diff git a/documentation/althoff_other.bib b/documentation/althoff_other.bib
deleted file mode 100644
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 a/documentation/althoff_other.bib
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@@ 1,10657 +0,0 @@
% This file was created with JabRef 2.7b.
% Encoding: UTF8

@INPROCEEDINGS{Gupta2004,
 author = {S. Gupta and B.~H. Krogh and R.~A. Rutenbar},
 title = {Towards Formal Verification of Analog Designs},
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 pages = {210217},
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 owner = {matthias},
 timestamp = {2015.01.13}
}

@INPROCEEDINGS{AlSammane2007,
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 title = {A Symbolic Methodology for the Verification of Analog and Mixed Signal
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 owner = {matthias},
 timestamp = {2015.01.13}
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 title = {Combining Symbolic Simulation and Interval Arithmetic for the Verification
 of {AMS} Designs},
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 year = {2007},
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 ee = {http://doi.ieeecomputersociety.org/10.1109/FMCAD.2007.17},
 owner = {matthias},
 timestamp = {2015.01.13}
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@PHDTHESIS{vanOsch2009,
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 author = {R.~P. {van Til} and W.~E. Schmitendorf},
 title = {Constrained Controllability of DiscreteTime Systems},
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@PHDTHESIS{Abate2007b,
 author = {A. Abate},
 title = {Probabilistic Reachability for Stochastic Hybrid Systems: Theory,
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 school = {University of California, Berkeley},
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 owner = {althoff},
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@INPROCEEDINGS{Abate2007,
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 title = {Computational Approaches to Reachability Analysis of Stochastic Hybrid
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 booktitle = {Hybrid Systems: Computation and Control},
 year = {2007},
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 owner = {althoff},
 timestamp = {2007.10.02}
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 volume = {44},
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 owner = {althoff},
 timestamp = {2009.12.15}
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 title = {Automatic Symbolic Verification of Embedded Systems},
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@INCOLLECTION{Hartong2004,
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 title = {Formal Verification for Nonlinear Analog Systems: Approaches to Model
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@INPROCEEDINGS{Heitmeyer2015,
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@INPROCEEDINGS{Heissing2003,
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@BOOK{Hendriks2008,
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@INBOOK{Henk2004,
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@INPROCEEDINGS{Henzinger1995,
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 title = {Algorithmic Analysis of Nonlinear Hybrid Systems},
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@ARTICLE{Henzinger1997,
 author = {T.~A. Henzinger and P.H. Ho and H. WongToi},
 title = {{HyTech}: A Model Checker for Hybrid Systems},
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@INPROCEEDINGS{Henzinger2000b,
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 title = {Beyond {HyTech}: Hybrid Systems Analysis Using Interval Numerical
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}

@INPROCEEDINGS{Henzinger2000c,
 author = {T.~A. Henzinger and R. Majumdar},
 title = {Symbolic Model Checking for Rectangular Hybrid Systems},
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 year = {2000},
 series = {LNCS 1785},
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 publisher = {Springer},
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 timestamp = {2012.09.24}
}

@INPROCEEDINGS{Henzinger2001,
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 title = {AssumeGuarantee Reasoning for Hierarchical Hybrid Systems},
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 year = {2001},
 series = {LNCS 2034},
 pages = {275290},
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 timestamp = {2014.02.06}
}

@ARTICLE{Henzinger1998,
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 year = {1998},
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@INPROCEEDINGS{Herceg2013,
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 and Systems},
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@PHDTHESIS{Hesse2013,
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 keywords = {analog/mixedsignal circuits;binary decision diagrams;model checker;satisfiability
 modulo theory solver;symbolic methods;symbolic model checking;Petri
 nets;binary decision diagrams;circuit analysis computing;formal verification;hardware
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 school = {University of Michigan},
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}

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@INPROCEEDINGS{Althoff2015b,
 author = {D. Althoff and M. Althoff and S. Scherer},
 title = {Online Safety Verification of Trajectories for Unmanned Flight with
 Offline Computed Robust Invariant Sets},
 booktitle = {Proc. of the IEEE/RSJ International Conference on Intelligent Robots
 and Systems},
 year = {2015},
 pages = {34703477},
 comment = {vCar},
 owner = {matthias},
 timestamp = {2015.06.29}
}

@INPROCEEDINGS{Althoff2010b,
 author = {D. Althoff and M. Althoff and D. Wollherr and M. Buss},
 title = {Probabilistic Collision State Checker for Crowded Environments},
 booktitle = {Proc. of the IEEE International Conference on Robotics and Automation},
 year = {2010},
 pages = {14921498},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2010/althoff2010ICRA.pdf:PDF;
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 owner = {matthias},
 timestamp = {2010.06.06}
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@INPROCEEDINGS{Althoff2015a,
 author = {M. Althoff},
 title = {An Introduction to {CORA} 2015},
 booktitle = {Proc. of the Workshop on Applied Verification for Continuous and
 Hybrid Systems},
 year = {2015},
 pages = {120151},
 comment = {UnCoVerCPS},
 owner = {matthias},
 timestamp = {2015.03.20}
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 title = {CORA 2015},
 author = {M. Althoff},
 organization = {Technische Universit\"at M\"unchen},
 year = {2015},
 owner = {matthias},
 timestamp = {2016.05.19}
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@ELECTRONIC{Althoff2015d,
 author = {M. Althoff},
 year = {2015},
 title = {On Computing the Minkowski Difference of Zonotopes},
 howpublished = {arXiv:1512.02794},
 url = {http://arxiv.org/abs/1512.02794},
 comment = {UnCoVerCPS},
 owner = {matthias},
 timestamp = {2015.10.05}
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@ARTICLE{Althoff2014c,
 author = {M. Althoff},
 title = {Formal and Compositional Analysis of Power Systems using Reachable
 Sets},
 journal = {IEEE Transactions on Power Systems},
 year = {2014},
 volume = {29},
 pages = {22702280},
 number = {5},
 owner = {matthias},
 timestamp = {2014.03.04}
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 author = {M. Althoff},
 title = {Reachability Analysis of Nonlinear Systems using Conservative Polynomialization
 and NonConvex Sets},
 booktitle = {Hybrid Systems: Computation and Control},
 year = {2013},
 pages = {173182},
 owner = {matthias},
 timestamp = {2012.12.23}
}

@PHDTHESIS{Althoff2010a,
 author = {M. Althoff},
 title = {Reachability Analysis and its Application to the Safety Assessment
 of Autonomous Cars},
 school = {Technische Universit\"{a}t M\"{u}nchen},
 year = {2010},
 type = {Dissertation},
 note = {http://nbnresolving.de/urn/resolver.pl?urn:nbn:de:bvb:91diss2010071596375214},
 filelinks = {[pdf]:media/New Volume/other/Jemdoc website/publications/2010/althoff2010thesis.pdf:PDF;
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 owner = {matthias},
 timestamp = {2010.08.03}
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@MASTERSTHESIS{Althoff2005,
 author = {M. Althoff},
 title = {Nonlinear Dynamics and Control of Integrally Actuated Helicopter
 Blades},
 school = {Technische Universit\"{a}t M\"{u}nchen},
 year = {2005},
 filelinks = {[pdf]:media/New Volume/other/Jemdoc website/publications/2005/althoff2005diplomaThesis.pdf:PDF},
 owner = {matthias},
 timestamp = {2011.05.22}
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 author = {M. Althoff and D. Althoff and D. Wollherr and M. Buss},
 title = {Safety Verification of Autonomous Vehicles for Coordinated Evasive
 Maneuvers},
 booktitle = {Proc. of the IEEE Intelligent Vehicles Symposium},
 year = {2010},
 pages = {10781083},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2010/althoff2010IV.pdf:PDF;
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 timestamp = {2010.06.06}
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 author = {M. Althoff and M. Cvetkovi{\'c} and M. Ili{\'c}},
 title = {Transient Stability Analysis by Reachable Set Computation},
 booktitle = {Proc. of the IEEE PES Conference on Innovative Smart Grid Technologies
 Europe},
 year = {2012},
 owner = {matthias},
 timestamp = {2012.10.22}
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 author = {M.~Althoff and J.~M.~Dolan},
 title = {Online Verification of Automated Road Vehicles Using Reachability
 Analysis},
 journal = {IEEE Transactions on Robotics},
 year = {2014},
 volume = {30},
 pages = {903918},
 number = {4},
 comment = {vCar},
 owner = {matthias},
 timestamp = {2014.01.14}
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 author = {M. Althoff and J.~M. Dolan},
 title = {Reachability Computation of LowOrder Models for the Safety Verification
 of HighOrder Road Vehicle Models},
 booktitle = {Proc. of the American Control Conference},
 year = {2012},
 pages = {35593566},
 owner = {matthias},
 timestamp = {2012.03.28}
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 author = {M. Althoff and J.~M. Dolan},
 title = {SetBased Computation of Vehicle Behaviors for the Online Verification
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 booktitle = {Proc. of the 14th IEEE Conference on Intelligent Transportation Systems},
 year = {2011},
 pages = {11621167},
 owner = {matthias},
 timestamp = {2011.09.06}
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 author = {M. Althoff and G. Frehse},
 title = {Combining Zonotopes and Support Functions for Efficient Reachability
 Analysis of Linear Systems},
 booktitle = {Proc. of the 55th IEEE Conference on Decision and Control},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.09.16}
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 author = {M. Althoff and D. Grebenyuk},
 title = {Implementation of Interval Arithmetic in {CORA} 2016},
 booktitle = {Proc. of the 3rd International Workshop on Applied Verification for
 Continuous and Hybrid Systems},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.07.20}
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 author = {M. Althoff and D. He{\ss} and F. Gambert},
 title = {Road Occupancy Prediction of Traffic Participants},
 booktitle = {Proc. of the 16th International IEEE Conference on Intelligent Transportation
 Systems},
 year = {2013},
 pages = {99105},
 owner = {matthias},
 timestamp = {2013.10.24}
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@ARTICLE{Althoff2014a,
 author = {M. Althoff and B.~H. Krogh},
 title = {Reachability Analysis of Nonlinear DifferentialAlgebraic Systems},
 journal = {IEEE Transactions on Automatic Control},
 year = {2014},
 volume = {59},
 pages = {371383},
 number = {2},
 owner = {matthias},
 timestamp = {2012.12.23}
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 title = {Avoiding Geometric Intersection Operations in Reachability Analysis
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 booktitle = {Hybrid Systems: Computation and Control},
 year = {2012},
 pages = {4554},
 owner = {matthias},
 timestamp = {2011.12.24}
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 author = {M. Althoff and B.~H. Krogh},
 title = {Zonotope Bundles for the Efficient Computation of Reachable Sets},
 booktitle = {Proc. of the 50th IEEE Conference on Decision and Control},
 year = {2011},
 pages = {68146821},
 owner = {matthias},
 timestamp = {2011.09.12}
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@INBOOK{Althoff2011b,
 chapter = {Analyzing Reachability of Linear Dynamic Systems with Parametric
 Uncertainties},
 pages = {6994},
 title = {Modeling, Design, and Simulation of Systems with Uncertainties},
 publisher = {Springer},
 year = {2011},
 editor = {A. Rauh and E. Auer},
 author = {M. Althoff and B.~H. Krogh and O. Stursberg},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2011/althoff2011MoDeSi.pdf:PDF},
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@INPROCEEDINGS{Althoff2011a,
 author = {M. Althoff and C. {Le Guernic} and B.~H. Krogh},
 title = {Reachable Set Computation for Uncertain TimeVarying Linear Systems},
 booktitle = {Hybrid Systems: Computation and Control},
 year = {2011},
 pages = {93102},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2011/althoff2011HSCC.pdf:PDF;
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 owner = {matthias},
 timestamp = {2011.01.11}
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@INPROCEEDINGS{Althoff2016b,
 author = {M. Althoff and R. L{\"o}sch},
 title = {Can Automated Road Vehicles Harmonize with Traffic Flow While Guaranteeing
 A Safe Distance?},
 booktitle = {Proc. of the 19th International IEEE Conference on Intelligent Transportation
 Systems},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.09.16}
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@ARTICLE{Althoff2016d,
 author = {M. Althoff and S. Magdici},
 title = {SetBased Prediction of Traffic Participants on Arbitrary Road Networks},
 journal = {IEEE Transactions on Intelligent Vehicles},
 year = {2016},
 volume = {1},
 pages = {187202},
 number = {2},
 owner = {matthias},
 timestamp = {2016.09.16}
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@ARTICLE{Althoff2011d,
 author = {M. Althoff and A. Mergel},
 title = {Comparison of {Markov} Chain Abstraction and {Monte Carlo} Simulation
 for the Safety Assessment of Autonomous Cars},
 journal = {IEEE Transactions on Intelligent Transportation Systems},
 year = {2011},
 volume = {12},
 pages = {12371247},
 number = {4},
 owner = {matthias},
 timestamp = {2011.05.22}
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@ARTICLE{Althoff2011g,
 author = {M. Althoff and M.~J. Patil and J.~P. Traugott},
 title = {Nonlinear Modeling and Control Design of Active Helicopter Blades},
 journal = {Journal of the American Helicopter Society},
 year = {2012},
 volume = {57},
 pages = {111},
 number = {1},
 owner = {matthias},
 timestamp = {2011.10.02}
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 title = {Nonlinear Modeling and Control Design of Active Helicopter Blades},
 booktitle = {Proc. of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics,
 and Materials Conference},
 year = {2006},
 note = {AIAA Paper 20062040},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2006/althoff2006ASC.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2006/althoff2006ASC_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2009.05.06}
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@ARTICLE{Althoff2013c,
 author = {M. Althoff and A. Rajhans and B.~H. Krogh and S. Yaldiz and X. Li
 and L. Pileggi},
 title = {Formal Verification of PhaseLocked Loops Using Reachability Analysis
 and Continuization},
 journal = {Communications of the ACM},
 year = {2013},
 volume = {56},
 pages = {97104},
 number = {10},
 owner = {matthias},
 timestamp = {2012.10.22}
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@INPROCEEDINGS{Althoff2011c,
 author = {M. Althoff and A. Rajhans and B.~H. Krogh and S. Yaldiz and X. Li
 and L. Pileggi},
 title = {Formal Verification of PhaseLocked Loops Using Reachability Analysis
 and Continuization},
 booktitle = {Proc. of the Int. Conference on Computer Aided Design},
 year = {2011},
 pages = {659666},
 owner = {matthias},
 timestamp = {2011.04.04}
}

@ARTICLE{Althoff2009b,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {Computing Reachable Sets of Hybrid Systems Using a Combination of
 Zonotopes and Polytopes},
 journal = {Nonlinear Analysis: Hybrid Systems},
 year = {2010},
 volume = {4},
 pages = {233249},
 number = {2},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2010/althoff2010nonAnalysis.pdf:PDF},
 owner = {althoff},
 timestamp = {2009.01.12}
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@ARTICLE{Althoff2009a,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {ModelBased Probabilistic Collision Detection in Autonomous Driving},
 journal = {IEEE Transactions on Intelligent Transportation Systems},
 year = {2009},
 volume = {10},
 pages = {299  310},
 number = {2},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2009/althoff2009ITSStrans.pdf:PDF},
 owner = {althoff},
 timestamp = {2009.01.12}
}

@INPROCEEDINGS{Althoff2009d,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {Safety Assessment for Stochastic Linear Systems using Enclosing Hulls
 of Probability Density Functions},
 booktitle = {Proc. of the European Control Conference},
 year = {2009},
 pages = {625630},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2009/althoff2009ECC.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2009/althoff2009ECC_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.07.11}
}

@INPROCEEDINGS{Althoff2009e,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {Safety Assessment of Driving Behavior in MultiLane Traffic for Autonomous
 Vehicles},
 booktitle = {Proc. of the IEEE Intelligent Vehicles Symposium},
 year = {2009},
 pages = {893900},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2009/althoff2009IV.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2009/althoff2009IV_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2009.02.18}
}

@INPROCEEDINGS{Althoff2008a,
 author = {M. Althoff AND O. Stursberg AND M. Buss},
 title = {Verification of Uncertain Embedded Systems by Computing Reachable
 Sets based on Zonotopes},
 booktitle = {Proc. of the 17th IFAC World Congress},
 year = {2008},
 pages = {51255130},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/althoff2008IFAC.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2008/althoff2008IFAC_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2007.10.09}
}

@INPROCEEDINGS{Althoff2008b,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {Stochastic Reachable Sets of Interacting Traffic Participants},
 booktitle = {Proc. of the IEEE Intelligent Vehicles Symposium},
 year = {2008},
 pages = {10861092},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/althoff2008IV.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2008/althoff2008IV_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.03.10}
}

@INPROCEEDINGS{Althoff2008c,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {Reachability Analysis of Nonlinear Systems with Uncertain Parameters
 using Conservative Linearization},
 booktitle = {Proc. of the 47th IEEE Conference on Decision and Control},
 year = {2008},
 pages = {40424048},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/althoff2008CDC.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2008/althoff2008CDC_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.04.28}
}

@INPROCEEDINGS{Althoff2008d,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {{O}nline{A}nalyse von {F}ahrstrategien kognitiver autonomer {F}ahrzeuge},
 booktitle = {Proc. of Automatisierungssysteme, Assistenzsysteme und eingebettete
 Systeme f\"ur Transportmittel (AAET)},
 year = {2008},
 pages = {314330},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/althoff2008AAET.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2008/althoff2008AAET_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.04.28}
}

@INPROCEEDINGS{Althoff2008e,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {{E}rreichbarkeitsanalyse von {V}erkehrsteilnehmern zur {V}erbesserung
 von {F}ahrerassistenzsystemen},
 booktitle = {Proc. of 3. Tagung Aktive Sicherheit durch Fahrerassistenz},
 year = {2008},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/althoff2008FAS.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2008/althoff2008FAS_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.04.28}
}

@ARTICLE{Althoff2008f,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {{S}icherheitsbewertung von {F}ahrstrategien kognitiver {A}utomobile},
 journal = {at  Automatisierungstechnik},
 year = {2008},
 volume = {56},
 pages = {653661},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/althoff2008AT.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.07.09}
}

@INPROCEEDINGS{Althoff2007a,
 author = {M. Althoff AND O. Stursberg AND M. Buss},
 title = {Safety Assessment of Autonomous Cars using Verification Techniques},
 booktitle = {Proc. of the American Control Conference},
 year = {2007},
 pages = {41544159},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2007/althoff2007ACC.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2007/althoff2007ACC_talk.pdf:PDF},
 owner = {althoff}
}

@INPROCEEDINGS{Althoff2007b,
 author = {M. Althoff and O. Stursberg and M. Buss},
 title = {Online Verification of Cognitive Car Decisions},
 booktitle = {Proc. of the IEEE Intelligent Vehicles Symposium},
 year = {2007},
 pages = {728733},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2007/althoff2007IV.pdf:PDF;
 [poster]:media/New Volume/other/Jemdoc website/posters/2007/althoff2007IV_poster.pdf:PDF},
 owner = {althoff},
 timestamp = {2007.06.26}
}

@INPROCEEDINGS{Althoff2007c,
 author = {M.~Althoff and O.~Stursberg and M.~Buss},
 title = {Reachability Analysis of Linear Systems with uncertain Parameters
 and Inputs},
 booktitle = {Proc. of the 46th IEEE Conference on Decision and Control},
 year = {2007},
 pages = {726732},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2007/althoff2007CDC.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2007/althoff2007CDC_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2007.08.29}
}

@INPROCEEDINGS{ElGuindy2017a,
 author = {A. ElGuindy and Y.~C. Chen and M. Althoff},
 title = {Compositional Transient Stability Analysis of Power Systems via the
 Computation of Reachable Sets},
 booktitle = {Proc. of the American Control Conference},
 year = {2017},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@ARTICLE{ElGuindy2016,
 author = {A. ElGuindy and D. Han and M. Althoff},
 title = {Formal Analysis of DrumBoiler Units to Maximize the LoadFollowing
 Capabilities of Power Plants},
 journal = {IEEE Transactions on Power Systems},
 year = {2016},
 note = {online available, not yet printed},
 owner = {matthias},
 timestamp = {2016.01.07}
}

