Abstract:On the basis of controlled Lagrangians, a controller design is proposed for underactuated mechanical systems with two degrees of freedom. A new kinetic energy equation (K-equation) independent of the gyroscopic forces is found due to the use of their property. As a result, the necessary and sufficient matching condition comprises the new K-equation and the potential energy equation (P-equation) cascaded, the regular condition, and the explicit gyroscopic forces. Further, for two classes of input decoupled systems that cover the main benchmark examples, the new K-equation, respectively, degenerates from a quasilinear partial differential equation (PDE) into an ordinary differential equation (ODE) under some choice and into a homogeneous linear PDE with two kinds of explicit general solutions. Benefiting from one of the general solutions, the obtained smooth state feedback controller for the Acrobots is of a more general form. Specifically, a constant fixed in a related paper by the system parameters is converted into a controller parameter ranging over an open interval along with some new nonlinear terms involved. Unlike what is mentioned in the related paper, some categories of the Acrobots cannot be stabilized with the existing interconnection and damping assignment passivity based control (IDA-PBC) method. As a contribution, the system can be locally asymptotically stabilized by the selection of the new controller parameter except for only one special case.
. Controller design for 2-DOF underactuated mechanical
systems based on controlled Lagrangians and application to Acrobot
control[J]. Front. Electr. Electron. Eng., 2009, 4(4): 417-439.
Maoqing LI, . Controller design for 2-DOF underactuated mechanical
systems based on controlled Lagrangians and application to Acrobot
control. Front. Electr. Electron. Eng., 2009, 4(4): 417-439.
Bloch A M, Leonard N E, Marsden J E. Stabilization of mechanical systems using controlledLagrangians. In: Proceedings of the 36thIEEE Conference on Decision and Control. 1997, 3: 2356―2361
Bloch A M, Leonard N E, Marsden J E. Controlled Lagrangians and the stabilization of mechanicalsystems I: the first matching theorem. IEEE Transactions on Automatic Control, 2000, 45(12): 2253―2270 doi: 10.1109/9.895562
Bloch A M, Chang D E, Leonard N E, Marsden J E. Controlled Lagrangians and the stabilization of mechanical systemsII: potential shaping. IEEE Transactionson Automatic Control, 2001, 46(10): 1556―1571 doi: 10.1109/9.956051
Hamberg J. General matching conditions in the theory of controlled Lagrangians. In: Proceedings of the 38th Conference on Decisionand Control. 1999, 3: 2519―2523
Auckly D, Kapitanski L. On the λ-equationsfor matching control laws. SIAM Journalon Control and Optimization, 2002, 41(5): 1372―1388 doi: 10.1137/S0363012901393304
Chang D E. Controlled Lagrangian and Hamiltonian systems. Dissertation for the Doctoral Degree. Pasadena, CA: California Instituteof Technology, 2002
Ortega R, van der Schaft A, Maschke B, Escobar G. Interconnection and damping assignment passivity-based control ofport-controlled Hamiltonian systems. Automatica, 2002, 38(4): 585―596 doi: 10.1016/S0005-1098(01)00278-3
Gomez-Estern F, Ortega R, Rubio F, Aracil J. Stabilizationof a class of underactuated mechanical systems via total energy shaping. In: Proceedings of the 40th IEEE Conference onDecision and Control. 2001, 2: 1137―1143
Ortega R, Spong M W, Gomez-Estern F, Blankenstein G. Stabilization of a class of underactuated mechanical systems viainterconnection and damping assignment. IEEE Transactions on Automatic Control, 2002, 47(8): 1218―1233 doi: 10.1109/TAC.2002.800770
Acosta J A, Ortega R, Astolfi A, Mahindrakar A D. Interconnection and damping assignment passivity based control ofmechanical systems with underactuation degree one. IEEE Transactions on Automatic Control, 2005, 50(12): 1936―1955 doi: 10.1109/TAC.2005.860292
Viola G, Ortega R, Banavar R, Acosta J A, Astolfi A. Total energy shaping control of mechanicalsystems: simplifying the matching equations via coordinate changes. IEEE Transactions on Automatic Control, 2007, 52(6): 1093―1099 doi: 10.1109/TAC.2007.899064
Blankenstein G, Ortega R, van der Schaft A J. The matching conditions of controlledLagrangians and IDA-passivity based control. International Journal of Control, 2002, 75(9): 645―665 doi: 10.1080/00207170210135939
Spong M W. The swing up control problem for the Acrobot. IEEE Control Systems Magazine, 1995, 15(1): 49―55 doi: 10.1109/37.341864
Hamberg J. Simplified conditions for matching and for generalized matching inthe theory of controlled Lagrangians. In: Proceedings of the 2000 American Control Conference. 2000, 6: 3918―3923
Zenkov D V, Bloch A M, Leonard N E, Marsden J E. Matching and stabilization of low-dimensional nonholonomic systems. In: Proceedings of the 39th IEEE Conference onDecision and Control. 2000, 2: 1289―1294
Bloch A M, Leonard N E, Marsden J E. Controlled Lagrangians and the stabilization of Euler-Poincarémechanical systems. International Journalof Robust and Nonlinear Control, 2001, 11(3): 191―214 doi: 10.1002/rnc.572
Mahindrakar A D, Astolfi A, Ortega R, Viola G. Further constructiveresults on interconnection and damping assignment control of mechanicalsystems: the Acrobot example. InternationalJournal of Robust and Nonlinear Control, 2006, 16(14): 671―685 doi: 10.1002/rnc.1088
Grizzle J W, Moog C H, Chevallereau C. Nonlinear control of mechanical systems with an unactuatedcyclic variable. IEEE Transactions on AutomaticControl, 2005, 50(5): 559―576 doi: 10.1109/TAC.2005.847057
Hauser J, Murray R M. Nonlinear controllers fornon-integrable systems: the Acrobot example. In: Proceedings of the 1990 American Control Conference. 1990, 669―671
Bortoff S A, Spong M W. Pseudolinearization of theAcrobot using spline functions. In: Proceedingsof the 31st IEEE Conference on Decision and Control. 1992, 1: 593―598
Bortoff S A. Approximate state feedback linearization using spline functions. Automatica, 1997, 33(8): 1449―1458 doi: 10.1016/S0005-1098(97)00070-8
Spong M W. Energy based control of a class of underactuated mechanical systems. In: Proceedings of IFAC World Congress. 1996, 431―435
Olfati-Saber R. Nonlinear control of underactuated mechanical systems with applicationto robotics and aerospace vehicles. Dissertationfor the Doctoral Degree. Cambridge,MA: Massachusetts Institute ofTechnology, 2001
Olfati-Saber R. Normal forms for underactuated mechanical systems with symmetry. IEEE Transactions on Automatic Control, 2002, 47(2): 305―308 doi: 10.1109/9.983365
Spong M W, Vidyasagar M. Robot Dynamics and Control. New York: John Wiley & Sons Inc., 1989