update notebook

Author: AtsushiSakai

PathTracking/model_predictive_speed_and_steer_control/notebook.ipynb

@@ -21,7 +21,7 @@
     "# Vehicle model linearization\n",
     "\n",
     "State vector is:\n",
-    "$$ x = [x, y, v,\\phi]$$ x: x-position, y:y-position, v:velocity, φ: yaw angle\n",
+    "$$ z = [x, y, v,\\phi]$$ x: x-position, y:y-position, v:velocity, φ: yaw angle\n",
     "\n",
     "Input vector is:\n",
     "$$ u = [a, \\delta]$$ a: accellation, δ: steering angle\n",
@@ -30,7 +30,20 @@
     "$$ \\dot{x} = vcos(\\phi)$$\n",
     "$$ \\dot{y} = vsin((\\phi)$$\n",
     "$$ \\dot{v} = a$$\n",
-    "$$ \\dot{\\phi} = \\frac{vtan(\\delta)}{L}$$"
+    "$$ \\dot{\\phi} = \\frac{vtan(\\delta)}{L}$$\n",
+    "\n",
+    "ODE is\n",
+    "\n",
+    "$$ \\dot{z} = Az+Bu$$\n",
+    "\n",
+    "and A, B are\n",
+    "\n",
+    "\n",
+    "\n",
+    "You can get a discrete-time mode with Forward Euler Discretization with sampling time dt.\n",
+    "\n",
+    "$$z_{k+1}=z_k+f(z_k,u_k)dt$$\n",
+    "\n"
    ]
   },
   {