HW2. Sensor Calibration¶

Due date: 2026-01-29 23:59.

Recommended Resources:

SF: Sensing and Filtering, Steven M. LaValle, PDF
Recorded Lectures and Slides

1D Sensor Calibration: Thermometer¶

In this part of the assignment, you will explore how sensor calibration can be expressed mathematically using linear algebra, the pseudoinverse, and the Singular Value Decomposition (SVD).

Problem Setup¶

Assume you have a simple thermometer that reports temperature in degrees Celsius. You measure the temperature on three consecutive days using:

a new thermometer you just bought, and
an old thermometer that you trust and treat as ground truth.

Let:

the measurements reported by the new (uncalibrated) thermometer be:

r_1, r_2, r_3. \\\

the true temperatures (reported by the trusted thermometer) be:

s_1, s_2, s_3. \\\

You notice that the new thermometer consistently reports different values, so you assume it has a linear calibration error. You model this error using a scale-and-offset model:

\begin{aligned} s_1 &= a r_1 + b, \\ s_2 &= a r_2 + b, \\ s_3 &= a r_3 + b. \\\ \end{aligned}

In the above equations, the following calibration parameters are introduced:

𝑎 represents the scale error.
𝑏 represents the offset error.

Your goal is to find 𝑎 and 𝑏, so that you can effectively use the new thermometer by applying these calibration parameters to any future measurement. In practice, the data may not satisfy the calibration model exactly, so 𝑎 and 𝑏 are instead estimated using least squares.

Download Visualization Tool¶

Download and run the following visualization tool for understanding linear calibration using least-squares and SVD:

Calibration1D

This tool visualizes the calibration process and the solution to the least-squares minimization problem by animating the following:

Parameter plane. The calibration parameters live in a 2D plane:

\beta = \begin{bmatrix} a \\ b \end{bmatrix} \in \mathbb{R}^2. \\\

Data matrix and mapping to the measurement space. The data matrix defines a linear mapping from the 2D parameter plane into the 3D measurement space:

M \beta \in \mathbb{R}^3. \\\

Ground truth vector. The ground truth vector also lives in the 3D measurement space:

s = \begin{bmatrix} s_1 \\ s_2 \\ s_3 \end{bmatrix} \in \mathbb{R}^3. \\\

Estimated calibration parameters: The goal of the calibration process is to compute the best-fit calibration parameters:

\hat{\beta} = \begin{bmatrix} a \\ b \end{bmatrix}. \\\

Least-squares solution: The estimated calibration parameters are the solution to the following least-squares problem:

\hat{\beta} = \arg\min_{\beta} || s - M \beta ||. \\\

Fitted point. The fitted point represents the best least-squares fit to the ground truth vector in the measurement space:

\hat{s} = M \hat\beta \in \mathbb{R}^3. \\\

Residual vector. The residual vector captures the remaining error after the calibration. It is defined as the difference between the ground truth vector and the fitted point. Geometrically, it represents the distance between the transformed parameter plane and the ground truth vector.

s - M \hat\beta. \\\

Geometrical interpretation. Geometrically, the fitted point is the orthogonal projection of the ground truth vector onto the transformed parameter plane. The calibration parameters 𝑎 and 𝑏 are, therefore, the coordinates of the point on this plane that produces the closest possible match to the ground truth measurements. Understanding the geometry of the calibration model is essential. You should also be able to reason about it analytically.

Question 1¶

Your first exercise is to put the calibration model into the matrix form. Start with the three calibration equations below:

\begin{aligned} s_1 &= a r_1 + b, \\ s_2 &= a r_2 + b, \\ s_3 &= a r_3 + b. \\\ \end{aligned}

You want to rewrite them in matrix form

s = M \beta, \\\

where the ground-truth temperature vector is

s = \begin{bmatrix} s_1 \\ s_2 \\ s_3 \end{bmatrix} \in \mathbb{R}^3, \\\

and the vector of calibration parameters is

\beta = \begin{bmatrix} a \\ b \end{bmatrix} \in \mathbb{R}^2. \\\

Which of the following is the correct data matrix?