@INPROCEEDINGS{Giusti2017,
 author = {A. Giusti and M. Althoff},
 title = {Efficient Computation of IntervalArithmeticBased Robust Controllers
 for Rigid Robots},
 booktitle = {Proc. of the First IEEE International Conference on Robotic Computing},
 year = {2017},
 owner = {matthias},
 timestamp = {2017.01.04}
}

@ARTICLE{Giusti2017b,
 author = {A. Giusti and M. Althoff},
 title = {OnTheFly Control Design of Modular Robot Manipulators},
 journal = {IEEE Transactions on Control Systems Technology},
 year = {2017},
 volume = {x},
 pages = {x},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Giusti2016a,
 author = {A. Giusti and M. Althoff},
 title = {Ultimate Robust Performance Control of Rigid Robot Manipulators Using
 Interval Arithmetic},
 booktitle = {Proc. of the American Control Conference},
 year = {2016},
 pages = {29953001},
 owner = {matthias},
 timestamp = {2016.03.02}
}

@INPROCEEDINGS{Giusti2015,
 author = {A. Giusti and M. Althoff},
 title = {Automatic Centralized Controller Design for Modular and Reconfigurable
 Robot Manipulators},
 booktitle = {Proc. of the IEEE/RSJ International Conference on Intelligent Robots
 and Systems},
 year = {2015},
 pages = {32683275},
 comment = {SMARTE},
 owner = {matthias},
 timestamp = {2015.06.29}
}

@INPROCEEDINGS{Giusti2017a,
 author = {A. Giusti and J. Malzahn and N.~G. Tsagarakis and M. Althoff},
 title = {Hybrid InverseDynamics/PassivityBased Control for Robots with Elastic
 Joints},
 booktitle = {Proc. of the IEEE International Conference on Robotics and Automation},
 year = {2017},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Goebl2008,
 author = {M. Goebl and M. Althoff and M. Buss and G. F\"arber and F. Hecker
 and B. Hei{\ss}ing and S. Kraus and R. Nagel and F. Puente Le{\'o}n
 and F. Rattei and M. Russ and M. Schweitzer and M. Thuy and C. Wang
 and H.J. W\"unsche},
 title = {Design and Capabilities of the {Munich} Cognitive Automobile},
 booktitle = {Proc. of the IEEE Intelligent Vehicles Symposium},
 year = {2008},
 pages = {11011107},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/goebl2008IV.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.07.11}
}

@INPROCEEDINGS{Han2016a,
 author = {D. Han and M. Althoff},
 title = {Power Systems Transient Stability Analysis via Optimal Rational Lyapunov
 Functions},
 booktitle = {IEEE Power \& Energy Society General Meeting},
 year = {2016},
 note = {accepted},
 owner = {matthias},
 timestamp = {2016.03.02}
}

@INPROCEEDINGS{Han2015,
 author = {D. Han and M. Althoff},
 title = {Control Synthesis for NonPolynomial Systems: A Domain of Attraction
 Perspective},
 booktitle = {Proc. of the 54th IEEE Conference on Decision and Control},
 year = {2015},
 pages = {11601167},
 comment = {ROCSGrid},
 owner = {matthias},
 timestamp = {2015.08.03}
}

@INPROCEEDINGS{Han2016c,
 author = {D. Han and A. ElGuindy and M. Althoff},
 title = {On Estimating the Robust Domain of Attraction for Uncertain NonPolynomial
 Systems: An LMI Approach},
 booktitle = {Proc. of the 55th IEEE Conference on Decision and Control},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Han2016d,
 author = {D. Han and A. ElGuindy and M. Althoff},
 title = {Estimating the Domain of Attraction based on the Invariance Principle},
 booktitle = {Proc. of the 55th IEEE Conference on Decision and Control},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Han2016b,
 author = {D. Han and A. Rizaldi and A. ElGuindy and M. Althoff},
 title = {On Enlarging the Backward Reachable Sets via Zonotopic Set Membership},
 booktitle = {IEEE MultiConference on Systems and Control},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.04.18}
}

@INPROCEEDINGS{Hassan2017,
 author = {H.~A. Hassan and E. Icer and K. ElAyat and M. Althoff},
 title = {Evolutionary Composition Synthesis of Modular Robots for a Given
 Task},
 booktitle = {Proc. of the IEEE International Conference on Robotics and Automation},
 year = {2017},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Hess2014,
 author = {D. He{\ss} and M. Althoff and T. Sattel},
 title = {Formal Verification of Maneuver Automata for Parameterized Motion
 Primitives},
 booktitle = {Proc. of the IEEE/RSJ International Conference on Intelligent Robots
 and Systems},
 year = {2014},
 pages = {14741481},
 comment = {vCar},
 owner = {matthias},
 timestamp = {2014.07.17}
}

@INPROCEEDINGS{Hess2013,
 author = {D. He{\ss} and M. Althoff and T. Sattel},
 title = {Comparison of Trajectory Tracking Controllers for Emergency Situations},
 booktitle = {Proc. of the IEEE Intelligent Vehicles Symposium},
 year = {2013},
 pages = {163170},
 owner = {matthias},
 timestamp = {2013.02.28}
}

@INPROCEEDINGS{Hess2013b,
 author = {D. He{\ss} and M. Althoff and T. Sattel},
 title = {Should Collision Avoidance Systems use Yaw Stabilization?},
 booktitle = {Proc. of the 16th International IEEE Conference on Intelligent Transportation
 Systems},
 year = {2013},
 pages = {20582062},
 owner = {matthias},
 timestamp = {2013.10.24}
}

@INPROCEEDINGS{Hess2016,
 author = {D. He{\ss} and M. Woehrle and J. Oehlerking and A. Biewer and M.
 Bottazzi and J. Sanchez and M. Althoff},
 title = {The {UnCoVerCPS} Verification Approach to Cooperative Automated Driving},
 booktitle = {Proc. of the 20th International Forum on Advanced Microsystems for
 Automotive Applications},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.05.13}
}

@INPROCEEDINGS{Icer2016,
 author = {E. Icer and M. Althoff},
 title = {A TaskDriven Algorithm for Configuration Synthesis of Modular Robots},
 booktitle = {Proc. of the IEEE International Conference on Robotics and Automation},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.03.02}
}

@INPROCEEDINGS{Icer2016b,
 author = {E. Icer and M. Althoff},
 title = {CostOptimal Composition Synthesis for Modular Robots},
 booktitle = {IEEE MultiConference on Systems and Control},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.04.18}
}

@INPROCEEDINGS{Kraus2009,
 author = {S. Kraus and M. Althoff and B. Hei{\ss}ing and M. Buss},
 title = {Cognition and Emotion in Autonomous Cars},
 booktitle = {Proc. of the IEEE Intelligent Vehicles Symposium},
 year = {2009},
 pages = {635640},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2009/kraus2009IV.pdf:PDF;
 [poster]:media/New Volume/other/Jemdoc website/posters/2009/kraus2009IV_poster.pdf:PDF},
 owner = {althoff},
 timestamp = {2009.03.27}
}

@INPROCEEDINGS{Lee2015,
 author = {H.S.~L. Lee and M. Althoff and S. Hoelldampf and M. Olbrich and
 E. Barke},
 title = {Automated Generation of Hybrid System Models for Reachability Analysis
 of Nonlinear Analog Circuits},
 booktitle = {Proc. of the 20th Asia and South Pacific Design Automation Conference},
 year = {2015},
 pages = {725730},
 comment = {ROCSGrid},
 owner = {matthias},
 timestamp = {2014.09.09}
}

@INPROCEEDINGS{Magdici2016,
 author = {S. Magdici and M. Althoff},
 title = {FailSafe Motion Planning of Autonomous Vehicles},
 booktitle = {Proc. of the 19th International IEEE Conference on Intelligent Transportation
 Systems},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Manzinger2017a,
 author = {S. Manzinger and M. Althoff},
 title = {Kooperative Bewegungsplanung autonomer Fahrzeuge unter Verwendung
 von Man{\"o}verTemplates},
 booktitle = {Proc. of Automatisierungssysteme, Assistenzsysteme und eingebettete
 Systeme f\"ur Transportmittel (AAET)},
 year = {2017},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Mesesan2015,
 author = {G. Mesesan and E. Icer and M. Althoff},
 title = {Hierarchical Genetic Path Planner for Highly Redundant Manipulators},
 booktitle = {Proc. of the Workshop on Task Planning for Intelligent Robots in
 Service and Manufacturing},
 year = {2015},
 comment = {SMARTE},
 owner = {matthias},
 timestamp = {2015.08.18}
}

@ARTICLE{Patil2011,
 author = {M.~J. Patil and M. Althoff},
 title = {Energyconsistent, {Galerkin} approach for the nonlinear dynamics
 of beams using intrinsic equations},
 journal = {Journal of Vibration and Control},
 year = {2011},
 volume = {17},
 pages = {17481758},
 number = {11},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2010/patil2010jvc.pdf:PDF},
 owner = {matthias},
 timestamp = {2011.02.14}
}

@INPROCEEDINGS{Patil2006,
 author = {M.~J. Patil and M. Althoff},
 title = {EnergyConsistent, {Galerkin} Approach for the Nonlinear Dynamics
 of Beams Using Mixed, Intrinsic Equations},
 booktitle = {Proc. of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics,
 and Materials Conference},
 year = {2006},
 note = {AIAA Paper 20061737},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2006/patil2006AIAA.pdf:PDF},
 owner = {althoff},
 timestamp = {2009.05.06}
}

@INPROCEEDINGS{Pek2017,
 author = {C. Pek and M. Werling and M. Althoff},
 title = {Evaluating Motion Safety of Selfdriving Vehicles using the Point
 of No Return},
 booktitle = {Proc. of the IEEE International Conference on Robotics and Automation},
 year = {2017},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Pereira2017a,
 author = {A. Pereira and M. Althoff},
 title = {Calculating Human Reachable Occupancy for Safe Robot Trajectory Planning},
 booktitle = {Proc. of the IEEE International Conference on Robotics and Automation},
 year = {2017},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@ARTICLE{Pereira2016a,
 author = {A. Pereira and M. Althoff},
 title = {Overapproximative, RealTime Human Arm Occupancy Prediction for HumanRobot
 Interaction},
 journal = {IEEE Transactions on Robotics},
 year = {2016},
 comment = {SMARTE},
 owner = {matthias},
 timestamp = {2015.10.05}
}

@INPROCEEDINGS{Pereira2016b,
 author = {A. Pereira and M. Althoff},
 title = {Overapproximative Arm Occupancy Prediction for HumanRobot Interaction
 Built from Archetypal Movements},
 booktitle = {International Conference on Robotics and Automation},
 year = {2016},
 comment = {SMARTE, possibly UnCoVerCPS},
 owner = {matthias},
 timestamp = {2015.10.05}
}

@INPROCEEDINGS{Pereira2015,
 author = {A. Pereira and M. Althoff},
 title = {Safety Control of Robots under Computed Torque Control using Reachable
 Sets},
 booktitle = {Proc. of the IEEE International Conference on Robotics and Automation},
 year = {2015},
 pages = {331  338},
 comment = {SMARTE},
 owner = {matthias},
 timestamp = {2015.02.05}
}

@INPROCEEDINGS{Rizaldi2015b,
 author = {A. Rizaldi and M. Althoff},
 title = {Formalising Traffic Rules for Accountability of Autonomous Vehicles},
 booktitle = {Proc. of the 18th IEEE International Conference on Intelligent Transportation
 Systems},
 year = {2015},
 pages = {1658  1665},
 comment = {PUMA},
 owner = {matthias},
 timestamp = {2015.06.16}
}

@INPROCEEDINGS{Rizaldi2016a,
 author = {A. Rizaldi and F. Immler and M. Althoff},
 title = {A Formally Verified Checker of the Safe Distance Traffic Rules for
 Autonomous Vehicles},
 booktitle = {8th NASA Formal Methods Symposium},
 year = {2016},
 pages = {175190},
 owner = {matthias},
 timestamp = {2016.04.11}
}

@INPROCEEDINGS{Rizaldi2015,
 author = {A. Rizaldi and S. S{\"o}ntges and M. Althoff},
 title = {On TimeMemory TradeOff for Collision Detection},
 booktitle = {Proc. of the IEEE Intelligent Vehicles Symposium},
 year = {2015},
 pages = {1173  1180},
 comment = {vCar, PUMA},
 owner = {matthias},
 timestamp = {2015.04.29}
}

@INPROCEEDINGS{Roehm2016b,
 author = {H. Roehm and J. Oehlerking and T. Heinz and M. Althoff},
 title = {{STL} Model Checking of Continuous and Hybrid Systems},
 booktitle = {Proc. of the 14th International Symposium on Automated Technology
 for Verification and Analysis},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.05.11}
}

@INPROCEEDINGS{Roehm2016,
 author = {H. Roehm and J. Oehlerking and M. Woehrle and M. Althoff},
 title = {Reachset Conformance Testing of Hybrid Automata},
 booktitle = {Proc. of Hybrid Systems: Computation and Control},
 year = {2016},
 pages = {277286},
 owner = {matthias},
 timestamp = {2015.12.17}
}

@INPROCEEDINGS{Rohrmueller2008,
 author = {F. Rohrm{\"u}ller and M. Althoff and D. Wollherr and M. Buss},
 title = {Probabilistic Mapping of Dynamic Obstacles Using {Markov} Chains
 for Replanning in Dynamic Environments},
 booktitle = {Proc. of the IEEE/RSJ International Conference on Intelligent Robots
 and Systems},
 year = {2008},
 pages = {25042510},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/rohrmueller2008IROS.pdf:PDF;
 [talk]:media/New Volume/other/Jemdoc website/talks/2008/rohrmueller2008IROS_talk.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.07.11}
}

@INPROCEEDINGS{Schuermann2016a,
 author = {B. Sch{\"u}rmann and M. Althoff},
 title = {ClosedForm Expressions of Convex Combinations for Controller Design},
 booktitle = {Proc. of American Control Conference},
 year = {2016},
 pages = {27952801},
 owner = {matthias},
 timestamp = {2015.12.17}
}

@INPROCEEDINGS{Schuermann2016b,
 author = {B. Sch{\"u}rmann and M. Althoff},
 title = {Convex Interpolation Control with Formal Guarantees},
 booktitle = {Proc. of Hybrid Systems: Computation and Control},
 year = {2016},
 note = {submitted},
 owner = {matthias},
 timestamp = {2015.12.17}
}

@ARTICLE{Soentges2017a,
 author = {S. S{\"o}ntges and M. Althoff},
 title = {Computing the Drivable Area of Autonomous Road Vehicles in Dynamic
 Road Scenes},
 journal = {IEEE Transactions on Intelligent Transportation Systems},
 year = {2017},
 volume = {x},
 pages = {x},
 owner = {matthias},
 timestamp = {2016.09.16}
}

@INPROCEEDINGS{Soentges2015,
 author = {S. S{\"o}ntges and M. Althoff},
 title = {Determining the Nonexistence of Evasive Trajectories for Collision
 Avoidance Systems},
 booktitle = {Proc. of the 18th IEEE International Conference on Intelligent Transportation
 Systems},
 year = {2015},
 pages = {956  961},
 comment = {vCar},
 owner = {matthias},
 timestamp = {2015.06.16}
}

@INPROCEEDINGS{Thuy2008,
 author = {M. Thuy and M. Goebl and F. Rattei and M. Althoff and F. Obermeier
 and S. Hawe and R. Nagel and S. Kraus and C. Wang and F. Hecker and
 M. Russ and M. Schweitzer and F. Puente Le{\'o}n and G. F{\"a}rber
 and M. Buss and K. Diepold and J. Ebersp{\"a}cher and B. Hei{\ss}ing
 and H.J. W{\"u}nsche},
 title = {{K}ognitive {A}utomobile: {N}eue {K}onzepte und {I}deen des {S}onderforschungsbereiches/{TR28}},
 booktitle = {Proc. of 3. Tagung Aktive Sicherheit durch Fahrerassistenz},
 year = {2008},
 filelinks = {[preprint]:media/New Volume/other/Jemdoc website/publications/2008/thuy2008FAS.pdf:PDF},
 owner = {althoff},
 timestamp = {2008.07.11}
}

@INPROCEEDINGS{Zeestraten2016,
 author = {M.~J.~A. Zeestraten and A. Pereira and M. Althoff and S. Calinon},
 title = {Online Motion Synthesis with Minimal Intervention Control and Formal
 Safety Guarantees},
 booktitle = {Proc. of IEEE International Conference on Systems, Man, and Cybernetics},
 year = {2016},
 owner = {matthias},
 timestamp = {2016.04.18}
}

@PROCEEDINGS{Frehse2016,
 title = {ARCH1415. 1st and 2nd International Workshop on Applied veRification
 for Continuous and Hybrid Systems},
 year = {2016},
 editor = {G. Frehse and M. Althoff},
 volume = {34},
 series = {EasyChair Proceedings in Computing},
 publisher = {EasyChair},
 owner = {matthias},
 timestamp = {2015.12.17}
}

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1 ExplicitGroup:Book chapter\;0\;;
}

diff git a/documentation/easychair.cls b/documentation/easychair.cls
deleted file mode 100644
index 9d7f61c07fdee9ad74405d9f230fda65af2c82ff..0000000000000000000000000000000000000000
 a/documentation/easychair.cls
+++ /dev/null
@@ 1,931 +0,0 @@
%
% Some credits
%

\def\easychairstyleauthor{easychair class style, by Serguei A. Mokhov and Andrei Voronkov <10 December 2015>}
\def\easychairstylerevision{CVS Revision: $Id: easychair.cls,v 3.4 2015/12/10 09:52 voronkov Exp $}
\def\easychairstylepurpose{Designed for easyChair.org, under guidelines and suggestions of}
\def\easychairstylevoronkov{\space\space\space\space\space\space\space\space\space\space\space\space\space Andrei Voronkov , and}
\def\easychairstylesutcliffe{\space\space\space\space\space\space\space\space\space\space\space\space\space Geoff Sutcliffe }
\def\easychairstylecopyright{Copyright terms are that of easychair.org}
\def\easychairstylebugs{For bug reports, please contact }

\everyjob{\typeout{\easychairstyleauthor}}
\everyjob{\typeout{\easychairstylerevision}}
\everyjob{\typeout{\easychairstylepurpose}}
\everyjob{\typeout{\easychairstylevoronkov}}
\everyjob{\typeout{\easychairstylesutcliffe}}
\everyjob{\typeout{\easychairstylecopyright}}
\everyjob{\typeout{\easychairstylebugs}}

\immediate\write10{\easychairstyleauthor}
\immediate\write10{\easychairstylerevision}
\immediate\write10{\easychairstylepurpose}
\immediate\write10{\easychairstylevoronkov}
\immediate\write10{\easychairstylesutcliffe}
\immediate\write10{\easychairstylecopyright}
\immediate\write10{\easychairstylebugs}

%
% Require LaTeX 2.09 or later
%

\NeedsTeXFormat{LaTeX2e}[1995/12/01]
\ProvidesClass{easychair}[2015/12/10 v3.4]
\def\@tempa#1#2\@nil{\edef\@classname{#1}}
\expandafter\@tempa\@currnamestack{}{}{}\@nil
\ifx\@classname\@empty \edef\@classname{\@currname}\fi

\RequirePackage{color}

%
% Debug
%

\def\easychairdebug#1{\gdef\@EasyDebug{#1}}
\def\@EasyDebug{}

\newif\ifdebug
\debugfalse
\DeclareOption{debug}{\debugtrue}

\newif\ifprocedia
\procediafalse
\DeclareOption{procedia}{\procediatrue}

% Procedia with the empty header
\newif\ifprocediaempty
\procediaemptyfalse
\DeclareOption{procediaempty}{\procediaemptytrue}

\newif\ifprocediafinal
\procediafinalfalse
\DeclareOption{procediafinal}{\procediafinaltrue}

\newif\ifEPiC
\EPiCfalse
\DeclareOption{EPiC}{\EPiCtrue}

% EPiC with the empty header
\newif\ifEPiCempty
\EPiCemptyfalse
\DeclareOption{EPiCempty}{\EPiCemptytrue}

