dmps.ipynb

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   "source": [
    "import numpy as np\n",
    "values = np.asarray([0.0473452,0.0520401,0.0591316,0.0680199,0.0785818,0.0880537,0.096684,0.102001,0.104691,0.10819,0.109185,0.108857,0.109138,0.1098,0.111327,0.112507,0.11667,0.122013,0.123026,0.119127,0.111823,0.102832,0.0929165,0.0849752,0.0785847,0.0736237,0.072608,0.0763474,0.0838251,0.0909338,-0.376493,-1.72607,-3.86127,-6.67567,-10.053,-13.8778,-18.0397,-22.4229,-26.9437,-31.4854,-35.9625,-40.2899,-44.3868,-48.1424,-51.4924,-54.3781,-56.718,-58.4685,-59.5793,-59.971,-59.9841,-51.0547,-27.0491,7.32804,46.7509,85.4871,117.319,136.255])#,139.495,139.647,139.264,138.504,137.396,135.946,134.205,132.242,130.083,127.735,125.192,122.442,119.495,116.356,113.047,109.579,105.972,102.232,98.3781,94.4288,90.3898,86.2794,82.1115,77.898,73.6556,69.399,65.1401,60.8899,56.6665,52.485,48.3592,44.303,40.3286,36.4504,32.6844,29.0431,25.5378,22.183,18.9923,15.9796,13.1595,10.5454,8.1497,5.98632,4.0697,2.4129,1.02636,-0.0779915,-0.885704,-1.38259,-1.55523,-1.56318,-1.57049,-1.57364,-1.57149,-1.56447,-1.55264,-1.53558,-1.51364,-1.48697,-1.45576,-1.42005,-1.38032,-1.33661,-1.28956,-1.23966,-1.18661,-1.13082,-1.07253,-1.01218,-0.949785,-0.886564,-0.821959,-0.757229,-0.693475,-0.629134,-0.727095,-0.505996,-0.451758,-0.62357,-0.499736,-0.490335,-0.429626,-0.376328,-0.305097,-0.250208,-0.19891,-0.155198,-0.117826,-0.0810051,-0.0407524,-0.00286102,0.0307188,0.0573716,0.0860987,0.108481,0.12431,0.136713,0.142861,0.148551,0.149808,0.149623,0.145592,0.137924,0.127851,0.116852,0.104792,0.0938349,0.0854578,0.0795684,0.0737286,0.0691791,0.0678396,0.0682497,0.0696268,0.0708208,0.0703773,0.0677452,0.0649438,0.0628977,0.0588546,0.0524254,0.0392332,0.0393424,0.0558257,0.0566998,0.0505705,0.0470505,0.0408559,0.0321574,0.0222287,0.0135694,0.00635195,-0.000571251,-0.00778723,-0.0153208,-0.0219254,-0.0288658,-0.0364332,-0.0446949,-0.0515456,-0.0563326,-0.0598893,-0.0475788,-0.0523763,-0.059269,-0.0621257,-0.0646868,-0.0652614,-0.0650749,-0.063282,-0.0609574,-0.0572343,-0.0538054,-0.0505271,-0.0456758,-0.04319,-0.0414267,-0.0416965,-0.0413237,-0.0402694,-0.0383334,-0.0348077,-0.0317602,-0.028295,-0.0250049,-0.0221276,-0.0187302,-0.0176058,-0.0158262,-0.0140343,-0.0125446,-0.0107398])"
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   "execution_count": 5,
   "metadata": {
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   "source": [
    "from matplotlib import pyplot as plt\n",
    "import numpy as np\n",
    "class SimpleDmp:\n",
    "    '''\n",
    "    A simple Ijspeert dmp with configurable forcing term\n",
    "    '''\n",
    "    def __init__(self, executionTime, startPos, startVel, goalPos):\n",
    "        self.T = executionTime\n",
    "        self.alpha = 25.0\n",
    "        self.beta = 6.25\n",
    "        self.g = goalPos\n",
    "        self.y = startPos\n",
    "        self.z = self.T * startVel;\n",
    "\n",
    "    def step(self, dt):\n",
    "        f = 0.0\n",
    "        zd = ((self.alpha * (self.beta * (self.g - self.y)- self.z)) / self.T) * dt\n",
    "        yd = self.z / self.T * dt\n",
    "        self.y += yd\n",
    "        self.z += zd\n",
    "\n",
    "    def run(self, dt, startT, endT):\n",
    "        '''\n",
    "        runs the whole dmp and returns ([ts], [ys], [yds])\n",
    "        '''\n",
    "        ts = []\n",
    "        ys = []\n",
    "        yds = []\n",
    "        t = startT\n",
    "        while t < endT:\n",
    "            ts.append(t)\n",
    "            ys.append(self.y)\n",
    "            yds.append(self.z / self.T)\n",
    "            t += dt\n",
    "            self.step(dt)\n",
    "        ts.append(t)\n",
    "        ys.append(self.y)\n",
    "        yds.append(self.z / self.T)\n",
    "\n",
    "        return (ts, ys, yds)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
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   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      ":0: FutureWarning: IPython widgets are experimental and may change in the future.\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import pylab\n",
    "#mpl.rcParams['figure.max_open_warning'] = 999999999\n",
    "from ipywidgets import StaticInteract, RangeWidget\n",
    "from IPython.html.widgets import interact\n",
    "pylab.rcParams['figure.figsize'] = (12.0, 6.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def plotPD(g = 5.0, y_start=0.0, yd_start= 0.0):\n",
    "    dt = 0.01    \n",
    "    T = 1.0\n",
    "    dmp = SimpleDmp(T, y_start, yd_start, g)\n",
    "    (ts, ys, yds) = dmp.run(dt, 0.0, T)\n",
    "    plt.plot(ts, ys)\n",
