Solution to IDA-PBC PDEs by Pfaffian Differential Equations

Finding the general solution of partial differential equations (PDEs) is essential for controller design in some methods. Interconnection and damping assignment passivity based control (IDA-PBC) is one of such methods in which the solution to corresponding PDEs is needed to apply it in practice. In this paper, such PDEs are transformed to corresponding Pfaffian differential equations. Furthermore, it is shown that upon satisfaction of the integrability condition, the solution to the corresponding third order Pfaffian differential equation may be obtained quite easily. The method is applied to the PDEs of IDA-PBC in some benchmark problems such as Magnetic levitation system, Pendubot and underactuated cable driven robot to verify its applicability

Energy shaping and partial feedback linearization of mechanical systems with kinematic constraints

Traditionally, the energy shaping for mechanical systems requires the elimination of holonomic and nonholonomic constraints. In recent years, it was argued that such elimination might be unnecessary, leading to a possible simplification of the matching conditions in energy shaping. On the other hand, the partial feedback linearization (PFL) approach has been widely applied to unconstrained mechanical systems, but there is no general result for the constrained case. In this regard, this paper formalizes the PFL for mechanical systems with kinematic constraints and extends the energy shaping of such systems by including systems with singular inertia matrix and non-workless constraint forces, which can arise from the coordinate selection and PFL. We validated the proposed methodology on a 5-DoF portal crane via simulation

Energy Shaping of Underactuated Systems via Interconnection and Damping Assignment Passivity-Based Control with Applications to Planar Biped Robots

The sought goal of this thesis is to show that total energy shaping is an effective and versatile tool to control underactuated mechanical systems. The performance of several approaches, rooted in the port-Hamiltonian formalism, are analyzed while tackling distinct control problems: i) equilibrium stabilization; ii) gait generation; iii) gait robustication. Firstly, a constructive solution to deal with interconnection and damping assignment passivity-based control (IDA-PBC) for underactuated two-degree-of-freedom mechanical systems is proposed. This strategy does not involve the resolution of any partial differential equation, since explicit solutions are given, while no singularities depending on generalized momenta are introduced by the controller. The methodology is applied to the stabilization of a translational oscillator with a rotational actuator system, as well as, to the gait generation for an underactuated compass-like biped robot (CBR). Then, the problem of gait generation is addressed using dissipative forces in the controller. In this sense, three distinct controllers are presented, namely simultaneous interconnection and damping assignment passivity-based control with dissipative forces, energy pumping-and-damping passivity-based control (EPD-PBC), and energy pumping-or-damping control. Finally, EPD-PBC is used to increase the robustness of the gait exhibited by the CBR over uncertainties on the initial conditions. The passivity of the system is exploited, as well as, its hybrid nature (using the hybrid zero dynamics method) to carry out the stability analysis. Besides, such an approach is applied to new gaits that are generated using IDA-PBC. Numerical case studies, comparisons, and critical discussions evaluate the performance of the proposed approaches

A Constructive Methodology for the IDA-PBC of Underactuated 2-DoF Mechanical Systems with Explicit Solution of PDEs

This paper presents a passivity-based control strategy dealing with underactuated two-degree-of-freedom (2-DoF) mechanical systems. Such a methodology, which is based on the interconnection and damping assignment passivity-based control (IDA-PBC), rooted within the port-controlled Hamiltonian framework, can be applied to a very large class of underactuated 2-DoF mechanical systems. The main contribution, compared to the previous literature, is that the new methodology does not involve the resolution of any partial differential equation, since explicit solutions are given, while no singularities depending on generalised momenta are introduced by the controller. The proposed strategy is applied to two case studies: a) the stabilisation of a translational oscillator with a rotational actuator (TORA) system; b) the gait generation for an underactuated compass-like biped robot. The performances of the presented solution are evaluated through numerical simulations

Speed Observation and Position Feedback Stabilization of Partially Linearizable Mechanical Systems

The problems of speed observation and position feedback stabilization of mechanical systems are addressed in this paper. Our interest is centered on systems that can be rendered linear in the velocities via a (partial) change of coordinates. It is shown that the class is fully characterized by the solvability of a set of partial differential equations (PDEs) and strictly contains the class studied in the existing literature on linearization for speed observation or control. A reduced order globally exponentially stable observer, constructed using the immersion and invariance methodology, is proposed. The design requires the solution of another set of PDEs, which are shown to be solvable in several practical examples. It is also proven that the full order observer with dynamic scaling recently proposed by Karagiannis and Astolfi obviates the need to solve the latter PDEs. Finally, it is shown that the observer can be used in conjunction with an asymptotically stabilizing full state-feedback interconnection and damping assignment passivity-based controller preserving asymptotic stability.</p

Control of Underactuated Mechanical Systems:Observer Design and Position Feedback Stabilization

We identify a class of mechanical systems for which a globally exponentially stable reduced order observer can be designed. The class is characterized by (the solvability of) a set of partial differential equations and contains all systems that can be rendered linear in (the unmeasurable) momenta via a (partial) change of coordinates. It is shown that this class is larger than the one reported in the literature of observer design and linearization. We also prove that, under very weak assumptions, the observer can be used in conjunction with an asymptotically stabilizing full state-feedback Interconnection and Damping Assignment Passivity-Based Controller, preserving stability.Caveat Emptor: This paper is a shortened version of the technical note [1] which can be obtained upon request from the authors.</p

Implicit IDA-PBC Design and Implementation for a Portal Crane System

Interconnection and damping assignment passivity-based control (IDA-PBC) is a wellknown technique which regulates the behavior of nonlinear systems, assigning a target port-Hamiltonian (pH) structure to the closed-loop. In underactuated mechanical systems (UMSs) its application requires the satisfaction of matching conditions, which in many cases demands to solve partial differential equations (PDEs). Only recently, the IDA-PBC has been extended to UMSs modeled implicitly, where the system dynamics in pH representation are described by a set of differential-algebraic equations (DAEs). In some system classes this implicit approach allows to circumvent the PDE problem and also to design an output-feedback law. The present thesis deals with the design and implementation of the total energy shaping implicit IDA-PBC on a portal crane system located at the laboratory of the Control Engineering Group at TU-Ilmenau. The implicit controller is additionally compared with a simplified (explicit) IDA-PBC [1]. This algorithm shapes the total energy and avoids the PDE problem. However, this thesis reveales a significant implementation flaw in the algorithm, which then could be solved.Interconnection and damping assignment passivity-based control (IDA-PBC) ist eine wohlbekannte Methode zur Regelung von nichtlinearen Systemen, die im geschlossenen Regelkreis eine gewünschte Port-Hamiltonian-Struktur (pH) haben. Die Anwendung auf unteraktuierte mechanische Systeme (UMS) erfordert die Erfüllung von sogenannten Matching Conditions, die meistens die Lösung partieller Differentialgleichungen (PDE) benötigt. Erst kürzlich wurde die IDA-PBC auf implizit modellierte UMS erweitert, bei denen die Systemdynamik in pH-Darstellungen durch Differentialalgebraische Gleichungen (DAE) beschrieben wird. Dieser implizite Ansatz ermöglicht bei einigen Systemklassen, das PDE-Problem zu umgehen und auch eine Ausgangsrückführung zu entwerfen. Die vorliegende Masterarbeit beschäftigt sich mit dem Entwurf und der Implementierung des impliziten IDA-PBC zur Gesamtenergievorgabe auf einem Portalkransystem im Labor des Fachgebiets Regelungstechnik der TU-Ilmenau. Der implizite Regler wird mit einem vereinfachten (expliziten) IDA-PBC verglichen [1]. Dieser Algorithmus gibt ebenso die Gesamtenergie vor und vermeidet das PDE-Problem. In der Masterarbeit wird in diesem Algorithmus ein wesentlicher Implementierungsfehler offengelegt und behoben.Tesi

Interconnection and Damping Assignment Passivity-Based Control for Gait Generation in Underactuated Compass-Like Robots

A compass-like biped robot can go down a gentle slope without the need of actuation through a proper choice of its dynamic parameter and starting from a suitable initial condition. Addition of control actions is requested to generate additional gaits and robustify the existing one. This paper designs an interconnection and damping assignment passivity- based control, rooted within the port-Hamiltonian framework, to generate further gaits with respect to state-of-the-art methodologies, enlarge the basin of attraction of existing gaits, and further robustify the system against controller discretization and parametric uncertainties. The performance of the proposed algorithm is validated through numerical simulations and comparison with existing passivity-based techniques