\newif\ifEPiCfinal
\EPiCfinalfalse
\DeclareOption{EPiCfinal}{\EPiCfinaltrue}

\def\easychairframe#1{\gdef\@EasyFrame{#1}}
\def\@EasyFrame{}

\newif\ifframe
\framefalse

\DeclareOption{frame}{\frametrue}

\def\easychairverbose#1{\gdef\@EasyVerbose{#1}}
\def\@EasyVerbose{}

\newif\ifverbose
\verbosefalse

\DeclareOption{verbose}{\verbosetrue}


%
% Thesis
% Perh Geoff, February 23, 2010 with support from Andrei
%

\def\easythesis#1{\gdef\@EasyThesis{#1}}
\def\@EasyThesis{}

\newif\ifthesis
\thesisfalse

\DeclareOption{thesis}{\thesistrue}


%
% Times New Roman or not
%

\def\easytimes#1{\gdef\@EasyTimes{#1}}
\def\@EasyTimes{}

\newif\ifnotimes
\notimesfalse

\DeclareOption{notimes}{\notimestrue}

\newif\ifwithtimes
\withtimesfalse

\DeclareOption{withtimes}{\withtimestrue}


%% Code added to use llncs style author list
\newcounter{@inst}
\newcounter{@auth}
\newcounter{auco}
\newdimen\instindent
\def\institute#1{\gdef\@institute{#1}}

\def\institutename{\par
 \begingroup
 \parskip=\z@
 \parindent=\z@
 \setcounter{@inst}{1}%
 \def\and{\par\stepcounter{@inst}%
 \noindent$^{\the@inst}$\enspace\ignorespaces}%
 \setbox0=\vbox{\def\thanks##1{}\@institute}%
 \ifnum\c@@inst=1\relax
 \gdef\fnnstart{0}%
 \else
 \xdef\fnnstart{\c@@inst}%
 \setcounter{@inst}{1}%
 \noindent$^{\the@inst}$\enspace
 \fi
 \ignorespaces
 \@institute\par
 \endgroup}

\def\inst#1{\unskip$^{#1}$}
\def\fnmsep{\unskip$^,$}
\def\email#1{{\tt#1}}

%%

\newif\ifauthorundefined
\authorundefinedtrue

\let\oldauthor=\author
\renewcommand
 {\author}
 [1]
 {%
 \ifauthorundefined
 \oldauthor{#1}
 \authorundefinedfalse
 \else
 \PackageWarning{easychair}{Another use of author ignored}
 \fi
 }

\newif\iftitleundefined
\titleundefinedtrue

\let\oldtitle=\title
\renewcommand
 {\title}
 [1]
 {
 \iftitleundefined
 \oldtitle{#1}
 \titleundefinedfalse
 \else
 \PackageWarning{easychair}{Another use of title ignored}
 \fi
 }


%
% Running heads definitions
%

%\def\titlerunning#1{\gdef\@titleRunning{#1}}
%\def\authorrunning#1{\gdef\@authorRunning{#1}}
%\titlerunning{easychair: Running title head is undefined.}
%\authorrunning{easychair: Running author head is undefined.}

\newif\iftitlerunningundefined
\titlerunningundefinedtrue

\newif\ifauthorrunningundefined
\authorrunningundefinedtrue

\gdef\@titleRunning{easychair: Running title head is undefined.}
\gdef\@authorRunning{easychair: Running author head is undefined.}

\def\titlerunning#1
{
 \iftitlerunningundefined
 \gdef\@titleRunning{#1}
 \titlerunningundefinedfalse
 \else
 \PackageWarning{easychair}{Another use of titlerunning ignored}
 \fi
}

\def\authorrunning#1
{
 \ifauthorrunningundefined
 \gdef\@authorRunning{#1}
 \authorrunningundefinedfalse
 \else
 \PackageWarning{easychair}{Another use of authorrunning ignored}
 \fi
}

%
% Affiliations
%

\newcommand{\affiliation}[1]{\footnotesize{#1}\vspace{3pt}}


%
% Decide between letter and A4 paper formats
% as well as orientation
%

% Default is 'letterpaper'
\def\paperformat#1{\gdef\@PaperFormat{#1}}
\def\@PaperFormat{letterpaper}

\newif\ifletterpaper
\newif\ifafourpaper
\newif\ifcustompaper

\letterpapertrue

\DeclareOption{letterpaper}{\paperformat{letterpaper}\afourpaperfalse\custompaperfalse}
\DeclareOption{a4paper}{\paperformat{a4paper}\afourpapertrue\letterpaperfalse\custompaperfalse}
\DeclareOption{custompaper}{\paperformat{letterpaper}\afourpaperfalse\letterpaperfalse\custompapertrue}
\ExecuteOptions{letterpaper}

\newlength{\@LMarginSize}
\newlength{\@RMarginSize}
\newlength{\@TMarginSize}
\newlength{\@BMarginSize}

\DeclareOption{lmargin}{}
\DeclareOption{rmargin}{}
\DeclareOption{tmargin}{}
\DeclareOption{bmargin}{}


% Default is portrait {}
\def\paperorientation#1{\gdef\@PaperOrientation{#1}}
\def\@PaperOrientation{}

\DeclareOption{portrait}{\paperorientation{}}
\DeclareOption{landscape}{\paperorientation{landscape}}

% Two sided running heads for titlerunning and author running
% twosided is the default
\newif\iftwosided
\twosidedfalse

\DeclareOption{onesided}{}
\DeclareOption{twosided}{\twosidedtrue}

%
% Decide between 1 or 2column formats
%

\def\columnCount#1{\gdef\@ColumnCount{#1}}
\def\@ColumnCount{onecolumn}

\DeclareOption{onecolumn}{}
\DeclareOption{twocolumn}{\columnCount{twocolumn}}

%
% Decide on line spacing
%

\def\lineSpacing#1{\gdef\@LineSpacing{#1}}
\def\@LineSpacing{1.0}

\DeclareOption{zerospacing}{\lineSpacing{0.0}}
\DeclareOption{singlespacing}{\lineSpacing{1.0}}
\DeclareOption{lineandhalfspacing}{\lineSpacing{1.5}}
\DeclareOption{doublespacing}{\lineSpacing{2.0}}

\DeclareOption{0.0}{\lineSpacing{0.0}}
\DeclareOption{1.0}{\lineSpacing{1.0}}
\DeclareOption{1.5}{\lineSpacing{1.5}}
\DeclareOption{2.0}{\lineSpacing{2.0}}

\DeclareOption{0.0pt}{\lineSpacing{0.0}}
\DeclareOption{1.0pt}{\lineSpacing{1.0}}
\DeclareOption{1.5pt}{\lineSpacing{1.5}}
\DeclareOption{2.0pt}{\lineSpacing{2.0}}

%
% Font point size; default is 10pt
%
% The \headheight will have to be changed later accordingly
% such that fancyhdr does not complain it is too small.
%

\def\baseFontSize#1{\gdef\@BaseFontSize{#1}}
\def\headHeightSize#1{\gdef\@HeadHeightSize{#1}}
\def\headSepSize#1{\gdef\@HeadSepSize{#1}}
\def\footSkipSize#1{\gdef\@FootSkipSize{#1}}

\def\@BaseFontSize{10pt}
\def\@HeadHeightSize{12.0pt}
\def\@HeadSepSize{16.0pt} % instead of the default 25pt
\def\@FootSkipSize{26.0pt} % instead of the default 30pt

\DeclareOption{8pt}{\PackageWarning{easychair}{Option '\CurrentOption' is not supported.}}
\DeclareOption{9pt}{\PackageWarning{easychair}{Option '\CurrentOption' is not supported.}}
\DeclareOption{10pt}{\baseFontSize{10pt}\headHeightSize{12.0pt}\headSepSize{16.0pt}\footSkipSize{26pt}}
\DeclareOption{11pt}{\baseFontSize{11pt}\headHeightSize{13.6pt}\headSepSize{23.0pt}\footSkipSize{28pt}}
\DeclareOption{12pt}{\baseFontSize{12pt}\headHeightSize{14.5pt}\headSepSize{25.0pt}\footSkipSize{30pt}}
\ExecuteOptions{10pt}

%
% Page sizing
%

%\newif\iffullpage
%\newif\ifsavetrees
%
%\DeclareOption{fullpage}{\fullpagetrue}
%\DeclareOption{savetrees}{\savetreestrue}

% Bark at any unknown package option
\DeclareOption*{\PackageWarning{easychair}{Unknown option '\CurrentOption'}}
\DeclareOption*{\PassOptionsToPackage{\CurrentOption}{geometry}}
%\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}

%\ExecuteOptions{centertags,portrait,10pt,twoside,onecolumn,final}
%\ExecuteOptions{}
\ProcessOptions\relax

%
% Required packages and classes.
%
% All must be standard as per most common LaTeX
% distributions.
%

\ifthesis
 \LoadClass[\@PaperFormat,\@PaperOrientation,\@ColumnCount,\@BaseFontSize,twoside]{report}
 \RequirePackage{makeidx}
\else
 % We are an article (more customized later)
 \LoadClass[\@PaperFormat,\@PaperOrientation,\@ColumnCount,\@BaseFontSize,twoside]{article}
 %\LoadClass[\@PaperFormat,\@PaperOrientation,\@ColumnCount,\@BaseFontSize]{article}
\fi

% Require UTF8 encoding, per Andrei Voronkov, to accomodate
% all sorts of author names.
%\RequirePackage[utf8]{inputenc}

% To ensure the footnotes are always at the bottom.
% IMPORTANT: footmisc should precede hyperref for the footnotes to hyperlink
% correctly to their pages where they are at instead of always at
% page 1. Per bug reports from a couple of users and a suggestion by
% Uwe Pfeiffer.
\RequirePackage[bottom]{footmisc}

% TOC/thumbnail LHS preview in the PDFs as well as active URLs and other crossrefs
% Newer versions of hyperref declare a4paper or letterpaper as obsolete and issue warnings
\definecolor{linkcolor}{RGB}{170,0,0} % aa0000
\definecolor{citecolor}{RGB}{44,160,46} % 2ca02d
\definecolor{urlcolor}{RGB}{0,51,153} % 003399
\RequirePackage[linktocpage,pdfcreator=easychair.cls3.4,colorlinks=True,urlcolor=urlcolor,citecolor=citecolor,linkcolor=linkcolor]{hyperref}

% Traditional graphics processing
\RequirePackage{graphicx}
%\RequirePackage{pdflscape}
%\RequirePackage{lscape}


%% Fonts, generally more compact but preserving point size

% Pick "Times Roman" as a base font unless explicitly told not to
\ifnotimes
 \ifwithtimes
 \PackageWarning{easychair}{Cannot really use 'notimes' and 'withtimes' together}
 \PackageWarning{easychair}{Defaulting to 'notimes'...}
 \else
 \PackageWarning{easychair}{'notimes' has been deprecated as it is the default in 2.0}
 \fi
\else
 \ifwithtimes
 \RequirePackage{mathptmx}
 \fi
\fi

% Pick "Helvetica" as a "SansSerif" font
%\RequirePackage[scaled=.85]{helvet}

% For algorithm and source code listings
\RequirePackage{listings}

%% Different Math and nonMath symbols and definitions

\RequirePackage{latexsym}
\RequirePackage{amsthm}
\RequirePackage{empheq}

%% Line spacing to be applied AFTER the above space saving packages

\renewcommand{\baselinestretch}{\@LineSpacing}

%% Final text printing area, per Geoff Sutcliffe

\RequirePackage{keyval}

\define@key{Ec}{lmargin}{\Ec@defbylen{lmargin}{#1}}

\newlength{\@MarginSize}
\setlength{\@MarginSize}{1in}

\setlength{\@LMarginSize}{\@MarginSize}
%\setlength{\@LMarginSize}{\Ec@lmargin}
\setlength{\@RMarginSize}{\@MarginSize}
\setlength{\@TMarginSize}{\@MarginSize}
\setlength{\@BMarginSize}{\@MarginSize}


% Head height is dependent on the font point size
\setlength{\headheight}{\@HeadHeightSize}
\setlength{\headsep}{\@HeadSepSize}
\setlength{\footskip}{\@FootSkipSize}

\ifletterpaper
 \immediate\write10{easychair: Selecting letter paper margin sizes.}
 \RequirePackage[%
 papersize={8.5in,11in},
 total={145mm,224mm},
 centering,
 twoside,
 includeheadfoot]{geometry}
\fi
\ifafourpaper
 \immediate\write10{easychair: Selecting A4 paper margin sizes.}
 \RequirePackage[%
 papersize={210mm,297mm},
 total={145mm,224mm},
 centering,
 twoside,
 includeheadfoot]{geometry}
\fi

\ifcustompaper
 \immediate\write10{easychair: Selecting custom paper margin sizes.}
 \RequirePackage[%
 papersize={189mm,246mm},
 total={145mm,224mm},
 top=9mm,
 left=24mm,
 twoside,
 includeheadfoot]{geometry}
 \headHeightSize{12.0pt}
 \headSepSize{16.0pt}
 \footSkipSize{26pt}
\fi

%\setlength{\textwidth}{16cm}
%\setlength{\textheight}{9in}


%
% Volume
%

\RequirePackage{lastpage}

\newif\ifvolumeundefined
\volumeundefinedtrue

% e.g.
% \volumeinfo
% {J. Bloe} % editor(s) #1
% {1} % No. of editors #2
% {LICS 2008} % event title #3
% {1} % volume number #4
% {4} % issue #5
% {134} % start page #6

\def\@EasyFontStyle{\footnotesize}
\newcommand{\headfootstyle}[1]{\def\@EasyFontStyle{#1}}

\def\@EasyVolumeInfo{}

\ifthesis
 \newcommand{\volumeinfo}[6]
 {\PackageWarning{easychair}{Cannot use volumeinfo with 'thesis' option. Ignoring...}}
\else
 \newcommand{\volumeinfo}[6]{%
 \ifvolumeundefined
 % \def\@makefntext##1{\noindent ##1}%
 \def\@EasyEdsNames{#1}%
 \def\@EasyEds{ed.}%
 \def\@EasyEvent{#3}%
 \def\@EasyVolume{}%
 \def\@EasyIssue{}%
 \def\@EasyFirstPage{#6}%
 \ifnum #2>1 \gdef\@EasyEds{eds.}\fi%
 \ifnum #4>0 \gdef\@EasyVolume{; Volume #4}\fi%
 \ifnum #5>0 \gdef\@EasyIssue{, issue: #5} \fi%
 % \footnotetext[0]{\sf \@EasyEdsNames (\@EasyEds); \@EasyEvent\@EasyVolume\@EasyIssue, pp. #6\pageref*{LastPage}}%
 % \def\@EasyVolumeInfo{\footnotesize{\sf\@EasyEdsNames~(\@EasyEds); \@EasyEvent\@EasyVolume\@EasyIssue, pp. \@EasyFirstPage\pageref*{LastPage}}}%
 \def\@EasyVolumeInfo{\@EasyFontStyle\@EasyEdsNames~(\@EasyEds); \@EasyEvent\@EasyVolume\@EasyIssue, pp. \@EasyFirstPage\pageref*{LastPage}}%
 %\def\@makefntext##1{\noindent\@makefnmark##1}%
 \setcounter{page}{\@EasyFirstPage}
 \volumeundefinedfalse
 \else
 {\PackageWarning{easychair}{May not redefine volumeinfo}}
 \fi
 }
\fi

\def\@EventInfo{}
\def\@VolumeInfo{}

%
% Allow for more space to place floats.
%

\renewcommand{\topfraction}{0.95}
\renewcommand{\bottomfraction}{0.95}
\renewcommand{\textfraction}{0.05}
\renewcommand{\floatpagefraction}{0.8}


%
% Running heads and ``foots''
%

\RequirePackage{fancyhdr}
\pagestyle{fancy}

\fancyhead{}
%\ifdebug
% \iftwosided
% \fancyhead[RE]{\overline{\@titleRunning}}
% \fancyhead[RO]{\overline{\@authorRunning}}
% \else
% \fancyhead[LO,LE]{\begin{math}\overline{\mbox{\@titleRunning}}\end{math}}
% \fancyhead[RO,RE]{\begin{math}\overline{\mbox{\@authorRunning}}\end{math}}
% \fi
%\else
 \iftwosided
 \fancyhead[RE]{{\@EasyFontStyle\@titleRunning}}
 \fancyhead[RO]{{\@EasyFontStyle\@authorRunning}}
 \else
 \fancyhead[LO,LE]{{\@EasyFontStyle\@titleRunning}}
 \fancyhead[RO,RE]{{\@EasyFontStyle\@authorRunning}}
 \fi
%\fi

\fancyfoot{}
\ifodd\c@page
 \fancyfoot[LO]{{\@EasyFontStyle\@VolumeInfo}}
 \fancyfoot[RE]{{\@EasyFontStyle\@EventInfo}}
\else
 \fancyfoot[RE]{{\@EasyFontStyle\@VolumeInfo}}
 \fancyfoot[LO]{{\@EasyFontStyle\@EventInfo}}
\fi
\ifodd\c@page
 \fancyfoot[RO]{{\normalsize\thepage}}
 \fancyfoot[LE]{{\normalsize\thepage}}
\else
 \fancyfoot[LE]{{\normalsize\thepage}}
 \fancyfoot[RO]{{\normalsize\thepage}}
\fi
 \renewcommand{\headrulewidth}{0pt}
 \renewcommand{\footrulewidth}{0pt}
%\fi

% Suppress the default date, per Geoff
\date{}

% For the first page
\fancypagestyle{plain}{%
\fancyhf{} % clear all header and footer fields
\ifodd\c@page
 \fancyfoot[L]{\@EasyVolumeInfo{}}%
 \fancyfoot[R]{}
\else
 \fancyfoot[R]{\@EasyVolumeInfo}%
 \fancyfoot[L]{}
\fi
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}}

\def\@maketitle{%
 %% Code added to use llncs style author list
\def\andname{and}
\def\lastandname{\unskip, and}
 \def\lastand{\ifnum\value{@inst}=2\relax
 \unskip{} \andname\
 \else
 \unskip \lastandname\
 \fi}%
 \def\and{\stepcounter{@auth}\relax
 \ifnum\value{@auth}=\value{@inst}%
 \lastand
 \else
 \unskip,
 \fi}%
 %%
 \newpage
 \null
 % Facelift a bit the title and make it bold, per Geoff
 \vspace{1cm}
 \begin{center}%
 \let\footnote\thanks%
 \ifwithtimes
 {\LARGE{\@title}\par}
 \else
 {\LARGE\@title\par}
 \fi
 \vskip \baselineskip
 \@date
 {\large
 \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
 \def\thanks##1{}\@author}%
 \global\value{@inst}=\value{@auth}%
 \global\value{auco}=\value{@auth}%
 \setcounter{@auth}{1}%
 {\lineskip .5em
 \noindent\ignorespaces
 \@author\vskip.35cm}}
 {\small\institutename}
 \end{center}%
}

\ifprocediafinal
\definecolor{volume}{RGB}{0,112,144}
\def\procediaSeries{Procedia Computer Science}
\def\procediaVolume{XXX}
\def\procediaYear{2016}
\def\procediaPages{Pages 1927}
\def\procediaConference{99th International Conference on Topics of Superb Significance (CATSS)}
\def\elsevierLogo{elsevier_24mm.jpg}
\def\procediaLogo{computer_science.jpg}
\fi

\ifEPiCfinal
\definecolor{volume}{RGB}{0,112,144}
\def\EPiCSeries{EPiC Series in Computer Science}
\def\EPiCVolume{XXX}
\def\EPiCYear{2016}
\def\EPiCPages{Pages 1927}
\def\EPiCConference{99th International Conference on Topics of Superb Significance (CATSS)}
\def\ChairLogo{logo_chair.png}
\def\EPiCLogo{logo_epic.png}
\fi

% 
% Change the maketitle command in EPiC
\let\oldmaketitle=\@maketitle
% 
\def\@maketitle{%
\ifprocediafinal
\vspace*{36pt}\noindent \hrulefill\\*[5pt]
\noindent
\begin{tabular}{lcr}%
 \vtop{\null\hbox{\includegraphics[height=73pt]{\elsevierLogo}}} &
\vtop{\null\hbox{%
\begin{minipage}[t]{261pt}\sf%
\begin{center}\vspace*{2pt}%
 {\Large \procediaSeries} \\[9pt]
 {\color{volume}Volume \procediaVolume}, \procediaYear, \procediaPages \\[15pt]
 \procediaConference
\end{center}
\end{minipage}}} &
 \vtop{\null\hbox{\includegraphics[height=73pt]{\procediaLogo}}}
 \end{tabular}

