update notebook

Author: AtsushiSakai

PathTracking/model_predictive_speed_and_steer_control/notebook.ipynb

@@ -43,19 +43,51 @@
     "ODE is\n",
     "\n",
     "$$ \\dot{z} =\\frac{\\partial }{\\partial z} z = A'z+B'u$$\n",
-    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "where\n",
     "\n",
     "\\begin{equation*}\n",
-    "A' = \n",
+    "A' =\n",
+    "\\begin{bmatrix}\n",
+    "\\frac{\\partial }{\\partial x}vcos(\\phi) & \n",
+    "\\frac{\\partial }{\\partial y}vcos(\\phi) & \n",
+    "\\frac{\\partial }{\\partial v}vcos(\\phi) &\n",
+    "\\frac{\\partial }{\\partial \\phi}vcos(\\phi)\\\\\n",
+    "\\frac{\\partial }{\\partial x}vsin(\\phi) & \n",
+    "\\frac{\\partial }{\\partial y}vsin(\\phi) & \n",
+    "\\frac{\\partial }{\\partial v}vsin(\\phi) &\n",
+    "\\frac{\\partial }{\\partial \\phi}vsin(\\phi)\\\\\n",
+    "\\frac{\\partial }{\\partial x}a& \n",
+    "\\frac{\\partial }{\\partial y}a& \n",
+    "\\frac{\\partial }{\\partial v}a&\n",
+    "\\frac{\\partial }{\\partial \\phi}a\\\\\n",
+    "\\frac{\\partial }{\\partial x}\\frac{vtan(\\delta)}{L}& \n",
+    "\\frac{\\partial }{\\partial y}\\frac{vtan(\\delta)}{L}& \n",
+    "\\frac{\\partial }{\\partial v}\\frac{vtan(\\delta)}{L}&\n",
+    "\\frac{\\partial }{\\partial \\phi}\\frac{vtan(\\delta)}{L}\\\\\n",
+    "\\end{bmatrix}\n",
+    "\\\\\n",
+    "　=\n",
     "\\begin{bmatrix}\n",
     "0 & 0 & cos(\\bar{\\phi}) & -\\bar{v}sin(\\bar{\\phi})\\\\\n",
     "0 & 1 & sin(\\bar{\\phi}) & \\bar{v}cos(\\bar{\\phi}) \\\\\n",
     "0 & 0 & 0 & 0 \\\\\n",
     "0 & 0 &\\frac{tan(\\bar{\\delta})}{L} & 0 \\\\\n",
     "\\end{bmatrix}\n",
     "\\end{equation*}\n",
-    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "\\begin{equation*}\n",
     "B' =\n",
     "\\begin{bmatrix}\n",