daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Fixed equations that didn't use x- for prediction.

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In a few places the KF equations didn't use the x^- nomenclature
for the predicted value of x.
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Roger Labbe 2014-08-30 14:45:37 -07:00
parent 428790378a
commit 7088156da0
1 changed files with 4 additions and 4 deletions
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8
Multidimensional_Kalman_Filters.ipynb
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@ -1,7 +1,7 @@
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@ -917,7 +917,7 @@
"\\mathbf{P } &:= \\mathbf{FP{F}}^T + \\mathbf{Q}\\;\\;\\;&(2) \\\\\n",
"\\\\\n",
"\\text{Update Step}\\\\\n",
"\\textbf{y} &:= \\mathbf{z} - \\mathbf{H x} \\;\\;\\;&(3)\\\\\n",
"\\textbf{y} &:= \\mathbf{z} - \\mathbf{H x^-} \\;\\;\\;&(3)\\\\\n",
"\\mathbf{K}&:= \\mathbf{PH}^T (\\mathbf{HPH}^T + \\mathbf{R})^{-1}\\;\\;\\;&(4) \\\\\n",
"\\mathbf{x}&:=\\mathbf{x}^- +\\mathbf{K\\textbf{y}} \\;\\;\\;&(5)\\\\\n",
"\\mathbf{P}&:= (\\mathbf{I}-\\mathbf{KH})\\mathbf{P}\\;\\;\\;&(6)\n",
@ -1145,9 +1145,9 @@
"\n",
"The Kalman filter equation that performs this step is:\n",
"\n",
"$$\\textbf{y} := \\mathbf{z} - \\mathbf{H x} \\;\\;\\;(3)$$\n",
"$$\\textbf{y} := \\mathbf{z} - \\mathbf{H x^-} \\;\\;\\;(3)$$\n",
"\n",
"where $\\textbf{y}$ is the residual, $\\textbf{z}$ is the measurement, and $\\textbf{H}$ is the measurement function. It is just a matrix that we multipy the state into to convert it into a measurement.\n",
"where $\\textbf{y}$ is the residual, $\\mathbf{x^-}$ is the predicted value for $\\mathbf{x}$, $\\textbf{z}$ is the measurement, and $\\textbf{H}$ is the measurement function. It is just a matrix that we multipy the state into to convert it into a measurement.\n",
"\n",
"For our dog tracking problem we have a sensor that measures position, but no sensor that measures velocity. So for a given state $\\mathbf{x}=\\begin{bmatrix}x & \\dot{x}\\end{bmatrix}^T$ we will want to multiply the position $x$ by 1 to get the corresponding measurement of the position, and multiply the velocity $\\dot{x}$ by 0 to get the corresponding measurement of velocity (of which there is none).\n",
"\n",