    "    plt.xlim(0, 1.5)\n",
    "    plt.ylim(-1.3, 6.3)\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"Position\")\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def plotPDT(g = 5.0, T = 1.0):\n",
    "    y_start = 0.0\n",
    "    yd_start = 0.0\n",
    "    dt = 0.01    \n",
    "    dmp = SimpleDmp(T, y_start, yd_start, g)\n",
    "    (ts, ys, yds) = dmp.run(dt, 0.0, T)\n",
    "    plt.plot(ts, ys)\n",
    "    plt.xlim(0, 1.5)\n",
    "    plt.ylim(-1.3, 6.3)\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"Position\")\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
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   "outputs": [],
   "source": [
    "from math import sin\n",
    "class SinDmp:\n",
    "    '''\n",
    "    A simple Ijspeert dmp with sin(t) forcing term\n",
    "    '''\n",
    "    def __init__(self, executionTime, startPos, startVel, goalPos, s):\n",
    "        self.T = executionTime\n",
    "        self.alpha = 25.0\n",
    "        self.beta = 6.25\n",
    "        self.g = goalPos\n",
    "        self.y = startPos\n",
    "        self.z = self.T * startVel;\n",
    "        self.t = 0.0\n",
    "        self.s = s\n",
    "        self.f = sin(self.t * 10) * self.s\n",
    "\n",
    "    def step(self, dt):\n",
    "        self.f = sin(self.t * 10) * self.s\n",
    "        zd = ((self.alpha * (self.beta * (self.g - self.y)- self.z) + self.f) / self.T) * dt\n",
    "        yd = self.z / self.T * dt\n",
    "        self.y += yd\n",
    "        self.z += zd\n",
    "        self.t += dt\n",
    "\n",
    "    def run(self, dt, startT, endT):\n",
    "        '''\n",
    "        runs the whole dmp and returns ([ts], [ys], [yds])\n",
    "        '''\n",
    "        ts = []\n",
    "        ys = []\n",
    "        yds = []\n",
    "        fs = []\n",
    "        t = startT\n",
    "        while t < endT:\n",
    "            ts.append(t)\n",
    "            ys.append(self.y)\n",
    "            yds.append(self.z / self.T)\n",
    "            fs.append(self.f)\n",
    "            t += dt\n",
    "            self.step(dt)\n",
    "        ts.append(t)\n",
    "        ys.append(self.y)\n",
    "        yds.append(self.z / self.T)\n",
    "        fs.append(self.f)\n",
    "\n",
    "        return (ts, ys, yds, fs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
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   "outputs": [],
   "source": [
    "def plotSin(K = 1.0):\n",
    "    g = 5.0\n",
    "    T = 1.0\n",
    "    dt = 0.01    \n",
    "    y_start = 0.0\n",
    "    yd_start = 0.0\n",
    "    dmp = SinDmp(T, y_start, yd_start, g, K)\n",
    "    (ts, ys, yds, fs) = dmp.run(dt, 0.0, T)\n",
    "    plt.plot(ts, ys)\n",
    "    plt.plot(ts, fs)\n",
    "    plt.xlim(0, 1.5)\n",
    "    plt.ylim(-0.3, 6.3)\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"Position\")\n",
    "    plt.legend([\"dmp\", \"sin\"])\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
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   "source": [
    "from math import log\n",
    "class CS:\n",
    "    def __init__(self, executionTime, lastPhaseValue = 0.01):\n",
    "        self.T = executionTime\n",
    "        self.alpha = -log(lastPhaseValue)\n",
    "        self.z = 1.0 #current phase value\n",
    "        self.t = 0.0#current time\n",
    "\n",
    "    def step(self, dt):\n",
    "        '''\n",
    "        initially the cs is at phase 1. the first call to step will move it.\n",
    "        '''\n",
    "        self.z += (-self.alpha * self.z / self.T) * dt\n",
    "        self.t += dt\n",
    "        return self.z\n",
    "\n",
    "    def get_phases(self, times):\n",
    "        return np.exp(-self.alpha / self.T * times)\n",
    "\n",
    "    def reset(self):\n",
    "        self.z = 1.0\n",
    "        self.t = 0.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plotCS(T = 1.0):\n",
    "    dt = 0.01\n",
    "    cs = CS(T)\n",
    "    times = []\n",
    "    values = []\n",
    "\n",
    "    t = 0.0\n",
    "    while(t < T):\n",
    "        times.append(t)\n",
    "        values.append(cs.z)\n",
    "        cs.step(dt)\n",
    "        t += dt\n",
    "    #get value from last step\n",
    "    times.append(t)\n",
    "    values.append(cs.z)\n",
    "    plt.plot(times, values)\n",
    "    plt.xlim(-0.1, 2.0)\n",
    "    plt.ylim(-0.1, 1.1)\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"Phase\")\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
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   "outputs": [],
   "source": [
    "from math import sin\n",
    "class SinDmpWithCS:\n",
    "    '''\n",