\vspace*{5pt}\noindent\hrulefill

\par\ \vspace*{12pt}\par
{\let\newpage\relax\oldmaketitle}%
\else \ifprocedia
\definecolor{grayheader}{RGB}{112,112,112}%
{\color{grayheader}\vspace*{36pt}\noindent \hrulefill\\
\noindent\centering
\raisebox{0pt}[47pt][47pt]{This space is reserved for the Procedia header, do not use it}%

\noindent\hrulefill}%
\\

{\let\newpage\relax\oldmaketitle}%
\else \ifprocediaempty
\raisebox{0pt}[47pt][47pt]{~}
\\

{\let\newpage\relax\oldmaketitle}%

% Change the maketitle command for EPiC series

\else \ifEPiCfinal

%\vspace*{10mm}\par
\definecolor{headerframe}{rgb}{0.473,0.676,0.656} % 79ada8
\definecolor{headerback}{rgb}{0.953125,0.95703125,0.9609375} % f4f5f6

\begin{tcolorbox}[
 enhanced,center,width=\linewidth2cm,
 underlay={
 \node[inner sep=0pt,outer sep=0pt,right] at ([xshift=1cm]frame.west)
 {\includegraphics[height=75pt]{\ChairLogo}};
 \node[inner sep=0pt,outer sep=0pt,left] at ([xshift=1cm]frame.east)
 {\includegraphics[height=75pt]{\EPiCLogo}};
 },
 before skip=0pt,% < before box
 after skip=6pt,% < after box
 left=71.54pt1cm+4mm,
 right=56.22423pt1cm+4mm,
 sharp corners,text fill,
 colback=headerback,colframe=headerframe,
 height=73pt,toprule=0.5pt,bottomrule=0.5pt,leftrule=0pt,rightrule=0pt,
 halign=center,valign=center,boxsep=0pt,
 top=2mm,% space to top border
 bottom=2mm,% space to bottom border
 ]%
{\Large \EPiCSeries}\par\vfill%
{\color{volume}Volume \EPiCVolume, \EPiCYear, \EPiCPages}\par\vfill%
\EPiCConference
\end{tcolorbox}

\par\ \vspace*{12pt}\par
{\let\newpage\relax\oldmaketitle}%
\else \ifEPiC
\definecolor{grayheader}{RGB}{112,112,112}%
{\color{grayheader}\vspace*{23pt}\noindent \hrulefill\\
\noindent\centering
\raisebox{0pt}[36.5pt][36.5pt]{This space is reserved for the EPiC Series header, do not use it}%

\noindent\hrulefill}%
\\

{\let\newpage\relax\oldmaketitle}%
\else \ifEPiCempty
\raisebox{0pt}[47pt][47pt]{~}
\\

{\let\newpage\relax\oldmaketitle}%

% otherwise the old maketitle

\else
\oldmaketitle
\fi\fi\fi\fi\fi\fi
}
% 

% 
% change the abstract command in Procedia
\let\old@abstract=\abstract
\let\old@endabstract=\endabstract
% 
\def\abstract{%
\ifprocedia
\noindent\hrulefill\par
\vspace*{5pt}\noindent\textbf{Abstract}\par\noindent\ignorespaces
\else \ifprocediafinal
\noindent\hrulefill\par
\vspace*{5pt}\noindent\textbf{Abstract}\par\noindent\ignorespaces
\else \ifprocediaempty
\noindent\hrulefill\par
\vspace*{5pt}\noindent\textbf{Abstract}\par\noindent\ignorespaces
\else
\old@abstract
\fi\fi\fi}
\def\endabstract{%
\ifprocedia
\vspace{5pt}\par\noindent
{\small\textit{Keywords:} \@keywords}
\par\vspace*{6pt}\noindent\hrulefill
\else \ifprocediafinal
\vspace{5pt}\par\noindent
{\small\textit{Keywords:} \@keywords}
\par\vspace*{6pt}\noindent\hrulefill
\old@endabstract
\else \ifprocediaempty
\vspace{5pt}\par\noindent
{\small\textit{Keywords:} \@keywords}
\par\vspace*{6pt}\noindent\hrulefill
\old@endabstract
\fi\fi\fi}
% 

% 
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\begin{document}

\mbox{} \vspace{0.8cm} \mbox{}
\begin{center} {\Huge\textbf{CommonRoad: Vehicle Models} \\[0.3cm] \huge(Version 2019b)} \end{center}
\vspace{0.2cm}
\begin{center} {\large Matthias Althoff} \\[0.2cm] Technische Universit\"at M\"unchen, 85748 Garching, Germany
\end{center}
\vspace{0.2cm}

\noindent

\begin{abstract}
This document presents models in \textit{CommonRoad} for vehicle dynamics ranging from simple to complicated: The simplest model is a pointmass model, while the most complicated one is a multibody model. All models are presented in statespace form to facilitate their implementation in standard solvers for ordinary differential equations. We further provide parameter sets and a precise initialization of the multibody model. To be able to compare the results with simpler models, it is presented how the initial states and the parameters of the multibody model can be transfered to simpler models. Implementation examples in MATLAB and Python are provided on the \textit{CommonRoad} website. Our repository also provide routines to convert initial states and parameters. Simulation examples demonstrate the advantages of more complicated models.
\end{abstract}


\setcounter{tocdepth}{2}
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\tableofcontents}


%\pagestyle{empty}

%
\section{Introduction}
\label{sec:introduction}

This document is part of the \textit{CommonRoad} benchmark repository for motion planning of road vehicles, alongside other documents for possible cost functions and road scenarios. It is assumed in this document that all vehicles have an underlying controller that can realize a commanded acceleration (positive and negative). Especially for adaptive cruise control, numerous works already exist that realize a commanded acceleration, see e.g. \cite{Shladover1991,Kim2007}. The effects on engine characteristics in terms of fuel consumption can be considered in the cost function as demonstrated in the document on cost functions.

The lateral dynamics, however, cannot be abstracted away to the same extent using controllers. Especially, when constraints such as the danger of rollover have to be considered in extreme maneuvers \cite{Odenthal1999, Gaspar2005}. For this reason, our models consider increasingly complicated lateral vehicle dynamics and tire models: pointmass model, kinematic singletrack model, singletrack model, and a multibody model. For each model, we (1) present the set of required ordinary differential equations, (2) convert them into statespace form so that common solvers can be used, and (3) provide typical parameters.

In \textit{CommonRoad}, we provide three types of vehicles:
\begin{itemize}
 \item a small vehicle (Ford Escort; vehicle ID: 1),
 \item a medium vehicle (BMW 320i; vehicle ID: 2),
 \item and a van (VW Vanagon; vehicle ID: 3).
\end{itemize}
Detailed parameters of these vehicles have been collected from \cite[Appendix A]{Allen1992} and other vehicle databases that are online available. For each vehicle, we provide the aforementioned models: pointmass model (Sec.~\ref{sec:PMmodel}), kinematic singletrack model (Sec.~\ref{sec:KSmodel}), singletrack model (Sec.~\ref{sec:STmodel}), and a multibody model (Sec.~\ref{sec:MBmodel}). After combining the vehicle identifier with the model type, one obtains the model ID. For instance, KS2 is a kinematic singletrack model using the parameters of the BMW 320i. In addition, we describe in Sec.~\ref{sec:conversion} how parameters and initial states can be converted from complicated to simpler models. Finally, in Sec.~\ref{sec:examples} we provide some numerical results.

\section{Changes Compared to Version 2019a}

The vehicle models did not change compared to version 2019a. Only the presentation for the kinematic singletrack model has changed: it is now clarified that the reference point is the center of the rear axle and not the center of mass.

\section{Steering and Acceleration Constraints} \label{sec:constraints}

With the exception of the pointmass model, all vehicle models respect steering and acceleration constraints. Since the point mass model only uses acceleration as an input, no steering constraints can be modeled. To formulate the constraints, let us first introduce the steering angle $\delta$, the velocity of the steering angle $v_\delta$, the velocity $v$, and the parameter $v_S$ describing the velocity above which the engine power is not sufficient to cause wheel slip. We denote by $\underline{\square}$ the minimum possible value and by $\overline{\square}$ the maximum possible value and by $\square_\mathtt{lat}$ the value of a variable in lateral direction and by $\square_\mathtt{long}$ the value in longitudinal direction.

The constraints on steering angle velocity, steering angle, and velocity are straightforward:
\begin{equation}
 v_\delta \in [\underline{v}_\delta, \overline{v}_\delta], \quad \delta \in [\underline{\delta}, \overline{\delta}], \quad v \in [\underline{v}, \overline{v}]. \label{eq:remainingRestrictions}
\end{equation}
Considering limited engine power and braking power results in the following constraint as detailed in \cite[Sec.~III.B]{Althoff2016d}:
\begin{equation}
 a_\mathtt{long} \in [\underline{a},\overline{a}(v)], \quad \overline{a}(v) = \begin{cases}
 a_\mathtt{max}\frac{v_S}{v} & \text{for } v > v_S, \\
 a_\mathtt{max} & \text{otherwise}.
 \end{cases} \label{eq:longitudinalRestriction}
\end{equation}
Finally, we consider the friction circle (aka Kamm's circle) limiting absolute acceleration:
\begin{equation}
\sqrt{a_\mathtt{long}^2 + (v\, \dot{\Psi})^2} \leq a_\mathtt{max} \qquad (a_\mathtt{lat} = v\, \dot{\Psi}) \label{eq:lateralRestriction}
\end{equation}
The constraints on steering, velocity, and acceleration can be directly considered by introducing a desired steering velocity $v_{\delta,d}$ and a desired acceleration $a_\mathtt{long,d}$ as well as choosing
\begin{flalign}
 v_{\delta} = f_{steer}(\delta,v_{\delta,d}) = & \begin{cases}
 0 & \text{ for } (\delta \leq \underline{\delta} \land v_{\delta,d} \leq 0) \lor (\delta \geq \overline{\delta} \land v_{\delta,d} \geq 0) \quad (C1), \\
 \underline{v}_\delta & \text{ for } \lnot C1 \land v_{\delta,d} \leq \underline{v}_\delta, \\
 \overline{v}_\delta & \text{ for } \lnot C1 \land v_{\delta,d} \geq \overline{v}_\delta, \\
 v_{\delta,d} & \text{ otherwise},
 \end{cases} \label{eq:steeringRestriction} \\
 a_\mathtt{long} = f_{acc}(v, a_\mathtt{long,d}) = & \begin{cases}
 0 & \text{ for } (v \leq \underline{v} \land a_\mathtt{long,d} \leq 0) \lor (v \geq \overline{v} \land a_\mathtt{long,d} \geq 0) \quad (C2), \\
 \underline{a} & \text{ for } \lnot C2 \land a_\mathtt{long,d} \leq \underline{a}, \\
 \overline{a}(v) & \text{ for } \lnot C2 \land a_\mathtt{long,d} \geq \overline{a}(v), \\
 a_\mathtt{long,d} & \text{ otherwise}.
 \end{cases} \label{eq:accelerationRestriction}
\end{flalign}
Constraint parameters for different vehicle models are listed in Tab.~\ref{tab:constraintParameters}.

\begin{table}[h]
\caption{Constraint Parameters (obtained from information on the Internet). }
{\small
\begin{center}\label{tab:constraintParameters}
\begin{tabular}{lllclll}
 \toprule
 \multicolumn{3}{c}{\textbf{vehicle parameter}} & \phantom{a} & \multicolumn{3}{c}{\textbf{vehicle identifier}} \\ \cmidrule{13} \cmidrule{57}
 \textbf{name} & \textbf{symbol} & \textbf{unit} && \textbf{1} & \textbf{2} & \textbf{3} \\ \hline
 minimum steering angle & $\underline{\delta}$ & [rad] && $0.910$ & $1.066$ & $1.023$ \\
 maximum steering angle & $\overline{\delta}$ & [rad] && $0.910$ & $1.066$ & $1.023$ \\
 minimum steering velocity & $\underline{v}_\delta$ & [rad/s] && $0.4$ & $0.4$ & $0.4$ \\
 maximum steering velocity & $\overline{v}_\delta$ & [rad/s] && $0.4$ & $0.4$ & $0.4$ \\
 min. velocity (also depending on traffic rules) & $\underline{v}$ & [m/s] && $13.9$ & $13.6$ & $11.2$ \\
 max. velocity (also depending on traffic rules) & $\overline{v}$ & [m/s] && $45.8$ & $50.8$ & $41.7$ \\
 switching velocity & $v_S$ & [m/s] && $4.755$ & $7.319$ & $4.824$ \\
 maximum acceleration & $a_\mathtt{max}$ & [m/s$^2$] && $11.5$ & $11.5$ & $11.5$ \\
 \bottomrule
\end{tabular}
\end{center}}
\end{table}

% 1989 Ford Escort: power: 67kW, top speed: 165 km/h, turning radius: 10.88m, length: 4.298m, width: 1.674m; reverseGearRatio: 3.27; velMax*gearRatio [m/s]: 45.4750; max. backwad velocity: 13.9067
% 1981 BMW 320i: Power: 92kW, top speed: 183 km/h, turning radius: 10.3m, length: 4.508m (with US bumpers), width: 1.610m; reverseGearRatio: 3.682; velMax*gearRatio [m/s]: 50.1161; max. backwad velocity: 13.9067
% 1984 VW Vanagon: Power: 82kW, top speed: 150 km/h, turning radius: 10.7m, length: 4.569m, width: 1.844m; reverseGearRatio: 3.67; velMax*gearRatio [m/s]: 40.9887; max. backwad velocity: 11.1686
% delta_max = arctan(l/R); only for bicycle model
% delta_max = arcsin(2*l/R); according to German Wiki entry
% v_s = P_max/(m*a_max)

We would also like to mention, that other works do not provide all the constraints presented in this document (which can be easily removed, but a removal should be stated since this simplifies motion planning).

\section{PointMass Model (PM)} \label{sec:PMmodel}

The pointmass model is the simplest model that is commonly used for motion planning, see e.g. \cite{TomasGabarron2013,Godbole1997}. This model abstracts the vehicle as a point mass that can be accelerated within bounds. This bound is typically chosen as a circle (Kamm's circle), which is also the bound chosen in this benchmark suite. Let us introduce $\square$ as the placeholder for a variable and $\square_x$ and $\square_y$ to denote the value of the corresponding variable in $x$ and $y$ direction, respectively. After further introducing position $s$, acceleration $a$, and maximum absolute acceleration $a_\mathtt{max}$, the dynamics of the point mass model is
\begin{flalign*}
 \ddot{s}_x = a_x, \quad \ddot{s}_y = a_y, \quad \sqrt{a_x^2 + a_y^2} \leq a_\mathtt{max}. &&
\end{flalign*}
The pointmass model ignores that vehicles have a minimum turning circle, which is considered in the kinematic singletrack model.

\subsection{State Space Model}

After introducing the state variables $x_i$ as
\begin{flalign*}
 x_1 = s_x, \quad x_2 = s_y, \quad x_3 = \dot{s}_x, \quad x_4 = \dot{s}_y &&
\end{flalign*}
and the input variables $u_i$ as
\begin{flalign*}
 u_1 = a_x, \quad u_2 = a_y, &&
\end{flalign*}
the system dynamics can be written as the linear system
\begin{flalign*}
 \dot{x} = Ax + Bu, \qquad A=\begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}. &&
\end{flalign*}
The only constraint in state space form is $\sqrt{u_1^2 + u_2^2} \leq a_\mathtt{max}$.

\subsection{Parameters}

The only parameter of this model is $a_\mathtt{max}$. Since in this version all vehicles use the same tire, we have $a_\mathtt{max} = 11.5$ [m/s$^2$] (see Tab.~\ref{tab:constraintParameters}).

\section{Kinematic SingleTrack Model (KS)} \label{sec:KSmodel}

The kinematic singletrack models a road vehicle with only two wheels, where the front and rear wheel pairs are each lumped into one wheel. This simplification is justified since the roll dynamics is not considered (see Fig.~\ref{fig:bicycleModel} and \cite[Sec.~2.2]{Rajamani2012}). This also explains the term \textit{singletrack model}. The kinematic singletrack model further does not consider any tire slip, so that the velocity vector $v$ at the center of the rear axle is always aligned with the link between the front and rear wheel as depicted in Fig.~\ref{fig:kinematicBicycleModel}. Similarly to the pointmass model, the kinematic singletrack model is used in many works for motion planning, e.g. \cite{Paden2016,Petti2005}. A simple example, where the benefit of a kinematic singletrack model is evident, is parking: a pointmass model is not sufficient since it would not consider the nonholonomic behavior and in particular the minimum turning radius.

In addition to the variables already introduced for the pointmass model and the already introduced velocity $v$, we additionally require the velocity of the steering angle $v_\delta$, the steering angle $\delta$, the heading $\Psi$, and the parameter $l_{wb}$ describing the wheelbase. The differential equations of the kinematic singletrack model as defined in this document are
\begin{flalign} \label{eq:KS}
 \dot{\delta} = & v_\delta, && \nonumber \\
 \dot{\Psi} = & \frac{v}{l_{wb}} \tan(\delta), \nonumber \\
 \dot{v} = & a_\mathtt{long}, \\
 \dot{s}_x = & v \cos(\Psi), \nonumber \\
 \dot{s}_y = & v \sin(\Psi), \nonumber
\end{flalign}
Please note that kinematic singletrack models slightly differ in publications, depending on whether one considers that (1) the steering angle or the steering velocity is an input or (2) the vehicle velocity or the vehicle acceleration is an input. For instance, in \cite[eq.~(8)]{Paden2016}, the vehicle velocity and the steering velocity are inputs.

\begin{figure}[h]
 \centering
 \footnotesize
 \psfrag{a}[c]{$\begin{bmatrix} s_x \\ s_y \end{bmatrix}$}
 \psfrag{b}[c]{$\Psi$}
 \psfrag{d}[c]{$l_{wb}$}
 \psfrag{f}[c]{$\delta$}
 \psfrag{x}[c]{$x$}
 \psfrag{y}[c]{$y$}
 \psfrag{v}[c]{$v$}
 \includegraphics[width=10cm]{./figures/kinematicBicycleModel.eps}
 \caption{Kinematic singletrack model.}
 \label{fig:kinematicBicycleModel}
\end{figure}

\subsection{State Space Model}

To write the kinematic singletrack model in statespace form, we introduce the following state variables:
\begin{flalign*}
 x_1 = s_x, \quad x_2 = s_y, \quad x_3 = \delta, \quad x_4 = v, \quad x_5 = \Psi. &&
\end{flalign*}
The input variables are
\begin{flalign} \label{eq:inputVariables}
 u_1 = v_\delta, \quad u_2 = a_\mathtt{long}. &&
\end{flalign}
Inserting the state and input variables into \eqref{eq:KS} and directly considering the constraints on steering, velocity, and acceleration, results in
\begin{flalign*}
 \dot{x}_1 = & x_4 \cos(x_5), \\
 \dot{x}_2 = & x_4 \sin(x_5), \\
 \dot{x}_3 = & f_{steer}(x_3, u_1), && \text{see \eqref{eq:steeringRestriction}} \\
 \dot{x}_4 = & f_{acc}(x_4, u_2), && \text{see \eqref{eq:accelerationRestriction}} \\
 \dot{x}_5 = & \frac{x_4}{l_{wb}} \tan(x_3).
\end{flalign*}
Because the constraints on steering, velocity, and acceleration are already considered by $f_{steer}(x_3, u_1)$ and $f_{acc}(x_4, u_2)$, it only remins to consider the friction circle:
\begin{equation} \label{eq:KSconstraints}
\begin{split}
 & \sqrt{u_2^2 + (x_4\, \dot{x}_5)^2} \leq a_\mathtt{max}.
\end{split}
\end{equation}

\subsection{Parameters}

The parameters of this model are listed in Tab.~\ref{tab:vehicleParametersKS} and the constraint parameters are presented in Tab.~\ref{tab:constraintParameters}.


