Allan Deviation in MEMS Sensors: Understanding Noise and Drift in Gyroscopes and Accelerometers

2. How to Read an Allan Deviation Plot

Now that we understand the basic principle, let’s examine the first Allan Deviation plot.

The graph illustrates how measurement stability changes as the averaging time increases. The same principles apply to both gyroscopes and accelerometers, although their physical measurement units differ.

Important: This is a conceptual illustration, not actual measurement data from a specific sensor.

Horizontal Axis – Averaging Time

The horizontal axis represents the averaging time, commonly denoted by the Greek letter τ (tau).

For example, when the averaging time is one second, the sensor data is divided into one-second intervals, and the average measurement is calculated for each interval.

When the averaging time increases to 100 seconds, the same calculation is performed using longer intervals.

This allows engineers to determine whether longer averaging periods improve measurement stability or whether slower noise processes and drift begin to dominate.

Vertical Axis – Allan Deviation

The vertical axis represents Allan Deviation, a statistical measure derived from the differences between the averages of adjacent measurement intervals.

A lower Allan Deviation value indicates greater measurement stability over the specified averaging time.

For a gyroscope measuring angular velocity, Allan Deviation may be expressed in degrees per second (°/s). For an accelerometer, it may be expressed in acceleration units such as mg or µg.

Both axes are typically displayed on logarithmic scales, allowing engineers to examine sensor behavior across a wide range of averaging times and noise levels.

The Three Main Regions of an Allan Deviation Plot

Region 1 – Short-Term Noise

On the left side of the plot, the Allan Deviation curve generally decreases as the averaging time increases.

In this region, high-frequency random noise often dominates the measurement. Averaging reduces the influence of this noise.

When white noise is the dominant mechanism, the curve exhibits an approximate slope of -1/2 on a logarithmic plot.

For gyroscopes, this region is associated with Angle Random Walk (ARW). For accelerometers, the corresponding noise parameter is Velocity Random Walk (VRW).

Region 2 – Minimum Region

As the averaging time increases, the downward slope gradually becomes less pronounced. The curve may approach a nearly horizontal region or reach a minimum.

At this stage, the influence of high-frequency white noise has decreased, while slower noise processes become more significant.

One important mechanism is Bias Instability, which describes slow statistical fluctuations in the sensor’s measurement bias.

This region is particularly valuable when evaluating sensor stability over extended periods.

However, the minimum point of an Allan Deviation curve does not necessarily represent pure Bias Instability. Accurate identification requires examining the curve’s slope, shape, and underlying noise mechanisms.

Region 3 – Long-Term Drift

On the right side of the plot, the Allan Deviation curve may begin to rise.

This indicates that increasing the averaging time no longer improves measurement stability.

At longer timescales, slow noise processes may dominate. In gyroscopes, these can include Rate Random Walk, while accelerometers may exhibit analogous low-frequency noise behavior.

Environmental influences, particularly temperature variations, can also contribute to long-term changes in sensor output.

It is therefore important to distinguish between the sensor’s intrinsic stochastic noise and environmental effects that may not have been fully controlled during testing.

The key takeaway: There is no single optimal averaging time for every sensor or application. An Allan Deviation plot helps engineers understand how measurement stability evolves over time and identify the dominant noise mechanisms at different timescales.

3. What Is Angle Random Walk (ARW) in a Gyroscope?

A gyroscope measures angular velocity, describing how quickly an object rotates around a particular axis.

To determine the angular displacement of an object, the measured angular velocity is integrated over time.

However, the gyroscope output also contains random noise.

Even when the sensor is completely stationary, integrating this noise introduces uncertainty into the calculated angle.

This effect is known as Angle Random Walk (ARW).

Imagine someone attempting to walk in a straight line while taking small, random steps to the left or right. Although these deviations have no consistent direction, the accumulated displacement from the intended path can increase over time.

Similarly, random noise in angular velocity measurements produces growing uncertainty in the estimated angular position.

How Should ARW Be Interpreted?

ARW is commonly expressed in degrees per square root of hour (°/√hr).

A lower ARW value indicates a smaller contribution from white noise to the accumulated angular uncertainty.

