Velocity State-Transition ModelΒΆ
The velocity propagation equation is based on the following first-order model:
\(\dot{\vec{v}}_{k-1}\) is an estimate of system acceleration (linear-acceleration corrected for gravity) and is formed from the accelerometer signal with estimated accelerometer-bias and gravity removed.
Substituting this expression (along with the noise term) into the velocity propagation equation, and explicitly stating the frames in which the readings are made, leads to:
where
The velocity process-noise vector resulting from accelerometer noise is:
leading to the final formulation for the velocity state-transition model:
The velocity process noise vector is used to compute the elements of the process covariance matrix (\(Q\)) related to the velocity estimate, as follows:
By making the assumption that all axes have the same noise characteristics (\({\sigma_{a}}^{2}\)) and manipulating the expression, the result can be simplified to the following: