Skip to main content
SpringerLink
Log in

Identification, Stability and Stabilization of Limit Cycles in a Compass-Gait Biped Model via a Hybrid Poincaré Map

  • Chapter
  • First Online:
Advances and Applications in Nonlinear Control Systems

Part of the book series: Studies in Computational Intelligence ((SCI,volume 635))

Abstract

This chapter focuses on identification, stability analysis and stabilization of hybrid limit cycle in the passive dynamic walking of the compass-gait biped robot as it goes down an inclined surface. The walking dynamics of such biped is described by an impulsive hybrid nonlinear system, which is composed of a nonlinear differential equation and a nonlinear algebraic equation. Under variation of the slope parameter, the passive biped robot displays symmetric (stable one-periodic) and asymmetric (unstable one-periodic and chaotic) behaviors. Then, the main objective of this chapter is to stabilize a desired asymmetric gait into a symmetric one by means of a control input, the hip torque. Nevertheless, the design of such control input using the impulsive hybrid dynamics is a complicated task. Then, to overcome this problem, we constructed a hybrid Poincaré map by linearizing the impulsive hybrid nonlinear dynamics around a desired unstable one-periodic hybrid limit cycle for some desired slope parameter. We stress that both the differential equation and the algebraic equation are linearized. The desired hybrid limit cycle is identified and analyzed first through the impulsive hybrid nonlinear dynamics via the fundamental solution matrix. We demonstrate that identification of the one-periodic fixed point of the designed hybrid Poincaré map and its stability depend only upon the nominal impact instant. We introduce a state-feedback controller in order to stabilize the linearized Poincaré map around the one-periodic fixed point. We show that the developed strategy for the design of the OGY-based controller has achieved the stabilization of the desired one-periodic hybrid limit cycle of the compass-gait biped robot.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Asano F (2015) Stability analysis of underactuated compass gait based on linearization of motion. Multibody Syst Dyn 33(1):93–111

    Article  MathSciNet  MATH  Google Scholar 

  2. Asano F, Luo Z-W, Yamakita M (2005) Biped gait generation and control based on a unified property of passive dynamic walking. IEEE Trans Robot 21(4):754–762

    Article  Google Scholar 

  3. Bououden S, Abdessemed F (2014) Walking control for a planar biped robot using 0-flat normal form. Robot Auton Syst 62:68–80

    Article  Google Scholar 

  4. Byl K, Tedrake R (1998) Intuitive control of a planar bipedal walking robot. In: Proceedings of the IEEE international conference on robotics and automation, pp 2014–2021

    Google Scholar 

  5. Byl K, Tedrake R (2008) Approximate optimal control of the compass gait on rough terrain. In: Proceedings of the IEEE international conference on robotics and automation, pp 1258–1263

    Google Scholar 

  6. Chaillet N, Abba G, Ostertag E (1994) Double dynamic modeling and computed torque control of a biped robot. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, pp 1149–1153

    Google Scholar 

  7. Chemori A (2009) A discrete-time control strategy for dynamic walking of a planar under-actuated biped robot. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, pp 3226–3231

    Google Scholar 

  8. Chemori A, Alamir M (2006) Multi-step limit cycle generation for rabbit’s walking based on a nonlinear low dimensional predictive control scheme. Mechatronics 16(5):259–277

    Article  Google Scholar 

  9. Chemori A, Loria A (2004) Control of a planar under-actuated biped on a complete walking cycle. IEEE Trans Autom Control 49(5):838–843

    Article  MathSciNet  Google Scholar 

  10. Cheng M-Y, Lin C-S (1996) Measurement of robustness for biped locomotion using a linearized poincaré map. Robolica 14:253–259

    Google Scholar 

  11. Chevallereau C (2003) Time-scaling control for an underactuated biped robot. IEEE Trans Robot Autom 19(2):362–368

    Article  Google Scholar 

  12. Chevallereau C, Djoudi D, Grizzle JW (2008) Stable bipedal walking with foot rotation through direct regulation of the zero moment point. IEEE Trans Robot 24(2):390–401

    Article  Google Scholar 

  13. Dardel M, Safartoobi M, Pashaei MH, Ghasemi MH, Navaei MK (2015) Finite difference method to find period-one gait cycles of simple passive walkers. Commun Nonlinear Sci Numer Simul 20(1):79–97