\begin{table}[h]
\caption{Vehicle Parameters for the kinematic singletrack model (values have been obtained according to Sec.~\ref{sec:parameterConversion}). }
{\small
\begin{center}\label{tab:vehicleParametersKS}
\begin{tabular}{lllclll}
 \toprule
 \multicolumn{3}{c}{\textbf{vehicle parameter}} & \phantom{a} & \multicolumn{3}{c}{\textbf{vehicle identifier}} \\ \cmidrule{13} \cmidrule{57}
 \textbf{name} & \textbf{symbol} & \textbf{unit} && \textbf{1} & \textbf{2} & \textbf{3} \\ \hline
 vehicle length & $l$ & [m] && $4.298$ & $4.508$ & $4.569$ \\
 vehicle width & $w$ & [m] && $1.674$ & $1.610$ & $1.844$ \\
 wheelbase & $l_{wb}$ & [m] && $2.391$ & $2.578$ & $2.471$ \\
 \bottomrule %LR
\end{tabular}
\end{center}}
\end{table}





\section{SingleTrack Model (ST)} \label{sec:STmodel}

Since the kinematic singletrack model does not consider tire slip, important effects such as understeer or oversteer are not considered \cite[Sec.~2.3]{Rajamani2012}. However, when the vehicle is not driving close to its physical capabilities, those effects are not dominant. The extension is the wellknown singletrack model, which is also known as the bicycle model. Works that perform planning of evasive maneuvers closer to physical limits require the singletrack model, see e.g. \cite{Jeon2011,Shiller1991}. We additionally consider the load transfer of the vehicle due to longitudinal acceleration $a_\mathtt{long}$ (neglecting suspension dynamics), such that the vertical forces on the front and rear axis $F_{z,f}$ and $F_{z,r}$ become
\begin{flalign*}
 F_{z,f} = \frac{mg l_r  m a_\mathtt{long} h_{cg}}{l_r+l_f}, \quad
 F_{z,r} = \frac{mg l_f + m a_\mathtt{long} h_{cg}}{l_r+l_f}, &&
\end{flalign*}
with parameters from Tab. \ref{tab:vehicleParametersST}. These forces are inserted into the derivation of the equations for the slip angle (at the center of gravity) $\beta$ and the yaw rate $\dot{\Psi}$ \cite[Sec.~2.3]{Rajamani2012}. Using the previously introduced variables and the parameters in Tab. \ref{tab:vehicleParametersST}, the singletrack model as defined in this work is
\begin{flalign} \label{eq:ST}
 \dot{\delta} = & v_\delta, \nonumber \\
 \dot{\beta} = & \frac{\mu}{v(l_r + l_f)} \Big( C_{S,f}(g l_r  a_\mathtt{long} h_{cg}) \delta
  (C_{S,r}(g l_f + a_\mathtt{long} h_{cg}) + C_{S,f}(g l_r  a_\mathtt{long} h_{cg})) \beta && \nonumber \\
 & + (C_{S,r}(g l_f + a_\mathtt{long} h_{cg})l_r  C_{S,f}(g l_r  a_\mathtt{long} h_{cg})l_f) \frac{\dot{\Psi}}{v} \Big)  \dot{\Psi}, \nonumber \\
 \ddot{\Psi} = & \frac{\mu m}{I_z(l_r + l_f)} \Big( l_f C_{S,f} (g l_r  a_\mathtt{long} h_{cg}) \delta \\
 & + \left( l_r C_{S,r}(g l_f + a_\mathtt{long} h_{cg})  l_f C_{S,f} (g l_r  a_\mathtt{long} h_{cg}) \right) \beta \nonumber \\
 &  \left(l_f^2 C_{S,f} (g l_r  a_\mathtt{long} h_{cg}) + l_r^2 C_{S,r}(g l_f + a_\mathtt{long} h_{cg}) \right) \frac{\dot{\Psi}}{v} \Big), \nonumber \\
 \dot{v} = & a_\mathtt{long}, \nonumber \\
 \dot{s}_x = & v \cos(\beta + \Psi), \nonumber \\
 \dot{s}_y = & v \sin(\beta + \Psi), \nonumber
\end{flalign}
under consideration of \eqref{eq:remainingRestrictions}\eqref{eq:lateralRestriction}. Please note that in contrast to this work, other works often only consider constant velocity when referring to a singletrack model (see e.g. \cite[Sec.~2.3]{Rajamani2012}). Also, the weight transfer between the front and rear axle is often neglected in singletrack models (see e.g. \cite{Jeon2011}). Note that we do not use the cornering stiffness $C$, as is typically done for singletrack models, but separate the effect of the friction coefficient $\mu$, the cornering stiffness coefficient $C_{S}$, and the vertical force $F_z$, such that $C_i = \mu C_{S,i} F_{z,i}$ and $i=\{f,r\}$ for the front and rear axle. This separation is done because the friction coefficient is the most dominant parameter modeling the influence of weather.

\begin{figure}[h]
 \centering
 \footnotesize
 \psfrag{a}[c]{$\begin{bmatrix} s_x \\ s_y \end{bmatrix}$}
 \psfrag{b}[c]{$\Psi$}
 \psfrag{c}[c]{$\beta$}
 \psfrag{d}[c]{$l_r$}
 \psfrag{e}[c]{$l_f$}
 \psfrag{f}[c]{$\delta$}
 \psfrag{x}[c]{$x$}
 \psfrag{y}[c]{$y$}
 \psfrag{v}[c]{$v$}
 \includegraphics[width=10cm]{./figures/bicycleModel.eps}
 \caption{Singletrack model.}
 \label{fig:bicycleModel}
\end{figure}



\subsection{State Space Model}

The singletrack model requires a few more state variables compared to the kinematic singletrack model. In order to share the constraint function in \eqref{eq:KSconstraints}, we keep the numbering of state variables shared with the kinematic singletrack model:
\begin{flalign*}
 x_1 = s_x, \quad x_2 = s_y, \quad x_3 = \delta, \quad x_4 = v, \quad x_5 = \Psi, \quad x_6 = \dot{\Psi}, \quad x_7 = \beta. &&
\end{flalign*}
The input variables are identical to \eqref{eq:inputVariables}. Inserting the state and input variables into \eqref{eq:ST} and directly considering the constraints on steering, velocity, and acceleration, results in the singletrack model for $\mathbf{x_4\geq0.1}$:
\begin{flalign}
 \dot{x}_1 = & x_4 \cos(x_5 + x_7), && \nonumber \\
 \dot{x}_2 = & x_4 \sin(x_5 + x_7), \nonumber \\
 \dot{x}_3 = & f_{steer}(x_3, u_1), && \text{see \eqref{eq:steeringRestriction}} \nonumber \\
 \dot{x}_4 = & f_{acc}(x_4, u_2), && \text{see \eqref{eq:accelerationRestriction}} \nonumber \\
 \dot{x}_5 = & x_6, \nonumber \\
 \dot{x}_6 = & \frac{\mu m}{I_z(l_r + l_f)} \Big(l_f C_{S,f} (g l_r  u_2 h_{cg}) x_3 + \left( l_r C_{S,r}(g l_f + u_2 h_{cg})  l_f C_{S,f} (g l_r  u_2 h_{cg}) \right) x_7 \nonumber \\
 &  \left(l_f^2 C_{S,f} (g l_r  u_2 h_{cg}) + l_r^2 C_{S,r}(g l_f + u_2 h_{cg}) \right) \frac{x_6}{x_4} \Big), \nonumber \\
 \dot{x}_7 = & \frac{\mu}{x_4(l_r + l_f)} \Big( C_{S,f}(g l_r  u_2 h_{cg}) x_3  (C_{S,r}(g l_f + u_2 h_{cg}) + C_{S,f}(g l_r  u_2 h_{cg})) x_7 \nonumber \\
 & + (C_{S,r}(g l_f + u_2 h_{cg})l_r  C_{S,f}(g l_r  u_2 h_{cg})l_f) \frac{x_6}{x_4} \Big)  x_6. \label{eq:slipAngle_ST}
\end{flalign}
Due to the special choice of state variables, the constraint is identical to \eqref{eq:KSconstraints}. The singletrack model becomes singular for small velocities. For this reason, we switch to the kinematic model for velocities below $0.1$m/s. The derivatives $\dot{x}_1$ up to $\dot{x}_5$ are obtained as for the kinematic model. The derivative $\dot{x}_6$ is obtained by computing the derivative of $\dot{x}_5$ of the kinematic model. The singletrack model for $\mathbf{x_4<0.1}$ is
\begin{flalign} \label{eq:smallVelocityModel}
 \dot{x}_1 = & x_4 \cos(x_5), && \nonumber \\
 \dot{x}_2 = & x_4 \sin(x_5), \nonumber \\
 \dot{x}_3 = & f_{steer}(x_3, u_1), && \text{see \eqref{eq:steeringRestriction}} \nonumber \\
 \dot{x}_4 = & f_{acc}(x_4, u_2), && \text{see \eqref{eq:accelerationRestriction}} \nonumber \\
 \dot{x}_5 = & \frac{x_4}{l_{wb}} \tan(x_3), \nonumber \\
 \dot{x}_6 = & \frac{f_{acc}(x_4, u_2)}{l_{wb}} \tan(x_3) + \frac{x_4}{l_{wb} \cos^2(x_3)} f_{steer}(x_3, u_1), \nonumber \\
 \dot{x}_7 = & 0
\end{flalign}
When switching to the kinematic model, the slip angle does not exist anymore, which would require to set $x_7=0$. While such a reset could be realized by a hybrid automaton \cite{Alur1995}, we have not implemented this for simplicity. Our implementation only ensures that $\dot{x}_7 = 0$ so that we can provide the model in the standard form $\dot{x} = f(x,u)$ as required by most solvers. Since $x_7$ is already small when the velocity is slightly above $0.1$ m/s, we argue that the reset can be safely ignored.

\subsection{Parameters}


The parameters of the singletrack model are listed in Tab.~\ref{tab:vehicleParametersST} and the constraint parameters are shown in Tab.~\ref{tab:constraintParameters}.


\begin{table}[h]
\caption{Vehicle Parameters for the singletrack model (values have been obtained according to Sec.~\ref{sec:parameterConversion}).}
{\small
\begin{center}\label{tab:vehicleParametersST}
\begin{tabular}{lllclll}
 \toprule
 \multicolumn{3}{c}{\textbf{vehicle parameter}} & \phantom{a} & \multicolumn{3}{c}{\textbf{vehicle identifier}} \\ \cmidrule{13} \cmidrule{57}
 \textbf{name} & \textbf{symbol} & \textbf{unit} && \textbf{1} & \textbf{2} & \textbf{3} \\ \hline
 vehicle length & $l$ & [m] && $4.298$ & $4.508$ & $4.569$ \\
 vehicle width & $w$ & [m] && $1.674$ & $1.610$ & $1.844$ \\
 total vehicle mass & $m$ & $10^3$[kg] && $1.225$ & $1.093$ & $1.478$ \\
 moment of inertia for entire mass about z axis & $I_z$ & $10^3$[kg\,m$^2$] && $1.538$ & $1.791$ & $2.473$ \\
 distance from center of gravity to front axle & $\a$ & [m] && $0.883$ & $1.156$ & $1.150$ \\
 distance from center of gravity to rear axle & $\b$ & [m] && $1.508$ & $1.422$ & $1.321$ \\
 center of gravity height of total mass & $h_{cg}$ & [m] && $0.557$ & $0.574$ & $0.747$ \\
 cornering stiffness coefficient (front) & $C_{S,f}$ & [1/rad] && $20.89$ & $20.89$ & $20.89$ \\
 friction coefficient & $\mu$ & [] && $1.048$ & $1.048$ & $1.048$ \\
 \bottomrule
\end{tabular}
\end{center}}
\end{table}




\section{MultiBody Model (MB)} \label{sec:MBmodel}

Although the previously introduced singletrack model considers already many important effects of vehicle dynamics, it does not consider the vertical load of all $4$ wheels due to roll, pitch, and yaw, their individual spin and slip, and nonlinear tire dynamics. An example where a multibody model is used for motion planning of a road vehicle is \cite{Bertolazzi2007}. Although many commercial multibody models for vehicle dynamics exist\footnote{\href{https://www.carsim.com/}{www.carsim.com}, \href{https://www.tesisdynaware.com}{www.tesisdynaware.com}, \href{http://www.mscsoftware.com/de/product/adams}{www.mscsoftware.com}}, those models are proprietary and thus not appropriate for a benchmark that requires public accessibility. Our multibody model is taken out of \cite[Appendix A]{Allen1992}, which is one of few detailed and accessible multibody dynamics descriptions. For easy use, we have translated the equations in \cite[Appendix A]{Allen1992} into a state space model, which is more suitable for implementation in ordinarydifferentialequation solvers. A MATLAB and a Python implementation can be found on \href{http://commonroad.in.tum.de}{commonroad.in.tum.de}.

The multibody dynamics is described by $3$ masses: The unsprung mass and the sprung mass of the front and rear axles. The forces between these masses are described by the dynamics of the suspension and the tire model. We consider all suspension forces in \cite[Appendix A]{Allen1992} originating from springs, dampers, and antiroll bars. We do not consider flexibilities in the steering system, bump stops, and squat/lift forces caused by the suspension geometry. All considered vehicles have an independent suspension so that we do not show the equations for solid axes. For the tire dynamics we use the PAC2002 MagicFormula tire model, which is widely used in industry \cite{Adams2011}. The combined lateral and longitudinal tire forces are computed from the slip angle, the camber angle, and the vertical tire force described in \cite[Appendix A]{Allen1992}. The tire parameters for all $4$ wheels are taken from the example of a PAC2002 tire property file in \cite{Adams2011}. Rewriting all equations as a state space model yields $29$ state variables. All state variables, including their initial values, are listed in Tab.~\ref{tab:fullModelInitialStates}, where the pairs $LF$, $RF$, $LR$, $RR$ indicate left/right and front/rear.

Compared to \cite[Appendix A]{Allen1992} the equations are presented in an order so that equations depend on previously computed results, making it possible to directly implement then; see our MATLAB and Python implementation on \href{http://commonroad.in.tum.de}{commonroad.in.tum.de}.

\subsection{State Variables}

We group the state variables into \textit{vehicle body}, \textit{front axle}, \textit{rear axle}, \textit{wheels}, and \textit{auxiliary}.

\paragraph{Vehicle body}

\begin{flalign*}
 x_1 =& s_x \quad (\text{xposition in a global coordinate system}), && \\
 x_2 =& s_y \quad (\text{yposition in a global coordinate system}), \\
 x_3 =& \delta \quad (\text{steering angle of front wheels}), \\
 x_4 =& v_x \quad (\text{velocity in longitudinal direction in the vehiclefixed coordinate system}), \\
 x_5 =& \Psi \quad (\text{yaw angle}), \\
 x_6 =& \dot{\Psi} \quad (\text{yaw rate}), \\
 x_7 =& \Phi_S \quad (\text{roll angle}), \\
 x_8 =& \dot{\Phi}_S \quad (\text{roll rate}), \\
 x_9 =& \Theta_S \quad (\text{pitch angle}), \\
 x_{10} =& \dot{\Theta}_S \quad (\text{pitch rate}), \\
 x_{11} =& v_y \quad (\text{velocity in lateral direction in the vehiclefixed coordinate system}), \\
 x_{12} =& s_z \quad (\text{zposition (height) from ground}), \\
 x_{13} =& v_z \quad (\text{velocity in vertical direction perpendicular to road plane}), \\
\end{flalign*}

\paragraph{Front axle}

\begin{flalign*}
 x_{14} =& \Phi_{UF} \quad (\text{roll angle front}), && \\
 x_{15} =& \dot{\Phi}_{UF} \quad (\text{roll rate front}), \\
 x_{16} =& v_{y,UF} \quad (\text{velocity in ydirection front}), \\
 x_{17} =& s_{z,UF} \quad (\text{zposition front}), \\
 x_{18} =& v_{z,UF} \quad (\text{velocity in zdirection front}),
\end{flalign*}


\paragraph{Rear axle}

\begin{flalign*}
 x_{19} =& \Phi_{UR} \quad (\text{roll angle rear}), && \\
 x_{20} =& \dot{\Phi}_{UR} \quad (\text{roll rate rear}), \\
 x_{21} =& v_{y,UR} \quad (\text{velocity in ydirection rear}), \\
 x_{22} =& s_{z,UR} \quad (\text{zposition rear}), \\
 x_{23} =& v_{z,UR} \quad (\text{velocity in zdirection rear}),
\end{flalign*}

\paragraph{Wheels}

\begin{flalign*}
 x_{24} =& \omega_{LF} \quad (\text{left front wheel angular velocity}), && \\
 x_{25} =& \omega_{RF} \quad (\text{right front wheel angular velocity}), \\
 x_{26} =& \omega_{LR} \quad (\text{left rear wheel angular velocity}), \\
 x_{27} =& \omega_{RR} \quad (\text{right rear wheel angular velocity}),
\end{flalign*}

\paragraph{Auxiliary}

\begin{flalign*}
 x_{28} =& \delta_{y,f} \quad (\text{front lateral displacement of sprung mass due to roll}), && \\
 x_{29} =& \delta_{y,r} \quad (\text{rear lateral displacement of sprung mass due to roll}).
\end{flalign*}





% x27 > x1 \\
% x28 > x2 \\
% > x3 (new) \\
% x7 > x4 \\
% x1 > x5 \\
% x2 > x6 \\
% x7 > (); delete \\
% x3 > x7 \\
% x4 > x8 \\
% x5 > x9 \\
% x6 > x10 \\
% x8 > x11 \\
% x9 > x12 \\
% x10 > x13 \\
% x11 > x14 \\
% x12 > x15 \\
% x15 > x16 \\
% x17 > x17 \\
% x18 > x18 \\
% x13 > x19 \\
% x14 > x20 \\
% x16 > x21 \\
% x19 > x22 \\
% x20 > x23 \\
% x21 > x24 \\
% x22 > x25 \\
% x23 > x26 \\
% x24 > x27 \\
% x25 > x28 \\
% x26 > x29 \\
% u1 > x3


\subsection{Auxiliary Variables} \label{sec:auxVariables}

\paragraph{Slip angle and velocity at center of gravity} These equations are derived by the author:
%
\begin{flalign*}
 \beta =& \arctan \left(\frac{x_{11}}{x_{4}}\right) && \\
 v_{CG} =& \sqrt{x_{4}^2 + x_{11}^2}
\end{flalign*}

\paragraph{Vertical tire forces} These equations are obtained from \cite[eq.~A48A51]{Allen1992}:
%
\begin{flalign*}
 F_{z,LF} =& (x_{17} + R_{w}(\cos(x_{14})  1)  \frac{1}{2}T_{f}\sin(x_{14}))K_{zt} && \\
 F_{z,RF} =& (x_{17} + R_{w}(\cos(x_{14})  1) + \frac{1}{2}T_{f}\sin(x_{14}))K_{zt} \\
 F_{z,LR} =& (x_{22} + R_{w}(\cos(x_{19})  1)  \frac{1}{2}T_{r}\sin(x_{19}))K_{zt} \\
 F_{z,RR} =& (x_{22} + R_{w}(\cos(x_{19})  1) + \frac{1}{2}T_{r}\sin(x_{19}))K_{zt}
\end{flalign*}


\paragraph{Individual tire velocities} These equations are derived from \cite[eq.~A56A59]{Allen1992} assuming that the rear wheels cannot be steered and by using $x_{4}\tan(\beta) = x_{11}$ from \cite[p.~A45]{Allen1992}:
%
\begin{flalign*}
 u_{w,LF} =& (x_{4} + \frac{1}{2}T_{f}\, x_{6})\cos(x_{3}) + (x_{11} + \a\, x_{6})\sin(x_{3}) && \\
 u_{w,RF} =& (x_{4}  \frac{1}{2}T_{f}\, x_{6})\cos(x_{3}) + (x_{11} + \a\, x_{6})\sin(x_{3}) \\
 u_{w,LR} =& x_{4} + \frac{1}{2}T_{r}\, x_{6} \\
 u_{w,RR} =& x_{4}  \frac{1}{2}T_{r}\, x_{6} \\
\end{flalign*}


