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.. sectionauthor:: Joseph S Motyka <jmotyka at aceinna.com>
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:
\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}
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