valentine.ipynb

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  {
   "cell_type": "code",
   "execution_count": 244,
   "id": "corrected-sharp",
   "metadata": {
    "tags": [
     "hide-cell"
    ]
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   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import fsolve\n",
    "plt.style.use('fivethirtyeight')\n",
    "from matplotlib import rc\n",
    "\n",
    "plt.rc('text', usetex=True)\n",
    "plt.rc('font', family='sans')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "acquired-print",
   "metadata": {},
   "source": [
    "# Happy Valentine's Linkage\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "shared-screw",
   "metadata": {},
   "source": [
    "In this notebook, you step through the process of defining the\n",
    "kinematics of a [four-bar\n",
    "linkage](https://en.wikipedia.org/wiki/Four-bar_linkage) that can draw a\n",
    "heart.\n",
    "[__Kinematics__](https://en.wikipedia.org/wiki/Kinematics#Kinematic_constraints)\n",
    "is the study of the geometry of motion. In this notebook, you predict\n",
    "and draw the geometry of three moving links. To create this solution you\n",
    "will:\n",
    "\n",
    "- solve a series of nonlinear equations using\n",
    "  [`fsolve`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fsolve.html)\n",
    "- use solutions to create 2D\n",
    "  [arrays](https://numpy.org/doc/stable/reference/generated/numpy.array.html) that vary in time and\n",
    "  location\n",
    "- plot and animate the motion of the four-bar linkage\n",
    "\n",
    "When you accomplish these steps you will learn:\n",
    "\n",
    "- How to set up and solve nonlinear equations\n",
    "- How to describe position and orientation\n",
    "- How to use NumPy, Scipy, and Matplotlib to create HTML animations\n",
    "- Why four-bar linkages are so cool\n",
    "\n",
    "## Background\n",
    "\n",
    "The four-bar linkages consist of 3 moving parts connected to a\n",
    "stationary support. Depending upon the desired output motion, you can\n",
    "vary the lengths of each moving arm, $l_1,~l_2,~and,l_3$, and the\n",
    "support locations, $d_x~and~d_y$. A four-bar linkage is an amazing\n",
    "__1-degree-of-freedom__ system. If one angle is fixed, all of the\n",
    "positions and angles have to be fixed too. Using some trigonometry, you\n",
    "can arrive at these two constraint equations that relate\n",
    "$\\theta_1,~\\theta_2,~and~\\theta_3$:\n",
    "\n",
    "![Diagram of a general four-bar linkage](./images/4bar-general.svg)\n",
    "\n",
    "1. $l_1\\sin\\theta_1+l_2\\sin\\theta_2-l_3\\sin\\theta_3 -d_y = 0$\n",
    "2. $l_1\\cos\\theta_1+l_2\\cos\\theta_2-l_3\\cos\\theta_3 -d_x = 0$\n",
    "\n",
    "If you have one of the angles, e.g. $\\theta_1$, you use equations 1 and 2\n",
    "to solve for the other two angles, $\\theta_2~and\\theta_3$. Here you can\n",
    "create a function and\n",
    "use `fsolve`. The function input is a vector with two values and the output is a\n",
    "vector with two values. \n",
    "\n",
    "$\\bar{f}(\\bar{x})= \\left[\\begin{array}{c}\n",
    "f_1(\\theta_2,~\\theta_3) \\\\ \n",
    "f_2(\\theta_2,~\\theta_3)\\end{array}\\right]=\\left[\\begin{array}{c}\n",
    "l_1\\sin\\theta_1+l_2\\sin\\theta_2-l_3\\sin\\theta_3 -d_y\\\\\n",
    "l_1\\cos\\theta_1+l_2\\cos\\theta_2-l_3\\cos\\theta_3 -d_x\n",
    "\\end{array}\\right]$\n",
    "\n",
    "## Defining your system\n",
    "\n",
    "The heart-drawing linkage system has the following properties:\n",
    "\n",
    "![diagram of the heart-drawing linkage system](./images/4bar-heart.svg)\n",
    "\n",
    "- link 1: $l_1 = 1~m$\n",
    "- link 2: $l_2 = 1~m$\n",
    "- link 3: $l_3 = 1~m$\n",
    "- support: $d_x=0.95~m~and~d_y=0~m$\n",
    "\n",
    "The constraint function is defined below as `Fbar`, a function of\n",
