daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Issue #95. Incorrect notation for process model.

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I used the Fundaental matrix F when I should have used the
system dynamics matrix A.
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Roger Labbe 2016-08-28 11:00:45 -07:00
parent d31721aa0f
commit cab54e11d6
1 changed files with 11 additions and 3 deletions
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11-Extended-Kalman-Filters.ipynb
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@ -321,13 +321,21 @@
"\n",
"For the linear filter we have these equations for the process and measurement models:\n",
"\n",
"$$\\begin{aligned}\\bar{\\mathbf x} &= \\mathbf{Ax} + \\mathbf{Bu} + w_x\\\\\n",
"$$\\begin{aligned}\\dot{\\mathbf x} &= \\mathbf{Ax} + w_x\\\\\n",
"\\mathbf z &= \\mathbf{Hx} + w_z\n",
"\\end{aligned}$$\n",
"\n",
"For the nonlinear model the linear expression $\\mathbf{Ax} + \\mathbf{Bu}$ is replaced by a nonlinear function $f(\\mathbf x, \\mathbf u)$, and the linear expression $\\mathbf{Hx}$ is replaced by a nonlinear function $h(\\mathbf x)$:\n",
"Where $\\mathbf A$ is the systems dynamic matrix. Using the state space methods covered in the **Kalman Filter Math** chapter these equations can be tranformed into \n",
"$$\\begin{aligned}\\bar{\\mathbf x} &= \\mathbf{Fx} \\\\\n",
"\\mathbf z &= \\mathbf{Hx}\n",
"\\end{aligned}$$\n",
"\n",
"$$\\begin{aligned}\\bar{\\mathbf x} &= f(\\mathbf x, \\mathbf u) + w_x\\\\\n",
"where $\\mathbf F$ is the *fundamental matrix*. The noise $w_x$ and $w_z$ terms are incorporated into the matrices $\\mathbf R$ and $\\mathbf Q$. This form of the equations allow us to compute the state at step $k$ given a measurement at step $k$ and the state estimate at step $k-1$. In earlier chapters I built your intuition and minimized the math by using problems describable with Newton's equations. We know how to design $\\mathbf F$ based on high school physics.\n",
"\n",
"\n",
"For the nonlinear model the linear expression $\\mathbf{Fx} + \\mathbf{Bu}$ is replaced by a nonlinear function $f(\\mathbf x, \\mathbf u)$, and the linear expression $\\mathbf{Hx}$ is replaced by a nonlinear function $h(\\mathbf x)$:\n",
"\n",
"$$\\begin{aligned}\\dot{\\mathbf x} &= f(\\mathbf x, \\mathbf u) + w_x\\\\\n",
"\\mathbf z &= h(\\mathbf x) + w_z\n",
"\\end{aligned}$$\n",
"\n",