Interconnection and damping assignment passivity-based control of an underactuated 2-DOF gyroscope

• Autorzy: Cordero G. , Santibáñez V. , Dzul A. , Sandoval J.

Identyfikatory

DOI

10.2478/amcs-2018-0051

• Języki publikacji: EN

Abstrakty

EN

In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2-DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme.

Słowa kluczowe

EN

gyroscope device; gyroscopic force; cyclic coordinate; generalized momenta; Routh’s equation; IDA-PBC

PL

żyroskop; siła żyroskopowa; współrzędna cykliczna; równanie Routha; IDA-PBC

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2018
• Tom: Vol. 28, no. 4
• Strony: 661--677
• Opis fizyczny: Bibliogr. 42 poz., rys., tab., wykr.

Twórcy

autor

Cordero G.

• Division of Graduate Studies and Research, National Institute of Technology of Mexico/Laguna Institute of Technology, Blvd. Revolución and Cuauhtemoc S/N, 27000 Torreón, Mexico

autor

Santibáñez V.

• Division of Graduate Studies and Research, National Institute of Technology of Mexico/Laguna Institute of Technology, Blvd. Revolución and Cuauhtemoc S/N, 27000 Torreón, Mexico

autor

Dzul A.

• Division of Graduate Studies and Research, National Institute of Technology of Mexico/Laguna Institute of Technology, Blvd. Revolución and Cuauhtemoc S/N, 27000 Torreón, Mexico

autor

Sandoval J.

• Division of Graduate Studies and Research, National Institute of Technology of Mexico/La Paz Institute of Technology, Blvd. Forjadores de BCS No. 4720, Apdo. Postal 43-B, 23080 La Paz, Mexico

Bibliografia

• [1] Acosta, J., Ortega, R., Astolfi, A. and Mahindrakar, A. (2005). Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one, IEEE Transactions on Automatic Control 50(12): 1936–1955.
• [2] Aguilar-Ibanez, C. (2016). Stabilization of the PVTOL aircraft based on a sliding mode and a saturation function, International Journal of Robust and Nonlinear Control 27(5): 843–859.
• [3] Antonio-Cruz, M., Hernandez-Guzman, V.M. and Silva-Ortigoza, R. (2018). Limit cycle elimination in inverted pendulums: Furuta pendulum and pendubot, IEEE Access 6: 30317–30332.
• [4] Bloch, A.M. (2003). Nonholonomic Mechanics and Control, Springer-Verlag, New York, NY.
• [5] Bloch, A.M., Reyhanoglu, M. and McClamroch, N.H. (1992). Control and stabilization of nonholonomic dynamic systems, IEEE Transactions on Automatic Control 37(11): 1746–1757.
• [6] Cannon, R.H. (1967). Dynamics of Physical Systems, McGraw-Hill, New York, NY.
• [7] Chang, D. (2014). On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems, Regular and Chaotic Dynamics 19(5): 556–575.
• [8] Chin, C.S., Lau, M.W.S., Low, E. and Seet, G.G.L. (2006). A robust controller design method and stability analysis of an underactuated underwater vehicle, International Journal of Applied Mathematics and Computer Science 16(3): 345–356.
• [9] Donaire, A., Mehra, R., Ortega, R., Satpute, S., Romero, J.G., Kazi, F. and Singh, N.M. (2015). Shaping the energy of mechanical systems without solving partial differential equations, American Control Conference, Chicago, IL, USA, pp. 1351–1356.
• [10] Donaire, A., Romero, J.G., Ortega, R., Siciliano, B. and Crespo, M. (2016). Robust IDA-PBC for underactuated mechanical systems subject to matched disturbances, International Journal of Robust and Nonlinear Control 27(6): 1000–1016.
• [11] Fantoni, I. and Lozano, R. (2002). Non-linear Control for Underactuated Mechanical Systems, Advances in Pattern Recognition, Springer, London.
• [12] Ferraz-Mello, S. (2007). Canonical Perturbation Theories, 1st Edn., Springer-Verlag, New York, NY.
• [13] Fulford, G., Forrester, P. and Jones, A. (1997). Modelling with Differential and Difference Equations, Australian Mathematical Society Lecture Series, Cambridge University Press, Cambridge.
• [14] Gómez-Estern, F. and van der Schaft, A. (2004). Physical damping in IDA-PBC controlled underactuated mechanical systems, European Journal of Control 10(5): 451–468.
• [15] Greenwood, D.T. (1977). Classical Dynamics, Dover, New York, NY.
• [16] Jiang, Z.P. (2010). Controlling underactuated mechanical systems: A review and open problems, in J. L´evine and P. Müllhaupt (Eds.), Advances in the Theory of Control, Signals and Systems with Physical Modeling, Lecture Notes in Control and Information Sciences, Vol. 407, Springer, Berlin, pp. 77–88.
• [17] Kang, W., Xiao, M. and Borges, C. (2003). New Trends in Nonlinear Dynamics and Control, and their Applications, 1st Edn., Springer-Verlag, Heidelberg/Berlin.
• [18] Kelly, R., Santibánez, V. and Loría, A. (2005). Control of Robot Manipulators in Joint Space, 1st Edn., Springer-Verlag, London.
• [19] Kotyczka, P. and Lohmann, B. (2009). Parametrization of IDA-PBC by assignment of local linear dynamics, Proceedings of the European Control Conference, Budapest, Hungary, pp. 4721–4726.
• [20] Kozlov, V.V. (1996). Symmetries, Topology and Resonances in Hamiltonian Mechanics, 1st Edn., Springer-Verlag, Heidelberg/Berlin.
• [21] Layek, G.C. (2015). An Introduction to Dynamical Systems and Chaos, 1st Edn., Springer, New Delhi.
• [22] Liu, Y. (2018). Control of a class of multibody underactuated mechanical systems with discontinuous friction using sliding-mode, Transactions of the Institute of Measurement and Control 40(2): 514–527.
• [23] Liu, Y. and Yu, H. (2013). A survey of underactuated mechanical systems, IET Control Theory and Applications 7(7): 921–931.
• [24] Mahindrakar, A.D., Astolfi, A., Ortega, R. and Viola, G. (2006). Further constructive results on interconnection and damping assignment control of mechanical systems: The acrobot example, International Journal of Robust and Nonlinear Control 16(14): 671–685.
• [25] Moreno-Valenzuela, J. and Aguilar-Avelar, C. (2018). Motion Control of Underactuated Mechanical Systems, 1st Edn., Springer International Publishing, Cham.
• [26] Nagata, W. and Namachchivaya, N.S. (2006). Bifurcation Theory and Spatio-Temporal Pattern Formation, Fields Institute Communications, Ontario.
• [27] Ortega, R., Spong, M., Gómez-Estern, F. and Blankeinstein, G. (2002). Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment, IEEE Transactions on Automatic Control 47(8): 1213–1233.
• [28] Ostrowski, J.P. and Burdick, J.W. (1997). Controllability for mechanical systems with symmetries and constraints, Applied Mathematics and Computer Science 7(2): 305–331.
• [29] Parks, R. (1999). Manual for model 750 control moment gyroscope, Technical document, Educational Control Products, Bell Canyon, CA, http://maecourses.ucsd.edu/ugcl/download/manuals/gyroscope.pdf.
• [30] Quanser (2009). Specialty plant: 3 DOF gyroscope. User manual, Document Number 806, Revision 1.0, https://www.quanser.com/products/3-dof-gyroscope.
• [31] Rana, N. and Joag, P. (1991). Classical Mechanics, Tata McGraw-Hill, New Delhi.
• [32] Reyhanoglu, M. and van de Loo, J. (2006a). Rest-to-rest maneuvering of a nonholonomic control moment gyroscope, IEEE International Symposium on Industrial Electronics, Montreal, Canada, pp. 160–165.
• [33] Reyhanoglu, M. and van de Loo, J. (2006b). State feedback tracking of a nonholonomic control moment gyroscope, Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 6156–6161.
• [34] Romero, J.G., Donaire, A., Ortega, R. and Borja, P. (2017). Global stabilisation of underactuated mechanical systems via PID passivity-based control, IFAC PapersOnLine 50(1): 9577–9582.
• [35] Sabattini, L., Secchi, C. and Fantuzzi, C. (2017). Collision avoidance for multiple Lagrangian dynamical systems with gyroscopic forces, International Journal of Advanced Robotic Systems 1(15): 1–15.
• [36] Sandoval, J., Kelly, R. and Santibanez, V. (2011). Interconnection and damping assignment passivity-based control of a class of underactuated mechanical systems with dynamic friction, International Journal of Robust and Nonlinear Control 21(7): 738–751.
• [37] Sandoval, J., Ortega, R. and Kelly, R. (2008). Interconnection and damping assignment passivity-based control of the pendubot, Proceedings of the 17th IFAC World Congress, Seoul, Korea, pp. 7700–7704.
• [38] Santibanez, V., Kelly, R. and Llama, M.A. (2004). Global asymptotic stability of a tracking sectorial fuzzy controller for robot manipulators, IEEE Transactions on Systems, Man, and Cybernetics B: Cybernetics 34(1): 710–718.
• [39] Seifried, R. (2014). Dynamics of Underactuated Multibody Systems: Modeling, Control and Optimal Design (Solid Mechanics and Its Applications), Springer, Cham.
• [40] Seyranian, A., Stoustrup, J. and Kliem, W. (1995). On gyroscopic stabilization, Journal of Applied Mathematics and Physics 46(2): 255–267.
• [41] She, J., Zhang, A., Lai, X. and Wu, M. (2012). Global stabilization of 2-DOF underactuated mechanical systems: An equivalent-input-disturbance approach, Nonlinear Dynamics 69(1–2): 495–509.
• [42] Wichlund, K.Y., Sordalen, O.J. and Egeland, O. (1995). Control of vehicles with second-order nonholonomic constraints: Underactuated vehicles, Proceedings of the 3rd European Control Conference, Rome, Italy, pp. 3086–3091.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).