Abstract
This paper presents a method to compute the largest sphere inside the position-workspace of a semi-regular Stewart platform manipulator , that is free of gain-type singularities. The sphere is specific to a given orientation of the moving platform, and is centred at a designated point of interest. The computation is performed in two parts; in the first part, a Computer Algebra System (CAS) is used to derive a set of exact symbolic expressions, which are then used further in a purely numerical manner for faster computation. The method thus affords high computation speed, while retaining the exactness and generic nature of the results. The numerical results are validated against those obtained from an established numerical algebraic geometry tool, namely, Bertini, and are illustrated via an example.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Gain-type singularities (also known as type-II singularities) occur when the forward kinematic solutions of a manipulator merge. See [1] and the references therein for more details.
- 2.
It may be noted that many different elimination sequences are possible. The one presented here resulted in relatively smaller degrees of the intermediate and final polynomials.
- 3.
The “size” of an expression in this context indicates the amount of memory required to store the expression in the internal format of the computer algebra system (CAS) used, namely, Mathematica.
References
Agarwal, A., Nasa, C., Bandyopadhyay, S.: Dynamic singularity avoidance for parallel manipulators using a task-priority based control scheme. Mech. Mach. Theory 96, Part 1, 107–126 (2016). doi:10.1016/j.mechmachtheory.2015.07.013
Bandyopadhyay, S., Ghosal, A.: Geometric characterization and parametric representation of the singularity manifold of a 6-6 Stewart platform manipulator. Mech. Mach. Theory 41(11), 1377–1400 (2006)
Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: software for numerical algebraic geometry. http://bertini.nd.edu. doi:10.7274/R0H41PB5
Egner, S.: Semi-numerical solution to 6/6-Stewart-platform kinematics based on symmetry. Appl. Algebra Eng. Commun. Comput. 7(6), 449–468 (1996). doi:10.1007/BF01293263
Ghosal, A.: Robotics: Fundamental Concepts and Analysis, 10th edn. Oxford University Press, New Delhi (2014)
Huynh, D.Q.: Metrics for 3D rotations: comparison and analysis. J. Math. Imaging Vis. 35(2), 155–164 (2009)
Jiang, Q., Gosselin, C.M.: Determination of the maximal singularity-free orientation workspace for the Gough-Stewart platform. Mech. Mach. Theory 44(6), 1281–1293 (2009)
Li, H., Gosselin, C.M., Richard, M.J.: Determination of the maximal singularity-free zones in the six-dimensional workspace of the general Gough-Stewart platform. Mech. Mach. Theory 42(4), 497–511 (2007). doi:10.1016/j.mechmachtheory.2006.04.006
Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994)
Ravani, B., Roth, B.: Motion synthesis using kinematic mappings. J. Mech. Transm. Autom. Design 105(3), 460–467 (1983)
Srivatsan, R.A., Bandyopadhyay, S.: Determination of the safe working zone of a parallel manipulator. In: Thomas, F., Perez Gracia, A. (eds.) Computational Kinematics: Proceedings of the 6th International Workshop on Computational Kinematics (CK2013), pp. 201–208. Springer, Netherlands (2014). doi:10.1007/978-94-007-7214-4
Wolfram, Research: Mathematica, version 10.4 edn. Wolfram Research, Inc., Champaign, Illinois (2016)
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
Nag, A., Reddy, V., Agarwal, S., Bandyopadhyay, S. (2018). Identifying Singularity-Free Spheres in the Position Workspace of Semi-regular Stewart Platform Manipulators. 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_44
Download citation
DOI: https://doi.org/10.1007/978-3-319-56802-7_44
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-56801-0
Online ISBN: 978-3-319-56802-7
eBook Packages: EngineeringEngineering (R0)