For example, consider a gyroscope with an ARW of 0.02°/√hr.

Assuming ideal white noise, the approximate RMS angular uncertainty caused by this noise component alone is:

These values are calculated theoretical examples, not experimental measurements.

They follow the relationship:

RMS Angular Error ≈ ARW × √Time

where time is expressed in hours when ARW is specified in °/√hr.

The calculation considers only the white-noise component represented by ARW. It excludes bias errors, temperature effects, calibration errors, and other sources of measurement uncertainty.

SX3 Low Noise – Why Does This Matter?

Gladiator Technologies has presented an ARW value below 0.02°/√hr for its next-generation SX3 Low Noise sensor technology.

This indicates a low level of angular random noise under the reported measurement conditions.

Such performance can be particularly valuable in applications requiring precise motion sensing, including electro-optical stabilization, inertial navigation, and advanced motion control systems.

However, ARW alone does not determine the overall accuracy of an inertial system.

Bias stability, sensor bandwidth, measurement latency, environmental conditions, and system-level calibration also influence actual performance.

Furthermore, technology-level measurement results should not automatically be interpreted as guaranteed specifications for every SX3-based product configuration.

4. What Is Velocity Random Walk (VRW) in an Accelerometer?

Accelerometers exhibit a similar noise-related phenomenon, but instead of accumulated angular uncertainty, the result is accumulated velocity uncertainty.

An accelerometer measures acceleration. When an inertial navigation system needs to estimate velocity, it integrates the acceleration measurements over time.

However, accelerometer output also contains random noise.

When this noise is integrated, it introduces uncertainty into the calculated velocity. For white acceleration noise, the RMS velocity uncertainty increases approximately with the square root of the integration time.

This phenomenon is known as Velocity Random Walk (VRW).

The fundamental difference between ARW and VRW lies in the physical quantity affected by the accumulated measurement noise.

In an inertial navigation system, velocity estimation errors can subsequently contribute to position errors when acceleration measurements are integrated a second time.

This is one reason why accelerometer noise performance is just as important as gyroscope noise performance in high-accuracy inertial navigation applications.

Why Does VRW Matter When Comparing Accelerometers?

Two accelerometers may offer the same measurement range, such as ±30g, while exhibiting significantly different noise characteristics.

An accelerometer with lower VRW generally contributes less white-noise-induced uncertainty to the estimated velocity, assuming comparable measurement conditions and other sensor characteristics.

However, VRW alone does not provide a complete picture of accelerometer performance.

To evaluate long-term measurement stability, engineers must also examine bias stability, temperature sensitivity, and Allan Deviation results.

An important distinction when comparing datasheets: Accelerometer manufacturers may express random noise performance using different units.

Acceleration noise density is commonly specified in µg/√Hz, whereas VRW can also be expressed in velocity-per-square-root-of-time units, such as m/s/√hr.

These quantities are related through the integration of acceleration noise, but their numerical values should not be compared directly without accounting for the units and the manufacturer’s definitions.

This distinction becomes particularly important when evaluating closed-loop accelerometers from different manufacturers.

5. From Theory to Real-World Performance – Gladiator Technologies Accelerometers

So far, we have explored the principles of Allan Deviation and the significance of random noise in gyroscopes and accelerometers.

Now, let’s examine how these parameters relate to the performance of real-world inertial sensors.

Gladiator Technologies develops advanced inertial sensing solutions, including gyroscopes, accelerometers, and Inertial Measurement Units (IMUs), for applications requiring low noise, high stability, and reliable performance under demanding environmental conditions.

Among its product families are closed-loop accelerometers, which use a feedback mechanism to apply a balancing force to the internal proof mass in response to external acceleration.

This force-rebalance approach can provide excellent measurement stability and support improved sensor performance, depending on the specific design and implementation.

Allan Deviation Measurements of a ±30g Closed-Loop Accelerometer

In its technical presentation, Gladiator Technologies provides Allan Deviation measurement results for a closed-loop accelerometer with a ±30g measurement range.

The graph shows Allan Deviation decreasing as the averaging time increases, followed by a region where the curve begins to flatten.

For the stability characteristic highlighted in this measurement, the manufacturer reports:

• Typical: 10 µg – Typical reported value.
• Maximum: 15 µg – Maximum value indicated in the presented data.