    Article  Google Scholar 

  14. Ding CT, Yang SX, Gan CB (2013) Input torque sensitivity to uncertain parameters in biped robot. Acta Mech Sin 29(3):452–461

    Article  MathSciNet  Google Scholar 

  15. Donde V, Hiskens IA (2006) Shooting methods for locating grazing phenomena in hybrid systems. Int J Bifurc Chaos 16(3):671–692

    Article  MathSciNet  MATH  Google Scholar 

  16. Dong H, Zhao M, Zhang N (2011) High-speed and energy-efficient biped locomotion based on virtual slope walking. Auton Robots 30(2):199–216

    Article  Google Scholar 

  17. Farshimi F, Naraghi M (2011) A passive-biped model with multiple routes to chaos. Acta Mech Sin 27(2):277–284

    Article  MathSciNet  MATH  Google Scholar 

  18. Flieller D, Riedinger P, Louis JP (2006) Computation and stability of limit cycles in hybrid systems. Nonlinear Anal: Theory Methods Appl 64(2):352–367

    Article  MathSciNet  MATH  Google Scholar 

  19. Freidovich LB, Shiriaev AS, Manchester IR (2008) Stability analysis and control design for an underactuated walking robot via computation of a transverse linearization. In: Proceedings of the 17th IFAC world congress, pp 10166–10171

    Google Scholar 

  20. Goswami A, Thuilot B, Espiau B (1998) Study of the passive gait of a compass-like biped robot: Symmetry and chaos. Int J Robot Res 17:1282–1301

    Article  Google Scholar 

  21. Gritli H (2014) Analyse et Contrôle du Chaos en Robotique: Cas des Robots Bipèdes Planaires. PhD thesis, Ecole Nationale d’Ingénieurs de Tunis, Tunisia

    Google Scholar 

  22. Gritli H (2015) Analyse et Contrôle du Chaos dans les Systèmes Mécaniques Impulsifs. Presses Académiques Francophones, Saarbrücken, Germany, Cas des Oscillateurs avec Impact et des Robots Bipèdes Planaires

    Google Scholar 

  23. Gritli H, Belghith S (2015) Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map. Chaos Solitons Fractals 81:172–183

    Article  MathSciNet  Google Scholar 

  24. Gritli H, Belghith S, Khraeif N (2012) Cyclic fold bifurcation and boundary crisis in dynamic walking of biped robots. Int J Bifurc Chaos 22(10):1250257. doi:10.1142/S0218127412502574

    Google Scholar 

  25. Gritli H, Belghith S, Khraeif N (2012) Intermittency and interior crisis as route to chaos in dynamic walking of two biped robots. Int J Bifurc Chaos 22(3):1250056. doi:10.1142/S0218127412500563

    Google Scholar 

  26. Gritli H, Belghith S, Khraeif N (2015a) OGY-based control of chaos in semi-passive dynamic walking of a torso-driven biped robot. Nonlinear Dyn 79(2):1363–1384

    Article  MATH  Google Scholar 

  27. Gritli H, Khraeif N, Belghith S (2011) Cyclic-fold bifurcation in passive bipedal walking of a compass-gait biped robot with leg length discrepancy. In: Proceedings of the IEEE international conference on mechatronics, pp 851–856

    Google Scholar 

  28. Gritli H, Khraeif N, Belghith S (2011) Falling of a passive compass-gait biped robot caused by a boundary crisis. In: Proceedings of the 4th chaotic modeling and simulation international conference, pp 155–162

    Google Scholar 

  29. Gritli H, Khraeif N, Belghith S (2011) Semi-passive control of a torso-driven compass-gait biped robot: Bifurcation and chaos. In: Proceedings of the international multi-conference on systems, signals and devices, pp 1–6

    Google Scholar 

  30. Gritli H, Khraeif N, Belghith S (2012c) Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot. Commun Nonlinear Sci Numer Simul 17(11):4356–4372

    Article  MathSciNet  MATH  Google Scholar 

  31. Gritli H, Khraeif N, Belghith S (2013) Chaos control in passive walking dynamics of a compass-gait model. Commun Nonlinear Sci Numer Simul 18(8):2048–2065

    Article  MathSciNet  MATH  Google Scholar 

  32. Gritli H, Khraeif N, Belghith S (2014) Further analysis of the period-three route to chaos in passive dynamic walking of a compass-gait biped robot. In: Proceedings of the copyright IPCO-2014, pp 123–130

