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Application of the Udwadia–Kalaba approach to tracking control of mobile robots

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Abstract

Udwadia–Kalaba approach which presents a new, general and explicit equation of motion for constrained mechanical systems with holonomic or nonholonomic constraints is applied to the trajectory tracking control of the mobile robot in this paper. Unlike any other nonlinear control methods, the inspiration for this methodology which does not make any linearization or approximations comes from a different, though closely allied, field, namely analytical dynamics. The control torques required to control the mobile robot so that it precisely satisfies the trajectory requirements which are represented by an arbitrary (sufficiently smooth) function of time are obtained explicitly and in closed form by solving Udwadia–Kalaba equation. Numerical simulations are performed to show the simplicity, efficacy and accuracy of this closed-form method.

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References

  1. Yang, J.M., Kim, J.H.: Sliding mode control for trajectory tracking of nonholonomic wheeled mobile robots. IEEE Trans. Robot. Autom. 15, 578–587 (1999)

    Article  Google Scholar 

  2. Rubagotti, M., Della Vedova, M.L., Ferrara, A.: Time-optimal sliding-mode control of a mobile robot in a dynamic environment. IET Control Theory Appl. 5, 1916–1924 (2011)

    Article  MathSciNet  Google Scholar 

  3. Xu, J., Guo, Z., Lee, T.: Design and implementation of integral sliding mode control on an underactuated two-wheeled mobile robot. IEEE Trans. Ind. Electron. 61, 3671–3681 (2014)

    Article  Google Scholar 

  4. Fukao, T., Nakagawa, H., Adachi, N.: Adaptive tracking control of a nonholonomic mobile robot. IEEE Trans. Robot. Autom. 16, 609–615 (2000)

    Article  Google Scholar 

  5. Park, B.S., Yoo, S.J., Park, J.B., Choi, Y.H.: Adaptive output-feedback control for trajectory tracking of electrically driven non-holonomic mobile robots. IET Control Theory Appl. 5, 830–838 (2011)

    Article  MathSciNet  Google Scholar 

  6. Biglarbegian, M.: A novel robust leader-following control design for mobile robots. J. Intell. Robot. Syst. 71, 391–402 (2013)

    Article  Google Scholar 

  7. Chen, C.Y., Li, T.H.S., Yeh, Y.C., Chang, C.C.: Design and implementation of an adaptive sliding-mode dynamic controller for wheeled mobile robots. Mechatronics 19, 156–166 (2009)

    Article  Google Scholar 

  8. Normey-Rico, J.E., Alcal, I., Gmez-Ortega, J., Camacho, E.F.: Mobile robot path tracking using a robust PID controller. Control Eng. Pract. 9, 1209–1214 (2001)

    Article  Google Scholar 

  9. Shojaei, K., Shahri, A.M.: Adaptive robust time-varying control of uncertain non-holonomic robotic systems. IET Control Theory Appl. 6, 90–102 (2012)

    Article  MathSciNet  Google Scholar 

  10. Xu, J.X., Guo, Z.Q., Lee, T.H.: Design and implementation of a Takagi–Sugeno-type fuzzy logic controller on a two-wheeled mobile robot. IEEE Trans. Ind. Electron. 60, 5717–5728 (2013)

    Article  Google Scholar 

  11. Chwa, D.: Fuzzy adaptive tracking control of wheeled mobile robots with state-dependent kinematic and dynamic disturbances. IEEE Trans. Fuzzy Syst. 20, 587–593 (2012)

    Article  Google Scholar 

  12. Mohareri, O., Dhaouadi, R., Rad, A.B.: Indirect adaptive tracking control of a nonholonomic mobile robot via neural networks. Neurocomputing 88, 54–66 (2012)

    Article  Google Scholar 

  13. Watanabe, K., Tang, J., Nakamura, M., et al.: A fuzzy-Gaussian neural network and its application to mobile robot control. IEEE Trans. Control Syst. Technol. 4, 193–199 (1996)

