6 081 Spring Semester 2007 Work for Week 12 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 081 Introduction to EECS I Spring Semester 2007 Work for Week 12 Software lab for Tuesday May 1 Pre lab tutor problems due Thursday May 3 Robot lab for Thursday May 3 Post lab written problems due Tuesday May 8 State estimation In the next two labs we ll use basic probabilistic modeling to build a system that estimates the robot s pose based on noisy sonar and odometry readings We ll start by building up your intuition for these ideas in a simple simulated world then next week we ll move on to using the real robots There will be two major parts of the work for this week working with the grid world simulator and writing your own state estimation code The plan is this Tuesday Work with your partner to start the grid world simulation lab Thursday Work with your partner to finish the grid world simulation lab when that s done Thursday Work on the state estimation code on your own Grid World Simulator Download the code for this lab in ps12 zip from the calendar page In it you will find StateEstimationNoisyNN py and StateEstLab py Don t use the Idle Python Shell for this lab You can use the Idle editor but you cannot run the code from inside Idle If you are a SoaR afficionado you can do this all from within SoaR Start SoaR open an editor window onto StateEstLab py choose Run in Interpreter from the SoaR menu and give commands by typing in the buffer at the bottom of the command window If not you can go to a terminal window and type python You can type commands to Python here python Python 2 4 4 1 Oct 18 2006 10 34 39 Type help copyright credits or license for more information Open StateEstLab py in Idle or your favorite other editor You ll see a command at the end of that file that says w51pp World 5 1 2 0 0 0 perfectSensorModel p erf ec tM ot io nM od el Go to your terminal window and type from StateEstLab import 6 081 Spring Semester 2007 Work for Week 12 2 As the lab goes along if you edit StateEstLab py then you ll need to go back to this window and type reload StateEstLab from StateEstLab import When you evaluate it you should see a window that looks like this This is a world with 5 possible states each of which is represented as a square with a colored border The possible colors of the borders are white black red green and blue In this example four squares are white and one is green There is a small orange rectangle representing the square that our simulated robot is actually occupying The inner part of each square is gray as we ll discuss in detail later how light the square is represents how likely the robot thinks it is that it s in that square The arguments to the World initialization function are The dimension of the world in x The dimension of the world in y Four tuples of coordinates each specifying the location of colored squares The first list gives the locations of black squares the second red the third green the last blue Squares unspecified in any of those lists are white 1 A pair of indices specifying the robot s initial location A model of how the sensors work A model of how the actions work So this world is 5 by 1 with one green square the robot initially at location 0 0 and perfect sensor and motion models You can issue commands to the robot by typing if w is a world w north w south w east w west w stay Question 1 Just to get the idea of how this works move the robot east and west a few times and just take a look at but don t be worried if you don t understand what is written on the screen It shows the actual numeric values associated with the robot s belief that it is in each of the squares This model is a slight variation on a hidden Markov models HMM that we saw during lecture The only difference is that the robot can select different actions like trying to move north or trying to move south and those different actions will cause different state transition distributions 1 The expression 2 0 makes a tuple that contains a single element which is itself a tuple that contains 2 and 0 The reason for the extra comma character is so that the outer parentheses are treated as a tuple constructor not just as grouping parentheses 6 081 Spring Semester 2007 Work for Week 12 3 The sensor model generally speaking is supposed to specify a probability distribution over what the robot sees given what state it is in P Ot ot St st In our case ot ranges over white black red green and blue and st ranges over all the possible states of the robot the different grid squares We have made a simplifying assumption which is that the robot s observation only depends on the color of the square it s standing in that is all white squares have the same distribution over possible observations So we really only need to specify P observedColor actualColor and then we can find the probability of observing each color in each state just by knowing the actual color of each state P Ot observedColor St st P Ot observedColor actualColor st In our program this set of distributions is specified as a tuple of tuples where each row corresponds to a different actual color So for example m actual observed would give the probability of observing observed in a square that was really colored actual Here s the model for perfect observations with no probability of error pe rf ec tS en so rM od el 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 Question 2 Give a sensor model in which red and blue are indistinguishable that is the robot is just as likely to see red as to see blue when it is in a red square or a blue square Question 3 Give a sensor model in which red always looks blue and blue always looks red Question 4 Which one of these models is more informative Question 5 Do the rows always have to sum to 1 The columns Why Question 6 Given an example situation in which our assumption that all squares of a given color have the same error model is unwarranted In a real world system we probably wouldn t ever want to have zeros anywhere in the sensor model it is important to concede the possibility of …
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