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COMP0130 Graph-based Optimisation and SLAM
创建日期：2024-04-29 16:20:51
标签： COMP0130
Graph-based Optimisation and SLAM

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COMP0130 Coursework 02:
Graph-based Optimisation and SLAM

• Weighting: 33% of module total
• Final Submission Format: a PDF file containing a report, and a zip file containing source
code.
2 Coursework Description
The goal of this coursework is to experiment with how a graphical-model based SLAM system
is implemented and assess its properties. The system will be implemented using the MiniSLAM
1
Figure 2.1: The scenario. The vehicle (small blue triangle,starting from (0,0) at bottom left)
drives a path (purple line) through an environment populated by landmarks (blue crosses). Var-
ious landmark estimates, with their associated covariance ellipses, are shown.
event-based framework you met in Topic 02 Lab 01 and the MATLAB port of the g2o port you
met in Topic 02 Lab 02. We assume you have familiarity with the questions and worked answers
to those workshops.
The scenario is shown in Figure 2.1. A wheeled robot (DriveBot) drives through a two-dimensional
environment which is populated by a set of landmarks. The number of landmarks and their lo-
cations are not known, and must be estimated by the SLAM system.
The vehicle is equipped with three types of sensors:
1. A vehicle odometry sensor, which records speed and heading.
2. A “GPS” sensor which measures vehicle position.
3. A 2D range bearing sensor which measures the range and bearing to the landmarks in 2D.
In addition, an external is provided to specify the initial position and orientation of the robot.
The mathematical models for these data sources and sensors are described in Appendix A.
2
A simulation system will create a stream of events which the SLAM system will need to pro-
cess. Data association has already been addressed. Therefore, each landmark measurement you
receive will have a unique (and correct) landmark ID.
The coursework consists of three questions:
Q1: Implement the prediction and update steps using the vehicle process model and the GPS
sensor. You will also investigate some general properties about how the graph optimizers
work. (28%)
Q2: Extend your implementation to undertake SLAM. (34%)
Q3: Discuss and implement a couple of ways improve the performance of the SLAM system
(38%).
Please note that different subparts of different questions have their own launcher scripts of
the form qx y.m. Although these scripts look very similar to one another, there are subtle
differences between them to support the different subquestions (for example, some datasets are
considerably longer than others). Furthermore, to mark your coursework, we will execute each
script in turn and so the changes required to answer each question should not interfere with us
being able to run the launcher code for a previous question.
3 Questions
1. In this question you will implement, within the DriveBotSLAMSystem class , the
prediction step (using the process model in Subsection A.2) and the GPS sensor (using the
observation model in Subsection A.3). Note that the GPS data arrives less frequently than
the vehicle odometry data, and you must ensure that timestamps are handled correctly.
a. Describe the structure of the graph which will be used to support predicting the robot
through time and updating its position periodically with the GPS sensor. Your de-
scription should list the different types of vertices and edges. Provide the equations
for the different types of edges and vertices.
Hint: Think about how angular discontinuities affect your solution and how these
are related to the ⊞ operator.
[6 marks]
3
b. Implement the prediction step in the DriveBotSLAMSystem.
i. In your report describe which methods you edited to complete the functionality.
In the code, please provide comments which explain what your implementation
does. These should be prefixed by “Q1b”.
[10 marks]
ii. Using the script q 1b.m, plot the graphs of the state error and the covariances
computed by the estimator. Describe the behaviours you see and explain why
you think they arise.
[7 marks]
c. Implement the GPS update step in the DriveBotSLAMSystem class.
i. In your report describe which methods you edited to complete the functionality.
In the code, please provide comments which explain what your implementation
does. These should be prefixed by “Q1c”.
[8 marks]
ii. Using the script q1 c.m generate plots of the state error and the covariance.
Present these graphs. Discuss the behaviours you see and why they arise.
[8 marks]
d. Modify the script q1 d.m so that the optimizer runs once every timestep instead of
at the end of the run. Plot the curves of the optimization time and the chi2 over the
run. Describe the behaviours you observe and propose an explanation for why they
arise.
[10 marks]
[Total 49 marks]
4
2. In this question you will extend the DriveBotSLAMSystem to support SLAM. The
observation model for these landmarks is given in Subsection A.4. The GPS sensor is not
used.
a. Describe the structure of the graph for the SLAM system. Your description should
include a discussion on the different types of edges and vertices and you should
include the error models and Jacobians needed to implement them.
Hint: Think about how angular discontinuities affect your solution and how these
are related to the ⊞ operator.
[11 marks]
b. Implement SLAM in DriveBotSLAMSystem.
i. In your report describe which methods you edited to complete the functionality.
In the code, please provide comments which explain what your implementation
does. These should be prefixed by “Q2b”.
[9 marks]
ii. Run the script q2 b.m which runs a SLAM example on a single closed loop.
Plot the vehicle covariances, chi2 values and the amount of time to complete
an optimization step. Discuss the behaviours you see and why you think they
arise.
[15 marks]
5
c. We will now look at some more details about the properties of the constructed graph.
i. By modifying the script q2 c.m, investigate the properties of the structure of
the graph. You should obtain information on: the number of vehicle poses
stored, the number of landmarks initalized, the average number of observations
made by a robot at each timestep, and the average number of observations re-
ceived by each landmark.
In your report, describe how you calculated these quantities. In the code, please
prefix your changes by “Q2c”.
Hint: You can obtain all the necessary information from querying the con-
structed graph, Matlab’s isa function, and using methods in the BaseVertex
and BaseEdge classes.
[7 marks]
ii. Present the results you obtained from analysing the graph and discuss what
these quantities imply.
[8 marks]
d. One of most important advantages of SLAM is the ability for it to perform loop
closure.
Using the script q2 d.m, generate figures which show the map, vehicle and land-
mark states just before and just after the loop closure event. Explain the form of
the covariance matrices you see on the landmark and vehicle estimates immediately
before and after the update. By looking at the determinant of the covariance of
each landmark, compute the amount by which the uncertainty has changed for each
landmark.
[10 marks]
6
[Total 60 marks]
e. This is a purely optional activity for those who are interested. No marks are awarded
for it.
Run the script q2 large map.m. This tests the code on a much more substantial
example. The landmark covariances matrices look like they are oriented as tangents
on a set of concentric circles. Where do you think the origin might be, and what do
you think might cause it?
7
3. This question explores several ways to improve the performance of a SLAM system by
deleting parts of the factor graph. Again, the GPS is not used.
a. One way to improve the performance of a SLAM system is to maintain all the sensor
observation data, but to remove all the vehicle prediction edges.
i. By referring to the structure of the factor graph, why do you think this might be
a sensible strategy to take? When do you think this is likely to be a successful
strategy? When is it likely to be unsuccessful?
[8 marks]
ii. Modify the DriveBotSLAMSystem system so that, when requested, it will
delete the vehicle edges before optimization. Your code should support two
cases: when all of the prediction edges are deleted, and when all but the first
prediction edge should be deleted.
In your report, describe which methods you edited to provide this functionality.
In the code, please provide comments which explain what your implementation
does. These should be prefixed by “Q3a”.
[4 marks]
iii. Using the script q3 a.m, present the three cases: all the prediction edges main-
tained, all prediction edges removed, and all but the first prediction edge is re-
moved. By referring to the structure of the graph, explain the differences in
performance and behaviour.
Hint: you should find that the algorithm will fail if all the prediction edges are
removed.
[8 marks]
iv. Plot the optimization time, covariance and chi2 values for the working case
where the edges have been removed. How do the results differ from the full
SLAM system? Given the difference in performance, discuss if you this strat-
egy on its own successful.
[8 marks]
8
b. Two approaches for simplifying graphs have been proposed: sparse keyframe based
SLAM, and graph pruning.
i. Describe how keyframe-based SLAM and graph pruning methods work. Dis-
cuss their strengths and weaknesses.
[8 marks]
ii. Chose one strategy — either sparse key frames or graph pruning — to imple-
ment. Explain which one you chose and why.
[3 marks]
iii. Implement your solution. You can either modify the existing code (but ensure
that it will run the earlier questions correctly) or you can create new classes to
implement it (either by subclassing by simply copying the existing code to cre-
ate a new class). To accompany it, you should write your own script q3 b.m,
following the pattern of the other scripts, to run your revised SLAM system
with the necessary settings.
In your report, describe how you implemented the functionality. In the code,
please provide comments which explains what your implementation does. These
should be prefixed by “Q3b”.
[15 marks]
iv. Evaluate the results of your algorithm. How you chose this will depend upon the
algorithm implemented. However, key characteristic to include are the graph
properties (such as number of vertices), chi2 performance, and error perfor-
mance. Your evaluation should show whether your changes have improved
performance, and provide an explanation of why you think you achieved the
results presented.
[12 marks]
[Total 66 marks]
[Coursework Total 175 marks]
9
A System Models
A.1 State Description
The vehicle state is described by its position and orientation in 2D:
xk =

