Skip to main content

Under-Actuated Systems: Nonlinear Control Showcase

  • Conference paper
  • First Online:
  • 2350 Accesses

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 465))

Abstract

Most of controllers for nonlinear systems are designed by using linearly approximated models and by applying linear control theory. In most of such cases, nonlinear control theory cannot improve control performance as long as we are controlling the systems in the vicinity of the equilibrium point. However, there are many under-actuated systems which are not stabilized with this linearization strategy: some sorts of singularity at the equilibrium point cause uncontrollability of the approximated linear model even though the system is controllable in nonlinear control theory. This note will present such nature of under-actuated systems and their control strategies.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   259.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   329.99
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

Learn about institutional subscriptions

References

  1. https://en.wikipedia.org/wiki/Inverted_pendulum

  2. https://en.wikipedia.org/wiki/Furuta_pendulum

  3. Brockett, R.W.: Asymptotic stability and feedback stabilization. Diff. Geom. Control theory 27, 181–191 (1983)

    MathSciNet  MATH  Google Scholar 

  4. Kolmanovsky, I., McClamroch, N.H.: Developments in nonholonomic control problems. IEEE Control Syst. Mag. 15(6), 20–36 (1995)

    Article  Google Scholar 

  5. De Wit, C.C., Sordalen, O.J.: Exponential stabilization of mobile robots with nonholonomic constraints. IEEE Trans. Autom. Control 37(11), 1791–1797 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Astolfi, A.: Discontinuous control of nonholonomic systems. Syst. Control Lett. 27(1), 37–45 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Pomet, J.-B.: Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift. Syst. Control Lett. 18(2), 147–158 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sampei, M.: A control strategy for a class of nonholonomic systems-time-state control form and its application. In: Proceedings of the 33rd IEEE Conference on Decision and Control, vol. 2. IEEE (1994)

    Google Scholar 

  9. Hoshi, Y., Sampei, M., Nakaura, S.: Stability analysis of hybrid time-state control for chained form systems. IFAC Proc. Vol. 37(13), 1223–1228 (2004)

    Article  Google Scholar 

  10. Sampei, M., Furuta, K.: On time scaling for nonlinear systems: application to linearization. IEEE Trans. Autom. Control 31(5), 459–462 (1986)

    Article  MATH  Google Scholar 

  11. Sampei, M., et al.: Arbitrary path tracking control of articulated vehicles using nonlinear control theory. IEEE Trans. Control Syst. Technol. 3(1), 125–131 (1995)

    Article  Google Scholar 

  12. Sampei, M., Kobayashi, T.: Applications of nonlinear control theory to path tracking control of articulated vehicles with double trailers. IFAC Proc. Vol. 26(2), 129–132 (1993)

    Article  Google Scholar 

  13. Sampei, M., Kiyota, H., Ishikawa, M.: Time-state control form and its application to a non-holonomic space robot. IFAC Proc. Vol. 28(14), 679–684 (1995)

    Article  Google Scholar 

  14. Nakagawa, T., et al.: An adaptive control of a nonholonomic space robot. In: Proceedings of the 36th IEEE Conference on Decision and Control, vol. 4. IEEE (1997)

    Google Scholar 

  15. Sampei, M., et al.: Position control of a rigid ball between two parallel plates. J. Robot. Soc. Jpn 14(8), 1237–1242 (1996). (in Japanese)

    Article  Google Scholar 

  16. Date, H., et al.: Simultaneous control of position and orientation for ball-plate manipulation problem based on time-state control form. IEEE trans. Robot. Autom. 20(3), 465–480 (2004)

    Article  Google Scholar 

  17. Kimura, S., et al.: Revived transformation for nonlinear systems subject to state constraints. In: 2015 IEEE 54th Annual Conference on Decision and Control (CDC). IEEE (2015)

    Google Scholar 

  18. Katsuyama, Y., et al.: Attitude controllability analysis of an underactuated satellite with two reaction wheels and its control. In: 2015 IEEE 54th Annual Conference on Decision and Control (CDC). IEEE (2015)

    Google Scholar 

  19. Sampei, M., Honda, T., Koga, M.: An attitude control of underactuated flying objects with initial angular momentum. J. Robot. Soc. Jpn 19(4), 461–467 (2001). (in Japanese)

