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Autonomous Robots

, Volume 41, Issue 6, pp 1277–1295 | Cite as

Orbital stabilization of an underactuated bipedal gait via nonlinear \({{\mathcal H}}_{\infty }\)-control using measurement feedback

  • Oscar Montano
  • Yury Orlov
  • Yannick Aoustin
  • Christine Chevallereau
Article

Abstract

The primary concern of the work is robust control of hybrid mechanical systems under unilateral constraints with underactuation degree one. Nonlinear \({{\mathcal H}}_{\infty }\) output feedback synthesis is developed in the hybrid setting, covering collision phenomena. Sufficient conditions are presented to ensure internal asymptotic stability while also attenuating external disturbances and plant uncertainties. The developed synthesis is applied to the orbital stabilization of an underactuated bipedal robot periodically touching the ground. Good performance of the closed-loop system is obtained not only in the presence of measurement noise and external disturbances, affecting the gait of the biped between collision time instants, but also under uncertainties at the velocity restitution when the ground collision occurs.

Keywords

Robust control Unilateral constraints Underactuated mechanical systems Orbital stabilization Bipedal robot Walking gait 

Notes

Acknowledgments

The authors acknowledge the financial support of Campus France grant Eiffel and CONACYT Grant No.165958.

References

  1. Alcaraz-Jiménez, J., Herrero-Pérez, D., & Martínez-Barberá, H. (2013). Robust feedback control of zmp-based gait for the humanoid robot nao. The International Journal of Robotics Research, 32(9–10), 1074–1088.CrossRefGoogle Scholar
  2. Ames, A., Galloway, K., & Grizzle, J. (2012). Control lyapunov functions and hybrid zero dynamics. In 2012 IEEE 51st Annual Conference on Decision and Control (CDC), IEEE (pp. 6837–6842).Google Scholar
  3. Angelosanto, G. (2008). Kalman filtering of imu sensor for robot balance control. PhD thesis, Massachusetts Institute of Technology.Google Scholar
  4. Aoustin, Y., & Formalsky, A. (2003). Control design for a biped: Reference trajectory based on driven angles as functions of the undriven angle. Journal of Computer and Systems Sciences International, 42(4), 645–662.MathSciNetzbMATHGoogle Scholar
  5. Aoustin, Y., Chevallereau, C., & Formalsky, A. (2006). Numerical and experimental study of the virtual quadrupedal walking robot-semiquad. Multibody System Dynamics, 16(1), 1–20.CrossRefzbMATHGoogle Scholar
  6. Aoustin, Y., Chevallereau, C., & Orlov, Y. (2010). Finite time stabilization of a perturbed double integrator-part ii: applications to bipedal locomotion. In 2010 49th IEEE Conference on Decision and Control (CDC) (pp. 3554–3559). IEEE, Piscataway.Google Scholar
  7. Arai, H., Tanie, K., & Shiroma, N. (1998). Time-scaling control of an underactuated manipulator. In 1998 IEEE International Conference on Robotics and Automation Proceedings (Vol. 3, pp. 2619–2626). IEEE, Piscataway.Google Scholar
  8. Basar, T., & Bernhard, P. (1995). \(\cal H_\infty \)-optimal control and related minimax design problems: A dynamic game approach. Boston: Birkhaeuser.Google Scholar
  9. Bezier, P. (1972). Numerical control: Mathematics and applications. London: Wiley and Sons.zbMATHGoogle Scholar
  10. Brogliato, B. (1999). Nonsmooth mechanics: Models, dynamics and control. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  11. Bullo, F., & Lynch, K. (2001). Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems. IEEE Transactions on Robotics and Automation, 17(4), 402–412.CrossRefGoogle Scholar
  12. Chevallereau, C., Abba, G., Aoustin, Y., Plestan, F., Westervelt, E., Canudas De Wit, C., et al. (2003). Rabbit: A testbed for advanced control theory. IEEE Control Systems Magazine, 23(5), 57–79.CrossRefGoogle Scholar
  13. Chevallereau, C., Grizzle, J., & Shih, C. (2009). Asymptotically stable walking of a five-link underactuated 3-d bipedal robot. IEEE Transactions on Robotics, 25(1), 37–50.CrossRefGoogle Scholar
  14. Dai, H., & Tedrake, R. (2012). Optimizing robust limit cycles for legged locomotion on unknown terrain. In 2012 IEEE 51st Annual Conference on Decision and Control (CDC) (pp. 1207–1213). IEEE, Piscataway.Google Scholar
  15. Dai, H., & Tedrake, R. (2013). \({\cal L}_{2}\)-gain optimization for robust bipedal walking on unknown terrain. In 2013 IEEE International Conference on Robotics and Automation (ICRA) (pp. 3116–3123). IEEE, Piscataway.Google Scholar