    "$\\theta_1$ and an array of $[\\theta_2,~\\theta_3]$ as such,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "id": "verified-round",
   "metadata": {
    "lines_to_next_cell": 0
   },
   "outputs": [],
   "source": [
    "l1 = 1\n",
    "l2 = 1\n",
    "l3 = 1\n",
    "a1 = np.pi/2\n",
    "dy = 0\n",
    "dx = 0.95\n",
    "Fbar = lambda a1,x: np.array([l1*np.sin(a1)+l2*np.sin(x[0])-l3*np.sin(x[1])-dy,\n",
    "                           l1*np.cos(a1)+l2*np.cos(x[0])-l3*np.cos(x[1])-dx])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "positive-approach",
   "metadata": {},
   "source": [
    "## Solve for one configuration\n",
    "\n",
    "Now, solve `Fbar` using\n",
    "[`scipy.optimize.fsolve`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fsolve.html).\n",
    "The inputs are a function, `Fbar`, and an initial guess, `x0`. You have\n",
    "to use `lambda` again to set the angle $\\theta_1$ as `a1 = np.pi/2`\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "id": "static-instruction",
   "metadata": {},
   "outputs": [],
   "source": [
    "a1 = np.pi/2\n",
    "x0 = np.array([0,np.pi/2])\n",
    "xsol = fsolve(lambda x: Fbar(a1, x), x0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "missing-transcription",
   "metadata": {},
   "source": [
    "The configuration for $\\theta_1=\\frac{\\pi}{2}$ is now saved in `xsol`.\n",
    "To look at the system in this state, define the positions of each hinge\n",
    "and the center of link 2 as such,"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "charitable-array",
   "metadata": {
    "lines_to_next_cell": 0
   },
   "source": [
    "__x- and y-locations of hinges__:\n",
    "* $rx = \\left[\\begin{array}~0\\\\l_1\\cos(\\theta_1)\\\\l_1\\cos(\\theta_1)+l_2\\cos(\\theta_2)\\\\ l_1\\cos(\\theta_1) + l_2\\cos(\\theta_2)-l_3\\cos(\\theta_3)\\end{array}\\right]$\n",
    "* $ry = \\left[\\begin{array}~0\\\\l_1\\sin(\\theta_1)\\\\l_1\\sin(\\theta_1)+l_2\\sin(\\theta_2)\\\\ l_1\\sin(\\theta_1)+l_2\\sin(\\theta_2)-l_3\\sin(\\theta_3)\\end{array}\\right]$\n",
    "\n",
    "__x- and y-location of point P__:\n",
    "* $rx = \\left[\\begin{array}~l_1\\cos(\\theta_1)+l_2\\cos(\\theta_2)\\end{array}\\right]$\n",
    "* $ry = \\left[\\begin{array}~l_1\\sin(\\theta_1)+l_2\\sin(\\theta_2)\\end{array}\\right]$\n",
    "\n",
    "In the Python cell, you define `rx` and `ry` to draw the three moving\n",
    "links and `rpx` and `rpy` that define point P as such,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 247,
   "id": "subtle-metro",
   "metadata": {
    "lines_to_next_cell": 0
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rx = np.array([0,\n",
    "               l1*np.cos(a1),\n",
    "               l1*np.cos(a1)+l2*np.cos(xsol[0]), \n",
    "               l1*np.cos(a1)+l2*np.cos(xsol[0])-l3*np.cos(xsol[1])])\n",
    "ry = np.array([0,\n",
    "               l1*np.sin(a1),\n",
    "               l1*np.sin(a1)+l2*np.sin(xsol[0]), \n",
    "               l1*np.sin(a1)+l2*np.sin(xsol[0])-l3*np.sin(xsol[1])])\n",
    "rpx = l1*np.cos(a1)+l2/2*np.cos(xsol[0])\n",
    "rpy = l1*np.sin(a1)+l2/2*np.sin(xsol[0])\n",
    "\n",
    "plt.plot(rx,ry,'o-')\n",
    "plt.plot(rpx,rpy,'rs')\n",
    "#plt.axis([-0.5, 1.5, -0.6, 0.6])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "manufactured-journey",
   "metadata": {},
   "source": [
    "## Solve for the whole cycle\n",
    "\n",
    "You have verified the solution works for one angle, `a1`. Now, you can\n",
    "create an array of `a1` and solve for the angles at each configuration\n",
    "as you rotate link 1.\n",
    "\n",
    "1. define `a1` as a\n",
    "[`linspace`](https://numpy.org/doc/stable/reference/generated/numpy.linspace.html)\n",
    "2. initialize `a2` and `a3` as zeros with `a1.shape`\n",
    "3. use a for-loop to `fsolve` each angle `a2` and `a3` given `a1`\n",
    "4. save the values of `a2` and `a3` in each step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "id": "native-inclusion",
   "metadata": {
