Multi-scale description of pedestrian collective dynamics with port-Hamiltonian systems

Authors

Antoine Tordeux, Claudia Totzeck

Abstract

Port-Hamiltonian systems (PHS) theory is a recent but already well-established modelling approach for non-linear physical systems. Some studies have shown lately that PHS frameworks are relevant for modelling and control of swarm and multi-agent systems. We identify in this contribution a general class of microscopic force-based pedestrian models that can be formulated as a port-Hamiltonian system. The pedestrian PHS has linear structure and dissipation components. Non-linear effects come from isotropic pedestrian interactions. Simulation results on a torus with disordered initial states show that the port-Hamiltonian pedestrian model can exhibit different types of dynamics. They range from relaxed speed models with no interaction, dynamical billiards, or crystallization dynamics to realistic pedestrian collective behaviors, including lane and strip formation for counter and crossing flow. The port-Hamiltonian framework is a natural multiscale description of pedestrian dynamics as the Hamiltonian turns out to be a generic order parameter that allows us to identify specific behaviours of the dynamics from a macroscopic viewpoint. Particular cases even enable through energy balance to determine the Hamiltonian behavior without requiring the tedious computation of the microscopic dynamics. Using PHS theory, we systematically identify a critical threshold value for the Hamiltonian, which relies only on exogenous input and can be physically interpreted.

Citation

• Journal: Networks and Heterogeneous Media
• Year: 2023
• Volume: 18
• Issue: 2
• Pages: 906–929
• Publisher: American Institute of Mathematical Sciences (AIMS)
• DOI: 10.3934/nhm.2023039

BibTeX

@article{Tordeux_2023,
  title={{Multi-scale description of pedestrian collective dynamics with port-Hamiltonian systems}},
  volume={18},
  ISSN={1556-1801},
  DOI={10.3934/nhm.2023039},
  number={2},
  journal={Networks and Heterogeneous Media},
  publisher={American Institute of Mathematical Sciences (AIMS)},
  author={Tordeux, Antoine and Totzeck, Claudia},
  year={2023},
  pages={906--929}
}

