A tunnel-driving autonomous vehicle, a surveying system moving over rough ground, and a stabilized payload section performing sharp turns all have one thing in common: an IMU that worked well on a test bench may not work as well after it has been installed. Such an IMU is subject to a wide variety of influences such as vibration, acceleration, thermal gradients, structural flexibility, and timing errors. Most of these have the potential to greatly decrease the quality of the data that is measured by the IMU.
Analyzing an Inertial Measurement Unit sensor for errors like bias-instability does not boil down to a comparison of figures. Also, one must analyze how errors will propagate in one’s application. One must also identify the conditions under which the application operates and in which errors will occur. Does the sensor’s calibration still make sense for the application?
At Andelu we view our product range of MEMS IMUs, gyroscope, accelerometers and complete navigation systems from a component-to-system perspective. Thus we apply a very practical evaluation principle: Choose the best sensing architecture according to the error budget of your mission and test it in a realistic installed configuration.
Why Static Accuracy Does Not Predict Dynamic Performance
A measurement model reveals the missing variables
A simplified gyroscope model is:
Here, S represents scale-factor errors, M represents misalignment and cross-axis coupling, b(T) is temperature-dependent bias, Gf represents acceleration sensitivity, and n represents noise. This is a first-order engineering model; additional nonlinear and history-dependent terms may be necessary.
A stationary test primarily exposes bias and noise under its particular conditions. Turning, braking, vibration, and temperature changes excite other terms. Consequently, a low static bias-instability value cannot establish dynamic accuracy.
Our IMU engineering material distinguishes individual-device calibration from system calibration. The latter must establish the relative orientation, scale factors, and biases of all sensing axes within a common coordinate frame. Packaging, mechanical support, electrical design, and temperature management influence whether those relationships remain stable during operation.
Translate sensor errors into application errors
For a constant accelerometer bias b_a, simplified unaided error growth is:
Gyroscope bias creates attitude error, which misprojects gravity into horizontal acceleration. Under a short-duration, small-angle approximation, with constant horizontal gyro bias and no aiding, this contribution becomes:
These relationships exclude several navigation effects and are not complete performance predictions. Their value is diagnostic: outage duration strongly affects the required bias performance. A sensor suitable for rapid attitude control may still be unsuitable for prolonged unaided positioning.
Parameters That Determine Dynamic Measurement Quality
Range, bandwidth, and delay must be evaluated together
For full-scale testing, it is necessary to have room for credible operating peaks and for all transients. In clipping information is removed before recovery by navigation. The ability of a system to survive shock is a separate property from that system’s ability to function after having been impacted. The latter does not guarantee any amount of accurate measurement while the impact is occurring.
The term bandwidth describes the frequency range which is transported through the measurement chain. A too low bandwidth of the signal condition system attenuates useful motion and introduces a phase shift. A too high bandwidth on the other hand allows superfluous noise and even vibrations to pass. The output data rate does not necessarily have to describe both cases.
The integration guidance given here for angular rate and specific force differs from that for angle and velocity. Amount, integration time and time-stamp for such quantities have to be specified by the buyer for the modules to be delivered. For example, the output rate for a quantity such as the angular rate does not specify the internal sampling rate, the amount of filtering etc. that is used for the computation of the quantity to be output as an increment.
For stabilization, check the phase and group delay as well as the amplitude response. For navigation, check if all filtering and transport delay are implemented in the estimator.
Noise and bias require different evidence
The Allan deviation is a measure to reveal stochastic processes averaged over different time spans. The application of the Allan deviation to the noise analysis and modeling of inertial sensors is described by El-Sheimy, Hou, and Niu. A stationary Allan-deviation curve is only a characterization of the test setup and environment and does not convey any information on vibration sensitivity or on thermal transients.
Please specify the duration of data acquisition, sample rate, filtering, temperature and other relevant parameters. In-run bias instability should be clearly distinguished from turn-on repeatability and residual thermal bias.
