Added robot steering example to UKF.

09_Unscented_Kalman_Filter.ipynb

@@ -289,9 +289,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x74c9e10>"
+       "<matplotlib.figure.Figure at 0x6ee7cf8>"
       ]
      },
      "metadata": {},
@@ -351,7 +351,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x525a358>"
+       "<matplotlib.figure.Figure at 0x44ae518>"
       ]
      },
      "metadata": {},
@@ -397,14 +397,14 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Difference in mean x=0.028, y=43.204\n"
+      "Difference in mean x=0.076, y=43.395\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8a7beb8>"
+       "<matplotlib.figure.Figure at 0x8572668>"
       ]
      },
      "metadata": {},
@@ -489,7 +489,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8a590b8>"
+       "<matplotlib.figure.Figure at 0x85570b8>"
       ]
      },
      "metadata": {},
@@ -527,7 +527,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x790ac18>"
+       "<matplotlib.figure.Figure at 0x789c6a0>"
       ]
      },
      "metadata": {},
@@ -572,7 +572,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x90abda0>"
+       "<matplotlib.figure.Figure at 0x78dad30>"
       ]
      },
      "metadata": {},
@@ -613,7 +613,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8a73080>"
+       "<matplotlib.figure.Figure at 0x7887cf8>"
       ]
      },
      "metadata": {},
@@ -726,7 +726,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa4f3198>"
+       "<matplotlib.figure.Figure at 0x8673208>"
       ]
      },
      "metadata": {},
@@ -765,14 +765,14 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Difference in mean x=-0.061, y=-0.124\n"
+      "Difference in mean x=0.032, y=0.245\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa673978>"
+       "<matplotlib.figure.Figure at 0x8b5a668>"
       ]
      },
      "metadata": {},
@@ -991,7 +991,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa55be80>"
+       "<matplotlib.figure.Figure at 0x907db00>"
       ]
      },
      "metadata": {},
@@ -1103,7 +1103,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa538cf8>"
+       "<matplotlib.figure.Figure at 0x8bf4358>"
       ]
      },
      "metadata": {},
@@ -1188,7 +1188,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa5c68d0>"
+       "<matplotlib.figure.Figure at 0x8bb4b38>"
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@@ -1366,7 +1366,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa584eb8>"
+       "<matplotlib.figure.Figure at 0x8b0a240>"
       ]
      },
      "metadata": {},
@@ -1376,7 +1376,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8a1b2b0>"
+       "<matplotlib.figure.Figure at 0x7a29c50>"
       ]
      },
      "metadata": {},
@@ -1386,7 +1386,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa678080>"
+       "<matplotlib.figure.Figure at 0x8551898>"
       ]
      },
      "metadata": {},
@@ -1492,7 +1492,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa8dddd8>"
+       "<matplotlib.figure.Figure at 0x8ba6550>"
       ]
      },
      "metadata": {},
@@ -1565,7 +1565,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x79124a8>"
+       "<matplotlib.figure.Figure at 0x788b518>"
       ]
      },
      "metadata": {},
@@ -1664,7 +1664,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa5842b0>"
+       "<matplotlib.figure.Figure at 0x7350630>"
       ]
      },
      "metadata": {},
@@ -1724,7 +1724,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAv8AAAEtCAYAAACf5Q+bAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa966860>"
+       "<matplotlib.figure.Figure at 0x8556518>"
       ]
      },
      "metadata": {},
@@ -1818,9 +1818,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7ff7a58>"
+       "<matplotlib.figure.Figure at 0x78b3b38>"
       ]
      },
      "metadata": {},
@@ -1872,7 +1872,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8a123c8>"
+       "<matplotlib.figure.Figure at 0x90a9e48>"
       ]
      },
      "metadata": {},
@@ -1974,7 +1974,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6ba630>"
+       "<matplotlib.figure.Figure at 0x90df278>"
       ]
      },
      "metadata": {},
@@ -2086,7 +2086,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x85e2cf8>"
+       "<matplotlib.figure.Figure at 0x86288d0>"
       ]
      },
      "metadata": {},
@@ -2127,7 +2127,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa59d940>"
+       "<matplotlib.figure.Figure at 0x6f1f518>"
       ]
      },
      "metadata": {},
@@ -2201,7 +2201,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6b7908>"
+       "<matplotlib.figure.Figure at 0x7a42a20>"
