1. What is Vibration Rectification Error (VRE)?
Vibration Rectification Error (VRE): is a systematic bias error in the output of an inertial sensor while the device is exposed to mechanical vibration. Unlike random noise, VRE occurs as a quasi-stationary offset that is a function of the vibration amplitude, frequency content as well as the non-linear properties of the sensor. It is often described as “the apparent change in gyro/accelerometer steady state bias with changing level of vibration”.
2. Physical Origins
| Source | Mechanism | Typical Impact |
|---|---|---|
| Scale-factor non-linearity | Transfer function of sensor is not perfectly linear and vibration alters this non-linear relationship and causes DC offset. | Dominant contributor of VRE of MEMS accelerometers and gyros. |
| Clipping & saturation | When vibration moves the sensor to the point of being out of measure the output is clipped and a bias is created which remains as the vibration subsides. | Offers proportional increase of bias to vibration amplitude. |
| Self-heating & integration errors | High frequency motion can increase internal temperature or numerical integration drift, both of which are manifested as a bias. | Phenomenon seen in low sample rate systems; can be minimized by increasing sampling rates. |
| Mechanical resonances | Vibration frequencies that are close to the resonant modes of the sensor amplify the motion occurring within the sensor, so that non linear effects are exaggerated. | Produces bias which changes with time (beat-frequency phenomenon). |
3. Why VRE is Important in High Dynamic Environments
- Cumulative navigation error – Since VRE acts like a bias, it accumulates with time which results in large position drift in dead reckoning navigation ins solutions.
- Platform stabilisation – In applications such as drill mast alignment or UAV attitude control a VRE induced bias is misinterpreted as a change in inclination which over-compensates.
- Bias-estimation limits – With traditional Kalman-filter bias-tracking techniques it is possible that they would not converge in the case that VRE is dominant (i.e. the error is not visible from the sensor output alone).
4. Quantifying VRE
VRE is usually represented by the manufacturer as a bias per unit of vibration, for example, °/h · g⁻² for gyroscopes or m/s² · g⁻¹ for accelerometers. A typical spec may read “VRE ≤ 0.5 °/h · g⁻²” where VRE is the value of vibration random error which means that it can introduce up to 0.5 °/h of bias when the random vibration is 1g.
5. Mitigation Methods
| Strategy | Principle | Practical Tips |
|---|---|---|
| Mechanical isolation | Minimize vibration amplitude that is transferred to the sensor. | Use Mount elastomeric (rubber), vacuum (air) housings, shock absorbers. |
| High-rate sampling | Capture vibration content that is above Nyquist which limits the alias induced bias. | Sample ≥50kHz with >100 Hz vibration components for environments. |
| Non-linearity calibration | Polynomial model of the sensor response is modeled and correction term is applied. | Perform multi tone vibration tests to fit a calibration matrix. |
| Real-time VRE compensation | Subtract induced bias from gyro output estimated from accelerometer based vibration measurements. | introduce the algorithm of EP 1930691 (coefficients stored in firmware). |
| Algorithmic filtering | Augment Kalman filter with a vibration – bias state that is excited by measured vibration RMS. | Requires a dedicated vibration sensor or use high frequency band of accelerometer. |
6. Test & Validation
- Controlled vibration table – Sinusoidal vibrations of different frequencies (e.g. 50 Hz, 100 Hz, 200 Hz) and amplitudes (0.1-2 g) can be applied.
- Record bias shift – Measure static output before, during and after vibration in order to extract VRE.
- Fit non – linear model – use least – squared to get a polynomial coefficients describing the rectification effect
- Cross-Validate – Validating the model on another vibration profile (random broadband) to ensure the robustness of the model.
7. IMU Checking List for IMU Manufacturers
- Giving VRE specification (bias per g) for both accelerometer and gyroscope axes.
- Record the scale-factors of non-linearities of the sensor and the sensor’s clipping limits.
- Provide a compensated calibrated table/firmware.
- Suggest minimum sampling rate for target vibration environment
- Integration manual should include mechanical isolation guidelines.
Bottom line: Vibration Rectification Error is a deterministic bias which is caused by the interaction between sensor non- non-larity and external vibration. In high dynamic applications it can be the major error contributor to the budget, therefore its reliable quantification, good calibration and intelligent mitigation of the mechanical/algorithmic errors are key to reliable IMU performance.