Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T06:07:33.793Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal periodic gain scheduling for bipedal walking with hybrid dynamics

Published online by Cambridge University Press:  09 December 2014

M. Harel ,
G. Agranovich  and
M. Brand
M. Harel
Affiliation:
Department of electrical engineering, Ariel University of Samaria, Ariel, Israel
G. Agranovich*
Affiliation:
Department of electrical engineering, Ariel University of Samaria, Ariel, Israel
M. Brand
Affiliation:
Department of mechanical engineering, Ariel University of Samaria, Ariel, Israel
*
*Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Summary

We present an optimal gain scheduling control design for bipedal walking with minimum tracking error. We obtained a linear approximation by linearizing the nonlinear hybrid dynamic model about a nominal periodic trajectory. This linearization allows us to identify the linear model as a linear periodic system. An optimal feedback was designed using Bellman's dynamic programming. The linear periodic system allows us to determine a linear quadratic regulator (LQR) for a single period and to set the Hamilton-Jacobi-Bellman (HJB) function in a linear quadratic form. In this way, the dynamic programming yielded an admissible continuous gain scheduling that was designed with regard to the hybrid dynamics of the system. We tuned the optimization parameters such that the tracking error and the average energy consumption are minimized. Due to linearization, we were able to examine the stability of the approximated periodic system achieved by the periodic gain according to Floquet's theory, by calculating the monodromy matrix of the closed-loop hybrid system. In addition to determining stability, the eigenvalues of this approximated monodromy matrix allowed us to evaluate the settling time of the system. This approach presents a direct method for optimal solution of locomotion control according to a given reference trajectory.

Keywords

Bipedal locomotionLinearizationFloquet theoryDynamic programmingLQRPeriodic linear system
Type
Articles
Information
Robotica , Volume 34 , Issue 8 , August 2016 , pp. 1811 - 1821
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. McGeer, T., “Passive dynamic walking,” Int. J. Robot. Res. 9 (2), 62–82 (1990).Google Scholar
2. Vukobratović, M. and Borovac, B., “Zero-moment point–-thirty five years of its life,” Int. J. Humanoid Robot. 1 (01), 157–173 (2004).Google Scholar
3. Chevallereau, C., Formal'sky, A. and Djoudi, D., “Tracking a joint path for the walk of an underactuated biped,” Robotica 22 (1), 15–28 (2004).Google Scholar
4. Djoudi, D., Chevallereau, C. and Aoustin, Y., “In Optimal Reference Motions for Walking of a Biped Robot,” Proceedings of the 2005 IEEE International Conference on, Robotics and Automation, 2005. ICRA 2005. (IEEE: 2005) pp. 2002–2007.Google Scholar
5. Plestan, F., Grizzle, J. W., Westervelt, E. R. and Abba, G., “Stable walking of a 7-DOF biped robot,” IEEE Trans. Robot. Autom. 19 (4), 653–668 (2003).Google Scholar
6. Braun, D. J. and Goldfarb, M., “A control approach for actuated dynamic walking in biped robots,” IEEE Trans. Robot. 25 (6), 1292–1303 (2009).Google Scholar
7. Jacobson, D., “Differential dynamic programming methods for solving bang-bang control problems,” IEEE Trans. Autom. Control 13 (6), 661–675 (1968).Google Scholar
8. Liu, C., Atkeson, C. G. and Su, J., “Neighboring optimal control for periodic tasks for systems with discontinuous dynamics,” Sci. China Inform. Sci. 54 (3), 653–663 (2011).Google Scholar
9. Todorov, E. and Tassa, Y., “In Iterative Local Dynamic Programming, Adaptive Dynamic Programming and Reinforcement Learning,” IEEE Symposium on, ADPRL'09,(IEEE: 2009) pp. 90–95.Google Scholar
10. Aoustin, Y. and Formal'sky, A., “In Design of Reference Trajectory to Stabilize Desired Nominal Cyclic Gait of a Biped,” Proceedings of the First Workshop on, Robot Motion and Control, RoMoCo'99, (IEEE: 1999) pp. 159–164.CrossRefGoogle Scholar
11. Goldstein, H., Poole, C. P. and Safko, J., Classical Mechanics, 3rd edn. (1980).Google Scholar
12. Westervelt, E. R., Grizzle, J. W., Chevallereau, C., Choi, J. H. and Morris, B., Feedback Control of Dynamic Bipedal Robot Locomotion (CRC Press, Boca Raton FL, 2007).Google Scholar
13. Kalman, R. E., “Contributions to the theory of optimal control,” Bol. Soc. Mat. Mexicana 5 (2), 102–119 (1960).Google Scholar
14. Lewis, F. L., Vrabie, D. and Syrmos, V. L., Optimal Control, (Wiley, New York, 2012).Google Scholar
15. Kirtley, C., “CGA Normative Gait Database,” Hong Kong Polytechnic University, (2005).Google Scholar
16. De Leva, P., “Adjustments to Zatsiorsky-Seluyanov's segment inertia parameters,” J. Biomech. 29 (9), 1223–1230 (1996).Google Scholar
17. Bittanti, S. and Colaneri, P., Periodic Systems: Filtering and Control, (Springer, Berlin, 2008) vol. 5108985.Google Scholar
18. Sun, W., Nagpal, K. M. and Khargonekar, P. P., “ H ∞ control and filtering for sampled-data systems,” IEEE Trans. Autom. Control 38 (8), 1162–1175 (1993).Google Scholar
19. Agranovich, G., “Some problems of discrete/continuous systems stabilization,” Funct. Differ. Equ. 10, 5–17 (2003).Google Scholar
20. Antsaklis, P. J. and Michel, A. N., Linear Systems, (Birkhäuser Boston: Cambridge, MA, 2005).Google Scholar
21. Slane, J. and Tragesser, S., “Analysis of periodic nonautonomous inhomogeneous systems,” Nonlinear Dyn. Syst. Theory 11 (2), 183–198 (2011).Google Scholar