Description
Dynamics and measurement functions (40 points)
In lab 2, you will be localizing the robot using different filters and different measurements. In this section
you will set up the required dynamics and measurement functions.
1. Given that the control action/inputs are the odometry information (distance traveled, angle turned),
what function would you use as g(xt−1, ut)? Explain. (Hint: you already wrote the function in
Homework 2).
2. Given that the measurement is a depth measurement, what function would you use as h(xt) to predict
the measurement? Explain. (Hint: you already wrote the function in Homework 2).
3. Given that the measurement is “GPS” like information (given by the overhead localization/true pose,
i.e. [x, y, θ]), explain and write a function hGPS.m that predict the measurement (This should be very
simple). Submit this function in the autograded assignment Homework4 hGPS.m on Canvas.
4. In order to use an EKF, one needs to calculate the Jacobian of the update and measurement functions at a given state. Write out the Jacobian Gt =
∂g
∂xt−1
. Write the function GjacDiffDrive.m
to output Gt for a particular pose x. Submit this function in the autograded assignment Homework4
GjacDiffDrive.m on Canvas.
5. Write the file HjacDepth.m to output Hdeptht at a particular xt. Explain how you calculate this.
(Hint: differentiating the function is difficult, think about using finite differences instead). Submit this
function in the autograded assignment Homework4 HjacDepth.m on Canvas.
6. Write out the Jacobian HGP St =
∂h
∂x . Write the function HjacGPS.m to output HGP St at a particular
xt (Again, this should be VERY simple). Submit this function in the autograded assignment Homework4
HjacGPS.m on Canvas.
Extended Kalman Filter(15 points)
In the lab, you will be using the EKF with different measurements. To make switching between measurement
types easy, in this section you will write a generic EKF that takes as input the measurement and update
functions and as such, can be reused.
Write the file EKF.m to perform one prediction & update step of the EKF. To make this file very generic,
the functions from the previous section should be inputs to this function and should be passed as anonymous
functions (details at the end of the problem set).
What are the inputs to the function EKF? what are the outputs?
This function is evaluated in the autograded assignment Homework4 testEKF.m on Canvas, where you will
have to call your filter to produce the estimated location at time t given all the information in time t − 1
(i.e. one time step).
Particle Filter(15 points)
Same as the EKF, it is best to write a generic particle filter that can be reused later on.
Write a function PF.m that takes in an initial particle set, odometry and depth measurements, prediction
and update functions and outputs a new particle set.
Assume there is an infinite wall at y = 1, the robot moved one meter forward (d = 1, φ = 0), and all
9 depth measurements are 2 meters. Create an initial particle set of size 30 with uniform distribution in
x ∈ [−1, 1], y ∈ [−3, 1] and θ = 0, plot the wall, the initial particle set (x,y position only for each particle),
and the particle set after one iteration of the particle filter on the same plot. Use one marker shape for the
initial particles and another marker shape for the final particles.
Running the filters in the simulator (115 points)
1. Write a function motionControl.m. This function should drive the robot in different directions while
reacting to the bump sensor (so that the robot does not try to drive through a wall). The control
should be independent of the location and deterministic. Furthermore, this function should:
(a) Call the sensor information (using the function readStoreSensorData.m provided in HW 1)
(b) Call integrateOdom.m and store the dead reckoning information in dataStore.deadReck
(c) Have a clearly identifiable and well commented section describing how to switch between
the three methods described below (EKF with GPS data, EKF with depth data, PF with depth
data). You will need to be able to switch between these three methods for lab 2. See Questions
4, 5, and 6 for more details.
2. In motionControl.m, create new fields datastore.ekfMu and datastore.ekfSigma (robot pose mean
and covariance, respectively).
3. Initialize dataStore.deadReck and datastore.ekfMu to the true initial pose (from dataStore.truthPose)
and datastore.ekfSigma to [2, 0, 0; 0, 2, 0; 0, 0, 0.1]. Define a new variable R (process noise covariance
matrix) and set it to 0.01I. Define a new variable Q (measurement noise covariance matrix) and set it
to 0.001I. What are the dimensions of R and Q?
4. EKF with GPS data:
(a) Create noisy GPS measurement by adding gaussian noise to the true pose. What should the noise
distribution be? Record this data in dataStore.GPS.
