6 01 Spring Semester 2008 Final exam announcement and practice final Revised May 12 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 01 Introduction to EECS I Spring Semester 2008 Final exam announcement and practice final Revised May 12 This is a sample final to help you study and to give you a sense of what kinds of problems to expect on the final The final will be given the afternoon of Wednesday May 21 You can pick it up in 34 501 the 6 01 lab at 1 30PM and you must turn it in by 7 30PM You may turn in your solution early You may not turn it in late Ground rules for taking the final These instructions apply to taking the real fine not this practice exam Read through the entire final and ask any questions you have before you leave If you find that you have a question after that try to pick your own answer and write down what assumption you re making Your paper must have all the sheets stapled together and must have your name on each sheet In the midterm some papers came in as separated sheets and without names We took the trouble then to identify the anonymous papers and reassemble the separated pieces We will not do that in grading the final we ll simply not grade any unidentified work For the problems that ask you to write Python code you should just think it through and write down the code Don t try to actually debug it it will take too long Don t spend time trying to be sure the syntax is correct It s more important to communicate your ability to think algorithmically than to worry about the fine grained details of syntax Be sure to add comments that explain what you re trying to do Your answers may be handwritten or typed they should be legible in either case We reserve the right to refuse to grade illegible papers You answers must include coherent explanations Simply writing down equations with no explanations or attaching pages of uncommented program output will get no credit In grading we will make no attempt to guess what you are thinking by trying to piece together fragments of phrases and equations and printout You may read anything you like labs books the Web during the exam you may not communicate with anyone else Take above prohibition on collaboration seriously On the midterm we found some papers a very small number where people had collaborated on problems We handled this by giving the students involved grades of zero for the midterm The consequences of collaborating on the final would be at least as severe and at a minimum result in failing the course Avoid temptation find some place to work on the final without other people in around You can use any of the software we ve provided or you ve written for lab But it s your responsibility to make sure that your computer runs the code Tales of woe of the form I couldn t get the code to run may receive sympathetic sighs but that won t make any difference in grading 6 01 Spring Semester 2008 Final exam announcement and practice final Revised May 12 2 1 Robot in a corridor For this problem please consider a long corridor of 100 alternating black and white squares labeled 0 through 99 Your robot is initially known to be in square 0 which is black and at the far left The robot can move only right When you ask the robot to move right one square it will actually move right one square only 50 of the time 40 of the time it will not move at all and 10 of the time it will move right two squares Suppose first that the robot can perfectly reliably observe whether it is on a black square or a white square a Suppose you ask the robot to move one square to the right twice but do not make any observations until after the second move is completed at which point the robot observes that it is on a black square What is the probability that the robot is on square 0 What is the entire belief state distribution for the robot s position b Suppose you want the robot to end up on square 10 A simple strategy would be to command the robot to move right ten times and hope for the best Devise a more reliable strategy that has the robot perform an observation after every move Describe your strategy in English c Write a Python program that implements your strategy from part b Assume you have procedures moveRight which commands the robot to move right and look which returns either black or white You do not need to debug your code d Suppose now that the robot s vision is faulty The observation model is this If the robot is actually on a black square it will see black with 80 probability and white with 20 probability If the robot is actually on a white square it will see black with 35 probability and white with 65 probability Suppose as in part a above the robot starts on square 0 and you command it to move one square to the right twice not making any observations until after the second move is completed If the robot now sees that it is on a black square what is the the probability that the robot is on square 0 What is the entire belief state distribution for the robot s position 6 01 Spring Semester 2008 Final exam announcement and practice final Revised May 12 3 2 Motor controller This problem is about using a computer to precisely control the speed of a spinning motor The motor has a tachometer that provides very precise speed measurements and you can control the voltage across the motor The motor speed S is given by S JV where J is a constant associated with the motor and V is the voltage across the motor Unfortunately not all all motors have the same J the nominal value of J is 10 but J can actually be as low as 9 and as high as 11 a Your computer samples the motor velocity 10 times a second and can change the motor voltage based on readings from previous samples One suggested control strategy is to determine the motor voltage using the nominal value of J J 10 1 V n 1 Sdesired 10 where Sdesired is a given desired motor speed In the worst case considering the variation in J how far off will the motor speed be from the desired speed b Suppose we use a feedback control system to reduce the difference between the actual motor speed and the desired speed as follows 1 V n Sdesired K Sdesired S n 1 10 where S is the actual motor speed as measured by the tachometer and K is the feedback gain Draw a delay adder gain block diagram that describes this control scheme as a system with …
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