\paragraph{Longitudinal slip} These equations are taken from \cite[eq.~A60]{Allen1992}:
%
\begin{flalign*}
 s_{LF} =& 1  \frac{R_{w}\, x_{24}}{u_{w,LF}} && \\
 s_{RF} =& 1  \frac{R_{w}\, x_{25}}{u_{w,RF}} \\
 s_{LR} =& 1  \frac{R_{w}\, x_{26}}{u_{w,LR}} \\
 s_{RR} =& 1  \frac{R_{w}\, x_{27}}{u_{w,RR}}
\end{flalign*}


\paragraph{Lateral slip angles} These equations are taken from \cite[eq.~A42A45]{Allen1992} assuming that the rear wheels cannot be steered:
%
\begin{flalign*}
 \alpha_{LF} =& \arctan \left(\frac{x_{11} + \a\, x_{6}  x_{15}(R_{w}  x_{17})}{x_{4} + \frac{1}{2}T_{f}\, x_{6}} \right)  x_{3} && \\
 \alpha_{RF} =& \arctan \left(\frac{x_{11} + \a\, x_{6}  x_{15}(R_{w}  x_{17})}{x_{4}  \frac{1}{2}T_{f}\, x_{6}} \right)  x_{3} \\
 \alpha_{LR} =& \arctan \left(\frac{x_{11}  \b\, x_{6}  x_{20}(R_{w}  x_{22})}{x_{4} + \frac{1}{2}T_{r}\, x_{6}} \right) \\
 \alpha_{RR} =& \arctan \left(\frac{x_{11}  \b\, x_{6}  x_{20}(R_{w}  x_{22})}{x_{4}  \frac{1}{2}T_{r}\, x_{6}} \right)
\end{flalign*}

\paragraph{Auxiliary suspension movement} These equations are taken from \cite[eq.~A23aA26a]{Allen1992} and \cite[eq.~A23bA26b]{Allen1992}:
%
\begin{flalign*}
 z_{S,LF} =& \frac{h_s  R_{w} + x_{17}  x_{12}}{\cos(x_{7})}  h_s + R_{w} + \a\, x_{9} + \frac{1}{2}(x_{7}  x_{14})T_{f} && \\
 z_{S,RF} =& \frac{h_s  R_{w} + x_{17}  x_{12}}{\cos(x_{7})}  h_s + R_{w} + \a\, x_{9}  \frac{1}{2}(x_{7}  x_{14})T_{f} \\
 z_{S,LR} =& \frac{h_s  R_{w} + x_{22}  x_{12}}{\cos(x_{7})}  h_s + R_{w}  \b\, x_{9} + \frac{1}{2}(x_{7}  x_{19})T_{r} \\
 z_{S,RR} =& \frac{h_s  R_{w} + x_{22}  x_{12}}{\cos(x_{7})}  h_s + R_{w}  \b\, x_{9}  \frac{1}{2}(x_{7}  x_{19})T_{r} \\
\end{flalign*}
%
\begin{flalign*}
 \dot{z}_{S,LF} =& x_{18}  x_{13} + \a\, x_{10} + \frac{1}{2}(x_{8}  x_{15})T_{f} && \\
 \dot{z}_{S,RF} =& x_{18}  x_{13} + \a\, x_{10}  \frac{1}{2}(x_{8}  x_{15})T_{f} \\
 \dot{z}_{S,LR} =& x_{23}  x_{13}  \b\, x_{10} + \frac{1}{2}(x_{8}  x_{20})T_{r} \\
 \dot{z}_{S,RR} =& x_{23}  x_{13}  \b\, x_{10}  \frac{1}{2}(x_{8}  x_{20})T_{r} \text{ ('$$' changed to '$+$' compared to \cite[eq.~A26b]{Allen1992})}
\end{flalign*}

\paragraph{Camber angles} These equations are taken from \cite[eq.~A27A30]{Allen1992}:
%
\begin{flalign*}
 \gamma_{LF} =& x_{7} + D_f\, z_{S,LF} + E_f(z_{S,LF})^2 && \\
 \gamma_{RF} =& x_{7}  D_f\, z_{S,RF}  E_f(z_{S,RF})^2 \\
 \gamma_{LR} =& x_{7} + D_r\, z_{S,LR} + E_r(z_{S,LR})^2 \\
 \gamma_{RR} =& x_{7}  D_r\, z_{S,RR}  E_r(z_{S,RR})^2
\end{flalign*}

\paragraph{Auxiliary movements/forces for compliant joint equations} These equations are taken from \cite[eq.~A61A68]{Allen1992}:

\begin{flalign*}
 \Delta z_{F} =& h_s  R_{w} + x_{17}  x_{12} && \\
 \Delta z_{R} =& h_s  R_{w} + x_{22}  x_{12} &&
\end{flalign*}
\begin{flalign*}
 \Delta \phi_{F} =& x_{7}  x_{14} &&\\
 \Delta \phi_{R} =& x_{7}  x_{19} &&
\end{flalign*}
\begin{flalign*}
 \Delta \dot{\phi}_{F} =& x_{8}  x_{15} &&\\
 \Delta \dot{\phi}_{R} =& x_{8}  x_{20} &&
\end{flalign*}
\begin{flalign*}
 \Delta \dot{z}_{F} =& x_{18}  x_{13} &&\\
 \Delta \dot{z}_{R} =& x_{23}  x_{13} &&
\end{flalign*}
\begin{flalign*}
 \Delta \dot{y}_{F} =& x_{11} + \a\, x_{6}  x_{16} &&\\
 \Delta \dot{y}_{R} =& x_{11}  \b\, x_{6}  x_{21} &&
\end{flalign*}
\begin{flalign*}
 \Delta_{F} =& \Delta z_{F}\sin(x_{7})  x_{28}\cos(x_{7})  (h_{RAF}  R_{w})\sin(\Delta \phi_{F}) &&\\
 \Delta_{R} =& \Delta z_{R}\sin(x_{7})  x_{29}\cos(x_{7})  (h_{RAR}  R_{w})\sin(\Delta \phi_{R}) &&
 \end{flalign*}
\begin{flalign*}
 \dot{\Delta}_{F} =& (\Delta z_{F}\cos(x_{7}) + x_{28}\sin(x_{7}))x_{8}
 + \Delta \dot{z}_{F}\sin(x_{7})  \Delta \dot{y}_{F}\cos(x_{7})  (h_{RAF}  R_{w})\cos(\Delta \phi_{F}) \Delta \dot{\phi}_{F} &&\\
 \dot{\Delta}_{R} =& (\Delta z_{R}\cos(x_{7}) + x_{29}\sin(x_{7}))x_{8}
 + \Delta \dot{z}_{R}\sin(x_{7})  \Delta \dot{y}_{R}\cos(x_{7})  (h_{RAR}  R_{w})\cos(\Delta \phi_{R}) \Delta \dot{\phi}_{R} &&
\end{flalign*}
\begin{flalign*}
 F_{RAF} =& \Delta_{F} \, K_{RAS} + \dot{\Delta}_{F} \, K_{RAD} &&\\
 F_{RAR} =& \Delta_{R} \, K_{RAS} + \dot{\Delta}_{R} \, K_{RAD} &&
\end{flalign*}

% \paragraph{Squat/lift forces depending on acceleration and deceleration} These equations are taken from \cite[eq.~A32A35]{Allen1992}:
% \begin{flalign*}
% F_{SQ,LF} =& \left(K_{SL,F} + \frac{z_{S,LF}}{L_{SA,F}}\right)(F_{y,LF}\cos(x_{7})  F_{z,LF}\sin(x_{7}))  F_{x,LF} \, K_{SAD,F} &&\\
% F_{SQ,RF} =& \left(K_{SL,F} + \frac{z_{S,RF}}{L_{SA,F}}\right)(F_{y,RF}\cos(x_{7}) + F_{z,RF}\sin(x_{7}))  F_{x,LR} \, K_{SAD,F} &&\\
% F_{SQ,LR} =& \left(K_{SL,R} + \frac{z_{S,LR}}{L_{SA,R}}\right)(F_{y,LR}\cos(x_{7})  F_{z,LR}\sin(x_{7})) + F_{x,LR} \, K_{SAD,R} &&\\
% F_{SQ,RR} =& \left(K_{SL,R} + \frac{z_{S,RR}}{L_{SA,R}}\right)(F_{y,RR}\cos(x_{7}) + F_{z,RR}\sin(x_{7})) + F_{x,RR} \, K_{SAD,R}
% \end{flalign*}

\paragraph{Auxiliary suspension forces (bump stop neglected; squat/lift forces neglected)} These equations are taken from \cite[eq.~A23A26]{Allen1992} and \cite[p.~A51]{Allen1992}:
\begin{flalign*}
 F_{S,LF} =& \frac{m_{s} \, g \, \b}{2(\a+\b)}  z_{S,LF} \, K_{S,F}  \dot{z}_{S,LF} \, K_{SD,F} + \frac{(x_{7}  x_{14}) K_{TS,F}}{T_{f}} &&\\
 F_{S,RF} =& \frac{m_{s} \, g \, \b}{2(\a+\b)}  z_{S,RF} \, K_{S,F}  \dot{z}_{S,RF} \, K_{SD,F}  \frac{(x_{7}  x_{14}) K_{TS,F}}{T_{f}} &&\\
 F_{S,LR} =& \frac{m_{s} \, g \, \a}{2(\a+\b)}  z_{S,LR} \, K_{S,R}  \dot{z}_{S,LR} \, K_{SD,R} + \frac{(x_{7}  x_{19}) K_{TS,R}}{T_{r}} &&\\
 F_{S,RR} =& \frac{m_{s} \, g \, \a}{2(\a+\b)}  z_{S,RR} \, K_{S,R}  \dot{z}_{S,RR} \, K_{SD,R}  \frac{(x_{7}  x_{19}) K_{TS,R}}{T_{r}} &&
\end{flalign*}


\paragraph{Auxiliary variables sprung mass} These equations are taken from \cite[eq.~A7A12]{Allen1992}:
\begin{flalign*}
 \sX =& F_{x,LR} + F_{x,RR} + (F_{x,LF} + F_{x,RF})\cos(x_{3})  (F_{y,LF} + F_{y,RF})\sin(x_{3}) &&\\
 \sN =& (F_{y,LF} + F_{y,RF})\a\cos(x_{3}) + (F_{x,LF} + F_{x,RF})\a\sin(x_{3}) \\
 & + (F_{y,RF}  F_{y,LF})\frac{1}{2}T_{f}\sin(x_{3}) + (F_{x,LF}  F_{x,RF})\frac{1}{2}T_{f}\cos(x_{3}) \\
 & + (F_{x,LR}  F_{x,RR})\frac{1}{2}T_{r}  (F_{y,LR} + F_{y,RR})\b &&\\
 \sY_{s} =& (F_{RAF} + F_{RAR})\cos(x_{7}) + (F_{S,LF} + F_{S,LR} + F_{S,RF} + F_{S,RR})\sin(x_{7}) &&\\
 \sL =& \frac{1}{2}F_{S,LF} \, T_{f} + \frac{1}{2}F_{S,LR} \, T_{r}  \frac{1}{2}F_{S,RF} \, T_{f}  \frac{1}{2}F_{S,RR} \, T_{r} \\
 &  \frac{F_{RAF}}{\cos(x_{7})}(h_s  x_{12}  R_{w} + x_{17}  (h_{RAF}  R_{w})\cos(x_{14})) \\
 &  \frac{F_{RAR}}{\cos(x_{7})}(h_s  x_{12}  R_{w} + x_{22}  (h_{RAR}  R_{w})\cos(x_{19})) &&\\
 \sZ_{s} =& (F_{S,LF} + F_{S,LR} + F_{S,RF} + F_{S,RR})\cos(x_{7})  (F_{RAF} + F_{RAR})\sin(x_{7}) &&\\
 \sM_{s} =& \a(F_{S,LF} + F_{S,RF})  \b(F_{S,LR} + F_{S,RR}) + ((F_{x,LF} + F_{x,RF})\cos(x_{3}) \\
 &  (F_{y,LF} + F_{y,RF})\sin(x_{3}) + F_{x,LR} + F_{x,RR})(h_s  x_{12})
\end{flalign*}


% \paragraph{Drag (Coment: needs discussion; why are not all formulas considered?)} These equations are based on \cite[eq.~A16]{Allen1992}:
% \begin{flalign*}
% X_A =& \frac{1}{2}\prho \pA C_{d}\, x_{4}^2 &&\\
% \sX =& \sX  X_A (\text{renaming of $\sX$ required})
% \end{flalign*}

\paragraph{Auxiliary variables unsprung mass} These equations are taken from \cite[eq.~A20A22]{Allen1992} assuming that only the front wheels can be steered:
\begin{flalign*}
 \sL_{uf} =& \frac{1}{2}F_{S,RF}\, T_{f}  \frac{1}{2}F_{S,LF}\, T_{f}  F_{RAF}(h_{RAF}  R_{w}) \\
 & + F_{z,LF}(R_{w}\sin(x_{14}) + \frac{1}{2}T_{f}\cos(x_{14})  K_{LT}\, F_{y,LF}) \\
 &  F_{z,RF}(R_{w}\sin(x_{14}) + \frac{1}{2}T_{f}\cos(x_{14}) + K_{LT}\, F_{y,RF}) \\
 &  ((F_{y,LF} + F_{y,RF})\cos(x_{3}) + (F_{x,LF} + F_{x,RF})\sin(x_{3}))(R_{w}  x_{17}) && \\
 \sL_{ur} =& \frac{1}{2}F_{S,RR}\, T_{r}  \frac{1}{2}F_{S,LR}\, T_{r}  F_{RAR}(h_{RAR}  R_{w}) \\
 & + F_{z,LR}(R_{w}\sin(x_{19}) + \frac{1}{2}T_{r}\cos(x_{19})  K_{LT}\, F_{y,LR}) \\
 &  F_{z,RR}(R_{w}\sin(x_{19}) + \frac{1}{2}T_{r}\cos(x_{19}) + K_{LT}\, F_{y,RR}) \\
 &  (F_{y,LR} + F_{y,RR})(R_{w}  x_{22}) && \\
 \sZ_{uf} =& F_{z,LF} + F_{z,RF} + F_{RAF}\sin(x_{7})  (F_{S,LF} + F_{S,RF})\cos(x_{7}) && \\
 \sZ_{ur} =& F_{z,LR} + F_{z,RR} + F_{RAR}\sin(x_{7})  (F_{S,LR} + F_{S,RR})\cos(x_{7}) && \\
 \sY_{uf} =& (F_{y,LF} + F_{y,RF})\cos(x_{3}) + (F_{x,LF} + F_{x,RF})\sin(x_{3}) \\
 &  F_{RAF}\cos(x_{7})  (F_{S,LF} + F_{S,RF})\sin(x_{7}) && \\
 \sY_{ur} =& (F_{y,LR} + F_{y,RR}) \\
 &  F_{RAR}\cos(x_{7})  (F_{S,LR} + F_{S,RR})\sin(x_{7})
\end{flalign*}


\subsection{Tire Formulas} \label{sec:tire}

We are using the Pacejka 2002 tire model \cite{Pacejka2002}, which is one of the most popular tire models. The exact parameters for a realistic tire are taken out of \cite{Adams2011}. For our particular model, we make the following assumptions:
\begin{itemize}
 \item Turn slip is neglected, so that $\forall i: \xi_i =1$;
 \item Effect of load increment is neglected so that $df_z=0$ (see \cite[PAC2002, eq.~16]{Adams2011});
 \item All scaling factors are set as $\forall i: \lambda_i = 1$.
\end{itemize}



\paragraph{Longitudinal tire forces using the magic formula for pure slip}
$\forall i\in \{LF, RF, LR, RR\}:$
\begin{flalign*}
 S_{Hx} =& p_{Hx1} && \text{(see \cite[PAC2002, eq.~27]{Adams2011})} \\
 S_{Vx,i} =& F_{z,i} \, p_{Vx1} && \text{(see \cite[PAC2002, eq.~28]{Adams2011})} \\
 \kappa_{i} =& s_{i} && \text{(coord. trans. \cite{Allen1992} $\to$ \cite{Adams2011})} \\
 \kappa_{x,i} =& \kappa_{i} + S_{Hx} && \text{(see \cite[PAC2002, eq.~19]{Adams2011})} \\
 \mu_{x,i} =& p_{Dx1}(1p_{Dx3} \, \gamma_{i}^2) && \text{(see \cite[PAC2002, eq.~23]{Adams2011})} \\
 C_x =& p_{Cx1} && \text{(see \cite[PAC2002, eq.~21]{Adams2011})} \\
 D_{x,i} =& \mu_x \, F_{z,i} && \text{(see \cite[PAC2002, eq.~22]{Adams2011})} \\
 E_x =& p_{Ex1} && \text{(see \cite[PAC2002, eq.~24]{Adams2011})} \\
 K_{x,i} =& F_{z,i} \, p_{Kx1} && \text{(see \cite[PAC2002, eq.~25]{Adams2011})} \\
 B_{x,i} =& \frac{K_{x,i}}{C_{x} \, D_{x,i}} && \text{(see \cite[PAC2002, eq.~26]{Adams2011})} \\
 F_{x0,i} =& D_{x,i}\sin(C_x \arctan(B_{x,i} \, \kappa_{x,i}  E_x(B_{x,i} \, \kappa_{x,i} \\
 &  \arctan(B_{x,i} \, \kappa_{x,i}))) + S_{Vx,i}) && \text{(see \cite[PAC2002, eq.~18]{Adams2011})} \\
\end{flalign*}

\paragraph{Lateral tire forces using the magic formula for pure slip}
$\forall i\in \{LF, RF, LR, RR\}:$
\begin{flalign}
 S_{Hy,i} =& \sign(\gamma_{i})(p_{Hy1} + p_{Hy3} \, \abs(\gamma_{i})) && \text{(see \cite[PAC2002, eq.~40]{Adams2011})} \nonumber \\
 S_{Vy,i} =& \sign(\gamma_{i}) F_{z,i}(p_{Vy1} + p_{Vy3} \, \abs(\gamma_{i})) && \text{(see \cite[PAC2002, eq.~41]{Adams2011})} \nonumber \\
 \alpha_{y,i} =& \alpha_{i} + S_{Hy,i} && \text{(see \cite[PAC2002, eq.~31]{Adams2011})} \nonumber \\
 \mu_{y,i} =& p_{Dy1}(1p_{Dy3}\gamma_{i}^2) && \text{(see \cite[PAC2002, eq.~35]{Adams2011})} \nonumber \\
 C_y =& p_{Cy1} && \text{(see \cite[PAC2002, eq.~33]{Adams2011})} \nonumber \\
 D_{y,i} =& \mu_{y,i} \, F_{z,i} && \text{(see \cite[PAC2002, eq.~34]{Adams2011})} \nonumber \\
 E_y =& p_{Ey1} && \text{(see \cite[PAC2002, eq.~36]{Adams2011})} \nonumber \\
 K_{y,i} =& F_{z,i} p_{Ky1} \qquad \text{(simplified $K_{y0}$ to $p_{Ky1} \, F_{z,i}$)} && \text{(see \cite[PAC2002, eq.~38]{Adams2011})} \nonumber \\
 B_{y,i} =& \frac{K_{y,i}}{C_y \, D_{y,i}} && \text{(see \cite[PAC2002, eq.~39]{Adams2011})} \nonumber \\
 F_{y0,i} =& D_{y,i}\sin(C_y \arctan(B_{y,i} \, \alpha_{y,i}  E_y(B_{y,i} \, \alpha_{y,i} \nonumber \\
 &  \arctan(B_{y,i}\alpha_{y,i})))) + S_{Vy,i} && \text{(see \cite[PAC2002, eq.~30]{Adams2011})} \label{eq:magicFormulaLateral}
\end{flalign}


\paragraph{Longitudinal tire forces for combined slip}
$\forall i\in \{LF, RF, LR, RR\}:$
\begin{flalign*}
 S_{Hx\alpha} =& r_{Hx1} && \text{(see \cite[PAC2002, eq.~65]{Adams2011})} \\
 \alpha_{s,i} =& \alpha_{i} + S_{Hx\alpha} && \text{(see \cite[PAC2002, eq.~60]{Adams2011})} \\
 B_{x\alpha,i} =& r_{Bx1}\cos(\arctan(r_{Bx2} \kappa_{i})) && \text{(see \cite[PAC2002, eq.~61]{Adams2011})} \\
 C_{x\alpha} =& r_{Cx1} && \text{(see \cite[PAC2002, eq.~62]{Adams2011})} \\
 E_{x\alpha} =& r_{Ex1} && \text{(see \cite[PAC2002, eq.~64]{Adams2011})} \\
 D_{x\alpha,i} =& F_{x0,i}/\cos\bigg(C_{x\alpha} \arctan\Big(B_{x\alpha,i} \, S_{Hx\alpha}  E_{x\alpha}\big(B_{x\alpha,i} \, S_{Hx\alpha} \\
 &  \arctan(B_{x\alpha,i} \, S_{Hx\alpha})\big)\Big)\bigg) && \text{(see \cite[PAC2002, eq.~63]{Adams2011})} \\
 F_{x,i} =& D_{x\alpha,i}\cos(C_{x\alpha} \arctan(B_{x\alpha,i} \, \alpha_{s,i}  E_{x\alpha}(B_{x\alpha} \, \alpha_{s,i} \\
 &  \arctan(B_{x\alpha,i} \, \alpha_{s,i})))) && \text{(see \cite[PAC2002, eq.~59]{Adams2011})} \\
\end{flalign*}


\paragraph{Lateral tire forces for combined slip}
$\forall i\in \{LF, RF, LR, RR\}:$
\begin{flalign*}
 S_{Hy\kappa} =& r_{Hy1} && \text{(see \cite[PAC2002, eq.~74]{Adams2011})} \\
 \kappa_{s,i} =& \kappa_{i} + S_{Hy\kappa} && \text{(see \cite[PAC2002, eq.~69]{Adams2011})} \\
 B_{y\kappa,i} =& r_{By1} \cos(\arctan(r_{By2}(\alpha_{i}r_{By3}))) && \text{(see \cite[PAC2002, eq.~70]{Adams2011})} \\
 C_{y\kappa} =& r_{Cy1} && \text{(see \cite[PAC2002, eq.~71]{Adams2011})} \\
 E_{y\kappa} =& r_{Ey1} && \text{(see \cite[PAC2002, eq.~73]{Adams2011})} \\
 D_{y\kappa} =& F_{y0,i}/\cos\bigg(C_{y\kappa} \arctan\Big(B_{y\kappa,i} \, S_{Hy\kappa}  E_{y\kappa}\big(B_{y\kappa} \, S_{Hy\kappa} \\
 &  \arctan(B_{y\kappa,i} \, S_{Hy\kappa})\big)\Big)\bigg) && \text{(see \cite[PAC2002, eq.~72]{Adams2011})} \\
 D_{vy\kappa,i} =& \mu_{y,i} \, F_{z,i}(r_{Vy1} + r_{Vy3} \gamma_{i}) \cos(\arctan(r_{Vy4} \, \alpha_{i})) && \text{(see \cite[PAC2002, eq.~76]{Adams2011})} \\
 S_{vy\kappa,i} =& D_{vy\kappa,i}\sin(r_{Vy5} \arctan(r_{Vy6} \, \kappa_{i})) && \text{(see \cite[PAC2002, eq.~75]{Adams2011})} \\
 F_{y,i} =& D_{y\kappa} \cos(C_{y\kappa} \arctan(B_{y\kappa,i} \, \kappa_{s,i}  E_{y\kappa} (B_{y\kappa,i} \kappa_{s,i} \\
 &  \arctan(B_{y\kappa,i} \, \kappa_{s,i})))) + S_{Vy\kappa} && \text{(see \cite[PAC2002, eq.~68]{Adams2011})} \\
\end{flalign*}


\subsection{Vehicle Dynamics}

Based on the auxiliary variables from Sec.~\ref{sec:auxVariables}, the tire forces from Sec.~\ref{sec:tire}, and steering constraints, we compute the right hand side of the vehicle dynamics $\dot{x} = f(x,u)$ in this subsection:

\paragraph{Dynamics common with singletrack model}
\begin{flalign*}
 \dot{x}_{1} =& v_{CG} \cos(\beta + x_{5}) && \text{(from \eqref{eq:ST})}\\
 \dot{x}_{2} =& v_{CG} \sin(\beta + x_{5}) && \text{(from \eqref{eq:ST})}\\
 \dot{x}_3 = & f_{steer}(x_3, u_1), && \text{see \eqref{eq:steeringRestriction}} \\
 \dot{x}_{4} =& \frac{1}{m} \sX + x_{6}\, x_{11} && \text{(from \cite[eq.~A1]{Allen1992})}\\
 \dot{x}_{5} =& x_{6} && \text{(trivial)} \\
 \dot{x}_{6} =& \frac{1}{I_{z}  \frac{I_{xz,s}^2}{I_{\phi,s}}}\left(\sN + \frac{I_{xz,s}}{I_{\phi,s}} \sL_{s} \right) && \text{(see below)}
\end{flalign*}

Derivation of $\dot{x}_6$:
\begin{flalign}
 I_{z} \dot{x}_6  I_{xz,s} \dot{x}_8 =& \sN && \text{(from \cite[eq.~A2]{Allen1992})} \label{eq:rollAndYaw_1}\\
 I_{\phi,s} \dot{x}_8  I_{xz,s} \dot{x}_6 =& \sL_{s} && \text{(from \cite[eq.~A4]{Allen1992})} \label{eq:rollAndYaw_2}
\end{flalign}
Multiplying \eqref{eq:rollAndYaw_2} with $\frac{I_{xz,s}}{I_{\phi,s}}$ and adding the result to \eqref{eq:rollAndYaw_1} yields
\begin{flalign*}
 \left(I_{z}  \frac{I_{xz,s}^2}{I_{\phi,s}}\right)\dot{x}_6 =& \sN + \frac{I_{xz,s}}{I_{\phi,s}} \sL_{s} &&
\end{flalign*}

\paragraph{Remaining sprung mass dynamics}
\begin{flalign*}
 \dot{x}_{7} =& x_{8} && \text{(trivial)} \\
 \dot{x}_{8} =& \frac{1}{(I_{\phi,s}  \frac{I_{xz,s}^2}{I_{z}})}\left(\frac{I_{xz,s}}{I_{z}} \sN + \sL\right) && \text{(see below)} \\
 \dot{x}_{9} =& x_{10} && \text{(trivial)} \\
 \dot{x}_{10} =& \frac{\sM_{s}}{I_{y,s}} && \text{(from \cite[eq.~A6]{Allen1992})} \\
 \dot{x}_{11} =& \frac{1}{m_{s}} \sY_{s}  x_{6}\, x_{4} && \text{(see below)} \\
 \dot{x}_{12} =& x_{13} && \text{(trivial)} \\
 \dot{x}_{13} =& g  \frac{1}{m_{s}} \sZ_{s} && \text{(from \cite[eq.~A5]{Allen1992})}
\end{flalign*}

Derivation of $\dot{x}_8$: \\
Multiplying \eqref{eq:rollAndYaw_1} with $\frac{I_{xz,s}}{I_{z}}$ and adding the result to \eqref{eq:rollAndYaw_2} yields
\begin{flalign*}
 \left(I_{\phi,s}  \frac{I_{xz,s}^2}{I_{z}}\right)\dot{x}_8 =& \frac{I_{xz,s}}{I_{z}}\sN + \sL_{s} &&
\end{flalign*}

Derivation of $\dot{x}_{11}$:
Using $a_y = \dot{x}_{11} + x_{6}\, x_{4}$ from \cite[eq.~A46]{Allen1992} and inserting it in \cite[eq.~A3]{Allen1992} results in
\begin{flalign} \label{eq:lateralAcceleration}
 m_s(\dot{x}_{11} + x_{6}\, x_{4}) =& \sY_{s} &&
\end{flalign}

\paragraph{Unsprung mass dynamics (front)}
\begin{flalign*}
 \dot{x}_{14} =& x_{15} && \text{(trivial)} \\
 \dot{x}_{15} =& \frac{\sL_{uf}}{I_{u,f}} && \text{(from \cite[eq.~A17]{Allen1992})} \\
 \dot{x}_{16} =& \frac{\sY_{uf}}{m_{u,f}}  x_{6}\, x_{4} && \text{(from \eqref{eq:lateralAcceleration} and \cite[eq.~A19]{Allen1992})} \\
 \dot{x}_{17} =& x_{18} && \text{(trivial)} \\
 \dot{x}_{18} =& g  \frac{\sZ_{uf}}{m_{u,f}} && \text{(from \cite[eq.~A18]{Allen1992})}
\end{flalign*}

\paragraph{Unsprung mass dynamics (rear)}
\begin{flalign*}
 \dot{x}_{19} =& x_{20} && \text{(trivial)} \\
 \dot{x}_{20} =& \frac{\sL_{ur}}{I_{u,r}} && \text{(from \cite[eq.~A17]{Allen1992})} \\
 \dot{x}_{21} =& \frac{\sY_{ur}}{m_{u,r}}  x_{6}\, x_{4} && \text{(from \eqref{eq:lateralAcceleration} and \cite[eq.~A19]{Allen1992})} \\
 \dot{x}_{22} =& x_{23} && \text{(trivial)} \\
 \dot{x}_{23} =& g  \frac{\sZ_{ur}}{m_{u,r}} && \text{(from \cite[eq.~A18]{Allen1992})}
\end{flalign*}

\paragraph{Convert acceleration input to brake and engine torque} This is an addition to \cite[Appendix A]{Allen1992}, which does not explicitly create a positive engine torque if the acceleration demand is positive and a braking torque if the acceleration demand is negative. We also consider maximum velocities and maximum engine power using $f_{acc}(x_4, u_2)$:
\begin{flalign*}
 u_2 := & f_{acc}(x_4, u_2), && \text{see \eqref{eq:accelerationRestriction}} \\
 T_B =& \begin{cases}
 0, \text{ for } u_{2}>0 \\
 m \, R_{w} \, u_{2}, \text{ otherwise}
 \end{cases} &&\\
 T_E =& \begin{cases}
 m \, R_{w} \, u_{2}, \text{ for } u_{2}>0 \\
 0, \text{ otherwise}
 \end{cases}
\end{flalign*}

\paragraph{Wheel dynamics} It is assumed that the brake torque $T_B$ in \cite[eq.~A55]{Allen1992} is split between the front and rear axle according to the newly introduced parameter $T_{s,b}$ (torque split, brake) and the engine torque $T_E$ in \cite[eq.~A55]{Allen1992} is split between the front and rear axle according to the newly introduced parameter $T_{s,e}$ (torque split, engine)
\begin{flalign*}
 \dot{x}_{24} =& \frac{1}{I_{y,w}}(R_{w} \, F_{x,LF} + \frac{1}{2}T_{s,b} \, T_B + \frac{1}{2}T_{s,e} \, T_E) && \text{(based on \cite[eq.~A55]{Allen1992})} \\
 \dot{x}_{25} =& \frac{1}{I_{y,w}}(R_{w} \, F_{x,RF} + \frac{1}{2}T_{s,b} \, T_B + \frac{1}{2}T_{s,e} \, T_E) && \text{(based on \cite[eq.~A55]{Allen1992})} \\
 \dot{x}_{26} =& \frac{1}{I_{y,w}}(R_{w} \, F_{x,LR} + \frac{1}{2}(1T_{s,b}) \, T_B + \frac{1}{2}(1T_{s,e}) T_E) && \text{(based on \cite[eq.~A55]{Allen1992})} \\
 \dot{x}_{27} =& \frac{1}{I_{y,w}}(R_{w} \, F_{x,RR} + \frac{1}{2}(1T_{s,b}) \, T_B + \frac{1}{2}(1T_{s,e}) T_E) && \text{(based on \cite[eq.~A55]{Allen1992})}
\end{flalign*}


\paragraph{Negative wheel spin forbidden} This is an addition to \cite[Appendix A]{Allen1992}, which forbids wheel spin in negative direction. When using brake torque, the wheels stay at rest when not moving anymore instead of accelerating in negative direction:
\begin{flalign*}
 \forall i\in\{24,...,27\}: \quad \dot{x}_{i} =& 0 \text{ for } x_{i}<0, \quad x_{i}:=0 \text{ for } x_{i}<0 &&\\
\end{flalign*}

\paragraph{Compliant joint equations}
\begin{flalign*}
 \dot{x}_{28} =& \Delta \dot{y}_{F} && \text{(trivial)} \\
 \dot{x}_{29} =& \Delta \dot{y}_{R} && \text{(trivial)}
\end{flalign*}

\paragraph{Small absolute velocities}

As for the singletrack model, the multibody model becomes singular for small absolute velocities. For this reason, we use the kineamtic model for $\dot{x}_{1}$$\dot{x}_{6}$ as presented in \eqref{eq:smallVelocityModel}. Further, all slip angles are set to zero: $s_{LF}=s_{RF}=s_{LR}=s_{RR}=\alpha_{LF}=\alpha_{RF}=\alpha_{LR}=\alpha_{RR}=0$.

\subsection{Parameters}

The multibody model requires in total $69$ parameters, of which $37$ specify the vehicle and $32$ the tires. The vehicle parameters of the multibody model can be found in Tab.~\ref{tab:vehicleParametersMB} and the ones for the tire model in Tab.~\ref{tab:tireParameters}. Please note that in the first version of this document we only consider one parameterization for the tires.


\begin{table}[h!tb]
\caption{Vehicle Parameters for the multibody model (see \cite[Table~E5.]{Allen1992}; values have been converted to SI units). Abbreviations: center of gravity (c.g.), moment of inertia (m.o.i.), suspension (susp.), auxiliary (aux.), damping (damp.).}
{\small
\begin{center}\label{tab:vehicleParametersMB}
\begin{tabular}{lllclll}
 \toprule
 \multicolumn{3}{c}{\textbf{vehicle parameter}} & \phantom{a} & \multicolumn{3}{c}{\textbf{vehicle identifier}} \\ \cmidrule{13} \cmidrule{57}
 \textbf{name} & \textbf{symbol} & \textbf{unit} && \textbf{1} & \textbf{2} & \textbf{3} \\ \hline
 vehicle length & $l$ & [m] && $4.298$ & $4.508$ & $4.569$ \\
 vehicle width & $w$ & [m] && $1.674$ & $1.610$ & $1.844$ \\
 total vehicle mass & $m$ & $10^3$[kg] && $1.225$ & $1.093$ & $1.478$ \\ %MASS
 sprung mass & $m_s$ & $10^3$[kg] && $1.094$ & $0.965$ & $1.316$ \\ %SMASS
 unsprung mass (front) & $m_{u,f}$ & [kg] && $65.67$ & $63.79$ & $81.14$ \\ %UMASSF
 unsprung mass (rear) & $m_{u,f}$ & [kg] && $65.67$ & $63.79$ & $81.14$ \\ %UMASSR
 distance from c.g. to front axle & $\a$ & [m] && $0.883$ & $1.156$ & $1.150$ \\ %LENA
 distance from c.g. to rear axle & $\b$ & [m] && $1.508$ & $1.422$ & $1.321$ \\ %LENB
 m.o.i. for $m_s$ in roll & $I_{\phi,s}$ & [kg\,m$^2$] && $244.0$ & $207.2$ & $479.8$ \\ %IXS
 m.o.i. for sprung mass about y axis & $I_{y,s}$ & $10^3$[kg\,m$^2$] && $1.342$ & $1.565$ & $2.204$ \\ %IYS
 m.o.i. for entire mass about z axis & $I_z$ & $10^3$[kg\,m$^2$] && $1.538$ & $1.791$ & $2.473$ \\ %IZZ
 cross product of inertia for $m_s$ (xz axis) & $I_{xz,s}$ & [kg\,m$^2$] && $0$ & $0$ & $0$ \\ %IXZ
 susp. spring rate at each wheel (front) & $K_{S,F}$ & $10^4$[N/m] && $ 2.189$ & $2.445$ & $3.357$ \\ %KSF
 susp. damping rate at each wheel (front) & $K_{SD,F}$ & $10^3$[N/m] && $1.459$ & $1.786$ & $2.405$ \\ %KSDF
 susp. spring rate at each wheel (rear) & $K_{S,R}$ & $10^4$[N/m] && $2.189$ & $1.963$ & $3.912$ \\ %KSR
 susp. damping rate at each wheel (rear) & $K_{SD,R}$ & $10^3$[N/m] && $1.459$ & $1.649$ & $2.769$ \\ %KSDR
 track width (front) & $T_f$ & [m] && $1.389$ & $1.386$ & $1.574$ \\ %TRWF
 track width (rear) & $T_r$ & [m] && $1.423$ & $1.364$ & $1.543$ \\ %TRWB
 lateral spring rate at compliant pin joint %between $M_s$ and $M_u$
 & $K_{RAS}$ & $10^5$[N/m] && $1.751$ & $1.751$ & $1.751$ \\ %KRAS
 aux. torsional roll stiffness per axle (front) & $K_{TS,F}$ & $10^4$[Nm/rad] && $1.28$ & $0.69$ & $3.39$ \\ %KTSF
 aux. torsional roll stiffness per axle (rear) & $K_{TS,R}$ & $10^3$[Nm/rad] && $0$ & $2.643$ & $7.731$ \\ %KTSR
 damp. rate at pin joint btw. $m_s$ and $m_u$ & $K_{RAD}$ & $10^4$[Ns/m] && $1.021$ & $1.021$ & $1.021$ \\ %KRADP
 vertical spring rate of tire & $K_{ZT}$ & $10^5$[N/m] && $1.897$ & $1.582$ & $2.126$ \\ %TSPRINGR
 c.g. height of total mass & $h_{cg}$ & [m] && $0.557$ & $0.574$ & $0.747$ \\ %HCG
 height of roll axis above ground (front) & $h_{RA,F}$ & [m] && $0$ & $0$ & $0$ \\ %HRAF
 height of roll axis above ground (rear) & $h_{RA,R}$ & [m] && $0$ & $0$ & $0$ \\ %HRAR
 $m_s$ c.g. height above ground & $h_s$ & [m] && $0.594$ & $0.613$ & $0.804$ \\ %HS
 m.o.i. for $m_{u,f}$ about xaxis (front) & $I_{u,f}$ & [kg\,m$^2$] && $32.53$ & $30.67$ & $50.27$ \\ %IXUF
 m.o.i. for $m_{u,r}$ about xaxis (rear) & $I_{u,r}$ & [kg\,m$^2$] && $32.53$ & $29.67$ & $48.34$ \\ %IXUR
 wheel inertia & $I_{y,w}$ & [kg\,m$^2$] && $1.700$ & $1.700$ & $1.700$ \\
 lateral compliance rate of tire, wheel, susp. & $K_{LT}$ & $10^{5}$[m/N] && $1.027$ & $1.643$ & $1.223$ \\
 effective tire radius (RR from \cite[PAC2002]{Adams2011}) & $R_w$ & [m] && $0.344$ & $0.344$ & $0.344$ \\
 torque split of brakes & $T_{s,b}$ & [] && $0.76$ & $0.66$ & $0.64$ \\
 torque split of engine & $T_{s,e}$ & [] && $1$ & $0$ & $0$ \\
 suspension parameter (front) & $D_f$ & [rad/m] && $0.62$ & $0.39$ & $0$ \\
 suspension parameter (rear) & $D_r$ & [rad/m] && $0.21$ & $0.90$ & $0$ \\
 suspension parameter (front) & $E_f$ & [rad/m$^2$] && $0$ & $0$ & $0$ \\
 suspension parameter (rear) & $E_r$ & [rad/m$^2$] && $0$ & $0$ & $0$ \\
 \bottomrule %LR
\end{tabular}
\end{center}}
\end{table}

% brake distribution formula: 1.2*l_r/(l_f+l_r)


