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Nonlinear optimal control for a spherical rolling robot

  • G. Rigatos
  • K. Busawon
  • J. Pomares
  • M. Abbaszadeh
Regular Paper
  • 52 Downloads

Abstract

The article presents a nonlinear H-infinity (optimal) control approach for the problem of the control of the spherical rolling robot. The solution of such a control problem is a nontrivial case due to underactuation and strong nonlinearities in the system’s state-space description. The dynamic model of the robot undergoes approximate linearization around a temporary operating point which is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the computation of the system’s Jacobian matrices. For the linearized dynamics of the spherical robot an H-infinity controller is designed. To compute the controller’s feedback gains an algebraic Riccati equation in solved at each iteration of the control algorithm. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, for the implementation of sensorless control for the spherical rolling robot, the H-infinity Kalman Filter is used as a robust state estimator.

Keywords

Spherical rolling robot Underactuation Nonlinear optimal control H-infinity control Jacobian matrices Riccati equation 

References

  1. Azizi, M.R., Keighbodi, J.: Point stabilization of nonholonomic spherical mobile robot using nonlinear model predictive control. Robot. Autonom. Syst. 98, 347–359 (2017)CrossRefGoogle Scholar
  2. Basseville, M., Nikiforov, I.: Detection of Abrupt Changes: Theory and Applications. Prentice-Hall, Upper Saddle River (1993)Google Scholar
  3. Bhattacharya, S., Agrawal, S.K.: Spherical rolling robot: a design and motion planning studies. IEEE Trans. Robot. Autom. 16(6), 835–899 (2000)CrossRefGoogle Scholar
  4. Bicchi, A., Balladi, A., Prattichizzo, D., Gorelli, A.: Introducingthe Sphericle: an experimental method for research and teaching in nonholonomy, IEEE ICRA 1997. In: Proc. of the 1997 IEEE Intl. Conf. on Robotics and Automation, Albuquerque, New Mexico (1997)Google Scholar
  5. Chen, J., Ye, P., Sun, H., Jin, Q.: Design and motion control of a spherical robot with control moment gyroscope. In: The 2016 3rd Intl. Conf. on Systems and Informatics, IEEE ICSAI 2016, Shanghai, China (2016)Google Scholar
  6. Chen, W.H., Chen, C.P., Tsai, J.S., Xiong, J., Lin, P.C.: Design and implementation of a ball-driven omnidirectional spherical robot. Mech. Mach. Theory 68, 35–48 (2013)CrossRefGoogle Scholar
  7. Chiu, C.H., Tsai, W.R.: Design and implementation of an omnidirectional spherical mobile platform. IEEE Trans. Ind. Electron. 62(3), 1619–1628 (2015)CrossRefGoogle Scholar
  8. Coricia, C., Conticelli, F., Bicchi, A.: Nonholonomic kinematics and dynamics of the Sphericle, IEEE IROS 2000. In: Proc. of the IRRR /RSJ2000 Intl. Conf. on Intelligent Robots and Systems, Takamatsu, Japan (2000)Google Scholar
  9. Gareshin, G.A., Keshmri, S., Shakli, D.: Nonlinear control based on H-infinity theory for autonomous aerial robots. In: 2017 Intl. Conf. of Unmanned Aircraft Systems, IETE ICUAJ 2017, Miami, Florida (2017)Google Scholar
  10. Gibbs, B.P.: Advanced Kalman Filtering, Least Squares and Modelling: A Practical Handbook. Wiley, Oxford (2011)CrossRefGoogle Scholar
  11. Gojbhiye, S., Banavar, R.M.: The Euleur-Poincaré equation for a spherical robot actuated by a pendulum. In: 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for nonlinear control. Bertinor, Italy (2012)Google Scholar
  12. Ishikawa, M., Kitayashi, R., Sugie, T.: Dynamic rolling locomotion by spherical mobile robot considering its generalized momentum, IEEE SICE 2010. Taipei, Taiwan (2010)Google Scholar
  13. Jia, Y.B.: Planning the initial motion fof a free sliding/rolling ball. IEEE Trans. Robot. 32(3), 566–584 (2016)CrossRefGoogle Scholar
  14. Kayacan, E., Bayraktaroglou, Z.Y., Saeyes, W.: Modelling and control of a spherical rolling robot: a decoupled dynamics approach. Robotica 30(4), 671–690 (2012)CrossRefGoogle Scholar
  15. Kayacan, E., Kayacan, E., Roman, H., Saeyes, W.: Adaptive neuro-fuzzy control of a spherical rolling robot using sliding-mode control-theory-based online learning algorithm. IEEE Trans. Cybern. 43(1), 170–179 (2013)CrossRefGoogle Scholar
  16. Kilin, A.A., Pivovarova, E.N., Ivanova, T.B.: Spherical Robot of Combined Type: Dynamics and Control, Regular and Chaotic Dynamics, pp. 716–728. Springer, Berlin (2015)zbMATHGoogle Scholar
  17. Lin, D., Sun, H.: Nonlinear sliding-mode control for motion of a spherical robot. In: Proc. of the 29th IEEE Chinese Control Conference, Beijing, China (2010)Google Scholar
  18. Liu, D., Sun, H., Jin, Q.: Stabilization and path following of a spherical robot. In: 2008 IEEE Conference on Robotics, Automation and Mechatronics, Changdu, China (2008)Google Scholar
  19. Liu, D., Sun, H.: Nonlinear sliding-mode control for motion of a spherical robot. In: Proc. of the 29th IEEE Chinese Control Conference, Beijing, China (2010)Google Scholar