       ]
      },
      "metadata": {},
@@ -2320,7 +2320,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtsAAAEWCAYAAAC37ltfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa704f28>"
+       "<matplotlib.figure.Figure at 0x7310e48>"
       ]
      },
      "metadata": {},
@@ -2381,9 +2381,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6bb048>"
+       "<matplotlib.figure.Figure at 0x8b5ac50>"
       ]
      },
      "metadata": {},
@@ -2439,7 +2439,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa4adcf8>"
+       "<matplotlib.figure.Figure at 0x8afb908>"
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@@ -2535,7 +2535,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa8c39b0>"
+       "<matplotlib.figure.Figure at 0x8ba62e8>"
       ]
      },
      "metadata": {},
@@ -2886,7 +2886,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa66c278>"
+       "<matplotlib.figure.Figure at 0x85f1128>"
       ]
      },
      "metadata": {},
@@ -2896,7 +2896,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x80365f8>"
+       "<matplotlib.figure.Figure at 0x90da1d0>"
       ]
      },
      "metadata": {},
@@ -2906,7 +2906,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8a1bf28>"
+       "<matplotlib.figure.Figure at 0x90727b8>"
       ]
      },
      "metadata": {},
@@ -2976,7 +2976,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa907828>"
+       "<matplotlib.figure.Figure at 0x8ac29b0>"
       ]
      },
      "metadata": {},
@@ -3022,9 +3022,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x85eddd8>"
+       "<matplotlib.figure.Figure at 0x8518780>"
       ]
      },
      "metadata": {},
@@ -3032,9 +3032,9 @@
     },
     {
      "data": {
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       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa8a8550>"
+       "<matplotlib.figure.Figure at 0x8b00908>"
       ]
      },
      "metadata": {},
@@ -3042,9 +3042,9 @@
     },
     {
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x847a780>"
+       "<matplotlib.figure.Figure at 0x8b0a048>"
       ]
      },
      "metadata": {},
@@ -3343,7 +3343,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa5957f0>"
+       "<matplotlib.figure.Figure at 0x9090e80>"
       ]
      },
      "metadata": {},
@@ -3442,7 +3442,9 @@
     "    steering_angle = u[1]\n",
     "    if abs(steering_angle) < 0.0001:\n",
     "        # approximate straight line with huge radius\n",
-    "        r = 1.e-30\n",
+    "        r = 1.e30\n",
+    "        steering_angle = 1e-5\n",
+    "        \n",
     "    b = dist / wheelbase * tan(steering_angle)\n",
     "    r = wheelbase / tan(steering_angle) # radius\n",
     "\n",
@@ -3672,7 +3674,10 @@
     "dt = 1.0\n",
     "wheelbase = 0.5\n",
     "\n",
-    "def run_localization(landmarks, sigma_vel, sigma_steer, sigma_range, sigma_bearing):\n",
+    "def run_localization(cmds, landmarks, sigma_vel, sigma_steer, sigma_range, \n",
+    "                     sigma_bearing, ellipse_step=1, step=10):\n",
+    "    \n",
+    "    \n",
     "    points = MerweScaledSigmaPoints(n=3, alpha=.00001, beta=2, kappa=0)\n",
     "    ukf = UKF(dim_x=3, dim_z=2, fx=fx, hx=Hx, dt=dt, points=points,\n",
     "             x_mean_fn=state_mean, z_mean_fn=z_mean,\n",
@@ -3689,16 +3694,18 @@
     "    # plot landmarks\n",
     "    plt.scatter(landmarks[:, 0], landmarks[:, 1], marker='s', s=60)\n",
     "    \n",
-    "    for i in range(200):\n",
+    "    track = []\n",
+    "    for i, u in enumerate(cmds):\n",
     "        \n",
-    "        sim_pos = move(sim_pos, u, dt/10., wheelbase) # simulate robot\n",
-    "        plt.plot(sim_pos[0], sim_pos[1], ',', color='g')\n",
+    "        sim_pos = move(sim_pos, u, dt/step, wheelbase) # simulate robot\n",
+    "        track.append(sim_pos)\n",
     "\n",
-    "        if i % 10 == 0:\n",
+    "        if i % step == 0:\n",
     "            ukf.predict(fx_args=u)\n",
     "\n",
-    "            plot_covariance_ellipse((ukf.x[0], ukf.x[1]), ukf.P[0:2, 0:2], std=6,\n",
-    "                                    facecolor='b', alpha=0.08)\n",
+    "            if i % ellipse_step == 0:\n",
+    "                plot_covariance_ellipse((ukf.x[0], ukf.x[1]), ukf.P[0:2, 0:2], std=6,\n",
+    "                                        facecolor='b', alpha=0.08)\n",
     "\n",
     "            x, y = sim_pos[0], sim_pos[1]\n",
     "            for lmark in landmarks:\n",
@@ -3708,9 +3715,11 @@
     "\n",
     "                ukf.update(z, hx_args=(lmark,))\n",
     "\n",
-    "            plot_covariance_ellipse((ukf.x[0], ukf.x[1]), ukf.P[0:2, 0:2], std=6,\n",
