Process Noise Covariance Matrix
The process covariance acts as a weighting matrix for the system process. It relates the covariance
between the \(i^{th}\) and \(j^{th}\) element of each process-noise vector. It is defined
as:
\[\Sigma_{ij} = cov{ \begin{pmatrix} {
\vec{x}_{i},\vec{x}_{j}
} \end{pmatrix}
}
= E{ \begin{bmatrix} {
{ \begin{pmatrix} { \vec{x}_{i}-\mu_{i} } \end{pmatrix} }
\cdot
{ \begin{pmatrix} { \vec{x}_{j}-\mu_{j} } \end{pmatrix} }
} \end{bmatrix}
}\]
A Kalman Filter can be viewed the combination of Gaussian distributions to form state estimates.
\(Q\) provides a measure of the width of the Gaussian distribution related to each noise state.
The wider the distribution, the more uncertainty exists in the process model. This leads to a
state-update that affects the state more than if the model had a tighter distribution, which results
in an update having less influence on the particular state.
Based on the state process-noise vectors, \(\vec{w}_{k}\) (found in previous sections), the
Process Noise Covariance Matrix is:
\[Q_{k} = {
\begin{bmatrix} {
\begin{array}{ccccc}
{\Sigma_{r}} &
{0_{3}} &
{0_{3 \times 4}} &
{0_{3}} &
{0_{3}}
\cr
{0_{3}} &
{\Sigma_{v}} &
{0_{3 \times 4}} &
{0_{3}} &
{0_{3}}
\cr
{0_{4 \times 3}} &
{0_{4 \times 3}} &
{\Sigma_{q}} &
{0_{4 \times 3}} &
{0_{4 \times 3}}
\cr
{0_{3}} &
{0_{3}} &
{0_{3 \times 4}} &
{\Sigma_{\omega b}} &
{0_{3}}
\cr
{0_{3}} &
{0_{3}} &
{0_{3 \times 4}} &
{0_{3}} &
{\Sigma_{ab}}
\end{array}
} \end{bmatrix}
}\]
The individual process covariance are repeated here:
\[\Sigma_{r} = {\begin{pmatrix} { \sigma_{a} \cdot {dt}^{2} } \end{pmatrix}}^{2} \cdot I_3\]
\[\Sigma_{v} = {\begin{pmatrix} { \sigma_{a} \cdot dt } \end{pmatrix}}^{2} \cdot I_3\]
\[\Sigma_{q} = { { \begin{pmatrix} {
{\sigma_{\omega} \cdot dt } \over {2}
} \end{pmatrix} }^{2}
}
\cdot
{
\begin{bmatrix} {
\begin{array}{cccc}
{1 - q_0^2} &
{-{q_0 \cdot q_1}} &
{-{q_0 \cdot q_2}} &
{-{q_0 \cdot q_3}}
\cr
{-{q_0 \cdot q_1}} &
{1 - q_1^2} &
{-{q_1 \cdot q_2}} &
{-{q_1 \cdot q_3}}
\cr
{-{q_0 \cdot q_2}} &
{-{q_1 \cdot q_2}} &
{1 - q_2^2} &
{-{q_2 \cdot q_3}}
\cr
{-{q_0 \cdot q_3}} &
{-{q_1 \cdot q_3}} &
{-{q_2 \cdot q_3}} &
{1 - q_3^2}
\end{array}
} \end{bmatrix}
}\]
\[\Sigma_{\omega b} = {\begin{pmatrix} { \sigma_{dd,\omega} \cdot dt } \end{pmatrix}}^{2} \cdot I_3\]
\[\Sigma_{ab} = {\begin{pmatrix} { \sigma_{dd,a} \cdot dt } \end{pmatrix}}^{2} \cdot I_3\]