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Research Papers

Introducing the Theory of Bonds for Stewart Gough Platforms With Self-Motions1

[+] Author and Article Information
Georg Nawratil

Institute of Discrete Mathematics and Geometry,
Vienna University of Technology,
Vienna 1040, Austria
e-mail: nawratil@geometrie.tuwien.ac.at

Dedicated to my newborn daughter and her mother on the occasion of her birth.

Recently an extended version of this paper was published (cf. [2]).

For $e1=e2=e3=0$ the rotation degenerates into a translation. This case is excluded.

Note that this is an analogy to the theory of bonds of overconstrained closed chains (cf. footnote 3).

We denote the ideal point of this direction by P.

It is worth noting that $B$ does not depend on the parameter β, which determines the geometry of the planar projective SG platform (cf. [19, 20]).

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received January 22, 2013; final manuscript received July 27, 2013; published online November 8, 2013. Assoc. Editor: J. M. Selig.

J. Mechanisms Robotics 6(1), 011004 (Nov 08, 2013) (9 pages) Paper No: JMR-13-1016; doi: 10.1115/1.4025623 History: Received January 22, 2013; Revised July 27, 2013

Abstract

We transfer the basic idea of bonds, introduced by Hegedüs, Schicho, and Schröcker for overconstrained closed chains with rotational joints, to the theory of self-motions of parallel manipulators of Stewart Gough (SG) type. Moreover, we present some basic facts and results on bonds and demonstrate the potential of this theory on the basis of several examples. As a by-product we give a geometric characterization of all SG platforms with a pure translational self-motion and of all spherical three-degrees of freedom (DOF) RPR manipulators with self-motions.

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Topics: Manipulators

References

Hegedüs, G., Schicho, J., and Schröcker, H.-P., 2012, “Bond Theory and Closed 5R Linkages,” Latest Advances in Robot Kinematics, J.Lenarcic and M.Husty, eds., Springer, New York, pp. 221–228.
Hegedüs, G., Schicho, J., and Schröcker, H.-P., 2013, “The Theory of Bonds: A New Method for the Analysis of Linkages,” Mech. Mach. Theory, 70, pp. 407–424.
Borel, E., 1908, “Mémoire sur les Déplacements à Trajectoires Sphériques,” Mémoire Présenteés par Divers Savants à l'Académie des Sciences de l'Institut National de France33(1), pp. 1–128.
Bricard, R., 1906, “Mémoire sur les Déplacements à Trajectoires Sphériques,” J. Ec. Polytech. (Paris), 11, pp. 1–96.
Husty, M., 2000, “E. Borel's and R. Bricard's Papers on Displacements With Spherical Paths and their Relevance to Self-Motions of Parallel Manipulators,” International Symposium on History of Machines and Mechanisms, M. Ceccarelli, ed., Kluwer, Dordrecht, Netherlands, pp. 163–172.
Karger, A., 2003, “Architecture Singular Planar Parallel Manipulators,” Mech. Mach. Theory, 38(11), pp. 1149–1164.
Nawratil, G., 2008, “On the Degenerated Cases of Architecturally Singular Planar Parallel Manipulators,” J. Geom. Graph., 12(2), pp. 141–149. Available at: http://www.heldermann.de/JGG/jggcover.htm
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Wohlhart, K., 2010, “From Higher Degrees of Shakiness to Mobility,” Mech. Mach. Theory, 45(3), pp. 467–476.
Karger, A., 2008, “Architecturally Singular Non-Planar Parallel Manipulators,” Mech. Mach. Theory, 43(3), pp. 335–346.
Nawratil, G., 2009, “A New Approach to the Classification of Architecturally Singular Parallel Manipulators,” Computational Kinematics, A.Kecskemethy and A.Müller, eds., Springer, New York, pp. 349–358.
Nawratil, G., 2012, “Review and Recent Results on Stewart Gough Platforms With Self-Motions,” Appl. Mech. Mater., 162, pp. 151–160.
Husty, M. L., 1996, “An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mech. Mach. Theory, 31(4), pp. 365–380.
Husty, M., Karger, A., Sachs, H., and Steinhilper, W., 1997, Kinematik und Robotik, Springer, New York.
Husty, M. L., and Karger, A., 2002, “Self Motions of Stewart-Gough Platforms: An Overview,” Proceedings of the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, C. M. Gosselin and I. Ebert-Uphoff, eds., Quebec, Canada, pp. 131–141.
Nawratil, G., 2012, “Self-Motions of Planar Projective Stewart Gough Platforms,” Latest Advances in Robot Kinematics, J.Lenarcic and M.Husty, eds., Springer, New York, pp. 27–34.
Borras, J., Thomas, F., and Torras, C., 2010, “Singularity-Invariant Leg Rearrangements in Doubly-Planar Stewart-Gough Platforms,” Proceedings of Robotics Science and Systems, Zaragoza, Spain.
Mielczarek, S., Husty, M. L., and Hiller, M., 2002, “Designing a Redundant Stewart-Gough Platform With a Maximal Forward Kinematics Solution Set,” Proceedings of the International Symposium of Multibody Simulation and Mechatronics, Mexico City, Mexico.
Nawratil, G., “On Elliptic Self-Motions of Planar Projective Stewart Gough Platforms,” Trans. Can. Soc. Mech. Eng. (in press).
Nawratil, G., 2013, “Non-Existence of Planar Projective Stewart Gough Platforms With Elliptic Self-Motions,” Computational Kinematics, F.Thomas and A.Perez Garcia, eds., Springer, New York, pp. 41–48.
Brunnthaler, K., Schröcker, H.-P., and Husty, M., 2006, “Synthesis of Spherical Four-Bar Mechanisms Using Spherical Kinematic Mapping,” Advances in Robot Kinematics: Mechanisms and Motion, J.Lenarcic and B.Roth, eds., Springer, New York, pp. 377–384.
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Nawratil, G., “Correcting Duporcq's Theorem,” (to be published).
Faugère, J. C., and Lazard, D., 1995, “Combinatorial Classes of Parallel Manipulators,” Mech. Mach. Theory, 30(6), pp. 765–776.

Figures

Fig. 1

A SG manipulator with planar platform and planar base (Ci = ci = 0 for i = 1,…,6) is called planar SG manipulator

Fig. 3

Sketch of the platform (left) and the base (right) of a planar SG manipulator with pairwise distinct anchor points, where Bi = ai = 0 for i = 1,…,4 and A5 = A6 = b5 = b6 = 0 hold

Fig. 2

Sketch of the platform (left) and the base (right) of the planar SG manipulator of Example 3, where M1 = M2 and m5 = m6 hold

Fig. 4

Left: spherical 3-DOF RPR manipulator. Right: w.l.o.g. we can assume that m1° and m3° coincide (after a perhaps necessary exchange of the platform and the base and reindexing of anchor points). In the configuration, where m1° = m3° coincides with M2°, the platform has a trivial rotational self-motion.

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