# American Institute of Mathematical Sciences

2014, 4(3): 181-191. doi: 10.3934/naco.2014.4.181

## Robust control design of autonomous bicycle kinematics

 1 Systems Engineering Department, King Fahd University of Petroleum and Minerals, P. O. Box 5067, Dhahran 31261, Saudi Arabia, Saudi Arabia

Received  August 2013 Revised  April 2014 Published  September 2014

In this paper, we provide a robust control approach for controlling the autonomous bicycle kinematics with the objective of stabilizing the bicycle steer $\delta$ and roll $\phi$ angles. The dynamical model is the so-called 'Whipples Bicycle Model', where the roll (lean) angle and the steer angle of the bicycle are the two outputs of the model and the torques across the roll and steer angle as the two control variables. Two control design methods are developed based on $H_\infty$ and $H_2$-norm optimization using dynamic output feedback. The ensuing results are compared with an adaptive control scheme. The autonomous bicycle was tested for varying velocities.
Citation: Magdi S. Mahmoud, Omar Al-Buraiki. Robust control design of autonomous bicycle kinematics. Numerical Algebra, Control and Optimization, 2014, 4 (3) : 181-191. doi: 10.3934/naco.2014.4.181
##### References:
 [1] K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control, IEEE Control Systems Magazine, 25 (2005), 26-47. doi: 10.1109/MCS.2005.1499389. [2] C. K. Chen and T. K. Dao, Speed-adaptive roll-angle-tracking control of an unmanned bicycle using fuzzy logic, Vehicle System Dynamics, 48 (2010), 133-147. [3] C. Cornejo and L. Alvarez-Icaza, Passivity based control of under-actuated mechanical systems with nonlinear dynamic friction, J. Vibration and Control, 18 (2012), 1025-1042. doi: 10.1177/1077546311408469. [4] M. L. Fair and S. L. Campbell, Active incipient fault detection in continuous time systems with multiple simultaneous faults, Numerical Algebra, Control and Optimization, 1 (2011), 211-224. doi: 10.3934/naco.2011.1.211. [5] L. Feng, Robust Control Design: An Optimal Control Approach, Wayne State University, USA and Tongji University, China, John Wiley and Sons Ltd, 2007. [6] N. H. Getz, Dynamic Inversion of Nonlinear Maps with Applications to Nonlinear Control and Robotics, Ph.D. Dissertation, University of California, 1995. [7] Y. Harata, Y. Banno and K. Taji, Parametric excitation based bipedal walking: Control method and optimization, Numerical Algebra, Control and Optimization, 1 (2011), 171-190. doi: 10.3934/naco.2011.1.171. [8] C. L. Hwang, H. M. Wu and C. L. Shih, Fuzzy sliding-mode underactuated control for autonomous dynamic balance of an electrical bicycle, IEEE Trans. Control Systems Technology, 17 (2009), 658-670. [9] N. H. K. Iuchi, H. Niki and T. Murakami, Attitude control of bicycle motion by steering angle and variable COG control, Proc. 31st Annual Conference of IEEE Industrial Electronics Society, IECON, (2005), 16-21. [10] R. N. Jazar, Mathematical theory of auto-driver for autonomous vehicles, J. Vibration and Control, 16 (2010), 253-279. doi: 10.1177/1077546309104467. [11] R. Khaled and N. G. Chalhoub, A dynamic model and a robust controller for a fully-actuated marine surface vessel, J. Vibration and Control, 17 (2011), 801-812. [12] L. Lujng, System Identification Theory for User, Linkopping University, Sweden. [13] M. S. Mahmoud, Computer-Operated Systems Control, Marcel Dekker Inc., New York, 1991. [14] M. S. Mahmoud, Robust control of blood gases during extracorporeal circulation, IET Control Theory and Applications, 5 (2011), 1577-1585. doi: 10.1049/iet-cta.2010.0665. [15] M. S. Mahmoud, Resilient L2 ⁄ L∞ filtering of polytopic systems with state delays, IET Control Theory And Applications, 1 (2007), 141-154. doi: 10.1049/iet-cta:20045281. [16] M. S. Mahmoud and A. Y. Al-Rayyah, Efficient parameterisation to stability and feedback synthesis of linear time-delay systems, IET control theory and applications, 3 (2009), 1107-1118. doi: 10.1049/iet-cta.2008.0152. [17] M. S. Mahmoud and Yuanqing Xia, Robust filter design for piecewise discrete-time systems with time-varying delays, International Journal of Robust and Nonlinear Control, 20 (2010), 544-560. doi: 10.1002/rnc.1447. [18] M. S. Mahmoud and M. M. Hussain, Design of linear systems with saturating actuators: A survey, Int. J. Numerical Algebra, Control and Optimization, 2 (2012), 413-435. doi: 10.3934/naco.2012.2.413. [19] J. Meijaard, J. Papadopoulos, A. Ruina and A. Schwab, Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review, Proc. the Royal Society A: Mathematical, Physical and Engineering Science, 463 (2007). doi: 10.1098/rspa.2007.1857. [20] K. Mendrok and Tadeusz Uhl, Load identification using a modified modal filter technique, J. Vibration and Control, 16 (2010), 89-105. doi: 10.1177/1077546309103274. [21] G. T. Michaltsos, Bouncing of a vehicle on an irregularity: A mathematical model, J. Vibration and Control, 16 (2010), 181-206. doi: 10.1177/1077546309104878. [22] H. Moradi, M. R. Movahhedy, and G. Vossoughi, Sliding mode control of machining chatter in the presence of tool wear and parametric uncertainties, J. Vibration and Control, 16 (2010), 231-251. [23] U. Nenner, R. Linker and P. Gutman, Robust feedback stabilization of an unmanned motorcycle, Control Engineering Practice, 2010. [24] Omar S. Al-Buraiki and El Ferik, Sami, Adaptive control of autonomous bicycle kinematics, Proc. 13th Automation and Systems (ICCAS), Gwangju, Korea, Oct. (2013), 20-23. [25] M. C. Pai, Sliding mode control of vibration in uncertain time-delay systems, J. Vibration and Control, 16 (2010),2131-2145. doi: 10.1177/1077546309350865. [26] H. Schttler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation, Int. J. Numerical Algebra, Control and Optimization, 2 (2012), 631-654. doi: 10.3934/naco.2012.2.631. [27] R. Sharp and D. Limebeer, A motorcycle model for stability and control analysis, Multi-body System Dynamics, 6 (2001), 123-142. [28] R. Sharp, Optimal preview speed-tracking control for motorcycles, Multi-body System Dynamics, 18 (2007), 397-411. [29] S. Sivrioglu, H∞ control for suppressing acoustic modes of a distributed structure using cluster sensing and actuation, J. Vibration and Control, 16 (2010), 439-453. [30] N. Umashankar and H. D. Sharma, Adaptive neuro-fuzzy controller for stabilizing autonomous bicycle, Proc. IEEE International Conference Robotics and Biometrics, ROBIO06, (2006), 1652-1657. [31] T. Yamaguchi, T. Shibata and T. Murakami, Self-sustaining approach of electric bicycle by acceleration control based backstepping, Proc. 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON, (2007), 2610-2614. [32] K. Zhou and J. C. Doyle, Essentials of Robust Control, NJ: Prentice Hall, 1998.

