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METR3100 – Prework for Prac 2 – Sem 1, 2023
This practical prework assignment has been designed to get you acquainted with the course material
required for the second practical assessment of METR3100.
The second practical will test the following Learning Objectives:
2.1 Characterise the operating principles of several sensors commonly used in the instrumentation
of control systems.
2.4 Instrument control systems comprising exteroceptive feedback e.g. LiDAR sensors.
The practical will use LiDAR sensors to locate objects.
Background:
I leant Tim my triangle. It was my favourite triangle. Tim lost my triangle. Thanks Tim.
In this practical prework, I’d like to ask for your help to try and recover my favourite triangle. It looks
like this:
Hopefully you can see why it’s my favourite. It has three vertices (as all good triangles do) and in the
triangle frame, they are located at:
𝑇0 = [
𝑇0,𝑥
𝑇0,𝑦
] = [
−
0
2
] 𝑇1 = [
𝑇1,𝑥
𝑇1,𝑦
] = [
3
1
] 𝑇2 = [
𝑇2,𝑥
𝑇2,𝑦
] = [
−
1
2
]
Now, Tim tells me that he was learning about LiDAR when he last saw my favourite triangle. In fact
he managed to collect some range measurements of my favourite triangle before he lost it. He
METR3100 – Prework for Prac 2 – Sem 1, 2023
collected the measurements with a very simple LiDAR (in fact it was one of the RPLiDAR A1 sensors
that you will use in the second practical). This LiDAR sensor reports range measurements at given
headings, i.e. polar coordinates.
The measurements are:
Heading (𝜃)
[°]
Range (𝑅)
[𝑚]
-12 6.51
-10 5.70
-8 5.07
-6 4.58
-4 4.17
-2 3.84
0 3.56
2 3.52
4 3.97
6 4.55
8 5.34
10 6.48
12 8.23
Part 1:
Determine and plot these measurements in the LiDAR coordinate frame. Zero degrees corresponds to
the X-axis. (5 marks).
Part 2:
Tim knows where the sensor was located when he took these measurements. Maybe he’s not that
useless after all. According to Tim, the LiDAR sensor was located in the World Coordinate system (this
is just a coordinate system that Tim and I like to communicate with) with the following 2D frame
transformation parameters:
Parameter Symbol (see Week 7B slides) Value [units]
Translation in X-axis 𝛿𝑥 1.0 [𝑚]
Translation in Y-axis 𝛿𝑦 2.0 [𝑚]
Rotation about Z-axis 𝜃 18.0 [°]
Produce another plot showing the LiDAR point cloud in the World frame. (5 marks).
METR3100 – Prework for Prac 2 – Sem 1, 2023
Part 3:
Tim can see how upset I am and he is now clearly clutching at straws on where he left my favourite
triangle. He says he left it in the World-frame at:
Parameter Symbol (see Week 7B slides) Value [units]
Translation in X-axis 𝛿𝑥 7.0 [𝑚]
Translation in Y-axis 𝛿𝑦 5.0 [𝑚]
Rotation about Z-axis 𝜃 -70.0 [°]
He even drew me a diagram depicting where it might be.
However, Tim has let me down far too often and I no longer trust him. Prove he is lying to me by
determining the 13 range errors that his hypothesis would produce. (25 marks).
Part 4:
Eureka! Tim remembers where he put it, but not how it was rotated. Classic Tim, job half done. My
favourite triangle is located at:
Parameter Symbol (see Week 7B slides) Value [units]
Translation in X-axis 𝛿𝑥 6.0 [𝑚]
Translation in Y-axis 𝛿𝑦 4.0 [𝑚]
Rotation about Z-axis 𝜃 ? [°]
Produce a plot showing the sum of range errors on the Y-axis and the rotation angle (𝜃) on the X-axis
to once and for all tell me where my favourite triangle is located. Non-intersecting rays can be given a
range error of 0m. (15 marks).