daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Fixed typo calling H F, and added some description of designing KF for air drag.

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Roger Labbe 2014-07-06 11:13:02 -07:00
parent d9e9d175a5
commit 0697383a4e
1 changed files with 50 additions and 3 deletions
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53
Designing_Kalman_Filters.ipynb
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@ -1,7 +1,7 @@
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@ -1232,7 +1232,7 @@
"\n",
"where\n",
"\n",
"$$\\mathbf{F} = \\begin{bmatrix}\n",
"$$\\mathbf{H} = \\begin{bmatrix}\n",
"1 & 0 & 0 & 0 & 0 \\\\\n",
"0 & 0 & 1 & 0 & 0\n",
"\\end{bmatrix}$$"
@ -1673,7 +1673,54 @@
"Notionally, the computation that $\\textbf{F}$ computes is\n",
"\n",
"$$x' = Fx$$\n",
"\n"
"\n",
"With no air drag, we had\n",
"\n",
"$$\n",
"\\mathbf{F} = \\begin{bmatrix}\n",
"1 & \\Delta t & 0 & 0 & 0 \\\\\n",
"0 & 1 & 0 & 0 & 0 \\\\\n",
"0 & 0 & 1 & \\Delta t & \\frac{1}{2}{\\Delta t}^2 \\\\\n",
"0 & 0 & 0 & 1 & \\Delta t \\\\\n",
"0 & 0 & 0 & 0 & 1\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"which corresponds to the Euler equations\n",
"\n",
"$$ \n",
"\\begin{aligned}\n",
"x &= x + v_x \\Delta t \\\\\n",
"v_x &= v_x \\\\\n",
"\\\\\n",
"y &= y + v_y \\Delta t + \\frac{a_y}{2} {\\Delta t}^2 \\\\\n",
"v_y &= v_y + a_y \\Delta t \\\\\n",
"a_y &= a_y\n",
"\\end{aligned}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"From the section above we know that our new Euler equations must be\n",
"\n",
"$$ \n",
"\\begin{aligned}\n",
"x &= x + v_x \\Delta t \\\\\n",
"v_x &= v_x \\\\\n",
"\\\\\n",
"y &= y + v_y \\Delta t + \\frac{a_y}{2} {\\Delta t}^2 \\\\\n",
"v_y &= v_y + a_y \\Delta t \\\\\n",
"a_y &= a_y\n",
"\\end{aligned}\n",
"$$"
]
},
{