Skip to main content

Reduced-Order Models

  • Reference work entry
  • First Online:
Humanoid Robotics: A Reference

Abstract

This chapter introduces reduced-order models that have been used in various studies of humanoid motion synthesis and control, particularly for balancing and locomotion. The models may be categorized into two groups. The first group derives physically explicit representations of the simplified system based on the centroidal dynamics embedded in the equation of motion. This category includes a number of successful models, which are intuitively easy to understand. The second group applies systematic order reduction techniques to the full-body dynamics. They make it easy to generalize the reduced-order model to different robot poses or tasks.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 899.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,099.99
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. D.C. Witt, A feasibility study on automatically-controlled powered lower-limb prostheses, 1970

    Google Scholar 

  2. M. Vukobratović, J. Stepanenko, On the stability of anthropomorphic systems. Math. Biosci. 15(1), 1–37 (1972)

    Article  Google Scholar 

  3. S. Kajita, K. Tani, Experimental study of biped dynamic walking in the linear inverted pendulum mode, in Proceeding of the IEEE International Conference on Robotics and Automation, 1995, pp. 2885–2819

    Google Scholar 

  4. R. Blickhan, The spring-mass model for running and hopping. J. Biomech. 22(11), 1217–1227 (1989)

    Article  Google Scholar 

  5. M. Vukobratović, A.A. Frank, D. Juričić, On the stability of biped locomotion. IEEE Trans. Biomed. Eng. BME 17(1), 25–36 (1970)

    Article  Google Scholar 

  6. C.K. Chow, D.H. Jacobson, Further studies of human locomotion: postural stability and control. Math. Biosci. 15(1), 93–108 (1972)

    Article  Google Scholar 

  7. T. Yamashita, M. Yamada, H. Inotani, A fundamental study of walking (in Japanese). Biomechanics 1, 226–234 (1972)

    Google Scholar 

  8. F. Gubina, H. Hemami, R.B. McGhee, On the dynamic stability of biped locomotion. IEEE Trans. Biomed. Eng. BME 21(2), 102–108 (1974)

    Article  Google Scholar 

  9. F. Miyazaki, S. Arimoto, A control theoretic study on dynamical biped locomotion. Trans. ASME J. Dyn. Syst. Measur. Control 102, 233–239 (1980)

    Article  MathSciNet  Google Scholar 

  10. J. Furusho, M. Masubuchi, Control of a dynamical biped locomotion system for steady walking. Trans. ASME J. Dyn. Syst. Measur. Control 108, 111–118 (1986)

    Article  Google Scholar 

  11. M.H. Raibert, Legged Robots That Balance (MIT Press, Cambridge, 1986)

    Article  Google Scholar 

  12. Y. Fujimoto, A. Kawamura, Simulation of an autonomous biped walking robot including environmental force interaction. IEEE Robot. Autom. Mag. 5(2), 33–41 (1998)

    Article  Google Scholar 

  13. D.E. Orin, A. Goswami, S.-H. Lee, Centroidal dynamics of a humanoid robot. Auton. Robot. 35, 161–176 (2013)

    Article  Google Scholar 

  14. R. Boulic, R. Mas, D. Thalmann, Inverse kinetics for center of mass position control and posture optimization, in Proceeding of the European Workshop on Combined Real and Synthetic Image Processing for Broadcast and Video Production, 1994

    Google Scholar 

  15. T. Sugihara, A Study of the realtime generation of legged motions on a whole-body humanoid robot (in Japanese). Masters thesis, The University of Tokyo, 2001

    Google Scholar 

  16. T. Sugihara, Y. Nakamura, H. Inoue, Realtime humanoid motion generation through ZMP manipulation based on inverted pendulum control, in Proceeding of the IEEE International Conference on Robotics and Automation, 2002, pp. 1404–1409

    Google Scholar 

  17. S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, H. Hirukawa, Resolved momentum control: humanoid motion planning based on the linear and angular momentum, in Proceeding of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2003, pp. 1644–1650

    Google Scholar 

  18. K. Mitobe, G. Capi, Y. Nasu, Control of walking robots based on manipulation of the zero moment point. Robotica 18, 651–657 (2000)

    Article  Google Scholar 

  19. M. Popovic, A. Goswami, H.M. Herr, Ground reference points in legged locomotion: definitions, biological trajectories and control implications. Int. J. Robot, Res. 24(12), 1013–1032 (2005)

    Google Scholar 

  20. K. Nagasaka, The whole body motion generation of humanoid robot using dynamics filter (in Japanese). Ph.D. thesis, 2000

    Google Scholar 

  21. S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, H. Hirukawa, Biped walking pattern generation by using preview control of zero-moment point, in Proceeding of the IEEE International Conference on Robotics and Automation, 2003, pp. 1620–1626

    Google Scholar 

  22. R.M. Alexander, Mechanics of bipedal locomotion, in Perspectives in Experimental Biology, vol. 1, ed. by P.S. Davies (Pergamon, Oxford, 1976), pp. 493–504

    Chapter  Google Scholar 

  23. R. Kato, M. Mori, Control method of biped locomotion giving asymptotic stability of trajectory. Automatica 20(4), 405–414 (1984)

    Article  Google Scholar 

  24. A. Goswami, B. Espiau, A. Keramane, Limit cycles and their stability in a passive bipedal gait, in Proceeding of the IEEE International Conference on Robotics and Automation, 1996, pp. 273–286

    Google Scholar 

  25. M. Garcia, A. Chatterjee, A. Ruina, M. Coleman, The simplest walking model: stability, complexity, and scaling. ASME J. Biomech. Eng. 120(2), 281–288 (1998)

    Article  Google Scholar 

  26. T. McGeer, Passive dynamic walking. Int. J. Robot. Res. 9(2), 62–82 (1990)

    Article  Google Scholar 

  27. J.H. Park, K.D. Kim, Biped robot walking using gravity-compensated inverted pendulum mode and computed torque control, in Proceeding of the IEEE International Conference on Robotics and Automation, 1998, pp. 3528–3533

    Google Scholar 

  28. Napoleon, S. Nakaura, M. Sampei, Balance control analysis of humanoid robot based on ZMP feedback control, in Proceeding of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2002, pp. 2437–2442

    Google Scholar 

  29. J. Pratt, J. Carff, S. Drakunov, A. Goswami, Capture point: a step toward humanoid push recovery, in Proceeding of the IEEE–RAS International Conference on Humanoid Robots, 2006, pp. 200–207

    Google Scholar 

  30. S. Lee, A. Goswami, Reaction mass pendulum (RMP): an explicit model for centroidal angular momentum of humanoid robots, in Proceeding of the IEEE International Conference on Robotics and Automation, 2007, pp. 4667–4672

    Google Scholar 

  31. A. Goswami, Kinematic and dynamic analogies between planar biped robots and the reaction mass pendulum (RMP) model, in Proceeding of the IEEE–RAS International Conference on Humanoid Robots, 2008, pp. 182–188

    Google Scholar 

  32. T. Sugihara, Y. Nakamura, Contact phase invariant control for humanoid robot based on variable impedant inverted pendulum model, in Proceeding of the IEEE International Conference on Robotics and Automation, 2003, pp. 51–56

    Google Scholar 

  33. O. Kwon, J.H. Park, Gait transitions for walking and running of biped robots, in Proceeding of the IEEE International Conference on Robotics and Automation, 2003, pp. 1350–1355

    Google Scholar 

  34. T. Sugihara, Y. Nakamura, Variable impedant inverted pendulum model control for a seamless contact phase transition on humanoid robot, in Proceeding of the IEEE–RAS International Conference on Humanoid Robots, 2003

    Google Scholar 

  35. K. Sadao, T. Hara, R. Yokokawa, Dynamic control of biped locomotion robot for disturbance on lateral plane (in Japanese), 1997, 10.37–10.38

    Google Scholar 

  36. S. Kajita, O. Matsumoto, M. Saigo, Real-time 3D walking pattern generation for a biped robot with telescopic legs, in Proceeding of the IEEE International Conference on Robotics and Automation, 2001, pp. 2299–2036

    Google Scholar 

  37. A.L. Hof, M.G.J. Gazendam, W.E. Sinke, The condition for dynamic stability. J. Biomech. 38, 1–8 (2005)

    Article  Google Scholar 

  38. J.E. Pratt, R. Tedrake, Velocity Based Stability Margins for Fast Bipedal Walking, 2006

    Google Scholar 

  39. T. Takenaka, T. Matsumoto, T. Yoshiike, Real time motion generation and control for biped robot – 1st Report, Walking Gait Pattern Generation, in Proceeding of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2009, pp. 1084–1091

    Google Scholar 

  40. J. Englsberger, C. Ott, A. Albu-Schäffer, Three-dimensional bipedal walking control based on divergent component of motion. IEEE Trans. Robot. 31(2), 355–368 (2015)

    Article  Google Scholar 

  41. F. Horak, L. Nashner, Central programming of postural movements: adaptation to altered support-surface configurations. J. Neurophys. 55(6), 1369–1381 (1986)

    Article  Google Scholar 

  42. K. Yamane, Systematic derivation of simplified dynamics for humanoid robots, in IEEE-RAS International Conference on Humanoid Robots, 2012, pp. 28–35

    Google Scholar 

  43. D.C. Hyland, D.S. Bernstein, The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore. IEEE Trans. Autom. Control AC 30(12), 1201–1211 (1985)

    Article  MathSciNet  Google Scholar 

  44. R. Tedrake, LQR-trees: feedback motion planning on sparse randomized trees, in Proceedings of Robotics: Science and Systems, 2009, pp. 17–24

    Google Scholar 

  45. K. Yamane, J.K. Hodgins, Simultaneous tracking and balancing of humanoid robots for imitating human motion capture data, in Proceedings of IEEE/RSJ International Conference on Intelligent Robot Systems, St. Louis, 2009, pp. 2510–2517

    Google Scholar 

  46. U. Nagarajan, K. Yamane, Automatic task-specific model reduction for humanoid robots, IEEE/RSJ International Conference on Intelligent Systems and Robots, 2013, pp. 2578–2585

    Google Scholar 

  47. M.G. Safonov, R.Y. Chiang, A Schur method for balanced model reduction. IEEE Trans. Autom. Control 34(7), 729–733 (1989)

    Google Scholar 

  48. D.G. Meyer, Fractional balanced reduction: model reduction via fractional representation. IEEE Trans. Autom. Control 35(12), 1341–1345 (1990)

    Article  MathSciNet  Google Scholar 

  49. K. Glover, All optimal Hankel norm approximation of linear multivariable systems and their L-error bounds. Int. J. Control 39(6), 1145–1193 (1984)

    Google Scholar 

  50. M.G. Safonov, R.Y. Chiang, D.J.N. Limebeer, Optimal Hankel model reduction for nonminimal systems. IEEE Trans. Autom. Control 35(4), 496–502 (1990)

    Article  MathSciNet  Google Scholar 

  51. W.L. Brogan, Modern Control Theory, 3rd edn. (Prentice Hall, Upper Saddle River, 1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Katsu Yamane .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature B.V.

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Sugihara, T., Yamane, K. (2019). Reduced-Order Models. In: Goswami, A., Vadakkepat, P. (eds) Humanoid Robotics: A Reference. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6046-2_56

Download citation

Publish with us

Policies and ethics