\begin{aligned} \mathsf{A.}\;\; &M = \begin{bmatrix} r_1 & b \\ r_2 & b \\ r_3 & b \end{bmatrix} \\\ \\\ \mathsf{B.}\;\; &M = \begin{bmatrix} a & 1 \\ a & 1 \\ a & 1 \end{bmatrix} \\\ \\\ \mathsf{C.}\;\; &M = \begin{bmatrix} s_1 & 1 \\ s_2 & 1 \\ s_3 & 1 \end{bmatrix} \\\ \\\ \mathsf{D.}\;\; &M = \begin{bmatrix} 1 & r_1 \\ 1 & r_2 \\ 1 & r_3 \end{bmatrix} \\\ \\\ \mathsf{E.}\;\; &M = \begin{bmatrix} s_1 & a \\ s_2 & b \\ s_3 & 1 \end{bmatrix} \\\ \\\ \mathsf{F.}\;\; &M = \begin{bmatrix} r_1 & 1 \\ r_2 & 1 \\ r_3 & 1 \end{bmatrix} \\\ \\\ \mathsf{G.}\;\; & \text{None of the above.} \end{aligned}

Question 2¶

Now consider a specific calibration experiment. Suppose the new (uncalibrated) thermometer reports the following temperatures:

r_1 = 0, r_2 = 1, r_3 = 2.5. \\\

The trusted thermometer reports the ground-truth temperatures:

s_1 = -2, s_2 = -1, s_3 = 0.5. \\\

Assume the same linear calibration model used above:

\begin{aligned} s_1 &= a r_1 + b, \\ s_2 &= a r_2 + b, \\ s_3 &= a r_3 + b. \\\ \end{aligned}

Refactor the equations to obtain the data matrix, and then use the Calibration1D visualization tool to estimate the calibration parameters using the pseudoinverse.

Which calibration parameters did the tool report?

Question 3¶

Which of the following pairs of measurements and ground-truth values results in the best calibration, meaning the smallest residual vector magnitude?

Question 4¶

Using the visualization tool, keep the data matrix 𝑀 fixed and keep changing the ground-truth vector 𝑠.

Which of the following quantities change as a result?

Question 5¶

Suppose the new thermometer reports the same value for different ground truth values:

r_1 = r_2 = r_3 = 1, \,\,\,\, s_1 = 0.1, s_2 = 1.1, s_3 = 2.1. \\\

In this case, the data matrix has rank 1, not 2. One of the singular values of the matrix is zero. However, the pseudoinverse can still be computed analytically using the SVD-based formula:

𝑀^{+} = VΣ^{+} U^{\top} \\\

The visualization tool still computes the matrices U, Σ, and V, but because Σ contains a zero singular value, the tool cannot proceed with animation or estimation of calibration parameters. Therefore, you cannot rely on the visualization tool in this case and must reason analytically.

Compute the pseudoinverse of M manually, and determine the estimated β̂ using the computed pseudoinverse. Which one is the resulting β̂ , you can round the decimals?

3D Sensor Calibration: Accelerometer¶

In this part of the assignment, you will extend the ideas from 1D thermometer calibration to the 3D calibration of an accelerometer.

An accelerometer measures acceleration along three orthogonal axes (𝑥, 𝑦, 𝑧). Due to manufacturing imperfections, the raw measurements typically contain scale errors, axis misalignment, and offset (bias) errors. Calibration is required to correct these errors before the sensor can be used reliably.

This calibration problem is a direct generalization of the 1D case. Instead of estimating two scalar parameters, you will now estimate a matrix of calibration parameters and an offset vector.

Sensor Data¶

Start by downloading raw acceleration measurements recorded by an uncalibrated accelerometer:

measurements.csv.

Each row corresponds to one accelerometer measurement and contains three values, representing the raw acceleration measured along the 𝑥, 𝑦, and 𝑧 axes:

\begin{aligned} &\dots \\\ r_{xi},\; &r_{yi},\; r_{zi}\\\ &\dots \end{aligned} \\\

Next, download the file containing acceleration measurements obtained from a trusted reference system:

groundtruth.csv.

This file contains measurements in the same format:

\begin{aligned} &\dots \\\ s_{xi},\; &s_{yi},\; s_{zi}\\\ &\dots \end{aligned} \\\

Both files contain thousands of measurements, making the calibration problem highly overdetermined.

