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Occurrence of Complex Behaviors in the Uncontrolled Passive Compass Biped Model

Year 2022, Volume 4, Issue 4, 246 - 266, 31.12.2022
https://doi.org/10.51537/chaos.1187427

Abstract

It is widely known that an appropriately built unpowered bipedal robot can walk down an inclined surface with a passive steady gait. The features of such gait are determined by the robot's geometry and inertial properties, as well as the slope angle. The energy needed to keep the biped moving steadily comes from the gravitational potential energy as it descends the inclined surface. The study of such passive natural motions could lead to ideas for managing active walking devices and a better understanding of the human locomotion. The major goal of this study is to further investigate order, chaos and bifurcations and then to demonstrate the complexity of the passive bipedal walk of the compass-gait biped robot by examining different bifurcation diagrams and also by studying the variation of the eigenvalues of the Poincaré map's Jacobian matrix and the variation of the Lyapunov exponents. We reveal also the exhibition of some additional results by changing the inertial and geometrical parameters of the bipedal robot model.

References

  • Added, E. and H. Gritli, 2020a Control of the passive dynamic gait of the bipedal compass-type robot through trajectory tracking. In 2020 20th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), pp. 155–162.
  • Added, E. and H. Gritli, 2020b Trajectory design and tracking-based control of the passive compass biped. In 2020 4th International Conference on Advanced Systems and Emergent Technologies (IC_ASET), pp. 417–424.
  • Added, E. and H. Gritli, 2022 Birth of the Neimark–Sacker bifurcation for the passive compass-gait walker. In Advances in Nonlinear Dynamics, edited by W. Lacarbonara, B. Balachandran, M. J. Leamy, J. Ma, J. A. Tenreiro Machado, and G. Stepan, pp. 683–697, Cham, Springer International Publishing.
  • Added, E. and H. Gritli, 2023 A further analysis of the passive compass-gait bipedal robot and its period-doubling route to chaos. In New Perspectives on Nonlinear Dynamics and Complexity, edited by D. Volchenkov and A. C. J. Luo, pp. 11–30, Cham, Springer International Publishing.
  • Added, E., H. Gritli, and S. Belghith, 2021a Additional complex behaviors, bifurcations and chaos, in the passive walk of the compass-type bipedal robot. IFAC-PapersOnLine 54: 111–116, 6th IFAC Conference on Analysis and Control of Chaotic Systems CHAOS 2021.
  • Added, E., H. Gritli, and S. Belghith, 2021b Further analysis of the passive dynamics of the compass biped walker and control of chaos via two trajectory tracking approaches. Complexity 2021: 5533451 (39 pages).
  • Added, E., H. Gritli, and S. Belghith, 2021c Further analysis of the passive walking gaits of the compass biped robot: Bifurcations and chaos. In 2021 18th International Multi-Conference on Systems, Signals & Devices (SSD), pp. 160–165.
  • Added, E., H. Gritli, and S. Belghith, 2022a Trajectory tracking-based control of the chaotic behavior in the passive bipedal compass-type robot. The European Physical Journal Special Topics 231: 1071–1084.
  • Added, E., H. Gritli, and S. Belghith, 2022b Trajectory tracking control of the compass-type bipedal robot gait via an improved PD+ controller. In 2022 5th International Conference on Advanced Systems and Emergent Technologies (IC_ASET), pp. 482–488.
  • Akgul, A., S. Hussain, and I. Pehlivan, 2016 A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik - International Journal for Light and Electron Optics 127: 7062 – 7071.
  • Andrievskii, B. R. and A. L. Fradkov, 2003 Control of chaos: Methods and applications. I. Methods. Automation and Remote Control 64: 673–713.
  • Andrievskii, B. R. and A. L. Fradkov, 2004 Control of chaos: Methods and applications. II. Applications. Automation and Remote Control 65: 505–533.
  • Aricioglu, M. A. and O. N. Berk, 2022 A comparative proposal on learning the chaos to understand the environment. Chaos Theory and Applications 4: 19 – 25.
  • Azar, A. T., C. Volos, N. A. Gerodimos, G. S. Tombras, V.-T. Pham, et al., 2017 A novel chaotic system without equilibrium: Dynamics, synchronization, and circuit realization. Complexity 2017: 7871467.
