Design And Construction Of An Inverted Pendulum

ABSTRACT

The Inverted Pendulum is a classical control theory problem that helps understand the importance of feedback control systems for a coupled plant. In this study, a custom built pendulum system is coupled with a linearly actuated cart and a control system is designed to show the stability of the pendulum. The three major objectives of this control system are to swing up the pendulum, balance the pendulum in the inverted position (i.e. 180^◦), and maintain the position of the cart.  The input to this system is the translational force applied to the cart using the rotation of the tires.

The main objective of this work is to design a control system that will help in balancing the pendulum while maintaining the position of the cart and implement it in a robot. The pendulum is made free rotating with the help of ball bearings and the angle of the pendulum is measured using an Inertial Measurement Unit (IMU) sensor. The cart is actuated by two Direct Current (DC) motors and the position of the cart is measured using encoders that generate pulse signals based on the wheel rotation. The control is implemented in a cascade format where an inner loop controller is used to stabilize and balance the pendulum in the inverted position and an outer loop controller is used to control the position of the cart. Both the inner loop and outer loop controllers follow the Proportional-Integral-Derivative (PID) control scheme with some modifications for the inner loop.

The system is first mathematically modeled using the Newton-Euler first principles method and based on this model, a controller is designed for specific closed-loop parameters. All of this is implemented on hardware with the help of an Arduino Due microcontroller which serves as the main processing unit for the system.

 

            COVER PAGE

TITLE PAGE

APPROVAL PAGE

DEDICATION

ACKNOWLEDGEMENT

ABSTRACT

ABBREVIATION

CHAPTER ONE

1.0      INTRODUCTION

1.1      BACKGROUND OF THE STUDY

• PROBLEM STATEMENT
• AIM AND OBJECTIVE OF THE STUDY
• APPLICATIONS OF THE PROJECT
• SCOPE OF THE STUDY
• LIMITATION OF THE STUDY

CHAPTER TWO

LITERATURE REVIEW

• INVERTED PENDULUM
• MODELING OF INVERTED PENDULUM ON A CART
• NEWTON EULER EQUATIONS
• LINEARIZATION

CHAPTER THREE

3.0      HARDWARE AND SYSTEM INTEGRATION

3.1      INTRODUCTION
3.2      SYSTEM HARDWARE

3.3      SYSTEM INTEGRATION

 

CHAPTER FOUR

4.1      PARAMETRIZATION OF FIRST PRINCIPLES MODEL

4.2      `RESULT

 

CHAPTER FIVE

• CONCLUSION
• RECOMMENDATION FOR FUTURE WORK

REFERENCES

 

 CHAPTER ONE

1.0                                                  INTRODUCTION

1.1                             BACKGROUND OF THE STUDY

The Inverted Pendulum has been a great platform for engineers to research and develop different types of applications like self-balancing scooters, segways etc. It is a classical problem used for learning the ideas of a coupled system. For a controls engineer, this system gives a very good insight into developing different feedback control mechanism.

The Inverted Pendulum system has a pendulum subsystem and a cart subsystem which are coupled together by mechanical design. Due to the coupled nature of this system, an effect on one subsystem also affects the other subsystem. In this thesis, the pendulum system on its own is considered as a rigid body that has a weight attached to one end while the other end is fixed to a pivot so that the pendulum can swing freely about the rotational axis of the pivot. The pendulum system always remains vertical when at rest due to the effect of gravitational force. This resting point of the pendulum is called the stable equilibrium of the system (i.e the system will always converge to this point in any given point of time). The pendulum will also have the same effect of the gravitational field when it is displaced 180^◦ from its resting point, this point is called the unstable equilibrium of the system because even a very small disturbance in the inverted position will make the system unstable. The main objective of this thesis is to swing up and stabilize the pendulum around its unstable equilibrium point which is similar to balancing a long rod with our hand where our hand plays an important role in keeping the rod steady at all times.

