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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Angular Acceleration And Fin Alpha\n",
      "\n",
      "Our control algorithm will give us a value of angular acceleration that we would like our rocket to have. In reality we change the roll rate of the rocket with small fins so we need a correlation between the fin angle and the angular acceleration.\n",
      "\n",
      "\n",
      "## Angular Acceleration:\n",
      "\n",
      "Angular acceleration is the result of tourqe on the rocket, where torque is the fin force (lift) and the lever arm to the fin, and the mass moment of inertia in the axis of rotation:\n",
      "\n",
      "$$\\begin{equation}\n",
      " \\ddot \\theta = \\frac{r \\times L}{I}\n",
      "\\end{equation}$$\n",
      "\n",
      "We design the rocket such that lift is perpedicular to the lever arm of the fin, so the cross product is just multiplication. Filling out with our lift equation:\n",
      "\n",
      "$$\\begin{equation}\n",
      " \\ddot \\theta = \\frac{2C_{L}\\rho v^2 A r}{I}\n",
      "\\end{equation}$$\n",
      "\n",
      "Where\n",
      "\n",
      " - $\\ddot \\theta$: Angular acceleration\n",
      " - $C_{L}$: Coeeficient of lift\n",
      " - $\\rho$: air density\n",
      " - $v$: velocity\n",
      " - $A$: fin area\n",
      " - $r$: fin lever arm\n",
      " - $I$: mass moment of inertia in rocket spin axis\n",
      "\n",
      "But will we have access to all these numbers during flight? Yes! Everything in this equation can be shown to be a function of either our **state vector** or **time**. Lets rewrite eveything showing them as a function of something we can know:\n",
      "\n",
      "$$\\begin{equation}\n",
      " \\ddot \\theta = \\frac{2C_{L}(\\alpha)\\rho(x) \\dot x^2 A r}{I(t)}\n",
      "\\end{equation}$$\n",
      "\n",
      "In particular we see $C_{L}$ is a function of fin angle $\\alpha$. Density is a function of altitude ($x$), I rewrote $v$ as $\\dot x$ to show that it's part of the state vector. $A$ and $r$ are constant and part of the rocket design. $I$ is a function of time."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Coefficient Of Lift\n",
      "\n",
      "We expand $C_{L}(\\alpha)$ to show it's dependence on angle of attack. The rocket spends a lot of time subsonic so let's treat it in the subsonic case:\n",
      "\n",
      "$$\\begin{equation}\n",
      "C_{L}(\\alpha) = K_P\\sin\\alpha\\cos^2\\alpha + K_V\\cos\\alpha\\sin^2\\alpha\n",
      "\\end{equation}$$\n",
      "\n",
      "Where $K_P$ and $K_V$ are known constants. Implemented in python:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import sin, cos, radians, degrees, exp, sqrt, fabs\n",
      "\n",
      "# Define PSAS:\n",
      "k_p = 2.45\n",
      "k_v = 3.2\n",
      "Cl_base = 3.2\n",
      "fin_area = 1.13e-3\n",
      "I_0 = 0.1\n",
      "I_1 = 0.01\n",
      "\n",
      "def C_L(a, v):\n",
      "    \"\"\"Find C_L for a given speed and angle of attack\n",
      "    \n",
      "    :param float a: Angle of attack, alpha in degrees\n",
      "    :param float v: velocity, v in m/s\n",
      "    \"\"\"\n",
      "    \n",
      "    # math is in radians\n",
      "    a = radians(a)\n",
      "    \n",
      "    # Subsonic case\n",
      "    def _subsonic():\n",
      "        sina = sin(a)\n",
      "        cosa = cos(a)\n",
      "        cl = k_p*sina*cosa*cosa\n",
      "        cl += k_v*cosa*sina*sina\n",
      "        return cl\n",
      "    \n",
      "    # Supersonic case\n",
      "    def _supersonic():\n",
      "        cl = a*Cl_base\n",
      "        return cl\n",
      "\n",
      "    if v <= 265:\n",
      "        return _subsonic()\n",
      "    elif v < 330:\n",
      "        # Intepolate between super and subsonic\n",
      "        y0 = _subsonic()\n",
      "        y1 = _supersonic()\n",
      "        x0 = 265\n",
      "        x1 = 330\n",
      "        cl = y0 + (y1-y0)*(v - x0)/(x1-x0)\n",
      "        return cl\n",
      "    else:\n",