  • Bekey, G. A. and K. Y. Goldberg, 2012 Neural networks in robotics, volume 202. Springer Science & Business Media.
  • Beritelli, F., E. Di Cola, L. Fortuna, and F. Italia, 2000 Multilayer chaotic encryption for secure communications in packet switching networks. In WCC 2000-ICCT 2000. 2000 International Conference on Communication Technology Proceedings (Cat. No. 00EX420), volume 2, pp. 1575–1582.
  • Boccaletti, S., C. Grebogi, Y.-C. Lai, H. Mancini, and D. Maza, 2000 The control of chaos: theory and applications. Physics Reports 329: 103–197.
  • Boubaker, O. and S. Jafari, 2019 Recent Advances in Chaotic Systems and Synchronization: From Theory to Real World Applications. Emerging Methodologies and Applications in Modelling, Identification and Control, Elsevier, first edition.
  • Breazeal, C. and B. Scassellati, 2002 Robots that imitate humans. Trends in cognitive sciences 6: 481–487.
  • Buscarino, A., C. Famoso, L. Fortuna, and M. Frasca, 2016 A new chaotic electro-mechanical oscillator. International Journal of Bifurcation and Chaos 26: 1650161.
  • Chevallereau, C., G. Bessonnet, G. Abba, and Y. Aoustin, 2009 Bipedal Robots: Modeling, Design and Walking Synthesis. JohnWiley & Sons, Wiley-ISTE, first edition.
  • Collins, S., A. Ruina, R. Tedrake, and M. Wisse, 2005 Efficient bipedal robots based on passive-dynamic walkers. Science 307: 1082–1085.
  • Croce, U. D., P. O. Riley, J. L. Lelas, and D. Kerrigan, 2001 A refined view of the determinants of gait. Gait & Posture 14: 79 – 84.
  • da Costa Barros, I. R. and T. P. Nascimento, 2021 Robotic mobile fulfillment systems: A survey on recent developments and research opportunities. Robotics and Autonomous Systems 137: 103729.
  • Deng, K., M. Zhao, and W. Xu, 2016 Level-ground walking for a bipedal robot with a torso via hip series elastic actuators and its gait bifurcation control. Robotics and Autonomous Systems 79: 58–71.
  • Deng, K., M. Zhao, and W. Xu, 2017a Bifurcation gait suppression of a bipedal walking robot with a torso based on model predictive control. Robotics and Autonomous Systems 89: 27–39.
  • Deng, K., M. Zhao, and W. Xu, 2017b Passive dynamic walking with a torso coupled via torsional springs. International Journal of Humanoid Robotics 13: 1650024.
  • Falco, J. A., J. A. Marvel, and R. J. Norcross, 2012 Collaborative robotics: Measuring blunt force impacts on humans. Chest 140: 45.
  • Fathizadeh, M., H. Mohammadi, and S. Taghvaei, 2019 A modified passive walking biped model with two feasible switching patterns of motion to resemble multi-pattern human walking. Chaos, Solitons & Fractals 127: 83 – 95.
  • Fathizadeh, M., S. Taghvaei, and H. Mohammadi, 2018 Analyzing bifurcation, stability and chaos for a passive walking biped model with a sole foot. International Journal of Bifurcation and Chaos 28: 1850113.
  • Ferreira, B. B., A. S. de Paula, and M. A. Savi, 2011 Chaos control applied to heart rhythm dynamics. Chaos, Solitons & Fractals 44: 587–599.
  • Firth, W., 1991 Chaos–predicting the unpredictable. BMJ: British Medical Journal 303: 1565.
  • Fradkov, A. L. and R. J. Evans, 2005 Control of chaos: Methods and applications in engineering. Annual Reviews in Control 29: 33–56.
  • Fradkov, A. L., R. J. Evans, and B. R. Andrievsky, 2006 Control of chaos: Methods and applications in mechanics. Philosophical Transactions of The Royal Society A 364: 2279–2307.
  • Garcia, M., A. Chatterjee, and A. Ruina, 2000 Efficiency, speed, and scaling of two-dimensional passive-dynamic walking. Dynamics and Stability of Sytems 15: 75–99.
  • Garcia, M., A. Chatterjee, A. Ruina, and M. Coleman, 1998 The simplest walking model: Stability, complexity, and scaling. Journal of Biomechanical Engineering 120: 281–288.
  • Goswami, A., B. Espiau, and A. Keramane, 1997 Limit cycles in a passive compass gait biped and passivity-mimicking control laws. Autonomous Robots 4: 273–286.
  • Goswami, A., B. Thuilot, and B. Espiau, 1996 Compass-like biped robot. Part I: Stability and bifurcation of passive gaits, volume 2996. Technical Report, INRIA.
  • Goswami, A., B. Thuilot, and B. Espiau, 1998 Study of the passive gait of a compass-like biped robot: Symmetry and chaos. International Journal of Robotics Research 17: 1282–1301.
  • Goswami, A. and P. Vadakkepat, 2019 Humanoid Robotics: A Reference. Springer Netherlands, first edition.
