José Ángel Acosta

Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one

We consider the problem of (asymptotic) stabilization of mechanical systems with underactuation degree one. A state-feedback design is derived applying the interconnection and damping assignment passivity-based control methodology. Its application relies on the possibility of solving a set of partial differential equations that identify the energy functions that can be assigned to the closed-loop. The following results are established: 1) identification - in terms of some algebraic inequalities - of a subclass of these systems for which the partial differential equations are trivially solved; 2) characterization of all systems which are feedback-equivalent to this subclass; and 3) introduction of a suitable parametrization of the assignable energy functions that provides the designer with a handle to address transient performance and robustness issues. An additional feature of our developments is that the open-loop system need not be described by a port-controlled Hamiltonian (or Lagrangian) model, a situation that arises often in applications due to model reductions or preliminary feedbacks that destroy the structure. The new result is applied to obtain an (almost) globally stabilizing controller for the inertia wheel pendulum, a controller for the chariot with pendulum system that can swing-up the pendulum from any position in the upper half plane and stop the chariot at any desired location, and an (almost) globally stabilizing scheme for the vertical takeoff and landing aircraft with strong input coupling. In all cases we obtain very simple and intuitive solutions that do not rely on, rather unnatural and technique-driven, linearization or decoupling procedures but instead endows the closed-loop system with a Hamiltonian structure with desired potential and kinetic energy functions.

Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations Via Coordinate Changes

Total energy shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function - that qualifies as Lyapunov function for the desired equilibrium. The success of the method relies on the possibility of solving two PDEs which identify the kinetic and potential energy functions that can be assigned to the closed loop. Particularly troublesome is the partial differential equation (PDE) associated to the kinetic energy which is nonlinear and inhomogeneous and the solution, that defines the desired inertia matrix, must be positive-definite. In this note, we prove that we can eliminate or simplify the forcing term in this PDE by modifying the target dynamics and introducing a change of coordinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coordinates, the possibility of simplifying the PDEs is determined by the interaction between the Coriolis and centrifugal forces and the actuation structure. The examples of pendulum on a cart and Furutas pendulum are used to illustrate the results.

Brief paper: A constructive solution for stabilization via immersion and invariance: The cart and pendulum system

Immersion and Invariance (II (ii) the construction of an invariant manifold such that the restriction of the system dynamics to this manifold coincides with the target dynamics; (iii) the design of a control law that renders the manifold attractive and ensures that all signals are bounded. The second step requires the solution of a partial differential equation (PDE) that may be difficult to obtain. In this short note we use the classical cart and pendulum system to show that by interlacing the first and second steps, and invoking physical considerations, it is possible to obviate the solution of the PDE. To underscore the generality of the proposed variation of I&I, we show that it is also applicable to a class of n-dimensional systems that contain, as a particular case, the cart and pendulum system.

A new swing-up law for the Furuta pendulum

In this paper the swing-up problem for the Furuta pendulum is solved applying Fradkovs speed-gradient (SG) method to a dimension 4 model of the system. The new law is compared with the conventional Åström-Furuta strategy, based on a dimension 2 model. A comparative analysis, including simulations and experiments, whereby the advantages and effectiveness of the new law for swinging the pendulum up are shown, is included.

Control of a multirotor outdoor aerial manipulator

This paper presents the design and control of a multirotor-based aerial manipulator developed for outdoor operation. The multi-rotor has eight rotors and large payload to integrate a 7-degrees of freedom arm and to carry sensors and processing hardware needed for outdoor positioning. The arm can also carry an end-effector and sensors to perform different missions. The paper focuses on the control design and implementation aspects. A stable backstepping-based controller for the multirotor that uses the coupled full dynamic model is proposed, and an admittance controller for the manipulator arm is outlined. Several experimental tests with the aerial manipulator are also presented. In one of the experiments, the performance of the pitch attitude controller is compared to a PID controller. Other experiments of the arm controller following an object with the camera are also presented.

