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4. We noted in the text that the EKF linearization is an approximation. To see how bad this approximation is, we ask you to work out an example. Suppose we have a mobile robot operating in a planar environment. Its state is its r-y-location and its global heading direction 8. Suppose we know x and y with high certainty, but the orientation is unknown. This is reflected by our initial estimate (000) and Σ = 0 0.01 0 0 0.01 0 0 0 10000 (a) Draw, graphically, your best model of the posterior over the robot pose after the robot moves d = 1 units forward. For this exercise, we assume the robot moves flawlessly without any noise. Thus, the expected lo- cation of the robot after motion will be (E) y + sin For your drawing, you can ignore and only draw the posterior in r-y-coordinates. (b) Now develop this motion into a prediction step for the EKF. For that, you have to define a state transition function and linearize it. You then have to generate a new Gaussian estimate of the robot pose using the linearized model. You should give the exact mathematical equations for each of these steps, and state the Gaussian that results. (c) Draw the uncertainty ellipse of the Gaussian and compare it with your intuitive solution. (d) Now incorporate a measurement. Our measurement shall be a noisy projection of the z-coordinate of the robot, with covariance Q = 0.01. Specify the measurement model. Now apply the measurement both to your intuitive posterior, and formally to the EKF estimate using the standard EKF machinery. Give the exact result of the EKF, and compare it with the result of your intuitive analysis. (e) Discuss the difference between your estimate of the posterior, and the Gaussian produced by the EKF. How significant are those differences? What can be changed to make the approximation more accurate? What would have happened if the initial orientation had been known, but not the robot's y-coordinate?

4. We noted in the text that the EKF linearization is an approximation. To see how bad this approximation is, we ask you to work out an example. Suppose we have a mobile robot operating in a planar environment. Its state is its r-y-location and its global heading direction 8. Suppose we know x and y with high certainty, but the orientation is unknown. This is reflected by our initial estimate (000) and Σ = 0 0.01 0 0 0.01 0 0 0 10000 (a) Draw, graphically, your best model of the posterior over the robot pose after the robot moves d = 1 units forward. For this exercise, we assume the robot moves flawlessly without any noise. Thus, the expected lo- cation of the robot after motion will be (E) y + sin For your drawing, you can ignore and only draw the posterior in r-y-coordinates. (b) Now develop this motion into a prediction step for the EKF. For that, you have to define a state transition function and linearize it. You then have to generate a new Gaussian estimate of the robot pose using the linearized model. You should give the exact mathematical equations for each of these steps, and state the Gaussian that results. (c) Draw the uncertainty ellipse of the Gaussian and compare it with your intuitive solution. (d) Now incorporate a measurement. Our measurement shall be a noisy projection of the z-coordinate of the robot, with covariance Q = 0.01. Specify the measurement model. Now apply the measurement both to your intuitive posterior, and formally to the EKF estimate using the standard EKF machinery. Give the exact result of the EKF, and compare it with the result of your intuitive analysis. (e) Discuss the difference between your estimate of the posterior, and the Gaussian produced by the EKF. How significant are those differences? What can be changed to make the approximation more accurate? What would have happened if the initial orientation had been known, but not the robot's y-coordinate?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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4. We noted in the text that the EKF linearization is an approximation. To see how bad this approximation is, we ask you to work out an example. Suppose we have a mobile robot operating in a planar environment. Its state is its r-y-location and its global heading direction 8. Suppose we know x and y with high certainty, but the orientation is unknown. This is reflected by our initial estimate (000) and Σ = 0 0.01 0 0 0.01 0 0 0 10000 (a) Draw, graphically, your best model of the posterior over the robot pose after the robot moves d = 1 units forward. For this exercise, we assume the robot moves flawlessly without any noise. Thus, the expected lo- cation of the robot after motion will be (E) y + sin For your drawing, you can ignore and only draw the posterior in r-y-coordinates. (b) Now develop this motion into a prediction step for the EKF. For that, you have to define a state transition function and linearize it. You then have to generate a new Gaussian estimate of the robot pose using the linearized model. You should give the exact mathematical equations for each of these steps, and state the Gaussian that results. (c) Draw the uncertainty ellipse of the Gaussian and compare it with your intuitive solution. (d) Now incorporate a measurement. Our measurement shall be a noisy projection of the z-coordinate of the robot, with covariance Q = 0.01. Specify the measurement model. Now apply the measurement both to your intuitive posterior, and formally to the EKF estimate using the standard EKF machinery. Give the exact result of the EKF, and compare it with the result of your intuitive analysis. (e) Discuss the difference between your estimate of the posterior, and the Gaussian produced by the EKF. How significant are those differences? What can be changed to make the approximation more accurate? What would have happened if the initial orientation had been known, but not the robot's y-coordinate?
Transcribed Image Text:4. We noted in the text that the EKF linearization is an approximation. To see how bad this approximation is, we ask you to work out an example. Suppose we have a mobile robot operating in a planar environment. Its state is its r-y-location and its global heading direction 8. Suppose we know x and y with high certainty, but the orientation is unknown. This is reflected by our initial estimate (000) and Σ = 0 0.01 0 0 0.01 0 0 0 10000 (a) Draw, graphically, your best model of the posterior over the robot pose after the robot moves d = 1 units forward. For this exercise, we assume the robot moves flawlessly without any noise. Thus, the expected lo- cation of the robot after motion will be (E) y + sin For your drawing, you can ignore and only draw the posterior in r-y-coordinates. (b) Now develop this motion into a prediction step for the EKF. For that, you have to define a state transition function and linearize it. You then have to generate a new Gaussian estimate of the robot pose using the linearized model. You should give the exact mathematical equations for each of these steps, and state the Gaussian that results. (c) Draw the uncertainty ellipse of the Gaussian and compare it with your intuitive solution. (d) Now incorporate a measurement. Our measurement shall be a noisy projection of the z-coordinate of the robot, with covariance Q = 0.01. Specify the measurement model. Now apply the measurement both to your intuitive posterior, and formally to the EKF estimate using the standard EKF machinery. Give the exact result of the EKF, and compare it with the result of your intuitive analysis. (e) Discuss the difference between your estimate of the posterior, and the Gaussian produced by the EKF. How significant are those differences? What can be changed to make the approximation more accurate? What would have happened if the initial orientation had been known, but not the robot's y-coordinate?
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