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Now that we understand the sensor noise, we want to estimate the robot's x-coordinate inside the warehouse. Two independent sets of measurements are available:
x_obs_A — 6 readings from Type A sensors
x_obs_B — 8 readings from Type B sensors
Each observation is an independent, noisy measurement of the robot's true x-coordinate.
During the same calibration run, the beacon also logged whether each transmitted echo pulse was successfully received or lost to background noise. These outcomes are stored in echo_successes.
[1 point] Specify a probabilistic model with a likelihood and a suitable prior distribution.
[1 point] Use RxInfer to infer a posterior distribution over the echo success probability. Store your result as found_reliability.
The x-coordinate uncertainty of pos_2d compared to x_pos_combined is...
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An autonomous logistics robot is navigating a large warehouse (40 m wide × 30 m tall). Because GPS signals are unreliable indoors, the robot uses ultrasonic beacons mounted at fixed, known positions on the walls.
Each beacon emits a short sound pulse and measures the time until it detects the returning echo. From this time-of-flight measurement, it computes a distance estimate to the robot.
Warehouse
The warehouse has two types of sensors:
Type A, mounted on the west wall (x = 0 m).
Type B, mounted on the east wall (x = 40 m).
Each sensor on the west wall measures the horizontal distance to the robot, giving a direct (noisy) estimate of the robot's x-coordinate. Each sensor on the east wall measures the distance from the far side, which after correction also gives an estimate of the robot's x-coordinate.
In this assignment you will:
Characterize the sensor noise and reliability using calibration data.
Estimate the x-coordinate of the robot from multiple independent sensor readings.
Estimate the 2D position of the robot using a multivariate probabilistic model.
Before deploying the sensors, we run a calibration test. The robot is placed at a known position, and a single Type A sensor repeatedly measures the distance to a reflector at a known distance of calib_distance = 8.0 m.
The calibration residuals (measured distance minus known distance) are stored in calib_residuals. A well-calibrated sensor should have residuals centred at zero. In practice, manufacturing tolerances and mounting geometry can introduce a small systematic bias$\delta_A$. We also want to know the spread of the measurements, i.e., the systematic variance$\sigma^2_A$.
calib_residuals — a vector of 200 calibration residuals (measurement − known distance) from a single Type A sensor pointed at a reflector at a known distance of calib_distance = 8.0 m.
x_obs_A — 6 x-coordinate estimates from the Type A sensors (noise $\sigma^2_A =$ ? m).
x_obs_B — 8 x-coordinate estimates from the Type B sensors (noise $\sigma^2_B = 0.25$ m).
pos_obs — 10 2D position estimates[x, y] from a combined positioning subsystem.
[2 points] Specify a probabilistic model with a multivariate Normal likelihood and use RxInfer to infer a posterior distribution over the 2D position vector μ_z = [x, y]. Store your result as pos_2d.
[1 point] Compare the x-coordinate uncertainty of pos_2d with that of x_pos_combined from Part 2. Which gives a more precise x-estimate, and why?
[1 point] Which point leads to the biggest drop in posterior uncertainty (expressed as determinant of the posterior covariance)?
The Type A sensors alone give a good estimate of x, but we also have readings from the Type B sensors. By combining both, we can improve our estimate further.
[1 point] Infer a posterior distribution over x using the Type B sensor ($\sigma^2_B =$ 0.25 m) observations as well (store as x_pos_from_B).
[2 point] Combinex_pos_from_A and x_pos_from_B into a single improved estimate using Bayes' rule. Store the combined result as x_pos_combined.
[2 point] Plot the three distributions (x_pos_from_A, x_pos_from_B, x_pos_combined). What can you say about the combined result?
[1 point] Specify a probabilistic model for calib_residuals that includes $\delta_A$ and $\sigma^2_A$.
[2 points] Infer a posterior distribution for the sensor bias $\delta_A$ and variance $\sigma^2_A$, and store your result as found_bias_distribution and found_variance_distribution.
[1 point] Explain how you chose your prior distribution.
