# The Kalman Filter Equations

1. Prediction:

$\begin{array}{ccc}\hfill {\stackrel{‾}{\mathbf{x}}}_{t}^{-}& \hfill =\hfill & {D}_{t}{\stackrel{‾}{\mathbf{x}}}_{t-\Delta t}^{+}\hfill \\ \hfill {\Sigma }_{t}^{-}& \hfill =\hfill & {\Sigma }_{\mathrm{D}t}+{D}_{t}{\Sigma }_{t-\Delta t}^{+}{D}_{t}^{T}\hfill \end{array}$
2. Correction:

$\begin{array}{ccc}\hfill {K}_{t}& \hfill =\hfill & {\Sigma }_{t}^{-}{M}_{t}^{T}{\left({M}_{t}{\Sigma }_{t}^{-}{M}_{t}^{T}+{\Sigma }_{\mathrm{M}t}\right)}^{-1}\hfill \\ \hfill {\stackrel{‾}{\mathbf{x}}}_{t}^{+}& \hfill =\hfill & {\stackrel{‾}{\mathbf{x}}}_{t}^{-}+{K}_{t}\left({\mathbf{y}}_{t}-{M}_{t}{\stackrel{‾}{\mathbf{x}}}_{t}^{-}\right)\hfill \\ \hfill {\Sigma }_{t}^{+}& \hfill =\hfill & \left(I-{K}_{t}{M}_{t}\right){\Sigma }_{t}^{-}\hfill \end{array}$

### Note

• D t = D t ( Δ t )

• Typically, D t = D ( Δ t ) and M t = M .

• We do not need “full” measurements; just manipulate M accordingly.