Constructive immersion and invariance stabilization for a class of underactuated mechanical systems

A constructive approach to stabilize a desired equilibrium for a class of underactuated mechanical systems, which obviates the solution of partial differential equations, is proposed. The Immersion & Invariance methodology is adopted, with the main result formulated in the Port-Hamiltonian framework, for both model and target dynamics. The procedure is applicable to mechanical systems with under-actuation degree larger than one, extending the results recently reported by some of the authors. The approach is successfully applied to two benchmark examples and some basic connections with the interconnection and damping assignment passivity-based control are revealed. An additional contribution of this work is the identification of a class of mechanical systems whose mechanical structure remains invariant under partial feedback linearization.

Stabilization of oscillations in the inverted pendulum

Abstract This paper addresses the problem of obtaining robust and stable oscillations in an electromechanical system. These oscillations are associated to a limit cycle that is born through a supercritical Hopf bifurcation. The method proposed in the paper works well for fully actuated systems, and even for certain underactuated ones. In order to illustrate the method, we have chosen an underactuated system that is well known in the literature and in control systems laboratories: the inverted pendulum. Actual stable and robust oscillations have been obtained experimentally in a rotating Furuta pendulum.

Furuta's pendulum: a conservative nonlinear model for theory and practise

Furutas pendulum has been an excellent benchmark for the automatic control community in the last years, providing, among others, a better understanding of model-based Nonlinear Control Techniques. Since most of these techniques are based on invariants and/or integrals of motion then, the dynamic model plays an important role. This paper describes, in detail, the successful dynamical model developed for the available laboratory pendulum. The success relies on a basic dynamical model derived from Classical Mechanics which has been augmented to compensate the non-conservative torques. Thus, the quasi-conservative “practical” model developed allows to design all the controllers as if the system was strictly conservative. A survey of all the nonlinear controllers designed and experimentally tested on the available laboratory pendulum is also reported.

Control of the longitudinal flight dynamics of an UAV using adaptive backstepping

An adaptive backstepping approach is used to control the longitudinal dynamics of an Unmanned Air Vehicle (UAV). The nonlinear controller designed makes the system follow references in the aerodynamic velocity and ight path angle, using the elevator deections and the thrust as actuators. Moreover, the (global) solution is valid for all the ight envelope, since it is based on a general nonlinear model. The adaptation scheme proposed allowed us to design an explicit controller with a minimal knowledge of the aircraft aerodynamics. Simulations are included for a realistic UAV model that includes actuator saturation.

Constructive feedback linearization of underactuated mechanical systems with 2-DOF

In the last years, the control community has developed several and powerful methods to control nonlinear systems, especially for underactuated mechanical systems. Thus, methods based on passivity, like Interconnection and damping assignment passivity–based control (IDA-PBC) and Controlled Lagrangians have solved many interesting control problems for particular full classes of systems. Usually, the solutions of these methods relies on solving a set of partial differential equations (PDEs), which is not always possible. This paper presents a constructive methodology to control underactuated mechanical systems with 2-DOF, by means of classical feedback linearization and Lyapunov design. The steps of the design are presented following a simple pseudo-code1, that allows us to redesign a proposed fictitious output in a constructive way. The methodology has been tested with three very well-known underactuated mechanical systems: the inertia wheel pendulum, the pendulum on a cart and the rotary pendulum. The obtained solution for the inertia wheel pendulum takes into account the friction, since recent works have shown that it cannot be neglected. In the case of the planar pendulum on a cart, the solution is quite similar to the one obtained by Controlled Lagrangians but with better performance, and, furthermore, our planar pendulum solution paves the way to obtain a new solution for the rotary pendulum, or the so–called Furuta pendulum, that, to the best of our knowledge, has the largest attraction basin presented and experimentally tested so far. The attraction basin tends to the whole upper half plane by increasing a control gain.