Note: This notebook will take a couple of minutes to start, because Julia is downloading and precompiling the packages that we use in this notebook (Plots, RxInfer, Distributions, BmlipTeachingTools).
You can check the progress in the "Status" tab in the bottom right.
This package installation and precompilation will happen only once. Afterwards, this notebook (and future notebooks that use the same package versions) will launch much more quickly.
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shortpathResit - robot localization.jlprocess_statusreadypathI/home/runner/work/course/course/assignments/Resit - robot localization.jlpluto_versionv0.20.19last_save_timeA}*~9cell_order 2$dd51677f-3a0c-4fa6-b896-174915fe7417$044f3410-2b9f-11f1-2fb2-ddeeb7049f22$021b20bc-d0ec-48d8-af25-26a29f9b4de4$abc102e0-5e58-4ac1-bbc8-b28dce742ae7$63f66b98-ded4-46dd-b42a-891989a39ea7$7f22dadc-39b5-417d-b29a-c91881a23097$69096420-5636-48e4-8380-24fe9aafab37$341618ea-677b-499e-b518-8a577e997e97$75fc1627-1e01-433f-9a44-791d982d4944$e47f428e-cc70-4a89-82a4-3e4dc563129a$98d51c6e-b70d-4874-84c9-007a51ae389d$103f53b2-be86-48e6-b380-fbc138c2cd33$6a26d308-3667-49a8-9f4a-0e6360230760$3d29e63b-8478-40cb-be8e-0516cfc59e0d$8bbddac3-d8a1-4787-bccf-02b7f4e5f055$4a699a86-e664-4049-8421-6b39a07ae521$7dcd02eb-7d02-4e2b-8610-936b0b19307e$9d14fd1c-e704-4858-9842-fe088f18ec9c$e55511e0-d714-4d37-91e9-0e1aa932541f$48e39a6f-249c-4ba1-a32e-061c08eba644$41c2a20b-a664-492c-8821-682ecd34647c$f6617439-fb86-4425-925a-6e705a78a2b3$a727a09c-7744-4f8b-a7c3-9f157e37ea26$ccf48c15-7aad-445e-86ef-b7c5b02abfde$02ee0c67-c31e-4db7-92a7-41aafa829d13$bb3362b6-6aa0-4796-b3b1-0385724feadd$1a940dbb-e1de-4c8f-8cb6-7d0b87f98b36$49e485c8-4b63-4fde-a7b6-6dc919e9bc38$767447b1-080b-411b-83d5-d0cb75e290d4$2064d975-d9ad-4ddd-9d81-87f64363f998$3f54c54b-7f29-44f8-b409-06e43e5f7d78$66deb52f-ee85-48d6-a56e-f6f43111230f$825db4bb-77aa-4686-82a7-0da72589b925$daac2bd0-cfd2-45b9-8675-38b0e5484771$b7aea8f0-a5ac-487f-a91b-372812c5353b$744d90e7-397e-45d5-ab11-fa1d5c8f9c2a$ef5c9636-42a4-49f8-8c66-68a590914f12$b90a0f2a-1117-4b71-a717-d711f5ea2223$f8c27934-94a2-4817-aaca-ab818a3e0273$e4f3e468-fec8-4c3f-941d-0003e24a20f6$b56a926e-7e4d-4d44-b399-bb042de8c071$3dac2b2c-494b-498a-a74b-a11349af54c3$8555f14e-df09-4f11-82c0-cd5935d66418$b185e8b7-4914-4f78-9f19-690c2a412ed4$7dc3b839-c454-486f-b8f7-747332a7e45c$2378b595-fa29-4a32-b428-1a5ecf4b05d7$3a9a632a-a3d5-4093-b9ed-ae7f6d90b46f$73b34659-be3d-4184-8fa5-adba70a7b2cd$705b06c6-600a-411a-b397-726e4d5acf3a$c4aa8122-4ada-4fb6-af5d-286992e42820published_objectsnbpkginstall_time_ns A#^instantiatedòinstalled_versionsLinearAlgebrastdlibDistributions0.25.123BmlipTeachingTools1.3.1Plots1.41.6StableRNGs1.0.4RxInfer4.7.1MarkdownstdlibStatsPlots0.15.8terminal_outputsnbpkg_sync
