From 0697383a4e91be1343637a221fe7de2952cbc0e5 Mon Sep 17 00:00:00 2001
From: Roger Labbe <rlabbejr@gmail.com>
Date: Sun, 6 Jul 2014 11:13:02 -0700
Subject: [PATCH] Fixed typo calling H F, and added some description of
 designing KF for air drag.

---
 Designing_Kalman_Filters.ipynb | 53 ++++++++++++++++++++++++++++++++--
 1 file changed, 50 insertions(+), 3 deletions(-)

diff --git a/Designing_Kalman_Filters.ipynb b/Designing_Kalman_Filters.ipynb
index bcf274e..aa74cfe 100644
--- a/Designing_Kalman_Filters.ipynb
+++ b/Designing_Kalman_Filters.ipynb
@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:dc7abfb3c31bceb18f38ed34d63ab344a0e937dfb0c1b339c16885b1cc985f8a"
+  "signature": "sha256:74efaa83634cbf732aa372565ab776312f5d3079a4732d7a7d797b8996bd3469"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -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",
+      "$$"
      ]
     },
     {