      "        return _supersonic()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Polynomial Aproximation\n",
      "\n",
      "We want to solve for $\\alpha$. Not so easy though... So we can replace the complicated parts with a much more simple polynomial aproximation in the form of \n",
      "\n",
      "$$\\begin{equation}\n",
      "C_{L}(\\alpha) = a_{f}\\alpha^2 + b_{f}\\alpha\n",
      "\\end{equation}$$\n",
      "\n",
      "And then we can solve for $\\alpha$. This will be quite close enough as long as our choices for $a_{f}$ and $b_{f}$ are reasonable.\n",
      "\n",
      "Lets pick:\n",
      "\n",
      " - $a_f = 0.00060$\n",
      " - $b_f = 0.045$\n",
      " \n",
      "In python:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def C_L_aprox(a, v):\n",
      "\n",
      "    # Subsonic case\n",
      "    def _subsonic():\n",
      "        af = 0.0006\n",
      "        bf = 0.045\n",
      "        cl = af*a**2 + bf*a\n",
      "        return cl\n",
      "    \n",
      "    # Supersonic case\n",
      "    def _supersonic():\n",
      "        cl = a*Cl_base\n",
      "        return cl\n",
      "\n",
      "    if v <= 265:\n",
      "        return _subsonic()\n",
      "    elif v < 330:\n",
      "        # Intepolate between super and subsonic\n",
      "        y0 = _subsonic()\n",
      "        y1 = _supersonic()\n",
      "        x0 = 265\n",
      "        x1 = 330\n",
      "        cl = y0 + (y1-y0)*(v - x0)/(x1-x0)\n",
      "        return cl\n",
      "    else:\n",
      "        return _supersonic()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import matplotlib.pyplot as plt\n",
      "from matplotlib import rcParams\n",
      "rcParams.update({'font.size': 16})"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# range of angle of attack to chart\n",
      "alpha = [a*0.5 for a in range(31)]\n",
      "CL = [C_L(a, 120) for a in alpha]\n",
      "CLaprox = [C_L_aprox(a, 120) for a in alpha]\n",
      "\n",
      "fig = plt.figure(figsize=(15,9))\n",
      "ax = fig.add_subplot(111)\n",
      "plt.title(\"Subsonic Angle of Attack and Aproximation\")\n",
      "plt.ylabel(r\"$C_{L\\alpha}$\")\n",
      "plt.xlabel(r\"$\\alpha$ [${}^0$]\")\n",
      "\n",
      "plt.plot(alpha, CL, label=\"Full Solution\")\n",
      "plt.plot(alpha, CLaprox, label=\"Polynomial Aproximation\")\n",
      "plt.legend(loc=2)\n",
      "ax.grid(color='grey', alpha=0.3, linestyle='dashed', linewidth=0.5)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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I2NiYnJyc6J9//in0WNPT0+nzzz8nOzs7ksvl1LBhQ1qzZo3Gx/Bo2ieR5lxc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lgmj9aHLVu2sH37dlxcXNKtfy9Mvw9kgQIF6N+/P+PGjbtleWBgIBMnTiQpKSnLz9W7d2/mzJnD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\nicQs/3vV3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       "text": [
        "<matplotlib.figure.Figure at 0x7fcefc06f990>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Solve For $\\alpha$\n",
      "\n",
      "Now we can solve for $\\alpha$. We'll still use two cases, super and subsonic.\n",
      "\n",
      "$$\\begin{equation}\n",
      "\\ddot\\theta I(t) = (a_f\\alpha^2 + bf\\alpha)2\\rho(x)\\dot x^2 Ar\n",
      "\\end{equation}$$\n",
      "\n",
      "Which we solve for $\\alpha$\n",
      "\n",
      "$$\\begin{equation}\n",
      "\\alpha(x,\\dot x,\\ddot\\theta,t) = \\frac{\\sqrt{\\frac{2\\ddot\\theta I(t) a_f}{\\rho(x)\\dot x^2 Ar} + b_f^2} - b_f}{2a_f}\n",
      "\\end{equation}$$\n",
      "\n",
      "And bring it together in callable function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def estimate_alpha(x, v, aa, t):\n",
      "    \"\"\"Return an estimated fin angle of attack for to\n",
      "    achieve the required angular acceleration.\n",
      "    \n",
      "    :param x: Altitude (meters, MSL)\n",
      "    :param v: Air velocity (m/s)\n",
      "    :param aa: Angular acceleration to compute alpha for (degrees/s/s)\n",
      "    :param t: Time (seconds since launch)\n",
      "    :returns fin angle:\n",
      "    \n",
      "    \"\"\"\n",
      "    # LV2.3 Constants\n",
      "    fin_area = 1.13e-3\n",
      "    fin_arm = 0.085\n",
      "    I_0 = 0.086\n",
      "    I_1 = 0.077\n",
      "    af = 0.0006\n",
      "    bf = 0.045\n",
      "    cl_super = 3.2\n",
      "    \n",
      "    \n",
      "    # compute rho\n",
      "    rho = 1.2250 * exp((-9.80665 * 0.0289644 * x)/(8.31432*288.15))\n",
      "    \n",
      "    # compute I\n",
      "    if t <= 0:\n",
      "        I = I_0\n",
      "    elif t < 5.6:\n",
      "        I = I_0 + (I_1-I_0)*t/5.6\n",
      "    else:\n",
      "        I = I_1\n",
      "\n",
      "        \n",
      "        \n",
      "    def _subsonic():\n",
      "        alpha = sqrt((2*aa*I*af)/(rho*v*v*fin_area*fin_arm) + bf*bf) - bf\n",
      "        alpha = alpha / (2*af)\n",
      "        return alpha\n",
      "    \n",
      "    def _supersonic():\n",
      "        alpha =  (aa*I)/(2*rho*v*v*fin_area*fin_arm*cl_super)\n",
      "        return degrees(alpha)\n",
      "    \n",
      "    \n",
      "    if v <= 265:\n",
      "        return _subsonic()\n",
      "    elif v < 330:\n",
      "        # Intepolate between super and subsonic\n",
      "        y0 = _subsonic()\n",
      "        y1 = _supersonic()\n",
      "        x0 = 265\n",
      "        x1 = 330\n",
      "        cl = y0 + (y1-y0)*(v - x0)/(x1-x0)\n",
      "        return cl\n",
      "    else:\n",
      "        return _supersonic()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Validation\n",
      "\n",
      "Let's see if we get good estimates from this technique by spot checking. Compute the desired alpha for a given angular acceleration, then recopmute the acceleration at that alpha with the more complete equation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def lift(a, v, alt):\n",
      "    \"\"\"Compute the lift of one fin at an angle of\n",
      "    attack, velocity, and altitdue\n",
      "    \n",
      "    :param float a: Angle of attack in degrees\n",
      "    :param float v: velocity in m/s\n",
      "    :param float alt: altitude MSL in meters\n",
      "    :returns float lift in Newtons:\n",
      "    \n",
      "    \"\"\"\n",
      "    \n",
      "    # get density of atmosphere with quick exponential model\n",
      "    rho = 1.2250 * exp((-9.80665 * 0.0289644 * alt)/(8.31432*288.15))\n",
      "\n",
      "    l = 0.5*C_L(a, v)*rho*v*v*fin_area\n",
      "    \n",
      "    return l\n",
      "\n",
      "def anglar_accel(a, x, v, t):\n",
      "    \"\"\"Compute angular accelation for a single point\n",
      "    \n",
      "    :param a: Fin alpha (degrees)\n",
      "    :param x: altitude (meters, MSL)\n",
      "    :param v: air velocity (m/s)\n",
      "    :param t: time since launch (seconds)\n",
      "    \"\"\"\n",
      "    \n",
      "    fin_arm = 0.085\n",
      "    I_0 = 0.086\n",
      "    I_1 = 0.077\n",
      "\n",
      "    # compute I\n",
      "    if t <= 0:\n",
      "        I = I_0\n",
      "    elif t < 5.6:\n",
      "        I = I_0 + (I_1-I_0)*t/5.6\n",
      "    else:\n",
      "        I = I_1\n",
      "    \n",
      "    \n",
      "    return (4*lift(a, v, x)*fin_arm)/I"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "worst = 0\n",
      "for aa in range(10,30,10):\n",
      "    for v in range(100,350,25):\n",
      "        for x in range(1000,5000,500):\n",
      "            for t in range(0,30,5):\n",
      "                alpha = estimate_alpha(x,v,aa,t)\n",
      "                aa_real = anglar_accel(alpha, x, v, t)\n",
      "                error = fabs((aa_real - aa)/aa_real)*100\n",
      "                if worst < error:\n",
      "                    worst = error\n",
      "\n",
      "print \"Estimate never off model by more than %0.1f%%\" % worst"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Estimate never off model by more than 5.8%\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": []
    }
   ],
   "metadata": {}
  }
 ]
}