Gyroscope/Gyrometer
Source: Phil’s Lab
Gyroscope model
$$ \begin{aligned} \mathbf{\omega}B &= \mathbf{\omega}\text{true} + \mathbf{\beta}(t) + \mathbf{\eta}(t) \end{aligned} $$
We need to transform these body rates to
Euler
rates!
$$
\begin{aligned}
\left[ \begin{array}{c} \dot{\phi} \ \dot{\theta} \end{array} \right]
&= \left[ \begin{array}{ccc}
1 & \sin\phi \tan\theta & \cos\phi\tan\theta\
0 & \cos\phi & -\sin\phi
\end{array} \right]
\left[ \begin{array}{c}p\q\r\end{array} \right]
\end{aligned}
$$
Problem: $\phi$ and $\theta$ need to be known!
–> integrate?
$$\hat{\phi} = \int_0^T \dot{\phi}(t) ~dt ~?$$
Direct integration not possible due to the presence of time-varying bias and noise; integration leads to gyroscope drift (we integrate the noise/error terms so we drift away from the true value)! $$\hat{\phi} = \int_0^T \dot{\phi}(t) + \beta_\phi(t) + \eta_\phi(t) ~dt$$
Conclusions
- Using only the gyroscope provides a good estimate over short periods of time (due to integration of bias terms)
- If using for a short period of time, the drift won’t affect us too much (not had time to accumulate)