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:

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