Composite state variable based nonlinear backstepping design for the underactuated TORA system

A nonlinear vibration controller is proposed for the translational oscillators with rotating actuator (TORA) system with the recursive technology. A composite state variable (CSV) is defined for the TORA system to start the recursive process. The design procedure treats the some state variables as virtual control inputs to design the virtual controllers step by step until the nonlinear vibration controller is obtained. The system stability is studied via a stability theorem and simulation results show the validity of the proposed controller

Nonlinear backstepping design for the underactuated TORA system

The nonlinear feedback cascade model of the underactuated translational oscillators with rotating actuator is obtained through a collocated partial feedback linearization and a global change of coordinates. A nonlinear controller is designed with the backsteping technology, which treats the state variables as virtual control inputs to design the virtual controllers step by step. The system stability is proved with the Lyapunov stability theorem. The simulation results show the system under any initial states can be asymptotically stabilized to the origin and the controller has a good control performance

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

Dynamics and control of a class of underactuated mechanical systems

This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable

PD control for global stabilization of an n-TORA system

This paper concerns a global stabilization problem for an n-TORA (Translational Oscillator with a Rotational Actuator) system which consists of n carts connected to the fixed walls and each other by n+1 linear springs with each cart having an eccentric rotational proof-mass actuator moving in the horizontal plane. First, this paper derives the motion equation of the n-TORA system. Then, by using Lyapunov stability theory and physical properties of mechanical parameters of the n-TORA system, this paper proves that the global stabilization of the n-TORA system can be achieved by the PD control of the angle of the rotational proof-mass of each TORA. This paper presents numerical simulation results for 2- and 3-TORA systems to validate the result of the global stabilization

Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (leaves 308-316).This thesis is devoted to nonlinear control, reduction, and classification of underactuated mechanical systems. Underactuated systems are mechanical control systems with fewer controls than the number of configuration variables. Control of underactuated systems is currently an active field of research due to their broad applications in Robotics, Aerospace Vehicles, and Marine Vehicles. The examples of underactuated systems include flexible-link robots, nobile robots, walking robots, robots on mobile platforms, cars, locomotive systems, snake-type and swimming robots, acrobatic robots, aircraft, spacecraft, helicopters, satellites, surface vessels, and underwater vehicles. Based on recent surveys, control of general underactuated systems is a major open problem. Almost all real-life mechanical systems possess kinetic symmetry properties, i.e. their kinetic energy does not depend on a subset of configuration variables called external variables. In this work, I exploit such symmetry properties as a means of reducing the complexity of control design for underactuated systems. As a result, reduction and nonlinear control of high-order underactuated systems with kinetic symmetry is the main focus of this thesis. By "reduction", we mean a procedure to reduce control design for the original underactuated system to control of a lowerorder nonlinear or mechanical system. One way to achieve such a reduction is by transforming an underactuated system to a cascade nonlinear system with structural properties. If all underactuated systems in a class can be transformed into a specific class of nonlinear systems, we refer to the transformed systems as the "normal form" of the corresponding class of underactuated systems. Our main contribution is to find explicit change of coordinates and control that transform several classes of underactuated systems, which appear in robotics and aerospace applications, into cascade nonlinear systems with structural properties that are convenient for control design purposes. The obtained cascade normal forms are three classes of nonlinear systems, namely, systems in strict feedback form, feedforward form, and nontriangular linear-quadratic form. The names of these three classes are due to the particular lower-triangular, upper-triangular, and nontriangular structure in which the state variables appear in the dynamics of the corresponding nonlinear systems. The triangular normal forms of underactuated systems can be controlled using existing backstepping and feedforwarding procedures. However, control of the nontriangular normal forms is a major open problem. We address this problem for important classes of nontriangular systems of interest by introducing a new stabilization method based on the solutions of fixed-point equations as stabilizing nonlinear state feedback laws. This controller is obtained via a simple recursive method that is convenient for implementation. For special classes of nontriangular nonlinear systems, such fixed-point equations can be solved explicitly ...by Reza Olfati-Saber.Ph.D

Dynamical Analysis and Stabilizing Control of Inclined Rotational Translational Actuator Systems

Rotational translational actuator (RTAC) system, whose motions occur in horizontal planes, is a benchmark for studying of control techniques. This paper presents dynamical analysis and stabilizing control design for the RTAC system on a slope. Based on Lagrange equations, dynamics of the inclined RTAC system is achieved by selecting cart position and rotor angle as the general coordinates and torque acting on the rotor as general force. The analysis of equilibriums and their controllability yields that controllability of equilibriums depends on inclining direction of the inclined RTAC system. To stabilize the system to its controllable equilibriums, a proper control Lyapunov function including system energy, which is used to show the passivity property of the system, is designed. Consequently, a stabilizing controller is achieved directly based on the second Lyapunov stability theorem. Finally, numerical simulations are performed to verify the correctness and feasibility of our dynamical analysis and control design

Model-Free Control of an Unmanned Aircraft Quadcopter Type System

A model-free control algorithm based on the sliding mode control method for unmanned aircraft systems is proposed. The mathematical model of the dynamic system is not required to derive the sliding mode control law for this proposed method. The knowledge of the system’s order, state measurements and control input gain matrix shape and bounds are assumed to derive the control law to track the required trajectories. Lyapunov’s Stability criteria is used to ensure closed-loop asymptotic stability and the error estimate between previous control inputs is used to stabilize the system. A smoothing boundary layer is introduced into the system to eliminate the high frequency chattering of the control input and the higher order states. The [B] matrix used in the model-free algorithm based on the sliding mode control is derived for a quadcopter system. A simulation of a quadcopter is built in Simulink and the model-free control algorithm based on sliding mode control is implemented and a PID control law is used to compare the performance of the model-free control algorithm based off of the RMS (Root-Mean-Square) of the difference between the actual state and the desired state as well as average power usage. The model-free algorithm outperformed the PID controller in all simulations with the quadcopter’s original parameters, double the mass, double the moments of inertia, and double both the mass and the moments of inertia while keep both controllers exactly the same for each simulation