% not considered: KSTR, KSCF, KSCB, DLADV, DYADV, DNADV, KTL, HCG, KBS, HBS, KRM, XACC, ZACC, DRAGC, LENS, LM, KBTF, KVB, KMB, KBPVL, swz, sww, KCF, LSO, KLAGV
% not considered: HF, HR, KSAF, KSAR, BF, BR, CF, CR, DF, DR, EF, ER, KSLF, KSLR, LSAF, LSAR, KSADF, KSADR, KSAD2F, KSAD2R, KACK
% to be discussed: DENSITY, REFAREA, CDX, AEROVEL (aerodynamics)

\begin{table}[h!tb]
\caption{Tire Parameters (see \cite[Sec.~PAC2002]{Adams2011}).}
{\small
\begin{center}\label{tab:tireParameters}
\begin{tabular}{lllclll}
 \toprule
 %\multicolumn{2}{c}{\textbf{tire parameter}} & \\ \hline
 \textbf{name} & \textbf{symbol} & \textbf{value} \\ \hline
 \multicolumn{3}{c}{\textbf{longitudinal parameters}} \\ \hline
 shape factor for longitudinal force & $p_{Cx1}$ & $1.6411$ \\
 longitudinal friction $\mu_x$ at $F_{z0}$ & $p_{Dx1}$ & $1.1739$ \\
 variation of friction $\mu_x$ with camber & $p_{Dx3}$ & $0$ \\
 longitudinal curvature at $F_{z0}$ & $p_{Ex1}$ & $0.4640$ \\
 longitudinal slip stiffness at $F_{z0}$ & $p_{Kx1}$ & $22.303$ \\
 horizontal shift at $F_{z0}$ & $p_{Hx1}$ & $1.2297\cdot 10^{3}$ \\
 vertical shift at $F_{z0}$ & $p_{Vx1}$ & $8.8098\cdot 10^{6}$ \\
 slope factor for combined slip $F_x$ reduction & $r_{Bx1}$ & $13.276$ \\
 variation of slope $F_x$ reduction with $\kappa$ & $r_{Bx2}$ & $13.778$ \\
 shape factor for combined slip $F_x$ reduction & $r_{Cx1}$ & $1.2568$ \\
 curvature factor of combined $F_x$ & $r_{Ex1}$ & $0.6522$ \\
 shift factor for combined slip $F_x$ reduction & $r_{Hx1}$ & $5.0722\cdot 10^{3}$ \\ \hline
 \multicolumn{3}{c}{\textbf{lateral parameters}} \\ \hline
 shape factor for lateral forces & $p_{Cy1}$ & $1.3507$ \\
 lateral friction $\mu_y$ & $p_{Dy1}$ & $1.0489$ \\
 variation of friction $\mu_y$ with squared camber & $p_{Dy3}$ & $2.8821$ \\
 lateral curvature at $F_{z0}$ & $p_{Ey1}$ & $7.4722\cdot 10^{3}$ \\
 maximum value of stiffness & $p_{Ky1}$ & $21.920$ \\
 horizontal shift at $F_{z0}$ & $p_{Hy1}$ & $2.6747\cdot 10^{3}$ \\
 variation of shift with camber & $p_{Hy3}$ & $3.1415\cdot 10^{2}$ \\
 vertical shift at $F_{z0}$ & $p_{Vy1}$ & $3.7318\cdot 10^{2}$ \\
 variation of vertical shift with camber & $p_{Vy3}$ & $0.3293$ \\
 slope factor for combined $F_y$ reduction & $r_{By1}$ & $7.1433$ \\
 variation of slope $F_y$ reduction with $\alpha$ & $r_{By2}$ & $9.1916$ \\
 shift term for $\alpha$ in slope $F_y$ reduction & $r_{By3}$ & $2.7856\cdot 10^{2}$ \\
 shape factor for combined $F_y$ reduction & $r_{Cy1}$ & $1.0719$ \\
 curvature factor of combined $F_y$ & $r_{Ey1}$ & $0.2757$ \\
 shift factor for combined $F_y$ reduction & $r_{Hy1}$ & $5.7448\cdot 10^{6}$ \\
 $\kappa$induced side force at $F_{z0}$ & $r_{Vy1}$ & $2.7825\cdot 10^{2}$ \\
 variation of $S_{Vy\kappa}/\mu_y \, F_z$ with camber & $r_{Vy3}$ & $0.2756$ \\
 variation of $S_{Vy\kappa}/\mu_y \, F_z$ with $\alpha$ & $r_{Vy4}$ & $12.120$ \\
 variation of $S_{Vy\kappa}/\mu_y \, F_z$ with $\kappa$ & $r_{Vy5}$ & $1.9$ \\
 variation of $S_{Vy\kappa}/\mu_y \, F_z$ with $\arctan(\kappa)$ & $r_{Vy6}$ & $10.704$ \\
 \bottomrule
\end{tabular}
\end{center}}
\end{table}


\section{Conversion of Initial States and Parameters} \label{sec:conversion}

As previously mentioned, we do not only like to provide different vehicle models of increasing complexity, but also would like to make results easily comparable. For this reason, we try to specify as many parameters sets for the complicated multibody model and convert them to simpler models. Similarly, we convert initial states across different models so that results can be compared in the best possible way. We start with converting parameters and afterwards discuss how initial states can be shared across models.

\subsection{Conversion of Parameters} \label{sec:parameterConversion}


\paragraph{From multibody model to singletrack model}

The singletrack model only requires $7$ parameters, see Tab.~\ref{tab:vehicleParametersST}. Out of those parameters, $6$ parameters are identical to the multibody model and do not require any conversion:
\begin{itemize}
 \item total vehicle mass $m$,
 \item moment of inertia for entire mass about z axis $I_z$,
 \item distance from center of gravity to front axle $l_f$,
 \item distance from center of gravity to rear axle $l_r$,
 \item height of center of gravity above ground $h_{cg}$,
 \item friction coefficient $\mu$, which is represented by the parameter $p_{Dy1}$ in \cite[Sec.~PAC2002]{Adams2011}.
\end{itemize}
Only the cornering stiffness coefficient has to be computed: As stated in Sec.~\ref{sec:STmodel}, we separate the effect of the friction coefficient $\mu$, the cornering stiffness coefficient $C_{S}$, and the vertical force $F_z$, such that the cornering stiffness becomes $C_i = \mu C_{S,i} F_{z,i}$ and $i=\{f,r\}$ for the front and rear axle. The cornering stiffness $C_{i}$ is by definition the linear approximation of the lateral tire forces. By linearizing the magic tire formula in \eqref{eq:magicFormulaLateral} at zero slip angle, one obtains the following value for the cornering stiffness:
\begin{flalign*}
C_{i} = B_y \, C_y \, D_y = \frac{K_{y,i}}{C_y \, D_{y,i}} C_y \, D_y = K_{y,i} = F_{z,i} \, p_{Ky1} &&
\end{flalign*}
so that
\begin{flalign*}
C_{S,i} = \frac{p_{Ky1}}{\mu} = \frac{p_{Ky1}}{p_{Dy1}}. &&
\end{flalign*}

%It remains to obtain the constraint parameters for the singletrack model:


\paragraph{From singletrack model to kinematic singletrack model}

This conversion is trivial: We only require the wheelbase $l_{wb} = l_f + l_r$.



\subsection{Conversion of Initial States}

We like to initialize all models such that their initial behavior is simliar. However, it is possible to initialize the model differently, but then this different initialization has to be explicitly stated. In order to facilitate switching between different models, we share the following initial values across all models:
\begin{itemize}
 \item initial xposition $s_{x,0}$,
 \item initial yposition $s_{y,0}$,
 \item initial steering angle $\delta_0$,
 \item initial velocity $v_0$,
 \item initial orientation $\Psi_0$,
 \item initial yaw rate $\dot{\Psi}_0$,
 \item initial slip angle $\beta_0$.
\end{itemize}

\paragraph{Multibody model}

Since the multibody model is tedious to initialize, we propose an initialization using the following auxiliary values:
\begin{itemize}
 \item $\omega_0 = \frac{v_{x,0}}{R}$ (no wheel spin, $R$: effective tire radius),
 \item $v_{x,0} = \cos(\beta_0) v_0$ (velocity in longitudinal direction from slip angle $\beta$),
 \item $v_{y,0} = \sin(\beta_0) v_0$ (velocity in lateral direction from slip angle $\beta$),
 \item $v_{yf,0} = v_{y,0} + l_f \dot{\Psi}_0$ (lateral velocity at front axle from velocity at c.g. and yaw rate),
 \item $v_{yr,0} = v_{y,0}  l_r \dot{\Psi}_0$ (lateral velocity at rear axle from velocity at c.g. and yaw rate),
 \item $z_{i,0} = \frac{F_{zi,0}}{2K_{zt}}$ ($i\in\{r,f\}$) (height over ground so that springs support weight),
% \item for $v_0\geq 0.1$: $\delta_0 = \frac{v_0(l_f + l_r)}{C_{S,f} \, g \, l_r \, \mu}\dot{\Psi}_0 + \frac{1}{C_{S,f} \, l_r}((C_{S,r} \, l_f + C_{S,f} \, l_r)\beta_0  (C_{S,r}  C_{S,f})l_r \, l_f \, \frac{\dot{\Psi}_0}{v_0}$ (initial steering angle from steady state of slip angle).
% \item for $v_0< 0.1$: $\delta_0 = \arctan\left(\frac{l_{wb}}{v} \dot{\Psi}\right)$ (slip angle neglected; steering angle follows from kinematic vehicle model).
\end{itemize}
% The initial steering angle is derived for $v_0\geq 1$ from demanding that the slip angle $\beta$ of the singletrack model is at steady state for $a_\mathtt{long} = 0$ (see \eqref{eq:slipAngle_ST}):
% \begin{flalign*}
% \frac{x_4(l_r + l_f)}{C_{S,f}g l_r \mu}x_6 + \frac{1}{C_{S,f} l_r} \Big( (C_{S,r} l_f + C_{S,f} l_r ) x_7  (C_{S,r}  C_{S,f}) \frac{l_f l_r x_6}{x_4} \Big) = & x_3 .
% \end{flalign*}
% For smaller initial velocities ($v_0< 0.1$), the above formula becomes singular and slip can be negelected. In that case we use the constraint from the kinematic model:
% \begin{flalign*}
% \dot{x}_5 = & \frac{x_4}{l_{wb}} \tan(x_3).
% \end{flalign*}
Inserting these values in Tab.~\ref{tab:fullModelInitialStates} initializes the multibody model as proposed in this document.

\begin{table}[h!tb]
\caption{Initial values of the multibody model.}
\begin{center}\label{tab:fullModelInitialStates}
\begin{tabular}{lllllllll}
 \hline
 \multicolumn{3}{c}{\textbf{sprung mass}} & \multicolumn{3}{c}{\textbf{unsprung mass}} & \multicolumn{3}{c}{\textbf{other}} \\ \hline
 & & \textbf{init.} & & & \textbf{init.} & & & \textbf{init.}\\
 \textbf{name} & \textbf{symb.} & \textbf{val.} & \textbf{name} & \textbf{symb.} & \textbf{val.} & \textbf{name} & \textbf{symb.} & \textbf{val.} \\ \hline
 yaw ang. & $x_{5,0}$ & $\Psi_0$ & roll ang. (f) & $x_{14,0}$ & $0$ & wheel speed (LF) & $x_{24,0}$ & $\omega_0$ \\
 yaw rate & $x_{6,0}$ & $\dot{\Psi}_0$ & roll rate (f) & $x_{15,0}$ & $0$ & wheel speed (RF) & $x_{25,0}$ & $\omega_0$ \\
 roll angle & $x_{7,0}$ & $0$ & roll ang. (r) & $x_{19,0}$ & $0$ & wheel speed (LR) & $x_{26,0}$ & $\omega_0$ \\
 roll rate & $x_{8,0}$ & $0$ & roll rate (r) & $x_{20,0}$ & $0$ & wheel speed (RR) & $x_{27,0}$ & $\omega_0$ \\
 pitch ang. & $x_{9,0}$ & $0$ & yvel. (f) & $x_{16,0}$ & $v_{yf,0}$ & pin joint diff. (f) & $x_{28,0}$ & $0$ \\
 pitch rate & $x_{10,0}$ & $0$ & yvel. (r) & $x_{21,0}$ & $v_{yr,0}$ & pin joint diff. (r) & $x_{29,0}$ & $0$ \\
 xvelocity & $x_{4,0}$ & $v_{x,0}$ & zpos. (f) & $x_{17,0}$ & $z_{f,0}$ & xposition & $x_{1,0}$ & $s_{x,0}$ \\
 yvelocity & $x_{11,0}$ & $v_{y,0}$ & zvel. (f) & $x_{18,0}$ & $0$ & yposition & $x_{2,0}$ & $s_{y,0}$ \\
 zposition & $x_{12,0}$ & $0$ & zpos. (r) & $x_{22,0}$ & $z_{r,0}$ & steering angle & $x_{3,0}$ & $\delta_0$ \\
 zvelocity & $x_{13,0}$ & $0$ & zvel. (r) & $x_{23,0}$ & $0$ \\ \hline
\end{tabular}
\end{center}
\end{table}



%x7 = ΦS roll angle
%x8 = ΦS roll rate
%x9 = ΘS pitch angle
%x10 = ΘS pitch rate
%x11 = v velocity in ydirection
%x12 = zS zposition
%x13 = w velocity in zdirection

%x14 = ΦUF roll angle front
%x15 = ΦUF roll rate front
%x16 = vUF velocity in ydirection front
%x17 = zUF zposition front
%x18 = wUF velocity in zdirection front

%x19 = ΦUR roll angle rear
%x20 = ΦUR roll rate rear
%x21 = vUR velocity in ydirection rear
%x22 = zUR zposition rear
%x23 = wUR velocity in zdirection rear

%x24 = ωLF left front wheel angular speed
%x25 = ωRF right front wheel angular speed
%x26 = ωLR left rear wheel angular speed
%x27 = ωRR right rear wheel angular speed

%x28 = delta_y_f
%x29 = delta_y_r

\paragraph{Singletrack model} The initialization of the singletrack model is straightforward:
\begin{flalign*}
 x_{1,0} = s_{x,0}, \quad x_{2,0} = s_{y,0}, \quad x_{3,0} = \delta_0, \quad x_{4,0} = v_0, \quad x_{5,0} = \Psi_0, \quad x_{6,0} = \dot{\Psi}_0, \quad x_{7,0} = \beta_0. &&
\end{flalign*}

\paragraph{Kinematic singletrack model} Similarly, the initialization of the kinematic singletrack model is straightforward:
\begin{flalign*}
 x_{1,0} = s_{x,0}, \quad x_{2,0} = s_{y,0}, \quad x_{3,0} = \delta_0, \quad x_{4,0} = v_0, \quad x_{5,0} = \Psi_0. &&
\end{flalign*}

\paragraph{Pointmass model} The initialization of the pointmass model only requires initial positions and velocities:
\begin{flalign*}
 x_{1,0} = s_{x,0}, \quad x_{2,0} = s_{y,0}, \quad x_{3,0} = v_0 \cos(\Psi_0) , \quad x_{4,0} = v_0 \sin(\Psi_0). &&
\end{flalign*}

\section{Examples} \label{sec:examples}

In this section, we perform numerical experiments based on the parameters of vehicle 2 (BMW 320i): First, we compare the kinematic singletrack model, the singletrack model and the multibody model in a left curve. Secod, we demonstrate understeering and oversteering for the multibody model during cornering. For all experiments we use the following initial states:
\begin{equation*}
 s_{x,0} = s_{y,0} = \delta_0 = \Psi_0 = \dot{\Psi}_0 = \beta_0 = 0, \quad v_0 = 15.
\end{equation*}
The simulation time for all tests is $1$~s.

\paragraph{Comparison of KS, ST and MB during cornering}

We perform a left curve by choosing $v_\delta = 0.15$~[rad/s]. Fig.~\ref{fig:corneringPosition} shows the paths of the kinematic singletrack model, the singletrack model and the multibody model. It can be easily seen that the kinematic singletrack model realizes the tightest bend since it does not consider tire slip; the singletrack model is a little wider due to considering tire slip. This effect is even stronger for the multibody model since its vehicle model considers saturation of tire forces. This can be even better seen when comparing the slip angles of the singletrack model and the multibody model in Fig.~\ref{fig:corneringSlipAngle}.

\begin{figure}[h!tb]
 \centering
 \footnotesize
 \psfrag{a}[c][c]{xposition $s_x$}
 \psfrag{b}[c][c]{yposition $s_y$}
 \psfrag{c}[r][c]{\shortstack{KS model}}
 \psfrag{d}[r][c]{\shortstack{ST model}}
 \psfrag{e}[r][c]{\shortstack{MB model}}
 %
 \subfigure[Path of center of gravity.]
 {\includegraphics[width=0.45\columnwidth]{./figures/cornering_position_015_b.eps} \label{fig:corneringPosition}} \hspace{0.1cm}
 %
 \psfrag{a}[c][c]{time $t$}
 \psfrag{b}[c][c]{slip angle $\beta$}
 %
 \subfigure[Slip angle.]
 {\includegraphics[width=0.45\columnwidth]{./figures/cornering_slipAngle_015_b.eps} \label{fig:corneringSlipAngle}}

 \caption{Comparing the kinematic singletrack (KS) model, the singletrack (ST) model and the multibody (MB) model during cornering.}
\end{figure}

\paragraph{Overstering and understeering of the multibody model}

During cornering, a vehicle tends to understeer when braking since typically more braking force is applied at the front brakes: Oversteering during braking would make a vehicle much less safe to drive. Oversteering can be achieved by accelerating with a rearwheeldrive vehicle during cornering. Fig.~\ref{fig:cornerBrakeAcc_position} shows the paths of the multibody model when using again $v_\delta = 0.15$~[rad/s] and in addition $a_\mathtt{long} = 0.7$~g for braking and $a_\mathtt{long} = 0.63$~g for acceleration. The tightest bend is realized by braking since the velocity drops and the widest bend is caused by accelerating since the velocity increases. It is evident that during braking we have understeer and during acceleration we have oversteer by observing the slip angle in Fig.~\ref{fig:cornerBrakeAcc_slipAngle}. This is also obvious from the orientation of the vehicle, where during acceleration, the vehicle turns into the corner as shown in Fig.~\ref{fig:cornerBrakeAcc_orientation}. Further, in Fig.~\ref{fig:cornerBrakeAcc_pitch} the pitch for braking shows that the vehicle is ``diving'' while the front lifts during acceleration. This plot also nicely shows the oscillation in the springmassdamper system since braking and acceleration is suddenly applied.

\begin{figure}[h!tb]
 \centering
 \footnotesize
 \psfrag{a}[c][c]{xposition $s_x$}
 \psfrag{b}[c][c]{yposition $s_y$}
 \psfrag{c}[r][c]{\shortstack{coasting}}
 \psfrag{d}[r][c]{\shortstack{braking}}
 \psfrag{e}[r][c]{\shortstack{accelerating}}
 %
 \subfigure[Path of center of gravity.]
 {\includegraphics[width=0.45\columnwidth]{./figures/position_cornerBrakeAcc_b.eps} \label{fig:cornerBrakeAcc_position}} \hspace{0.1cm}
 %
 \psfrag{a}[c][c]{time $t$}
 \psfrag{b}[c][c]{slip angle $\beta$}
 %
 \subfigure[Slip angle.]
 {\includegraphics[width=0.45\columnwidth]{./figures/slipAngle_cornerBrakeAcc_b.eps} \label{fig:cornerBrakeAcc_slipAngle}} \\
 %
 \psfrag{a}[c][c]{time $t$}
 \psfrag{b}[c][c]{orientation $\Psi$}
 %
 \subfigure[Orientation.]
 {\includegraphics[width=0.45\columnwidth]{./figures/orientation_cornerBrakeAcc_b.eps} \label{fig:cornerBrakeAcc_orientation}} \hspace{0.1cm}
 %
 \psfrag{a}[c][c]{time $t$}
 \psfrag{b}[c][c]{pitch $\Theta_S$}
 %
 \subfigure[Pitch.]
 {\includegraphics[width=0.45\columnwidth]{./figures/pitch_cornerBrakeAcc_b.eps} \label{fig:cornerBrakeAcc_pitch}}

 \caption{Investigating oversteering and understeering for the multibody model.}
\end{figure}


\section{Conclusions}

This document describes four models for motion planning of automated vehicles as part of the \textit{CommonRoad} benchmark suite: pointmass model, kinematic singletrack model, singletrack model, and a multibody model. To easily exchange models, we also present how to convert parameters and initial states from the multibody model to simpler models. The sources of all equations are carefully referenced in this work and all models are available as MATLAB and Python code. Numerical experiments provide further insight into what effects certain models can show.


\section*{Acknowledgment}

\vspace{0.2cm}
The author gratefully acknowledge financial support by the Free State of Bavaria.



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