Download the bib file

References

• van der Schaft A (2007) Port-Hamiltonian systems: an introductory survey. In: Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006. EMS Press, pp 1339–1365 – 10.4171/022-3/65
• van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2-3):173–378. https://doi.org/10.1561/2600000002 – 10.1561/2600000002
• van der Schaft A (1981) Symmetries and conservation laws for Hamiltonian systems with inputs and outputs: A generalization of Noether’s theorem. Systems & Control Letters 1(2):108–115. https://doi.org/10.1016/s0167-6911(81)80046-1 – 10.1016/s0167-6911(81)80046-1
• Maschke BM, Van Der Schaft AJ, Breedveld PC (1992) An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute 329(5):923–966. https://doi.org/10.1016/s0016-0032(92)90049-m – 10.1016/s0016-0032(92)90049-m
• Maschke BM, van der Schaft AJ (1993) PORT-CONTROLLED HAMILTONIAN SYSTEMS: MODELLING ORIGINS AND SYSTEMTHEORETIC PROPERTIES. In: Nonlinear Control Systems Design 1992. Elsevier, pp 359–365 – 10.1016/b978-0-08-041901-5.50064-6
• Schaft AJ (2004) Port-Hamiltonian Systems: Network Modeling and Control of Nonlinear Physical Systems. In: Advanced Dynamics and Control of Structures and Machines. Springer Vienna, Vienna, pp 127–167 – 10.1007/978-3-7091-2774-2_9
• van der Schaft AJ, Maschke BM (2013) Port-Hamiltonian Systems on Graphs. SIAM J Control Optim 51(2):906–937. https://doi.org/10.1137/110840091 – 10.1137/110840091
• Rashad R, Califano F, van der Schaft AJ, Stramigioli S (2020) Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information 37(4):1400–1422. https://doi.org/10.1093/imamci/dnaa018 – 10.1093/imamci/dnaa018
• Knorn S, Chen Z, Middleton RH (2016) Overview: Collective Control of Multiagent Systems. IEEE Trans Control Netw Syst 3(4):334–347. https://doi.org/10.1109/tcns.2015.2468991 – 10.1109/tcns.2015.2468991
• Knorn S, Ahlén A (2016) Deviation bounds in multi agent systems described by undirected graphs. Automatica 67:205–210. https://doi.org/10.1016/j.automatica.2016.01.038 – 10.1016/j.automatica.2016.01.038
• Knorn S, Ahlén A (2016) Deviation bounds in multi agent systems described by undirected graphs. Automatica 67:205–210. https://doi.org/10.1016/j.automatica.2016.01.038 – 10.1016/j.automatica.2016.01.038
• Cristofaro A, Giunta G, Giordano PR (2022) Fault-Tolerant Formation Control of Passive Multi-Agent Systems Using Energy Tanks. IEEE Control Syst Lett 6:2551–2556. https://doi.org/10.1109/lcsys.2022.3169308 – 10.1109/lcsys.2022.3169308
• van der Schaft AJ, Maschke BM (2010) Port-Hamiltonian Dynamics on Graphs: Consensus and Coordination Control Algorithms. IFAC Proceedings Volumes 43(19):175–178. https://doi.org/10.3182/20100913-2-fr-4014.00012 – 10.3182/20100913-2-fr-4014.00012
• Xue D, Hirche S, Cao M (2020) Opinion Behavior Analysis in Social Networks Under the Influence of Coopetitive Media. IEEE Trans Netw Sci Eng 7(3):961–974. https://doi.org/10.1109/tnse.2019.2894565 – 10.1109/tnse.2019.2894565
• Sharf M, Zelazo D (2019) Analysis and Synthesis of MIMO Multi-Agent Systems Using Network Optimization. IEEE Trans Automat Contr 64(11):4512–4524. https://doi.org/10.1109/tac.2019.2908258 – 10.1109/tac.2019.2908258
• Matei I, Mavridis C, Baras JS, Zhenirovskyy M (2019) Inferring Particle Interaction Physical Models and Their Dynamical Properties. In: 2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, pp 4615–4621 – 10.1109/cdc40024.2019.9029524
• Ma Y, Chen J, Wang J, Xu Y, Wang Y (2022) Path-Tracking Considering Yaw Stability With Passivity-Based Control for Autonomous Vehicles. IEEE Trans Intell Transport Syst 23(7):8736–8746. https://doi.org/10.1109/tits.2021.3085713 – 10.1109/tits.2021.3085713
• Knorn S, Donaire A, Agüero JC, Middleton RH (2014) Passivity-based control for multi-vehicle systems subject to string constraints. Automatica 50(12):3224–3230. https://doi.org/10.1016/j.automatica.2014.10.038 – 10.1016/j.automatica.2014.10.038
• Dai S, Koutsoukos X (2017) Safety analysis of integrated adaptive cruise control and lane keeping control using discrete-time models of port-Hamiltonian systems. In: 2017 American Control Conference (ACC). IEEE, pp 2980–2985 – 10.23919/acc.2017.7963404