In addition to the two above issues, scale-factor error and non-orthogonality must be addressed. Large rotations in particular multiply scale-factor errors. Moreover, cross-talk between the sensor axes can transfer strong excitation into apparently ‘quiet’ axes. Thus, in addition to checking for rotation in both directions, it is essential to check for a number of different input magnitudes.
Vibration, Mounting, and Temperature: The Coupled Failure Mechanisms
Vibration can create a false mean measurement
Vibration rectification refers to nonlinear response of a system that converts input oscillations into a constant amplitude output with nonzero mean. For capacitive MEMS accelerometers the amount of rectification can depend on the excitation frequency and on the dynamics of the device.
A broadband noise can be reduced with filters but a rectified bias occupies the low frequency part of a measurement and cannot be removed by downstream filtering because it represents wanted motion.
For useful vibration evaluation measurements have to be taken before the test, during the test and after the test. Changes of mean value, noise and saturation as well as changes of the calibration have to be measured. For the evaluation, measurements should cover as much of the platform’s vibration spectrum as possible. A vibration level does not necessarily indicate a harmful resonance.
The production mounting arrangement belongs in the test
Our calibration guidance describes potential differences of assembled performance outside of our laboratory and identifies soldering stress, housing attachment, mounting torque and installation-angle deviations as potential reasons for such differences. Consequently, also the production bracket, fasteners, cable arrangement and enclosure have to be tested.
ISO 5348:2021 defines the mechanical mounting of accelerometers and describes the influence of the mounting on the measurement accuracy. Note that this standard is not intended for the complete qualification of an IMU.
The mechanism isolation also creates a tradeoff. It can reduce the amount of vibration that is transmitted but allow for other types of movement such as relative motion or change in alignment. This transfer response needs to be measured after the mechanism has been installed to assess if it has improved the user’s ability to navigate or control the system.
Temperature compensation must survive thermal history
This thermal-calibration material describes multi-temperature bias and scale-factor characterization and discusses topics such as polynomial compensation, use of lookup tables, and heating-versus-cooling hysteresis. It also discusses thermal gradients and package stress and explains why equilibrium calibration is not suitable for fast changes in temperature.
A practical evaluation should start with a warm-up, then perform a temperature ramp, stabilized dwells and repeated heating and cooling cycles. The residual error of the model after compensation should be recorded rather than only the fitted calibration curve.
It’s important to distinguish between a unit’s environmental survival and its operational accuracy. The environmental-testing guidance provided here delineates between simple storage testing, normal use testing, and rigorous ‘cycling’ testing – all of which have conditions determined for specific applications and product specifications. Even a unit that passes environmental testing can exceed the measurement error permissible for a given mission.
What Common Solutions Solve—and What They Leave Unresolved
More filtering cannot recover missing information
Noise filtering, even when focused to specific noise components, generally introduces some delay. Useful signals also get reduced, and once data has been clipped or even aliased due to too high frequency exceeding the sampled data band, it cannot be restored by any amount of filtering.
In addition to the final output rate, also specify the internal sampling rate and anti-alias filter settings. Compare suitable measurement devices with identical filter settings. It would be unfair to mistakenly purchase lower noise with unacceptable delay.
Better calibration cannot remove every changing error
Calibration addresses repeatable relations. Its reliability decreases as the mechanical or thermal deployed state deviates from the calibration state.
Our material on IMU describes how the choice of sensor, packaging, thermal design and calibration all impact the performance of a finished module. So the key thing to ask a supplier for is the residual performance after compensation has been applied and to verify this after final assembly of the module into your system.
Fusion works only while its assumptions remain valid
GNSS, wheel-speed measurements, and vehicle-motion constraints can limit inertial drift. Our integrated-navigation material describes combining complementary measurements through state estimation rather than treating external aiding as a universal correction.
Slipping wheels violate the odometer assumption. Vehicles that deviate laterally from the estimator’s predicted path violate vehicle constraints. Poor reception of GNSS signals due to blockage or reflected signals can cause poor quality signals. These cases must be tested in transitions from non-aiding to aiding and back to non-aiding. The estimator must be able to reject such signals, downweight them and eventually bring the signal back to normal use.