  16. Djoudi, D., Chevallereau, C., & Aoustin, Y. (2005). Optimal reference motions for walking of a biped robot. In Proceedings of the 2005 IEEE International Conference on Robotics and Automation, 2005. ICRA 2005 (pp 2002–2007). IEEE, Piscataway.Google Scholar
  17. Freidovich, L., & Shiriaev, A. (2009). Transverse linearization for mechanical systems with passive links, impulse effects, and friction forces. In Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009 (pp. 6490–6495). IEEE, Piscataway.Google Scholar
  18. Freidovich, L., Shiriaev, A., & Manchester, I. (2008). Stability analysis and control design for an underactuated walking robot via computation of a transverse linearization. In Proceedings of 17th IFAC World Congress, Seoul (pp. 10–166).Google Scholar
  19. Goebel, R., Sanfelice, R., & Teel, A. (2009). Hybrid dynamical systems. IEEE Control Systems, 29(2), 28–93.MathSciNetCrossRefGoogle Scholar
  20. Grizzle, J., Abba, G., & Plestan, F. (2001). Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Transactions on Automatic Control, 46(1), 51–64.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Haddad, W., Kablar, N., Chellaboina, V., & Nersesov, S. (2005). Optimal disturbance rejection control for nonlinear impulsive dynamical systems. Nonlinear Analysis: Theory, Methods & Applications, 62(8), 1466–1489.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Hamed, K., & Grizzle, J. (2013). Robust event-based stabilization of periodic orbits for hybrid systems: Application to an underactuated 3D bipedal robot. In Proceedings of the 2013 American Control Conference.Google Scholar
  23. Hamed, K., & Grizzle, J. (2014). Event-based stabilization of periodic orbits for underactuated 3-d bipedal robots with left-right symmetry. IEEE Transactions on Robotics, 30(2), 365–381.CrossRefGoogle Scholar
  24. Hamed, K., Buss, B., & Grizzle, J. (2014). Continuous-time controllers for stabilizing periodic orbits of hybrid systems: Application to an underactuated 3d bipedal robot. In Proceedings of the 53rd IEEE Conderence on Decision and Control.Google Scholar
  25. Hobbelen, D., & Wisse, M. (2007). A disturbance rejection measure for limit cycle walkers: The gait sensitivity norm. IEEE Transactions on Robotics, 23(6), 1213–1224.CrossRefGoogle Scholar
  26. Isidori, A., & Astolfi, A. (1992). Disturbance attenuation and \({\cal H}_\infty \)-control via measurement feedback in nonlinear systems. IEEE Transactions on Automatic Control, 37(9), 1283–1293.MathSciNetCrossRefzbMATHGoogle Scholar
  27. La Hera, P., Shiriaev, A., Freidovich, L., Mettin, U., & Gusev, S. (2013). Stable walking gaits for a three-link planar biped robot with one actuator. IEEE Transactions on Robotics, 29(3), 589–601.CrossRefGoogle Scholar
  28. Leonov, G. (2006). Generalization of the andronov-vitt theorem. Regular and Chaotic Dynamics, 11(2), 281–289.MathSciNetCrossRefzbMATHGoogle Scholar
  29. Manamani, N., Gauthier, N., & MSirdi, N. (1997). Sliding mode control for pneumatic robot leg. In Proceedings European Control Conference.Google Scholar
  30. Mettin, U., La Hera, P., Freidovich, L., & Shiriaev, A. (2007). Planning human-like motions for an underactuated humanoid robot based on the virtual constraints approach. Proceedings of 13th International Conference on Advanced Robotics, Jeju, Korea (pp. 585–590).Google Scholar
  31. Meza-Sanchez, I., Aguilar, L., Shiriaev, A., Freidovich, L., & Orlov, Y. (2011). Periodic motion planning and nonlinear \({\cal H}_\infty \) tracking control of a 3-dof underactuated helicopter. International Journal of Systems Science, 42(5), 829–838.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Miossec, S., & Aoustin, Y. (2005). A simplified stability study for a biped walk with underactuated and overactuated phases. The International Journal of Robotics Research, 24(7), 537–551.CrossRefGoogle Scholar
  33. Miossec, S., & Aoustin, Y. (2006). Dynamical synthesis of a walking cyclic gait for a biped with point feet. In Fast motions in biomechanics and robotics, Springer, Berlin (pp. 233–252).Google Scholar
  34. Montano, O., Orlov, Y., & Aoustin, Y. (2014). Nonlinear \({\cal H}_\infty \)-control of mechanical systems under unilateral constraints. In Proceedings of the 19th World Congress of the International Federation of Automatic Control, IFAC, pp 3833–3838 (extended journal version was submitted to Control Engineering Practice under the title “Nonlinear \({\cal H}_\infty \)-stabilization of fully actuated bipedal locomotion under unilateral constraints”).Google Scholar
