add some docs

Localization/extended_kalman_filter/extended_kalman_filter_localization.ipynb

@@ -25,7 +25,32 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Kalman Filter"
+    "### Filter Design\n",
+    "\n",
+    "In this simulation, the robot has a state vector includes 4 states at time $t$.\n",
+    "\n",
+    "$$\\textbf{x}_t=[x_t, y_t, \\theta_t, v_t]$$\n",
+    "\n",
+    "x, y are a 2D x-y position, $\\theta$ is orientation, and v is velocity.\n",
+    "\n",
+    "\n",
+    "The robot has a speed sensor and a gyro sensor.\n",
+    "\n",
+    "So, the input vecor can be used as each time step\n",
+    "\n",
+    "$$\\textbf{u}_t=[v_t, \\omega_t]$$\n",
+    "\n",
+    "Also, the robot has a GNSS sensor, it means that the robot can observe x-y position at each time.\n",
+    "\n",
+    "$$\\textbf{z}_t=[x_t,y_t]$$\n",
+    "\n",
+    "The input and observation vector includes sensor noise.\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n"
    ]
   },
   {