Nonlinear Dynamics

, Volume 95, Issue 1, pp 175–194 | Cite as

Vehicle motion control under equality and inequality constraints: a diffeomorphism approach

  • Hui YinEmail author
  • Ye-Hwa Chen
  • Dejie Yu
Original Paper


This study addresses the problem of vehicle lateral and yaw motion control when both equality and inequality (i.e., bilateral and unilateral) constraints are involved. By using the Udwadia–Kalaba approach, the explicit equation of vehicle motion with equality constraints is established, and the corresponding control inputs can be obtained from the equation. The equality constraints aim to render the vehicle to move along the desired trajectory. However, as the initial conditions of vehicle motion may take values leading the vehicle to violate the road-bound lines, it is necessary to impose an additional constraint to constrain the vehicle to move within the road-bound lines, which is an inequality constraint. As the inequality constraint cannot be handled by the original Udwadia–Kalaba approach, a creative diffeomorphism approach is proposed to integrate the inequality constraint into the equality constraints, and thus it creatively enables the Udwadia–Kalaba approach to deal with both equality and inequality constraints. By solving the equation established based on the Udwadia–Kalaba approach and diffeomorphism approach, the control inputs that can render the vehicle to move along the desired trajectory without violating the road-bound lines are obtained. The effectiveness of the proposed method is demonstrated by numerical simulation results.


Udwadia–Kalaba approach Vehicle motion control Equality constraint Inequality constraint Diffeomorphism approach 



The paper is supported by National Natural Science Foundation of China (No. 11572121), Independent Research Projects of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body in Hunan University (Grant No. 71375004) and the China Scholarship Council (201606130100).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Nam, K., Fujimoto, H., Hori, H.: Lateral stability control of in-wheel-motor-driven electric vehicles based on sideslip angle estimation using lateral tire force sensors. IEEE Trans. Veh. Technol. 61(5), 1972–1985 (2012)CrossRefGoogle Scholar
  2. 2.
    Weir, D.H., McRuer, D.T.: Dynamics of driver vehicle steering control. Automatica 6(1), 87–98 (1970)CrossRefGoogle Scholar
  3. 3.
    Chen, C., Tomizuka, M.: Dynamic modeling of tractor-semitrailer vehicles in automated highway systems. In: California Partners for Advanced Transit and Highways (PATH) (1995)Google Scholar
  4. 4.
    Chen, C., Tomizuka, M.: Lateral control of tractor-semitrailer vehicles in automated highway systems. In: California Partners for Advanced Transit and Highways (PATH) (1996)Google Scholar
  5. 5.
    Lee, T., Chen, Y.H., Chuang, C.: Control for tractor-semitrailer vehicle systems: a Lyapunov minimax approach. Dyn. Control 9(1), 21–37 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Matsumoto, N., Tomizuka, M.: Vehicle lateral velocity and yaw rate control with two independent control inputs. J. Dyn. Syst.-Trans. ASME 114, 606–613 (1992)CrossRefGoogle Scholar
  7. 7.
    Besselink, B.C.: Computer controlled steering system for vehicles having two independently driven wheels. Comput. Electron. Agric. 39(3), 209–226 (2003)CrossRefGoogle Scholar
  8. 8.
    Mutoh, N., Kazama, T., Takita, K.: Driving characteristics of an electric vehicle system with independently driven front and rear wheels. IEEE Trans. Ind. Electron. 53(3), 803–813 (2006)CrossRefGoogle Scholar
  9. 9.
    Kalaba, R.E., Udwadia, F.E.: Analytical dynamics with constraint forces that do work in virtual displacements. Appl. Math. Comput. 121(2), 211–217 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Udwadia, F.E., Kalaba, R.E.: On the foundations of analytical dynamics. Int. J. Nonlinear Mech. 37(6), 1079–1090 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  12. 12.
    Sun, H., Zhao, H., Zhen, S.: Application of the Udwadia–Kalaba approach to tracking control of mobile robots. Nonlinear Dyn. 83, 389–400 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Schutte, A.D., Dooley, B.A.: Constrained motion of tethered satellites. J. Aerosp. Eng. 18(4), 242–250 (2005)CrossRefGoogle Scholar
  14. 14.
    Pappalardo, C.M.: A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems. Nonlinear Dyn. 81(4), 1841–1869 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Chen, Y.H.: Constraint-following servo control design for mechanical systems. J. Vib. Control 15(3), 369–389 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chen, Y.H., Zhang, X.: Adaptive robust approximate constraint-following control for mechanical systems. J. Frankl. Inst. 347(1), 69–86 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Huang, Q., Chen, Y.H., Cheng, A.: Adaptive robust control for fuzzy mechanical systems: constraint-following and redundancy in constraints. IEEE Trans. Fuzzy Syst. 23(4), 1113–1126 (2015)CrossRefGoogle Scholar
  18. 18.
    Sun, H., Zhao, H., Huang, K., Zhen, S., et al.: Adaptive robust constraint-following control for satellite formation flying with system uncertainty. J. Guid. Control Dyn. 40(6), 1–7 (2017)CrossRefGoogle Scholar
  19. 19.
    Jacobson, D., Lele, M.: A transformation technique for optimal control problems with a state variable inequality constraint. IEEE Trans. Autom. Control 14(5), 457–464 (1969)CrossRefGoogle Scholar
  20. 20.
    Itiki, C., Kalaba, R.E., Udwadia, F.E.: Inequality constraints in the process of jumping. Appl. Math. Comput. 78(2–3), 163–173 (1996)zbMATHGoogle Scholar
  21. 21.
    Abramova, I., Latyshev, S.: Using the fundamental equation of constrained motion with inequality constraints. Appl. Math. Comput. 215(8), 2858–2876 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Loxton, R.C., Teo, K.L., Rehbock, V., et al.: Optimal control problems with a continuous inequality constraint on the state and the control. Automatica 45(10), 2250–2257 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Liu, X., Hu, Y., Feng, J., et al.: A novel penalty approach for nonlinear dynamic optimization problems with inequality path constraints. IEEE Trans. Autom. Control 59(10), 2863–2867 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Prajna, S., Jadbabaie, A., Pappas, G.J.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Autom. Control 52(8), 1415–1428 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Tee, K.P., Ge, S.S., Tay, E.H.: Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4), 918–927 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Liu, Y.J., Tong, S.: Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems. Automatica 76, 143–152 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Rossetter, E.J., Gerdes, J.C.: Lyapunov based performance guarantees for the potential field lane-keeping assistance system. J. Dyn. Syst. Meas. Control 128(3), 510–522 (2006)CrossRefGoogle Scholar
  28. 28.
    Wieland, P., Allgöwer, F.: Constructive safety using control barrier functions. IFAC Proc. 40(12), 462–467 (2007)CrossRefGoogle Scholar
  29. 29.
    Ames, A.D., Xu, X., Grizzle, J.W., et al.: Control barrier function based quadratic programs for safety critical systems. IEEE Trans. Autom. Control 62(8), 3861–3876 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Xu, X., Grizzle, J.W., Tabuada, P., et al.: Correctness guarantees for the composition of lane keeping and adaptive cruise control. IEEE Trans. Sci. Eng. (2017). CrossRefGoogle Scholar
  31. 31.
    Xu, X.: Constrained control of input–output linearizable systems using control sharing barrier functions. Automatica 87, 195–201 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Acosta, J.Á., Dòria-Cerezo, A., Fossas, E.: Diffeomorphism-based control of nonlinear systems subject to state constraints with actual applications. In: Control Applications (CCA), IEEE Conference, pp. 923–928 (2014)Google Scholar
  33. 33.
    Kimura, S., Nakamura, H., Ibuki, T., et al.: Revived transformation for nonlinear systems subject to state constraints. In: Decision and Control (CDC), IEEE 54th Annual Conference, pp. 7554–7559 (2015)Google Scholar
  34. 34.
    Khalil, H.K.: Nonlinear Control. Prentice-Hall, Englewood Cliffs (2014)Google Scholar
  35. 35.
    Boyce, W.E., DiPrima, R.C., Meade, D.B.: Elementary Differential Equations and Boundary Value Problems. Wiley, New York (1992)zbMATHGoogle Scholar
  36. 36.
    Udwadia, F.E.: A new perspective on the tracking control of nonlinear structural and mechanical systems. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 459(2035), 1783–1800 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Noble, B., Daniel, J.W.: Applied Linear Algebra. Prentice-Hall, Englewood Cliffs (1988)zbMATHGoogle Scholar
  38. 38.
    Nagai, M., Hirano, Y., Yamanaka, S.: Integrated control of active rear wheel steering and direct yaw moment control. Veh. Syst. Dyn. 27(5–6), 357–370 (1997)CrossRefGoogle Scholar
  39. 39.
    Chen, Y., Hedrick, J.K., Guo, K.: A novel direct yaw moment controller for in-wheel motor electric vehicles. Veh. Syst. Dyn. 51(6), 925–942 (2013)CrossRefGoogle Scholar
  40. 40.
    Mousavinejad, E., Han, Q., Yang, F., et al.: Integrated control of ground vehicles dynamics via advanced terminal sliding mode control. Veh. Syst. Dyn. 55(2), 268–294 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle BodyHunan UniversityChangshaPeople’s Republic of China
  2. 2.The George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations