checking out different implementation

notebooks/1.1-symbolic-system-equations.ipynb

@@ -151,14 +151,49 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Ackermann"
+    "## Ackermann Kinematics model\n",
+    "\n",
+    "### Jacobians"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 5,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & \\cos{\\left(\\theta \\right)} & - v \\sin{\\left(\\theta \\right)}\\\\0 & 0 & \\sin{\\left(\\theta \\right)} & v \\cos{\\left(\\theta \\right)}\\\\0 & 0 & 0 & 0\\\\0 & 0 & \\frac{\\tan{\\left(\\delta \\right)}}{L} & 0\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "Matrix([\n",
+       "[0, 0,   cos(theta), -v*sin(theta)],\n",
+       "[0, 0,   sin(theta),  v*cos(theta)],\n",
+       "[0, 0,            0,             0],\n",
+       "[0, 0, tan(delta)/L,             0]])"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\\\1 & 0\\\\0 & \\frac{v \\left(\\tan^{2}{\\left(\\delta \\right)} + 1\\right)}{L}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "Matrix([\n",
+       "[0,                       0],\n",
+       "[0,                       0],\n",
+       "[1,                       0],\n",
+       "[0, v*(tan(delta)**2 + 1)/L]])"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "x,y,theta,v,delta,L,a = sp.symbols(\"x y theta v delta L a\")\n",
     "\n",
@@ -175,39 +210,21 @@
     "U = sp.Matrix([a,delta])\n",
     "\n",
     "#B\n",
-    "B=gs.jacobian(U)#.subs({x:0,y:0,theta:0})B="
+    "B=gs.jacobian(U)\n",
+    "display(A)\n",
+    "display(B)"
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": 6,
+   "cell_type": "markdown",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\\\1 & 0\\\\0 & \\frac{v \\left(\\tan^{2}{\\left(\\delta \\right)} + 1\\right)}{L}\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "Matrix([\n",
-       "[0,                       0],\n",
-       "[0,                       0],\n",
-       "[1,                       0],\n",
-       "[0, v*(tan(delta)**2 + 1)/L]])"
-      ]
-     },
-     "execution_count": 6,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
-    "B"
+    "### Discretized and Linearized model"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
@@ -223,22 +240,9 @@
        "[0, 0, dt*tan(delta)/L,                1]])"
       ]
      },
-     "execution_count": 7,
      "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "#A LIN\n",
-    "DT = sp.symbols(\"dt\")\n",
-    "sp.eye(4)+A*DT"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 8,
-   "metadata": {},
-   "outputs": [
+     "output_type": "display_data"
+    },
     {
      "data": {
       "text/latex": [
@@ -252,20 +256,9 @@
        "[ 0, dt*v*(tan(delta)**2 + 1)/L]])"
       ]
      },
-     "execution_count": 8,
      "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "B*DT"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "metadata": {},
-   "outputs": [
+     "output_type": "display_data"
+    },
     {
      "data": {
       "text/latex": [
@@ -279,13 +272,16 @@
        "[-delta*dt*v*(tan(delta)**2 + 1)/L]])"
       ]
      },
-     "execution_count": 9,
      "metadata": {},
-     "output_type": "execute_result"
+     "output_type": "display_data"
     }
    ],
    "source": [
-    "DT*(gs - A*X - B*U)"
+    "DT = sp.symbols(\"dt\")\n",
+    "\n",
+    "display(sp.eye(4)+A*DT)\n",
+    "display(B*DT)\n",
+    "display(DT*(gs - A*X - B*U))"
    ]
   },
   {