update docs

Localization/Kalmanfilter_basics_2.ipynb

@@ -11,7 +11,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    " ### Probabilistic Generative Laws\n",
+    "### Probabilistic Generative Laws\n",
     " \n",
     "#### 1st Law:\n",
     "The belief representing the state $x_{t}$, is conditioned on all past states, measurements and controls. This can be shown mathematically by the conditional probability shown below:\n",

PathTracking/model_predictive_speed_and_steer_control/Model_predictive_speed_and_steering_control.ipynb

@@ -128,7 +128,7 @@
    "source": [
     "where\n",
     "\n",
-    "$$\\begin{equation*}\n",
+    "\\begin{equation*}\n",
     "A' =\n",
     "\\begin{bmatrix}\n",
     "\\frac{\\partial }{\\partial x}vcos(\\phi) & \n",
@@ -156,7 +156,7 @@
     "0 & 0 & 0 & 0 \\\\\n",
     "0 & 0 &\\frac{tan(\\bar{\\delta})}{L} & 0 \\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{equation*}$$\n",
+    "\\end{equation*}\n",
     "\n"
    ]
   },
@@ -164,7 +164,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$$\\begin{equation*}\n",
+    "\\begin{equation*}\n",
     "B' =\n",
     "\\begin{bmatrix}\n",
     "\\frac{\\partial }{\\partial a}vcos(\\phi) &\n",
@@ -184,7 +184,7 @@
     "1 & 0 \\\\\n",
     "0 & \\frac{\\bar{v}}{Lcos^2(\\bar{\\delta})} \\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{equation*}$$\n",
+    "\\end{equation*}\n",
     "\n"
    ]
   },
@@ -212,7 +212,7 @@
     "\n",
     "where,\n",
     "\n",
-    "$$\\begin{equation*}\n",
+    "\\begin{equation*}\n",
     "A = (I + dtA')\\\\\n",
     "=\n",
     "\\begin{bmatrix} \n",
@@ -221,14 +221,14 @@
     "0 & 0 & 1 & 0 \\\\\n",
     "0 & 0 &\\frac{tan(\\bar{\\delta})}{L}dt & 1 \\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{equation*}$$"
+    "\\end{equation*}"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$$\\begin{equation*}\n",
+    "\\begin{equation*}\n",
     "B = dtB'\\\\\n",
     "=\n",
     "\\begin{bmatrix} \n",
@@ -237,14 +237,14 @@
     "dt & 0 \\\\\n",
     "0 & \\frac{\\bar{v}}{Lcos^2(\\bar{\\delta})}dt \\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{equation*}$$"
+    "\\end{equation*}"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$$\\begin{equation*}\n",
+    "\\begin{equation*}\n",
     "C = (f(\\bar{z},\\bar{u})-A'\\bar{z}-B'\\bar{u})dt\\\\\n",
     "= dt(\n",
     "\\begin{bmatrix} \n",
@@ -275,7 +275,7 @@
     "0\\\\\n",
     "-\\frac{\\bar{v}\\bar{\\delta}}{Lcos^2(\\bar{\\delta})}dt\\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{equation*}$$"
+    "\\end{equation*}"
    ]
   },
   {

docs/modules/Kalmanfilter_basics_2.rst

@@ -2,7 +2,8 @@
 KF Basics - Part 2
 ------------------
 
-### Probabilistic Generative Laws
+Probabilistic Generative Laws
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 1st Law:
 ^^^^^^^^

docs/modules/Model_predictive_speed_and_steering_control.rst

@@ -94,61 +94,57 @@ ODE is
 
 where
 
-.. math::
+:raw-latex:`\begin{equation*}
+A' =
+\begin{bmatrix}
+\frac{\partial }{\partial x}vcos(\phi) & 
+\frac{\partial }{\partial y}vcos(\phi) & 
+\frac{\partial }{\partial v}vcos(\phi) &
+\frac{\partial }{\partial \phi}vcos(\phi)\\
+\frac{\partial }{\partial x}vsin(\phi) & 
+\frac{\partial }{\partial y}vsin(\phi) & 
+\frac{\partial }{\partial v}vsin(\phi) &
+\frac{\partial }{\partial \phi}vsin(\phi)\\
+\frac{\partial }{\partial x}a& 
+\frac{\partial }{\partial y}a& 
+\frac{\partial }{\partial v}a&
+\frac{\partial }{\partial \phi}a\\
+\frac{\partial }{\partial x}\frac{vtan(\delta)}{L}& 
+\frac{\partial }{\partial y}\frac{vtan(\delta)}{L}& 
+\frac{\partial }{\partial v}\frac{vtan(\delta)}{L}&
+\frac{\partial }{\partial \phi}\frac{vtan(\delta)}{L}\\
+\end{bmatrix}
+\\
+　=
+\begin{bmatrix}
+0 & 0 & cos(\bar{\phi}) & -\bar{v}sin(\bar{\phi})\\
+0 & 0 & sin(\bar{\phi}) & \bar{v}cos(\bar{\phi}) \\
+0 & 0 & 0 & 0 \\
+0 & 0 &\frac{tan(\bar{\delta})}{L} & 0 \\
+\end{bmatrix}
+\end{equation*}`
 