  20. Lublin, L., Athans, M.: An experimental comparison of and designs for interferometer testbed. In: Francis, B., Tannenbaum, A. (eds.) Lectures Notes in Control and Information Sciences: Feedback Control, Nonlinear Systems and Complexity, pp. 150–172. Springer, Berlin (1995)zbMATHGoogle Scholar
  21. Madhashani, T.W.U., Maithripola, D.H.S., Berg, J.M.: Feedback regularization and geometric PID control for trajectory tracking of mechanical systems: hoop robots on an inclined plane. In: 2017 American Control Conference. Seattle, USA (2017)Google Scholar
  22. Madhashani, T.W.U., Maithripola, D.H.S., Wijayakalasooriya, J.V., Berg, J.M.: Semi-globally exponential trajectory tracking for a class of spherical robots. Automatica 85, 327–339 (2017)MathSciNetCrossRefGoogle Scholar
  23. Maralidharan, V., Mahindrakar, A.D.: Geometric controllability and stabilization of spherical robot dynamics. IEEE Trans. Autom. Control 60(10), 2762–2767 (2015)MathSciNetCrossRefGoogle Scholar
  24. Meng, Y., Baoyin, L.: Disturbance adaptive control for an underactuated spherical robot based on hierarchical sliding-mode technology. In: Proc. of the 31st Chinese Control Conference, Hefei, China, (2012)Google Scholar
  25. Michaud, F., Caron, S.: Roball, the Rolling Robot, Autonomous Robots, vol. 12, pp. 211–222. Springer, Berlin (2002)zbMATHGoogle Scholar
  26. Morinaga, A., Svinin, M., Yamamoto, M.: A motion planning strategy for a spherical rolling robot driven by two internal motors. IEEE Trans. Robot. 30(4), 993–1002 (2014)CrossRefGoogle Scholar
  27. Niu, X., Saherlan, A.P., Soh, G.S., Feong, S., Word, K., Otta, K.: Mechanical development and control of a miniature nonholonomic spherical rolling robot. In: IEEE ICARCV 2014, 13th Intl. Conf. on Control, Automation, Robotics and Vision, Singapore (2014)Google Scholar
  28. Rigatos, G.G.: Modelling and Control for Intelligent Industrial Systems: Adaptive Algorithms in Robotics and Industrial Engineering. Springer, Berlin (2011)CrossRefGoogle Scholar
  29. Rigatos, G.: Nonlinear Control and Filtering Using Differential Flatness Approaches: Applications to Electromechanicsl Systems. Springer, Berlin (2015)CrossRefGoogle Scholar
  30. Rigatos, G., Busawon, K.: Robotic Manipulators and Vehicles: Control, Estimation and Filtering. Springer, Berlin (2017)zbMATHGoogle Scholar
  31. Rigatos, G.G., Tzafestas, S.G.: Extended Kalman filtering for fuzzy modelling and multi-sensor fusion. Math. Comput. Model. Dyn. Syst. 13, 251–266 (2007)MathSciNetCrossRefGoogle Scholar
  32. Rigatos, G., Zhang, Q.: Fuzzy model validation using the local statistical approach. Fuzzy Sets Syst. 60(7), 882–904 (2009)MathSciNetCrossRefGoogle Scholar
  33. Rigatos, G., Siano, P., Cecati, C.: A New Nonlinear H-infinity Feedback Control Approach for Three-phase Voltage Source Converters, Electric Power Components and Systems. Taylor and Francis, Routledge (2015)Google Scholar
  34. Roozegar, M., Mahjoob, M.J.: Modelling and control of a nonholonomic pendulum-driven spherical robot moving on an inclined plane: simulation and experimental results. IEEE Control Theory Appl. 11(4), 541–549 (2017)CrossRefGoogle Scholar
  35. Rouzeger, M., Mahjoob, M.J.: Modelling and control of non-holonomic pendulum-driven spherical robot moing on an inclined plane: simulation and experimental results. IET Control Theory Appl. 11(4), 541–549 (2017)MathSciNetCrossRefGoogle Scholar
  36. Simon, D.: A game theory approach to constrained minimax state estimation. IEEE Trans. Signal Process. 54(2), 405–412 (2006)CrossRefGoogle Scholar
  37. Svinin, M., Marinaga, A., Yamamoto, M.: On the geometric phase approach to motion planning for a spherical rolling robot in dynamic formulation. In: 2013 IEEE/RST Intl. Conf. on Intelligent Robots and Systems, IROS 2013, Tokyo, Japan (2013)Google Scholar
  38. Toussaint, G.J., Basar, T., Bullo, F.: \(H_{\infty }\) optimal tracking control techniques for nonlinear underactuated systems. In: Proc. IEEE CDC 2000, 39th IEEE Conference on Decision and Control, Sydney Australia (2000)Google Scholar
  39. Urakawa, T., Monno, M., Mackawa, S., Tamaki, H.: Dynamic modelling and controller design for a spherical rolling robot equiped with a gyro. IEEE Trans. Control Syst. Technol. 24(5), 1669–1679 (2016)CrossRefGoogle Scholar
  40. Ylikorpi, T., Forsman, P., Halme, A., Saarinen, J.: Unified representation of decoupled dynamic models for pendulum-driven ball-shaped robots. In: Proc. of the 28th European Conference on Modelling and Simulation (2014)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Unit of Industrial Automation Industrial Systems InstituteRion PatrasGreece
  2. 2.Nonlinear Control Group Northumbria UniversityNewcastleUK
  3. 3.Department of Systems EngineeringUniversity of AlicanteAlicanteSpain
  4. 4.GE Global Research General Electric Co.NiskayunaUSA

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