  • Grebogi, C., Y.-C. Lai, and S. Hayes, 1997 Control and applications of chaos. Journal of the Franklin Institute 334: 1115–1146, Visions of Nonlinear Mechanics in the 21st Century.
  • Gritli, H. and S. Belghith, 2017aWalking dynamics of the passive compass-gait model under OGY-based control: Emergence of bifurcations and chaos. Communications in Nonlinear Science and Numerical Simulation 47: 308–327.
  • Gritli, H. and S. Belghith, 2017b Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Analysis of local bifurcations via the hybrid poincaré map. Chaos, Solitons & Fractals 98: 72 – 87.
  • Gritli, H. and S. Belghith, 2018a Diversity in the nonlinear dynamic behavior of a one-degree-of-freedom impact mechanical oscillator under ogy-based state-feedback control law: order, chaos and exhibition of the border-collision bifurcation. Mechanism and Machine Theory 124: 1–41.
  • Gritli, H. and S. Belghith, 2018bWalking dynamics of the passive compass-gait model under OGY-based state-feedback control: Rise of the Neimark–Sacker bifurcation. Chaos, Solitons & Fractals 110: 158 – 168.
  • Gritli, H., S. Belghith, and N. Khraeif, 2012a Intermittency and interior crisis as route to chaos in dynamic walking of two biped robots. International Journal of Bifurcation and Chaos 22: 1250056.
  • Gritli, H., S. Belghith, and N. Khraeif, 2015 OGY-based control of chaos in semi-passive dynamic walking of a torso-driven biped robot. Nonlinear Dynamics 79: 1363–1384.
  • Gritli, H., N. Khraeif, and S. Belghith, 2012b Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot. Communications in Nonlinear Science and Numerical Simulation 17: 4356–4372.
  • Gritli, H., N. Khraeif, and S. Belghith, 2013 Chaos control in passive walking dynamics of a compass-gait model. Communications in Nonlinear Science and Numerical Simulation 18: 2048–2065.
  • Gritli, H., N. Khraief, and S. Belghith, 2018 Complex walking behaviours, chaos and bifurcations of a simple passive compass-gait biped model suffering from leg length asymmetry. International Journal of Simulation and Process Modelling 13: 446–462.
  • Grizzle, J.W., G. Abba, and F. Plestan, 2001 Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Transaction on Automatic Control 46: 51–64.
  • Grizzle, J. W., C. Chevallereau, R. W. Sinnet, and A. D. Ames, 2014 Models, feedback control, and open problems of 3d bipedal robotic walking. Automatica 50: 1955–1988.
  • Guanrong, C., 2021 Chaos theory and applications: a new trend. Chaos Theory and Applications 3: 1–2.
  • Harrison, R. C., A. OLDAG, E. PERK˙INS, et al., 2022 Experimental validation of a chaotic jerk circuit based true random number generator. Chaos Theory and Applications 4: 64–70.
  • Huang, Y., Q. Huang, and Q. Wang, 2016 Chaos and bifurcation control of torque-stiffness-controlled dynamic bipedal walking. IEEE Transactions on Systems, Man, and Cybernetics: Systems 47: 1229–1240.
  • Iqbal, S., X. Z. Zang, Y. H. Zhu, and J. Zhao, 2014 Bifurcations and chaos in passive dynamic walking: A review. Robotics and Autonomous Systems 62: 889–909.
  • Jimenez, A., E. N. Sanchez, G. Chen, and J. P. Perez, 2009 Realtime chaotic circuit stabilization via inverse optimal control. International Journal of Circuit Theory and Applications 37: 887–898.
  • Jun, M., 2022 Chaos theory and applications: the physical evidence, mechanism are important in chaotic systems. Chaos Theory and Applications 4: 1–3.
  • Khraief Haddad, N., S. Belghith, H. Gritli, and A. Chemori, 2017 From hopf bifurcation to limit cycles control in underactuated mechanical systems. International Journal of Bifurcation and Chaos 27: 1750104.
  • Kuo, A. D., 2007 The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective. Human Movement Science 26: 617 – 656.
  • Li, T.-Y. and J. A. Yorke, 2004 Period three implies chaos. In The theory of chaotic attractors, pp. 77–84, Springer.
  • Lorenz, E. N., 1960 Energy and numerical weather prediction. Tellus 12: 364–373.
  • Makarenkov, O., 2020 Existence and stability of limit cycles in the model of a planar passive biped walking down a slope. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476: 20190450.
  • McGeer, T., 1990 Passive dynamic walking. International Journal of Robotics Research 9: 62–82.
  • Meng, M. Q.-H. and R. Song, 2022 Legged mobile robots for challenging terrains. Biomimetic Intelligence and Robotics 2: 100034.
