4), 765–772 (2008)
Macho, E., Altuzarra, O., Pinto, C., Hernandez, A.: Transitions between multiple solutions of the direct kinematic problem. Advances in Robot Kinematics: Analysis and Design, pp. 301–310. Springer, Berlin (2008)
Coste, M., Chablat, D., Wenger, P.: Nonsingular change of assembly mode without any cusp. Advances in Robot Kinematics, pp. 105–112. Springer, Berlin (2014)
Peidro, A., Marin, J.M., Gil, A., Reinoso, O.: Performing nonsingular transitions between assembly modes in analytic parallel manipulators by enclosing quadruple solutions. J. Mech. Design 137(12), 122302–122302-11 (2015)
Whitney, H.: On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane. Ann. Math. 62, 374–410 (1955)
Wenger, P., Chablat, D.: Uniqueness domains in the workspace of parallel manipulators. In: I.F.A.C-Symposium on Robot Control (SYROCO97), pp. 431–436 (1997)
Callahan, J.: Singularities and plane maps. Am. Math. Mon. 81, 211–240 (1974)
Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. 1. Birkhäuser, Basel (1985)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Coste, M., Wenger, P., Chablat, D. (2018). Hidden Cusps. In: Lenarčič, J., Merlet, JP. (eds) Advances in Robot Kinematics 2016. Springer Proceedings in Advanced Robotics, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-56802-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-56802-7_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-56801-0
Online ISBN: 978-3-319-56802-7
eBook Packages: EngineeringEngineering (R0)