Project #1: Inverted Pendulum Cart
2005502016 - Adnan Alper DEMIRCI (aalperdemirci [at] gmail.com)
2006502071 - Omer Faruk CILESIZ (farukcilesiz [at] hotmail.com)
Nowadays, there are some considerable applications of inverted pendulum. The dynamics of inverted pendulum simulates the dynamics of robotic arm in the condition when the center of pressure lies below the centre of gravity for the arm so that the system is also unstable. Robotic arm behaves very much like inverted pendulum under this condition. Also, the inverted pendulum is widely accepted as an adequate model of a human standing still (quiet standing). Inverted pendulum uses at in the design of several early Seismometers, too. In addition to the applications, The Segway PT is a two-wheeled, self-balancing transportation machine. The dynamics of the Segway PT are similar to a classic control problem, the inverted pendulum. Like these applications, the dynamics of inverted pendulum can be used at different applications.
This report is about a robotics project that tries to realize an inverted pendulum.
The inverted pendulum system is a popular demonstration of using feedback control to stabilize an open-loop unstable system. It is simply a stick on a moving surface and desired to be in balance perpendicular to its surface. It is often useful to demonstrate concepts in linear control such as the stabilization of unstable systems. Different mechanisms can be constructed to demonstrate the inverted pendulum, therefore the moving cart is selected in this project to realize the inverted pendulum.
In this system, an inverted pendulum is attached to a cart equipped with a DC motor that drives it along a circular track. The cart tries to hold the inverted pendulum in a desired interval of angle by moving left or right on the circle.
Aim of the project is to control of an inverted pendulum. There are some reasons to select an inverted pendulum which are it is the most available system for laboratory usage and It is a nonlinear system, which can be treated to be linear, without much error, for quite a wide range of variation and an inverted pendulum provides a good practice for prospective control engineers. This project contains theoretical and experimental analyses of the system dynamics and control methods. In this system, there is a bar which behaves as an inverted pendulum. This bar is attached to a cart which is controlled by a PIC microcontroller and a pivotal system to make inverted pendulum stable.
Figure 1.1: A block diagram to the system.
2.1. Mathematical Analyses of Inverted Pendulum and Cart
In this section, the suitable mathematical model for an inverted pendulum system is derived. System model is shown below in Figure 2.1.
Figure 2.1: Free body diagrams of the system.
When the forces in the Free Body Diagram of the cart in the horizontal direction are summed, the following equation of motion will be found:
Also, we can say that forces in vertical direction are unnecessary because of that summing of forces in vertical direction is not being added to mathematical model for an inverted pendulum.
The force exerted in the horizontal direction due to the moment on the pendulum is determined as follows:
Component of this force in the direction of N is
The component of the centripetal force acting along the horizontal axis is as follows:
Component of this force in the direction of N is
When the forces in the Free Body Diagram of the cart in the horizontal direction are summed, the following equation is founded for N:
If this equation (6) is substituted into the first equation (1), the first equation of motion for this system is founded.
To get the second equation of motion, the forces perpendicular to the pendulum is summed. The vertical components of those forces are considered here to get the following equation:
To get rid of the P and N terms in the equation above, sum the moments around the centered of the pendulum to get the following equation:
The second dynamic equation is founded with combining these last two equations:
The set of equations completely defining the dynamics of the inverted pendulum are:
These two equations are non-linear and need to be linearized for the operating range. Since the pendulum is being stabilized at an unstable equilibrium position, which is Pi radians from the stable equilibrium position, this set of equations should be linearized about theta = Pi. Assume that theta = Pi + ø, (where ø represents a small angle from the vertical upward direction).
Therefore, cos (theta) = -1, sin (theta) = -ø, and (d(theta)/dt)^2 = 0.
After linearization the two equations of motion become (where u represents the input):
Laplace transform of the system equations are taken to obtain the transfer function of the linearized system equations analytically the. The Laplace transforms are:
2.2. Transfer Function of Inverted Pendulum and Cart
After finding the transfer function, initial conditions are assumed to be zero. The transfer function is the ratio of desired position [Output] with the force on the cart [Input].
If the angle Phi is as the output, the first equation for X(s) will be as following equation,
Then, substituting into the second equation will yield:
where, q = (M +m)(I +ml 2) - (ml)2.
If the transfer function will be simple, the following equation is obtained:
The transfer function can thus be simplified as:
If we NEGLECT THE FRICTION in the system, that is, we take the coefficient of friction b=0, then
2.3. State Space of Inverted Pendulum and Cart
There are several different ways to describe a system of linear differential equations. The state-space representation is given by the equations:
Where x is an n by 1 vector representing the state (commonly position and velocity variable in mechanical systems), u is a scalar representing the input (commonly a force or torque in mechanical systems), and y is a scalar representing the output. The matrices A (n by n), B (n x 1), and C (1 x n) determine the relationships between the state and input and output variable. Note that there are n first-order differential equations. State space representation can also be used for systems with multiple inputs and outputs (MIMO), but we will only use single-input, single-output (SISO) systems in these tutorials.
The linearized system equations can also be represented in state-space form:
The C matrix is 2 by 4, because both the cart's position and the pendulum's position are part of the output. For the state-space design problem we will be controlling a multi-output system so we will be observing the cart's position from the first row of output and the pendulum's with the second row.