  • Miladi, Y., A. Chemori, and M. Feki, 16-19 March 2015 The compass-like biped robot revisited: Nonlinear control of the disturbed passive dynamic walking. In 2015 IEEE 12th International Multi-Conference on Systems, Signals Devices (SSD15), pp. 1–7, Mahdia, Tunisia.
  • Miladi, Y., N. Derbel, and M. Feki, 2021 Optimal control based on multiple models approach of chaotic switched systems, application to a stepper motor. International Journal of Automation and Control 15: 240–258.
  • Montazeri Moghadam, S., M. Sadeghi Talarposhti, A. Niaty, F. Towhidkhah, and S. Jafari, 2018 The simple chaotic model of passive dynamic walking. Nonlinear Dynamics 93: 1183–1199.
  • Nourian Zavareh, M., F. Nazarimehr, K. Rajagopal, and S. Jafari, 2018 Hidden attractor in a passive motion model of compass-gait robot. International Journal of Bifurcation and Chaos 28: 1850171.
  • Ott, E., C. Grebogi, and J. A. Yorke, 1990 Controlling chaos. Physical review letters 64: 1196–1199.
  • Reher, J. and A. D. Ames, 2021 Dynamic walking: Toward agile and efficient bipedal robots. Annual Review of Control, Robotics, and Autonomous Systems 4: 535–572.
  • Sambas, A., S. Vaidyanathan, M. Mamat, W. Sanjaya, and D. S. Rahayu, 2016 A 3-d novel jerk chaotic system and its application in secure communication system and mobile robot navigation. In Advances and applications in Chaotic systems, pp. 283–310, Springer.
  • Sprott, J., 2020 Do we need more chaos examples? Chaos Theory and Applications 2: 49 – 51.
  • Taghvaei, S. and R. Vatankhah, 2016 Detection of unstable periodic orbits and chaos control in a passive biped model. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering 40: 303–313.
  • Taylor, R. H., 2006 A perspective on medical robotics. Proceedings of the IEEE 94: 1652–1664.
  • Vaidyanathan, S., M. Feki, A. Sambas, and C.-H. Lien, 2018 A new biological snap oscillator: its modelling, analysis, simulations and circuit design. International Journal of Simulation and Process Modelling 13: 419–432.
  • Vaidyanathan, S., A. Sambas, M. Mamat, and W. M. Sanjaya, 2017 A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot. Archives of Control Sciences 27: 541–554.
  • Volos, C., A. Akgul, V.-T. Pham, I. Stouboulos, and I. Kyprianidis, 2017 A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme. Nonlinear Dynamics 89: 1047–1061.
  • Volos, C. K., I. M. Kyprianidis, and I. N. Stouboulos, 2012 A chaotic path planning generator for autonomous mobile robots. Robotics and Autonomous Systems 60: 651–656.
  • Volos, C. K., I. M. Kyprianidis, and I. N. Stouboulos, 2013 Experimental investigation on coverage performance of a chaotic autonomous mobile robot. Robotics and Autonomous Systems 61: 1314–1322.
  • Walter, S., 2014 Poincaré on clocks in motion. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 47: 131–141.
  • Westervelt, E. R., J. W. Grizzle, C. Chevallereau, J.-H. Choi, and B. Morris, 2007 Feedback control of dynamic bipedal robot locomotion. Taylor & Francis/CRC, London.
  • Xiaoting, Y., Y. Liguo, and W. Zhouchao, 2022 Stability and hopf bifurcation analysis of a fractional-order leslie-gower prey-predator-parasite system with delay. Chaos Theory and Applications 4: 71–81.
  • Yang, D. and J. Zhou, 2014 Connections among several chaos feedback control approaches and chaotic vibration control of mechanical systems. Communications in Nonlinear Science and Numerical Simulation 19: 3954–3968.
  • Yang, X., H. She, H. Lu, T. Fukuda, and Y. Shen, 2017 State of the art: Bipedal robots for lower limb rehabilitation. Artificial Life and Robotics 7: 1182.
  • Znegui,W., H. Gritli, and S. Belghith, 2020a Design of an explicit expression of the Poincaré map for the passive dynamic walking of the compass-gait biped model. Chaos, Solitons & Fractals 130: 109436.
  • Znegui, W., H. Gritli, and S. Belghith, 2020b Stabilization of the passive walking dynamics of the compass-gait biped robot by developing the analytical expression of the controlled Poincaré map. Nonlinear Dynamics 101: 1061–1091.
  • Znegui, W., H. Gritli, and S. Belghith, 2021 A new Poincaré map for analysis of complex walking behavior of the compass-gait biped robot. Applied Mathematical Modelling 94: 534–557.
  • Öztürk, H., 2020 A novel chaos application to observe performance of asynchronous machine under chaotic load. Chaos Theory and Applications 2: 90 – 97.

