Rate and Acceleration Bias State-Transition Models

The process models for the bias terms are based on the assumption that bias is made up of two components:

1. A constant bias offset (\(\vec{\omega}_{offset}^{B}\))
2. A randomly varying component superimposed on the offset (\(\vec{\omega}_{drift}^{B}\)) based on the measured bias-instability value of the sensor

For the rate-sensor, the bias model is

\[\vec{\omega}_{bias}^{B} = \vec{\omega}_{offset}^{B} + \vec{\omega}_{drift}^{B}\]

The drift model follows a random-walk process[1], i.e. the drift value wanders according to a Gaussian distribution.

\[\vec{\omega}_{drift,k}^{B} = \vec{\omega}_{drift,k-1}^{B} + \dot{\vec{\omega}}_{drift,k-1}^{B} \cdot dt\]

where

\[\dot{\vec{\omega}}_{drift,k-1}^{B} = N \begin{pmatrix} { 0,\sigma_{dd,\omega}^{2} } \end{pmatrix}\]

Note

The subscript dd stands for “drift-dot”.

Based on this model, the process variance for \(\vec{\omega}_{drift}^{B}\) at time, t, is given by:

\[\sigma_{d,\omega}^{2}(t) = \begin{bmatrix} { (\sigma_{dd,\omega} \cdot \sqrt{dt}) \cdot \sqrt{t} } \end{bmatrix} ^{2}\]

An empirical study related \(\sigma_{dd,\omega}\) to the BI and ARW values as follows:

\[\sigma_{dd,\omega} = {{2 \cdot \pi} \over {ln(2)}} \cdot {{{BI}^{2}} \over {ARW}}\]

To find the rate-bias process-noise covariance, set \(t = dt\) in the process-variance model (above), resulting in:

\[\Sigma_{\omega b} = \sigma_{d,\omega}^{2} (dt) \cdot I_3 = {\begin{pmatrix} { \sigma_{dd,\omega} \cdot dt } \end{pmatrix}}^{2} \cdot I_3\]

The accelerometer drift model mirrors this formulation and results in:

\[\Sigma_{ab} = \sigma_{d,a}^{2} (dt) \cdot I_3 = {\begin{pmatrix} { \sigma_{dd,a} \cdot dt } \end{pmatrix}}^{2} \cdot I_3\]

[1]	This is not a perfect assumption as the output of the model is unbounded while the actual process is not.