-   \begin{equation*}
-   A' =
-   \begin{bmatrix}
-   \frac{\partial }{\partial x}vcos(\phi) & 
-   \frac{\partial }{\partial y}vcos(\phi) & 
-   \frac{\partial }{\partial v}vcos(\phi) &
-   \frac{\partial }{\partial \phi}vcos(\phi)\\
-   \frac{\partial }{\partial x}vsin(\phi) & 
-   \frac{\partial }{\partial y}vsin(\phi) & 
-   \frac{\partial }{\partial v}vsin(\phi) &
-   \frac{\partial }{\partial \phi}vsin(\phi)\\
-   \frac{\partial }{\partial x}a& 
-   \frac{\partial }{\partial y}a& 
-   \frac{\partial }{\partial v}a&
-   \frac{\partial }{\partial \phi}a\\
-   \frac{\partial }{\partial x}\frac{vtan(\delta)}{L}& 
-   \frac{\partial }{\partial y}\frac{vtan(\delta)}{L}& 
-   \frac{\partial }{\partial v}\frac{vtan(\delta)}{L}&
-   \frac{\partial }{\partial \phi}\frac{vtan(\delta)}{L}\\
-   \end{bmatrix}
-   \\
-   　=
-   \begin{bmatrix}
-   0 & 0 & cos(\bar{\phi}) & -\bar{v}sin(\bar{\phi})\\
-   0 & 0 & sin(\bar{\phi}) & \bar{v}cos(\bar{\phi}) \\
-   0 & 0 & 0 & 0 \\
-   0 & 0 &\frac{tan(\bar{\delta})}{L} & 0 \\
-   \end{bmatrix}
-   \end{equation*}
-
-.. math::
-
-   \begin{equation*}
-   B' =
-   \begin{bmatrix}
-   \frac{\partial }{\partial a}vcos(\phi) &
-   \frac{\partial }{\partial \delta}vcos(\phi)\\
-   \frac{\partial }{\partial a}vsin(\phi) &
-   \frac{\partial }{\partial \delta}vsin(\phi)\\
-   \frac{\partial }{\partial a}a &
-   \frac{\partial }{\partial \delta}a\\
-   \frac{\partial }{\partial a}\frac{vtan(\delta)}{L} &
-   \frac{\partial }{\partial \delta}\frac{vtan(\delta)}{L}\\
-   \end{bmatrix}
-   \\
-   　=
-   \begin{bmatrix}
-   0 & 0 \\
-   0 & 0 \\
-   1 & 0 \\
-   0 & \frac{\bar{v}}{Lcos^2(\bar{\delta})} \\
-   \end{bmatrix}
-   \end{equation*}
+:raw-latex:`\begin{equation*}
+B' =
+\begin{bmatrix}
+\frac{\partial }{\partial a}vcos(\phi) &
+\frac{\partial }{\partial \delta}vcos(\phi)\\
+\frac{\partial }{\partial a}vsin(\phi) &
+\frac{\partial }{\partial \delta}vsin(\phi)\\
+\frac{\partial }{\partial a}a &
+\frac{\partial }{\partial \delta}a\\
+\frac{\partial }{\partial a}\frac{vtan(\delta)}{L} &
+\frac{\partial }{\partial \delta}\frac{vtan(\delta)}{L}\\
+\end{bmatrix}
+\\
+　=
+\begin{bmatrix}
+0 & 0 \\
+0 & 0 \\
+1 & 0 \\
+0 & \frac{\bar{v}}{Lcos^2(\bar{\delta})} \\
+\end{bmatrix}
+\end{equation*}`
 