• Dai S, Koutsoukos X (2020) Safety analysis of integrated adaptive cruise and lane keeping control using multi-modal port-Hamiltonian systems. Nonlinear Analysis: Hybrid Systems 35:100816. https://doi.org/10.1016/j.nahs.2019.100816 – 10.1016/j.nahs.2019.100816
• Bansal H, Schulze P, Abbasi MH, Zwart H, Iapichino L, Schilders WHA, Wouw N van de (2021) Port-Hamiltonian formulation of two-phase flow models. Systems & Control Letters 149:104881. https://doi.org/10.1016/j.sysconle.2021.104881 – 10.1016/j.sysconle.2021.104881
• Rashad R, Califano F, Schuller FP, Stramigioli S (2021) Port-Hamiltonian modeling of ideal fluid flow: Part I. Foundations and kinetic energy. Journal of Geometry and Physics 164:104201. https://doi.org/10.1016/j.geomphys.2021.104201 – 10.1016/j.geomphys.2021.104201
• (1975) INTERNATIONAL CONFERENCES. Medical Journal of Australia 2(10):409–411. https://doi.org/10.5694/j.1326-5377.1975.tb105931.x – 10.5694/j.1326-5377.1975.tb105931.x
• HENDERSON LF (1971) The Statistics of Crowd Fluids. Nature 229(5284):381–383. https://doi.org/10.1038/229381a0 – 10.1038/229381a0
• Martinez-Gil F, Lozano M, García-Fernández I, Fernández F (2017) Modeling, Evaluation, and Scale on Artificial Pedestrians. ACM Comput Surv 50(5):1–35. https://doi.org/10.1145/3117808 – 10.1145/3117808
• Chraibi M, Tordeux A, Schadschneider A, Seyfried A (2018) Modelling of Pedestrian and Evacuation Dynamics. In: Encyclopedia of Complexity and Systems Science. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 1–22 – 10.1007/978-3-642-27737-5_705-1
• Bellomo N, Dogbe C (2011) On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives. SIAM Rev 53(3):409–463. https://doi.org/10.1137/090746677 – 10.1137/090746677
• Albi G, Bellomo N, Fermo L, Ha S-Y, Kim J, Pareschi L, Poyato D, Soler J (2019) Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives. Math Models Methods Appl Sci 29(10):1901–2005. https://doi.org/10.1142/s0218202519500374 – 10.1142/s0218202519500374
• Mishra P, Wrzosek D (2021) Repulsive chemotaxis and predator evasion in predator–prey models with diffusion and prey-taxis. Math Models Methods Appl Sci 32(01):1–42. https://doi.org/10.1142/s0218202522500014 – 10.1142/s0218202522500014
• Fischer M, ,Radon Institute for Applied and Computational Mathematics, Altenbergerstr. 694040, Linz, Austria, Jankowiak G, Wolfram M-T, ,University of Warwick, Mathematics Institute, Gibbet Hill Road, CV47AL Coventry, UK, ,Radon Institute for Applied and Computational Mathematics, Altenbergerstr. 694040, Linz, Austria (2020) Micro- and macroscopic modeling of crowding and pushing in corridors. Networks & Heterogeneous Media 15(3):405–426. https://doi.org/10.3934/nhm.2020025 – 10.3934/nhm.2020025
• Barré J, ,Institut Denis Poisson, Université d’Orléans, CNRS, Université de Tours, B.P. 6759, 45067 Orléans cedex 2, France, Institut Universitaire de France, Paris, France, Degond P, Peurichard D, Zatorska E, ,Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom, ,INRIA Team Mamba, INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris, France, Sorbonne Université, UMR 7598 LJLL, BC187, 4, Place de Jussieu, F-75252 Paris Cedex 5, France, ,Department of Mathematics, University College London, Gower Street London WC1E 6BT, UK, United Kingdom (2020) Modelling pattern formation through differential repulsion. Networks & Heterogeneous Media 15(3):307–352. https://doi.org/10.3934/nhm.2020021 – 10.3934/nhm.2020021
• Burger M, Kreusser LM, Totzeck C (2021) Mean-field optimal control for biological pattern formation. ESAIM: COCV 27:40. https://doi.org/10.1051/cocv/2021034 – 10.1051/cocv/2021034
• Burger M, Pinnau R, Totzeck C, Tse O (2021) Mean-Field Optimal Control and Optimality Conditions in the Space of Probability Measures. SIAM J Control Optim 59(2):977–1006. https://doi.org/10.1137/19m1249461 – 10.1137/19m1249461
• Helbing D, Molnár P (1995) Social force model for pedestrian dynamics. Phys Rev E 51(5):4282–4286. https://doi.org/10.1103/physreve.51.4282 – 10.1103/physreve.51.4282
• Chraibi M, ,Jülich Supercomputing Centre, Forschungszentrum Jülich, 52425 Jülich, Kemloh U, Schadschneider A, Seyfried A, ,Institute for Theoretical Physics, University of Cologne, 50937 Köln (2011) Force-based models of pedestrian dynamics. Networks & Heterogeneous Media 6(3):425–442. https://doi.org/10.3934/nhm.2011.6.425 – 10.3934/nhm.2011.6.425
• van den Berg J, Ming Lin, Manocha D (2008) Reciprocal Velocity Obstacles for real-time multi-agent navigation. In: 2008 IEEE International Conference on Robotics and Automation. IEEE, pp 1928–1935 – 10.1109/robot.2008.4543489
• van den Berg J, Guy SJ, Lin M, Manocha D (2011) Reciprocal n-Body Collision Avoidance. In: Springer Tracts in Advanced Robotics. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 3–19 – 10.1007/978-3-642-19457-3_1
• van Toll W, Pettré J (2021) Algorithms for Microscopic Crowd Simulation: Advancements in the 2010s. Computer Graphics Forum 40(2):731–754. https://doi.org/10.1111/cgf.142664 – 10.1111/cgf.142664
• Chraibi M, Ezaki T, Tordeux A, Nishinari K, Schadschneider A, Seyfried A (2015) Jamming transitions in force-based models for pedestrian dynamics. Phys Rev E 92(4):042809. https://doi.org/10.1103/physreve.92.042809 – 10.1103/physreve.92.042809
• Nowak S, Schadschneider A (2012) Quantitative analysis of pedestrian counterflow in a cellular automaton model. Phys Rev E 85(6):066128. https://doi.org/10.1103/physreve.85.066128 – 10.1103/physreve.85.066128
• Kowalska M, Woźniak M, Kijek M, Mitrosz P, Szakiel J, Turek P (2022) Management of validation of HPLC method for determination of acetylsalicylic acid impurities in a new pharmaceutical product. Sci Rep 12(1):1. https://doi.org/10.1038/s41598-021-99269-x – 10.1038/s41598-021-99269-x
• Totzeck C, ,University of Mannheim, School of Business Informatics and Mathematics, 68159 Mannheim, Germany (2020) An anisotropic interaction model with collision avoidance. Kinetic & Related Models 13(6):1219–1242. https://doi.org/10.3934/krm.2020044 – 10.3934/krm.2020044
• (2001) Putting energy back in control. IEEE Control Syst 21(2):18–33. https://doi.org/10.1109/37.915398 – 10.1109/37.915398
• (2001) Putting energy back in control. IEEE Control Syst 21(2):18–33. https://doi.org/10.1109/37.915398 – 10.1109/37.915398
• Sametoglu F, Celikel O (2011) Note: Design and characterization of an optical light source based on mixture of white and near-ultraviolet light emitting diode spectra. Review of Scientific Instruments 82(4):046111. https://doi.org/10.1063/1.3584870 – 10.1063/1.3584870
• Tordeux A, Chraibi M, Seyfried A (2016) Collision-Free Speed Model for Pedestrian Dynamics. In: Traffic and Granular Flow ‘15. Springer International Publishing, Cham, pp 225–232 – 10.1007/978-3-319-33482-0_29
• U. Wilensky, Netlogo. Evanston, IL: Center for connected learning and computer-based modeling, Northwestern University, (1999).
• Boltes M, Zhang J, Tordeux A, Schadschneider A, Seyfried A (2018) Empirical Results of Pedestrian and Evacuation Dynamics. In: Encyclopedia of Complexity and Systems Science. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 1–29 – 10.1007/978-3-642-27737-5_706-1
• Schadschneider A, Chraibi M, Seyfried A, Tordeux A, Zhang J (2018) Pedestrian Dynamics: From Empirical Results to Modeling. In: Modeling and Simulation in Science, Engineering and Technology. Springer International Publishing, Cham, pp 63–102 – 10.1007/978-3-030-05129-7_4
• Rex M, Löwen H (2007) Lane formation in oppositely charged colloids driven by an electric field: Chaining and two-dimensional crystallization. Phys Rev E 75(5):051402. https://doi.org/10.1103/physreve.75.051402 – 10.1103/physreve.75.051402
• Nowak S, Schadschneider A (2012) Quantitative analysis of pedestrian counterflow in a cellular automaton model. Phys Rev E 85(6):066128. https://doi.org/10.1103/physreve.85.066128 – 10.1103/physreve.85.066128
• Xu Q, Chraibi M, Tordeux A, Zhang J (2019) Generalized collision-free velocity model for pedestrian dynamics. Physica A: Statistical Mechanics and its Applications 535:122521. https://doi.org/10.1016/j.physa.2019.122521 – 10.1016/j.physa.2019.122521