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enabled÷restart_recommended_msgrestart_required_msgbusy_packageswaiting_for_permission,waiting_for_permission_but_probably_disabled«cell_inputs 2$b185e8b7-4914-4f78-9f19-690c2a412ed4cell_id$b185e8b7-4914-4f78-9f19-690c2a412ed4codemd"""
## Submission
To submit this notebook, please use the **"Export to HTML"** feature from Pluto ([documentation](https://plutojl.org/en/docs/export-html/)) to get the HTML file to submit. You can also submit a `.jl` Julia file. **PDF is not accepted.**
"""metadatashow_logsèdisabled®skip_as_script«code_folded$e55511e0-d714-4d37-91e9-0e1aa932541fcell_id$e55511e0-d714-4d37-91e9-0e1aa932541fcode(exercise_statement("Sensor Reliability")metadatashow_logsèdisabled®skip_as_script«code_folded$3f54c54b-7f29-44f8-b409-06e43e5f7d78cell_id$3f54c54b-7f29-44f8-b409-06e43e5f7d78code>exercise_statement("Sensor Fusion", header_level=2, big=true)
metadatashow_logsèdisabled®skip_as_script«code_folded$02ee0c67-c31e-4db7-92a7-41aafa829d13cell_id$02ee0c67-c31e-4db7-92a7-41aafa829d13codemd"""
# Part 2: Estimating the x-Position
Now that we understand the sensor noise, we want to **estimate the robot's x-coordinate** inside the warehouse. Two independent sets of measurements are available:
- `x_obs_A` — $(length(x_obs_A)) readings from Type A sensors
- `x_obs_B` — $(length(x_obs_B)) readings from Type B sensors
Each observation is an independent, noisy measurement of the robot's true x-coordinate.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$c4aa8122-4ada-4fb6-af5d-286992e42820cell_id$c4aa8122-4ada-4fb6-af5d-286992e42820codeBfunction show_warehouse(; title_str="Logistics Warehouse (40 m × 30 m)")
p = plot(;
xlim=(-2, 42), ylim=(-2, 32),
aspect_ratio=1,
title=title_str,
xlabel="x position (m)", ylabel="y position (m)",
)
plot!(p, [0,40,40,0,0], [0,0,30,30,0]; lw=2, c=:black, label="Walls")
scatter!(p, zeros(length(sensor_pos_A)), sensor_pos_A;
c=:blue, marker=:ltriangle, ms=8, label="Type A sensors (west wall)")
scatter!(p, fill(40.0, length(sensor_pos_B)), sensor_pos_B;
c=:darkorange, marker=:rtriangle, ms=8, label="Type B sensors (east wall)")
plot!(legend=:inside)
p
end;metadatashow_logsèdisabled®skip_as_script«code_folded$767447b1-080b-411b-83d5-d0cb75e290d4cell_id$767447b1-080b-411b-83d5-d0cb75e290d4code### YOUR CODE HEREmetadatashow_logsèdisabled®skip_as_script«code_folded$daac2bd0-cfd2-45b9-8675-38b0e5484771cell_id$daac2bd0-cfd2-45b9-8675-38b0e5484771code### YOUR CODE HEREmetadatashow_logsèdisabled®skip_as_script«code_folded$ccf48c15-7aad-445e-86ef-b7c5b02abfdecell_id$ccf48c15-7aad-445e-86ef-b7c5b02abfdecodeFreliability_prior_comment = md"""
I chose this prior because...