 You can get a discrete-time mode with Forward Euler Discretization with
 sampling time dt.
@@ -167,66 +163,60 @@ So,
 
 where,
 
-.. math::
+:raw-latex:`\begin{equation*}
+A = (I + dtA')\\
+=
+\begin{bmatrix} 
+1 & 0 & cos(\bar{\phi})dt & -\bar{v}sin(\bar{\phi})dt\\
+0 & 1 & sin(\bar{\phi})dt & \bar{v}cos(\bar{\phi})dt \\
+0 & 0 & 1 & 0 \\
+0 & 0 &\frac{tan(\bar{\delta})}{L}dt & 1 \\
+\end{bmatrix}
+\end{equation*}`
 
-   \begin{equation*}
-   A = (I + dtA')\\
-   =
-   \begin{bmatrix} 
-   1 & 0 & cos(\bar{\phi})dt & -\bar{v}sin(\bar{\phi})dt\\
-   0 & 1 & sin(\bar{\phi})dt & \bar{v}cos(\bar{\phi})dt \\
-   0 & 0 & 1 & 0 \\
-   0 & 0 &\frac{tan(\bar{\delta})}{L}dt & 1 \\
-   \end{bmatrix}
-   \end{equation*}
+:raw-latex:`\begin{equation*}
+B = dtB'\\
+=
+\begin{bmatrix} 
+0 & 0 \\
+0 & 0 \\
+dt & 0 \\
+0 & \frac{\bar{v}}{Lcos^2(\bar{\delta})}dt \\
+\end{bmatrix}
+\end{equation*}`
 
-.. math::
-
-   \begin{equation*}
-   B = dtB'\\
-   =
-   \begin{bmatrix} 
-   0 & 0 \\
-   0 & 0 \\
-   dt & 0 \\
-   0 & \frac{\bar{v}}{Lcos^2(\bar{\delta})}dt \\
-   \end{bmatrix}
-   \end{equation*}
-
-.. math::
-
-   \begin{equation*}
-   C = (f(\bar{z},\bar{u})-A'\bar{z}-B'\bar{u})dt\\
-   = dt(
-   \begin{bmatrix} 
-   \bar{v}cos(\bar{\phi})\\
-   \bar{v}sin(\bar{\phi}) \\
-   \bar{a}\\
-   \frac{\bar{v}tan(\bar{\delta})}{L}\\
-   \end{bmatrix}
-   -
-   \begin{bmatrix} 
-   \bar{v}cos(\bar{\phi})-\bar{v}sin(\bar{\phi})\bar{\phi}\\
-   \bar{v}sin(\bar{\phi})+\bar{v}cos(\bar{\phi})\bar{\phi}\\
-   0\\
-   \frac{\bar{v}tan(\bar{\delta})}{L}\\
-   \end{bmatrix}
-   -
-   \begin{bmatrix} 
-   0\\
-   0 \\
-   \bar{a}\\
-   \frac{\bar{v}\bar{\delta}}{Lcos^2(\bar{\delta})}\\
-   \end{bmatrix}
-   )\\
-   =
-   \begin{bmatrix} 
-   \bar{v}sin(\bar{\phi})\bar{\phi}dt\\
-   -\bar{v}cos(\bar{\phi})\bar{\phi}dt\\
-   0\\
-   -\frac{\bar{v}\bar{\delta}}{Lcos^2(\bar{\delta})}dt\\
-   \end{bmatrix}
-   \end{equation*}
+:raw-latex:`\begin{equation*}
+C = (f(\bar{z},\bar{u})-A'\bar{z}-B'\bar{u})dt\\
+= dt(
+\begin{bmatrix} 
+\bar{v}cos(\bar{\phi})\\
+\bar{v}sin(\bar{\phi}) \\
+\bar{a}\\
+\frac{\bar{v}tan(\bar{\delta})}{L}\\
+\end{bmatrix}
+-
+\begin{bmatrix} 
+\bar{v}cos(\bar{\phi})-\bar{v}sin(\bar{\phi})\bar{\phi}\\
+\bar{v}sin(\bar{\phi})+\bar{v}cos(\bar{\phi})\bar{\phi}\\
+0\\
+\frac{\bar{v}tan(\bar{\delta})}{L}\\
+\end{bmatrix}
+-
+\begin{bmatrix} 
+0\\
+0 \\
+\bar{a}\\
+\frac{\bar{v}\bar{\delta}}{Lcos^2(\bar{\delta})}\\
+\end{bmatrix}
+)\\
+=
+\begin{bmatrix} 
+\bar{v}sin(\bar{\phi})\bar{\phi}dt\\
+-\bar{v}cos(\bar{\phi})\bar{\phi}dt\\
+0\\
+-\frac{\bar{v}\bar{\delta}}{Lcos^2(\bar{\delta})}dt\\
+\end{bmatrix}
+\end{equation*}`
 
 This equation is implemented at