Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment
TL;DR: This work describes a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction, given in terms of solvability of certain partial differential equations.
Abstract: We consider the application of a formulation of passivity-based control (PBC), known as interconnection and damping assignment (IDA) to the problem of stabilization of underactuated mechanical systems, which requires the modification of both the potential and the kinetic energies. Our main contribution is the characterization of a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction. The class is given in terms of solvability of certain partial differential equations. One important feature of IDA-PBC, stemming from its Hamiltonian formulation, is that it provides new degrees of freedom for the solution of these equations. Using this additional freedom, we are able to show that the method of "controlled Lagrangians"-in its original formulation-may be viewed as a special case of our approach. As illustrations we design asymptotically stabilizing IDA-PBCs for the classical ball and beam system and a novel inertia wheel pendulum.
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TL;DR: In this article, the authors show that standard PBC is stymied by the presence of unbounded energy dissipation, hence it is applicable only to systems that are stabilizable with passive controllers.
Abstract: Energy is one of the fundamental concepts in science and engineering practice, where it is common to view dynamical systems as energy-transformation devices. This perspective is particularly useful in studying complex nonlinear systems by decomposing them into simpler subsystems that, upon interconnection, add up their energies to determine the full system's behavior. The action of a controller may also be understood in energy terms as another dynamical system. The control problem can then be recast as finding a dynamical system and an interconnection pattern such that the overall energy function takes the desired form. This energy-shaping approach is the essence of passivity-based control (PBC), a controller design technique that is very well known in mechanical systems. Our objectives in the article are threefold. First, to call attention to the fact that PBC does not rely on some particular structural properties of mechanical systems, but hinges on the more fundamental (and universal) property of energy balancing. Second, to identify the physical obstacles that hamper the use of standard PBC in applications other than mechanical systems. In particular, we show that standard PBC is stymied by the presence of unbounded energy dissipation, hence it is applicable only to systems that are stabilizable with passive controllers. Third, to revisit a PBC theory that has been developed to overcome the dissipation obstacle as well as to make the incorporation of process prior knowledge more systematic. These two important features allow us to design energy-based controllers for a wide range of physical systems.
865 citations
Book•
01 Jan 2005
TL;DR: In this article, a comprehensive set of modeling, analysis and design techniques for a class of simple mechanical control systems is presented, that is, systems whose Lagrangian is kinetic energy minus potential energy.
Abstract: This talk will outline a comprehensive set of modeling, analysis and design techniques for a class of mechanical systems. We concern ourselves with simple mechanical control systems, that is, systems whose Lagrangian is kinetic energy minus potential energy. Example devices include robotic manipulators, aerospace and underwater vehicles, and mechanisms that locomote exploiting nonholonomic constraints. Borrowing techniques from nonlinear control and geometric mechanics, we propose a coordinateinvariant control theory for this class of systems. First, we take a Riemannian geometric approach to modeling systems dened on smooth manifolds, subject to nonholonomic constraints, external forces and control forces. We also model mechanical systems on groups and symmetries. Second, we analyze some control-theoretic properties of this class of systems, including controllability, averaged response to oscillatory controls, and kinematic reductions. Finally, we exploit the modeling and analysis results to tackle control design problems. Starting from controllability and kinematic reduction assumptions we propose some algorithms for generating and tracking trajectories.
848 citations
TL;DR: The fundamental theory, main new results and practical applications of this control system design approach are reviewed as well as to discuss the current open problems and future directions.
581 citations
Book•
01 Jan 2005
TL;DR: In this article, a sampling of design methodologies Linear and nonlinear potential shaping for stabilization and tracking for fully actuated systems Stabilization and tracking using oscillatory controls Motion planning for underactuated systems Appendices Timedependent vector fields Some proofs.
Abstract: Part I: Modeling of mechanical systems Introductory examples and problems Linear and multilinear algebra Differential geometry Simple mechanical control systems Lie groups, systems on groups, and symmetries.- Part II: Analysis of mechanical control systems Stability Controllability Low-order controllability and kinematic reduction Perturbation analysis.- Part III: A sampling of design methodologies Linear and nonlinear potential shaping for stabilization Stabilization and tracking for fully actuated systems Stabilization and tracking using oscillatory controls Motion planning for underactuated systems Appendices Time-dependent vector fields Some proofs.
