MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 081 Introduction to EECS I Spring Semester 2007 Work for Week 5 Issued Tuesday March 6 This handout contains Software Lab for March 6th Prelab exercises due Thursday March 8th before lab Post lab exercises due Tuesday March 13th in Lecture Dynamic Feedback Control In many of the labs so far you developed progressively more sophisticated approaches for measuring the robot s environment as well as progressively more intelligent approaches to controlling the robot s actions based on those measurements In those labs our primary goal was to have you learn a little about the robot as well as have you investigate some ideas in software engineering In this lab we will also be using measurements to control the robot also known as feedback control but we will be investigating a situation in which detailed analysis is needed to achieve a desired performance The problem we are asking you to investigate is how to control the robot so that it drives down the center of a narrow corridor at a constant forward velocity Such a task is similar to what an automobile driver does when keeping a car on the road We will suggest a simple feedback scheme to keep the robot in the center of corridor and ask you to develop and analyze a mathematical model based on difference equations that explains the behavior of that simple feedback system In next week s lab you will use more sophisticated mathematical tools to develop improvements to the simple feedback scheme suggested in this lab The summary of tasks for this week are Post lecture software lab on solving second order difference equations Pre lab tutor exercises to practice solving difference equations Robot lab on implementing and analyzing a simple feedback scheme Post lab writeup and exercises due Tuesday in lecture Tuesday s Software Lab Solving second order difference equations For this lab you will be writing a python program that solves arbitrary second order homogeneous difference equations analytically by computing natural frequencies Your program should take as inputs the initial values x 0 and x 1 as well as coefficients a1 and a2 for the difference equation in the form y n a1 y n 1 a2 y n 2 Your program should print out the solution to the difference equation and also return a procedure that when called with n returns y n For example if the difference equation is y n 2 y n 1 2 y n 2 and the initial conditions are y 0 0 y 1 1 your program should print something equivalent to y n 2 77555756156e 017 0 5j 1 1j n 2 77555756156e 017 0 5j 1 1j n and your program should ALSO return a function which can be used to evaluate y n for any n Getting Started To save time you should use our implementation of the polynomial manipulation program from the recent post lab exercises in your second order difference equation solver Please download importPolynomial py and polynomial pyc Executing the module importPolynomial py in IDLE will import our version of the polynomial manipulation routines implemented in the class Polynomial You can see the class interface by typing help Polynomial at the IDLE command line Note that the Polynomial class has functions to add multiply print and find the roots of polynomials Demonstrate to your LA that you understand the Polynomial class Then before you write any code describe your approach to the difference equation solver to your LA A note on complex numbers For this problem your procedure will sometimes need to compute z n where z is a complex number Python understands complex numbers to some extent but you have to write them in the form a bj where a is the real part and b is the imaginary part Note that Python uses the convention that j 1 This should not surprise you as entering EECS students you already appreciate that the variable i must be reserved for electrical current So for example the Python command 4 0 5 will produce an error but the Python command 4 0j 0 5 produces 1 2246063538223773e 016 2j Why the very small real part Note use y 0 5 to compute the square root of a number The python sqrt function does not understand complex numbers Demonstrate that your program works Use your program to solve the difference equation y n 2 y n 1 2 y n 2 using the initial conditions y 0 0 y 1 1 Demonstrate to your LA that the program prints the correct information and also returns the correct procedure Use your program to find difference equations with given properties Use your program to find a difference equation whose solution oscillates rapidly but the amplitude of the oscillation decays slowly Use your program to find a difference equation whose solution oscillates rapidly but the amplitude of the oscillation remains constant What to turn in For this software lab hand in a brief few page lab report along with your post lab report on March 13th Also hand in a copy of your carefully commented python code with an explanation written in actual English sentences of how and why it works Homework in preparation for Thursday s lab This section includes problems to complete with the online tutor These are due before lab on Thursday The examples are to help you solidify your understanding of difference equations Read the entire document Please read the entire section on the lab before coming to lab Exercises with the online tutor Use the online tutor to complete the problems due Thursday March 8th Thursday s Lab As we are sure you have come to expect Thursday s lab will end with a nano quiz that is based both on tutorial problems due on thursday and on material covered this week It should be no great surprise that you will be asked a question about the solution to a difference equation Please keep in mind that the point of the quizzes is as a diagnostic for us did we succeed in teaching you and to encourage you to do the tutorial problems before coming to lab and participate in lab Figure 1 Robot in corridor Robot Feedback System In this lab you will be controlling the robot to drive down a narrow corridor as shown in Figure 1 Notice that in the figure we have denoted the forward velocity of the robot V the distance to the left wall dlef t the distance to the right wall dright and the angle the robot is making with respect to the parallel walls Consider the problem of trying to steer the robot down the center of the corridor while keeping it moving forward with a constant velocity For example if the robot is in the center of the corridor
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