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7dcd02eb-7d02-4e2b-8610-936b0b19307ecell_id$7dcd02eb-7d02-4e2b-8610-936b0b19307ecodeBbegin
if !@isdefined(found_bias_distribution)
var_not_defined(:found_bias_distribution)
elseif !(found_bias_distribution isa Distribution{Univariate, <:Any})
keep_working(md"`found_bias_distribution` should be a univariate `Distribution`.")
elseif found_bias_distribution == Normal(0.0, 1.0)
md"> Fill in your result above to see the estimated bias."
else
correct(md"You have entered a $(typeof(found_bias_distribution)). Estimated bias: mean = $(round(mean(found_bias_distribution), digits=4)) m, std = $(round(std(found_bias_distribution), digits=4)) m.")
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$4a699a86-e664-4049-8421-6b39a07ae521cell_id$4a699a86-e664-4049-8421-6b39a07ae521code6# 👉 Uncomment and fill this in:
# found_bias_distribution = ... your code here
# found_variance_distribution = ... your code here
# If you want to continue without completing Part 1, leave the default:
begin
found_bias_distribution = Normal(0.0, 1.0)
found_variance_distribution = InverseGamma(1/2,1.)
end;metadatashow_logsèdisabled®skip_as_script«code_folded$8bbddac3-d8a1-4787-bccf-02b7f4e5f055cell_id$8bbddac3-d8a1-4787-bccf-02b7f4e5f055code### YOUR CODE HEREmetadatashow_logsèdisabled®skip_as_script«code_folded$63f66b98-ded4-46dd-b42a-891989a39ea7cell_id$63f66b98-ded4-46dd-b42a-891989a39ea7codebegin
__name_id_check = if !@isdefined(name)
var_not_defined(:name)
elseif !@isdefined(student_id)
var_not_defined(:student_id)
elseif !(name isa String && student_id isa Integer)
danger("Check that `name` is a String and `student_id` is an Int.")
elseif name == "Pietje Puk" || student_id == 0123456
warning_box("Fill in your name and student ID [above](#name).")
else
correct("Looks like you filled in your name and ID correctly, thanks $(name)!")
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$a727a09c-7744-4f8b-a7c3-9f157e37ea26cell_id$a727a09c-7744-4f8b-a7c3-9f157e37ea26codebegin
if !@isdefined(found_reliability)
var_not_defined(:found_reliability)
elseif !(found_reliability isa Distribution{Univariate, <:Any})
keep_working(md"`found_reliability` should be a univariate `Distribution`.")
elseif found_reliability == Beta(1.0, 1.0)
md"> Fill in your result above."
else
correct(md"You entered a $(typeof(found_reliability)). Estimated detection probability: mean = $(round(mean(found_reliability), digits=3)), std = $(round(std(found_reliability), digits=4)).")
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$48e39a6f-249c-4ba1-a32e-061c08eba644cell_id$48e39a6f-249c-4ba1-a32e-061c08eba644codemd"""
During the same calibration run, the beacon also logged whether each transmitted echo pulse was **successfully received** or **lost** to background noise. These outcomes are stored in `echo_successes`.
- [1 point] Specify a **probabilistic model** with a likelihood and a suitable prior distribution.
- [1 point] Use RxInfer to **infer a posterior distribution** over the echo success probability. Store your result as `found_reliability`.
- [1 point] What **prior** did you choose, and why?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$021b20bc-d0ec-48d8-af25-26a29f9b4de4cell_id$021b20bc-d0ec-48d8-af25-26a29f9b4de4codename = "Pietje Puk"metadatashow_logsèdisabled®skip_as_script«code_folded$b56a926e-7e4d-4d44-b399-bb042de8c071cell_id$b56a926e-7e4d-4d44-b399-bb042de8c071codemcomparison_comment = md"""
The x-coordinate uncertainty of `pos_2d` compared to `x_pos_combined` is...
"""metadatashow_logsèdisabled®skip_as_script«code_folded$341618ea-677b-499e-b518-8a577e997e97cell_id$341618ea-677b-499e-b518-8a577e997e97codeshow_warehouse()metadatashow_logsèdisabled®skip_as_script«code_folded$2064d975-d9ad-4ddd-9d81-87f64363f998cell_id$2064d975-d9ad-4ddd-9d81-87f64363f998codeDprior_choice_motivation = md"""
I chose this prior because...
