# 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.