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Further constructive results on interconnection and damping assignment control of mechanical systems: the Acrobot example
Author(s) -
D. Mahindrakar Arun,
Astolfi Alessandro,
Ortega Romeo,
Viola Giuseppe
Publication year - 2006
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.1088
Subject(s) - control theory (sociology) , sylvester's law of inertia , passivity , lyapunov function , underactuation , constructive , inertia , mechanical system , interconnection , hamiltonian system , mathematics , partial differential equation , hamiltonian (control theory) , computer science , mathematical optimization , robot , control (management) , nonlinear system , mathematical analysis , symmetric matrix , engineering , physics , process (computing) , artificial intelligence , operating system , computer network , eigenvalues and eigenvectors , quantum mechanics , classical mechanics , electrical engineering
Interconnection and damping assignment passivity‐based control is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed‐loop system with a Hamiltonian structure with a desired energy function—that qualifies as Lyapunov function for the desired equilibrium. The assignable energy functions are characterized by a set of partial differential equations that must be solved to determine the control law. A class of underactuation degree one systems for which the partial differential equations can be explicitly solved —making the procedure truly constructive—was recently reported by the authors. In this brief note, largely motivated by the interesting Acrobot example, we pursue this investigation for two degrees‐of‐freedom systems where a constant inertia matrix can be assigned. We concentrate then our attention on potential energy shaping and give conditions under which an explicit solution of the associated partial differential equation can be obtained. Using these results we show that it is possible to swing‐up the Acrobot from some configuration positions in the lower half plane, provided some conditions on the robot parameters are satisfied. Copyright © 2006 John Wiley & Sons, Ltd.