    "    A simple Ijspeert dmp with sin(t) forcing term\n",
    "    '''\n",
    "    def __init__(self, executionTime, startPos, startVel, goalPos, s):\n",
    "        self.T = executionTime\n",
    "        self.alpha = 25.0\n",
    "        self.beta = 6.25\n",
    "        self.g = goalPos\n",
    "        self.y = startPos\n",
    "        self.z = self.T * startVel;\n",
    "        self.t = 0.0\n",
    "        self.s = s\n",
    "        self.cs = CS(executionTime)\n",
    "\n",
    "    def step(self, dt):\n",
    "        f = sin(self.t * 10) * self.s\n",
    "        phase = self.cs.step(dt)\n",
    "        f *= phase\n",
    "        zd = ((self.alpha * (self.beta * (self.g - self.y)- self.z) + f) / self.T) * dt\n",
    "        yd = self.z / self.T * dt\n",
    "        self.y += yd\n",
    "        self.z += zd\n",
    "        self.t += dt\n",
    "\n",
    "    def run(self, dt, startT, endT):\n",
    "        '''\n",
    "        runs the whole dmp and returns ([ts], [ys], [yds])\n",
    "        '''\n",
    "        ts = []\n",
    "        ys = []\n",
    "        yds = []\n",
    "        t = startT\n",
    "        while t < endT:\n",
    "            ts.append(t)\n",
    "            ys.append(self.y)\n",
    "            yds.append(self.z / self.T)\n",
    "            t += dt\n",
    "            self.step(dt)\n",
    "        ts.append(t)\n",
    "        ys.append(self.y)\n",
    "        yds.append(self.z / self.T)\n",
    "\n",
    "        return (ts, ys, yds)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def plotSinCS(sin_scale = 1.0, T = 1.0, g=5.0):\n",
    "    dt = 0.01    \n",
    "    y_start = 0.0\n",
    "    yd_start = 0.0\n",
    "    s = sin_scale\n",
    "    dmp = SinDmpWithCS(T, y_start, yd_start, g, s)\n",
    "    (ts, ys, yds) = dmp.run(dt, 0.0, T)\n",
    "    plt.plot(ts, ys)\n",
    "    plt.xlim(0, 1.5)\n",
    "    plt.ylim(-0.3, 6.3)\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"Position\")\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from math import sin\n",
    "class SinDmpWithCS2:\n",
    "    '''\n",
    "    A simple Ijspeert dmp with sin(z) forcing term\n",
    "    '''\n",
    "    def __init__(self, executionTime, startPos, startVel, goalPos, s):\n",
    "        self.T = executionTime\n",
    "        self.alpha = 25.0\n",
    "        self.beta = 6.25\n",
    "        self.g = goalPos\n",
    "        self.y = startPos\n",
    "        self.z = self.T * startVel;\n",
    "        self.s = s\n",
    "        self.cs = CS(executionTime)\n",
    "\n",
    "    def step(self, dt):\n",
    "        phase = self.cs.step(dt)\n",
    "        f = sin((1.0 - phase) * 10) * self.s\n",
    "        f *= phase\n",
    "        zd = ((self.alpha * (self.beta * (self.g - self.y)- self.z) + f) / self.T) * dt\n",
    "        yd = self.z / self.T * dt\n",
    "        self.y += yd\n",
    "        self.z += zd\n",
    "\n",
    "    def run(self, dt, startT, endT):\n",
    "        '''\n",
    "        runs the whole dmp and returns ([ts], [ys], [yds])\n",
    "        '''\n",
    "        ts = []\n",
    "        ys = []\n",
    "        yds = []\n",
    "        t = startT\n",
    "        while t < endT:\n",
    "            ts.append(t)\n",
    "            ys.append(self.y)\n",
    "            yds.append(self.z / self.T)\n",
    "            t += dt\n",
    "            self.step(dt)\n",
    "        ts.append(t)\n",
    "        ys.append(self.y)\n",
    "        yds.append(self.z / self.T)\n",
    "\n",
    "        return (ts, ys, yds)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def plotSinCS2(sin_scale = 1.0, T = 1.0, g=5.0):\n",
    "    dt = 0.01    \n",
    "    y_start = 0.0\n",
    "    yd_start = 0.0\n",
    "    s = sin_scale\n",
    "    dmp = SinDmpWithCS2(T, y_start, yd_start, g, s)\n",
    "    (ts, ys, yds) = dmp.run(dt, 0.0, T)\n",
    "    plt.plot(ts, ys)\n",
    "    plt.xlim(0, 1.5)\n",
    "    plt.ylim(-0.3, 6.3)\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"Position\")\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "class Rbf:\n",
    "    '''\n",
    "    A simple radial basis function approximator\n",
    "    '''\n",
    "    def __init__(self, cs, executionTime, numWeights = 30, overlap = 0.1):\n",
    "        '''\n",
    "        The cs is only needed to spread the centers.\n",
    "        ExecutionTime is only needed to spread the centers. needs to be the original executionTime the the weights have been learned with\n",
    "        '''\n",
    "        self.numWeights = numWeights\n",
    "        self.cs = cs\n",
    "        self.executionTime = executionTime\n",
    "        #evenly spread the centers throughout the execution time\n",