xk
yk
ψk
 . (A.1)
A standard right handed coordinate system is used: x points to the right, y points up and a
positive rotation is in an anticlockwise direction. Angles are in radians.
Landmarks are in 2D. The state of the ith landmark is given by
mi =
xi
yi
 . (A.2)
A.2 Process Model
The vehicle process model is the similar to the one you saw in Workshop 04. The model has
the form,
xk+1 = xk +∆TkMk (uk + vk) . (A.3)
The matrix
Mk =

cosψk − sinψk 0
sinψk cosψk 0
0 0 1

is the rotation from the vehicle-fixed frame to the world-fixed frame and ∆Tk is the length of
the prediction interval. Note that the prediction interval is governed by when various sensors
are available, and can vary through the simulation.
The control input consists of the speed of the vehicle (which is oriented along a body-fixed x
axis) together with an angular velocity term,
uk =

sk
0
ψ˙k
 .
10
The process noise is zero mean, Gaussian and additive on all three dimensions,
vk =

vk
vy
vψ˙
 .
The noise in the body-fixed y direction allows for slip and the fact, as discussed in the lectures,
that the velocity is related to the front wheel and not the body orientation. The process noise
covariance Qk is diagonal.
A.3 GPS Observation Model
The GPS sensor is a highly idealized one which provides direct measurements of the position
of the robot. The observation model is
zGk =
xk
yk
+wGk
The covariance of wGk , R
G
k is diagonal and constant.
A.4 Landmark Observation Model
The landmark observation model measures the range, azimuth and elevation of a landmark
relative to the platform frame,
zLk =
rik
βik
+wLk ,
where
rik =
√
(xi − xk)2 + (yi − yk)2
βik = tan
−1
(
yi − yk
xi − xk
)
− ϕk
The covariance of wLk , R
L
k is diagonal and is assumed to be time invariant and the same for all
landmarks.
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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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