"""metadatashow_logsèdisabled®skip_as_script«code_folded$e4f3e468-fec8-4c3f-941d-0003e24a20f6cell_id$e4f3e468-fec8-4c3f-941d-0003e24a20f6code### YOUR CODE HEREmetadatashow_logsèdisabled®skip_as_script«code_folded$8555f14e-df09-4f11-82c0-cd5935d66418cell_id$8555f14e-df09-4f11-82c0-cd5935d66418code!feedback = md"""
This assignment took me ... minutes.
_I enjoyed this assignment?_ (yes / no / neutral) ...
_I found the assignment clear?_ (yes / no / neutral) ...
Did I use LLM tools (ChatGPT, Claude, Copilot, etc.) to help with this assignment? (yes / no) ... Was it helpful? ...
"""metadatashow_logsèdisabled®skip_as_script«code_folded$49e485c8-4b63-4fde-a7b6-6dc919e9bc38cell_id$49e485c8-4b63-4fde-a7b6-6dc919e9bc38codemd"""
- [1 point] Specify a probabilistic model for the Type A sensor observations.
- [1 point] Use RxInfer to **infer a posterior distribution** over the robot's x-coordinate. Store your result as `x_pos_from_A`.
- [1 point] What **prior** did you use for x, and why?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7dc3b839-c454-486f-b8f7-747332a7e45ccell_id$7dc3b839-c454-486f-b8f7-747332a7e45ccode__name_id_checkmetadatashow_logsèdisabled®skip_as_script«code_folded$3dac2b2c-494b-498a-a74b-a11349af54c3cell_id$3dac2b2c-494b-498a-a74b-a11349af54c3code8md"""
# Feedback
Please fill in the feedback below.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$103f53b2-be86-48e6-b380-fbc138c2cd33cell_id$103f53b2-be86-48e6-b380-fbc138c2cd33code### YOUR CODE HEREmetadatashow_logsèdisabled®skip_as_script«code_folded$69096420-5636-48e4-8380-24fe9aafab37cell_id$69096420-5636-48e4-8380-24fe9aafab37codemd"""
# Introduction: robot localization
An autonomous logistics robot is navigating a large **warehouse** (40 m wide × 30 m tall). Because GPS signals are unreliable indoors, the robot uses **ultrasonic beacons** mounted at fixed, known positions on the walls.
Each beacon emits a short sound pulse and measures the time until it detects the returning echo. From this time-of-flight measurement, it computes a distance estimate to the robot.
## Warehouse
The warehouse has two types of sensors:
- Type A, mounted on the west wall (x = 0 m).
- Type B, mounted on the east wall (x = 40 m).
Each sensor on the west wall measures the horizontal distance to the robot, giving a direct (noisy) estimate of the robot's x-coordinate. Each sensor on the east wall measures the distance from the far side, which after correction also gives an estimate of the robot's x-coordinate.
In this assignment you will:
1. **Characterize** the sensor noise and reliability using calibration data.
2. **Estimate the x-coordinate** of the robot from multiple independent sensor readings.
3. **Estimate the 2D position** of the robot using a multivariate probabilistic model.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$b7aea8f0-a5ac-487f-a91b-372812c5353bcell_id$b7aea8f0-a5ac-487f-a91b-372812c5353bcode9fusion_comment = md"""
The combined estimate is...
"""metadatashow_logsèdisabled®skip_as_script«code_folded$f6617439-fb86-4425-925a-6e705a78a2b3cell_id$f6617439-fb86-4425-925a-6e705a78a2b3codeٳ# 👉 Uncomment and fill this in:
# found_reliability = ... your code here
# If you want to continue without this exercise, leave the default:
found_reliability = Beta(1.0, 1.0)metadatashow_logsèdisabled®skip_as_script«code_folded$e47f428e-cc70-4a89-82a4-3e4dc563129acell_id$e47f428e-cc70-4a89-82a4-3e4dc563129acodemd"""
# Part 1: Sensor Calibration
Before deploying the sensors, we run a **calibration test**. The robot is placed at a known position, and a single Type A sensor repeatedly measures the distance to a reflector at a known distance of `calib_distance` = $(calib_distance) m.