    "        centers_in_time = np.linspace(start=0.0, stop=executionTime, num=numWeights)\n",
    "        #and move them to phase space\n",
    "        self.centers = np.exp(-cs.alpha / executionTime * centers_in_time)\n",
    "        self.weights = np.zeros_like(self.centers)\n",
    "\n",
    "        #set the widths of each rbf according to overlap\n",
    "        self.widths = np.ndarray(numWeights)\n",
    "        log_overlap = -log(overlap)\n",
    "        for i in range(1, numWeights):\n",
    "            self.widths[i - 1] = log_overlap / ((self.centers[i] - self.centers[i - 1])**2)\n",
    "        self.widths[numWeights - 1] = self.widths[numWeights - 2];\n",
    "\n",
    "\n",
    "    def set_weights(self, weights):\n",
    "        assert(len(weights) == self.numWeights)\n",
    "        self.weights = weights\n",
    "\n",
    "    def evaluate(self, z):\n",
    "        psi = np.exp(-self.widths * (z - self.centers)**2)\n",
    "        nom = np.dot(self.weights, psi) * z\n",
    "        denom = np.sum(psi)\n",
    "        return nom / denom\n",
    "\n",
    "    def plot_gaussians(self):\n",
    "        points_in_time = np.linspace(0.0, executionTime * 1.3, num=600)\n",
    "        points_in_phase = self.cs.get_phases(points_in_time)\n",
    "        for i in range(self.numWeights):#for each guassian\n",
    "            values = np.exp(-self.widths[i] * (points_in_phase - self.centers[i])**2)\n",
    "            plt.plot(points_in_phase, values)\n",
    "            plt.plot(points_in_time, values)\n",
    "\n",
    "    def plot_function(self):\n",
    "        points_in_time = np.linspace(0.0, executionTime * 1.3, num=600)\n",
    "        points_in_phase = self.cs.get_phases(points_in_time)\n",
    "        values = []\n",
    "        for z in points_in_phase:\n",
    "            values.append(self.evaluate(z))\n",
    "        plt.plot(points_in_time, values)\n",
    "\n",
    "    def psi(self, i, phases):\n",
    "        '''\n",
    "        evaluates the i'th gaussian at the specified phases and returns the results as vector\n",
    "        '''\n",
    "        assert(i < self.numWeights)\n",
    "        assert(i >= 0)\n",
    "        return np.exp(-self.widths[i] * (phases - self.centers[i])**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "class DmpWithImitation:\n",
    "    '''\n",
    "    A simple Ijspeert dmp with forcing term\n",
    "    '''\n",
    "    def __init__(self, executionTime, startPos, startVel, goalPos, cs, numWeights, overlap):\n",
    "        self.T = executionTime\n",
    "        self.cs = cs\n",
    "        self.alpha = 25.0\n",
    "        self.beta = 6.25\n",
    "        self.g = goalPos\n",
    "        self.y = startPos\n",
    "        self.startPos = startPos\n",
    "        self.z = self.T * startVel;\n",
    "        self.startZ = self.z\n",
    "        self.rbf = Rbf(cs, executionTime, numWeights, overlap)\n",
    "\n",
    "    def step(self, dt):\n",
    "        z = self.cs.step(dt)\n",
    "        f = self.rbf.evaluate(z)\n",
    "            \n",
    "        zd = ((self.alpha * (self.beta * (self.g - self.y)- self.z) + f) / self.T) * dt\n",
    "        yd = self.z / self.T * dt\n",
    "        self.y += yd\n",
    "        self.z += zd\n",
    "\n",
    "    def imitate(self, times, positions, dt):\n",
    "        '''\n",
    "        first position at t=0 and last position at t = executionTime\n",
    "        dt = sampling dt\n",
    "        '''\n",
    "        velocities = np.gradient(positions, dt)\n",
    "        accelerations = np.gradient(velocities, dt)\n",
    "        goal = positions[len(positions) - 1]\n",
    "        references = self.T**2 * accelerations - self.alpha * (self.beta * (goal - positions) - self.T * velocities)\n",
    "        phases = self.cs.get_phases(times)\n",
    "        weights = np.ndarray(self.rbf.numWeights)\n",
    "        for i in range(self.rbf.numWeights):\n",
    "            psi = self.rbf.psi(i, phases)\n",
    "            psiD = np.diag(psi)\n",
    "            weights[i] = np.linalg.inv([[np.dot(phases.T, np.dot(psiD, phases))]]) * np.dot(phases.T, np.dot(psiD, references))\n",
    "        self.rbf.set_weights(weights)\n",
    "\n",
    "\n",
    "    def run(self, dt, startTime = 0.0, endTime = None):\n",
    "        '''\n",
    "        runs the whole dmp and returns ([ts], [ys], [yds])\n",
    "        '''\n",
    "        ts = []\n",
    "        ys = []\n",
    "        yds = []\n",
    "        t = startTime\n",
    "        if endTime is None:\n",
    "            endTime = self.T\n",
    "        while t < endTime:\n",
