From 8d5215ffc863b0a84938773a33ed2e0e8a956caf Mon Sep 17 00:00:00 2001
From: Peter Schneider <peteryschneider@gmail.com>
Date: Sat, 29 Jul 2017 01:27:21 -0700
Subject: [PATCH] fixed derivation of EKF propagation equations

---
 11-Extended-Kalman-Filters.ipynb | 13 +++----------
 1 file changed, 3 insertions(+), 10 deletions(-)

diff --git a/11-Extended-Kalman-Filters.ipynb b/11-Extended-Kalman-Filters.ipynb
index 77b0ed5..cf46013 100644
--- a/11-Extended-Kalman-Filters.ipynb
+++ b/11-Extended-Kalman-Filters.ipynb
@@ -382,29 +382,22 @@
     "\n",
     "$$\\begin{aligned}m &= f'(x=1.5) \\\\&= 2(1.5) - 2 \\\\&= 1\\end{aligned}$$ \n",
     "\n",
-    "Linearizing systems of differential equations is similar. We linearize $f(\\mathbf x, \\mathbf u)$, and $h(\\mathbf x)$ by taking the partial derivatives of each to evaluate $\\mathbf A$ and $\\mathbf H$ at the point $\\mathbf x_t$ and $\\mathbf u_t$. We call the partial derivative of a matrix the [*Jacobian*](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant). This gives us the the system dynamics matrix and measurement model matrix:\n",
+    "Linearizing systems of differential equations is similar. We linearize $f(\\mathbf x, \\mathbf u)$, and $h(\\mathbf x)$ by taking the partial derivatives of each to evaluate $\\mathbf F$ and $\\mathbf H$ at the point $\\mathbf x_t$ and $\\mathbf u_t$. We call the partial derivative of a matrix the [*Jacobian*](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant). This gives us the the discrete state transition matrix and measurement model matrix:\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
-    "\\mathbf A \n",
+    "\\mathbf F \n",
     "&= {\\frac{\\partial{f(\\mathbf x_t, \\mathbf u_t)}}{\\partial{\\mathbf x}}}\\biggr|_{{\\mathbf x_t},{\\mathbf u_t}} \\\\\n",
     "\\mathbf H &= \\frac{\\partial{h(\\bar{\\mathbf x}_t)}}{\\partial{\\bar{\\mathbf x}}}\\biggr|_{\\bar{\\mathbf x}_t} \n",
     "\\end{aligned}\n",
     "$$\n",
     "\n",
-    "Finally, we find the discrete state transition matrix $\\mathbf F$ by using the Taylor series expansion of $e^{\\mathbf A \\Delta t}$:\n",
-    "\n",
-    "$$\\mathbf F = e^{\\mathbf A\\Delta t} = \\mathbf{I} + \\mathbf A\\Delta t  + \\frac{(\\mathbf A\\Delta t)^2}{2!} + \\frac{(\\mathbf A\\Delta t)^3}{3!} + ... $$\n",
-    "\n",
-    "Alternatively, you can use one of the other techniques we learned in the **Kalman Math** chapter. \n",
-    "\n",
     "This leads to the following equations for the EKF. I put boxes around the differences from the linear filter:\n",
     "\n",
     "$$\\begin{array}{l|l}\n",
     "\\text{linear Kalman filter} & \\text{EKF} \\\\\n",
     "\\hline \n",
-    "& \\boxed{\\mathbf A = {\\frac{\\partial{f(\\mathbf x_t, \\mathbf u_t)}}{\\partial{\\mathbf x}}}\\biggr|_{{\\mathbf x_t},{\\mathbf u_t}}} \\\\\n",
-    "& \\boxed{\\mathbf F = e^{\\mathbf A \\Delta t}} \\\\\n",
+    "& \\boxed{\\mathbf F = {\\frac{\\partial{f(\\mathbf x_t, \\mathbf u_t)}}{\\partial{\\mathbf x}}}\\biggr|_{{\\mathbf x_t},{\\mathbf u_t}}} \\\\\n",
     "\\mathbf{\\bar x} = \\mathbf{Fx} + \\mathbf{Bu} & \\boxed{\\mathbf{\\bar x} = f(\\mathbf x, \\mathbf u)}  \\\\\n",
     "\\mathbf{\\bar P} = \\mathbf{FPF}^\\mathsf{T}+\\mathbf Q  & \\mathbf{\\bar P} = \\mathbf{FPF}^\\mathsf{T}+\\mathbf Q \\\\\n",
     "\\hline\n",