482 citations
TL;DR: In this article, a simple proportional-derivative (PD) controller is used to asymptotically regulate the overhead crane position and the payload angle, and two nonlinear controllers are presented to increase the coupling between the planar gantry position and payload angle.
Abstract: In this paper, we consider the regulation control problem for an underactuated overhead crane system. Motivated by recent passivity-based controllers for underactuated systems, we design several controllers that asymptotically regulate the planar gantry position and the payload angle. Specifically, utilizing LaSalle's invariant set theorem, we first illustrate how a simple proportional-derivative (PD) controller can be utilized to asymptotically regulate the overhead crane system. Motivated by the desire to achieve improved transient performance, we then present two nonlinear controllers that increase the coupling between the planar gantry position and the payload angle. Experimental results are provided to illustrate the improved performance of the nonlinear controllers over the simple PD controller.
318 citations
References
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Book•
01 Jan 1989
TL;DR: This self-contained introduction to practical robot kinematics and dynamics includes a comprehensive treatment of robot control, providing background material on terminology and linear transformations and examples illustrating all aspects of the theory and problems.
Abstract: From the Publisher:
This self-contained introduction to practical robot kinematics and dynamics includes a comprehensive treatment of robot control. Provides background material on terminology and linear transformations, followed by coverage of kinematics and inverse kinematics, dynamics, manipulator control, robust control, force control, use of feedback in nonlinear systems, and adaptive control. Each topic is supported by examples of specific applications. Derivations and proofs are included in many cases. Includes many worked examples, examples illustrating all aspects of the theory, and problems.
3,736 citations
"Stabilization of a class of underac..." refers methods in this paper
...The dynamic equations of the device can be written in standard form using the EL formulation [27] as...
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2,725 citations
Book•
16 Aug 1996TL;DR: In this article, a small gain and passivity of input-output maps are discussed. But the authors focus on the Hamiltonian system as passive systems and do not consider the Hamilton-Jacobi Inequalities.
Abstract: 1 Input-Output Stability.- 2 Small-gain and Passivity of Input-Output Maps.- 3 Dissipative Systems Theory.- 4 Hamiltonian Systems as Passive Systems.- 5 Passivity by Feedback.- 6 Factorizations of Nonlinear Systems.- 7 Nonlinear H? Control.- 8 Hamilton-Jacobi Inequalities.
1,909 citations
"Stabilization of a class of underac..." refers background or methods in this paper
...To overcome this problem we have developed in [18] (see also [19, 28 ]) a new PBC design methodology called interconnection and damping assignment or IDA–PBC....
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...As explained below, to preserve the energy interpretation of the stabilization mechanism we also require the closed–loop system to be in port– controlled Hamiltonian form [ 28 ]....
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...5 See [18, 28 ] for the physical and analytical justification of this choice....
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TL;DR: A new PBC theory is developed which extends to a broader class of systems the aforementioned energy-balancing stabilization mechanism and the structure invariance and considers instead port-controlled Hamiltonian models, which result from the network modelling of energy-conserving lumped-parameter physical systems with independent storage elements, and strictly contain the class of EL models.
1,444 citations
TL;DR: In this paper, it was shown that weakly minimum phase nonlinear systems with relative degree one can be globally asymptotically stabilized by smooth state feedback, provided that suitable controllability-like rank conditions are satisfied.
Abstract: Conditions under which a nonlinear system can be rendered passive via smooth state feedback are derived. It is shown that, as in the case of linear systems, this is possible if and only if the system in question has relative degree one and is weakly minimum phase. It is proven that weakly minimum phase nonlinear systems with relative degree one can be globally asymptotically stabilized by smooth state feedback, provided that suitable controllability-like rank conditions are satisfied. This result incorporates and extends a number of stabilization schemes recently proposed for global asymptotic stabilization of certain classes of nonlinear systems. >
1,379 citations