    "lines_to_next_cell": 0
   },
   "outputs": [],
   "source": [
    "a1 = np.linspace(-np.pi/2,3*np.pi/2,500)\n",
    "a2 = np.zeros(a1.shape) # initialize\n",
    "a3 = np.zeros(a1.shape) # initialize\n",
    "for i, a in enumerate(a1):\n",
    "    xsol = fsolve(lambda x: Fbar(a,x), xsol) # solve\n",
    "    a2[i] = xsol[0] # save value for a2\n",
    "    a3[i] = xsol[1] # save value for a3\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "welsh-detroit",
   "metadata": {},
   "source": [
    "### Verify cycle solution\n",
    "\n",
    "Now, you can see how the functions, $\\theta_2=f_{\\theta_2}(\\theta_1)$ and\n",
    "$\\theta_3=f_{\\theta_3}(\\theta_1)$. This is another verification step, to\n",
    "see if there are any discontinuities in the solution that could cause\n",
    "problems. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "id": "successful-astrology",
   "metadata": {
    "lines_to_next_cell": 0
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'output: angles (degrees)')"
      ]
     },
     "execution_count": 251,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(a1*180/np.pi, a2*180/np.pi, label = r'$\\theta_2$')\n",
    "plt.plot(a1*180/np.pi, a3*180/np.pi, label = r'$\\theta_3$')\n",
    "plt.legend()\n",
    "plt.xlabel(r'input: $\\theta_1$ (degrees)')\n",
    "plt.ylabel('output: angles (degrees)')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "welcome-macintosh",
   "metadata": {},
   "source": [
    "## Time to animate\n",
    "\n",
    "Now, you are ready to animate. You have the angles for the entire cycle\n",
    "of motion for the four-bar linkage. Now, you need to import\n",
    "[`matplotlib.animation`](https://matplotlib.org/stable/api/animation_api.html)\n",
    "and\n",
    "[`IPython.display.HTML`](https://ipython.readthedocs.io/en/stable/api/generated/IPython.display.html#IPython.display.HTML). \n",
    "\n",
    "### Import functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "id": "innovative-niagara",
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import animation\n",
    "from IPython.display import HTML"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "every-stake",
   "metadata": {},
   "source": [
    "These two functions allow you:\n",
    "1. create an animation\n",
    "2. display it in a browser\n",
    "\n",
    "### Define lines and paths\n",
    "\n",
    "Then, use the solutions for `a1`, `a2`, and `a3` to plot the hinge\n",
    "locations and point P as such"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 253,
   "id": "bearing-thumb",
   "metadata": {
    "lines_to_next_cell": 2
   },
   "outputs": [],
   "source": [
    "rx = np.array([np.zeros(a1.shape),\n",
    "               l1*np.cos(a1),\n",
    "               l1*np.cos(a1)+l2*np.cos(a2), \n",
    "               l1*np.cos(a1)+l2*np.cos(a2)-l3*np.cos(a3)])\n",
    "ry = np.array([np.zeros(a1.shape),\n",
    "               l1*np.sin(a1),\n",
    "               l1*np.sin(a1)+l2*np.sin(a2), \n",
    "               l1*np.sin(a1)+l2*np.sin(a2)-l3*np.sin(a3)])\n",
    "\n",
    "rpx = l1*np.cos(a1)+l2/2*np.cos(a2)\n",
    "rpy = l1*np.sin(a1)+l2/2*np.sin(a2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 254,
   "id": "physical-conversation",
   "metadata": {
    "lines_to_next_cell": 0
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fdfe5250e80>]"
      ]
     },
     "execution_count": 254,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(rpx,rpy)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "piano-executive",
   "metadata": {},
   "source": [
    "### Set up figure and axes\n",
    "Plotting just the solution for point P's path, you see the heart shape\n",
    "is upside-down. In the animation, you can reverse the y-axis to flip the\n",
    "drawing. Here, you set up the figure to create the animation:\n",
    "\n",
    "* `ax`: axis for plotting the lines\n",
    "* `line1`: lines that draw the three moving links `rx` and `ry`\n",
    "* `line2`: line that updates the heart drawing as links move along paths"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 255,
   "id": "unavailable-demonstration",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fdfe51f5580>]"
      ]
     },
     "execution_count": 255,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "#ax.set_xlim(( -30, 30))\n",
    "ax.set_ylim((1.5, -1.5))\n",
    "ax.set_xlabel('x-position (m)')\n",
    "ax.set_ylabel('y-position (m)')\n",
    "\n",
    "line1, = ax.plot([], [],'bo-')\n",
    "line2, = ax.plot([], [],'r')\n",
    "ax.plot(rx[1,:],ry[1,:],'g--', alpha=0.5)\n",
    "ax.plot(rx[2,:],ry[2,:],'g--', alpha=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "destroyed-trail",
   "metadata": {},
   "source": [
    "### Define initializing and animation functions\n",
    "\n",
    "Create an initializing (`init`) function that clears the previous\n",
    "lines"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 256,
   "id": "passing-commander",
   "metadata": {},
   "outputs": [],
   "source": [
    "def init():\n",
    "    line1.set_data([], [])\n",
    "    line2.set_data([], [])\n",
    "    return (line1, line2, )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "antique-nancy",
   "metadata": {},
   "source": [
    "Create an animating (`animate`) function that updates the lines. The\n",
    "links should be drawn for a given column, i.e. `line1` is defined as\n",
    "`rx[:, i]` and `ry[:, i]`, but you want the path of the heart up to the\n",
    "current frame i.e. line2 is defined as `rpx[:i]` and `rpy[:i]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 257,
   "id": "treated-surname",
   "metadata": {
    "lines_to_next_cell": 0
   },
   "outputs": [],
   "source": [
    "def animate(i):\n",
    "    '''function that updates the line and marker data\n",
    "    arguments:\n",
    "    ----------\n",
    "    i: index of timestep\n",
    "    outputs:\n",
    "    --------\n",
    "    line: the line object plotted in the above ax.plot(...)\n",
    "    '''\n",
    "    line1.set_data(rx[:, i], ry[:,i])\n",
    "    line2.set_data(rpx[:i], rpy[:i])\n",
    "    return (line1, line2 )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "stupid-attraction",
   "metadata": {},
   "source": [
    "### Create the animation and display\n",
    "\n",
    "Create an animation (`anim`) variable using the `animation.FuncAnimation`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 258,
   "id": "described-wildlife",
   "metadata": {},
   "outputs": [],
   "source": [
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=range(0,len(a1)), interval=10, \n",
    "                               blit=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 259,
   "id": "absent-netscape",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "  <source type=\"video/mp4\" src=\"data:video/mp4;base64,AAAAIGZ0eXBNNFYgAAACAE00ViBpc29taXNvMmF2YzEAAAAIZnJlZQABsWZtZGF0AAACrgYF//+q\n",
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       "b3JnL3gyNjQuaHRtbCAtIG9wdGlvbnM6IGNhYmFjPTEgcmVmPTMgZGVibG9jaz0xOjA6MCBhbmFs\n",
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       "  Your browser does not support the video tag.\n",
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     "execution_count": 259,
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  {
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   "id": "direct-communication",
   "metadata": {},
   "source": [
    "## Wrapping up\n",
    "\n",
    "There you have it, a heart-drawing four-bar linkage. This solution\n",
    "required arrays, nonlinear solutions, and some trigonometry to\n",
    "build locations of hinges and links. \n",
    "\n",
    "Try changing the lengths the different links or changing the fixed\n",
    "positions. What else can you draw?\n"
   ]
  }
 ],
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