@inproceedings{msolve,
  TITLE = {{msolve: A Library for Solving Polynomial Systems}},
  AUTHOR = {Berthomieu, J{\'e}r{\'e}my and Eder, Christian and {Safey El Din}, Mohab},
  URL = {https://hal.sorbonne-universite.fr/hal-03191666},
  BOOKTITLE = {{2021 International Symposium on Symbolic and Algebraic Computation}},
  ADDRESS = {Saint Petersburg, Russia},
  SERIES = {46th International Symposium on Symbolic and Algebraic Computation},
  YEAR = {2021},
  MONTH = Jul,
  DOI = {10.1145/3452143.3465545},
  PDF = {https://hal.sorbonne-universite.fr/hal-03191666v2/file/main.pdf},
  HAL_ID = {hal-03191666},
  HAL_VERSION = {v2},
}

@article{SparseFGLM,
title = "Sparse {FGLM} algorithms",
journal = "Journal of Symbolic Computation",
volume = "80",
pages = "538--569",
year = "2017",
issn = "0747-7171",
doi = "https://doi.org/10.1016/j.jsc.2016.07.025",
url = "http://www.sciencedirect.com/science/article/pii/S0747717116300700",
author = "Jean-{\relax Ch}arles Faug{\`e}re and Chenqi Mou",
keywords = "Gröbner bases, Change of ordering, Zero-dimensional ideal, Sparse matrix, Wiedemann algorithm,  algorithm",
abstract = "Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical FGLM algorithm. Combining all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over 60000. Such an implementation outperforms the Magma and Singular ones, as shown by our experiments. First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is O(D(N1+nlog⁡(D)2)), where N1 is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX Gröbner basis of I via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp--Massey--Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided. Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove that its construction is free. With the asymptotic analysis of such sparsity, we are able to show that for generic systems the complexity above becomes O(6/nπD2+n−1n)."
}


@book {CLO,
    AUTHOR = {Cox, David A. and Little, John and O'Shea, Donal},
     TITLE = {Ideals, varieties, and algorithms},
    SERIES = {Undergraduate Texts in Mathematics},
   EDITION = {Fourth},
      NOTE = {An introduction to computational algebraic geometry and
              commutative algebra},
 PUBLISHER = {Springer, Cham},
      YEAR = {2015},
     PAGES = {xvi+646},
      ISBN = {978-3-319-16721-3},
   MRCLASS = {13P10 (13-01 14-01 14Qxx 68W30)},
  MRNUMBER = {3330490},
       DOI = {10.1007/978-3-319-16721-3},
       URL = {https://doi.org/10.1007/978-3-319-16721-3},
}

@inproceedings{Collins,
  title={Quantifier elimination for real closed fields by cylindrical algebraic decompostion},
  author={Collins, George E},
  booktitle={Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern, May 20--23, 1975},
  pages={134--183},
  year={1975},
  organization={Springer}
}

@inproceedings{caro2012non,
  title={Non-singular assembly mode changing trajectories of a 6-DOF parallel robot},
  author={Caro, St{\'e}phane and Wenger, Philippe and Chablat, Damien},
  booktitle={ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference},
  pages={1245--1254},
  year={2012},
  organization={American Society of Mechanical Engineers}
}


@inproceedings{FaSaSp12,
  title={Critical points and Gr{\"o}bner bases: the unmixed case},
  author={Faug{\`e}re, Jean-Charles and {Safey El Din}, Mohab and Spaenlehauer, Pierre-Jean},
  booktitle={Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation},
  pages={162--169},
  year={2012},
  organization={ACM}
}

@article{HaSo12,
  title={Algorithm 921: alphaCertified: certifying solutions to polynomial systems},
  author={Hauenstein, Jonathan D and Sottile, Frank},
  journal={ACM Transactions on Mathematical Software (TOMS)},
  volume={38},
  number={4},
  pages={28},
  year={2012},
  publisher={ACM}
}

@article{HSSV18,
  title={Solving determinantal systems using homotopy techniques},
  author={Hauenstein, Jonathan D and {Safey El Din}, Mohab and Schost, {\'E}ric and Vu, Thi Xuan},
  journal={arXiv preprint arXiv:1802.10409},
  year={2018}
}


@article{SaSc18,
  title={Bit complexity for multi-homogeneous polynomial system solving—Application to polynomial minimization},
  author={{Safey El Din}, Mohab and Schost, {\'E}ric},
  journal={Journal of Symbolic Computation},
  volume={87},
  pages={176--206},
  year={2018},
  publisher={Elsevier}
}

@article{Hau13,
  title={Numerically computing real points on algebraic sets},
  author={Hauenstein, Jonathan D},
  journal={Acta applicandae mathematicae},
  volume={125},
  number={1},
  pages={105--119},
  year={2013},
  publisher={Springer}
}

@InProceedings{F5,
  author =       {Faug\`ere, J.-C.},
  title =        {A new
    efficient algorithm for computing Gr\"obner bases without
    reduction to zero (F5)},
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  booktitle = {Proceedings of the 2002 {I}nternational
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}

@Article{F4,
  author =       {Faug\`ere, J.-C.},
  title =        {A new efficient algorithm for computing Gr\"obner bases (F4)},
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@article{SchSha,
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@Article{CoMo15,
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  year =         {2015},
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  ALTeditor =    {},
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}



