Trajectory Exploration and Maneuver Regulation of the Pendubot

The pendulum provides a seemingly inexhaustible source of practical applications and interesting problems which have motivated research in a variety of disciplines. In this thesis, we study equations that described a driven pendulum with odd-periodic driving. The equations also describe the under-actuated, double pendulum system called the pendubot. Techniques for trajectory exploration are developed. For the inverted pendulum, we first wrote the problem as a two point boundary value problem with Dirichlet boundary conditions. Then, we develop an equivalent linear operator that combines a Nemitski operator (or superposition operator) with the linear operator for the unstable harmonic oscillator. By exploring the properties of the Green’s function for the unstable harmonic oscillator with Dirichlet boundary conditions, we developed bounds on various norms that prove useful for determining which parameter values will satisfy invariance and contraction conditions. With a direct application of the Schauder fixed point theorem, we showed that our family of equations representing an inverted pendulum always possessed an odd-periodic solution. Using the Banach fixed point theorem we showed that there is a unique solution within an invariant region of the space of possible solution curves. When there is a unique solution, successive approximations can be used to compute the solution trajectory. To illustrate the power and application of these ideas, we apply them to a pendubot with the inner arm moving at a constant velocity. For non-inverted trajectories of the pendubot, we presented a necessary condition for trajectories to exist with general periodic forcing. For odd-periodic periodic driving functions this condition is always satisfied. For a driving function of A sin(wt), we found multiple solutions for the outer link. With the trajectories in hand, we demonstrated through simulation and/or physical implementation, the usefulness of maneuver regulation for providing orbital stabilization

Safe experimentation dynamics algorithm for data-driven PID controller of a class of underactuated systems

In recent decades, various control strategies for underactuated mechanical systems (UMS) have been widely reported which are derived from the systems’ model. Due to the problem of the unmodeled dynamics, there is a significant disparity between the theory of control and its actual applications, which makes the model-based controller difficult to apply. In recent years, control researchers have been switching to the method of data-driven control in order to eliminate this disparity. The control performance of this method is independent of the plant’s model accuracy to attain the control objective. This is because its controller’s design is founded only on the input-output (I/O) data measurement of the actual plants. In the industry, the proportional-integral-derivative (PID) controller is the control method that has been widely implemented because of its simplicity, the fact that it is more understandable and more reliable to be used for industrial purposes. So far, the tuning methods used for data-driven PID for the underactuated systems are mostly based on the multi-agent-based optimization, which means that the design requires substantial computation time and make it not practical for on-line tuning applications. Therefore, it is necessary to develop a tuning strategy that requires less computation time. Previously, a stochastic approximation based method such as the norm-limited simultaneous perturbation stochastic approximation (NL-SPSA) and global NL-SPSA (G-NL-SPSA) have shown successful results as tools for the data-driven PID tuning. Notably, the SPSA and GSPSA based methods only produced the optimal design parameter at the final iteration while it may keep a better design parameter during the tuning process if it has a memory feature. Hence, a memory-based optimization tool has good potential to retain the optimal design parameter during the PID tuning process. This can overcome the existing memory-based algorithms such as random search (RS) and simulated annealing (SA) which currently produce less control accuracy due to the local minimum problem. Motivated by the limitations of the current methods, there is an advantage to using safe experimentation dynamics (SED) as a tool for optimization. SED offers memory-based features and effectiveness to perform with lesser computation time to overcome a range of optimization problems, even for high-dimensional parameter tuning. Moreover, other than the memory-based feature, SED algorithm has fewer design parameters to be addressed and the independence of the gain sequence in the tuning process. Previously, SED algorithm has been applied in to control scheme of wind farm to optimize the total power production but has yet to be applied in PID tuning. Therefore, it is good to study the effectiveness of SED in PID tuning. In this study, the efficiency of the proposed approach is tested by applying the PID controller tuning to the slosh control system, double-pendulum-type overhead crane (DPTOC) control system and multi-input-multi-output (MIMO) crane control system. The performance was evaluated using numerical examples in terms of tracking performance and control input energy. Thirty trials have been performed to evaluate the SED, norm limited SPSA (NL-SPSA), global norm limited SPSA (G-NL-SPSA), and RS algorithms in each example. Next, when the pre-stated termination condition is fitted, each method is evaluated based on the statistical analysis involving the objective function, the total norm of the error and total norm of the input. Then, the rise time, settling time, and percentage of overshoot of the one best trial out of the 30 trials were observed for each method. In the DPTOC control system, we also present the examples with disturbance. The performance comparison was made only between the SED based method and G-NL-SPSA based method. In addition, the average percentage of the control objective improvement retrieved from the 30 trials for each method was also observed

Control Global del Péndulo con Rueda de Reacción Empleado Redes Neuronales Artificiales y Linealización Extendida

