Kinematic Models of Doubly Generalized N-trailer Systems

  • Ignacy Duleba
Open Access


In this paper a Pfaff matrix for doubly generalized N-trailer systems is derived when not only lateral, as in generalized N-trailer systems, but also longitudinal constraints are respected. Based on the matrix, kinematic models are presented for doubly generalized N-trailer systems parameterized with a vector composed of codes of active constraints at each axle. For all constraints active, a closed-form formula for kinematics is derived while for other models – a recursive one is proposed. It is shown how to construct analytically a null space for two types of possible Pfaff matrices and some examples are provided to illustrate introduced formulas. The kinematic models can be used either to test algorithms of motion planning (control) for a broad class of easy parameterizable models or to design or verify wheeled systems.


Model Kinematics Generalized N-trailer Nonholonomic systems 



  1. 1.
    Altafini, C.: Some properties of the general N-trailer. Int. J. Control. 74(4), 409–424 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayle, B., Renaud, M., Fourquet, J.Y.: Nonholonomic mobile manipulators: kinematics, velocities and redundancies. J. Intell. Robot. Syst. 36(1), 45–63 (2003)CrossRefGoogle Scholar
  3. 3.
    Bicchi, A., Marigo, A.: Dexterous grippers: putting nonholonomy to work for fine manipulation. Int. J. Robot. Res. 21(5–6), 427–442 (2002)CrossRefGoogle Scholar
  4. 4.
    Bryant, J., Sangwin, C.: How Round is Your Circle. Where Engineering and Mathematics Meet. Princeton University Press, Princeton (2005)zbMATHGoogle Scholar
  5. 5.
    Chow, W.L.: Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117 (1), 98–105 (1939)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Duleba, I.: Algorithms of Motion Planning for Nonholonomic Robots. Publ. House of Wroclaw Univ of Technology (1998)Google Scholar
  7. 7.
    Fliess, M., Rouchon, P., Lévine, J., Martin, P.: Flatness, motion planning and trailer systems. In: 1993 IEEE Conference on Decision and Control (CDC), pp. 2700–2705, San Antonio (1993)Google Scholar
  8. 8.
    Nakamura, Y., Savant, S.: Nonholonomic motion control of an autonomous underwater vehicle. In: IEEE/RSJ Int. Workshop on Intelligent Robots and Systems Osaka, pp. 1254–1259 (1991)Google Scholar
  9. 9.
    Michalek, M.: Non-minimum-phase property of N-trailer kinematics resulting from off-axle interconnections. Int. J. Control. 86(4), 740–758 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Michalek, M., Kielczewski, M., Jedwabny, T.: Cascaded VFO control for non-standard N-trailer robots. J. Intell. Robot. Syst. 77(3–4), 415–432 (2015)CrossRefGoogle Scholar
  11. 11.
    Nakamura, Y., Chung, W., Sordalen, O.J.: Design and control of the nonholonomic manipulator. IEEE Trans. Robot. Autom. 17(1), 48–59 (2001)CrossRefGoogle Scholar
  12. 12.
    De Luca, A., Oriolo, G.: Modelling and control of nonholonomic mechanical systems. In: Angeles, J., Kecskemethy, A. (eds.) Kinematics and Dynamics of Multi-body Systems, CISM Courses and Lectures, vol. 360, pp. 277–342. Springer (1995)Google Scholar
  13. 13.
    Pasillas-Lepine, W., Respondek, W.: Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus. Int. J. Control. 74(6), 628–637 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences,, A000079
  15. 15.
    Sordalen, O.J.: Conversion of the kinematics of a car with N trailers into a chained form. In: 1993 IEEE International Conference on Robotics and Automation (ICRA), vol. 1, pp. 382–387, Atlanta (1993)Google Scholar
  16. 16.
    Nakamura, Y., Chung, W., Sordalen, O.J.: Design and control of the nonholonomic manipulator. IEEE Trans. Robot. Autom. 17(1), 48–59 (2001)CrossRefGoogle Scholar
  17. 17.
    Tchon, K., Jakubiak, J., Zadarnowska, K.: Doubly nonholonomic mobile manipulators. In: 2004 IEEE International Conference on Robotics and Automation (ICRA), pp. 4590–4595, New Orleans (2004)Google Scholar
  18. 18.
    Tchon, K., Zadarnowska, K., Juszkiewicz, Ł., Arent, K.: Modeling and control of a skid-steering mobile platform with coupled side wheels. Bull. Acad. Pol. Sci. Tech. Sci. 63(3), 807–818 (2015)Google Scholar
  19. 19.
    Tilbury, D., Murray, R.M., Sastry, S.S.: Trajectory generation for the N-Trailer problem using goursat normal form. IEEE Trans. Autom. Control 40(5), 802–819 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vafa, Z., Dubowsky, S.: The kinematics and dynamics of space manipulators: the virtual manipulator approach. Int. J. Robot. Res. 9(4), 3–21 (1990)CrossRefGoogle Scholar
  21. 21.
    Zadarnowska, K.: Switched modeling and Task-Priority motion planning of wheeled mobile robots subject to slipping. J. Intell. Robot Syst. 85, 449–469 (2017)CrossRefGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Chair of Cybernetics and RoboticsWroclaw University of Science and TechnologyWroclawPoland

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