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

2024

[Qi]

A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero

Jiayue Qi

Journal of Symbolic Computation 122(102253), pp. -. 2024. ISSN: 0747-7171. [doi]
[bib]
@article{RISC6774,
author = {Jiayue Qi},
title = {{A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero}},
language = {english},
journal = {Journal of Symbolic Computation},
volume = {122},
number = {102253},
pages = {--},
isbn_issn = {ISSN: 0747-7171},
year = {2024},
refereed = {yes},
length = {52},
url = {https://doi.org/10.1016/j.jsc.2023.102253}
}
[Schicho]

Calibrating figures

N. Lubbes, J. Schicho

Comp. Aided Geom. Design 112, pp. -. 2024. ISSN 1879-2332. [doi]
[bib]
@article{RISC7122,
author = {N. Lubbes and J. Schicho},
title = {{Calibrating figures}},
language = {english},
journal = {Comp. Aided Geom. Design},
volume = {112},
pages = {--},
isbn_issn = {ISSN 1879-2332},
year = {2024},
refereed = {yes},
length = {0},
url = {https://doi.org/10.1016/j.cagd.2024.102365}}
}

2023

[Hoxhaj]

Using Algebraic Geometry to Reconstruct a Darboux Cyclide from a Calibrated Camera Picture

E. Hoxhaj, J.-M. Menjanahary, J. Schicho

J. AAECC, pp. -. 2023. 1432-0622. to appear. [doi]
[bib]
@article{RISC6879,
author = {E. Hoxhaj and J.-M. Menjanahary and J. Schicho},
title = {{Using Algebraic Geometry to Reconstruct a Darboux Cyclide from a Calibrated Camera Picture}},
language = {english},
journal = {J. AAECC},
pages = {--},
isbn_issn = {1432-0622},
year = {2023},
note = {to appear},
refereed = {yes},
length = {19},
url = {https://doi.org/10.1007/s00200-023-00600-y}
}
[Koutschan]

Representing piecewise linear functions by functions with small arity

C. Koutschan, B. Moser, A. Ponomarchuk, J. Schicho

J. AAECC, pp. -. 2023. 1432-0622.
[bib]
@article{RISC6880,
author = {C. Koutschan and B. Moser and A. Ponomarchuk and J. Schicho},
title = {{Representing piecewise linear functions by functions with small arity}},
language = {english},
journal = {J. AAECC},
pages = {--},
isbn_issn = {1432-0622},
year = {2023},
refereed = {yes},
length = {0}
}
[Schicho]

Trilinear birational maps in dimension three

L. Buse, P. Gonz'alez-Maz'on, J. Schicho

M. Comp. 92, pp. 1837-1866. 2023. 0025-5718.
[bib]
@article{RISC6878,
author = {L. Buse and P. Gonz'alez-Maz'on and J. Schicho},
title = {{Trilinear birational maps in dimension three}},
language = {english},
journal = {M. Comp.},
volume = {92},
pages = {1837--1866},
isbn_issn = {0025-5718},
year = {2023},
refereed = {yes},
length = {30}
}
[Schicho]

Apollonian-de Casteljau type algorithms for complex rational Bezier curves

B. J�ttler, J. Schicho, Z. Sir

CAGD 107, pp. -. 2023. 0167-8396.
[bib]
@article{RISC6881,
author = {B. J�ttler and J. Schicho and Z. Sir},
title = {{Apollonian-de Casteljau type algorithms for complex rational Bezier curves}},
language = {english},
journal = {CAGD},
volume = {107},
pages = {--},
isbn_issn = {0167-8396},
year = {2023},
refereed = {yes},
length = {0}
}

2022

[Grasegger]

Flexible placements of graphs with rotational symmetry

Sean Dewar, Georg Grasegger, Jan Legerský

In: 2nd IMA Conference on Mathematics of Robotics, W. Holderbaum, J.M. Selig (ed.), Springer Proceedings in Advanced Robotics 21, pp. 89-97. 2022. 978-3-030-91351-9. [doi]
[bib]
@inproceedings{RISC6387,
author = {Sean Dewar and Georg Grasegger and Jan Legerský},
title = {{Flexible placements of graphs with rotational symmetry}},
booktitle = {{2nd IMA Conference on Mathematics of Robotics}},
language = {english},
series = {Springer Proceedings in Advanced Robotics},
volume = {21},
pages = {89--97},
isbn_issn = {978-3-030-91351-9},
year = {2022},
editor = {W. Holderbaum and J.M. Selig},
refereed = {yes},
length = {9},
url = {https://doi.org/10.1007/978-3-030-91352-6_9}
}
[Grasegger]

Zero-Sum Cycles in Flexible Non-triangular Polyhedra

Matteo Gallet, Georg Grasegger, Jan Legerský, Josef Schicho

In: 2nd IMA Conference on Mathematics of Robotics, W. Holderbaum, J.M. Selig (ed.), Springer Proceedings in Advanced Robotics 21, pp. 137-143. 2022. 978-3-030-91351-9. [doi]
[bib]
@inproceedings{RISC6388,
author = {Matteo Gallet and Georg Grasegger and Jan Legerský and Josef Schicho},
title = {{Zero-Sum Cycles in Flexible Non-triangular Polyhedra}},
booktitle = {{2nd IMA Conference on Mathematics of Robotics}},
language = {english},
series = {Springer Proceedings in Advanced Robotics},
volume = {21},
pages = {137--143},
isbn_issn = {978-3-030-91351-9},
year = {2022},
editor = {W. Holderbaum and J.M. Selig},
refereed = {yes},
length = {7},
url = {https://doi.org/10.1007/978-3-030-91352-6_14}
}
[Grasegger]

Zero-sum Cycles in Flexible Polyhedra

M. Gallet, G. Grasegger, J. Legersky, J. Schicho

Bull. LMS 54, pp. 112-125. 2022. 1469-2120.
[bib]
@article{RISC6662,
author = {M. Gallet and G. Grasegger and J. Legersky and J. Schicho},
title = {{Zero-sum Cycles in Flexible Polyhedra}},
language = {english},
journal = {Bull. LMS},
volume = {54},
pages = {112--125},
isbn_issn = {1469-2120},
year = {2022},
refereed = {yes},
length = {14}
}
[Koutschan]

Approximation of convex polygons by polygons

C. Koutschan, A. Ponomarchuk, J. Schicho

In: 23rd International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), C. Schneider et al. (ed.), pp. 91-98. 2022. IEEE, 978-1-6654-0650-5.
[bib]
@inproceedings{RISC6877,
author = {C. Koutschan and A. Ponomarchuk and J. Schicho },
title = {{Approximation of convex polygons by polygons}},
booktitle = {{23rd International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)}},
language = {english},
pages = {91--98},
publisher = {IEEE},
isbn_issn = {978-1-6654-0650-5},
year = {2022},
editor = {C. Schneider et al.},
refereed = {yes},
length = {8}
}
[Schicho]

And Yet it Moves - Paradoxically Moving Linkages in Kinematics

J. Schicho

Bulletin AMS 59, pp. 59-95. 2022. ISSN 0273-0979. [doi]
[bib]
@article{RISC6474,
author = {J. Schicho},
title = {{And Yet it Moves -- Paradoxically Moving Linkages in Kinematics}},
language = {english},
journal = {Bulletin AMS},
volume = {59},
pages = {59--95},
isbn_issn = {ISSN 0273-0979},
year = {2022},
refereed = {yes},
length = {37},
url = {https://doi.org/10.1090/bull/1721}
}
[Schicho]

Classification of higher mobility closed-loop linkages

T. Duarte Guerreiro, Z. Li, J. Schicho

Annali di Matematica Pura et Applicata, pp. -. 2022. 0373-3114.
[bib]
@article{RISC6663,
author = {T. Duarte Guerreiro and Z. Li and J. Schicho},
title = {{Classification of higher mobility closed-loop linkages}},
language = {english},
journal = {Annali di Matematica Pura et Applicata},
pages = {--},
isbn_issn = {0373-3114},
year = {2022},
refereed = {yes},
length = {0}
}
[Schicho]

A new line-symmetric mobile infinity-pod

M. Gallet, J. Schicho

Confl. Math. 14, pp. 35-47. 2022. 1793-7442.
[bib]
@article{RISC6664,
author = {M. Gallet and J. Schicho},
title = {{A new line-symmetric mobile infinity-pod}},
language = {english},
journal = {Confl. Math.},
volume = {14},
pages = {35--47},
isbn_issn = {1793-7442},
year = {2022},
refereed = {yes},
length = {13}
}
[Schicho]

