Hostname: page-component-89b8bd64d-dvtzq Total loading time: 0 Render date: 2026-05-08T14:42:26.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Biped walking control using a trajectory library

Published online by Cambridge University Press:  25 May 2012

Chenggang Liu ,
Christopher G. Atkeson  and
Jianbo Su
Chenggang Liu*
Affiliation:
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
Christopher G. Atkeson
Affiliation:
Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213, USAcga@cmu.edu
Jianbo Su
Affiliation:
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
*
*Corresponding author. E-mail: cgliu2008@gmail.com
Get access Rights & Permissions [Opens in a new window]

Summary

This paper presents biped walking control using a library of optimal trajectories. Biped walking control is formulated as an optimal control problem. We use a parametric trajectory optimization method to find the periodic steady-state walking trajectory. As a second stage, we use Differential Dynamic Programming to generate a library of optimal trajectories and locally linear models of the optimal control law, which are used to construct a more global control law. The proposed controller is compared with a trajectory tracking controller using optimal gains. The utility and performance of the proposed method are evaluated using simulated walking control of a planar five-link biped robot.

Keywords

Trajectory optimizationTrajectory libraryRobot control

Information

Type
Articles
Information
Robotica , Volume 31 , Issue 2 , March 2013 , pp. 311 - 322
Copyright
Copyright © Cambridge University Press 2012

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.)