Calibration Model¶

We assume a linear calibration model that accounts for scale, cross-axis coupling, and offset errors. The calibration parameters consist of:

a 3×3 calibration matrix, 𝑨,
a 3×1 offset vector, 𝑏.

A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \qquad b = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \\\

The calibration model is:

s_i = A r_i + b, \qquad s_i = \begin{bmatrix} s_{xi} \\ s_{yi} \\ s_{zi} \end{bmatrix}, \qquad r_i = \begin{bmatrix} r_{xi} \\ r_{yi} \\ r_{zi} \end{bmatrix}. \\\

You can also re-write it component-wise for each pair of measurements 𝑠ᵢ and 𝑟ᵢ:

\begin{aligned} & \dots\\ s_{xi} &= a_{11} r_{xi} + a_{12} r_{yi} + a_{13} r_{zi} + b_1 \\ s_{yi} &= a_{21} r_{xi} + a_{22} r_{yi} + a_{23} r_{zi} + b_2 \\ s_{zi} &= a_{31} r_{xi} + a_{32} r_{yi} + a_{33} r_{zi} + b_3\\ & \dots\\ \end{aligned} \\\

Your goal is to estimate the calibration parameters 𝑨 and 𝑏 using least-squares.

Data Matrix Formulation¶

Your task is to refactor all calibration equations into a single linear system of the form:

s = M \beta \\\

where:

the vector 𝑠 contains all ground-truth acceleration values stacked into a single column vector,
the vector 𝛽 contains all the unknown calibration parameters,
the matrix 𝑀 is a large data matrix constructed from the raw accelerometer measurements.

Because the number of measurements is much larger than the number of unknown parameters, this system is highly overdetermined and must be solved using the pseudoinverse.

Question 6¶

Which of the following correctly represents the vector of unknown calibration parameters for the 3D accelerometer, as described in lecture?

𝛽 = [ 𝑎₁ 𝑎₂ 𝑎₃ 𝑏₁ 𝑏₂ 𝑏₃ ]ᵀ

𝛽 = [ 𝑎₁₁ 𝑎₁₂ 𝑎₁₃ 𝑏₁ 𝑏₂ 𝑏₃ ]

𝛽 = [ 𝑎₁₁ 𝑎₁₂ 𝑎₁₃ 𝑎₂₁ 𝑎₂₂ 𝑎₂₃ 𝑎₃₁ 𝑎₃₂ 𝑎₃₃ ]ᵀ

𝛽 = [ 𝑎₁₁ 𝑎₁₂ 𝑎₁₃ 𝑎₂₁ 𝑎₂₂ 𝑎₂₃ 𝑎₃₁ 𝑎₃₂ 𝑎₃₃ ]

𝛽 = [ 𝑎₁₁ 𝑎₁₂ 𝑎₁₃ 𝑏₁ 𝑎₂₁ 𝑎₂₂ 𝑎₂₃ 𝑏₂ 𝑎₃₁ 𝑎₃₂ 𝑎₃₃ 𝑏₃ ]ᵀ

𝛽 = [ 𝑎₁₁ 𝑎₁₂ 𝑎₁₃ 𝑏₁ 𝑎₂₁ 𝑎₂₂ 𝑎₂₃ 𝑏₂ 𝑎₃₁ 𝑎₃₂ 𝑎₃₃ 𝑏₃ ]

𝛽 = [ 𝑎₁₁ 𝑎₁₂ 𝑎₁₃ 0 0 0 𝑎₂₁ 𝑎₂₂ 𝑎₂₃ 0 0 0 𝑎₃₁ 𝑎₃₂ 𝑎₃₃ 0 0 0 𝑏₁ 𝑏₂ 𝑏₃ ]

𝛽 = [ 𝑎₁₁ 𝑎₁₂ 𝑎₁₃ 0 0 0 𝑎₂₁ 𝑎₂₂ 𝑎₂₃ 0 0 0 𝑎₃₁ 𝑎₃₂ 𝑎₃₃ 0 0 0 𝑏₁ 𝑏₂ 𝑏₃ ]ᵀ

None of the above.

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Question 7¶

Which of the following correctly describes the size of the data matrix 𝑀?

Question 8¶

Using the two provided files (measurements.csv and groundtruth.csv), which of the following options correctly lists the first four rows of the data matrix 𝑀?