The **calibration residuals** (measured distance minus known distance) are stored in `calib_residuals`. A well-calibrated sensor should have residuals centred at zero. In practice, manufacturing tolerances and mounting geometry can introduce a small **systematic bias** ``\delta_A``. We also want to know the spread of the measurements, i.e., the **systematic variance** ``\sigma^2_A``.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$41c2a20b-a664-492c-8821-682ecd34647ccell_id$41c2a20b-a664-492c-8821-682ecd34647ccode### YOUR CODE HEREmetadatashow_logsèdisabled®skip_as_script«code_folded$825db4bb-77aa-4686-82a7-0da72589b925cell_id$825db4bb-77aa-4686-82a7-0da72589b925codechint(md"""
The combined posterior is proportional to the product of the individual posteriors.
""")metadatashow_logsèdisabled®skip_as_script«code_folded$75fc1627-1e01-433f-9a44-791d982d4944cell_id$75fc1627-1e01-433f-9a44-791d982d4944codemd"""
## Data overview
The following data are available to you:
- `calib_residuals` — a vector of **$(length(calib_residuals)) calibration residuals** (measurement − known distance) from a single Type A sensor pointed at a reflector at a known distance of `calib_distance` = $(calib_distance) m.
- `x_obs_A` — **$(length(x_obs_A)) x-coordinate estimates** from the Type A sensors (noise ``\sigma^2_A = `` ? m).
- `x_obs_B` — **$(length(x_obs_B)) x-coordinate estimates** from the Type B sensors (noise ``\sigma^2_B = 0.25`` m).
- `pos_obs` — **$(length(pos_obs)) 2D position estimates** `[x, y]` from a combined positioning subsystem.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$abc102e0-5e58-4ac1-bbc8-b28dce742ae7cell_id$abc102e0-5e58-4ac1-bbc8-b28dce742ae7codestudent_id = 0123456metadatashow_logsèdisabled®skip_as_script«code_folded$2378b595-fa29-4a32-b428-1a5ecf4b05d7cell_id$2378b595-fa29-4a32-b428-1a5ecf4b05d7codecmd"""
# Appendix
The following cells are not important for the understanding of this assigment.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$705b06c6-600a-411a-b397-726e4d5acf3acell_id$705b06c6-600a-411a-b397-726e4d5acf3acode%begin
# Sensor layout (y-positions on each wall, used for warehouse visualisation only)
sensor_pos_A = [5.0, 12.0, 19.0, 24.0, 28.0] # west wall
sensor_pos_B = [6.0, 15.0, 26.0] # east wall
# Noise standard deviations (given to students in the problem description)
sigma_A = 0.20 # m (Type A sensor)
sigma_B = 0.50 # m (Type B sensor)
sigma_z = 0.35 # m (2D positioning subsystem, isotropic)
Λ_z = inv(sigma_z*diageye(2))
# ── Part 1: calibration data ──────────────────────────────────────────────
calib_distance = 8.0 # m (true distance to reflector)
rng_calib = StableRNG(1)
calib_measurements = [
calib_distance + secret_bias + sigma_A * randn(rng_calib)
for _ in 1:200
]
calib_residuals = calib_measurements .- calib_distance
# ── Part 1b: echo pulse reliability data ─────────────────────────────────
const secret_true_p = 0.82 # true detection probability
rng_echo = StableRNG(4)
echo_successes = rand(rng_echo, Bernoulli(secret_true_p), 300)
# ── Part 2: x-position measurements ──────────────────────────────────────
rng_x = StableRNG(2)
x_obs_A = [secret_true_x + sigma_A * randn(rng_x) for _ in 1:6]
x_obs_B = [secret_true_x + sigma_B * randn(rng_x) for _ in 1:8]
# ── Part 3: 2D position measurements ─────────────────────────────────────
rng_2d = StableRNG(3)
pos_obs = [
[secret_true_x + sigma_z * randn(rng_2d),
secret_true_y + sigma_z * randn(rng_2d)]
for _ in 1:10
]
end;metadatashow_logsèdisabled®skip_as_script«code_folded$3a9a632a-a3d5-4093-b9ed-ae7f6d90b46fcell_id$3a9a632a-a3d5-4093-b9ed-ae7f6d90b46fcodeٰbegin
using Plots; default(label="")
using Distributions
using RxInfer
using StatsPlots
using StableRNGs
using LinearAlgebra
using BmlipTeachingTools
using Markdown
endmetadatashow_logsèdisabled®skip_as_script«code_folded$6a26d308-3667-49a8-9f4a-0e6360230760cell_id$6a26d308-3667-49a8-9f4a-0e6360230760codeOcalib_histogram_comment = md"""
Looking at the histogram, I find that...
