Hybrid System Identification via Switched System Optimal Control for Bipedal Robotic Walking

  • Ram VasudevanEmail author
Part of the Springer Tracts in Advanced Robotics book series (STAR, volume 100)


While the goal of robotic bipedal walking to date has been the development of anthropomorphic gait, the community as a whole has been unable to agree upon an appropriate model to generate such gait. In this paper, we describe a method to segment human walking data in order to generate a robotic model capable of human-like walking. Generating the model requires the determination of the sequence of contact point enforcements which requires solving a combinatorial scheduling problem. We resolve this problem by transforming the detection of contact point enforcements into a constrained switched system optimal control problem for which we develop a provably convergent algorithm. We conclude the paper by illustrating the performance of the algorithm on identifying a model for robotic bipedal walking.


Contact Point Optimal Control Problem Bipedal Robot Optimal Control Scheme Unilateral Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I would like to thank Sam Burden for the many conversations on system identification, Humberto Gonzalez and Maryam Kamgarpour for their help with the development of the switched system optimal control algorithm, Ryan Sinnet for his help with robotic model considered in the Application Section. I am also grateful to Aaron Ames for the many insightful discussions on bipedal walking and system identification and Ruzena Bajcsy for her willingness to try to solve hard problems. This research is supported in part by NSF Awards CCR-0325274, CNS-0953823, ECCS-0931437, IIS-0703787, IIS-0724681, IIS-0840399 and NHARP Award 000512-0184-2009.