-    "                                    facecolor='g', alpha=0.4)\n",
-    "\n",
+    "            if i % ellipse_step == 0:\n",
+    "                plot_covariance_ellipse((ukf.x[0], ukf.x[1]), ukf.P[0:2, 0:2], std=6,\n",
+    "                                        facecolor='g', alpha=0.4)\n",
+    "    track = np.array(track)\n",
+    "    plt.plot(track[:, 0], track[:,1], color='k', lw=2)\n",
     "    plt.axis('equal')\n",
     "    plt.title(\"UKF Robot localization\")\n",
     "    plt.show()\n",
@@ -3727,9 +3736,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8a1ac50>"
+       "<matplotlib.figure.Figure at 0x8bf6e10>"
       ]
      },
      "metadata": {},
@@ -3745,8 +3754,8 @@
    ],
    "source": [
     "landmarks = np.array([[5, 10], [10, 5], [15, 15]])\n",
-    "\n",
-    "ukf = run_localization(landmarks, sigma_vel=0.1, sigma_steer=np.radians(1),\n",
+    "cmds = [np.array([1.1, .01])]*200\n",
+    "ukf = run_localization(cmds, landmarks, sigma_vel=0.1, sigma_steer=np.radians(1),\n",
     "                      sigma_range=0.3, sigma_bearing=0.1)\n",
     "print(ukf.P.diagonal())"
    ]
@@ -3758,6 +3767,68 @@
     "The rest of the code runs the simulation and plots the results, and shouldn't need too much comment by now. I create a variable `landmarks` that contains the coordinates of the landmarks. I update the simulated robot position 10 times a second, but run the EKF only once. This is for two reasons. First, we are not using Runge Kutta to integrate the differental equations of motion, so a narrow time step allows our simulation to be more accurate. Second, it is fairly normal in embedded systems to have limited processing speed. This forces you to run your Kalman filter only as frequently as absolutely needed."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Steering the Robot\n",
+    "\n",
+    "While the robot was turning in the example above, we really can't say it was being driven realistically. The velocity and steering angles never changed, which doesn't pose much of a problem for the Kalman filter. We could implement a complicated PID controlled robot simulation, but I will just generate varying steering commands using NumPy's `linspace` method. I'll also add more landmarks as the robot will be travelling much further than in the first example."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 58,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x9156208>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "landmarks = np.array([[5, 10], [10, 5], [15, 15], [20, 5],\n",
+    "                      [0, 30], [50, 30], [40, 10]])\n",
+    "\n",
+    "\n",
+    "dt = 0.1\n",
+    "wheelbase = 0.5\n",
+    "sigma_range=0.3\n",
+    "sigma_bearing=0.1\n",
+    "\n",
+    "# accelerate from a stop\n",
+    "cmds = [[v, .0] for v in np.linspace(0.001, 1.1, 30)]\n",
+    "cmds.extend([cmds[-1]]*50)\n",
+    "\n",
+    "# turn left\n",
+    "v = cmds[-1][0]\n",
+    "cmds.extend([[v, a] for a in np.linspace(0, np.radians(2), 15)])\n",
+    "cmds.extend([cmds[-1]]*100)\n",
+    "\n",
+    "#turn right\n",
+    "cmds.extend([[v, a] for a in np.linspace(np.radians(2), -np.radians(2), 15)])\n",
+    "cmds.extend([cmds[-1]]*200)\n",
+    "\n",
+    "cmds.extend([[v, a] for a in np.linspace(-np.radians(2), 0, 15)])\n",
+    "cmds.extend([cmds[-1]]*50)\n",
+    "\n",
+    "cmds.extend([[v, a] for a in np.linspace(0, np.radians(1), 25)])\n",
+    "cmds.extend([cmds[-1]]*100)\n",
+    "\n",
+    "ukf = run_localization(cmds, landmarks, sigma_vel=0.1, sigma_steer=np.radians(1),\n",
+    "                      sigma_range=0.3, sigma_bearing=0.1, step=1,\n",
+    "                      ellipse_step=20)\n"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -3779,22 +3850,6 @@
     "\n",
     "- [3] Cholesky decomposition. Wikipedia. http://en.wikipedia.org/wiki/Cholesky_decomposition"
    ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "**orphan text**: I have found the literature on choosing values for $\\alpha$, $\\beta$, and $\\kappa$ to be rather lacking. Julier's original paper suggests a value of $n+\\kappa=3$ if the distribution is Gaussian, and if not a \"different choice of $\\kappa$ might be more appropriate\"[2]. He also describes it as a \"fine tuning\" parameter. So let's just explore what it does. I will let $n=1$ just to minimize the size of the arrays we need to look at."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {
@@ -3813,7 +3868,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.4.3"
+   "version": "3.4.1"
   }
  },
  "nbformat": 4,