Year 2022, Volume 4, Issue 4, 246 - 266, 31.12.2022
https://doi.org/10.51537/chaos.1187427

Abstract

References

  • Added, E. and H. Gritli, 2020a Control of the passive dynamic gait of the bipedal compass-type robot through trajectory tracking. In 2020 20th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), pp. 155–162.
  • Added, E. and H. Gritli, 2020b Trajectory design and tracking-based control of the passive compass biped. In 2020 4th International Conference on Advanced Systems and Emergent Technologies (IC_ASET), pp. 417–424.
  • Added, E. and H. Gritli, 2022 Birth of the Neimark–Sacker bifurcation for the passive compass-gait walker. In Advances in Nonlinear Dynamics, edited by W. Lacarbonara, B. Balachandran, M. J. Leamy, J. Ma, J. A. Tenreiro Machado, and G. Stepan, pp. 683–697, Cham, Springer International Publishing.
  • Added, E. and H. Gritli, 2023 A further analysis of the passive compass-gait bipedal robot and its period-doubling route to chaos. In New Perspectives on Nonlinear Dynamics and Complexity, edited by D. Volchenkov and A. C. J. Luo, pp. 11–30, Cham, Springer International Publishing.
  • Added, E., H. Gritli, and S. Belghith, 2021a Additional complex behaviors, bifurcations and chaos, in the passive walk of the compass-type bipedal robot. IFAC-PapersOnLine 54: 111–116, 6th IFAC Conference on Analysis and Control of Chaotic Systems CHAOS 2021.
  • Added, E., H. Gritli, and S. Belghith, 2021b Further analysis of the passive dynamics of the compass biped walker and control of chaos via two trajectory tracking approaches. Complexity 2021: 5533451 (39 pages).
  • Added, E., H. Gritli, and S. Belghith, 2021c Further analysis of the passive walking gaits of the compass biped robot: Bifurcations and chaos. In 2021 18th International Multi-Conference on Systems, Signals & Devices (SSD), pp. 160–165.
  • Added, E., H. Gritli, and S. Belghith, 2022a Trajectory tracking-based control of the chaotic behavior in the passive bipedal compass-type robot. The European Physical Journal Special Topics 231: 1071–1084.
  • Added, E., H. Gritli, and S. Belghith, 2022b Trajectory tracking control of the compass-type bipedal robot gait via an improved PD+ controller. In 2022 5th International Conference on Advanced Systems and Emergent Technologies (IC_ASET), pp. 482–488.
  • Akgul, A., S. Hussain, and I. Pehlivan, 2016 A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik - International Journal for Light and Electron Optics 127: 7062 – 7071.
  • Andrievskii, B. R. and A. L. Fradkov, 2003 Control of chaos: Methods and applications. I. Methods. Automation and Remote Control 64: 673–713.
  • Andrievskii, B. R. and A. L. Fradkov, 2004 Control of chaos: Methods and applications. II. Applications. Automation and Remote Control 65: 505–533.
  • Aricioglu, M. A. and O. N. Berk, 2022 A comparative proposal on learning the chaos to understand the environment. Chaos Theory and Applications 4: 19 – 25.
  • Azar, A. T., C. Volos, N. A. Gerodimos, G. S. Tombras, V.-T. Pham, et al., 2017 A novel chaotic system without equilibrium: Dynamics, synchronization, and circuit realization. Complexity 2017: 7871467.
  • Bekey, G. A. and K. Y. Goldberg, 2012 Neural networks in robotics, volume 202. Springer Science & Business Media.
  • Beritelli, F., E. Di Cola, L. Fortuna, and F. Italia, 2000 Multilayer chaotic encryption for secure communications in packet switching networks. In WCC 2000-ICCT 2000. 2000 International Conference on Communication Technology Proceedings (Cat. No. 00EX420), volume 2, pp. 1575–1582.
  • Boccaletti, S., C. Grebogi, Y.-C. Lai, H. Mancini, and D. Maza, 2000 The control of chaos: theory and applications. Physics Reports 329: 103–197.
  • Boubaker, O. and S. Jafari, 2019 Recent Advances in Chaotic Systems and Synchronization: From Theory to Real World Applications. Emerging Methodologies and Applications in Modelling, Identification and Control, Elsevier, first edition.
  • Breazeal, C. and B. Scassellati, 2002 Robots that imitate humans. Trends in cognitive sciences 6: 481–487.
  • Buscarino, A., C. Famoso, L. Fortuna, and M. Frasca, 2016 A new chaotic electro-mechanical oscillator. International Journal of Bifurcation and Chaos 26: 1650161.