    Article  Google Scholar 

  14. Su, K.H., Chen, Y.Y., Su, S.F.: Design of neural-fuzzy-based controller for two autonomously driven wheeled robot. Neurocomputing 73, 2478–2488 (2010)

    Article  Google Scholar 

  15. Park, B.S., Yoo, S.J., Park, J.B., et al.: Adaptive neural sliding mode control of nonholonomic wheeled mobile robots with model uncertainty. IEEE Trans. Control Syst. Technol. 17, 207–214 (2009)

    Article  Google Scholar 

  16. Udwadia, F.E., Kalaba, R.E.: A new perspective on constrained motion. Proc. R. Soc. 439, 407–410 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach. Cambridge University Press, Cambridge (1996)

  18. Udwadia, F.E., Kalaba, R.E.: Explicit equations of motion for mechanical systems with non-ideal constraints. J. Appl. Mech. 68, 462–467 (2001)

    Article  MATH  Google Scholar 

  19. Udwadia, F.E.: On constrained motion. Appl. Math. Comput. 164, 313–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Udwadia, F.E., Kalaba, R.E.: On the foundations of analytical dynamics. Int. J. Non-linear Mech. 37, 1079–1090 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Udwadia, F.E., Kalaba, R.E.: What is the general form of the explicit equations of motion for constrained mechanical systems? J. Appl. Mech. 69, 335–339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Udwadia, F.E., Phohomsiri, P.: Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proc. R. Soc. A Math. Phys. Eng. Sci. 462, 2097–2117 (2006)

  23. Schutte, A.D., Dooley, B.A.: Constrained motion of tethered satellites. J. Aerosp. Eng. 18, 242–250 (2005)

    Article  Google Scholar 

  24. Pennestri, E., Valentini, P.P., de Falco, D.: An application of the Udwadia–Kalaba dynamic formulation to flexible multibody systems. J. Frankl. Inst. 347, 173–194 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Huang, Q.M., Chen, Y.H., Nie, X.: The closed-form equation of motion of a human body with joint friction. In: ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers (2013)

  26. Zhao, H., Zhen, S.C., Chen, Y.H.: Dynamic modeling and simulation of multi-body systems using the Udwadia–Kalaba theory. Chin. J. Mech. Eng. 26, 839–850 (2013)

    Article  Google Scholar 

  27. Huang, J., Chen, Y.H., Zhong, Z.H.: Udwadia–Kalaba Approach for Parallel Manipulator Dynamics. J Dyn. Syst. Meas. Control 135, 061003 (2013)

    Article  Google Scholar 

  28. Zhang, B.Z., Zhen, S.C., Zhao, H., et al.: A novel study on Kepler’s law and inverse square law of gravitation. Eur. J. Phys. 36, 035018 (2015)

    Article  MATH  Google Scholar 

  29. Zhen, S.C., Huang, K., Zhao, H., et al.: Why can a free-falling cat always manage to land safely on its feet? Nonlinear Dyn. 79, 2237–2250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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, 1783–1800 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Udwadia, F.E.: Optimal tracking control of nonlinear dynamical systems. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 464, 2341–2363 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Udwadia, F.E., Schutte, A.D.: A unified approach to rigid body rotational dynamics and control. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 468, 395–414 (2011)

  33. Cho, H., Udwadia, F.E.: Explicit solution to the full nonlinear problem for satellite formation-keeping. Acta Astronaut. 67, 369–387 (2010)

    Article  Google Scholar 

  34. Udwadia, F.E., Cho, H.: New solutions to the exact formation-keeping control of satellites with attitude constraints. Earth Space 10, 1423–1432 (2012)

    Google Scholar 

  35. Cho, H., Udwadia, F.E.: Explicit control force and torque determination for satellite formation-keeping with attitude requirements. J. Guid. Control Dyn. 36, 589–605 (2013)

    Article  Google Scholar 

  36. Udwadia, F.E., Wanichanon, T.: A closed-form approach to tracking control of nonlinear uncertain systems using the fundamental equation. Earth Space 10, 1339–1348 (2012)

    Google Scholar 

  37. Udwadia, F.E., Wanichanon, T.: Control of uncertain nonlinear multibody mechanical systems. J. Appl. Mech. 281, 041020 (2014)

    Google Scholar 

  38. Wanichanon, T., Cho, H., Udwadia, F.E.: An approach to the dynamics and control of uncertain multi-body systems. Procedia IUTAM 13, 43–52 (2015)

    Article  Google Scholar 

  39. Udwadia, F.E., Wanichanon, T., Cho, H.: Methodology for satellite formation-keeping in the presence of system uncertainties. J. Guid. Control Dyn. 37, 1611–1624 (2014)

    Article  Google Scholar 

  40. Udwadia, F.E., Koganti, P.B.: Dynamics and control of a multi-body planar pendulum. Nonlinear Dyn. 81, 845–866 (2015)

    Article  MathSciNet  Google Scholar 

  41. Braun, D.J., Goldfarb, M.: A control approach for actuated dynamic walking in biped robots. IEEE Trans. Robot. 25, 1292–1303 (2009)

    Article  Google Scholar 

  42. Lam, T.: New approach to mission design based on the fundamental equations of motion. J. Aerosp. Eng. 19, 59–67 (2006)

    Article  Google Scholar 

  43. Chen, Y.H.: Artificial swarm system: boundedness, convergence, and control. J. Aerosp. Eng. 21, 288–293 (2008)

    Article  Google Scholar 

  44. Chen, Y.H., Zhang, X.R.: Adaptive robust approximate constraint-following control for mechanical systems. J. Frankl. Inst. 347, 69–86 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  45. Chen, Y.H.: Adaptive robust control of artificial swarm systems. Appl. Math. Comput. 217, 980–987 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Xu, J., Chen, Y.H., Guo, H.G.: A new approach to control design for constraint-following for fuzzy mechanical systems. J. Optim. Theory Appl. 165, 1022–1049 (2015)

  47. Schutte, A.D.: Permissible control of general constrained mechanical systems. J. Frankl. Inst. 347, 208–227 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Gau, C.F.: Uber ein neues allgemeines Grundgesetz der Mechanik. Journal fur die reine und angewandte Mathematik 4, 232–235 (1829)

  49. Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1, 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  50. Chen, Y.H.: Constraint-following servo control design for mechanical systems. J. Vib. Control 15, 369–389 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Here we show thanks and appreciations sincerely to Associate Professor Qi Chen of Hefei University of Technology (a visiting scholar in the Ohio State University, USA) for his help during the process of research. The research is also supported in part by the Science and Technology Research Project of Anhui Province of China under Grant No. 1301021003.

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Authors and Affiliations

  1. School of Mechanical and Automotive Engineering, Hefei University of Technology, Hefei, 230009, Anhui, People’s Republic of China

    Hao Sun, Han Zhao, Shengchao Zhen, Kang Huang & Fumin Zhao

  2. Chuzhou Automobile and Home Appliance Research Institute, Chuzhou, 239000, Anhui, People’s Republic of China

    Kang Huang

  3. College of Engineering Science, University of Science and Technology of China, Hefei, 230027, Anhui, People’s Republic of China

    Xianmin Chen

  4. The George W Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Ye-Hwa Chen

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  1. Hao Sun
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  2. Han Zhao
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  3. Shengchao Zhen
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  4. Kang Huang
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  5. Fumin Zhao
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Correspondence to Shengchao Zhen.

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Sun, H., Zhao, H., Zhen, S. et al. Application of the Udwadia–Kalaba approach to tracking control of mobile robots. Nonlinear Dyn 83, 389–400 (2016). https://doi.org/10.1007/s11071-015-2335-3

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  • DOI: https://doi.org/10.1007/s11071-015-2335-3

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