Review and Recent Results on Stewart Gough Platforms with Self-Motions

Abstract:

Article Preview

In this paper we give a detailed review on Stewart Gough (SG) platforms with self-motions and the related Borel Bricard problem. Moreover, we report about recent results achieved by the author on this topic (SG platforms with type II DM self-motions). In context of these results, we also present two new theorems, which open the way for addressed future work. Finally, we give some remarks on planar SG platforms with type I DM self-motions and formulate a central conjecture.

Info:

Periodical:

Edited by:

Grigore Gogu, Inocentiu Maniu, Erwin-Christian Lovasz, Jean-Christophe Fauroux and Valentin Ciupe

Pages:

151-160

Citation:

G. Nawratil "Review and Recent Results on Stewart Gough Platforms with Self-Motions", Applied Mechanics and Materials, Vol. 162, pp. 151-160, 2012

Online since:

March 2012

Authors:

Export:

Price:

$38.00

[1] J. -P. Merlet, Singular Configurations of Parallel Manipulators and Grassmann Geometry, J. Robot. Res. 8: 5 (1992) 45-56.

[2] E. Borel, Mémoire sur les déplacements à trajectoires sphériques, Mém. présenteés par divers savants, Paris 2: 33 (1906) 1-128.

[3] R. Bricard, Mémoire sur les déplacements à trajectoires sphériques, J. École Polyt. 2: 11 (1906) 1-96.

[4] M. Husty, E. Borel's and R. Bricard's Papers on Displacements with Spherical Paths and their Relevance to Self-Motions of Parallel Manipulators, in: M. Ceccarelli (Ed. ), International Symposium on History of Machines and Mechanisms, Kluwer, 2000, pp.163-172.

DOI: https://doi.org/10.1007/978-94-015-9554-4_19

[5] H. Vogler, Bemerkungen zu einem Satz von W. Blaschke und zur Methode von Borel-Bricard, Grazer Math. Ber. 352 (2008) 1-16.

[6] M. Chasles, Sur les six droites qui peuvent étre les directions de six forces en équilibre, Comptes Rendus des Séances de l'Académie des Sciences 52 (1861) 1094-1104.

DOI: https://doi.org/10.3406/crai.1864.66930

[7] R. Bricard, Mouvement d'un solide dont tous les points décrivent des lignes sphériques, Comptes Rendus des Séances de l'Académie des Sciences 123 (1896) 39.

[8] R. Bricard, Mémoire sur la théorie de l'octaèdre articulé, J. de Mathématiques pures et appliquées 3 (1897) 113-148.

[9] E. Duporcq, Sur la correspondance quadratique et rationnelle de deux figures planes et sur un déplacement remarquable, Comptes Rendus des Séances de l'Académie des Sciences 126 (1898) 1405-1406.

DOI: https://doi.org/10.5962/bhl.part.2250

[10] A. Karger, Parallel manipulators and Borel-Bricard's problem, Comput. Aided Geom. Des. 27: 8 (2010) 669-680.

DOI: https://doi.org/10.1016/j.cagd.2010.07.002

[11] E. Duporcq, Sur le déplacement le plus général d'une droite dont tous les points décrivent des trajectoires sphériques, J. de Mathématiques pures et appliquées 5 (1898) 121-136.

[12] A. Mannheim, Principes et Développements de Géometrié cinématique, Gauthier-Villars, Paris, 1894.

[13] G. Kœnigs, Leçons de Cinématique, Librairie Scientifique A. Hermann, Paris, 1897.

[14] J. Krames, Zur Bricardschen Bewegung, deren sämtliche Bahnkurven auf Kugeln liegen (Über symmetrische Schrotungen II), Mh. Math. Phys. 45 (1937) 407-417.

DOI: https://doi.org/10.1007/bf01708004

[15] J. Krames, Die Borel-Bricard-Bewegung mit punktweise gekoppelten orthogonalen Hyper-boloiden (Über symmetrische Schrotungen VI), Mh. Math. Phys. 46 (1937) 172-195.

DOI: https://doi.org/10.1007/bf01792673

[16] J.M. Selig, M. Husty, Half-turns and line symmetric motions, Mech. Mach. Theory 46: 2 (2011) 156-167.

DOI: https://doi.org/10.1016/j.mechmachtheory.2010.10.001

[17] O. Ma, J. Angeles, Architecture Singularities of Parallel Manipulators, Int. J. Robot. Automat. 7: 1 (1992) 23-29.

[18] A. Karger, Architecture singular planar parallel manipulators, Mech. Mach. Theory 38: 11 (2003) 1149-1164.

DOI: https://doi.org/10.1016/s0094-114x(03)00064-8

[19] G. Nawratil, On the degenerated cases of architecturally singular planar parallel manipulators, J. Geom. Graphics 12: 2 (2008) 141-149.

[20] O. Röschel, S. Mick, Characterisation of architecturally shaky platforms, in: J. Lenarcic, M. Husty (Eds. ), Advances in Robot Kinematics: Analysis and Control, Kluwer, 1998, pp.465-474.

DOI: https://doi.org/10.1007/978-94-015-9064-8_47

[21] K. Wohlhart, From higher degrees of shakiness to mobility, Mech. Mach. Theory 45: 3 (2010) 467-476.

DOI: https://doi.org/10.1016/j.mechmachtheory.2009.10.006

[22] A. Karger, Architecturally singular non-planar parallel manipulators, Mech. Mach. Theory 43: 3 (2008) 335-346.

DOI: https://doi.org/10.1016/j.mechmachtheory.2007.03.006

[23] G. Nawratil, A new approach to the classification of architecturally singular parallel manipulators, in: A. Kecskemethy, A. Müller (Eds. ), Computational Kinematics, Springer, pp.349-358.

DOI: https://doi.org/10.1007/978-3-642-01947-0_43

[24] M. Husty, A. Karger, Self-motions of Griffis-Duffy type platforms, Proceedings of IEEE conference on Robotics and Automation, San Francisco, USA, 2000, pp.7-12.

[25] M. Husty, A. Karger, Architecture singular parallel manipulators and their self-motions, in: J. Lenarcic, M.M. Stanisic (Eds. ), Advances in Robot Kinematics, Kluwer, 2000, pp.355-364.

DOI: https://doi.org/10.1007/978-94-011-4120-8_37

[26] X. Kong, C. Gosselin, Generation of Architecturally Singular 6-SPS Parallel Manipulators with Linearly Related Planar Platforms, Electronic Journal of Computational Kinematics 1: 1 (2002) 9 pages.

[27] A.J. Sommese, C.W. Wampler, Numerical solution of systems of polynomials arising in engineering and science, World Scientific Publishing, Singapore, (2005).

[28] M. Husty, P. Zsombor-Murray, A Special Type of Singular Stewart Gough platform, in: J. Lenarcic, B. Ravani (Eds. ), Advances in Robot Kinematics and Computational Geometry, Kluwer, 1994, pp.439-449.

DOI: https://doi.org/10.1007/978-94-015-8348-0_45

[29] K. Wohlhart, Architectural Shakiness or Architectural Mobility of Platforms, in: J. Lenarcic, M.M. Stanisic (Eds. ), Advances in Robot Kinematics, Kluwer, 2000, pp.365-374.

DOI: https://doi.org/10.1007/978-94-011-4120-8_38

[30] P. Zsombor-Murray, H. Husty, D. Hartmann, Singular Stewart-Gough Platforms with Sphero- cylindrical and Spheroconical Hip Joint Trajectories, Proceedings of 9th IFToMM World Congress on the Theory of Machines and Mechanisms, Milano, Italy, 1995, pp.1886-1890.

[31] M. Husty, A. Karger, Self motions of Stewart-Gough platforms: an overview, Proceedings of the workshop on fundamental issues and future research directions for parallel mechanisms and manipulators, Quebec City, Canada, 2002, pp.131-141.

[32] A. Karger, Singularities and self-motions of a special type of platforms, in: J. Lenarcic, F. Thomas (Eds. ), Advances in Robot Kinematics: Theory and Applications, Springer, 2002, pp.155-164.

DOI: https://doi.org/10.1007/978-94-017-0657-5_17

[33] A. Karger, Singularities and self-motions of equiform platforms, Mech. Mach. Theory 36: 7 (2001) 801-815.

DOI: https://doi.org/10.1016/s0094-114x(01)00027-1

[34] A. Karger, Parallel Manipulators with Simple Geometrical Structure, in: M. Ceccarelli (Ed. ), Proceedings of the 2nd European Conference on Mechanism Science, Springer, 2008, pp.463-470.

