Nonlinear Dynamics

, Volume 81, Issue 3, pp 1475–1487 | Cite as

Feedback stabilization of a nonholonomic system with potential fields: application to a two-wheeled mobile robot among obstacles

  • Takateru Urakubo
Original Paper


In this paper, a feedback controller for a nonholonomic system with three states and two inputs is derived using an artificial potential function that has no local minima, and the stability of equilibria of the system is analyzed. Although the system with the controller has an infinite number of equilibria due to the nonholonomic constraint, those equilibria except the critical points of the potential function are unstable because of a skew-symmetric component of the controller. When the potential function has critical points of saddle type, the saddles may be stable equilibria in addition to the stable equilibrium at the minimum of the function. The controller is applied to a two-wheeled mobile robot among obstacles and modified by using a time-varying potential function in order to avoid convergence to the saddles. As a result, with the controller, the mobile robot converges to a desired position and orientation without collision with obstacles.


Nonholonomic system Feedback control Lyapunov stability Potential field method 


  1. 1.
    Khatib, O.: Real-time obstacle avoidance for manipulators and mobile robots. Int. J. Robot. Res. 5(1), 90–98 (1986)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kim, J.-O., Khosla, P.K.: Real-time obstacle avoidance using harmonic potential functions. IEEE Trans. Robot. Autom. 8(5), 338–349 (1992)CrossRefGoogle Scholar
  3. 3.
    Guldner, J., Utkin, V.I.: Sliding mode control for gradient tracking and robot navigation using artificial potential fields. IEEE Trans. Robot. Autom. 11(2), 247–254 (1995)CrossRefGoogle Scholar
  4. 4.
    Ge, S.S., Cui, Y.J.: New potential functions for mobile robot path planning. IEEE Trans. Robot. Autom. 16(5), 615–620 (2000)CrossRefGoogle Scholar
  5. 5.
    Rimon, E., Koditschek, D.E.: Exact robot navigation using artificial potential functions. IEEE Trans. Robot. Autom. 8(5), 501–518 (1992)CrossRefGoogle Scholar
  6. 6.
    Rimon, E., Koditschek, D.E.: The construction of analytic diffeomorphisms for exact robot navigation on star worlds. Trans. Am. Math. Soc. 327(1), 71–116 (1991)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lamiraux, F., Sekhavat, S., Laumond, J.P.: Motion planning and control for Hilare pulling a trailer. IEEE Trans. Robot. Autom. 15(4), 640–652 (1999)CrossRefGoogle Scholar
  8. 8.
    Sekhavat, S., Laumond, J.-P.: Topological property for collision-free nonholonomic motion planning: the case of sinusoidal inputs for chained form systems. IEEE Trans. Robot. Autom. 14(5), 671–680 (1998)CrossRefGoogle Scholar
  9. 9.
    Laumond, J.-P., Jacobs, P.E., Taïx, M., Murray, R.M.: A motion planner for nonholonomic mobile robots. IEEE Trans. Robot. Autom. 10(5), 577–593 (1994)CrossRefGoogle Scholar
  10. 10.
    Que, Z., Wang, J., Plaisted, C.E.: A new analytical solution to mobile robot trajectory generation in the presence of moving obstacles. IEEE Trans. Robot. 20(6), 978–993 (2004)CrossRefGoogle Scholar
  11. 11.
    Papadopoulos, E., Poulakakis, I., Papadimitriou, I.: On path planning and obstacle avoidance for nonholonomic platforms with manipulators: a polynomial approach. Int. J. Robot. Res. 21(4), 367–383 (2002)CrossRefGoogle Scholar
  12. 12.
    Lamiraux, F., Bonnafous, D., Lefebvre, O.: Reactive path deformation for nonholonomic mobile robots. IEEE Trans. Robot. 20(6), 967–977 (2004)CrossRefGoogle Scholar
  13. 13.
    Divelbiss, A.W., Wen, J.T.: A path space approach to nonholonomic motion planning in the presence of obstacles. IEEE Trans. Robot. Autom. 13(3), 443–451 (1997)CrossRefGoogle Scholar
  14. 14.
    Weir, M.K., Bott, M.P.: High quality goal connection for nonholonomic obstacle navigation allowing for drift using dynamic potential fields. In: Proceedings of the 2010 IEEE International Conference on Robotics and Automation, pp. 3221–3226 (2010)Google Scholar
  15. 15.
    Jacobs, P., Canny, J.: Planning smooth paths for mobile robots. In: Li, Z., Canny, J.F. (eds.) Nonholonomic Motion Planning, pp. 271–342. Kluwer, Dordrecht (1993)CrossRefGoogle Scholar
  16. 16.
    Yang, S.X., Meng, M.Q.-H.: Real-time collision-free motion planning of a mobile robot using a neural dynamics-based approach. IEEE Trans. Neural Netw. 14(6), 1541–1552 (2003)CrossRefGoogle Scholar
  17. 17.
    Rigatos, G.G., Tzafestas, S.G., Evangelidis, G.J.: Reactive parking control of nonholonomic vehicles via a fuzzy learning automaton. IEE Proc. Control Theory Appl. 148(2), 169–179 (2001)CrossRefGoogle Scholar
  18. 18.