Article purchase

Temporarily unavailable

References

1.Vukobratovic, M., Borovac, B., Surla, D. and Stokic, D., Biped Locomotion: Dynamics, Stability, Control and Application (Scientific Fundamentals of Robotics) (Springer, New York, 1990).CrossRefGoogle Scholar
2.Hirai, K., Hirose, M., Haikawa, Y. and Takenaka, T., “The Development of Honda Humanoid Robot,” In: Proceedings of the IEEE International Conference on Robotics and Automation vol. 2 (May 1998) pp. 1321–1326.CrossRefGoogle Scholar
3.Huang, Q., Yokoi, K., Kajita, S., Kaneko, K., Arai, H., Koyachi, N. and Tanie, K., “Planning walking patterns for a biped robot,” IEEE Trans. Robot. Autom. 17 (3), 280–289 (2001).CrossRefGoogle Scholar
4.Kajita, S. and Tani, K., “Experimental study of biped dynamic walking,” IEEE Control Syst. Mag. 16 (1), 13–19 (1996).Google Scholar
5.Kajita, S., Kanehiro, F., Kaneko, K., Fujiwara, K., Harada, K., Yokoi, K. and Hirukawa, H., “Biped walking pattern generation by using preview control of zero-moment point,” Proc. IEEE Int. Conf. Robot. Autom. 2, 1620–1626 (2003).Google Scholar
6.Kajita, S., Kanehiro, F., Kaneko, K., Yokoi, K. and Hirukawa, H., “The 3D linear inverted pendulum mode: A simple modeling for a biped walking pattern generation,” Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. 1, 239–246 (2001).Google Scholar
7.Kajita, S., Matsumoto, O. and Saigo, M., “Real-time 3D walking pattern generation for a biped robot with telescopic legs,” Proc. IEEE Int. Conf. Robot. Autom. 3, 2299–2306 (2001).Google Scholar
8.Roussel, L., Canudas, C. and Goswami, A., “Generation of Energy Optimal Complete Gait Cycles for Biped Robots,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Leuven, Belgium (May, 1998) pp. 2036–2041.Google Scholar
9.Hardt, M., “Multibody Dynamical Algorithms, Numerical Optimal Control, with Detailed Studies in the Control of Jet Engine Compressors and Biped Walking,” Ph.D. dissertation, (San Diego, CA: University of California, 1999).Google Scholar
10.Djoudi, D., Chevallereau, C. and Aoustin, Y., “Optimal Reference Motions for Walking of a Biped Robot,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Barcelona, Spain (Apr. 2005) pp. 2002–2007.Google Scholar
11.Bessonnet, G., Seguin, P. and Sardain, P., “A parametric optimization approach to walking pattern synthesis,” Int. J. Robot. Res. 24 (7), 523–536 (2005).CrossRefGoogle Scholar
12.Saidouni, T. and Bessonnet, G., “Generating globally optimised sagittal gait cycles of a biped robot,” Robotica 21 (2), 199–210 (2003).CrossRefGoogle Scholar
13.Chevallereau, C. and Aoustin, Y., “Optimal reference trajectories for walking and running of a biped robot,” Robotica 19 (5), 557–569 (2001).CrossRefGoogle Scholar
14.Chevallereau, C., Djoudi, D. and Grizzle, J., “Stable bipedal walking with foot rotation through direct regulation of the zero moment point,” IEEE Trans. Robot. 24 (2), 390–401 (Apr. 2008).CrossRefGoogle Scholar
15.Loffler, K., Gienger, M., Pfeiffer, F. and Ulbrich, H., “Sensors and control concept of a biped robot,” IEEE Trans. Ind. Electron. 51 (5), 972–980 (2004).CrossRefGoogle Scholar
16.Komura, T., Leung, H., Kudoh, S. and Kuffner, J., “A Feedback Controller for Biped Humanoids that Can Counteract Large Perturbations During Gait,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Barcelona, Spain (Apr. 2005) pp. 1989–1995.Google Scholar
17.Wieber, P. B. and Chevallereau, C., “Online adaptation of reference trajectories for the control of walking systems,” Robot. Auton. Syst. 54 (7), 559–566 (2006).CrossRefGoogle Scholar
18.Bellman, R., Dynamic Programming (Princeton University Press, Princeton, NJ, 1957).Google ScholarPubMed
19.Whitman, E. C. and Atkeson, C. G., “Control of Instantaneously Coupled Systems Applied to Humanoid Walking,” Proceedings of the IEEE International Conference on Humanoid Robotics, Nashville, TN (Dec. 2010).Google Scholar
20.Mandersloot, T., Wisse, M. and Atkeson, C. G., “Controlling Velocity in Bipedal Walking: A Dynamic Programming Approach,” In: Proceedings of the IEEE International Conference on Humanoid Robots, Genova, Italy (Dec. 2006) pp. 124–130.Google Scholar
21.Stilman, M., Atkeson, C. G., Kuffner, J. and Zeglin, G., “Dynamic Programming in Reduced Dimensional Spaces: Dynamic Planning for Robust Biped Locomotion,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Barcelona, Spain (Apr. 2005) pp. 2399–2404.Google Scholar
22.Bryson, A. E. and Ho, Y. C., Applied Optimal Control: Optimization, Estimation, and Control (Hemisphere, Washington, DC, 1975).Google Scholar
23.Dyer, P. and McReynolds, S. R., The Computation and Theory of Optimal Control (Academic Press, Maryland Heights, MO, 1970).Google Scholar
24.Jacobson, D. H. and Mayne, D. Q., Modern Analytic and Computational Methods in Science and Mathematics, Vol. 24: Differential Dynamic Programming. (Elsevier, Maryland Heights, MO, 1970).Google Scholar
25.Morimoto, J. and Atkeson, C., “Minimax Differential Dynamic Programming: An Application to Robust Biped Walking,” Proceedings of the Sixteenth Annual Conference on Advances in the Neural Information Processing Systems, Vancouver, Canada (2002).Google Scholar
26.Atkeson, C. G. and Morimoto, J., “Nonparametric Representation of Policies and Value Functions: A Trajectory-Based Approach,” In: Proceedings of the Annual Conference on Advances in Neural Information Processing Systems (2003) pp. 1643–1650.Google Scholar
27.Tassa, Y., Erez, T. and Smart, W., “Receding horizon differential dynamic programming,” Adv. Neural Inf. Proc. Syst. 20, 1465–1472 (2008).Google Scholar
28.Erez, T., Tassa, Y. and Todorov, E., “Infinite-Horizon Model Predictive Control for Periodic Tasks with Contacts,” Proceedings of Robotics: Science and Systems, University of Southern California Los Angeles, CA, USA (2011).Google Scholar
29.Liu, C., Atkeson, C. G. and Su, J., “Neighboring optimal control for periodic tasks for systems with discontinuous dynamics,” Sci. China Inf. Sci. 54 (3), 653–663 (2011).CrossRefGoogle Scholar
30.Liu, C. and Su, J., “Biped Walking Control Using Offline and Online Optimization,” Proceedings of the 30th Chinese Control Conference, Yantai, China (2011).Google Scholar
31.Betts, J. T., Practical Methods for Optimal Control Using Nonlinear Programming (Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2001).Google Scholar
32.Hargraves, C. R. and Paris, S. W., “Direct trajectory optimization using nonlinear programming and collocation,” J. Guid. Control Dyn. 10 (4), 338–342 (1987).CrossRefGoogle Scholar
33.Stryk, O. Von, “Numerical solution of optimal control problems by direct collocation,” Optim. control Calculus Variations Optim. Control Theory Numer. Methods 111 (4), 129–143 (1993).Google Scholar
34.Elnagar, G., Kazemi, M. and Razzaghi, M., “The pseudospectral Legendre method for discretizing optimal control problems,” IEEE Trans. Autom. Control 40 (10), 1793–1796 (Oct. 1995).CrossRefGoogle Scholar
35.Vlassenbroeck, J. and Van Dooren, R., “A Chebyshev technique for solving nonlinear optimal control problems,” IEEE Trans. Autom. Control 33 (4), 333–340 (Apr. 1988).CrossRefGoogle Scholar
36.Gill, P. E., Murray, W. and Saunders, M. A., “SNOPT: An SQP algorithm for large-scale constrained optimization,” SIAM J. Optim. 12, 979–1006 (1997).CrossRefGoogle Scholar
37.Stryk, O. V. and Bulirsch, R., “Direct and indirect methods for trajectory optimization,” Ann. Oper. Res. 37 (1), 357–373 (1992).CrossRefGoogle Scholar
38.Atkeson, C. G. and Stephens, B., “Random sampling of states in dynamic programming,” IEEE Trans. Syst. Man Cybern. B 38 (4), 924–929 (2008).CrossRefGoogle ScholarPubMed
39.Stolle, M. and Atkeson, C. G., “Finding and transferring policies using stored behaviors,” Auton. Robots 29 (2), 169–200 (2010).CrossRefGoogle Scholar
40.Breitner, M., “Robust optimal onboard reentry guidance of a space shuttle: dynamic game approach and guidance synthesis with neural networks,” J. Optim. Theory Appl. 107 (3), 481–503 (2000).CrossRefGoogle Scholar
41.Murray, J. J., Cox, C., Lendaris, G. G. and Saeks, R., “Adaptive dynamic programming,” IEEE Trans. Syst. Man Cybern. C 32 (2), 140–153 (2002).CrossRefGoogle Scholar
42.Safonova, A. and Hodgins, J. K., “Construction and optimal search of interpolated motion graphs,” ACM Trans. Graph. 26 (3), 106 (2007).CrossRefGoogle Scholar
43.Nishiwaki, K., Sugihara, T., Kagami, S., Inaba, M. and Inoue, H., “Online Mixture and Connection of Basic Motions for Humanoid Walking Control by Footprint Specification,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Seoul, Korea (May 2001) pp. 4110–4115.Google Scholar
44.Denk, J. and Schmidt, G., “Synthesis of Walking Primitive Databases for Biped Robots in 3D-Environments,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Taipei, Taiwan (Sep. 2003) pp. 1343–1349.Google Scholar
45.Kavraki, L. E., Svestka, P., Latombe, J. C. and Overmars, M., “Probabilistic roadmaps for path planning in high-dimensional configuration spaces,” IEEE Trans. Robot. Autom. 12 (4), 566–580 (1996).CrossRefGoogle Scholar
46.Atkeson, C. G., “Using local trajectory optimizers to speed up global optimization in dynamic programming,” In: Advances in Neural Information Processing Systems, vol. 6 (Cowan, J. D., Tesauro, G., and Alspector, J., eds.). (Morgan Kaufmann, Massachusetts, 1994) pp. 663–670.Google Scholar
47.Tedrake, R., “LQR-trees: Feedback Motion Planning on Sparse Randomized Trees,” Proceedings of Robotics: Science and Systems, Seattle, USA (Jun. 2009).Google Scholar
48.Reist, P. and Tedrake, R., “Simulation-Based LQR-Trees with Input and State Constraints,” Proceedings of the IEEE International Conference on Robotics and Automation, Anchorage, Alaska, USA (May 2010).Google Scholar
49.Liu, C. and Atkeson, C. G., “Standing Balance Control Using a Trajectory Library,” In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and System, St. Louis, USA (Oct. 2009) pp. 3031–3036.Google Scholar
50.Bentivegna, D., Atkeson, C. and Kim, J.-Y., “Compliant Control of a Hydraulic Humanoid Joint,” In: Proceedings of the IEEE-RAS International Conference on Humanoid Robots, Pittsburgh, PA, USA (Nov. 2007) pp. 483–489.Google Scholar
51.Grizzle, J., Abba, G. and Plestan, F., “Asymptotically stable walking for biped robots: Analysis via systems with impulse effects,” IEEE Trans. Autom. Control 46 (1), 51–64 (2001).CrossRefGoogle Scholar
52.Liao, L.-Z. and Shoemaker, C., “Convergence in unconstrained discrete-time differential dynamic programming,” IEEE Trans. Autom. Control 36 (6), 692–706 (Jun. 1991).CrossRefGoogle Scholar
53.Tassa, Y., “Theory and Implementation of Bio-Mimetic Motor Controllers,” Ph.D. dissertation, Hebrew University of Jerusalem. Available at: URL: http://www.cs.washington.edu/homes/todorov/papers/TassaThesis.pdf (Feb. 2011).Google Scholar
54.Stolle, M. and Atkeson, C. G., “Policies Based on Trajectory Libraries,” Proceedings of the IEEE International Conference on Robotics and Automation, Orlando, FL, USA (May 2006).Google Scholar