  • Chevallereau, C., G. Bessonnet, G. Abba, and Y. Aoustin, 2009 Bipedal Robots: Modeling, Design and Walking Synthesis. JohnWiley & Sons, Wiley-ISTE, first edition.
  • Collins, S., A. Ruina, R. Tedrake, and M. Wisse, 2005 Efficient bipedal robots based on passive-dynamic walkers. Science 307: 1082–1085.
  • Croce, U. D., P. O. Riley, J. L. Lelas, and D. Kerrigan, 2001 A refined view of the determinants of gait. Gait & Posture 14: 79 – 84.
  • da Costa Barros, I. R. and T. P. Nascimento, 2021 Robotic mobile fulfillment systems: A survey on recent developments and research opportunities. Robotics and Autonomous Systems 137: 103729.
  • Deng, K., M. Zhao, and W. Xu, 2016 Level-ground walking for a bipedal robot with a torso via hip series elastic actuators and its gait bifurcation control. Robotics and Autonomous Systems 79: 58–71.
  • Deng, K., M. Zhao, and W. Xu, 2017a Bifurcation gait suppression of a bipedal walking robot with a torso based on model predictive control. Robotics and Autonomous Systems 89: 27–39.
  • Deng, K., M. Zhao, and W. Xu, 2017b Passive dynamic walking with a torso coupled via torsional springs. International Journal of Humanoid Robotics 13: 1650024.
  • Falco, J. A., J. A. Marvel, and R. J. Norcross, 2012 Collaborative robotics: Measuring blunt force impacts on humans. Chest 140: 45.
  • Fathizadeh, M., H. Mohammadi, and S. Taghvaei, 2019 A modified passive walking biped model with two feasible switching patterns of motion to resemble multi-pattern human walking. Chaos, Solitons & Fractals 127: 83 – 95.
  • Fathizadeh, M., S. Taghvaei, and H. Mohammadi, 2018 Analyzing bifurcation, stability and chaos for a passive walking biped model with a sole foot. International Journal of Bifurcation and Chaos 28: 1850113.
  • Ferreira, B. B., A. S. de Paula, and M. A. Savi, 2011 Chaos control applied to heart rhythm dynamics. Chaos, Solitons & Fractals 44: 587–599.
  • Firth, W., 1991 Chaos–predicting the unpredictable. BMJ: British Medical Journal 303: 1565.
  • Fradkov, A. L. and R. J. Evans, 2005 Control of chaos: Methods and applications in engineering. Annual Reviews in Control 29: 33–56.
  • Fradkov, A. L., R. J. Evans, and B. R. Andrievsky, 2006 Control of chaos: Methods and applications in mechanics. Philosophical Transactions of The Royal Society A 364: 2279–2307.
  • Garcia, M., A. Chatterjee, and A. Ruina, 2000 Efficiency, speed, and scaling of two-dimensional passive-dynamic walking. Dynamics and Stability of Sytems 15: 75–99.
  • Garcia, M., A. Chatterjee, A. Ruina, and M. Coleman, 1998 The simplest walking model: Stability, complexity, and scaling. Journal of Biomechanical Engineering 120: 281–288.
  • Goswami, A., B. Espiau, and A. Keramane, 1997 Limit cycles in a passive compass gait biped and passivity-mimicking control laws. Autonomous Robots 4: 273–286.
  • Goswami, A., B. Thuilot, and B. Espiau, 1996 Compass-like biped robot. Part I: Stability and bifurcation of passive gaits, volume 2996. Technical Report, INRIA.
  • Goswami, A., B. Thuilot, and B. Espiau, 1998 Study of the passive gait of a compass-like biped robot: Symmetry and chaos. International Journal of Robotics Research 17: 1282–1301.
  • Goswami, A. and P. Vadakkepat, 2019 Humanoid Robotics: A Reference. Springer Netherlands, first edition.
  • Grebogi, C., Y.-C. Lai, and S. Hayes, 1997 Control and applications of chaos. Journal of the Franklin Institute 334: 1115–1146, Visions of Nonlinear Mechanics in the 21st Century.
  • Gritli, H. and S. Belghith, 2017aWalking dynamics of the passive compass-gait model under OGY-based control: Emergence of bifurcations and chaos. Communications in Nonlinear Science and Numerical Simulation 47: 308–327.
  • Gritli, H. and S. Belghith, 2017b Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Analysis of local bifurcations via the hybrid poincaré map. Chaos, Solitons & Fractals 98: 72 – 87.
  • Gritli, H. and S. Belghith, 2018a Diversity in the nonlinear dynamic behavior of a one-degree-of-freedom impact mechanical oscillator under ogy-based state-feedback control law: order, chaos and exhibition of the border-collision bifurcation. Mechanism and Machine Theory 124: 1–41.
  • Gritli, H. and S. Belghith, 2018bWalking dynamics of the passive compass-gait model under OGY-based state-feedback control: Rise of the Neimark–Sacker bifurcation. Chaos, Solitons & Fractals 110: 158 – 168.
