Integration

Sensor outputs usually arrive at a regular sampling rate. For example, the Oculus Rift gyroscope provides a measurement every $1$ms (yielding a $1000$Hz sampling rate). Let $\hat{\omega}[k]$ refer to the $k$th sample, which arrives at time $k \Delta t$.

The orientation $\theta(t)$ at time $t = k\Delta t$ can be estimated by integration as:

$$\hat{\theta}[k] = \theta(0) + \sum_{i=1}^k \hat{\omega}[i] \Delta t .$$ (9.8)

Each output $\hat{\omega}[i]$ causes a rotation of $\Delta\theta[i] = \hat{\omega}[i] \Delta t$. It is sometimes more convenient to write (9.8) in an incremental form, which indicates the update to $\hat{\theta}$ after each new sensor output arrives:

$$\hat{\theta}[k] = \hat{\omega}[k] \Delta t + \hat{\theta}[k-1].$$ (9.9)

For the first case, $\hat{\theta}[0] = \theta(0)$.

If $\omega(t)$ varies substantially between $\theta(k \Delta t)$ and $\theta((k+1)\Delta t)$, then it is helpful to know what $\hat{\omega}[k]$ corresponds to. It could be angular velocity at the start of the interval $\Delta t$, the end of the interval, or an average over the interval. If it is the start or end, then a trapezoidal approximation to the integral may yield less error over time [134].