"""metadatashow_logsèdisabled®skip_as_script«code_folded$f8c27934-94a2-4817-aaca-ab818a3e0273cell_id$f8c27934-94a2-4817-aaca-ab818a3e0273codemd"""
- [2 points] Specify a probabilistic model with a **multivariate Normal likelihood** and use RxInfer to **infer a posterior distribution** over the 2D position vector `μ_z = [x, y]`. Store your result as `pos_2d`.
- [1 point] **Compare** the x-coordinate uncertainty of `pos_2d` with that of `x_pos_combined` from Part 2. Which gives a more precise x-estimate, and why?
- [1 point] Which point leads to the biggest drop in posterior uncertainty (expressed as determinant of the posterior covariance)?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$044f3410-2b9f-11f1-2fb2-ddeeb7049f22cell_id$044f3410-2b9f-11f1-2fb2-ddeeb7049f22codetitle("Resit Assignment")metadatashow_logsèdisabled®skip_as_script«code_folded$73b34659-be3d-4184-8fa5-adba70a7b2cdcell_id$73b34659-be3d-4184-8fa5-adba70a7b2cdcodebegin
# Well found!! 😅
const secret_true_x = 22.5 # m (robot's true x-position)
const secret_true_y = 14.8 # m (robot's true y-position)
const secret_bias = 0.12 # m (sensor mounting bias in calibration)
end;metadatashow_logsèdisabled®skip_as_script«code_folded$7f22dadc-39b5-417d-b29a-c91881a23097cell_id$7f22dadc-39b5-417d-b29a-c91881a23097codeTableOfContents()metadatashow_logsèdisabled®skip_as_script«code_folded$b90a0f2a-1117-4b71-a717-d711f5ea2223cell_id$b90a0f2a-1117-4b71-a717-d711f5ea2223code@exercise_statement("2D Localization", header_level=2, big=true)
metadatashow_logsèdisabled®skip_as_script«code_folded$66deb52f-ee85-48d6-a56e-f6f43111230fcell_id$66deb52f-ee85-48d6-a56e-f6f43111230fcodevmd"""
The Type A sensors alone give a good estimate of x, but we also have readings from the Type B sensors. By combining both, we can improve our estimate further.
- [1 point] Infer a posterior distribution over x using the **Type B sensor** (``\sigma^2_B = `` $(sigma_B^2) m) observations as well (store as `x_pos_from_B`).
- [2 point] **Combine** `x_pos_from_A` and `x_pos_from_B` into a single improved estimate using **Bayes' rule**. Store the combined result as `x_pos_combined`.
- [2 point] Plot the three distributions (`x_pos_from_A`, `x_pos_from_B`, `x_pos_combined`). What can you say about the combined result?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3d29e63b-8478-40cb-be8e-0516cfc59e0dcell_id$3d29e63b-8478-40cb-be8e-0516cfc59e0dcode|md"""
- [1 point] Specify a **probabilistic model** for `calib_residuals` that includes ``\delta_A`` and ``\sigma^2_A``.
