Due to a result from M. Chasles (1861) it is known, that planar SGPs, where the platform anchor points and corresponding base anchor points are
related by a nonsingular projectivity, are architecturally singular, if the anchor points are located on a conic.
Moreover, it is known that all architecturally singular SGPs are redundant. Therefore, they possess a selfmotion in each pose.
It can easily be seen by the above given example, that this only holds over the complex number field. In the given pose the platform and the base
coincide as well as the centers of the two circles (projectivity is a similarity).

Stewart Gough Platforms with SelfMotions
The main aim of the project is the systematic determination, investigation and classification of Stewart Gough platforms with selfmotions.
This project is funded by the Austrian Science Fund (FWF).
FWF grant no. P 24927N25
Duration: 20132018 (extended for 24 months without additional costs)
FWFFunding: € 319 347.00
Project leader: Georg Nawratil
Aims and Scope
A Stewart Gough platform (SGP) is a parallel manipulator, consisting of a moving platform, which is connected via six sphericalprismaticspherical legs with the base,
where only the prismatic joints are active. If the geometry of the SGP and the six leg lengths are given, then the manipulator is in general rigid, but under particular conditions
it can perform an nparametric motion (n>0), which is called selfmotion.
Moreover, all selfmotions are solutions to the still unsolved problem posed by the French Academy of Science for the Prix Vaillant of the year 1904,
which is also known as Borel Bricard problem and reads as follows: "Determine and study all displacements of a rigid body in which distinct points of the body move on spherical paths."
Project Publications

G. Nawratil: Correcting Duporcq's theorem.
Mechanism and Machine Theory 73 282–295 (2014) DOI 10.1016/j.mechmachtheory.2013.11.012
[Open Access]

G. Nawratil: Introducing the theory of bonds for Stewart Gough platforms with selfmotions.
ASME Journal of Mechanisms and Robotics 6 (1) 011004 (2014) DOI 10.1115/1.4025623
[Preprint]

G. Nawratil: On Stewart Gough manipulators with multidimensional selfmotions.
Computer Aided Geometric Design 31 (78) 582594 (2014) DOI 10.1016/j.cagd.2014.02.012
[Corresponding Technical Report]

G. Nawratil: Congruent Stewart Gough platforms with nontranslational selfmotions.
Proc. of 16th International Conference on Geometry and Graphics (H.P. Schröcker, M. Husty eds.), Innsbruck, August 48 2014, Austria, pages 204215, ISBN 9783902936462
[Preprint]
Supplementary data: Exemplary selfmotions, Animation
1a,
1b,
1c,
2a,
2b,
2c,
3a,
3b,
3c,
4a,
4b,
4c,
5a,
5b,
5c

G. Nawratil: On equiform Stewart Gough platforms with selfmotions.
Journal for Geometry and Graphics 17 (2) 163175 (2013)
[Preprint]
Supplementary data: Exemplary selfmotions, Animation
1a,
1b,
1c,
2a,
2b,
2c,
3a,
3b,
3c

G. Nawratil: Kinematic Mapping of SE(4) and the Hypersphere Condition.
Advances in Robot Kinematics (J. Lenarcic, O. Khatib eds.), pages 1119, Springer, 2014, ISBN 9783319066974,
DOI 10.1007/9783319066981_2
[Open Access,
Erratum]

M. Gallet, G. Nawratil and J. Schicho:
Bond theory for pentapods and hexapods. Journal of Geometry 106 (2) 211228 (2015)
DOI 10.1007/s0002201402431
[arXiv:1404.2149]

M. Gallet, G. Nawratil and J. Schicho:
Möbius Photogrammetry. Journal of Geometry 106 (3) 421439 (2015)
DOI 10.1007/s000220140255x
[arXiv:1408.6716]
Erratum: Journal of Geometry 106 (3) 441442 (2015) DOI 10.1007/s0002201502978

G. Nawratil and J. Schicho:
Pentapods with Mobility 2. ASME Journal of Mechanisms and Robotics 7 (3) 031016 (2015)
DOI 10.1115/1.4028934
[arXiv:1406.0647]
Supplementary data: Maple 17 Worksheets of the General Case
[mws,
pdf]
and the Special Case
[mws,
pdf]
discussed in the Appendix

G. Nawratil and J. Schicho:
Selfmotions of pentapods with linear platform. Robotica 35 (4) 832860 (2017) DOI 10.1017/S0263574715000843
[arXiv:1407.6126]
Supplementary data: Animations of
Example 1 and
Example 2

G. Nawratil:
On the SelfMobility of PointSymmetric Hexapods. In special issue
Rigidity and Symmetry (B. Schulze ed.),
Symmetry 6 (4) 954974 (2014) DOI 10.3390/sym6040954
[Open Access]

G. Nawratil:
Fundamentals of quaternionic kinematics in Euclidean 4space.
Advances in Applied Clifford Algebras 26 (2) 693717 (2016) DOI 10.1007/s0000601506132
[Fulltext Viewonly Version,
Corresponding Technical Report]

G. Nawratil:
Quaternionic approach to equiform kinematics and lineelements of Euclidean 4space and 3space.
Computer Aided Geometric Design 47 150162 (2016) DOI 10.1016/j.cagd.2016.06.003
[Preprint]

M. Gallet, G. Nawratil and J. Schicho:
Liaison Linkages. Journal of Symbolic Computation 79 (1) 6598 (2017) DOI 10.1016/j.jsc.2016.08.006
[arXiv:1510.01127]
Supplementary data:
Animation of the given Example and
Maple Worksheets of the proofs of Proposition 3.1
[mws,
pdf]
and Proposition 3.2
[mws,
pdf]

G. Nawratil:
On the linesymmetry of selfmotions of linear pentapods.
Advances in Robot Kinematics 2016 (J. Lenarcic, J.P. Merlet eds.), pages 149159, Springer, 2017, ISBN 9783319568010,
DOI 9783319568027_16
[HAL version,
Extended version on arXiv:1510.03567]
Supplementary data: Maple Worksheet of the example given in the Appendix of the extended version
[mws,
pdf]

B. Aigner and G. Nawratil:
Planar Stewart Gough platforms with quadratic singularity surface.
New Trends in Mechanisms Science – Theory and Industrial Applications (P. Wenger, P. Flores eds.),
pages 93102, Springer, 2016, ISBN 9783319441559,
DOI 10.1007/9783319441566_10
[Corresponding Technical Report]

G. Nawratil and J. Schicho:
Duporcq Pentapods. ASME Journal of Mechanisms and Robotics 9 (1) 011001 (2017)
DOI 10.1115/1.4035085
[Corresponding arXiv paper]

M. Gallet, G. Nawratil, J. Schicho and J.M. Selig:
Mobile Icosapods. Advances in Applied Mathematics 88 125 (2017) DOI 10.1016/j.aam.2016.12.002
[arXiv:1603.07304]
Supplementary data:
Animation of the given example and
Maple code
for generating statistical data on the number of real legs

G. Nawratil:
Pointmodels for the set of oriented lineelements – a survey.
Mechanism and Machine Theory 111 118134 (2017)
DOI 10.1016/j.mechmachtheory.2017.01.008 [Preprint]

G. Nawratil:
Parallel manipulators in terms of dual CayleyKlein parameters.
Computational Kinematics (S. Zeghloul et al. eds.), pages 265273, Springer, 2017, ISBN 9783319608662,
DOI 9783319608679_30
[Preprint]

A. Rasoulzadeh and G. Nawratil:
Rational Parametrization of Linear Pentapod's Singularity Variety and the Distance to it.
Computational Kinematics (S. Zeghloul et al. eds.), pages 516524, Springer, 2017, ISBN 9783319608662,
DOI 9783319608679_59
[Extended version on arXiv:1701.09107]

G. Nawratil and A. Rasoulzadeh:
Kinematically Redundant Octahedral Motion Platform for Virtual Reality Simulations.
New Advances in Mechanism and Machine Science (I. Doroftei et al. eds.),
pages 387400, Springer, 2018, ISBN 9783319791104, DOI 10.1007/9783319791111_39
[arXiv:1704.04677]
Supplementary data:
Animation of the example given in Fig. 3

G. Nawratil:
Kinematic interpretation of the Study quadric's ambient space.
Advances in Robot Kinematics 2018 (J. Lenarcic, V. ParentiCastelli eds.),
pages 311, Springer, 2019, ISBN 9783319931876, DOI 10.1007/9783319931883_1
[Extended version on arXiv:1708.02622]

G. Nawratil:
Alternative interpretation of the
Plücker quadric's ambient space and its application.
ICGG 2018 – Proc. of the 18th International Conference on Geometry and Graphics (L. Cocchiarella ed.),
pages 918929, Springer, 2019, ISBN 9783319955872, DOI 10.1007/9783319955889_79
[Preprint]

G. Nawratil:
Hexapods with planesymmetric selfmotions. In special issue
Kinematics and Robot Design (R. Di Gregorio ed.),
Robotics 7 (2) 27 (2018) DOI 10.3390/robotics7020027 [Open Access]

G. Nawratil:
The Plücker Quadric's Ambient Space: Alternative Interpretation and its Application.
Journal for Geometry and Graphics 22 (2) 229244 (2018)
[Corresponding arXiv paper]

A. Rasoulzadeh and G. Nawratil:
Linear Pentapods with a Simple Singularity Variety – Part I: Determination and Redundant Designs.
Advances in Mechanism and Machine Science –
Proc. of the 15th IFToMM World Congress on Mechanism and Machine Science (T. Uhl ed.), pages 689698, Springer, 2019, ISBN 9783030201319,
DOI 10.1007/9783030201319_69
[Preprint,
Part of arXiv:1712.06952]

A. Rasoulzadeh and G. Nawratil:
Linear Pentapods with a Simple Singularity Variety – Part II: Computation of SingularityFree Balls.
Advances in Mechanism and Machine Science –
Proc. of the 15th IFToMM World Congress on Mechanism and Machine Science (T. Uhl ed.), pages 699708, Springer, 2019, ISBN 9783030201319,
DOI 10.1007/9783030201319_70
[Preprint,
Part of arXiv:1712.06952]

Z. Li^{*}, G. Nawratil^{*}, F. Rist and M. Hensel:
Invertible Paradoxic Loop Structures for Transformable Design.
Computer Graphics Forum 39 (2) 261275 (2020) DOI 10.1111/cgf.13928
[Open Access,
Video]
^{*}joint first author
Project Talks

G. Nawratil:
Conference on Geometry: Theory and Applications, Ljubljana June 2428 2013, Slovenia, Title:
Introducing the theory of bonds for Stewart Gough platforms with selfmotions.
[Abstract,
Slides]

M. Gallet:
XIV Encuentro de Algebra Computacional y Aplicaciones (EACA 2014), Barcelona June 1820 2014, Spain, Title:
Bond theory for pentapods and hexapods.
[Extended Abstract]

G. Nawratil:
14th International Symposium on Advances in Robot Kinematics (ARK'14), Ljubljana June 29July 3 2014, Slovenia, paper presentation.
[Slides]

G. Nawratil:
16th International Conference on Geometry and Graphics (ICGG'14), Innsbruck August 48 2014, Austria, paper presentation.
[Extended Abstract,
Slides]

G. Nawratil:
Conference on Geometry: Theory and Applications, Kefermarkt June 812 2015, Austria, Title:
Fundamentals of quaternionic kinematics in Euclidean 4space.
[Abstract,
Slides]

M. Gallet:
Effective Methods in Algebraic Geometry (MEGA 2015), Trento June 1519 2015, Italy, Title:
Construction of movable hexapods via Möbius photogrammetry.
[Abstract]

G. Nawratil:
2. IFToMM DACH Konferenz, Innsbruck February 2526 2016, Austria, Title:
Kinematische Abbildungen für die Menge orientierter Linienelemente.

G. Nawratil:
Workshop on geometric rigidity and applications, Edinburgh May 30June 3 2016, Scotland, Title:
A necessary geometric criterion for the mobility of npods.
[Slides]

G. Nawratil:
15th International Symposium on Advances in Robot Kinematics (ARK'16), Grasse June 2630 2016, France, paper presentation.
[Slides]

G. Nawratil:
6th European Conference on Mechanism Science (EuCoMeS), Nantes September 2023 2016, France, paper presentation.
[Slides]

G. Nawratil:
7th IFToMM International Workshop on Computational Kinematics (CK), FuturoscopePoitiers May 2224 2017, France, paper presentation.
[Slides]

A. Rasoulzadeh:
7th IFToMM International Workshop on Computational Kinematics (CK), FuturoscopePoitiers May 2224 2017, France, paper presentation.

G. Nawratil:
Conference on Geometry: Theory and Applications, Pilsen June 2630 2017, Czech Republic, Title:
On the set of oriented lineelements: pointmodels, metrics and applications.
[Abstract,
Slides]

G. Nawratil:
12th IFToMM International Symposium on Science of Mechanisms and Machines (SYROM'2017), Iasi November 23 2017, Romania,
Plenary Talk, Title: Kinematically Redundant Octahedral Motion Platform for Virtual Reality Simulations.

A. Rasoulzadeh:
4. IFToMM DACH Konferenz, Lausanne February 1516 2018, Switzerland, Title:
Lineare Pentapoden mit einfacher Singularitätsvarietät.
[Abstract]

G. Nawratil:
16th International Symposium on Advances in Robot Kinematics (ARK'18), Bologna July 15 2018, Italy, paper presentation.
[Slides]

G. Nawratil:
18th International Conference on Geometry and Graphics (ICGG'18), Milano August 37 2018, Italy, paper presentation.
[Extended Abstract,
Slides]

G. Nawratil:
39. Fortbildungstagung für Geometrie,
Strobl November 810 2018, Austria, Title: Symmetric Rollings with Spherical Trajectories.
[Abstract,
Slides]

A. Rasoulzadeh: 15th IFToMM World Congress,
Krakow June 30July 4 2019, Poland, paper presentations.

Z. Li: Eurographics & Eurovis,
Norrköping May 2529 2020, Sweden, virtual paper presentation (due to COVID19).
[Talk on YouTube,
Slides]
