6 01 Fall Semester 2007 Assignment 6 Issued Tuesday Oct 9 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 01 Introduction to EECS I Fall Semester 2007 Assignment 6 Issued Tuesday Oct 9 To do this week No Tuesday software lab before the lab Thursday 10 11 1 Do the on line Tutor problems for week 6 that are due on Thursday Part 6 1 2 Do the writeup for the 10 2 and 10 4 software labs in a previous handout providing written answers including code and test cases for every numbered question in this handout in lab on Thursday 10 11 1 Work on the robot steering lab described below 2 Do the nanoquiz it will be based on the material in the lecture notes and the on line Tutor problems due on Thursday before lecture Tuesday 10 16 1 Do the writeup for the 10 11 design lab In this handout On Athena or the lab laptops make sure you execute athrun 6 01 update so that you can get the Desktop 6 01 lab4 directory which has the files mentioned in this handout You need the file polynomial pyc You need the file feedback py You need the file differenceEquationWithInput py You will need the z transform manipulation software you wrote You need the file soar graph py for the software lab 6 01 Fall Semester 2007 Assignment 6 Issued Tuesday Oct 9 2 Figure 1 Robot in corridor Thursday Design Lab More sophisticated robot control Just as in the previous 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 speed of the robot V the distance to the left wall dleft the distance to the right wall dright and the angle the robot is making with respect to the parallel walls Unlike two weeks ago when you tried to steer the robot to drive straight down the center of the hallway this week you will be trying to steer the robot to stay a desired distance ddesired from the center of the hallway if ddesired 0 the problem is equivalent to last week s problem Note that the distance from the center is d 0 5 dright dleft Question 1 Briefly explain why the distance to the left of center is 0 5 dright dleft In the last lab you used a very simple control strategy to keep the robot in the center of the corridor ddesired 0 by adjusting the rotational speed based on the current value of d n You discovered that a model of such a controller predicted robot behavior that would oscillate and eventually hit one of the walls This time you will design a better controller Using Z transforms to analyze the simple controller Consider trying to steer the robot down a hallway so as to maintain a desired distance ddesired from the center of the hallway while moving forward with a constant speed Let the error at step n e n be defined as e n ddesired n d n where d n is the measured distance at time n from the center of the hallway Then the objective would be to keep the magnitude of e n as small as possible We can use almost the same difference equation as last lab to model this more general case of an adjustable desired displacement In particular the center displacement will still be related to the robot angle by d n 1 d n V t n 6 01 Fall Semester 2007 Assignment 6 Issued Tuesday Oct 9 3 where t is the system s time between samples and V is the robot forward speed For this lab use V 0 1 meters per second If we use the same control strategy as in the last lab then the robot angle will satisfy a slightly modified difference equation for this more general input case n 1 n K te n where K is the gain of the feedback You experimented with different values of K in the last lab BE CAREFUL ABOUT THE SIGN OF K which depends on the definition of e Notice that if ddesired 0 then Ke n Kd n Question 2 Determine by hand a symbolic expression for the transfer function H z in D z H z D desired z as a function of the gain K Determine and use the numerical value for t and use 0 1 for the forward velocity V Question 3 Pick a numerical value K and demonstrate that you can use your transfer function manipulation program to combine the above difference equations Show that you can both generate an H z and a difference equation that relates ddesired n to d n Question 4 Demonstrate that you can use your program to compute the system s natural frequencies given a numerical value for K and verify that you program can reproduce your results from lab 4 Checkpoint 10 45 Demonstrate your answers to the above questions to an LA Analyzing a more sophisticated controller In order to design a better controller one can process the error e n in a more sophisticated way For example one could adjust the rotational speed using some combination of the present and previous values of the displacement error The robot angle would then satisfy the difference equation n 1 n t K1 e n K2 e n 1 6 01 Fall Semester 2007 Assignment 6 Issued Tuesday Oct 9 4 Question 5 Determine by hand a symbolic expression for the transfer function H z of the new controller D z H z D desired z as a function of the gains K1 and K2 Determine and use the numerical value for t and use 0 1 for the forward velocity V Question 6 Pick numerical values for K1 and K2 and for demonstrate that you can use your transfer function manipulation program to combine the difference equations for the more complicated controller Show that you can both generate an H z and a difference equation that relates ddesired n to d n Question 7 Demonstrate that you can use your program to compute the system s natural frequencies given numerical values for K1 and K2 Designing the controller by placing the natural frequencies As you have no doubt discovered the more sophisticated controller generates a transfer function whose denominator is a cubic polynomial In addition two of the denominator polynomial coefficients are functions of K1 and K2 Since the roots of the denominator polynomial are the natural frequencies of the feedback system your design problem is to pick values of K1 and K2 so that the natural frequencies are less than one in magnitude One approach to determining values of K1 and K2 is to use your transfer function program to perform a brute force search for values of K1 and K2 Before trying brute force consider a strategy of starting by specifying a set of desired natural frequencies and then determining the associated feedback gains Question 8 Using your expression for the transfer function can you determine values …
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