$\begin{array}{l}\hfill \begin{array}{cc}{\mathbit{P}}_{k|k}& ={\mathbit{P}}_{k|k-1}-{\mathbit{K}}_{k}\mathbit{C}{\mathbit{P}}_{k|k-1}-{\mathbit{P}}_{k|k-1}{\mathbit{C}}^{T}{\mathbit{K}}_{k}^{T}+{\mathbit{K}}_{k}{\mathbit{S}}_{k}{\mathbit{K}}_{k}^{T}\\ ={\mathbit{P}}_{k|k-1}-{\mathbit{K}}_{k}\mathbit{C}{\mathbit{P}}_{k|k-1}-{\mathbit{P}}_{k|k-1}{\mathbit{C}}^{T}{\mathbit{K}}_{k}^{T}+{\mathbit{P}}_{k|k-1}{\mathbit{C}}^{T}{\mathbit{K}}_{k}^{T}\\ ={\mathbit{P}}_{k|k-1}-{\mathbit{K}}_{k}\mathbit{C}{\mathbit{P}}_{k|k-1}\end{array}\end{array}$

Example 6.19

There is a mobile robot that drives around on a plane and measures its position with a GPS sensor 10 times in a second (sampling time Ts = 0.1 s). The position measurement is disturbed with zero-mean Gaussian noise with variance 10 m2. The mobile robot drives with the speed of 1 m/s in the x direction and with the speed of 0 m/s in the y direction. The variance of Gaussian noise of the velocity is 0.1 m2/s2. At the beginning of observation, the mobile robot is situated in the origin x = [0, 0]T, but our initial estimate about the ...

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