# Position State-Transition Model¶

The position process model is based on the following first-order model:

$\vec{r}_{k} = \vec{r}_{k-1} + \dot{\vec{r}}_{k-1} \cdot dt$

where $$\dot{\vec{r}}_{k-1}$$ is the estimated velocity state, $$\vec{v}_{k-1}$$. Substituting in the velocity term (including noise) results in:

$\vec{r}_{k} = \vec{r}_{k-1} + \vec{v}_{k-1} \cdot dt + \vec{w}_{r,k-1}$

$$\vec{w}_{r,k-1}$$ is the process noise associated with the position state-transition model, which is directly related to the velocity process noise:

$\begin{split}\vec{w}_{r,k-1} &= {\vec{w}_{v,k-1}} \cdot dt\\ {\hspace{5mm}} \\ &= {^{N}{R}_{k-1}^{B}} \cdot {\vec{a}_{noise}^{B}} \cdot {dt}^{2}\end{split}$

Like the previous process models, this expression is used to compute the elements of the process covariance matrix (Q) related to the position estimate:

$\Sigma_{r} = {\vec{w}_{r,k-1}} \cdot {\vec{w}_{r,k-1}}^{T}$

By making the assumption that all axes have the same noise characteristics ($${\sigma_{a}}^{2}$$), $$\Sigma_{r}$$ simplifies to:

$\Sigma_{r} = ({\sigma_{a} \cdot dt}^{2} )^{2} \cdot I_3$