(b) In the simulator, load cornerMap and place the robot somewhere near [−4; 2; 0]. Create and load
a config file with noise on the depth measurements and the odomerty. What values did you choose
for the noise? Make sure the values make sense with respect to R and Q.
(c) In motionControl.m, call the EKF with the noisy GPS data and the appropriate functions
(dynamics and measurement).
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(d) Run motionControl.m in the simulator. On the same figure, plot the map and three final trajectories (x,y): dataStore.truthPose (the data from overhead localization), dataStore.deadReck
(from integrating the odometry only), and datastore.ekfMu (estimate from the EKF). Also plot
the 1-Σ ellipses from datastore.ekfSigma. Note, you only need to plot the x,y position and
not the orientation.(You may use the provided function plotCovEllipse.m or any other function
that plots ellipses. If you do, make sure to cite the source).
(e) Place the robot somewhere near [−4; 2; 0] again. Repeat 4(b–d) using a different initialization
for the filter (we want to see what happens to the estimate if the initial belief is different):
µ0 = [−2; −1.5; π/2] and Σ0 = [4, 0, 0; 0, 4, 0; 0, 0, 0.02]. (Initialize datastore.deadreckon to the
same initial pose estimate µ0 here in (e).)
(f) Compare the results of 4(d) and 4(e) – How does the initial estimate affect the robot’s estimate?
5. EKF with depth data: In motionControl.m, call the EKF with the depth data and the appropriate
functions (dynamics and measurement). Make sure to write comments in motionControl.m on how
to switch between using the GPS and depth measurements.
(a) Repeat 3 and 4(b,d). Make sure the dimension of R,Q are correct.
(b) Repeat 5(a) using R= 0.001I and Q= 0.01I.
(c) Compare the results from 5(a) and 5(b). How do the process and measurement noise covariances
affect the robot’s estimate?
(d) Does the filter behave as you’d expect? Why or why not?
(e) What are some of the problems with using an EKF with depth measurements?
(f) What errors do you expect to see when you run the EKF on the physical robot in the lab?
6. PF with depth data: In motionControl.m, create new field datastore.particles. Generate an
initial particle set of size 20, with x-location sampled from a uniform distribution between [−5, 0],
y-location sampled from a uniform distribution between [−5, 5], and θ sampled from a uniform distribution between [−0.2, 0.2]. Initialize the particle weights to be uniform. Call the PF with the depth
data and the appropriate functions (dynamics and measurement). Make sure to write comments in
motionControl.m on how to switch between using the EKF and using the PF.
(a) In the simulator, load cornerMap and place the robot somewhere near [−4; 2; 0]. Load your noisy
config file and run motionControl.m.
(b) Plot the initial particles on the map (plot the x,y positions as well as headings).
(c) Plot the final particles on the map (x,y position only).
(d) For each time step, choose the particle with the highest weight and plot it. You will get the “best”
trajectory. Do this for 5 additional particles and plot the 5 trajectories on the same map. Also
plot the true trajectories (from datastore.truthPose).
(e) Repeat 6(a–d) using a particle set of size 500.
(f) Does the filter behave as you’d expect? Why or why not?
(g) How does the size of the particle set affect the robot’s estimate?
(h) How else could you represent the robot’s estimate (other than just looking at the highest weight
particle)? Plot the trajectory of that estimate. Is it closer to the truth?
Comparing Filters (30 points)
Fill out the following table:
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Filter Assumptions about This filter is Advantages Disadvantages
the system appropriate when
Kalman filter
Extended Kalman filter
Particle filter
Using functions as inputs
As previously mentioned, for re-usability, we can send the dynamics, measurement and Jacobian functions
as parameters to our filter. There are two ways to do this, but only one that is accepted with Autograder
on Canvas. Below is a simple example and link for extra information (assume robot2global is defined
elsewhere):
Anonymous functions –
r2g = @(pose, p_xy) robot2global(pose, p_xy);
MyComplicatedFilter(r2g)
function MyComplicatedFilter(converter)
% some calculations to get my_pose, my_xyR
y = converter(my_pose, my_xyR);
return y
More information: https://www.mathworks.com/help/matlab/matlab_prog/anonymous-functions.html
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