From 7088156da0e3bd321b270699690cf319070b8138 Mon Sep 17 00:00:00 2001
From: Roger Labbe <rlabbejr@gmail.com>
Date: Sat, 30 Aug 2014 14:45:37 -0700
Subject: [PATCH] Fixed equations that didn't use x- for prediction.

In a few places the KF equations didn't use the x^- nomenclature
for the predicted value of x.
---
 Multidimensional_Kalman_Filters.ipynb | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/Multidimensional_Kalman_Filters.ipynb b/Multidimensional_Kalman_Filters.ipynb
index 694a4b2..ff215d6 100644
--- a/Multidimensional_Kalman_Filters.ipynb
+++ b/Multidimensional_Kalman_Filters.ipynb
@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:b7aa6703b660eebd93c7ab43089d54976dac12f5f99b40d113490e15932faa46"
+  "signature": "sha256:b02cf3016d12a2b3dbf885298a5c67f74ad1fe1b799eebfb3d479a95d4f2362e"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -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",