    Google Scholar 

  33. Gritli H, Khraeif N, Belghith S (2015) Handbook of research on advanced intelligent control engineering and automation, chapter Further investigation of the period-three route to chaos in the passive compass-gait biped model. Advances in computational intelligence and robotics (ACIR). IGI Global, USA, pp 279–300

    Google Scholar 

  34. Grizzle JW, Abba G, Plestan F (2001) Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Trans Autom Control 46:51–64

    Article  MathSciNet  MATH  Google Scholar 

  35. Hardt M, Kreutz-Delgado K, Helton J (1998) Minimal energy control of a biped robot with numerical methods and a recursive symbolic dynamic model. In: Proceedings of the IEEE international conference on decision and control, pp 413–416

    Google Scholar 

  36. Hiskens IA (2001) Stability of hybrid system limit cycles: application to the compass gait biped robot. In: Proceedings of the IEEE international conference on decision control, pp 774–779

    Google Scholar 

  37. Hiskens IA, Pai MA (2000) Trajectory sensitivity analysis of hybrid systems. IEEE Trans Circuits Syst I 47:204–220

    Article  Google Scholar 

  38. Holm JK, Spong MW (2008) Kinetic energy shaping for gait regulation of underactuated bipeds. In Proceedings of the IEEE International conference on control applications, part of the IEEE multi-conference on systems and control, pp 1232–1238

    Google Scholar 

  39. Hu Y, Yan G, Lin Z (2011a) Feedback control of planar biped robots with regulable step length and walking speed. IEEE Trans Robot 27(1):162–169

    Article  Google Scholar 

  40. Hu Y, Yan G, Lin Z (2011b) Gait generation and control for biped robots with underactuation degree one. Automatica 47(8):1605–1616

    Article  MathSciNet  MATH  Google Scholar 

  41. Huang Y, Wang Q-N, Gao Y, Xie G-M (2012) Modeling and analysis of passive dynamic bipedal walking with segmented feet and compliant joints. Acta Mech Sin 28(3):1457–1465

    Article  MathSciNet  Google Scholar 

  42. Iida F, Tedrake R (2010) Minimalistic control of biped walking in rough terrain. Auton Robots 28(3):355–368

    Article  Google Scholar 

  43. Ijspeert AJ (2008) Central pattern generators for locomotion control in animals and robots: A review. Neural Netw 21(4):642–653

    Article  Google Scholar 

  44. Iqbal S, Zang XZ, Zhu YH, Zhao J (2014) Bifurcations and chaos in passive dynamic walking: A review. Robot Auton Syst 62(6):889–909

    Article  Google Scholar 

  45. Katič D, Vukobratovič M (2003) Survey of intelligent control techniques for humanoid robots. J Intell Rob Syst 37(2):117–141

    Article  Google Scholar 

  46. Kim Y-D, Lee B-J, Ryu J-H, Kim J-H (2007) Landing force control for humanoid robot by time-domain passivity approach. IEEE Trans Robot 23(6):1294–1301

    Article  Google Scholar 

  47. Lee JH, Okamoto S, Koike H, Tani K (2014) Development and motion control of a biped walking robot based on passive walking theory. Artif Life Robot 19(1):68–75

    Article  Google Scholar 

  48. Li Q, Guo J, Yang XS (2013) New bifurcations in the simplest passive walking model. Chaos: Interdiscip J Nonlinear Sci 23:043110

    Article  MathSciNet  MATH  Google Scholar 

  49. Li Q, Yang XS (2012) New walking dynamics in the simplest passive bipedal walking model. Appl Math Model 36(11):5262–5271

    Article  MathSciNet  MATH  Google Scholar 

  50. Li Q, Yang XS (2014) Bifurcation and chaos in the simple passive dynamic walking model with upper body. Chaos: Interdiscip J Nonlinear Sci 24:033114

    Article  MathSciNet  Google Scholar 

  51. Liu N, Li J, Wang T (2008) Passive walker that can walk down steps: simulations and experiments. Acta Mech Sin 24:569–573

    Article  MATH  Google Scholar 

  52. Liu Z, Zhou C, Zhang P, Tian Y (2007) Robotic welding, intelligence and automation. Lecture notes in control and information sciences, chapter Anti-phase synchronization control scheme of passive biped robot, vol 362. Springer, pp 529–539

