Symbolic Methods in Kinematics

Designing mechanical devices, called linkages, that perform a prescribed motion has been a topic that interested engineers and mathematicians for hundreds of years.

We apply techniques from algebraic geometry and symbolic computation to various problems in this area, for example the classification of closed 6R linkages, the study of pentapods and hexapods, or the construction of planar linkages. In this context, the concept of motion polynomial and the theory of bonds have been developed, which are powerful tools to answer questions arising in kinematics.

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Ongoing Projects

Publications

2018

Rational General Solutions of Systems of First-Order Partial Differential Equations

Georg Grasegger, Alberto Lastra, J. Rafael Sendra, Franz Winkler

Journal of Computational and Applied Mathematics 331, pp. 88-103. 2018. ISSN: 0377-0427.
[bib]
@article{RISC5509,
author = {Georg Grasegger and Alberto Lastra and J. Rafael Sendra and Franz Winkler},
title = {{Rational General Solutions of Systems of First-Order Partial Differential Equations}},
language = {english},
journal = {Journal of Computational and Applied Mathematics},
volume = {331},
pages = {88--103},
isbn_issn = {ISSN: 0377-0427},
year = {2018},
refereed = {yes},
length = {16}
}

Deciding the Existence of Rational General Solutions for First-Order Algebraic ODEs

N.T. Vo, G. Grasegger, F. Winkler

Journal of Symbolic Computation 87, pp. 127-139. 2018. ISSN 0747-7171.
[bib]
@article{RISC5589,
author = {N.T. Vo and G. Grasegger and F. Winkler},
title = {{Deciding the Existence of Rational General Solutions for First-Order Algebraic ODEs}},
language = {english},
journal = {Journal of Symbolic Computation},
volume = {87},
pages = {127--139},
isbn_issn = {ISSN 0747-7171},
year = {2018},
refereed = {yes},
length = {12}
}

The Number of Realizations of a Laman Graph

Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, Josef Schicho

SIAM Journal on Applied Algebra and Geometry 2(1), pp. 94-125. 2018. 2470-6566. [url]
[bib]
@article{RISC5700,
author = {Jose Capco and Matteo Gallet and Georg Grasegger and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{The Number of Realizations of a Laman Graph}},
language = {english},
journal = {SIAM Journal on Applied Algebra and Geometry},
volume = {2},
number = {1},
pages = {94--125},
isbn_issn = {2470-6566},
year = {2018},
refereed = {yes},
length = {32},
url = {https://doi.org/10.1137/17M1118312}
}

On sets defining few ordinary circles

A. Lin, M. Makhul, H. N. Mojarrad, J. Schicho, K. Swanepoel, F. de Zeeuw

Discrete and Computational Geometry 59, pp. 59-87. 2018. 0179-5376.
[bib]
@article{RISC5535,
author = {A. Lin and M. Makhul and H. N. Mojarrad and J. Schicho and K. Swanepoel and F. de Zeeuw},
title = {{On sets defining few ordinary circles}},
language = {english},
journal = {Discrete and Computational Geometry},
volume = {59},
pages = {59--87},
isbn_issn = {0179-5376},
year = {2018},
refereed = {yes},
length = {29}
}

2017

Generalizing some Results in Field Theory for Rings

Jose Capco

Communications in Algebra, pp. 0-12. 2017. 0092-7872. Preprint. [pdf]
[bib]
@article{RISC5375,
author = {Jose Capco},
title = {{Generalizing some Results in Field Theory for Rings}},
language = {english},
journal = {Communications in Algebra},
pages = {0--12},
isbn_issn = {0092-7872},
year = {2017},
note = {Preprint},
refereed = {no},
length = {13}
}

Planar linkages following a prescribed motion

M. Gallet, C. Koutschan, Z. Li, G. Regensburger, J. Schicho, N. Villamizar

Math. Comp. 87, pp. 473-506. 2017. ISSN: 1088-6842. [url]
[bib]
@article{RISC5410,
author = {M. Gallet and C. Koutschan and Z. Li and G. Regensburger and J. Schicho and N. Villamizar},
title = {{Planar linkages following a prescribed motion}},
language = {english},
journal = {Math. Comp.},
volume = {87},
pages = {473--506},
isbn_issn = {ISSN: 1088-6842},
year = {2017},
refereed = {yes},
length = {34},
url = {http://www.ams.org/journals/mcom/2017-86-303/S0025-5718-2016-03120-8/}
}

Computing the number of realizations of a Laman graph

Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

In: Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017), Vadim Lozin (ed.), Proceedings of The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)61, pp. 207-213. 2017. ISSN 1571-0653. [url]
[bib]
@inproceedings{RISC5478,
author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{Computing the number of realizations of a Laman graph}},
booktitle = {{Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017)}},
language = {english},
abstract = {Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.},
volume = {61},
pages = {207--213},
isbn_issn = {ISSN 1571-0653},
year = {2017},
editor = {Vadim Lozin},
refereed = {yes},
keywords = {Laman graph; minimally rigid graph; tropical geometry; euclidean embedding; graph realization},
length = {7},
conferencename = {The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)},
url = {http://www.koutschan.de/data/laman/}
}

Mobile Icosapods

M. Gallet, G. Nawratil, J. Schicho, J. Selig

Adv. Appl. Math. 88, pp. 1-25. 2017. ISSN: 0196-8858. [url]
[bib]
@article{RISC5411,
author = {M. Gallet and G. Nawratil and J. Schicho and J. Selig},
title = {{Mobile Icosapods}},
language = {english},
journal = {Adv. Appl. Math.},
volume = {88},
pages = {1--25},
isbn_issn = {ISSN: 0196-8858},
year = {2017},
refereed = {yes},
length = {25},
url = {http://dx.doi.org/10.1016/j.aam.2016.12.002}
}

New developments in geometry. Theory and applications

U. Hertrich-Jeromin, B. Jüttler, J. Schicho

Comp. Aided Geom. Des. 47, pp. 1-2. 2017. ISSN: 0167-8396. [url]
[bib]
@article{RISC5413,
author = {U. Hertrich-Jeromin and B. Jüttler and J. Schicho},
title = {{New developments in geometry. Theory and applications}},
language = {english},
journal = {Comp. Aided Geom. Des.},
volume = {47},
pages = {1--2},
isbn_issn = {ISSN: 0167-8396},
year = {2017},
refereed = {yes},
length = {2},
url = {http://dx.doi.org/10.1016/j.cagd.2016.08.002}
}

Liaison linkages

M. Gallet, G. Nawratil, J. Schicho

J. Symb. Comp. 79, pp. 65-98. 2017. ISSN: 0747-7171. [url]
[bib]
@article{RISC5414,
author = {M. Gallet and G. Nawratil and J. Schicho},
title = {{Liaison linkages}},
language = {english},
journal = {J. Symb. Comp.},
volume = {79},
pages = {65--98},
isbn_issn = {ISSN: 0747-7171},
year = {2017},
refereed = {yes},
length = {34},
url = {http://dx.doi.org/10.1016/j.jsc.2016.08.006}
}

2016

Algebraic Geometry methods in Kinematics: Mobile Pods

Matteo Gallet

Johannes Kepler University. PhD Thesis. October 2016.
[bib]
@phdthesis{RISC5393,
author = {Matteo Gallet},
title = {{Algebraic Geometry methods in Kinematics: Mobile Pods}},
language = {English},
year = {2016},
month = {October},
translation = {0},
school = {Johannes Kepler University},
length = {82}
}

The rational motion of minimal dual quaternion degree with prescribed trajectory

Zijia Li, , J. Schicho, Hans-Peter Schröcker

Comp. Aided Geom. Des. 41, pp. 1-9. 2016. ISSN: 0167-8396. [url]
[bib]
@article{RISC5408,
author = {Zijia Li and and J. Schicho and Hans-Peter Schröcker},
title = {{The rational motion of minimal dual quaternion degree with prescribed trajectory}},
language = {english},
journal = {Comp. Aided Geom. Des.},
volume = {41},
pages = {1--9},
isbn_issn = {ISSN: 0167-8396},
year = {2016},
refereed = {yes},
length = {9},
url = {http://dx.doi.org/10.1016/j.cagd.2015.10.002}
}

A survey on the theory of bonds

Zijia Li , Josef Schicho, Hans-Peter Schröcker

IMA Journal of Mathematical Control and Information, pp. -. 2016. ISSN: 1471-6887. online. [url]
[bib]
@article{RISC5409,
author = {Zijia Li and Josef Schicho and Hans-Peter Schröcker},
title = {{A survey on the theory of bonds}},
language = {english},
journal = {IMA Journal of Mathematical Control and Information},
pages = {--},
isbn_issn = {ISSN: 1471-6887},
year = {2016},
note = {online},
refereed = {yes},
length = {20},
url = {https://doi.org/10.1093/imamci/dnw048}
}