  • Gritli, H., S. Belghith, and N. Khraeif, 2012a Intermittency and interior crisis as route to chaos in dynamic walking of two biped robots. International Journal of Bifurcation and Chaos 22: 1250056.
  • Gritli, H., S. Belghith, and N. Khraeif, 2015 OGY-based control of chaos in semi-passive dynamic walking of a torso-driven biped robot. Nonlinear Dynamics 79: 1363–1384.
  • Gritli, H., N. Khraeif, and S. Belghith, 2012b Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot. Communications in Nonlinear Science and Numerical Simulation 17: 4356–4372.
  • Gritli, H., N. Khraeif, and S. Belghith, 2013 Chaos control in passive walking dynamics of a compass-gait model. Communications in Nonlinear Science and Numerical Simulation 18: 2048–2065.
  • Gritli, H., N. Khraief, and S. Belghith, 2018 Complex walking behaviours, chaos and bifurcations of a simple passive compass-gait biped model suffering from leg length asymmetry. International Journal of Simulation and Process Modelling 13: 446–462.
  • Grizzle, J.W., G. Abba, and F. Plestan, 2001 Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Transaction on Automatic Control 46: 51–64.
  • Grizzle, J. W., C. Chevallereau, R. W. Sinnet, and A. D. Ames, 2014 Models, feedback control, and open problems of 3d bipedal robotic walking. Automatica 50: 1955–1988.
  • Guanrong, C., 2021 Chaos theory and applications: a new trend. Chaos Theory and Applications 3: 1–2.
  • Harrison, R. C., A. OLDAG, E. PERK˙INS, et al., 2022 Experimental validation of a chaotic jerk circuit based true random number generator. Chaos Theory and Applications 4: 64–70.
  • Huang, Y., Q. Huang, and Q. Wang, 2016 Chaos and bifurcation control of torque-stiffness-controlled dynamic bipedal walking. IEEE Transactions on Systems, Man, and Cybernetics: Systems 47: 1229–1240.
  • Iqbal, S., X. Z. Zang, Y. H. Zhu, and J. Zhao, 2014 Bifurcations and chaos in passive dynamic walking: A review. Robotics and Autonomous Systems 62: 889–909.
  • Jimenez, A., E. N. Sanchez, G. Chen, and J. P. Perez, 2009 Realtime chaotic circuit stabilization via inverse optimal control. International Journal of Circuit Theory and Applications 37: 887–898.
  • Jun, M., 2022 Chaos theory and applications: the physical evidence, mechanism are important in chaotic systems. Chaos Theory and Applications 4: 1–3.
  • Khraief Haddad, N., S. Belghith, H. Gritli, and A. Chemori, 2017 From hopf bifurcation to limit cycles control in underactuated mechanical systems. International Journal of Bifurcation and Chaos 27: 1750104.
  • Kuo, A. D., 2007 The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective. Human Movement Science 26: 617 – 656.
  • Li, T.-Y. and J. A. Yorke, 2004 Period three implies chaos. In The theory of chaotic attractors, pp. 77–84, Springer.
  • Lorenz, E. N., 1960 Energy and numerical weather prediction. Tellus 12: 364–373.
  • Makarenkov, O., 2020 Existence and stability of limit cycles in the model of a planar passive biped walking down a slope. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476: 20190450.
  • McGeer, T., 1990 Passive dynamic walking. International Journal of Robotics Research 9: 62–82.
  • Meng, M. Q.-H. and R. Song, 2022 Legged mobile robots for challenging terrains. Biomimetic Intelligence and Robotics 2: 100034.
  • Miladi, Y., A. Chemori, and M. Feki, 16-19 March 2015 The compass-like biped robot revisited: Nonlinear control of the disturbed passive dynamic walking. In 2015 IEEE 12th International Multi-Conference on Systems, Signals Devices (SSD15), pp. 1–7, Mahdia, Tunisia.
  • Miladi, Y., N. Derbel, and M. Feki, 2021 Optimal control based on multiple models approach of chaotic switched systems, application to a stepper motor. International Journal of Automation and Control 15: 240–258.
  • Montazeri Moghadam, S., M. Sadeghi Talarposhti, A. Niaty, F. Towhidkhah, and S. Jafari, 2018 The simple chaotic model of passive dynamic walking. Nonlinear Dynamics 93: 1183–1199.
  • Nourian Zavareh, M., F. Nazarimehr, K. Rajagopal, and S. Jafari, 2018 Hidden attractor in a passive motion model of compass-gait robot. International Journal of Bifurcation and Chaos 28: 1850171.
  • Ott, E., C. Grebogi, and J. A. Yorke, 1990 Controlling chaos. Physical review letters 64: 1196–1199.
