6 01 Fall Semester 2007 Assignment 12 Issued Tuesday Nov 20 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 01 Introduction to EECS I Fall Semester 2007 Assignment 12 Issued Tuesday Nov 20 To do this week in Tuesday software lab 1 Start writing code and test cases for the numbered questions in the software lab Paste all your code including your test cases into the box provided in the Software Lab problem on the on line Tutor What you submit to the tutor will not be graded what you hand in next Tuesday will before the start of lecture next Tuesday 1 Do the lab writeup providing written answers including code and test cases for every numbered question in this handout On Athena machines make sure you do athrun 6 01 update so that you can get the Desktop 6 01 lab12 directory which has the files mentioned in this handout You need the whole lab12 distribution for the software lab use one of our laptops for the software lab 6 01 Fall Semester 2007 Assignment 12 Issued Tuesday Nov 20 2 State estimation In the next two software and two robot 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 we ll move on to using the real robots This week we ll start by working with a non deterministic grid world simulator You should work with your partner on this Grid World Simulator 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 In the lab12 folder you will find StateEstSM py Open this file in Emacs Idle or whatever editor you like If you use Idle as an editor do not try to evaluate the python commands in the Idle Python Shell Then open a Terminal window if you re on Athena remember to do add f 6 01 then connect to the lab12 director cd Desktop 6 01 lab12 and type python python Python 2 5 r25 51918 Sep 19 2006 08 49 13 GCC 4 0 1 Apple Computer Inc build 5341 on darwin Type help copyright credits or license for more information You can type commands to Python here In this Python type from StateEstSM import As the lab goes along if you edit StateEstSM py then you ll need to go back to this window and type reload StateEstSM from StateEstSM import You ll see a command at the end of the StateEstSM py file that says tw makeGridSim 5 1 2 0 0 0 perfectSensorModel pe rf ec tM ot io nM od el Execute this command in Python on your Terminal window When you evaluate it you should see a window that looks like this 6 01 Fall Semester 2007 Assignment 12 Issued Tuesday Nov 20 3 This is a world with 5 possible states each of which is represented as a colored square on the left The possible colors of the states 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 On the right are five squares that start out being black the color in those squares represents how likely the robot thinks it is that it s in that square This is called the belief state The colors illustrating the belief state are black when the value is what the uniform distribution would assign 0 2 for 5 states shades of green when the probability is higher than the uniform and shades of red when it is below the uniform value The arguments to the makeGridSim function are The dimension of the world in x The dimension of the world in y Four lists 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 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 grid 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 tw run and clicking on the buttons in the simulator window The buttons correspond to the five available actions Stay West East South and North Clicking on the Done button returns control to Python The window will remain after you click Done Do not kill the window until you are completely done with it You can type tw reset tw run to start interacting with the window again after you ve typed Done Question 1 Just to get the idea of how this works move the robot east and west a few times and write down 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 You should be able to relate this at least roughly to the HMM example in the notes 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 6 01 Fall Semester 2007 Assignment 12 Issued Tuesday Nov 20 4 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 we specify the observation model by a function which takes as input the trueColor and returns an instance of the Dist class The Dist class which we saw earlier in the term specifies a probability distribution over a set of values as a list of value probability pairs The probabilities must sum to one Any possible value that is not mentioned in the distribution is assumed to have probability equal to 0 0 Here s the model for perfect observations with no probability of error def pe rf ec tSe ns or Mo de l trueColor …
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