In this paper describes the design and simulation of a two-stage hybrid controller for the inverted reaction wheel pendulum (RWP) is presented. In the first stage, the general pendulum arm modeling is performed through a nonlinear model to determine the stored energy in the plant, and using a strategy known as swing up energy regulation, the data required for the training of an artificial neural network are obtained. In the second stage, via a soft switching system, neuronal control is exchanged by a controller based on the extended linearization of the state variables, whereby the permanence of the pendulum in the region of operation is guaranteed. The control strategy proposed shows excellent performance against external disturbance phenomena and ensures the overall operation of the physical system. En este artículo se presenta el diseño y la simulación de un controlador híbrido de dos etapas para el péndulo invertido con rueda de reacción (RWP). En la primera fase se realiza el modelado general del brazo pendular a través de un modelo no lineal que determina la energía almacenada en la planta, y usando una estrategia de balance conocida como regulación de energía (RE), se obtienen los datos necesarios para el entrenamiento de una red neuronal artificial (RNA). En la segunda etapa, por medio de un sistema de conmutación suave, se intercambia el control neuronal por un controlador basado en la linealización extendida de las variables de estado (LEVE), con lo cual se garantiza la permanencia del péndulo en la región de operación. La estrategia de control propuesta, muestra un excelente desempeño ante fenómenos de perturbación externa y garantiza la operatividad global del sistema físico

Intelligent model-based control of complex three-link mechanisms

The aim of this study is to understand the complexity and control challenges of the locomotion of a three-link mechanism of a robot system. In order to do this a three-link robot gymnast (Robogymnast) has been built in Cardiff University. The Robogymnast is composed of three links (one arm, one torso, one leg) and is powered by two geared DC motors. Currently the robot has three potentiometers to measure the relative angles between adjacent links and only one tachometer to measure the relative angular position of the first link. A mathematical model for the robot is derived using Lagrange equations. Since the model is inherently nonlinear and multivariate, it presents more challenges when modelling the Robogymnast and dealing with control motion problems. The proposed approach for dealing with the design of the control system is based on a discrete-time linear model around the upright position of the Robogymnast. To study the swinging motion of the Robogymnast, a new technique is proposed to manipulate the frequency and the amplitude of the sinusoidal signals as a means of controlling the motors. Due to the many combinations of the frequency and amplitude, an optimisation method is required to find the optimal set. The Bees Algorithm (BA), a novel swarm-based optimisation technique, is used to enhance the performance of the swinging motion through optimisation of the manipulated parameters of the control actions. The time taken to reach the upright position at its best is 128 seconds. Two different control methods are adopted to study the balancing/stablising of the Robogymnast in both the downward and upright configurations. The first is the optimal control algorithm using the Linear Quadratic Regulator (LQR) technique with integrators to help achieve and maintain the set of reference trajectories. The second is a combination of Local Control (LC) and LQR. Each controller is implemented via reduced order state observer to estimate the unmeasured states in terms of their relative angular velocities. From the identified data in the relative angular positions of the upright balancing control, it is reported that the maximum amplitude of the deviation in the relative angles on average are approximately 7.5° for the first link and 18° for the second link. It is noted that the third link deviated approximately by 2.5° using only the LQR controller, and no significant deviation when using the LQR with LC. To explore the combination between swinging and balancing motions, a switching mechanism between swinging and balancing algorithm is proposed. This is achieved by dividing the controller into three stages. The first stage is the swinging control, the next stage is the transition control which is accomplished using the Independent Joint Control (IJC) technique and finally balancing control is achieved by the LQR. The duration time of the transition controller to track the reference trajectory of the Robogymnast at its best is found to be within 0.4 seconds. An external disturbance is applied to each link of the Robogymnast separately in order to study the controller's ability to overcome the disturbance and to study the controller response. The simulation of the Robogymnast and experimental realization of the controllers are implemented using MATLAB® software and the C++ program environment respectively

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

Dynamic balancing of underactuated robots

This thesis presents the control of planar underactuated systems that have one less control input than the number of degrees of freedom. The underactuated robots are studied to achieve dynamically stable motions commonly encountered during robot locomotion. This work emphasizes the relation between the underactuated systems and biped locomotion and builds on the previous works in the literature on underactuated robot locomotion. Two planar system models are treated: an acrobatic robot and a compass biped with torso. The dynamic stability of fast periodic trajectories of these systems are regulated by designing asymptotically stable feedback controllers. The resulting internal dynamics of the systems are analyzed and shaped to achieve energy efficiency and robustness of the closed-loop system trajectories. In particular, Bézier polynomial approximations and parameter optimization methods are used to systematically construct the internal dynamics of the systems. Simulation results are presented for dynamically stable orbits of the acrobatic robot and the compass biped with torso

Taps (2001)

https://tigerprints.clemson.edu/yearbooks/1098/thumbnail.jp