  • Reher, J. and A. D. Ames, 2021 Dynamic walking: Toward agile and efficient bipedal robots. Annual Review of Control, Robotics, and Autonomous Systems 4: 535–572.
  • Sambas, A., S. Vaidyanathan, M. Mamat, W. Sanjaya, and D. S. Rahayu, 2016 A 3-d novel jerk chaotic system and its application in secure communication system and mobile robot navigation. In Advances and applications in Chaotic systems, pp. 283–310, Springer.
  • Sprott, J., 2020 Do we need more chaos examples? Chaos Theory and Applications 2: 49 – 51.
  • Taghvaei, S. and R. Vatankhah, 2016 Detection of unstable periodic orbits and chaos control in a passive biped model. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering 40: 303–313.
  • Taylor, R. H., 2006 A perspective on medical robotics. Proceedings of the IEEE 94: 1652–1664.
  • Vaidyanathan, S., M. Feki, A. Sambas, and C.-H. Lien, 2018 A new biological snap oscillator: its modelling, analysis, simulations and circuit design. International Journal of Simulation and Process Modelling 13: 419–432.
  • Vaidyanathan, S., A. Sambas, M. Mamat, and W. M. Sanjaya, 2017 A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot. Archives of Control Sciences 27: 541–554.
  • Volos, C., A. Akgul, V.-T. Pham, I. Stouboulos, and I. Kyprianidis, 2017 A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme. Nonlinear Dynamics 89: 1047–1061.
  • Volos, C. K., I. M. Kyprianidis, and I. N. Stouboulos, 2012 A chaotic path planning generator for autonomous mobile robots. Robotics and Autonomous Systems 60: 651–656.
  • Volos, C. K., I. M. Kyprianidis, and I. N. Stouboulos, 2013 Experimental investigation on coverage performance of a chaotic autonomous mobile robot. Robotics and Autonomous Systems 61: 1314–1322.
  • Walter, S., 2014 Poincaré on clocks in motion. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 47: 131–141.
  • Westervelt, E. R., J. W. Grizzle, C. Chevallereau, J.-H. Choi, and B. Morris, 2007 Feedback control of dynamic bipedal robot locomotion. Taylor & Francis/CRC, London.
  • Xiaoting, Y., Y. Liguo, and W. Zhouchao, 2022 Stability and hopf bifurcation analysis of a fractional-order leslie-gower prey-predator-parasite system with delay. Chaos Theory and Applications 4: 71–81.
  • Yang, D. and J. Zhou, 2014 Connections among several chaos feedback control approaches and chaotic vibration control of mechanical systems. Communications in Nonlinear Science and Numerical Simulation 19: 3954–3968.
  • Yang, X., H. She, H. Lu, T. Fukuda, and Y. Shen, 2017 State of the art: Bipedal robots for lower limb rehabilitation. Artificial Life and Robotics 7: 1182.
  • Znegui,W., H. Gritli, and S. Belghith, 2020a Design of an explicit expression of the Poincaré map for the passive dynamic walking of the compass-gait biped model. Chaos, Solitons & Fractals 130: 109436.
  • Znegui, W., H. Gritli, and S. Belghith, 2020b Stabilization of the passive walking dynamics of the compass-gait biped robot by developing the analytical expression of the controlled Poincaré map. Nonlinear Dynamics 101: 1061–1091.
  • Znegui, W., H. Gritli, and S. Belghith, 2021 A new Poincaré map for analysis of complex walking behavior of the compass-gait biped robot. Applied Mathematical Modelling 94: 534–557.
  • Öztürk, H., 2020 A novel chaos application to observe performance of asynchronous machine under chaotic load. Chaos Theory and Applications 2: 90 – 97.

Details

Primary Language English
Subjects Engineering, Multidisciplinary
Journal Section Research Articles
Authors

Essia ADDED>
National Engineering School of Tunis
0000-0002-0680-2755
Tunisia


Hassène GRİTLİ> (Primary Author)
Higher Institute of Information and Communication Technologies
0000-0002-5643-134X
Tunisia


Safya BELGHİTH>
National Engineering School of Tunis
0000-0001-7408-7848
Tunisia

Publication Date December 31, 2022
Published in Issue Year 2022, Volume 4, Issue 4

Cite

APA Added, E. , Gritli, H. & Belghith, S. (2022). Occurrence of Complex Behaviors in the Uncontrolled Passive Compass Biped Model . Chaos Theory and Applications , Dissemination and Research in the Study of Complex Systems and Their Applications (EDIESCA 2022) , 246-266 . DOI: 10.51537/chaos.1187427

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 23830