2.4. PID Controller
A proportionalintegralderivative controller (PID controller) is a generic control loop feedback mechanism widely used in industrial control systems. A PID is the most commonly used feedback controller. A PID controller calculates an "error" value as the difference between a measured process variable and a desired set point. The controller attempts to minimize the error by adjusting the process control inputs.
The PID controller calculation (algorithm) involves three separate constant parameters, and is accordingly sometimes called three-term control: the proportional, the integral and derivative values, denoted P, I, and D. Heuristically, these values can be interpreted in terms of time: P depends on the present error, I on the accumulation of past errors, and D is a prediction of future errors, based on current rate of change. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve, or the power supplied to a heating element.
In the absence of knowledge of the underlying process, a PID controller is the best controller. By tuning the three parameters in the PID controller algorithm, the controller can provide control action designed for specific process requirements. The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the setpoint and the degree of system oscillation. Note that the use of the PID algorithm for control does not guarantee optimal control of the system or system stability.
Some applications may require using only one or two actions to provide the appropriate system control. This is achieved by setting the other parameters to zero. A PID controller will be called a PI, PD, P or I controller in the absence of the respective control actions. PI controllers are fairly common, since derivative action is sensitive to measurement noise, whereas the absence of an integral term may prevent the system from reaching its target value due to the control action.
Figure 2.2. A block diagram of a PID controller
2.4.1. Proportional term
The proportional term makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp, called the proportional gain.
The proportional term is given by:
A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable. In contrast, a small gain results in a small output response to a large input error, and a less responsive or less sensitive controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances. Tuning theory and industrial practice indicate that the proportional term should contribute the bulk of the output change.
2.4.2. Integral term
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. The integral in a PID controller is the sum of the instantaneous error over time and gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain (Ki) and added to the controller output.
The integral term is given by:
The integral term accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a pure proportional controller. However, since the integral term responds to accumulated errors from the past, it can cause the present value to overshoot the set point value
2.4.3. Derivative term
The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain Kd. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, Kd.
The derivative term is given by:
The derivative term slows the rate of change of the controller output. Derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, the derivative term slows the transient response of the controller. Also, differentiation of a signal amplifies noise and thus this term in the controller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large. Hence an approximation to a differentiator with a limited bandwidth is more commonly used. Such a circuit is known as a phase-lead compensator.
3.1. How It Works
An inverted pendulum stick is connected to the cart with a potentiometer that is used for converting the resistance value to the angle information of the inverted pendulum. The output of the potentiometer is an analog signal, it is converted to digital at the built-in ADC of PIC microcontroller. Then, according to the angle value, it is decided whether there is a disturbance on the stick. If there is a disturbance, the cart is moved on the direction which is same as the disturbance. This movement creates a force on the stick and it goes towards the normal reference position. This is repeated until the balance of the stick is provided.
3.2. Used Parts
A wooden, round table (radius is 35cm) is constructed to prepare the base for the cart. The cart is basically in triangular shape, and it is connected to the table centrically. There are two wheels on the other corners of the cart. Therefore, the car makes a pivotal motion around the table. The PCB card is mounted on the cart.
The inverted pendulum is attached to the cart via a feedback potentiometer and it is 60cm long. PID constants (K, Ki and Kd) are represented by three potentiometers on the device. This allows the constants to be adjusted so the PID control can be tuned for optimal performance.
A 120rpm DC motor with 100:1 gear ratio is used in order to move the cart. It is attached to one wheel.
PIC 16F887 is used in the project as a microcontroller to control the data in overall system. Its built-in ADC and ECCP (Enhanced Capture/Compare/PWM) modules have been very useful for this project. Angle detection is provided by the potentiometer which is connected to the inverted pendulum. The output voltage of the potentiometer is sent to the ADC of the microcontroller. After it is digitized, it is converted to the corresponding value of angle. Motor driving is also done with the microcontroller using the ECCP module.
Figure 3.1: Pin descriptions of PIC16F887.
The algorithm used to steady the pendulum is very susceptible to high frequency noise from the feedback potentiometer. In order to prevent this, a low pass filter was placed between this potentiometer and the A/D input port on the microcontroller. The filter has a cutoff frequency at about 60Hz.
The H-bridge is simply a collection of electronic switches to drive the motor and very popular in robotics. It allows the one to move motors forward, backward and with varying speeds through PWM (pulse width modulation). The switches on the H-bridge are built by TC4428, 1.5A dual high speed power MOSFET drivers.
Figure 3.2: H-Bridge schematic.
At the end of the project, the hardware and mechanical parts are obtained and tested for proving that they are working. But the overall inverted pendulum system could not reach to work successfully because of struggles with software programming and time constraints.
 XK-DLB1 Inverted Pendulum Model, http://sdxk.en.alibaba.com/product/493712072-210752993/XK_DLB1_Inverted_Pendulum_Model.html
 Inverted Pendulum Tutorial Web Page, http://www.library.cmu.edu/ctms/ctms/examples/pend/invpen.htm
 KATSUHIKO OGATA, Modern Control Engineering, 4 Edition, Pearson Education International, U.S.A, 2002. pp 86-90, 681-703.
 A block diagram of a PID controller, http://radhesh.wordpress.com/2008/05/11/pid-controller-simplified/
You can download the ISIS schematic here.
You can download the PCB layout here.
You can download the program codes here.