Experimental control of a ‎f‎lexible ‎l‎ink ‎by ‎the ‎method ‎of‎ ‎Controlled Lagrangian

Document Type: Full Length Article


1 Department of Mechanical Engineering, Amirkabir University of technology, Tehran, Iran

2 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

3 School of Mechanical Engineering, Shiraz University, Shiraz, Iran



The Controlled Lagrangian method is a branch of energy shaping methods that is designed to control underactuated mechanical systems. The method employs the mechanical energy (kinetic energy plus potential energy) of an artificial Lagrangian system, that generates similar equations of motion to the original underactuated system, as the Lyapunov function. This paper presents an application of the Controlled Lagrangian method to control an underactuated flexible link, and the results of a theoretical study through simulations confirmed by the results from an experimental setup. It is shown that the method’s performance is acceptable from a practical point of view as well as theoretical perspective. The simulations and the experimental results are presented in the sequel to
validate the theoretical studies. The effect of changing controller gains on the designed controller performance is studied in more detail under the terms of the system’s mechanical energy. Moreover, gain tuning is also performed to attain high quality performance in the experimental study by the aid of their influence in the system’s nergy.
Comparison of the proposed method with the partial feedback linearization method shows the degree of robustness of the proposed method. The simplicity of the gain tuning shows that the method can be implemented conveniently to control mechanical systems


  • A nonlinear method is used to control vibration of flexible links.
  • The method belongs to the family of energy-shaping methods for mechanical systems.
  • Effect of the controller gains on performance is studied from energy viewpoint.  
  • Reasons of discrepancy between simulation and experiments are discussed.
  • The negative effect of uncertainties in experiments is eliminated via tuning.


Main Subjects

[1] J.M. Martins, Z. Mohamed, M.O. Tokhi, J.S.a. Da Costa, M.A. Botto, Approaches for dynamic modelling of flexible manipulator systems, IEE Proceedings-Control Theory and Applications, 150 (2003) 401-411.

[2] G. Naganathan, A. Soni, An analytical and experimental investigation of flexible manipulator performance, in:  Robotics and Automation. Proceedings. 1987 IEEE International Conference on, IEEE, 1987, pp. 767-773.

[3] E. Soltani, M. Naraghi, Inversion-based nonlinear end-tip control of flexible arm in presence of large model uncertainties, in:  Robotics, Automation and Mechatronics, 2006 IEEE Conference on, IEEE, 2006, pp. 1-6.

[4] D.P. Magee, W.J. Book, Eliminating multiple modes of vibration in a flexible manipulator, in, Georgia Institute of Technology, 1993.

[5] Z.H. Luo, Direct strain feedback control of flexible robot arms: new theoretical and experimental results, IEEE Transactions on Automatic Control, 38 (1993) 1610-1622.

[6] A.M. Bloch, N.E. Leonard, J.E. Marsden, Stabilization of mechanical systems using controlled Lagrangians, in:  Decision and Control, 1997., Proceedings of the 36th IEEE Conference on, IEEE, 1997, pp. 2356-2361.

[7] A.M. Bloch, N.E. Leonard, J.E. Marsden, Matching and stabilization by the method of controlled Lagrangians, in:  Decision and Control, 1998. Proceedings of the 37th IEEE Conference on, IEEE, 1998, pp. 1446-1451.

[8] A.M. Bloch, N.E. Leonard, J.E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem, IEEE Transactions on automatic control, 45 (2000) 2253-2270.

[9] A.M. Bloch, D.E. Chang, N.E. Leonard, J.E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping, IEEE Transactions on Automatic Control, 46 (2001) 1556-1571.

[10] D. Auckly, L. Kapitanski, W. White, Control of nonlinear underactuated systems, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 53 (2000) 354-369.

[11] D. Auckly, L. Kapitanski, On the λ-equations for matching control laws, SIAM Journal on control and optimization, 41 (2002) 1372-1388.

[12] F. Andreev, D. Auckly, S. Gosavi, L. Kapitanski, A. Kelkar, W. White, Matching, linear systems, and the ball and beam, Automatica, 38 (2002) 2147-2152.