\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \begin{aligned} \mathsf{A.}\; & {\scriptsize \begin{bmatrix} 0.44228 & -0.094418 & -12.6990 & 1 & 1 & 1 \\ 0.41223 & -0.77859 & -11.5290 & 1 & 1 & 1 \\ 0.21464 & -0.53937 & -9.6857 & 1 & 1 & 1 \\ -0.63297 & -0.84337 & -9.8479 & 1 & 1 & 1\\ \end{bmatrix} } \\\ \mathsf{B.}\; & {\scriptsize \begin{bmatrix} 0.44228 & 0 & 0 & -0.094418 & 0 & 0 & -12.6990 & 0 & 0 \\ 0 & 0.44228 & 0 & 0 & -0.094418 & 0 & 0 & -12.6990 & 0 \\ 0 & 0 & 0.44228 & 0 & 0 & -0.094418 & 0 & 0 & -12.6990 \\ 0.41223 & 0 & 0 & -0.77859 & 0 & 0 & -11.5290 & 0 & 0\\ \end{bmatrix} } \\\ \mathsf{C.}\; & {\scriptsize \begin{bmatrix} 0.44228 & -0.094418 & -12.6990 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.44228 & -0.094418 & -12.6990 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0.44228 & -0.094418 & -12.6990 & 1 \\ 0.41223 & -0.77859 & -11.5290 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ \cdots&\cdots &\cdots &\cdots &\cdots & \cdots &\cdots &\cdots &\cdots & \cdots&\cdots & \cdots\\ \end{bmatrix} } \\\ \mathsf{D.}\; & {\scriptsize \begin{bmatrix} 1 & 0.44228 & -0.094418 & -12.6990 & 0 & 0 & 0 \\ 1 & 0.41223 & -0.77859 & -11.5290 & 0 & 0 & 0 \\ 1 & 0.21464 & -0.53937 & -9.6857 & 0 & 0 & 0 \\ 1 & -0.63297 & -0.84337 & -9.8479 & 0 & 0 & 0\\ \end{bmatrix} } \\\ \mathsf{E.}\; & {\scriptsize \begin{bmatrix} 0.44228 & -0.094418 & -12.6990 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.44228 & -0.094418 & -12.6990 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0& 0.44228 & -0.094418 & -12.6990 \\ 0.41223 & -0.77859 & -11.5290 & 0 & 0 & 0 & 0 & 0 & 0 \\\ \cdots&\cdots &\cdots &\cdots &\cdots & \cdots &\cdots &\cdots &\cdots \\ \end{bmatrix} } \\\ \mathsf{F.}\; &\text{More than one above.}\\ \mathsf{G.}\; &\text{None of the above.} \end{aligned}

Question 9¶

Using the two provided files (measurements.csv and groundtruth.csv), which of the following correctly represents the ground-truth vector 𝑠?

[0.24493 0.19181 0.004109 −0.85458 0.19367 −0.48706 −0.24062 −0.51548 −10.5870 −9.6089 −8.0760 −8.2087 ...]ᵀ

[ 0.24493, 0.19367, -10.587 0.19181, -0.48706, -9.6089 0.0041088,-0.24062, -8.076 -0.85458,-0.51548, -8.2087 ...]ᵀ

[ 0.24493 0.19181 0.0041088 -0.85458 0.19367 -0.48706 -0.24062 -0.51548 ...]ᵀ

[ 0.24493 0.19367 −10.5870 0.21464 −0.53937 −9.6857 −0.63297 0.41223 −0.77859 −11.5290 −0.84337 −9.8479 ...]ᵀ

[0.24493 0.19181 0.0041088 -0.85458 -0.53762 0.013582 0.020564 -0.24833 -0.28456 ...]ᵀ

None of the above.

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Question 10¶

Using the two provided files (measurements.csv and groundtruth.csv), compute the estimated calibration parameters using pseudoinverse. You can pick any numerical tool you like (for example: Python, NumPy, MATLAB, or similar).

Report:

the estimated calibration matrix 𝑨,
the offset vector 𝑏,
and the tool or software used.

Report numerical values with reasonable precision (for example, 3–4 significant digits).

Authors¶

Anna LaValle.

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