# 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:

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

where

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

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:

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