⚡ electrical engineering

Parametric and State Estimation of Stationary MEMS-IMUs: A Tutorial

This tutorial paper analyzes the benefits of using multiple stationary MEMS-IMUs for parametric and state estimation, demonstrating through both theoretical modeling and experimental validation that sensor arrays significantly improve signal accuracy, noise rejection, and navigation state estimation by mitigating inherent sensor errors.

Original authors: Daniel Engelsman, Yair Stolero, Itzik Klein

Published 2026-05-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniel Engelsman, Yair Stolero, Itzik Klein

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Why One Sensor Isn't Enough

Imagine you are trying to guess the exact temperature of a room. You have one thermometer, but it's a bit "jittery" (it has random noise) and it's slightly off the mark (it has a bias). If you rely on just that one thermometer, your guess will be a little wrong, and over time, if you use that wrong guess to calculate how far you've walked, your error will grow bigger and bigger.

This paper is about Inertial Navigation Systems (INS)—the technology inside your phone or a drone that tells it where it is without using GPS. These systems use tiny sensors called accelerometers (to feel movement) and gyroscopes (to feel spinning).

The authors argue that instead of using just one of these sensors, we should use a whole team of them (an array). They wanted to prove mathematically and experimentally that if you put many sensors together and average their answers, you get a much better result than using just one.

The Two Main Problems: The "Drunk" and the "Jittery"

The paper identifies two types of errors that mess up these sensors:

The Drunk (Bias): Imagine a compass that is slightly bent. No matter how many times you look at it, it always points 5 degrees to the left. This is a bias. It's a consistent mistake.
The Jittery (Noise): Imagine a compass that is perfectly straight, but someone is shaking your hand while you look at it. Sometimes it points left, sometimes right, randomly. This is noise.

The Solution: The "Crowd Wisdom" Analogy

The core idea of the paper is simple: If you ask 10 people to guess the weight of a pumpkin, and you average their answers, you will get a much better guess than if you ask just one person.

The Drunk (Bias) gets canceled out: If one sensor is biased high and another is biased low, when you average them, the errors cancel each other out.
The Jittery (Noise) gets smoothed out: Random shaking in one sensor is unlikely to happen at the exact same time as random shaking in another. When you average them, the random shakes cancel out, leaving a smooth, steady signal.

What They Did (The Experiment)

The researchers set up a stationary platform (it didn't move) in a lab. They used 10 identical sensors (specifically, Xsens-DOT units) sitting right next to each other.

They turned them on and let them sit for about 100 seconds.
They measured the errors: They looked at how much the single sensors were "drunk" (biased) and "jittery" (noisy).
They compared: They compared the results of using 1 sensor vs. 4 sensors vs. all 10 sensors.

The Results: The Magic of Math

The paper proves two main things using some heavy math (which they simplified for us):

More Sensors = Less Noise: If you double the number of sensors, the noise doesn't just go down by half; it goes down by the square root of the number of sensors.
- Analogy: If you have 10 sensors, the noise is reduced by a factor of roughly $\sqrt{10}$ (about 3.16). The "jitter" becomes much smoother.
More Sensors = Less Drift: Because the "drunk" bias is also averaged out, the system doesn't drift away from the truth as quickly.
- Analogy: If one sensor thinks you are moving when you are standing still, a single sensor will make you think you are flying away. Ten sensors will mostly agree that you are standing still, keeping you in the right place.

The "Cheat Sheet" of Findings

The authors created a table to show how much better things get:

Time helps, but sensors help more: If you wait longer (collect more data over time), the error goes down. But if you add more sensors, the error goes down even faster.
The 10x Rule: They found that using 10 sensors instead of 1 reduced the error significantly. The math showed that the uncertainty (how unsure the system is) dropped by about 3 times (which is $\sqrt{10}$ ).

Important Limitations (What They Didn't Do)

To be fair, the authors are very honest about what their study covers:

They didn't move: The sensors were sitting still on a table. They didn't test this on a moving car or a flying drone (though the math suggests it would work there too).
They used identical twins: All 10 sensors came from the same factory batch. They were almost identical. They didn't test mixing a "cheap" sensor with an "expensive" one.
They didn't use GPS: They only looked at the sensors themselves, without help from outside signals like GPS.

The Bottom Line

This paper is a tutorial proving that quantity has a quality all its own when it comes to sensors.