    Google Scholar 

  53. Luo X, Zhu L, Xia L (2015) Principle and method of speed control for dynamic walking biped robots. Robot Auton Syst 66:129–144

    Article  Google Scholar 

  54. Manchester IR (2011) Transverse dynamics and regions of stability for nonlinear hybrid limit cycles. In: Proceedings of the 18th IFAC world congress, pp 6285–6290

    Google Scholar 

  55. Manchester IR, Mettin U, Iida F, Tedrake R (2009) Stable dynamic walking over rough terrain: Theory and experiment. In: Proceedings of the international symposium on robotics research, pp 1–16

    Google Scholar 

  56. Manchester IR, Tobenkin MM, Levashov M, Tedrake R (2011) Regions of attraction for hybrid limit cycles of walking robots. In: Proceedings of the 18th IFAC world congress, pp 5801–5806

    Google Scholar 

  57. McGeer T (1990) Passive dynamic walking. Int J Robot Res 9(2):62–68

    Article  Google Scholar 

  58. McMahon T (1984) Mechanics of locomotion. Int J Robot Res 3(2):4–28

    Article  Google Scholar 

  59. Morimoto J, Atkeson CG (2007) Learning biped locomotion. IEEE Robot Autom Mag 14(2):41–51

    Article  Google Scholar 

  60. Morimoto J, Atkeson CG (2009) Nonparametric representation of an approximated poincaré map for learning biped locomotion. Auton Robots 27:131–144

    Article  Google Scholar 

  61. Morris B, Grizzle JW (2005) A restricted poincaré map for determining exponentially stable periodic orbits in systems with impulse effects: application to bipedal robots. In: Proceedings of the IEEE conference on decision and control, and the european control conference, pp 4199–4206

    Google Scholar 

  62. Muller PC (1995) Calculation of lyapunov exponents for dynamic systems with discontinuities. Chaos Solitons Fractals 5:1671–1681

    Article  MathSciNet  MATH  Google Scholar 

  63. Ning L, Junfeng L, Tianshu W (2009) The effects of parameter variation on the gaits of passive walking models: Simulations and experiments. Robotica 27(4):511–528

    Article  Google Scholar 

  64. Owaki D, Osuka K, Ishiguro A (2009) Understanding the common principle underlying passive dynamic walking and running. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, pp 3208–3213

    Google Scholar 

  65. Park JH (2001) Impedance control for biped robot locomotion. IEEE Trans Robot Autom 17(6):870–882

    Article  Google Scholar 

  66. Parker TS, Chua LO (1989) Practical numerical algorithms for chaotic systems. Springer, New York

    Book  MATH  Google Scholar 

  67. Qi F, Wang T, Li J (2011) The elastic contact influences on passive walking gaits. Robotica 29(5):787–796

    Article  Google Scholar 

  68. Safa AT, Alasty A, Naraghi M (2015) A different switching surface stabilizing an existing unstable periodic gait: An analysis based on perturbation theory. Nonlinear Dyn 81(4):2127–2140

    Article  MathSciNet  Google Scholar 

  69. Safa AT, Naraghi M (2015) The role of walking surface in enhancing the stability of the simplest passive dynamic biped. Robotica 33(1):195–207

    Article  Google Scholar 

  70. Safa AT, Saadat MG, Naraghi M (2007) Passive dynamic of the simplest walking model: Replacing ramps with stairs. Mech Mach Theory 42(10):1314–1325

    Article  MathSciNet  MATH  Google Scholar 

  71. Safartoobi M, Dardel M, Ghasemi MH, Daniali HM (2014) Stabilization and walking control for a simple passive walker using computed torque method. Int J Eng 27(11):1025–2495

    Google Scholar 

  72. Saglam CO, Byl K (2014) Stability and gait transition of the five-link biped on stochastically rough terrain using a discrete set of sliding mode controllers. In: Proceedings of the IEEE international conference on robotics and automation, pp 5655–5662

    Google Scholar 

  73. Shiriaev AS, Freidovich LB (2009) Transverse linearization for impulsive mechanical systems with one passive link. IEEE Trans Autom Control 54(12):2882–2888

    Article  MathSciNet  Google Scholar 

  74. Shiriaev AS, Freidovich LB, Manchester IR (2008) Periodic motion planning and analytical computation of transverse linearizations for hybrid mechanical systems. In: Proceedings of the IEEE conference on decision and control, pp 4326–4331