  1. 1.
    R. Ambrose, H. Aldridge, R. Askew, R. Burridge, W. Bluethmann, M. Diftler, C. Lovchik, D. Magruder, F. Rehnmark, Robonaut: Nasa’s space humanoid. IEEE Intell. Syst. Appl. 15(4), 57–63 (2000)CrossRefGoogle Scholar
  2. 2.
    A. Ames, R. Vasudevan, R. Bajcsy, Human-data based cost of bipedal robotic walking, in Proceedings of the 14th International Conference on Hybrid Systems: Computation and Control (ACM, 2011), pp. 153–162Google Scholar
  3. 3.
    S. Au, P. Dilworth, H. Herr, An ankle-foot emulation system for the study of human walking biomechanics, in Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006, ICRA 2006 (IEEE, 2006), pp. 2939–2945Google Scholar
  4. 4.
    G. Bergmann, G. Deuretzbacher, M. Heller, F. Graichen, A. Rohlmann, J. Strauss, G. Duda, Hip contact forces and gait patterns from routine activities. J. Biomech. 34(7), 859–871 (2001)CrossRefGoogle Scholar
  5. 5.
    D.J. Braun, M. Goldfarb, A control approach for actuated dynamic walking in bipedal robots. IEEE Trans. Rob. 25, 1–12 (2009)CrossRefGoogle Scholar
  6. 6.
    J.H. Choi, J.W. Grizzle, Planar bipedal walking with foot rotation. pp. 4909–4916, Portland, OR (2005)Google Scholar
  7. 7.
    S.H. Collins, M. Wisse, A. Ruina, A 3-d passive dynamic walking robot with two legs and knees. Int. J. Robot. Res. 20, 607–615 (2001)CrossRefGoogle Scholar
  8. 8.
    J. Duysens, H. Van de Crommert, Neural control of locomotion; part 1: the central pattern generator from cats to humans. Gait & Posture 7(2), 131–141 (1998)CrossRefGoogle Scholar
  9. 9.
    H. Gonzalez, R. Vasudevan, M. Kamgarpour, S. Sastry, R. Bajcsy, C. Tomlin, A numerical method for the optimal control of switched systems, in 2010 49th IEEE Conference on Decision and Control (CDC) (IEEE, 2010), pp. 7519–7526Google Scholar
  10. 10.
    H. Gonzalez, R. Vasudevan, M. Kamgarpour, S. Sastry, R. Bajcsy, C. Tomlin, A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems, in Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control (ACM, 2010), pp. 51–60Google Scholar
  11. 11.
    A. Goswami, B. Thuilot, B. Espiau, Compass-like biped robot part I : Stability and bifurcation of passive gaits. Rapport de recherche de l’INRIA (1996)Google Scholar
  12. 12.
    J.W. Grizzle, G. Abba, F. Plestan, Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Autom Control 46, 51–64 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J.W. Grizzle, C. Chevallereau, A.D. Ames, R.W. Sinnet, 3d bipedal robotic walking: models, feedback control, and open problems, in NOLCOS, Bologna, Italy (2010)Google Scholar
  14. 14.
    T. McGeer, Passive walking with knees, in IEEE International Conference on Robotics and Automation, Cincinnati, OH (1990)Google Scholar
  15. 15.
    R.M. Murray, Z. Li, S.S. Sastry, A Mathematical Introduction to Robotic Manipulation, Boca Raton, FL (1993)Google Scholar
  16. 16.
    J. Perry, J. Burnfield, Gait Analysis: Normal and Pathological Function, SLACK (2010)Google Scholar
  17. 17.
    E. Polak, Optimization: Algorithms and Consistent Approximations (Springer, 1997)Google Scholar
  18. 18.
    M. Rodgers, Dynamic biomechanics of the normal foot and ankle during walking and running. Phys. Ther. 68(12), 1822 (1988)Google Scholar
  19. 19.
    T. Schaub, M. Scheint, M. Sobotka, W. Seiberl, M. Buss, Effects of compliant ankles on bipedal locomotion, in IROS, St. Louis, Missouri, USA (2009)Google Scholar
  20. 20.
    A. Seireg, R. Arvikar, The prediction of muscular load sharing and joint forces in the lower extremities during walking. J. Biomech. 8(2), 89–102 (1975)CrossRefGoogle Scholar
  21. 21.
    R. Sinnet, M. Powell, R. Shah, A. Ames, A human-inspired hybrid control approach to bipedal robotic walking, in International Federation on Automatic Control (2011)Google Scholar
  22. 22.
    R.W. Sinnet, A.D. Ames, 2D bipedal walking with knees and feet: a hybrid control approach, in 48th IEEE Conference on Decision and Control, Shanghai, P.R. China (2009)Google Scholar
  23. 23.
    S. Srinivasan, E. Westervelt, A. Hansen, A low-dimensional sagittal-plane forward dynamic model for asymmetric gait and its application to study the gait of transtibial prosthesis users. J. Biomech. Eng. 131, 031003 (2009)CrossRefGoogle Scholar
  24. 24.
    R. Tedrake, T. Zhang, H. Seung, Learning to walk in 20 minutes, in Proceedings of the Fourteenth Yale Workshop on Adaptive and Learning Systems, New Haven, Connecticut, USA (2005)Google Scholar
  25. 25.
    D. Tlalolini, C. Chevallereau, Y. Aoustin, Comparison of different gaits with rotation of the feet for planar biped. Robot. Auton. Syst. 57, 371–383 (2009)CrossRefGoogle Scholar
  26. 26.
    R. Vasudevan, A. Ames, R. Bajcsy, Persistent homology for automatic determination of human-data based cost of bipedal walking, in Nonlinear Analysis: Hybrid Systems (2013), pp. 101–115Google Scholar
  27. 27.
    R. Vasudevan, H. Gonzalez, R. Bajcsy, S. S. Sastry, Consistent approximations for the optimal control of constrained switched systems---part 1: A conceptual algorithm. SIAM J. Control Optim. 51(6), 4463–4483 (2013)Google Scholar
  28. 28.
    R. Vasudevan, H. Gonzalez, R. Bajcsy, S. S. Sastry, Consistent approximations for the optimal control of constrained switched systems---Part 2: An implementable algorithm. SIAM J. Control Optim. 51(6), 4484–4503 (2013)Google Scholar
  29. 29.
    E.R. Westervelt, J.W. Grizzle, C. Chevallereau, J. Choi, B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion. Control and Automation. Boca Raton, FL (2007)Google Scholar
  30. 30.
    J. Yang, D. Winter, R. Wells, Postural dynamics of walking in humans. Biol. Cybern. 62(4), 321–330 (1990)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of MichiganAnn ArborUSA

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