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##### References:
 [1] K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control, IEEE Control Systems Magazine, 25 (2005), 26-47. doi: 10.1109/MCS.2005.1499389. [2] C. K. Chen and T. K. Dao, Speed-adaptive roll-angle-tracking control of an unmanned bicycle using fuzzy logic, Vehicle System Dynamics, 48 (2010), 133-147. [3] C. Cornejo and L. Alvarez-Icaza, Passivity based control of under-actuated mechanical systems with nonlinear dynamic friction, J. Vibration and Control, 18 (2012), 1025-1042. doi: 10.1177/1077546311408469. [4] M. L. Fair and S. L. Campbell, Active incipient fault detection in continuous time systems with multiple simultaneous faults, Numerical Algebra, Control and Optimization, 1 (2011), 211-224. doi: 10.3934/naco.2011.1.211. [5] L. Feng, Robust Control Design: An Optimal Control Approach, Wayne State University, USA and Tongji University, China, John Wiley and Sons Ltd, 2007. [6] N. H. Getz, Dynamic Inversion of Nonlinear Maps with Applications to Nonlinear Control and Robotics, Ph.D. Dissertation, University of California, 1995. [7] Y. Harata, Y. Banno and K. Taji, Parametric excitation based bipedal walking: Control method and optimization, Numerical Algebra, Control and Optimization, 1 (2011), 171-190. doi: 10.3934/naco.2011.1.171. [8] C. L. Hwang, H. M. Wu and C. L. Shih, Fuzzy sliding-mode underactuated control for autonomous dynamic balance of an electrical bicycle, IEEE Trans. Control Systems Technology, 17 (2009), 658-670. [9] N. H. K. Iuchi, H. Niki and T. Murakami, Attitude control of bicycle motion by steering angle and variable COG control, Proc. 31st Annual Conference of IEEE Industrial Electronics Society, IECON, (2005), 16-21. [10] R. N. Jazar, Mathematical theory of auto-driver for autonomous vehicles, J. Vibration and Control, 16 (2010), 253-279. doi: 10.1177/1077546309104467. [11] R. Khaled and N. G. Chalhoub, A dynamic model and a robust controller for a fully-actuated marine surface vessel, J. Vibration and Control, 17 (2011), 801-812. [12] L. Lujng, System Identification Theory for User, Linkopping University, Sweden. [13] M. S. Mahmoud, Computer-Operated Systems Control, Marcel Dekker Inc., New York, 1991. [14] M. S. Mahmoud, Robust control of blood gases during extracorporeal circulation, IET Control Theory and Applications, 5 (2011), 1577-1585. doi: 10.1049/iet-cta.2010.0665. [15] M. S. Mahmoud, Resilient L2 ⁄ L∞ filtering of polytopic systems with state delays, IET Control Theory And Applications, 1 (2007), 141-154. doi: 10.1049/iet-cta:20045281. [16] M. S. Mahmoud and A. Y. Al-Rayyah, Efficient parameterisation to stability and feedback synthesis of linear time-delay systems, IET control theory and applications, 3 (2009), 1107-1118. doi: 10.1049/iet-cta.2008.0152. [17] M. S. Mahmoud and Yuanqing Xia, Robust filter design for piecewise discrete-time systems with time-varying delays, International Journal of Robust and Nonlinear Control, 20 (2010), 544-560. doi: 10.1002/rnc.1447. [18] M. S. Mahmoud and M. M. Hussain, Design of linear systems with saturating actuators: A survey, Int. J. Numerical Algebra, Control and Optimization, 2 (2012), 413-435. doi: 10.3934/naco.2012.2.413. [19] J. Meijaard, J. Papadopoulos, A. Ruina and A. Schwab, Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review, Proc. the Royal Society A: Mathematical, Physical and Engineering Science, 463 (2007). doi: 10.1098/rspa.2007.1857. [20] K. Mendrok and Tadeusz Uhl, Load identification using a modified modal filter technique, J. Vibration and Control, 16 (2010), 89-105. doi: 10.1177/1077546309103274. [21] G. T. Michaltsos, Bouncing of a vehicle on an irregularity: A mathematical model, J. Vibration and Control, 16 (2010), 181-206. doi: 10.1177/1077546309104878. [22] H. Moradi, M. R. Movahhedy, and G. Vossoughi, Sliding mode control of machining chatter in the presence of tool wear and parametric uncertainties, J. Vibration and Control, 16 (2010), 231-251. [23] U. Nenner, R. Linker and P. Gutman, Robust feedback stabilization of an unmanned motorcycle, Control Engineering Practice, 2010. [24] Omar S. Al-Buraiki and El Ferik, Sami, Adaptive control of autonomous bicycle kinematics, Proc. 13th Automation and Systems (ICCAS), Gwangju, Korea, Oct. (2013), 20-23. [25] M. C. Pai, Sliding mode control of vibration in uncertain time-delay systems, J. Vibration and Control, 16 (2010),2131-2145. doi: 10.1177/1077546309350865. [26] H. Schttler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation, Int. J. Numerical Algebra, Control and Optimization, 2 (2012), 631-654. doi: 10.3934/naco.2012.2.631. [27] R. Sharp and D. Limebeer, A motorcycle model for stability and control analysis, Multi-body System Dynamics, 6 (2001), 123-142. [28] R. Sharp, Optimal preview speed-tracking control for motorcycles, Multi-body System Dynamics, 18 (2007), 397-411. [29] S. Sivrioglu, H∞ control for suppressing acoustic modes of a distributed structure using cluster sensing and actuation, J. Vibration and Control, 16 (2010), 439-453. [30] N. Umashankar and H. D. Sharma, Adaptive neuro-fuzzy controller for stabilizing autonomous bicycle, Proc. IEEE International Conference Robotics and Biometrics, ROBIO06, (2006), 1652-1657. [31] T. Yamaguchi, T. Shibata and T. Murakami, Self-sustaining approach of electric bicycle by acceleration control based backstepping, Proc. 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON, (2007), 2610-2614. [32] K. Zhou and J. C. Doyle, Essentials of Robust Control, NJ: Prentice Hall, 1998.
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