- [2 points] Infer a **posterior distribution** for the sensor bias ``\delta_A`` and variance ``\sigma^2_A``, and store your result as `found_bias_distribution` and `found_variance_distribution`.
- [1 point] Explain how you chose your prior distribution.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$ef5c9636-42a4-49f8-8c66-68a590914f12cell_id$ef5c9636-42a4-49f8-8c66-68a590914f12codeٻlet
scatter(
first.(pos_obs), last.(pos_obs);
label="2D observations", c=:purple, ms=6,
xlabel="x (m)", ylabel="y (m)",
title="2D position measurements",
aspect_ratio=1,
)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$98d51c6e-b70d-4874-84c9-007a51ae389dcell_id$98d51c6e-b70d-4874-84c9-007a51ae389dcode٣md"""
$(exercise_statement("Sensor Noise Model"))
- [1 point] Plot a **histogram** of `calib_residuals`. What can you say about the shape of the distribution?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$dd51677f-3a0c-4fa6-b896-174915fe7417cell_id$dd51677f-3a0c-4fa6-b896-174915fe7417codemd"""
> #### Loading?
> Note: This notebook will **take a couple of minutes to start**, because Julia is downloading and precompiling the packages that we use in this notebook (Plots, RxInfer, Distributions, BmlipTeachingTools).
>
> You can check the progress in the **"Status" tab** in the bottom right.
>
> This package installation and precompilation will happen only once. Afterwards, this notebook (and future notebooks that use the same package versions) will launch much more quickly.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$744d90e7-397e-45d5-ab11-fa1d5c8f9c2acell_id$744d90e7-397e-45d5-ab11-fa1d5c8f9c2acodemd"""
# Part 3: 2D Position Estimation
So far, we have only estimated the robot's **x-coordinate**. We now also want to know its **y-coordinate** to fully localise the robot in the warehouse.
Suppose we have 2D measurements from a separate positioning subsystem. Each measurement is a 2-element vector $[x, y]$ with **isotropic** Gaussian noise:
$$z[i] \sim \mathcal{N}\!\left(\boldsymbol{\mu_z},\; \Lambda_z^{-1} \right)$$
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9d14fd1c-e704-4858-9842-fe088f18ec9ccell_id$9d14fd1c-e704-4858-9842-fe088f18ec9ccodekbegin
if !@isdefined(found_variance_distribution)
var_not_defined(:found_variance_distribution)
elseif !(found_variance_distribution isa Distribution{Univariate, <:Any})
keep_working(md"`found_variance_distribution` should be a univariate `Distribution`.")
elseif found_variance_distribution == InverseGamma(1/2, 1.)
md"> Fill in your result above to see the estimated bias."
else
correct(md"You have entered a $(typeof(found_variance_distribution)). Estimated variance: mode = $(round(mode(found_variance_distribution), digits=4)) m, std = $(round(std(found_variance_distribution), digits=4)) m.")
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$bb3362b6-6aa0-4796-b3b1-0385724feaddcell_id$bb3362b6-6aa0-4796-b3b1-0385724feaddcodelet
p = scatter(x_obs_A;
label="Type A", c=:blue, marker=:ltriangle, ms=7,
xlabel="Sensor index", ylabel="x estimate (m)",
title="x-position measurements",
ylim=(mean(x_obs_A) - 3, mean(x_obs_A) + 3),
)
scatter!(p, x_obs_B;
label="Type B", c=:darkorange, marker=:rtriangle, ms=7,
)
hline!(p, [mean([x_obs_A; x_obs_B])]; lw=2, ls=:dash, c=:gray, label="Data mean")
p
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1a940dbb-e1de-4c8f-8cb6-7d0b87f98b36cell_id$1a940dbb-e1de-4c8f-8cb6-7d0b87f98b36codeQexercise_statement("Estimating x from Type A Sensors", header_level=2, big=true)
metadatashow_logsèdisabled®skip_as_script«code_foldedënotebook_id$443b7d04-4562-11f1-80ba-997e31e6b12cin_temp_dir¨metadata