[13] D.E. Chang, A.M. Bloch, N.E. Leonard, J.E. Marsden, C.A. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002) 393-422.

[14] A. Donaire, R. Mehra, R. Ortega, S. Satpute, J.G. Romero, F. Kazi, N.M. Singh, Shaping the energy of mechanical systems without solving partial differential equations, in:  American Control Conference (ACC), 2015, IEEE, 2015, pp. 1351-1356.

[15] A. Albu-Schäffer, C. Ott, F. Petit, Energy Shaping Control for a Class of Underactuated Euler-Lagrange Systems, in:  SyRoCo, 2012, pp. 567-575.

[16] J. Xie, B. Sun, W. Wei, Z. Liu, Application Kinetic Energy Shaping to Controlling and Anticontrolling Chaotic Gait of Underactuated Compass-Gait Bipedal Robot, in:  ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, 2017.

[17] G. Lv, R.D. Gregg, Towards total energy shaping control of lower-limb exoskeletons, in:  American Control Conference (ACC), 2017, IEEE, 2017, pp. 4851-4857.

[18] R. Ortega, M.W. Spong, F. Gomez-Estern, G. Blankenstein, Stabilization of underactuated mechanical systems via interconnection and damping assignment, IEEE Trans. Aut. Control, (2000).

[19] R. Ortega, M.W. Spong, F. Gómez-Estern, G. Blankenstein, Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment, IEEE transactions on automatic control, 47 (2002) 1218-1233.

[20] R. Ortega, A. Van Der Schaft, B. Maschke, G. Escobar, Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems, Automatica, 38 (2002) 585-596.

[21] C.F. Aguilar‐Ibañez, O.O.G. Frias, A simple model matching for the stabilization of an inverted pendulum cart system, International Journal of Robust and Nonlinear Control: IFAC‐Affiliated Journal, 18 (2008) 688-699.

[22] A. Sanz, V. Etxebarria, Interconnection and damping assignment passivity-based experimental control of a single-link flexible robot arm, in:  Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006 IEEE, IEEE, 2006, pp. 2504-2509.

[23] N.K. Haddad, A. Chemori, S. Belghith, Robustness enhancement of IDA-PBC controller in stabilising the inertia wheel inverted pendulum: theory and real-time experiments, International Journal of Control, (2017) 1-16.

[24] A. Donaire, J.G. Romero, R. Ortega, B. Siciliano, M. Crespo, Robust IDA‐PBC for underactuated mechanical systems subject to matched disturbances, International Journal of Robust and Nonlinear Control, 27 (2017) 1000-1016.

[25] J. Ferguson, A. Donaire, R. Ortega, R.H. Middleton, Matched disturbance rejection for energy-shaping controlled underactuated mechanical systems, in:  Decision and Control (CDC), 2017 IEEE 56th Annual Conference on, IEEE, 2017, pp. 1484-1489.

[26] J. José, E. Saletan, Classical dynamics: a contemporary approach, in, First ed. Cambridge University Press, UK.

, 2000.

[27] J.E. Marsden, Lectures on mechanics, Cambridge University Press, 1992.

[28] F. Bellezza, L. Lanari, G. Ulivi, Exact modeling of the flexible slewing link, in:  Robotics and Automation, 1990. Proceedings., 1990 IEEE International Conference on Robotics and Automation, IEEE, 1990, pp. 734-739.

[29] Quanser Consulting Inc, SRV02-Series: Flexible Link Handout, in, Ontario, Canada, 2009.

[30] M. Hemmasian Ettefagh, M. Naraghi, M. Mahzoon, Robustness of Controlled Lagrangian Method to the Structured Uncertainties, AUT Journal of Mechanical Engineering, 2 (2018) 61-72.

[31] A. Arisoy, M. Gokasan, O. Bogosyan, Partial feedback linearization control of a single flexible link robot manipulator, in:  Recent Advances in Space Technologies, 2005. RAST 2005. Proceedings of 2nd International Conference on Recent Advances in Space Technologies, IEEE, 2005, pp. 282-287.