    "            ts.append(t)\n",
    "            ys.append(self.y)\n",
    "            yds.append(self.z / self.T)\n",
    "            t += dt\n",
    "            self.step(dt)\n",
    "        ts.append(t)\n",
    "        ys.append(self.y)\n",
    "        yds.append(self.z / self.T)\n",
    "        return (ts, ys, yds)\n",
    "\n",
    "    def reset(self, cs, goal, executionTime, start):\n",
    "        self.cs = cs\n",
    "        self.g = goal\n",
    "        self.T = executionTime\n",
    "        self.y = start\n",
    "        self.z = self.startZ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2fe4950>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotImitateDmp>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def plotImitateDmp(g=values[-1], T=0.01 * len(values), y0=values[0]):\n",
    "    dt = 0.01\n",
    "    executionTime = dt * len(values)\n",
    "\n",
    "    lastPhaseValue = 0.1\n",
    "    numWeights = 100\n",
    "    overlap = 0.2\n",
    "    endTime = executionTime\n",
    "    times, step = np.linspace(start=0.0, stop=executionTime, num = len(values), retstep=True)\n",
    "    endPos = values[-1]\n",
    "\n",
    "\n",
    "    cs = CS(executionTime, lastPhaseValue)\n",
    "    dmp = DmpWithImitation(executionTime, values[0], (values[1] - values[0]) / dt, values[len(values) - 1], cs, numWeights, overlap)\n",
    "    dmp.imitate(times, values, dt)\n",
    "    \n",
    "    cs2 = CS(T, lastPhaseValue)\n",
    "    dmp.reset(cs2, g, T, y0)\n",
    "    (ts, ys, yds) = dmp.run(0.001)\n",
    "\n",
    "    plt.plot(times, values,\"r\")\n",
    "    plt.plot(ts, ys)\n",
    "    plt.legend([\"Demo\", \"Imitated\"], loc=\"upper left\")\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"X Position\")\n",
    "    plt.xlim(0.0, 0.75)\n",
    "    plt.ylim(-100, 200)\n",
    "    plt.show()\n",
    "    \n",
    "interact(plotImitateDmp, g=(-10, 200, 1), T=(0.1, 1.5, 0.1), y0=(-30, 30, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "class DmpWithImitationScale:\n",
    "    '''\n",
    "    A simple Ijspeert dmp with forcing term\n",
    "    '''\n",
    "    def __init__(self, executionTime, startPos, startVel, goalPos, cs, numWeights, overlap):\n",
    "        self.T = executionTime\n",
    "        self.cs = cs\n",
    "        self.alpha = 25.0\n",
    "        self.beta = 6.25\n",
    "        self.g = goalPos\n",
    "        self.y = startPos\n",
    "        self.startPos = startPos\n",
    "        self.z = self.T * startVel;\n",
    "        self.startZ = self.z\n",
    "        self.rbf = Rbf(cs, executionTime, numWeights, overlap)\n",
    "        self.amplitude = 0\n",
    "\n",
    "    def step(self, dt):\n",
    "        z = self.cs.step(dt)\n",
    "        f = self.rbf.evaluate(z) * (self.g - self.startPos) / self.amplitude\n",
    "            \n",
    "        zd = ((self.alpha * (self.beta * (self.g - self.y)- self.z) + f) / self.T) * dt\n",
    "        yd = self.z / self.T * dt\n",
    "        self.y += yd\n",
    "        self.z += zd\n",
    "\n",
    "    def imitate(self, times, positions, dt):\n",
    "        '''\n",
    "        first position at t=0 and last position at t = executionTime\n",
    "        dt = sampling dt\n",
    "        '''\n",
    "        self.amplitude = positions[-1] - positions[0]\n",
    "        velocities = np.gradient(positions, dt)\n",
    "        accelerations = np.gradient(velocities, dt)\n",
    "        goal = positions[len(positions) - 1]\n",
    "        references = self.T**2 * accelerations - self.alpha * (self.beta * (goal - positions) - self.T * velocities)\n",
    "        phases = self.cs.get_phases(times)\n",
    "        weights = np.ndarray(self.rbf.numWeights)\n",
    "        for i in range(self.rbf.numWeights):\n",
    "            psi = self.rbf.psi(i, phases)\n",
    "            psiD = np.diag(psi)\n",
    "            weights[i] = np.linalg.inv([[np.dot(phases.T, np.dot(psiD, phases))]]) * np.dot(phases.T, np.dot(psiD, references))\n",
    "        self.rbf.set_weights(weights)\n",
    "\n",
    "\n",
    "    def run(self, dt, startTime = 0.0, endTime = None):\n",
    "        '''\n",
    "        runs the whole dmp and returns ([ts], [ys], [yds])\n",
    "        '''\n",
    "        ts = []\n",
    "        ys = []\n",
    "        yds = []\n",
    "        t = startTime\n",
    "        if endTime is None:\n",
    "            endTime = self.T\n",
    "        while t < endTime:\n",
    "            ts.append(t)\n",
    "            ys.append(self.y)\n",
    "            yds.append(self.z / self.T)\n",
    "            t += dt\n",
    "            self.step(dt)\n",
    "        ts.append(t)\n",
    "        ys.append(self.y)\n",
    "        yds.append(self.z / self.T)\n",
    "        return (ts, ys, yds)\n",
    "\n",
    "    def reset(self, cs, goal, executionTime, start):\n",
    "        self.cs = cs\n",
    "        self.g = goal\n",
    "        self.T = executionTime\n",
    "        self.y = start\n",
    "        self.z = self.startZ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
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       "rkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x3229890>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotImitateScaleDmp>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def plotImitateScaleDmp(g=values[-1], T=0.01 * len(values), y0=values[0]):\n",
    "    dt = 0.01\n",
    "    executionTime = dt * len(values)\n",
    "\n",
    "    lastPhaseValue = 0.1\n",
    "    numWeights = 100\n",
    "    overlap = 0.2\n",
    "    endTime = executionTime\n",
    "    times, step = np.linspace(start=0.0, stop=executionTime, num = len(values), retstep=True)\n",
    "    endPos = values[-1]\n",
    "\n",
    "\n",
    "    cs = CS(executionTime, lastPhaseValue)\n",
    "    dmp = DmpWithImitationScale(executionTime, values[0], (values[1] - values[0]) / dt, values[len(values) - 1], cs, numWeights, overlap)\n",
    "    dmp.scale = True\n",
    "    dmp.imitate(times, values, dt)\n",
    "    \n",
    "    cs2 = CS(T, lastPhaseValue)\n",
    "    dmp.reset(cs2, g, T, y0)\n",
    "    (ts, ys, yds) = dmp.run(0.001)\n",
    "\n",
    "    plt.plot(times, values,\"r\")\n",
    "    plt.plot(ts, ys)\n",
    "    plt.legend([\"Demo\", \"Imitated\"], loc=\"upper left\")\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"X Position\")\n",
    "    plt.xlim(0.0, 0.75)\n",
    "    plt.ylim(-100, 200)\n",
    "    plt.show()\n",
    "    \n",
    "interact(plotImitateScaleDmp, g=(-10, 200, 1), T=(0.1, 1.5, 0.1), y0=(-30, 30, 1))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x3215490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotPerturbateDmp>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def plotPerturbateDmp(p_loc=0.2, p_strength=1.0):\n",
    "    dt = 0.01\n",
    "    executionTime = dt * len(values)\n",
    "\n",
    "    lastPhaseValue = 0.1\n",
    "    numWeights = 100\n",
    "    overlap = 0.2\n",
    "    endTime = executionTime\n",
    "    times, step = np.linspace(start=0.0, stop=executionTime, num = len(values), retstep=True)\n",
    "    endPos = values[-1]\n",
    "\n",
    "\n",
    "    cs = CS(executionTime, lastPhaseValue)\n",
    "    dmp = DmpWithImitation(executionTime, values[0], (values[1] - values[0]) / dt, values[len(values) - 1], cs, numWeights, overlap)\n",
    "    dmp.imitate(times, values, dt)\n",
    "    \n",
    "    (ts, ys, _) = dmp.run(0.001, endTime=p_loc)\n",
    "    dmp.y += p_strength\n",
    "    (ts2, ys2, _) = dmp.run(0.001, startTime=p_loc)\n",
    "    ts.extend(ts2)\n",
    "    ys.extend(ys2)\n",
    "    plt.plot(times, values,\"r\")\n",
    "    plt.plot(ts, ys)\n",
    "    plt.legend([\"Demo\", \"Imitated\"], loc=\"upper left\")\n",
    "    plt.xlabel(\"Time\")\n",
    "    plt.ylabel(\"X Position\")\n",
    "    plt.xlim(0.0, 0.75)\n",
    "    plt.ylim(-100, 200)\n",
    "    plt.show()\n",
    "    \n",
    "interact(plotPerturbateDmp, p_loc=(0.01, 0.5, 0.01), p_strength=(-200, 200, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       "table,td,tr,th {border:none!important}\n",
       "</style>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%html\n",
    "<style>\n",
    "table,td,tr,th {border:none!important}\n",
    "</style>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#Introduction to Discrete Dynamic Movement Primitives\n",
    "  \n",
    "  \n",
    "\n",
    "### Arne Böckmann "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##What are DMPs\n",
    "<table>\n",
    "  <tr>\n",
    "    <td>\n",
    "    <ul style=\"list-style-type:disc\">\n",
    "        <li>Dynamical systems</li>\n",
    "        <li>Represent goal directed movements in joint or task space </li>\n",
    "        <li>Guaranteed to converge to the goal</li>\n",
    "        <li>Scale spatial and temporal</li>\n",
    "        <li>Robust against pertubations</li>\n",
    "        <li>Adaptable online</li>\n",
    "        <li>Extendable</li>\n",
    "        <li>Learnable from demonstration</li>\n",
    "    </ul>\n",
    "    </td>\n",
    "    <td><img src=\"dmp.png\"></td>\n",
    "  </tr>\n",
    "</table>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Basic Idea\n",
    "\n",
    "* Use well understood stable dynamical system with convenient properties\n",
    "* Add nonlinear terms to achieve the desired movement behavior\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Damped Spring Model\n",
    "\\begin{align}\n",
    "\\dot{z} &= \\alpha_z(\\beta_z(g-y)-z)\\\\\n",
    "\\dot{y} &= z\n",
    "\\end{align}\n",
    "\n",
    "$y$ - position, $g$ - goal position\n",
    "\n",
    "$\\alpha_z$, $\\beta_z$ - Dampening constants. Use $\\beta_z = \\alpha_z/4$ for critical dampening.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Damped Spring Model\n",
    "Difference Equations\n",
    "\\begin{align}\n",
    "z_t &= z_{t-1} + \\alpha_z(\\beta_z(g-y_{t-1})-z_{t-1}) * \\Delta t\\\\\n",
    "y_t &= y_{t-1} + z_{t-1} * \\Delta t\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Damped Spring Model (Demo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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       "yQAAAIAdSjIAAABgh5IMAAAA2Pn/bDmk6BOXccEAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x33b7810>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(plotPD, g=(-2.0, 6.0, 0.1), y_start=(-1.0, 2.0, 0.1), yd_start=(-50.0, 50.0, 5.0))     "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Adding temporal scaling\n",
    "* Add temporal scaling factor $\\color{red}\\tau$\n",
    "\\begin{align}\n",
    "\\color{red}\\tau\\dot{z} &= \\alpha_z(\\beta_z(g-y)-z)\\\\\n",
    "\\color{red}\\tau\\dot{y} &= z\\\\\\\\\n",
    "z_t &= z_{t-1} + \\frac{\\alpha_z(\\beta_z(g-y_{t-1})-z_{t-1}) * \\Delta t}{\\color{red}\\tau}\\\\\n",
    "y_t &= y_{t-1} + \\frac{z_{t-1} * \\Delta t}{\\color{red}\\tau}\n",
    "\\end{align}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Adding temporal scaling (Demo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x3217790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(plotPDT, g=(-2.0, 6.0, 0.1), T=(0.01, 1.5, 0.1))     "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Shaping the Trajectory (The Transformation System)\n",
    "Introduce the forcing term $\\color{red}f$\n",
    "\\begin{align}\n",
    "z_t &= z_{t-1} + \\frac{\\alpha_z(\\beta_z(g-y_{t-1})-z_{t-1} + \\color{red}f) * \\Delta t}{\\tau}\\\\\n",
    "\\end{align}\n",
    "* 'forces' the system into a certain shape.\n",
    "* could be a function of time: $f(t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "###Shaping the Trajectory (Demo)\n",
    "* set $f(t) = K * sin(t)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "AElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2fe2450>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(plotSin, K=(0.1, 1000.0, 0.1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Restoring Goal Convergence\n",
    "* Influence of $f$ should diminish over time\n",
    "* Introducing the phase $\\color{red}s$\n",
    "\\begin{align}\n",
    "z_t &= z_{t-1} + \\frac{\\alpha_z(\\beta_z(g-y_{t-1})-z_{t-1} + \\color{red}sf) * \\Delta t}{\\tau}\\\\\n",
    "\\end{align}\n",
    "\n",
    "* $\\color{red}s$ starts at 1 and exponentially decays to 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Restoring Goal Convergence (The Canonical System)\n",
    "$s$ can be generated by the following system:\n",
    "\\begin{align}\n",
    "\\tau \\dot{s} = -\\alpha_s s\\\\\\\\\n",
    "s_t = s_{t-1} + \\frac{-\\alpha_s \\Delta t}{T}\n",
    "\\end{align}\n",
    "* $\\alpha_s$ defines how fast the system decays.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Canonical System (Demo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x3391510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotCS>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(plotCS, T=(0.1, 2.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Sin Forcing Term and Canonical System (Demo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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       "/h8bU6eWIMDvDQAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x3207390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotSinCS>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(plotSinCS, sin_scale=(-1000, 1000, 1.0), T=(0.1, 1.5, 0.1), g=(0.0, 6.0, 0.1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Restoring Temporal Scaling\n",
    " * Because $f$ is time dependent\n",
    " * Solution: Replace the time dependency with a phase dependency\n",
    " * $f = f(s)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Restoring Temporal Scaling (Demo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x322f8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotSinCS2>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(plotSinCS2, sin_scale=(-1000, 1000, 1.0), T=(0.1, 1.5, 0.1), g=(0.0, 6.0, 0.1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Imitating arbitrary Trajectories\n",
    "* Set $f(s) = \\frac{\\sum_{i=1}^N \\psi_i(s)w_i}{\\sum_{i=1}^N \\psi_i(s)}$\n",
    " * $\\psi_i$ are fixed basis functions (e.g. radial basis functions)\n",
    " * $N$ is the number of basis functions/weights\n",
    " * $f$ can be shaped by adjusting the $w_i$\n",
    " * $f$ can imitate any given trajectory using locally weighted regression (not part of this talk)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "####Imitating arbitrary Trajectories\n",
    "This is a part of B-Human's current kick trajectory"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x3694950>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x33b9490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.linspace(start=0.0, stop=0.01 * len(values), num = len(values)), values); plt.xlabel(\"Time\"); plt.ylabel(\"X Position\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Imitating arbitrary Trajectories (Demo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x3215c90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotImitateDmp>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(plotImitateDmp, g=(-10, 200, 1), T=(0.1, 1.5, 0.1), y0=(-30, 30, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Adding Spatial Scaling\n",
    "* $w_i$ are learned for one specific goal position\n",
    " * Are too strong/weak for other goal positions\n",
    "* Solution: Scale the forcing term according to the difference $\\color{red}d$ between the original trajectory and the modified one\n",
    "\\begin{align}\n",
    "\\color{red}d &= (y_e - y_0)/(y_{de} - y_{d0})\\\\\n",
    "z_t &= z_{t-1} + \\frac{\\alpha_z(\\beta_z(g-y_{t-1})-z_{t-1} + \\color{red}dsf) * \\Delta t}{\\tau}\\\\\n",
    "\\end{align}\n",
    "\n",
    "   * $y_{de}$ and $y_{d0}$ - End and start position of the demonstrated trajectory"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Spatial Scaling (Demo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "AAAAN8BgDQAAAADgBhisAQAAAADcAIM1AAAAAIAbYLAGAAAAAHCDvwHfIGMNmC4DKgAAAABJRU5E\n",
       "rkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x369b950>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotImitateScaleDmp>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(plotImitateScaleDmp, g=(-10, 200, 1), T=(0.1, 1.5, 0.1), y0=(-30, 30, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Perturbation robustness"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x36bd550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotPerturbateDmp>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(plotPerturbateDmp, p_loc=(0.01, 0.5, 0.01), p_strength=(-200, 200, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Extension to multiple dimensions\n",
    "* As long as the dimensions are independent\n",
    " * Use one Transformation System per dimension\n",
    " * Use the same Canonical System for all dimensions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Extensions\n",
    "* Oscillating movements\n",
    "* Enforcing a goal velocity\n",
    "* Obstacle avoidance\n",
    "* Closed-loop perception-action control\n",
    "* Movement synchroization\n",
    "* Rotation Dmps (quaternion based and rotation matrix based)\n",
    "* ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Recap\n",
    "* Dmps are a toolbox to imitate movements and modify them on the fly\n",
    "* Robust against perturbations\n",
    "* Always converge to the goal in time\n",
    "* Can be executed and modified in real-time\n",
    "* Can be extended to multiple dimensions\n",
    "* Can be extended using arbitrary coupling terms to achieve any desired behavior"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Recap\n",
    "#### Parameters\n",
    "* $\\alpha_z$ and $\\beta_z$ are constant $\\rightarrow$ can be ignored\n",
    "* $\\alpha_s$ can be calculated automatically\n",
    "* Good values for the parameters of $f$ can be found automatically\n",
    "* The weights of $f$ can be learned from demonstration using locally weighted regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##Examples\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"400\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/Ge0GduY1rtE\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x320c290>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo('Ge0GduY1rtE')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"400\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/SH3bADiB7uQ\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x320ca90>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo('SH3bADiB7uQ')  "
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.3"
  }
 },
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}