Eksik Tahrikli Tekerlekli Sarkaç Sisteminin Tasarımı Ve Kontrolü

Abstract

Literatürde “Reaction Wheel Pendulum” veya “Inertia Wheel Pendulum” olarak geçen eksik tahrikli tekerlekli sarkaç sistemi ilk olarak 1999 yılında şu anda University of Texas at Dallas’da olan Prof. Mark W. Spong tarafından ortaya konulmuştur. Deney düzeneği, eyleyicisiz serbest dönebilen bir sarkaç koluna akuple edilmiş bir motorla sürülen yüksek simetriye sahip bir tekerlekten oluşmaktadır. Sistemin kontrolü motor ile sürülen tekerlek ile sağlanır. Sürülen tekerlek ivmelenerek tüm sistemde eylemsizliğinden dolayı bir tork oluşturur ve oluşan bu tork sistemin salınıma geçmesini sağlar. Sistemin yüksek simetriye sahip olması dinamik modellemesinin daha rahat çıkarılmasına ve daha kolay analiz edilmesine sebep olur. Dinamik modellemesinin daha rahat yapılmasının yanı sıra sisteme ileri düzey kontrol metotları (geribeslemeli lineerleştirme metodu, enerji tabanlı salınım kontrolü ve hibrid kontrol gibi) da uygulanabilmektedir. Ayrıca, lineer olmayan bir davranış sergilediğinden ve eksik tahrikli olduğundan literatürdeki çalışmaların ilgi odağı olmaktadır. Bu çalışmada, ilk olarak tekerlekli sarkaç ile ilgili genel bilgiler verilmiş ve daha sonra tekerlekli sarkaç sistemi ile ilgili literatür taramaları incelenmiş ve tezin genel hatları ile ilgili bilgi verilmiştir. İkinci bölümde, sarkacın imalat kısmı, sarkaçtaki her parçanın üç boyutlu katı modelleme ve analiz yazılımı olan CATIA ile tasarımı ve mukavemet analizi yapılarak imalat edilebilirliği irdelenmiş ve sonuç olarak imal edilebileceği kararı verilmiştir. Üçüncü bölümde sarkacın ideal ortam için dinamik modeli çıkartılmış ve ardından sürtünmenin sisteme nasıl dahil edilebileceği hakkında yorum yapılmıştır. Dördüncü bölümde sarkacın salınım kontrolü ve üst denge noktasındaki kontrolü için kontrolörler tasarlanmıştır. Üst denge noktasındaki kontrolü için giriş-çıkış geribeslemeli lineerleştirme metodu kullanılmış olup, salınım kontrolü için enerji tabanlı ve anahtarlamalı salınım kontrolleri tasarlanıp bunların detayları verilmiştir. Daha sonra salınım kontrolü ile denge kontrolü hibrid hale getirilerek sistemin salınımından üst denge noktasına ulaşana kadarki kontrol algoritması geliştirilmiştir. Çalışmanın son bölümü olan gelecek çalışması için ise eksik tahrikli tek sarkaç sistemi yerine aralarında elastik bir yük bulunan birlikte çalışan çift sarkaç sistemi ele alınıp, bu sistemin imalatından bahsedilerek, dinamik modeli çıkarılmış ve MATLAB-Simulink modeli oluşturularak sistem analiz edilmiştir.