The most useful acceptance criterion for this application would be to check for bounded error in application during the time the system is degraded and then return to normal when reliable information is available again.
A Mission-Based Evaluation Protocol
1. Define acceptance limits before selecting hardware
Document maximum rates and accelerations, vibration spectra, temperature histories, outage duration, maximum allowed latency, and required accuracy of attitude, velocity, position, etc.
The stabilization performance for a payload is best evaluated using the system’s phase response and short-term angular noise. Tunnel navigation is best determined by the system’s outage drift and available vehicle-aiding. Vibration measurement is best determined by the system’s spectral fidelity and its mounting response. Such applications should not have an undifferentiated “high-accuracy” requirement.
2. Use complementary tests
| Evaluation | Evidence to collect | Engineering decision |
| Stationary records | Noise, bias repeatability, Allan deviation | Establish stochastic models |
| Multi-position and rate-table tests | Scale factor, misalignment, asymmetry | Verify deterministic compensation |
| Installed vibration tests | Mean shift, spectral response, clipping | Assess platform compatibility |
| Thermal ramps and cycles | Warm-up, hysteresis, compensated residuals | Validate thermal robustness |
| Representative trajectories | State errors, aiding transitions, recovery | Confirm mission performance |
Our environmental and calibration material supports the characterization with controlled tests and verification by application specific tests. No single test can determine the full dynamic suitability of a system.
3. Control the reference and configuration
Note down the firmware version, serial number, filter settings, coordinate system, time stamps, temperature, power supply and mounting. Also take into account the uncertainty of the reference system.
Time alignment test cases should be written to test for this apparent sensor error during for example a rapid maneuver. Also, test cases should be written for the spatial separation between the IMU and the reference point and the corresponding lever-arm compensation.
4. Compare architecture-level tradeoffs
Andelu sensing components and engineering evaluation modules are offered to potential customers. Choices between sensors and modules are made based on whether a raw inertial solution, attitude solution or an aided full navigation solution is required for a particular application.
Specific evidence for range, bandwidth, residual calibration error, environmental use, interfaces and aiding capability for the selected model should be requested. Portfolio descriptions are not substitute for guaranteed product specifications.
Select for Demonstrated Mission Performance
A reliable NASE for a particular application is a unit that has errors that can be confined to an acceptable value for that application, even when subject to vibration, temperature variations, loss of reference and subsequent recovery.
Begin with an error budget, differentiate between det. and stoch. behavior, verify your complete measurement chain. The best purchasing decision is based on traceable config data plus sufficient testing – not on one single favorable value.
Our component-to-system approach at Andelu allows us to discuss these requirements on the basis of our component IMUs, inertial sensors and complete navigation systems.
References
[1] Andelu. IMU. Technical material, Sections 1.3.3–1.3.4: system integration, packaging, and calibration.
[2] Andelu. Inertial Navigation Error and Calibration. Technical material, Sections 2.3.1–2.3.9: error models, propagation, thermal compensation, installation effects and calibration.
[3] Andelu. System Integration and Interface Technologies. Technical material, Sections 2.6.3–2.6.4 and related synchronization guidance.
[4] El-Sheimy N., Hou H., Niu X., “Analysis and Modeling of Inertial Sensors Using Allan Variance”, IEEE Transactions on Instrumentation and Measurement, vol. 57, no. 1, pp. 140–149, 2008. https://doi.org/10.1109/TIM.2007.908635
[5] “Analysis of the Frequency-Dependent Vibration Rectification Error in Area-Variation-Based Capacitive MEMS Accelerometers” Micromachines, vol. 15, no. 1, 2024, Art. No. 65. https://doi.org/10.3390/mi15010065
[6] ISO 5348:2021. Mechanical vibration and shock—Mechanical mounting of accelerometers.
[7] Andelu. Environmental Reliability and Testing. Technical material, Sections 2.7.1–2.7.2 and environmental-test guidance.
[8] Andelu. Integrated Navigation Technology. Technical material, Section 2.5.