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2012 | OriginalPaper | Chapter

Basic Result on Type II DM Self-Motions of Planar Stewart Gough Platforms

Author : G. Nawratil

Published in: Mechanisms, Transmissions and Applications

Publisher: Springer Netherlands

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Abstract

In a recent publication [10] the author showed that self-motions of general planar Stewart Gough platforms can be classified into two so-called Darboux Mannheim (DM) types (I and II). Moreover, in [10] the author was able to compute the set of equations yielding a type II DM self-motion explicitly. Based on these equations we present a basic result for this class of self-motions.

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Footnotes
1
For \(e_0e_2-e_1e_3\neq 0\) this can be done w.l.o.g., as this factor belongs to the denominator of f i.
 
2
Therefore we are looking for a common factor of Ω and Π, which depends on e 0.
 
3
\(\varOmega: \sum\nolimits_{i=0}^3 c_i e_i^2 + c_4e_0e_3+c_5e_1e_2\) where \(c_0,\ldots,c_5\) only depend on the geometry of the SG platform.
 
Literature
1.
Borel, E.: Mémoire sur les déplacements à trajectoires sphériques, Mém. présenteés par divers savants, Paris(2), 33, 1–128 (1908).
2.
Borras, J., Thomas, F., Torras, C.: Singularity-invariant leg rearrangements in doubly-planar Stewart-Gough platforms, In Proc. of Robotics Science and Systems, Zaragoza, Spain (2010).
3.
Bricard, R.: Mémoire sur les déplacements à trajectoires sphériques, Journ. École Polyt.(2), 11, 1–96 (1906).
4.
Husty, M.: E. Borel’s and R. Bricard’s Papers on Displacements with Spherical Paths and their Relevance to Self-Motions of Parallel Manipulators, Int. Symp. on History of Machines and Mechanisms (M. Ceccarelli ed.), 163–172, Kluwer (2000).
5.
Husty, M., Mielczarek, S., Hiller, M.: A redundant spatial Stewart-Gough platform with a maximal forward kinematics solution set, Advances in Robot Kinematics: Theory and Applications (J. Lenarcic, F. Thomas eds.), 147–154, Kluwer (2002).
6.
Karger, A.: Architecture singular planar parallel manipulators, Mechanism and Machine Theory 38 (11) 1149–1164 (2003).MATHCrossRef
7.
Karger, A.: New Self-Motions of Parallel Manipulators, Advances in Robot Kinematics: Analysis and Design (J. Lenarcic, P. Wenger eds.), 275–282, Springer (2008).
8.
Karger, A.: Self-motions of Stewart-Gough platforms, Computer Aided Geometric Design, Special Issue: Classical Techniques for Applied Geometry (B. Jüttler, O. Röschel, E. Zagar eds.) 25 (9) 775–783 (2008).MathSciNetMATHCrossRef
9.
Mielczarek, S., Husty, M.L., Hiller, M.: Designing a redundant Stewart-Gough platform with a maximal forward kinematics solution set, In Proc. of the International Symposion of Multibody Simulation and Mechatronics (MUSME), Mexico City, Mexico, September 2002.
10.
Nawratil, G.: Types of self-motions of planar Stewart Gough platforms, under review.
11.
Nawratil, G.: Basic result on type II DM self-motions of planar Stewart Gough platforms, Technical Report No. 215, Geometry Preprint Series, TU Vienna (2011).
12.
Vogler, H.: Bemerkungen zu einem Satz von W. Blaschke und zur Methode von Borel-Bricard, Grazer Mathematische Berichte 352 1–16 (2008).MathSciNetMATH
Metadata
Title
Basic Result on Type II DM Self-Motions of Planar Stewart Gough Platforms
Author
G. Nawratil
Copyright Year
2012
Publisher
Springer Netherlands
Book
Mechanisms, Transmissions and Applications

Print ISBN: 978-94-007-2726-7

Electronic ISBN: 978-94-007-2727-4

Copyright Year: 2012

DOI
https://doi.org/10.1007/978-94-007-2727-4_21

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