Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-27T20:03:11.189Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear Optimal Control for the Wheeled Inverted Pendulum System

Published online by Cambridge University Press:  16 April 2019

G. Rigatos ,
K. Busawon ,
J. Pomares  and
M. Abbaszadeh
G. Rigatos*
Affiliation:
Unit of Industrial Automation, Industrial Systems Institute, Rion, Patras 26504, Greece
K. Busawon
Affiliation:
Nonlinear Dynamics Group, University of Northumbria, Newcastle NE1 8ST, UK. E-mail: [email protected]
J. Pomares
Affiliation:
Department of Systems Engineering, University of Alicante, 03080 Alicante, Spain. E-mail: [email protected]
M. Abbaszadeh
Affiliation:
GE Global Research, General Electric, Niskayuna, NY, 12309, USA. E-mail: [email protected]
*
*Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Summary

The article proposes a nonlinear optimal control method for the model of the wheeled inverted pendulum (WIP). This is a difficult control and robotics problem due to the system’s strong nonlinearities and due to its underactuation. First, the dynamic model of the WIP undergoes approximate linearization around a temporary operating point which is recomputed at each time step of the control method. The linearization procedure makes use of Taylor series expansion and of the computation of the associated Jacobian matrices. For the linearized model of the wheeled pendulum, an optimal (H-infinity) feedback controller is developed. The controller’s gain is computed through the repetitive solution of an algebraic Riccati equation at each iteration of the control algorithm. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, by using the H-infinity Kalman Filter as a robust state estimator, the implementation of a state estimation-based control scheme becomes also possible.