[35] G. Nawratil, Self-motion of TSSM manipulators with two parallel rotary axes, ASME J. Mech. Robot. 3: 3 (2011) 031007.

DOI: https://doi.org/10.1115/1.4004030

[36] G. Nawratil, Self-motions of parallel manipulators associated with flexible octahedra, in: M. Hofbaur, M. Husty (Eds. ), Proceedings of the Austrian Robotics Workshop, Hall in Tyrol, Austria, 2011, p.232–248.

[37] G. Nawratil, Flexible octahedra in the projective extension of the Euclidean 3-space, J. Geom. Graphics 14: 2 (2010) 147-169.

[38] G. Nawratil, A remarkable set of Schönflies-singular planar Stewart Gough platforms, Comput. Aided Geom. Des. 27: 7 (2010) 503-513.

DOI: https://doi.org/10.1016/j.cagd.2010.06.005

[39] A. Karger, M. Husty, Classification of all self-motions of the original Stewart-Gough platform, Comput. Aided Des. 30: 3 (1998) 205-215.

DOI: https://doi.org/10.1016/s0010-4485(97)00059-6

[40] A. Karger, New Self-Motions of Parallel Manipulators, in: J. Lenarcic, P. Wenger (Eds. ), Advances in Robot Kinematics: Analysis and Design, Springer, 2008, pp.275-282.

DOI: https://doi.org/10.1007/978-1-4020-8600-7_29

[41] A. Karger, Self-motions of Stewart-Gough platforms, Comput. Aided Geom. Des. 25: 9 (2008) 775-783.

DOI: https://doi.org/10.1016/j.cagd.2008.09.003

[42] F. Geiß, F. -O. Schreyer, A family of exceptional Stewart-Gough mechanisms of genus 7, in: D.J. Bates, G. Besana, S. Di Rocco, C.W. Wampler (Eds. ), Interactions of Classical and Numerical Algebraic Geometry, Contemporary Mathematics 496, American Mathematical Society, 2009, pp.221-234.

DOI: https://doi.org/10.1090/conm/496/09725

[43] G. Nawratil, Types of self-motions of planar Stewart Gough platforms: submitted to Meccanica.

DOI: https://doi.org/10.1007/s11012-012-9659-6

[44] G. Nawratil, Basic result on type II DM self-motions of planar Stewart Gough platforms, in: E. Ch. Lovasz, B. Corves (Eds. ), Mechanisms, Transmissions and Applications, Springer, 2011, pp.235-244.

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

[45] G. Nawratil, Necessary conditions for type II DM self-motions of planar Stewart Gough platform: submitted to special issue Recent Advances in Applied Geometry, of Comput. Aided Geom. Des.

[46] G. Nawratil, Stewart Gough platforms with a type II DM self-motion, accepted for publication in J. Geometry.

[47] S. Mielczarek, M. Husty, M. Hiller, 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, 2002, 12 pages.

DOI: https://doi.org/10.1007/978-94-017-0657-5_16

[48] J. Borras, F. Thomas, C. Torras, Singularity-invariant leg rearrangements in doubly-planar Stewart Gough platforms, Proceedings of Robotics Science and Systems, Zaragoza, Spain, 2010, 8 pages.

DOI: https://doi.org/10.15607/rss.2010.vi.014

[49] O. Bottema, B. Roth, Theoretical Kinematics, Series in applied mathematics and mechanics Vol. 24, North-Holland Publishing Company, (1979).

[50] M.G. Darboux, Sur les déplacements d'une figure invariable, Comptes Rendus des Séances de l'Académie des Sciences 92 (1881) 118-121.

[51] A. Mannheim, Etude d'un déplacement particulier d'une figure de forme invariable, Rendic. Circ. Math. Palermo 3 (1889) 131-144.

DOI: https://doi.org/10.1007/bf03011514

[52] G. Nawratil, Self-motions of planar projective Stewart Gough platforms: submitted to Advances in Robot Kinematics, Innsbruck, Austria, (2012).

DOI: https://doi.org/10.1007/978-94-007-4620-6_4

[53] A. Karger, Self-motions of 6-3 Stewart-Gough type parallel manipulators, in: J. Lenarcic, M.M. Stanisic (Eds. ), Advances in Robot Kinematics: Motion in Man and Machine, Springer, 2010, pp.359-366.

DOI: https://doi.org/10.1007/978-90-481-9262-5_38

Fetching data from Crossref.
This may take some time to load.