#### Integration

Sensor outputs usually arrive at a regular sampling rate. For example, the Oculus Rift gyroscope provides a measurement every ms (yielding a Hz sampling rate). Let refer to the th sample, which arrives at time .

The orientation at time can be estimated by integration as:

 (9.8)

Each output causes a rotation of . It is sometimes more convenient to write (9.8) in an incremental form, which indicates the update to after each new sensor output arrives:

 (9.9)

For the first case, .

If varies substantially between and , then it is helpful to know what corresponds to. It could be angular velocity at the start of the interval , 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 [133].

Steven M LaValle 2016-12-31