Velocity State-Transition Model

The velocity propagation equation is based on the following first-order model:

\[\vec{v}_{k} = \vec{v}_{k-1} + \dot{\vec{v}}_{k-1} \cdot dt\]

\(\dot{\vec{v}}_{k-1}\) 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.

\[\vec{a}_{motion,k-1} = \vec{a}_{meas,k-1} - \vec{a}_{bias,k-1} - \vec{a}_{grav}\]

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:

\[\vec{v}_{k}^N = \vec{v}_{k-1}^N + \begin{pmatrix} { \vec{a}_{motion,k-1}^N - {^{N}{R}_{k-1}^{B}} \cdot \vec{a}_{noise}^{B} } \end{pmatrix} \cdot {dt}\]


\[\vec{a}_{motion,k-1}^N = {^{N}{R}_{k-1}^{B}} \cdot \begin{pmatrix} { \vec{a}_{meas,k-1}^B - \hat{a}_{bias,k-1}^B } \end{pmatrix} - \vec{a}_{grav}^{N}\]

The velocity process-noise vector resulting from accelerometer noise is:

\[\vec{w}_{v,k-1}^{N} = -{^{N}{R}_{k-1}^{B}} \cdot \vec{a}_{noise}^{B} \cdot {dt}\]

leading to the final formulation for the velocity state-transition model:

\[\vec{v}_{k}^N = \vec{v}_{k-1}^N + \vec{a}_{motion,k-1}^N \cdot dt + \vec{w}_{v,k-1}^{N}\]

The velocity process noise vector is used to compute the elements of the process covariance matrix (\(Q\)) related to the velocity estimate, as follows:

\[\Sigma_{v} = {\vec{w}_{v,k-1}} \cdot {\vec{w}_{v,k-1}}^{T}\]

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:

\[\Sigma_{v} = { \begin{pmatrix} { \sigma_{a} \cdot dt } \end{pmatrix} }^{2} \cdot I_3\]