Keywords

Wheeled inverted pendulumNonlinear optimal controlH-infinity controlApproximate linearizationTaylor series expansionJacobian matricesRiccati equationLyapunov analysisGlobal stability
Type
Articles
Information
Robotica , Volume 38 , Issue 1 , January 2020 , pp. 29 - 47
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Huang, J., Ri, S., Liu, L., Wang, Y., Kim, J. and Rek, G., “Nonlinear disturbance observer-based dynamic surface control of mobile wheeled inverted pendulum,” IEEE Trans. Control Syst. Technol. 23(6), 2400–2407 (2015).10.1109/TCST.2015.2404897CrossRefGoogle Scholar
Ri, S., Huang, J., Tao, C., Ri, M., Ri, Y. and Han, D., “A High-Order Disturbance Observer Based Sliding Mode Velocity Control of Mobile Wheeled Inverted Pendulum Systems,” IEEE 12th International World Congress on Intelligent Contorl and Automation, IEEE WCICA 2016, Guilin, China (2014, June).10.1109/WCICA.2016.7578328CrossRefGoogle Scholar
Yosida, K., Sekikawa, M. and Hosomi, K., “Nonlinear analysis on purely mechanical stabilization of a wheeled inverted pendulum on a slope,” Nonlinear Dyn. 83(1–2), 905–917 (2016).10.1007/s11071-015-2376-7CrossRefGoogle Scholar
Yokoyama, K. and Takahashi, M., “Dynamics-based nonlinear acceleration control and energy shaping for a mobile inverted pendulum with a slider mechanism,” IEEE Trans. Control Syst. Technol. 24(1), 40–55 (2016).10.1109/TCST.2015.2417499CrossRefGoogle Scholar
Li, Z. and Yang, C., “Neural-adaptive output feedback control of a class of transportation vehicles based on wheeled inverted pendulum models,” IEEE Trans. Control Syst. Technol. 20(6), 1583–1591 (2012).10.1109/TCST.2011.2168224CrossRefGoogle Scholar
Tokei, T., Imamura, R. and Yuta, S., “Baggage transportation and navigation by a wheeled inverted pendulum mobile robot,” IEEE Trans. Ind. Electron. 56(10), 3985–3994 (2009).10.1109/TIE.2009.2027252CrossRefGoogle Scholar
Pathak, K., Franch, J. and Agrawal, S. K., “Velocity and position control of a wheeled inverted pendulum by partial feedback linearization,” IEEE Trans. Rob. 21(3), 505–513 (2005).10.1109/TRO.2004.840905CrossRefGoogle Scholar
Brissila, R. M. and Sankarannarayanan, V., “Nonlinear control of mobile inverted pendulum,” Rob. Auton. Syst. 70, 145–155 (2015).10.1016/j.robot.2015.02.012CrossRefGoogle Scholar
Fukushima, H., Kakue, M., Kon, K. and Matsuno, F., “Transformation control of an inverted pendulum for a mobile robot with wheel-arms using partial feedback linearization and polytopic model set,” IEEE Trans. Rob. 23(3), 774–783 (2013).10.1109/TRO.2013.2239555CrossRefGoogle Scholar
Maralidharan, V. and Mahindrakar, A. D., “Position stabilization and waypoint tracking control of mobile inverted pendulum robot,” IEEE Trans. Control Syst. Technol. 22(6), 2360–2367 (2014).10.1109/TCST.2014.2300171CrossRefGoogle Scholar
Xu, J. Y., Guo, Z. G. and Lee, T. H., “Design and implementation of integral sliding-mode control on an underactuated two-wheeled mobile robot,” IEEE Trans. Ind. Electron. 61(7), 3671–3681 (2014).10.1109/TIE.2013.2282594CrossRefGoogle Scholar
Huang, J., Ding, F., Fukuda, T., and Matsuno, T., “Modelling and velocity control for a novel narrow vehicle based on mobile wheeled inverted pendulum,” IEEE Trans. Control Syst. Technol. 21(5), 1607–1617 (2013).10.1109/TCST.2012.2214439CrossRefGoogle Scholar
Huang, J., Guan, Z. H., Matsuno, T., Fukuda, T. and Sekiyama, K., “Sliding-mode velocity control of mobile-wheeled inverted-pendulum systems,” IEEE Trans. Rob. 26(4), 750–758 (2010).10.1109/TRO.2010.2053732CrossRefGoogle Scholar
Dai, F., Gao, X., Jiang, S., Guo, W. and Liu, Y., “A two-wheeled inverted pendulum robot with friction compensation,” Mechatronics 30, 116–125 (2015).10.1016/j.mechatronics.2015.06.011CrossRefGoogle Scholar
Zhou, Y. and Wang, Z., “Motion controller design of wheeled inverted pendulum with an input delay via optimal control theory,” J. Optim. Theory Appl. 138(2), 625–645 (2016).10.1007/s10957-015-0759-zCrossRefGoogle Scholar
Yang, C., Li, Z., Cui, R. and Xu, B., “Neural network-based motion control of underactuated wheeled inverted pendulum models,” IEEE Trans. Neural Networks Learn. Syst. 25(11), 2004–2016 (2014).10.1109/TNNLS.2014.2302475CrossRefGoogle Scholar
Tsai, C. C., Huang, H. C. and Lin, S. C., “Adaptive neural network control of a self-balancing two-wheeled scooter,” IEEE Trans. Ind. Electron. 57(4), 1420–1428 (2010).10.1109/TIE.2009.2039452CrossRefGoogle Scholar
Ravichandran, M. T. and Mahindrakar, A. D., “Robust stabilization of a class of underactuated mechanical systems using time-scaling and Lyapunov redesign,” IEEE Trans. Ind. Electron. 58(3), 4299–4213 (2011).10.1109/TIE.2010.2102318CrossRefGoogle Scholar
Li, Z. and Luo, J., “Adaptive robust dynamic balance and motion controls of mobile wheeled inverted pendulums,” IEEE Trans. Control Syst. Technol. 17(1), 233–241 (2009).Google Scholar
Ye, W., Li, Z., Yang, C., Sun, J., Su, C. Y. and Lu, R., “Vision-based human tracking control of a wheeled inverted pendulum robot,” IEEE Trans. Cybern. 46(1), 2423–2434 (2016).10.1109/TCYB.2015.2478154CrossRefGoogle ScholarPubMed
Yue, M., Wang, S. and Sun, J. Z., “Simultaneous balancing and trajectory tracking control for two-wheeled inverted pendulum vehicles: a composite control approach,” Neurocomputing 101, 44–54 (2016).10.1016/j.neucom.2016.01.008CrossRefGoogle Scholar
Yue, M., An, C., Du, Y. and Sun, J., “Indirect adaptive fuzzy control for a nonholonomic/underactuated wheeled inverted pendulum vehicle based on a data-driven trajectory planning,” Fuzzy Sets Syst. 290, 158–177 (2016).10.1016/j.fss.2015.08.013CrossRefGoogle Scholar
Rigatos, G., “Modelling and Control for Intelligent Industrial Systems,” In: Adaptive Algorithms in Robotics and Industrial Engineering (Springer, Berlin, Heidelberg, 2011).Google Scholar
Rigatos, G., Nonlinear Control and Filtering Using Differential Flatness Approaches: Applications to Electromechanicsl Systems (Springer, Switzerland, 2015).10.1007/978-3-319-16420-5CrossRefGoogle Scholar
Rigatos, G., Intelligent Renewable Energy systems: Modelling and Control (Springer, Switzerland, 2017).Google Scholar
Rigatos, G., Siano, P. and Cecati, C., “A new nonlinear H-infinity feedback control approach for three-phase voltage source converters,” Electr. Power Compon. Syst. 44(3), 302–312 (2015).10.1080/15325008.2015.1092056CrossRefGoogle Scholar
Rigatos, G. G. and Tzafestas, S. G., “Extended Kalman filtering for fuzzy modelling and multi-sensor fusion,” Math. Comput. Modell. Dyn. Syst. 13, 251–266 (2007).10.1080/01443610500212468CrossRefGoogle Scholar
Basseville, M. and Nikiforov, I., Detection of Abrupt Changes: Theory and Applications (Prentice-Hall, Upper Saddle River, New Jersey, USA, 1993).Google Scholar
Rigatos, G. and Zhang, Q., “Fuzzy model validation using the local statistical approach,” Fuzzy Sets Syst. 60(7), 882–904 (2009).10.1016/j.fss.2008.07.008CrossRefGoogle Scholar
Toussaint, G. J., Basar, T. and Bullo, F., “H ∞ Optimal Tracking Control Techniques for Nonlinear Underactuated Systems,” Proceedings of IEEE CDC 2000, 39th IEEE Conference on Decision and Control, Sydney, Australia (2000, December).Google Scholar
Lublin, L. and Athans, M., “An Experimental Comparison of and Designs for Interferometer Testbed,” In: Feedback Control, Nonlinear Systems and Complexity (Francis, B. and Tannenbaum, A., eds.) (Springer, Berlin, Heidelberg, 1995) pp. 150–172.10.1007/BFb0027676CrossRefGoogle Scholar
Gibbs, B. P., Advanced Kalman Filtering, Least Squares and Modelling: A Practical Handbook (J. Wiley, Hoboken, New Jersey, USA, 2011).10.1002/9780470890042CrossRefGoogle Scholar
Simon, D., “A game theory approach to constrained minimax state estimation,” IEEE Trans. Signal Process. 54(2), 405–412 (2006).10.1109/TSP.2005.861732CrossRefGoogle Scholar
Wang, Z. J. and Wang, F. C., “The Development and Control of a Two-Wheeled Inverted Pendulum,” Proceedings of the IEEE SICE 2017 International Conference, Kanazawa, Japan (2017, September).10.23919/SICE.2017.8105588CrossRefGoogle Scholar
Phogat, K. S., Banavar, R. and Chatterjee, D., “Structure Preserving Discrete-Time Optimal Maneuvers of a Wheeled Inverted Pendulum,” 6th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2018, Valparaiso, Chile (2018, May).10.1016/j.ifacol.2018.06.042CrossRefGoogle Scholar
Huang, J., Ri, M., Wu, D. and Ri, S., “Interval type-2 fuzzy logic modeling and control of a mobile two-wheeled inverted pendulum,” IEEE Trans. Fuzzy Syst. 26(4), 2039–2038 (2018).10.1109/TFUZZ.2017.2760283CrossRefGoogle Scholar
Yue, M., An, C., and Li, Z., “Constrained adaptive robust trajectory tracking for WIP vehicles using model predictive control and extended state observer,” IEEE Trans. Syst. Man Cybern. 48(5), 733–742 (2018).10.1109/TSMC.2016.2621181CrossRefGoogle Scholar
Sihite, E. and Bewley, T., “Attitude Estimation of a High-Yaw-Rate Mobile Inverted Pendulum; Comparison of Extended Kalman Filtering, Complementary Filtering, and Motion Captureitalic,” IEEE ACC 2018, Annual American Control Conference (ACC) Wisconsin Center, Milwaukee, USA (2018, June).10.23919/ACC.2018.8431624CrossRefGoogle Scholar
Zhang, Y., Zhang, L., Wang, W., Li, Y. and Zhang, Q., “Design and implementation of a two-wheel and hopping robot with a linkage mechanism,” IEEE Access 6, 42422–42430 (2018).10.1109/ACCESS.2018.2859840CrossRefGoogle Scholar
Weiss, A., Hadida, E. and Hanan, U. B., “Optimizing step climbing by two connected wheeled inverted pendulum models,” Procedia Manuf. 21, 236–242 (2018).10.1016/j.promfg.2018.02.116CrossRefGoogle Scholar
Han, S. and Lee, J. M., “Balancing and velocity control of a unicycle robot based on the dynamic,” IEEE Trans. Ind. Electron. 68(1), 405–413 (2015).10.1109/TIE.2014.2327562CrossRefGoogle Scholar
Dai, F., Gao, X., Jiang, S., Guo, W. and Liu, Y., “A two-wheeled inverted pendulum robot with friction compensation,” Mechatronics 30, 116–125 (2015).10.1016/j.mechatronics.2015.06.011CrossRefGoogle Scholar