Journal of Applied Mathematics

Dynamical Analysis and Stabilizing Control of Inclined Rotational Translational Actuator Systems

Bingtuan Gao and Fei Ye

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Rotational translational actuator (RTAC) system, whose motions occur in horizontal planes, is a benchmark for studying of control techniques. This paper presents dynamical analysis and stabilizing control design for the RTAC system on a slope. Based on Lagrange equations, dynamics of the inclined RTAC system is achieved by selecting cart position and rotor angle as the general coordinates and torque acting on the rotor as general force. The analysis of equilibriums and their controllability yields that controllability of equilibriums depends on inclining direction of the inclined RTAC system. To stabilize the system to its controllable equilibriums, a proper control Lyapunov function including system energy, which is used to show the passivity property of the system, is designed. Consequently, a stabilizing controller is achieved directly based on the second Lyapunov stability theorem. Finally, numerical simulations are performed to verify the correctness and feasibility of our dynamical analysis and control design.

Article information

J. Appl. Math., Volume 2014 (2014), Article ID 598384, 9 pages.

First available in Project Euclid: 2 March 2015

Permanent link to this document

Digital Object Identifier


Gao, Bingtuan; Ye, Fei. Dynamical Analysis and Stabilizing Control of Inclined Rotational Translational Actuator Systems. J. Appl. Math. 2014 (2014), Article ID 598384, 9 pages. doi:10.1155/2014/598384.

Export citation


  • M. W. Spong, “Underactuated mechanical systems,” in Control Problems in Robotics and Automation, B. Siciliano and K. P. Valavanis, Eds., pp. 135–150, Springer, London, UK, 1998.
  • M. Reyhanoglu, A. van der Schaft, N. H. McClamroch, and I. Kolmanovsky, “Dynamics and control of a class of underactuated mechanical systems,” IEEE Transactions on Automatic Control, vol. 44, no. 9, pp. 1663–1671, 1999.
  • I. Fantoni and R. Lozano, Nonlinear Control for Underactuated Mechanical Systems, Springer, London, UK, 2002.
  • C.-J. Wan, D. S. Bernstein, and V. T. Coppola, “Global stabilization of the oscillating eccentric rotor,” Nonlinear Dynamics, vol. 10, no. 1, pp. 49–62, 1996.
  • R. T. Bupp, D. S. Bernstein, and V. T. Coppola, “A benchmark problem for nonlinear control design,” International Journal of Robust and Nonlinear Control, vol. 8, no. 4-5, pp. 307–310, 1998.
  • M. Jankovic, D. Fontaine, and P. V. Kokotović, “TORA example: cascade- and passivity-based control designs,” IEEE Transactions on Control Systems Technology, vol. 4, no. 3, pp. 292–297, 1996.
  • R. T. Bupp, D. S. Bernstein, and V. T. Coppola, “Experimental implementation of integrator backstepping and passive nonlinear controllers on the RTAC testbed,” International Journal of Robust and Nonlinear Control, vol. 8, no. 4-5, pp. 435–457, 1998.
  • C.-H. Lee and S.-K. Chang, “Experimental implementation of nonlinear TORA system and adaptive backstepping controller design,” Neural Computing and Applications, vol. 21, no. 4, pp. 785–800, 2012.
  • P. Tsiotras, M. Corless, and M. A. Rotea, “An $L2$ disturbance attenuation solution to the nonlinear benchmark problem,” International Journal of Robust and Nonlinear Control, vol. 8, no. 4-5, pp. 311–330, 1998.
  • Z. Petres, P. Baranyi, and H. Hashimoto, “Approximation and complexity trade-off by TP model transformation in controller design: a case study of the TORA system,” Asian Journal of Control, vol. 12, no. 5, pp. 575–585, 2010.
  • J. She, A. Zhang, X. Lai, and M. Wu, “Global stabilization of 2-DOF underactuated mechanical systems–-an equiavlent-input-disturbance approach,” Nonlinear Dynamics, vol. 69, no. 1-2, pp. 495–509, 2012.
  • T. Burg and D. Dawson, “Additional notes on the TORA example: a filtering approach to eliminate velocity measurements,” IEEE Transactions on Control Systems Technology, vol. 5, no. 5, pp. 520–523, 1997.
  • G. Escobar, R. Ortega, and H. Sira-Ramírez, “Output-feedback global stabilization of a nonlinear benchmark system using a saturated passivity-based controller,” IEEE Transactions on Control Systems Technology, vol. 7, no. 2, pp. 289–293, 1999.
  • F. Celani, “Output regulation for the TORA benchmark via rotational position feedback,” Automatica, vol. 47, no. 3, pp. 584–590, 2011.
  • B. T. Gao, “Dynamical modeling and energy-based control design for TORA,” Acta Automatica Sinica, vol. 34, no. 9, pp. 1221–1224, 2008.
  • J. M. Avis, S. G. Nersesov, R. a.. Nathan, and K. R. Muske, “A comparison study of nonlinear control techniques for the RTAC system,” Nonlinear Analysis, vol. 11, no. 4, pp. 2647–2658, 2010.
  • B. Gao, X. Zhang, H. Chen, and J. Zhao, “Energy-based control design of an underactuated 2-dimensional TORA system,” in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS '09), pp. 1296–1301, St. Louis, Mo, USA, October 2009.
  • B. Gao, J. Xu, J. Zhao, and X. Huang, “Stabilizing control of an underactuated 2-dimensional tora with only rotor angle measurement,” Asian Journal of Control, vol. 15, no. 5, pp. 1477–1488, 2013.
  • R. Ortega, A. Loria, and P. Nicklasson, Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications, Springer, Berlin, Germany, 1998.
  • B. Gao, Y. Bao, J. Xie, and L. Jia, “Passivity-based control of two-dimensional translational oscillator with rotational actuator,” Transactions of the Institute of Measurement and Control, vol. 36, pp. 185–190, 2014. \endinput