Projective isomorphisms between rational surfaces

B. J�ttler, N. Lubbes, J. Schicho

J. Algebra 54, pp. 112-125. 2022. 0021-8693.
[bib]
@article{RISC6668,
author = {B. J�ttler and N. Lubbes and J. Schicho},
title = {{Projective isomorphisms between rational surfaces}},
language = {english},
journal = {J. Algebra},
volume = {54},
pages = {112--125},
isbn_issn = { 0021-8693},
year = {2022},
refereed = {yes},
length = {14}
}

2021

[Grasegger]

Combinatorics of Bricard's octahedra

M. Gallet, G. Grasegger, J. Legerský, J. Schicho

Comptes Rendus. Mathématique 359(1), pp. 7-38. 2021. Académie des sciences, Paris, ISSN 1631-073X. [doi]
[bib]
@article{RISC6288,
author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},
title = {{Combinatorics of Bricard's octahedra}},
language = {english},
journal = {Comptes Rendus. Mathématique},
volume = {359},
number = {1},
pages = {7--38},
publisher = {Académie des sciences, Paris},
isbn_issn = {ISSN 1631-073X},
year = {2021},
refereed = {yes},
length = {32},
url = {https://doi.org/10.5802/crmath.132}
}
[Grasegger]

On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere

M. Gallet, G. Grasegger, J. Legerský, J. Schicho

SIAM Journal on Discrete Mathematics 35(1), pp. 325-361. 2021. ISSN 0895-4801. [doi]
[bib]
@article{RISC6290,
author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},
title = {{On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere}},
language = {english},
journal = {SIAM Journal on Discrete Mathematics},
volume = {35},
number = {1},
pages = {325--361},
isbn_issn = {ISSN 0895-4801},
year = {2021},
refereed = {yes},
length = {37},
url = {https://doi.org/10.1137/19M1289467}
}
[Grasegger]

Bracing frameworks consisting of parallelograms

Georg Grasegger, Jan Legerský

The Art of Discrete and Applied Mathematics Accepted Manuscripts, pp. --. 2021. 2590-9770. [doi]
[bib]
@article{RISC6389,
author = {Georg Grasegger and Jan Legerský},
title = {{Bracing frameworks consisting of parallelograms}},
language = {english},
journal = {The Art of Discrete and Applied Mathematics},
volume = {Accepted Manuscripts},
pages = {----},
isbn_issn = {2590-9770},
year = {2021},
refereed = {yes},
length = {0},
url = {https://doi.org/10.26493/2590-9770.1379.7a4}
}
[Legersky]

On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2

E. Bartzos, I.Z. Emiris, J. Legerský, E. Tsigaridas

Journal of Symbolic Computation 102, pp. 189-208. 2021. ISSN 0747-7171. [doi]
[bib]
@article{RISC5992,
author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},
title = {{On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2}},
language = {english},
journal = {Journal of Symbolic Computation},
volume = {102},
pages = {189--208},
isbn_issn = {ISSN 0747-7171},
year = {2021},
refereed = {yes},
length = {20},
url = {https://doi.org/10.1016/j.jsc.2019.10.015}
}
[Qi]

How to avoid collisions in 3D-realizations for moving graphs

Jiayue Qi

Mechanism and Machine Theory 162(104337), pp. -. 2021. ISSN 0094-114X. [doi]
[bib]
@article{RISC6772,
author = {Jiayue Qi},
title = {{How to avoid collisions in 3D-realizations for moving graphs}},
language = {english},
journal = {Mechanism and Machine Theory },
volume = {162},
number = {104337},
pages = {--},
isbn_issn = {ISSN 0094-114X},
year = {2021},
refereed = {yes},
length = {15},
url = {https://doi.org/10.1016/j.mechmachtheory.2021.104337}
}

2020

[Capco]

Two remarks on sums of squares with rational coefficients

Jose Capco, Claus Scheiderer

Banach Center Publications 121, pp. 25-36. 2020. 0137-6934. [doi]
[bib]
@article{RISC5700,
author = {Jose Capco and Claus Scheiderer},
title = {{ Two remarks on sums of squares with rational coefficients}},
language = {english},
journal = {Banach Center Publications},
volume = {121},
pages = {25--36},
isbn_issn = {0137-6934},
year = {2020},
refereed = {yes},
length = {11},
url = {https://doi.org/10.4064/bc121-2}
}

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