If you want a navigation system that is super accurate and doesn't drift away, don't just buy one expensive sensor. Buy ten cheaper ones, put them together, and let their "group vote" decide the answer. The paper shows that this "team approach" makes the system much more reliable, accurate, and precise, turning a shaky, drifting guess into a solid, trustworthy measurement.

Technical Summary: Parametric and State Estimation of Stationary MEMS-IMUs

Problem Statement
Inertial Navigation Systems (INS) rely on Micro-Electro-Mechanical Systems (MEMS) inertial measurement units (IMUs) to estimate position, velocity, and orientation. However, inherent sensor errors—specifically stochastic noise and deterministic bias—propagate through integration, causing state estimates to drift and degrade over time. While high-grade hardware can mitigate these issues, it is often cost-prohibitive. Software-based approaches, such as filtering and data fusion, offer alternatives but often require complex initialization or external aiding. This paper addresses the specific challenge of mitigating instrumental errors in stationary MEMS-IMUs by analyzing the theoretical and empirical benefits of employing multiple sensors (sensor arrays) rather than a single unit.

Methodology
The study develops a simplified analytical framework to model the relationship between estimation error, time, and the number of sensors ( $K$ ).

Sensor Modeling: The authors model inertial sensor outputs as linear combinations of true signals, gain matrices, biases, and white Gaussian noise. They assume Wide-Sense Stationarity (WSS) and that sensors are uncorrelated.
Parametric Estimation: The paper derives the statistical properties of the sample mean for both a single sensor and an array of $K$ sensors. It proves that while the sample mean is an unbiased estimator, its variance decreases inversely proportional to the number of samples ( $N$ ) and the number of sensors ( $K$ ). Specifically, the variance of the mean for $K$ sensors is reduced by a factor of $1/(NK)$.
State Estimation: The framework extends to the navigation state vector (position, velocity, misalignment, and biases). Using a linear state-space model, the authors propagate the error covariance matrix over time. They demonstrate that the process noise covariance and the resulting state uncertainties also scale inversely with the number of sensors ( $1/K$ ).
Experimental Validation: To validate the analytical models, the authors conducted experiments using ten identical Xsens-DOT MEMS-IMUs arranged in a stationary, leveled configuration. Data was collected over 100 seconds at 100 Hz. The study compared the performance of a single sensor against an array of ten sensors, analyzing bias reduction, noise density (RMS), and the propagation of state errors and uncertainties.

Key Contributions
The paper presents three primary contributions:

User-Friendly Overview: It breaks down complex estimation topics into a step-by-step analysis, making the relationship between error, time, and sensor count accessible to both newcomers and advanced practitioners.
Comprehensive Validation: The study provides a rigorous comparison between the hypothesized analytical models and experimental results, confirming the theoretical predictions regarding error reduction.
Open-Source Access: All data and code used in the study are made publicly available to facilitate seamless integration and reproducibility.

Results
The experimental results align closely with the analytical hypotheses:

Parametric Level: As the number of sensors ( $K$ ) increased, the standard deviation of the estimated bias and noise decreased. The results confirmed that the error standard deviation scales with $1/\sqrt{K}$ . For $K=10$ , the reduction was approximately $1/\sqrt{10}$ , corresponding to a -5 dB improvement.
State Level: In the absence of external aiding, the INS solution drifts due to uncompensated biases. The study showed that using a sensor array significantly reduces both the magnitude of the state errors (position, velocity, misalignment) and the associated state uncertainties (covariance).
Covariance Reduction: The state uncertainty covariance for the 10-sensor array was observed to be approximately $1/10$ of the single-sensor uncertainty, consistent with Proposition 5.
Visual Confirmation: Error ellipsoids representing position uncertainty after 100 seconds demonstrated a dramatic reduction in volume and centroid displacement when using the sensor array compared to a single sensor.

Significance and Claims
The authors claim that their findings substantiate the validity of using multiple MEMS-IMUs as a "force multiplier" for inertial navigation. The study demonstrates that increasing the number of sensors acts as a robust method to mitigate instrumental errors and outliers, leading to improved accuracy (bias cancellation) and precision (noise reduction).

The paper posits that this approach offers practical implications for the design of high-precision systems, particularly in scenarios where non-ideal conditions (such as contamination or limited averaging time) might otherwise degrade performance. By adding more sensors, operational constraints regarding initialization time and accuracy can be met more effectively without the need for expensive high-grade hardware. The study concludes that while the analysis was conducted under stationary conditions, the fundamental principles of error reduction through averaging provide a strong theoretical basis for extending these benefits to dynamic scenarios and optimal INS integration strategies.