Skip to main content
SpringerLink
Log in

Construction of invariant ellipsoids in the stabilization problem for a wheeled robot following a curvilinear path

  • Determinate Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

The synthesis control problem for the plane motion of a wheeled robot is studied. The goal of the control is to bring the robot to an assigned curvilinear trajectory and to stabilize its motion along it in the presence of phase and control constraints. For a synthesized control law, invariant ellipsoids—quadratic approximations of the attraction domains of the target trajectory—are constructed, which allow one to check in the course of the robot motion whether the control law can stabilize motion along the current trajectory segment. To take into account constraints on the control, methods of absolute stability theory are applied. The construction of the invariant ellipsoids reduces to solving a system of linear matrix inequalities.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Samson, C., Control of Chained Systems Application to Path Following and Time-Varying Point-Stabilization of Mobile Robots, IEEE Trans. Automat. Contr., 1995, vol. 40, no. 1, pp. 64–77.

    Article  MATH  MathSciNet  Google Scholar 

  2. Kolmanovsky, I. and McClamroch, N.H., Developments in Nonholonomic Control Problems, IEEE Control Syst., 1995, no. 12, pp. 20–36.

  3. Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.L., Nonlinear and Adaptive Control of Complex Systems, St. Petersburg: Nauka, 2000.

    MATH  Google Scholar 

  4. Rapoport, L.B., Estimation of Attraction Domain in a Wheeled Robot Control Problem, Avtom. Telemekh., 2006, no. 9, pp. 69–89.

  5. Thuilot, B., Cariou, C., Martinet, P., and Berducat, M., Automatic Guidance of a Farm Tractor Relying on a Single CP-DGPS, Autonomous Robots, 2002, no. 13, pp. 53–61.

  6. Cordesses, L., Cariou, C., and Berducat, M., Combine Harvester Control Using Real Time Kinematic GPS, Precision Agriculture, 2000, no. 2, pp. 147–161.

  7. Gilimyanov, R.F., Pesterev, A.V., and Rapoport, L.B., Motion Control for a Wheeled Robot Following a Curvilinear Path, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2008, no. 6, pp. 209–216.

  8. Pesterev, A.V, Rapoport, L.B., and Gilimyanov, R.F., Control of a Wheeled Robot Following a Curvilinear Path, Sixth EUROMECH Nonlinear Dynamics Conference, St. Petersburg, 2008.

  9. Polyak, B.T. and Shcherbakov, P.S., Robast Stability and Controllability, Moscow: Nauka, 2002.

    Google Scholar 

  10. Kamenetskii, V.A. and Rapoport, L.B., Stabilization of a Wheeled Robot with Regard to Control Constraints, Third Int. Conf. on Control Problems, Selected Papers, Moscow, 2006, pp. 139–146.

  11. Pesterev, A.V. and Gilimyanov, R.F., Path Planning for a Wheeled Robot, Trudy ISA RAN, Proc. of Inst. for System Analysis, Moscow: KomKniga, 2006, vol. 25.

    Google Scholar 

  12. Gilimyanov, R.F., Pesterev, A.V., and Rapoport, L.B., Smoothing Curvature of Trajectories Constructed by Noisy Measurements in Path Planning Problems for Wheeled Robots, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2008, no. 5, pp. 152–159.

  13. Pyatnitskii, E.S., Absolute Stability of Nonstationary Nonlinear Systems, Avtom. Telemekh., 1970, no. 1, pp. 5–15.

  14. Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Stability of Nonlinear Systems with a Nonunique Equilibrium State, Moscow: Nauka, 1978.

    MATH  Google Scholar 

  15. Formal’skii, A.M., Controllability and Stability of Systems with Limited Resources, Moscow: Nauka, 1974.

    Google Scholar 

  16. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    A. V. Pesterev & L. B. Rapoport

Authors
  1. A. V. Pesterev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. B. Rapoport
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © A.V. Pesterev, L.B. Rapoport, 2009, published in Avtomatika i Telemekhanika, 2009, No. 2, pp. 52–67.

This work was supported by the Presidium of Russian Academy of Sciences (Program 22) and by the State Program of Support of Leading Scientific Schools, project no. NSh-1676.2008.1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pesterev, A.V., Rapoport, L.B. Construction of invariant ellipsoids in the stabilization problem for a wheeled robot following a curvilinear path. Autom Remote Control 70, 219–232 (2009). https://doi.org/10.1134/S0005117909020040

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117909020040

PACS number

Search

Navigation