    Kallen, V., Komoroski, A.T., Kumar, V.: Sequential composition for navigating a nonholonomic cart in the presence of obstacles. IEEE Trans. Robot. 27(6), 1152–1159 (2011)CrossRefGoogle Scholar
  19. 19.
    Xiang, X., Lapierre, L., Jouvencel, B.: Guidance based collision avoidance of coordinated nonholonomic autonomous vehicles. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 6064–6069 (2010)Google Scholar
  20. 20.
    Huang, W.H., Fajen, B.R., Fink, J.R., Warren, W.H.: Visual navigation and obstacle avoidance using a steering potential function. Robot. Auton. Syst. 54(4), 288–299 (2006)CrossRefGoogle Scholar
  21. 21.
    Ramírez, G., Zeghloul, S.: Collision-free path planning for nonholonomic mobile robots using a new obstacle representation in the velocity space. Robotica 19(5), 543–555 (2001)CrossRefGoogle Scholar
  22. 22.
    Patel, S., Jung, S.-H., Ostrowski, J.P., Rao, R., Taylor, C.J.: Sensor based door navigation for a nonholonomic vehicle. In: Proceedings of the 2002 IEEE International Conference on Robotics and Automation, pp. 3081–3086 (2002)Google Scholar
  23. 23.
    Kolmanovsky, I., McClamroch, N.H.: Developments in nonholonomic control problems. IEEE Control Syst. 15(6), 20–36 (1995)CrossRefGoogle Scholar
  24. 24.
    Brockett, R.W.: Asymptotic stability and feedback stabilization. In: Brockett, R.W., Millman, R.S., Sussman, H.J. (eds.) Differential Geometric Control Theory, pp. 181–191. Birkhäuser, Switzerland (1983)Google Scholar
  25. 25.
    Samson, C.: Control of chained systems application to path following and time-varying point-stabilization of mobile robots. IEEE Trans. Autom. Control 40(1), 64–77 (1995)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    M’Closkey, R.T., Murray, R.M.: Exponential stabilization of driftless nonlinear control systems using homogeneous feedback. IEEE Trans. Autom. Control 42(5), 614–628 (1997)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Khennouf, H., Canudas de Wit, C.: Quasi-continuous exponential stabilizers for nonholonomic systems. In: Proceedings of IFAC 13th World Congress, 2b-174 (1996)Google Scholar
  28. 28.
    Astolfi, A.: Exponential stabilization of a wheeled mobile robot via discontinuous control. ASME J. Dyn. Syst. Meas. Control 121(1), 121–125 (1999)Google Scholar
  29. 29.
    Tayebi, A., Tadjine, M., Rachid, A.: Invariant manifold approach for the stabilization of nonholonomic chained systems: application to a mobile robot. Nonlinear Dyn. 24(2), 167–181 (2001)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tanner, H.G., Loizou, S., Kyriakopoulos, K.J.: Nonholonomic stabilization with collision avoidance for mobile robots. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1220–1225 (2001)Google Scholar
  31. 31.
    Tanner, H.G., Loizou, S.G., Kyriakopoulos, K.J.: Nonholonomic navigation and control of cooperating mobile manipulators. IEEE Trans. Robot. Autom. 19(1), 53–64 (2003)CrossRefGoogle Scholar
  32. 32.
    Valbuena, L., Tanner, H.G.: Hybrid potential field based control of differential drive mobile robots. J. Intell. Robot. Syst. 68(3–4), 307–322 (2012)CrossRefMATHGoogle Scholar
  33. 33.
    Karydis K., Valbuena L., Tanner, H.G.: Model predictive navigation for position and orientation control of nonholonomic vehicles. In: Proceedings of the 2012 IEEE International Conference on Robotics and Automation, pp. 3206–3211 (2012)Google Scholar
  34. 34.
    LaValle, S.M., Kuffner, J.J.: Randomized kinodynamic planning. Int. J. Robot. Res. 20(5), 378–400 (2001)CrossRefGoogle Scholar
  35. 35.
    LaValle, S.M., Konkimalla, P.: Algorithms for computing numerical optimal feedback motion strategies. Int. J. Robot. Res. 20(9), 729–752 (2001)CrossRefGoogle Scholar
  36. 36.
    Tsuchiya, K., Urakubo, T., Tsujita, K.: Motion control of a nonholonomic system based on the Lyapunov control method. J. Guidance Control Dyn. 25(2), 285–290 (2002)CrossRefGoogle Scholar
  37. 37.
    Urakubo, T., Tsuchiya, K., Tsujita, K.: Motion control of a two-wheeled mobile robot. Adv. Robot. 15(7), 711–728 (2001)CrossRefGoogle Scholar
  38. 38.
    Urakubo, T., Okuma, K., Tada, Y.: Feedback control of a two wheeled mobile robot with obstacle avoidance using potential functions. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2428–2433 (2004)Google Scholar
  39. 39.
    Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. American Mathematical Society, Providence (1972)MATHGoogle Scholar
  40. 40.
    Zhu, H.P., Yu, A.B.: A contribution to the stability of nonholonomic systems. Mech. Res. Commun. 29(5), 307–314 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Systems ScienceKobe UniversityKobeJapan

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