Reaction Wheel Pendulum or Inertia Wheel Pendulum as known in literature was first mentioned in 1999 by Prof. Mark W. Spong. In experiment, unactuated pendulum coupled to highly symmetric wheel driven by a DC motor. The unactuated pendulum is controlled by this actuated wheel. Actuation of the wheel creates angular acceleration due to torque generated by the DC motor. This torque causes the pendulum to swing up. Dynamic model needs to be obtained to control the system. In addition to this, advanced control methods such as feedback linearization, energy based control and hybrid control can be applied. Moreover, reaction wheel pendulum has non-linear property and it is an underactuated system. Thus, reaction wheel pendulum has been an attraction center in literature. In this study, general information is given about underactuated systems and the reaction wheel pendulum. Moreover, the literature investigation and general scheme of thesis are displayed in introduction. In second chapter, product design and manufacture are discussed. Later on, chapter three explains how to derive dynamic model of reaction wheel pendulum with Euler-Lagrange method that depends on system energy. Subsequently, control methods of reaction pendulum are given. Then, coorperating reaction wheel pendulums with elastic load is shown in future work. Finally, conclusion of this study is given. In introduction, basic information about underactuated systems, such as what it is and why it is important, is given. Besides, definition of the reaction wheel pendulum is declared. For instance, what the reaction wheel pendulum is, how it works and why it is important. Afterwards, literature investigations which give knowledge about control, linearization and stability analysis methods of underactuated pendulums are explained. First, swing up control with Lyapunov funtion is mentioned. Basic of this method is to select approptiate Lyapunov function. This selection is usually done depending on total energy of the system. Its aim is to swing up the pendulum from down to up equilibrium point. Second, some linarization methods are given for underactuated pendulums. One of them is a conventional method. With this method, dynamic model of the pendulum is linearized using Taylor series. According to this, pendulum can be controled in up equilibrium point. The other method is feedback linearization. In this method, proper output function is selected. Then, between input and output functions are looked for a relation with derivative of output function. When the relation is provided, feedback control function is generated to control pendulum in up equilibrium point. Third, stability analysis is exposed with Lyapunov function and LaSalle’s invariant stability analysis. They are given to determine system stability. Finally, schedule of thesis is found. In second section, manufacturing and design of pendulum are discussed. Pendulum design is done using CATIA, which is three dimentional solid modelling and analysis software program. First idea of pendulum design, is to have materials as far as light and durable such as delrin and aluminum. Therefore, delrin is used for wheel and pendulum arm. Alimunium is used for stable parts of pendulum. However, shafts are manufactured with steel. What’s more, detailed information about pendulum’s parts are also included in this section. To determine whether the pendulum parts can be manufactured or not, strength analyses is investigated using CATIA. Strength analysis forces are defined from simple dynamic model of pendulum which includes point charge and negligible rope. Point charge include all mass of the pendulum and rope lenght is the same as center of mass. Then, magnitudes of forces are derived with Newton equations. Pursuant to these forces, strength analyses are done with CATIA. Position graph is shown using pendulum’s dynamic equations. Related to the position graph, it is proved that the designed pendulum swings 2 times which is desired for this study to reach its up equilibrium point. These two approaches are demonstrated that needed pendulum can be manufactured. At the end of this chapter full product is presented. In third section, dynamic models is derived for ideal environment with Euler-Lagrange method. Furthermore, some advices are given about how to add frictions to the pendulum’s dynamic model using MATLAB-Simulink. Euler-Lagrange method is useful to derive dynamic equations of a system. It depends on kinetic and potential energy of entire system. In order to define kinetic energy of pendulum, moment of inertia, is necessary to be found out. After finding moment of inertia and energies, Lagrange function is able to be determined. Thanks to Lagrange function, Euler-Lagrange equation can be finally solved. The solution of Euler-Lagrange equation gives dynamic model of the pendulum. Whereby the solution of the equation, dynamic model is found. According to dynamic model, simulation is performed by using MATLAB-Simulink. As a result of this simulation, positions and angular velocities can be determined for wheel and pendulum arm. In spite of the fact that dynamic equations are determined for ideal environment, in practice, friction forces have effects over the pendulum. Therefore, some advices are given how to determine those friction forces. For example, if pendulum arm is free fell from 90 degrees, then its motion can be monitored via encoder. The plot of motion can be compared to Simulink Coulomb and Viscous friction block’s result, which is added to pendulum’s dynamic model. With respect to this comparison, Coulomb and Viscous friction block’s gains are able to be determined. Similarly, wheel frictions gains can be found, as well. Wheel can be driven until its maximum limit, after given energy is stopped and wheel can be allowed to free rotation. Then, plot of free rotation can be compared to dynamic model with Coulomb and Viscous friction block. In conclusion, block gains are provided. In fourth section, three different control methods and their hybrid control are mentioned. First of all, their aims are discussed. Two of three different control methods are to swing up pendulum from down to up right (unstable equlibrium) point. The other one is to balance pendulum in up right possition. Swing up control methods can not be used to control the pendulum in up righ possiton. Therefore, third control method which is known as input-output feedback linearization (feedback linearization) method is used. Before using feedback linearization, controllability of pendulum is shown via state-space model. To use state-space model, dynamic model is linearized using Taylor series. Feedback linearization method’s fundamental is to determine a suitable output function. Using derivative of selected output function, a relation can be supplied with input function. When the relation is obvious, then new control signal can be created which is used to control pendulum. Second, the pendulum requires Lyapunov function method to be swinged. This method is called as “energy based swing up control method” because system’s total energy is used in Lyapunov function. An appropriate Lyapunov function is selected as related to total energy of the pendulum. According to Lyapunov function, a control signal is selected to swing up the pendulum until up righ point. In addition, passivity of pendulum is explained to be able to use Lyapunov function. Third, the main control method for this study is given to swing up the pendulum. This method depends on switching the motor power. Pendulum is started with initial conditions with a maximum torque, it reaches to a maximum point and stops due to loss of energy. Then, the motor is switched to transfer more energy to the system. Hence, pendulum reaches higher point at the end of the first period. This switching continues until reaching up right position zone. The important idea is to stop transfered energy, when the motor is stopped, at right moment. Because of this, end swing angle and cut off torque angle are found. After determining control inputs of swing up and balance, their hybrid controls are created. Finally, these hybrid controls are compared. Fifth section is designed as future work. Cooperating reaction wheel pendulums are included in future work. Begining of the section, cooperating reaction wheel pendulums’ manufacture are exhibited. In product, there are two identical reaction wheel pendulum and their junction is supplied with elastic load. Afterwards, its dynamic model is derived using Newton-Euler approach which gives the same result as Euler-Lagrange with different way. This method is a suitable way for manipulator and it depends on force and torque balance of each link. Therefore, cooperating reaction wheel pendulums are modeled as a manipulator system. Then, its kinematic model is derived for each part. Related to this kinematic model, dynamic model is presented for ideal environment. Its simulation is performed by using the found dynamic model in MATLAB-Simulink and its result is shown and analysed. After, control scenarios of cooperating system are explained. In final section, general results of thesis are declareted and some control methods such as feedback linearization are compared to literature investigations. In addtion to this, general summary about the thesis is given.