  35. Montano, O., Orlov, Y., & Aoustin, Y. (2015a). Nonlinear output feedback \(\cal H_\infty \)-control of mechanical systems under unilateral constraints. In Proceedings of the 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems (pp 284–289).Google Scholar
  36. Montano, O., Orlov, Y., Aoustin, Y., & Chevallereau, C. (2015b). Nonlinear orbital \(\cal H_\infty \)-stabilization of underactuated mechanical systems with unilateral constraints. In Proceedings of the 14th European Control Conference (pp 800–805).Google Scholar
  37. Morris, B., & Grizzle, J. (2005). A restricted poincaré map for determining exponentially stable periodic orbits in systems with impulse effects: Application to bipedal robots. In 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05 (pp. 4199–4206). IEEE, Piscataway.Google Scholar
  38. Naldi, R., & Sanfelice, R. G. (2013). Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts. Automatica, 49(5), 1104–1116.MathSciNetCrossRefzbMATHGoogle Scholar
  39. Nešić, D., Zaccarian, L., & Teel, A. (2008). Stability properties of reset systems. Automatica, 44(8), 2019–2026.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Nešić, D., Teel, A. R., Valmorbida, G., & Zaccarian, L. (2013). Finite-gain stability for hybrid dynamical systems. Automatica, 49(8), 2384–2396.MathSciNetCrossRefzbMATHGoogle Scholar
  41. Nikkhah, M., Ashrafiuon, H., & Fahimi, F. (2007). Robust control of underactuated bipeds using sliding modes. Robotica, 25(03), 367–374.CrossRefGoogle Scholar
  42. Orlov, Y., & Aguilar, L. (2014). Advanced \({\cal H} _\infty \) control-towards nonsmooth theory and applications. Boston: Birkhauser.Google Scholar
  43. Orlov, Y., Acho, L., & Solis, V. (1999). Nonlinear \(\cal H_\infty \)-control of time-varying systems. In Proceedings of the 38th IEEE Conference on Decision and Control, 1999 (Vol. 4, pp. 3764–3769). IEEE, Piscataway.Google Scholar
  44. Oza, H., Orlov, Y., Spurgeon, S., Aoustin, Y., & Chevallereau, C. (2014). Finite time tracking of a fully actuated biped robot with pre-specified settling time: a second order sliding mode synthesis. In 2014 IEEE International Conference on Robotics and Automation (ICRA) (pp. 2570–2575). IEEE, Piscataway.Google Scholar
  45. Raibert, M., Tzafestas, S., & Tzafestas, C. (1993). Comparative simulation study of three control techniques applied to a biped robot. In International Conference on Systems, Man and Cybernetics, 1993’.Systems Engineering in the Service of Humans’, Conference Proceedings (pp 494–502). IEEE, Piscataway.Google Scholar
  46. Shiriaev, A., Perram, J., & Canudas-de Wit, C. (2005). Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach. IEEE Transactions on Automatic Control, 50(8), 1164–1176.MathSciNetCrossRefGoogle Scholar
  47. Shiriaev, A., Freidovich, L., & Manchester, I. (2008). Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems. Annual Reviews in Control, 32(2), 200–211.CrossRefGoogle Scholar
  48. Shiriaev, A. S., & Freidovich, L. B. (2009). Transverse linearization for impulsive mechanical systems with one passive link. IEEE Transactions on Automatic Control, 54(12), 2882–2888.MathSciNetCrossRefGoogle Scholar
  49. Tlalolini, D., Chevallereau, C., & Aoustin, Y. (2011). Human-like walking: Optimal motion of a bipedal robot with toe-rotation motion. IEEE/ASME Transactions on Mechatronics, 16(2), 310–320.CrossRefGoogle Scholar
  50. Van Der Schaft, A. (1991). On a state space approach to nonlinear h control. Systems & Control Letters, 16(1), 1–8.MathSciNetCrossRefzbMATHGoogle Scholar
  51. Westervelt, E., Buche, G., & Grizzle, J. (2004). Experimental validation of a framework for the design of controllers that induce stable walking in planar bipeds. The International Journal of Robotics Research, 23(6), 559–582.CrossRefGoogle Scholar
  52. Westervelt, E., Grizzle, J., Chevallereau, C., Choi, J., & Morris, B. (2007). Feedback control of dynamic bipedal robot locomotion. Boca Raton: CRC Press.CrossRefGoogle Scholar
  53. Westervelt, E. R., Grizzle, J. W., & Koditschek, D. E. (2003). Hybrid zero dynamics of planar biped walkers. IEEE Transactions on Automatic Control, 48(1), 42–56.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Oscar Montano
    • 1
  • Yury Orlov
    • 1
  • Yannick Aoustin
    • 2
  • Christine Chevallereau
    • 2
  1. 1.CICESEEnsenadaMexico
  2. 2.L’IRCCyNNantes Cedex 3France

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