Duporcq pentapods

G. Nawratil, J. Schicho

J. Mechanisms Robotics 9, pp. -. 2016. ISSN: 1942-4302. [url]
[bib]
@article{RISC5412,
author = {G. Nawratil and J. Schicho},
title = {{Duporcq pentapods}},
language = {english},
journal = {J. Mechanisms Robotics},
volume = {9},
pages = {--},
isbn_issn = {ISSN: 1942-4302},
year = {2016},
refereed = {yes},
length = {0},
url = {http://mechanismsrobotics.asmedigitalcollection.asme.org/article.aspx?articleid=2579747}
}

2015

Ambiguities in a Problem in Planar Geodesy

Josef Schicho, Matteo Gallet

SIGMA(008), pp. 1-13. 2015. 1815-0659. [url]
[bib]
@article{RISC5107,
author = {Josef Schicho and Matteo Gallet},
title = {{Ambiguities in a Problem in Planar Geodesy}},
language = {english},
abstract = {This is a study of a problem in geodesy with methods from complex algebraic geometry: for a fixed number ofmeasure points and target points at unknown position in the Euclidean plane, we study the problem of determining theirrelative position when the viewing angles between target points seen from measure points are known.In particular, we determine all situations in which there is more than one solution.},
journal = {SIGMA},
number = {008},
pages = {1--13},
isbn_issn = {1815-0659},
year = {2015},
refereed = {yes},
length = {13},
url = {http://www.emis.de/journals/SIGMA/2015/008/}
}

The theory of bonds II: closed 6R linkages with maximal genus

G. Hegedüs, Z. Li, J. Schicho, H.-P. Schroecker

J. Symb. Comp. 68, pp. 167-180. 2015. 0747-7171.
[bib]
@article{RISC5225,
author = {G. Hegedüs and Z. Li and J. Schicho and H.-P. Schroecker},
title = {{The theory of bonds II: closed 6R linkages with maximal genus}},
language = {english},
journal = {J. Symb. Comp.},
volume = {68},
pages = {167--180},
isbn_issn = {0747-7171},
year = {2015},
refereed = {yes},
length = {14}
}

An algebraic study of linkages with helical joints

H. Ahmadinezhad, Z. Li, J. Schicho

J. Pure and Appl. Algebra 219, pp. 2245-2259. 2015. 0022-4049.
[bib]
@article{RISC5226,
author = {H. Ahmadinezhad and Z. Li and J. Schicho},
title = {{An algebraic study of linkages with helical joints}},
language = {english},
journal = {J. Pure and Appl. Algebra},
volume = {219},
pages = {2245--2259},
isbn_issn = {0022-4049},
year = {2015},
refereed = {yes},
length = {15}
}

Pentapods with Mobility 2

G. Nawratil, J. Schicho

ASME Journnal of Mechanasms and Robotics J. Mechanisms Robotics, pp. -. 2015. 1942-4302.
[bib]
@article{RISC5227,
author = {G. Nawratil and J. Schicho},
title = {{Pentapods with Mobility 2}},
language = {english},
journal = {ASME Journnal of Mechanasms and Robotics},
volume = {J. Mechanisms Robotics},
pages = {--},
isbn_issn = {1942-4302},
year = {2015},
refereed = {yes},
length = {0}
}

A technique for deriving equational conditions on the Denavit-Hartenberg parameters of a 6R linkage that are necessary for movability

Z. Li, J. Schicho

Mech. Mach. Theory 94, pp. 1-8. 2015. 0094-114X.
[bib]
@article{RISC5228,
author = {Z. Li and J. Schicho},
title = {{A technique for deriving equational conditions on the Denavit-Hartenberg parameters of a 6R linkage that are necessary for movability}},
language = {english},
journal = {Mech. Mach. Theory},
volume = {94},
pages = {1--8},
isbn_issn = {0094-114X},
year = {2015},
refereed = {yes},
length = {8}
}

Spatial straight-line linkages by factorization of motion polynomials

Z. Li, J. Schicho, H.-P. Schroecker

J. Mechanisms Robotics, pp. -. 2015. 1942-4302.
[bib]
@article{RISC5229,
author = {Z. Li and J. Schicho and H.-P. Schroecker},
title = {{Spatial straight-line linkages by factorization of motion polynomials}},
language = {english},
journal = {J. Mechanisms Robotics},
pages = {--},
isbn_issn = {1942-4302},
year = {2015},
refereed = {yes},
length = {0}
}

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