Similarly, a cart subsystem is designed and coupled with the pendulum subsystem so that it can balance the pendulum when it is in the inverted position. The cart subsystem is constructed with the help of two high powered DC motors which are used for actuation of the system. To balance the pendulum, the cart has to accelerate in one direction which provides a torque to swing the pendulum. Therefore, a controller designed for this type of system must have an inverse response due to the effect of right half plane zeros present in the system (i.e For the cart to go right, it must first move left and unbalance the pendulum).

This system can be either modeled using the Newton-Euler method or the Euler-Lagrange method. In this thesis, the Newton-Euler approach of first principles modeling is used to derive the equations of motion. Based on the equations of motion Force is the only input given into the system. But, Force can only be used in a simulation environment as it cannot be measured accurately in a real-world system. Hence, the equations of motion should replace the Force input with the dynamics of the two DC motors used so that a better approximate model of the system is derived. Several research papers show different methods to build this system and the motivation for this thesis is the design, model, and construction of a robot that balances a free rotating pendulum with the help of digital sensors like accelerometer and gyroscope which are used to estimate the angle of the pendulum and a motor encoder sensor to estimate the position of the robot. Using these sensors to balance an inverted pendulum while maintaining the position of the cart makes this problem more interesting and difficult to control. This work is aimed at building an inverted pendulum on a cart.

 

1.2          PROBLEM STATEMENT

The demand of industrial revolution has become an unavoidable challenge (Barnett, 2013). Regarding to these circumstances, it is necessary to reorient the curriculum to be more relevant for a 21st-century world, especially with the massive technological developments (Schaefer, 2016). The current technology is not just about turning on/off the machine power but more on controls that can adapt to the surrounding environment. For instance, development of self-driving car or often called as smart car. This kind of vehicle has various automatic sensors that can detect the surrounding environment to operate as a human driver. It urges academicians and researchers to master up-to-date technology to have global competitiveness. To boost technological literacy, a proper model is truly used to represent this requirement.

Inverted Pendulum is one model that represents a simple adaptive control. To learn control in this model, it requires some innovation that involves mathematical functions. This control is commonly used in industries, such as Proportional Integral Derivative (PID) controller, and Sliding Mode Controller (SMC). This control system has been the basis of applications in present technology, like rockets, airplanes, unicycles (single-wheeled vehicles), modern motor vehicles, wheelchairs, satellite position control, games (Cubic Li) and others.

1.3                                                     AIM AND OBJECTIVES OF THE PROJECT

The main goal of this thesis is to develop complete equipment with a working control system for an inverted pendulum. This includes making a good description and technical report for the next master thesis, so that they can modify our solution to a fault tolerant system. The objectives of the study are:

1. To develop an inverted pendulum on a cart
2. To learn the basic concepts of adaptive control in the form of simple modeling

1.4                                 APPLICATION OF THE PROJECT

This control is commonly used in industries, such as Proportional Integral Derivative (PID) controller, and Sliding Mode Controller (SMC). This control system has been the basis of applications in present technology, like rockets, airplanes, unicycles (single-wheeled vehicles), modern motor vehicles, wheelchairs, satellite position control, games (Cubic Li) and others.

1.5                                 SCOPE OF THE STUDY

In this work, a system and controller were designed to balance the pendulum for about 3-5 seconds before reaching the edge of the track. Applying small “taps” to the pendulum in the opposite direction would allow for much longer control. The final result of this design process can be deemed a success since a working mechanical system was developed with an optimal controller designed by minimizing the Linear Quadratic Regulator cost equation, given the maximum desired angle, displacement, and control effort.

1.6                                            LIMITATION OF THE PROJECT

During the design process, many setbacks were encountered. The most important and time consuming setbacks all related to broken motors. When the first motor broke, a faster and higher torque motor was obtained because the original one was not able to react fast enough in order to balance the pendulum. Since the new motor had different dimensions than the first, the whole mechanical system needed to be redesigned. The next two motors broke when the control effort was too high and changing direction rapidly, putting too much stress on the internal gears. To fix this, the controller was redesigned with a tighter constraint on the control effort.