@Article{TII2018,
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  year =         {2018},
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  author =       {Bock, H.G. and Plitt, K.J.},
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@InProceedings{BoLi86,
  author =       {Borrel, P. and Liegeois, A.},
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@Book{angeles,
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@Book{MLS,
  author =    {Murray, R.M. and Li, Z. and Sastry, S.S.},
  ALTeditor =    {},
  title =        {A Mathematical Introduction to Robotic Manipulation},
  publisher =    {CRC Press Taylor \& Francis Group},
  year =         {1994},
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@InProceedings{Nawratil10,
  author =       {Nawratil, G.},
  title =        {Stewart Gough platforms with linear singularity surface},
  OPTcrossref =  {},
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  booktitle = {Proc. of 19th IEEE International Workshop on Robotics in Alpe-Adria-Danube Region (RAAD'10)},
  year =      {2010},
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@Book{selig,
  author =    {Selig, J.M.},
  editor =    {Gries, D. and Schneider, F.B.},
  title =        {Geometric Fundamentals of Robotics},
  publisher =    {Springer},
  year =         {2005},
  OPTkey =       {},
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  OPTnumber =    {},
  series =    {Monographs in Computer Science},
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}

@Article{NaSch17,
  author =       {Nawratil, G. and Schicho, J.},
  title =        {Self-motions of pentapods with linear platform},
  journal =      {Robotica},
  year =         {2017},
  OPTkey =       {},
  volume =    {35},
  number =    {4},
  pages =     {832-860},
  OPTmonth =     {},
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}

@article{SaSc17,
  title={A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets},
  author={{Safey El Din}, Mohab and Schost, {\'E}ric},
  journal={Journal of the ACM (JACM)},
  volume={63},
  number={6},
  pages={48},
  year={2017},
  publisher={ACM}
}

@Article{wenger,
  author =       {Wenger, P.},
  title =        {Cuspidal and noncuspidal robot manipulators},
  journal =      {Special issue of Robotica on Geometry in Robotics and Sensing},
  year =         {2007},
  OPTkey =       {},
  volume =    {25},
  number =    {6},
  pages =     {677-690},
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}

@article{SaSc11,
  title={A baby steps/giant steps probabilistic algorithm for computing roadmaps in smooth bounded real hypersurface},
  author={{Safey El Din}, Mohab and Schost, {\'E}ric},
  journal={Discrete \& Computational Geometry},
  volume={45},
  number={1},
  pages={181--220},
  year={2011},
  publisher={Springer}
}

@Article{parenti,
  author =       {Innocenti, C. and {Parenti-Castelli}, V.},
  title =        {Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators, Kinematic mapping based assmbly mode evaluation of planar four-bar mechanisms},
  journal =      {ASME Journal of Mechanical Design},
  year =         {1998},
  OPTkey =       {},
  volume =    {120},
  OPTnumber =    {},
  pages =     {73-99},
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}



@Article{SHM,
  author =       {Schr\"ocker, H.-P. and Husty, M. and McCarthy, J.M.},
  title =        {Kinematic mapping based assembly mode evaluation of planar four-bar mechanisms},
  journal =      {ASME Journal of Mechanical Design},
  year =         {2006},
  OPTkey =       {},
  volume =    {129},
  OPTnumber =    {9},
  pages =     {924-929},
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}

@Article{GNSS,
  author =       {Gallet, M. and Nawratil, G. and Schicho, J. and Selig, J.M.},
  title =        {Mobile Icosapods},
  journal =      {Advances in Applied Mathematics},
  year =         {2017},
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@Article{BRSS,
  author =       {Basu, S. and Roy, M.-F. and {Safey El Din}, M. and Schost, \'E.},
  title =        {A Baby Step-Giant Step Roadmap Algorithm for General Algebraic Sets},
  journal =      {Foundations of Computational Mathematics},
  year =         {2014},
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@Book{merlet,
  author =    {Merlet, J.-P.},
  ALTeditor =    {},
  title =        {Parallel robots},
  publisher =    {Springer},
  year =         {2006},
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  edition =   {2nd},
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}

@Misc{irb,
  OPTkey =       {},
  author =    {{IRB5500 Datasheet}},
  title =     {ABB Ltd, Affolternstrasse 44, 8050 Zurich, Switzerland},
  howpublished = {\href{https://library.e.abb.com/public/86aad4124d6ad6dec12579f4002ab48b/DatenblattIRB5500_web.pdf}{https://library.e.abb.com/public/86aad4124d6ad6dec12579f4002ab48b/DatenblattIRB5500\\ \_web.pdf}},
  OPTmonth =     {},
  year =      {2017},
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}

@Misc{ur5,
  OPTkey =       {},
  author =    {{UR5 Datasheet}},
  title =     {Universal Robots A/S Energivej 25 DK-5260},
  howpublished = {\url{http://media1.limitless.dk/UR_Tech_Spec/UR5_GB.pdf}},
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}

@Article{LaKu2,
  author =       {LaValle, S.M. and Kuffner, J.J.},
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@InProceedings{KuLa1,
  author =       {Kuffner, J.J. and LaValle, S.M.},
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  booktitle = {IEEE International Conference on Robotics and Automation},
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}