experiments/ukfloc.py

@@ -44,7 +44,8 @@ def move(x, u, dt, wheelbase):
 
     if abs(steering_angle) < 0.0001:
         # approximate straight line with huge radius
-        r = 1.e-30
+        r = 1.e30
+        steering_angle = 1.e-5
     b = dist / wheelbase * tan(steering_angle)
     r = wheelbase / tan(steering_angle) # radius
 
@@ -90,13 +91,14 @@ def z_mean(sigmas, Wm):
 sigma_r = .3
 sigma_h =  .1#np.radians(1)
 sigma_steer =  np.radians(.01)
-dt = 1.0
+dt = 0.1
 wheelbase = 0.5
 
 m = array([[5, 10],
            [10, 5],
            [15, 15],
-           [20, 5]])
+           [20, 5],
+           [0, 30], [50, 30], [40, 10]])
 
 
 def fx(x, dt, u):
@@ -114,47 +116,84 @@ def Hx(x, landmark):
     Hx = array([dist, atan2(py - x[1], px - x[0]) - x[2]])
     return Hx
 
-points = MerweScaledSigmaPoints(n=3, alpha=1.e-3, beta=2, kappa=0)
+points = MerweScaledSigmaPoints(n=3, alpha=.1, beta=2, kappa=0)
 ukf= UKF(dim_x=3, dim_z=2, fx=fx, hx=Hx, dt=dt, points=points,
          x_mean_fn=state_mean, z_mean_fn=z_mean,
          residual_x=residual_x, residual_z=residual_h)
 ukf.x = array([2, 6, .3])
 ukf.P = np.diag([.1, .1, .2])
 ukf.R = np.diag([sigma_r**2, sigma_h**2])
-ukf.Q = np.zeros((3,3))
+ukf.Q = np.eye(3)*0.001
 
 
-u = array([1.1, .01])
+u = array([1.1, .0000001])
 
 xp = ukf.x.copy()
 
+
 plt.figure()
 plt.scatter(m[:, 0], m[:, 1])
 
-for i in range(200):
-    xp = move(xp, u, dt/10., wheelbase) # simulate robot
-    plt.plot(xp[0], xp[1], ',', color='g')
+cmds = [[v, .0] for v in np.linspace(0.001, 1.1, 30)]
+cmds.extend([cmds[-1]]*50)
 
-    if i % 10 == 0:
-        ukf.predict(fx_args=u)
+v = cmds[-1][0]
+cmds.extend([[v, a] for a in np.linspace(0, np.radians(2), 15)])
+cmds.extend([cmds[-1]]*100)
 
+cmds.extend([[v, a] for a in np.linspace(np.radians(2), -np.radians(2), 15)])
+cmds.extend([cmds[-1]]*200)
+
+cmds.extend([[v, a] for a in np.linspace(-np.radians(2), 0, 15)])
+cmds.extend([cmds[-1]]*50)
+
+cmds.extend([[v, a] for a in np.linspace(0, -np.radians(1), 25)])
+
+
+
+cmds = np.array(cmds)
+
+cindex = 0
+u = cmds[0]
+
+track = []
+
+while cindex < len(cmds):
+    u = cmds[cindex]
+    xp = move(xp, u, dt, wheelbase) # simulate robot
+    track.append(xp)
+
+
+
+    ukf.predict(fx_args=u)
+
+    if cindex % 20 == 0:
         plot_covariance_ellipse((ukf.x[0], ukf.x[1]), ukf.P[0:2, 0:2], std=3,
-                                facecolor='b', alpha=0.08)
+                                facecolor='b', alpha=0.18)
 
-        for lmark in m:
-            d = sqrt((lmark[0] - xp[0])**2 + (lmark[1] - xp[1])**2)  + randn()*sigma_r
-            a = atan2(lmark[1] - xp[1], lmark[0] - xp[0]) - xp[2] + randn()*sigma_h
-            z = np.array([d, a])
+    for lmark in m:
+        d = sqrt((lmark[0] - xp[0])**2 + (lmark[1] - xp[1])**2)  + randn()*sigma_r
+        a = atan2(lmark[1] - xp[1], lmark[0] - xp[0]) - xp[2] + randn()*sigma_h
+        z = np.array([d, a])
 
-            ukf.update(z, hx_args=(lmark,))
+        ukf.update(z, hx_args=(lmark,))
 
+    if cindex % 20 == 0:
         plot_covariance_ellipse((ukf.x[0], ukf.x[1]), ukf.P[0:2, 0:2], std=3,
                                 facecolor='g', alpha=0.4)
+    cindex += 1
 
 
-    #plt.plot(ekf.x[0], ekf.x[1], 'x', color='r')
+track = np.array(track)
+plt.plot(track[:, 0], track[:,1], color='k')
 
 plt.axis('equal')
 plt.title("UKF Robot localization")
 plt.show()
-print(ukf.P.diagonal())
+print(ukf.P.diagonal())
+
+
+
+
+
+