To put these figures into perspective, 10 micro-g corresponds to just 0.00001g, or approximately 0.000098 m/s².

This is an extremely small acceleration level, illustrating the measurement sensitivity and stability required in high-performance inertial sensing applications.

However, this value should not be interpreted as a guarantee that the accelerometer provides an absolute accuracy of 10 µg under all operating conditions.

It represents a stability characteristic derived from the manufacturer’s reported measurements, not a comprehensive specification of total measurement error.

GA100, GA200, and GA300 Closed-Loop Accelerometer Families

Gladiator Technologies offers three families of single-axis closed-loop accelerometers designed for different acceleration measurement ranges.

These figures refer to the D versions and are based on preliminary performance specifications published by Gladiator Technologies.

Technical note: The manufacturer identifies the 20 µg/√Hz parameter as VRW. These units are also commonly used to express acceleration noise density. When comparing accelerometers from different manufacturers, engineers should verify the definitions, measurement conditions, and units used in each datasheet.

Another important distinction concerns the stability specifications.

For the D versions, Gladiator Technologies lists the following In-Run / Short-Term (5hr) Stability values:

These values represent different stability characteristics as defined in the manufacturer’s preliminary specifications. They should not be interpreted as interchangeable measurements or as guarantees of total sensor accuracy.

The differences between these accelerometer families demonstrate why selecting an inertial sensor should involve more than simply choosing the highest available measurement range.

For example, a system designed to measure accelerations up to ±15g does not necessarily require an accelerometer capable of measuring ±70g.

Engineers must also consider noise performance, bias stability, vibration conditions, bandwidth, and the overall requirements of the application.

The engineering advantage lies in selecting an accelerometer that matches the application’s motion profile and accuracy requirements, rather than relying solely on maximum acceleration range.

6. What Is Bias Instability, and Why Does It Matter?

Until now, we have focused primarily on random noise, whose influence can often be reduced through averaging.

However, inertial sensors face another challenge that can be equally important, or even more significant in certain applications:

The sensor’s measurement bias does not necessarily remain constant over time.

This behavior is closely related to Bias Instability.

First, What Is Sensor Bias?

Consider a gyroscope resting on a stationary surface.

Ideally, its output should indicate an angular velocity of zero degrees per second.

However, suppose the sensor consistently reports 0.01°/s even though no rotation is occurring.

This represents a measurement bias of 0.01°/s.

If the bias were perfectly constant, it could be measured and compensated for through calibration.

In practice, however, sensor bias may change during operation.

For example, the gyroscope may initially report a bias of 0.01°/s, but this value may gradually fluctuate even while the sensor remains stationary.

These slow variations make it more difficult for the navigation or control system to distinguish actual motion from changes originating within the sensor itself.

Bias Instability vs. In-Run Bias Stability

Both terms describe aspects of measurement bias stability, but they do not necessarily represent the same performance parameter.

Bias Instability is a statistical measure associated with slow fluctuations in sensor bias, commonly characterized through Allan Deviation analysis.

In particular, it is often associated with low-frequency flicker noise, although the extraction method depends on the noise model and analysis procedure.

In-Run Bias Stability, on the other hand, describes the stability of sensor bias during continuous operation.

Its exact definition and calculation method may vary between manufacturers and datasheets. Depending on the specification, it may be derived from statistical analysis, direct measurements over a defined operating period, or other test procedures.

Therefore, when comparing inertial sensors, it is essential to confirm that the reported values are based on comparable definitions, measurement methods, and test conditions.

Why Can a Small Bias Cause a Large Navigation Error?

Consider a gyroscope with a constant, uncompensated bias of 0.01°/s.

If an inertial navigation system integrates this angular velocity error over one minute, the resulting angular error will be:

0.01°/s × 60 s = 0.6°

After ten minutes, the accumulated angular error will reach:

0.01°/s × 600 s = 6°

This example illustrates the accumulation of angular error caused by a constant, uncompensated bias.

It is not a calculation of Bias Instability.

The distinction is important because Bias Instability describes stochastic fluctuations in bias, rather than a fixed offset that produces a predictable linear drift.

Nevertheless, the example demonstrates why even small bias errors can become significant in inertial navigation systems operating without continuous external corrections.

SX3 Low Noise – Combining Low Random Noise with Improved Bias Stability

For its next-generation SX3 Low Noise technology, Gladiator Technologies has presented a best-case measured Bias Instability value of 0.25°/hr, together with an Angle Random Walk value below 0.02°/√hr.

These reported results highlight two important performance objectives:

• Reducing high-frequency random noise in angular velocity measurements.
• Improving the stability of the sensor’s measurement bias over time.

Together, these characteristics can help reduce uncertainty in inertial measurements, particularly in applications where accurate angular information is required over extended operating periods.

However, it is important to distinguish between Bias Instability, expressed in degrees per hour, and accumulated angular error.

A Bias Instability value of 0.25°/hr does not necessarily mean that the angular error will be 0.25° after one hour.

The actual error depends on the underlying noise processes, operating conditions, integration time, and the correction methods implemented by the navigation system.

Furthermore, the reported SX3 values represent performance results presented for the technology and should not automatically be treated as guaranteed specifications for every product configuration.

In advanced inertial navigation and stabilization systems, low ARW and improved bias stability can contribute to more accurate motion estimation, especially when combined with appropriate control-system design, signal processing, temperature compensation, and sensor calibration.

7. In-Run Bias Stability – What Happens After Several Hours of Operation?

We have already explored how Allan Deviation helps identify different noise characteristics and how Bias Instability describes slow statistical fluctuations in sensor bias.

However, from an engineering perspective, it is equally important to understand how an inertial sensor behaves during continuous operation.

Consider an accelerometer installed in an Inertial Navigation System (INS) operating for several hours.

Even if the sensor remains stationary, its output may gradually change over time.

Such changes can result from internal stabilization processes, temperature variations, or low-frequency noise mechanisms.

One way to evaluate this behavior is through In-Run Bias Stability measurements.

How to Interpret an In-Run Bias Stability Plot

Unlike an Allan Deviation plot, which shows a statistical stability measure as a function of averaging time, a time-history plot used to assess In-Run Bias Stability shows how the sensor output evolves during the actual test period.

Two important characteristics can often be observed:

1. Short-Term Fluctuations Around the Measured Value

These fluctuations may be associated with short-term measurement noise.

They appear as rapid, relatively small variations in the sensor output, giving the measurement curve a slightly irregular appearance.

2. Gradual Changes in the Measured Value

When the measurement curve gradually shifts upward or downward, this may indicate a change in the sensor’s bias over time.

Such behavior can result from temperature effects, internal stabilization, or other slow processes.

Both characteristics matter, but they affect the system differently.

High-frequency noise may influence signal quality and control-system performance, while slow bias variations can contribute to accumulated errors in systems that integrate inertial measurements over time.

A time-history plot helps visualize these effects, although a numerical In-Run Bias Stability specification must still be interpreted according to the manufacturer’s defined calculation method.

Stability During Continuous Operation – Gladiator Technologies Accelerometers

In measurements presented by Gladiator Technologies for a ±30g closed-loop accelerometer, the acceleration output can be observed over a test period lasting several hours.

The measurement curves show relatively small fluctuations, together with gradual changes in the measured acceleration.

These results illustrate why evaluating sensor stability over time is important, rather than focusing exclusively on instantaneous noise levels.

According to the preliminary performance specifications for the GA200-030-100D accelerometer, Gladiator Technologies reports:

• In-Run Stability: 7 µg
• Short-Term Stability (5 hr): 60 µg

These are two distinct stability parameters presented by the manufacturer.

They should not be treated as interchangeable values, and neither should be interpreted independently as a guarantee of total accelerometer accuracy.

For inertial navigation applications, the significance of these measurements depends on the manufacturer’s test methodology, environmental conditions, and the way sensor errors propagate through the navigation system.

8. How to Compare Gyroscopes and Accelerometers Correctly

One of the most common mistakes when selecting an inertial sensor is focusing on a single performance specification.

For example, two gyroscopes may offer the same angular rate measurement range while exhibiting significant differences in noise, bias stability, bandwidth, and latency.

Similarly, two accelerometers with identical ±30g measurement ranges may be suitable for entirely different applications.

A meaningful engineering comparison must therefore consider several performance characteristics together.

1. Random Noise – ARW and VRW

For gyroscopes, a lower Angle Random Walk (ARW) value indicates a smaller contribution from white noise to accumulated angular uncertainty.

For accelerometers, a lower Velocity Random Walk (VRW) value indicates a smaller contribution from white acceleration noise to accumulated velocity uncertainty.

However, comparisons must be performed using consistent definitions and units.

Manufacturers may express gyroscope and accelerometer noise performance differently, so values should be converted to equivalent units before drawing conclusions.

2. Bias Instability

A lower Bias Instability value may indicate better long-term bias stability over the relevant observation timescales.

However, the reported values must be interpreted carefully.

Different manufacturers may use different measurement procedures, estimation methods, and environmental conditions.

For a meaningful comparison, engineers should verify that the underlying definitions and test methods are compatible.

3. In-Run Bias Stability

In-Run Bias Stability is particularly important in systems operating continuously for extended periods.

When comparing specifications, engineers should examine:

• The duration of the measurement.
• Temperature conditions and thermal stabilization.
• The statistical method used to calculate the reported value.
• Whether the specification represents typical performance, a measured result, or a guaranteed maximum.

A single stability figure without its measurement context may provide an incomplete picture of real-world performance.

4. Bandwidth and Measurement Latency

Low noise does not automatically guarantee good performance in a high-speed control system.

Electro-optical stabilization systems, for example, may require both high sensor bandwidth and low measurement latency.

Aggressive digital filtering can reduce measurement noise, but it may also introduce additional delay and affect the dynamic response of the control loop.

Therefore, noise performance must be evaluated alongside bandwidth, latency, and the overall dynamic requirements of the application.

5. Environmental Conditions

Allan Deviation measurements are generally performed under controlled conditions.

In real-world applications, however, inertial sensors may be exposed to vibration, mechanical shock, temperature variations, and other environmental disturbances.

Sensor performance under these conditions may differ significantly from its behavior during a stationary laboratory measurement.

For this reason, Allan Deviation analysis does not replace environmental qualification testing or system-level performance evaluation.

Practical Example – Selecting an Inertial Sensor for Different Applications

Consider two systems with very different performance requirements.

Application 1 – Electro-Optical Camera Stabilization

An electro-optical stabilization system must detect rapid angular motion and respond with minimal delay.

In this application, important sensor characteristics include:

• Low gyroscope noise.
• Sufficient measurement bandwidth.
• Low latency.
• Appropriate dynamic response for the stabilization control loop.

Although bias stability remains relevant, fast and accurate motion detection may be a particularly important design priority.

Application 2 – Inertial Navigation Without Continuous External Corrections

An inertial navigation system estimates motion by integrating gyroscope and accelerometer measurements over time.

When external position or attitude corrections are unavailable for extended periods, sensor errors can accumulate.

In this application, both random noise and bias stability become critical factors affecting navigation accuracy.

Long-term bias behavior may have a particularly significant influence on accumulated navigation errors.

Both applications rely on inertial sensors, but their performance priorities are not necessarily the same.

The best sensor is not always the one with the lowest noise figure. It is the sensor whose overall performance characteristics best match the requirements of the application.

SX3 Low Noise – Combining Low Noise with High Bias Stability

Gladiator Technologies has presented the following performance results for its next-generation SX3 Low Noise inertial sensor technology:

• Angle Random Walk (ARW): Below 0.02°/√hr
• Bias Instability: 0.25°/hr in the best reported measurement

These results highlight the company’s focus on reducing random noise while improving bias stability.

For engineers developing advanced navigation, stabilization, and motion-control systems, both characteristics are important.

However, they must be evaluated alongside other essential parameters, including sensor bandwidth, latency, environmental performance, and application-specific requirements.

It is also important to distinguish between technology-level measurement results and guaranteed product specifications.

The reported SX3 performance values should not be interpreted as a guarantee of identical performance across all operating conditions or product configurations.

Final sensor selection should always be based on the latest manufacturer specifications for the specific product and configuration under consideration.