    Article  Google Scholar 

  20. Takahashi, Y., Nakaura, S., Sampei, M.: Position control of surface vessel with unknown disturbances. In: 2007 46th IEEE Conference on Decision and Control. IEEE (2007)

    Google Scholar 

  21. Arai, H., Tanie, K., Shiroma, N.: Nonholonomic control of a three-DOF planar underactuated manipulator. IEEE Trans. Robot. Autom. 14(5), 681–695 (1998)

    Article  MATH  Google Scholar 

  22. Sagami, T., Nakaura, S., Sampei, M.: Control of 1generator high-order nonholonomic system through input-dependent coordinate transform. Trans. Soc. Instrum. Control Eng. 43(12), 1144–1150 (2007). (in Japanese)

    Article  MATH  Google Scholar 

  23. Ishikawa, M., Sampei, M.: On equilibria set and feedback stabilizability of nonlinear control systems. IFAC Proc. Vol. 31(17), 609–614 (1998)

    Article  Google Scholar 

  24. Kanazawa, M., Nakaura, S., Sampei, M.: Inverse optimal control problem for bilinear systems: application to the inverted pendulum with horizontal and vertical movement. In: Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference, CDC/CCC 2009. IEEE (2009)

    Google Scholar 

  25. Ibuki, T., et al.: 3D inverted pendulum stabilization on a quadrotor via bilinear system approximations. In: 2015 IEEE Conference on Control Applications (CCA). IEEE (2015)

    Google Scholar 

  26. Shimizu, E., Sampei, M., Koga, M.: Design of a nonlinear H/sub/spl infin//state feedback controller for bilinear systems with nonlinear weight. In: Proceedings of the 36th IEEE Conference on Decision and Control, vol. 3. IEEE (1997)

    Google Scholar 

  27. Ohsaku, S., et al.: Nonlinear H∞ control for semi-active suspension. JSAE Rev. 20(4), 447–452 (1999)

    Article  Google Scholar 

  28. Shoji, T., et al.: Throwing motion control of the springed pendubot. IEEE Trans. Control Syst. Technol. 21(3), 950–957 (2013)

    Article  Google Scholar 

  29. Sekiguchi, K., Sampei, M., Nakaura, S.: Parameterization of the output with respect to the relative degree. SICE J. Control Meas. Syst. Integr. 3(2), 137–143 (2010)

    Article  Google Scholar 

  30. Nakaura, S., et al.: Enduring rotary motion control of devil stick. IFAC Proc. 37(13), 805–810 (2004)

    Article  Google Scholar 

  31. Shimizu, T., Nakaura, S., Sampei, M.: The control of a bipedal running robot based on output zeroing considered rotation of the ankle joint. In: 2006 45th IEEE Conference on Decision and Control. IEEE (2006)

    Google Scholar 

  32. Isobe, S., Nakaura, S., Sampei, M.: Continuous rolling motion control for the Acrobot composed of rounded links. In: 47th IEEE Conference on Decision and Control, CDC 2008. IEEE (2008)

    Google Scholar 

  33. Murakami, N., Sugimori, A., Ibuki, T., Sampei, M.: Rolling motion control introducing back bending for Acrobot composed of rounded links. In: Preprints of the 20th World Congress IFAC 2017 (2017)

    Google Scholar 

  34. Westervelt, E.R., Grizzle, J.W., Koditschek, D.E.: Hybrid zero dynamics of planar biped walkers. IEEE Trans. Autom. Control 48(1), 42–56 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hirose, S., Yamada, H.: Snake-like robots [tutorial]. IEEE Robot. Autom. Mag. 16(1), 88–98 (2009)

    Article  Google Scholar 

  36. Date, H., et al.: Locomotion control of a snake robot with constraint force attenuation. In: Proceedings of the 2001 American Control Conference, vol. 1. IEEE (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mitsuji Sampei .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Sampei, M. (2018). Under-Actuated Systems: Nonlinear Control Showcase. In: Duy, V., Dao, T., Zelinka, I., Kim, S., Phuong, T. (eds) AETA 2017 - Recent Advances in Electrical Engineering and Related Sciences: Theory and Application. AETA 2017. Lecture Notes in Electrical Engineering, vol 465. Springer, Cham. https://doi.org/10.1007/978-3-319-69814-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-69814-4_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-69813-7

  • Online ISBN: 978-3-319-69814-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics