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IMU Noise Model

In our visual-inertial fusion algorithm, we model, as traditionally done, the stochastic part of the IMU error as including three components:

  • Turn-on bias
  • Bias random walk
  • White noise

The noise along any of the axis of the inertial sensor is described as the sum of those contributions:

xksampled=xktrue+nk0turn-on bias+nkbias random walk+nkwhite noise x^{\text{sampled}}_k = x^{\text{true}}_k + n^{\text{turn-on bias}}_{k0} + n^{\text{bias random walk}}_k + n^{\text{white noise}}_k

Where:

  • xktruex^{\text{true}}_k is the real value of the quantity measured projected on the sensor sensitive axis.
  • xksampledx^{\text{sampled}}_k is the sampled value seen as a random variable.
  • nk0turn-on biasn^{\text{turn-on bias}}_{k0} is a random variable draw from a Gaussian when the device is turned-on (denoted here as step k0k_0).
  • nkwhite noisen^{\text{white noise}}_{k} is the time-independent noise components and draw from a 0-centered Gaussian at each step kk.
    • The covariance of the Gaussian is usually characterized by the continuous noise σc\sigma_c strength, which needs to be multiplied by the bandwidth (BW)h in Hz to get the distribution of the sample noise: σc2⋅BW\sigma^2_c \cdot BW.
    • σc\sigma_c is sometimes called Angle Random walk in rad/s/Hzrad/s/\sqrt{Hz} or rad/srad/\sqrt{s} for the gyrometer and the Velocity Random Walk in m/s2/Hzm/s^2/\sqrt{Hz} or m/s/(s)m/s/\sqrt(s) for the accelerometer.
  • nkbias random walkn^{\text{bias random walk}}_k is drawn from a random walk process.
Provenance of noise parameters

The noise parameters listed below are inherited from Project Aria Gen1. They were derived from Allan Variance analysis performed on Gen1 hardware, using data collected over a 24-hour period in a temperature-stable environment. Because Gen2 uses the same IMU components as Gen1, the same parameters apply. No independent Allan Variance analysis has been performed on Gen2 hardware to re-derive these values.

Table 1: White noise and bias instability parameters of Aria IMU sensors (1 gee = 9.81m/s^2)

accel-leftaccel-rightgyro-leftgyro-right
white noise (σc\sigma_c)0.9×10−4 gees/Hz0.9\times 10^{-4}\ \text{gees}/\sqrt{Hz}0.8×10−4 gees/Hz0.8\times 10^{-4}\ \text{gees}/\sqrt{Hz}5×10−3 dps/Hz5\times 10^{-3}\ \text{dps}/\sqrt{Hz}1×10−2 dps/Hz1\times10^{-2}\ \text{dps}/\sqrt{Hz}
bias instability2.8×10−5 gees2.8\times 10^{-5}\ \text{gees}3.5×10−5 gees3.5\times 10^{-5}\ \text{gees}1×10−3 dps1\times 10^{-3}\ \text{dps}1.3×10−3 dps1.3\times10^{-3}\ \text{dps}

With our sensor configuration for bandwidth, this leads to the following values for the sample noise:

Table 2: Sample noise of Aria IMU sensors

accel-left (BW: 343Hz)accel-right (BW: 353Hz)gyro-left (BW:300Hz)gyro-right (BW:116Hz)
σkwhite noise\sigma^{\text{white noise}}_k16×10−3 m/s216\times 10^{-3}\ m/s^215×10−3 m/s215\times 10^{-3}\ m/s^21.5×10−3 rad/s1.5\times10^{-3}\ \text{rad/s}1.8×10−3 rad/s1.8\times10^{-3}\ \text{rad/s}

The data duration used for Allan Variance was not enough to capture the bias random walk confidently. This is because real MEMS sensors are not that well modelled by this stochastic model. To address this, we tune the parameter of the bias random walk used for sensor-fusion. We start from the bias instability measured on the Allan Variance of the sensor (the floor of the curve) and inflate it by a tuning factor.