    Google Scholar 

  75. Song G, Zefran M (2006) Stabilization of hybrid periodic orbits with application to bipedal walking. In: Proceedings of the american control conference, pp 2504–2509

    Google Scholar 

  76. Spong MW, Bullo F (2005) Controlled symmetries and passive walking. IEEE Trans Autom Control 50:1025–1031

    Article  MathSciNet  Google Scholar 

  77. Spong MW, Holm JK, Dongjun L (2007) Passivity-based control of bipedal locomotion. IEEE Robot Autom Mag 14(2):30–40

    Article  Google Scholar 

  78. Sugimoto Y, Osuka K (2005) Stability analysis of passive-dynamic-walking focusing on the inner structure of Poincaré map. In: Proceedings of the international conference on advanced robotics, pp 236–241

    Google Scholar 

  79. Sugimoto Y, Osuka K (2007) Hierarchical implicit feedback structure in passive dynamic walking. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, pp 2217–2222

    Google Scholar 

  80. Tzafestas S, Raibert M, Tzafestas C (1996) Robust sliding-mode control applied to a 5-link biped robot. Int J Intell Robot Syst 15:67–133

    Article  Google Scholar 

  81. Vallejos P, del Solar JR, Swett F (2011) A new methodology for the design of passive biped robots: Determining conditions on the robot’s parameters for the existence of stable walking cycles. J Intell Rob Syst 63:503–523

    Article  Google Scholar 

  82. Westervelt ER, Grizzle JW, Canudas C (2003) Switching and pi control of walking motions of planar biped walkers. IEEE Trans Autom Control 48(2):308–312

    Article  MathSciNet  Google Scholar 

  83. Westervelt ER, Grizzle JW, Chevallereau C, Choi J-H, Morris B (2007a) Feedback control of dynamic bipedal robot locomotion. Taylor & Francis/CRC, London

    Book  Google Scholar 

  84. Westervelt ER, Morris B, Farrell KD (2007b) Analysis results and tools for the control of planar bipedal gaits using hybrid zero dynamics. Auton Robots 23:131–145

    Article  Google Scholar 

  85. Wisse M, van der Linde RQ (2007) Delft Pneumatic Bipeds. Springer, Berlin

    Book  Google Scholar 

  86. Wu B, Zhao M (2014) Bifurcation and chaos of a biped robot driven by coupled elastic actuation. In: Proceedings of the world congress on intelligent control and automation, pp 1905–1910

    Google Scholar 

  87. Yazdi EA, Aria A (2008) Stabilization of biped walking robot using the energy shaping method. J Comput Nonlinear Dyn 3(4):1–8

    Article  Google Scholar 

  88. Zhang P, Zhou C, Zhang L, Tian Y, Liu Z (2009) Adaptive compliant control of humanoid biped foot with elastic energy storage. In: Proceedings of the IEEE/ASME international conference on advanced intelligent mechatronics, pp 928–933

    Google Scholar 

  89. Zhao J, Wu X-G, Zang X-Z, Yan J-H (2012) Analysis of period doubling bifurcation and chaos mirror of biped passive dynamic robot gait. Chin Sci Bull 57(14):1743–1750

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Supérieur des Technologies de l’Information et de la Communication, Université de Carthage, 1164, Borj Cedria, Tunis, Tunisia

    Hassène Gritli

  2. Department of Electrical Engineering, Ecole Nationale D’Ingénieurs de Tunis, Université de Tunis El-Manar, BP. 37, 1002, Le Belvédère, Tunis, Tunisia

    Safya Belghith

Authors
  1. Hassène Gritli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Safya Belghith
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hassène Gritli .

Editor information

Editors and Affiliations

  1. Chennai, India

    Sundarapandian Vaidyanathan

  2. Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Christos Volos

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Gritli, H., Belghith, S. (2016). Identification, Stability and Stabilization of Limit Cycles in a Compass-Gait Biped Model via a Hybrid Poincaré Map. In: Vaidyanathan, S., Volos, C. (eds) Advances and Applications in Nonlinear Control Systems. Studies in Computational Intelligence, vol 635. Springer, Cham. https://doi.org/10.1007/978-3-319-30169-3_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-30169-3_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30167-9

  • Online ISBN: 978-3-319-30169-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation