Fixed diverging UKF filter.

The UKF for the circle tracking was diverging. Also, I have partially started the section on the ILS, but much remains to be done on it.
2015-02-27 16:23:53 -08:00
1 parent 457a938ce4
commit 458b12956e
@@ -1,7 +1,7 @@
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 "metadata": {
  "name": "",
-  "signature": "sha256:5123227f02317fd83d71aa3ea353fa57c3d4ca1d4df752502464a6f416fc2d92"
+  "signature": "sha256:f703c39ccf197ee4e8df8f635110033a7e0d4b951edd2c89832e946c004b5836"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
@@ -260,7 +260,7 @@
       "output_type": "pyout",
       "prompt_number": 1,
       "text": [
-        "<IPython.core.display.HTML at 0x7f35736aca58>"
+        "<IPython.core.display.HTML at 0x7f3e8034d048>"
       ]
      }
     ],
@@ -300,9 +300,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556efc160>"
+        "<matplotlib.figure.Figure at 0x7f3e63c4aa90>"
       ]
      }
     ],
@@ -348,7 +348,7 @@
       "output_type": "display_data",
       "png": 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       "text": [
-        "<matplotlib.figure.Figure at 0x7f35736ae9e8>"
+        "<matplotlib.figure.Figure at 0x7f3e803539b0>"
       ]
      }
     ],
@@ -429,16 +429,16 @@
      {
       "metadata": {},
       "output_type": "display_data",
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truncated
+       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f35528a6fd0>"
+        "<matplotlib.figure.Figure at 0x7f3e5bc49780>"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
-        "Difference in mean x=-0.019, y=43.196\n"
+        "Difference in mean x=-0.029, y=43.180\n"
       ]
      }
     ],
@@ -482,7 +482,7 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f35529486d8>"
+        "<matplotlib.figure.Figure at 0x7f3e5bceaf60>"
       ]
      }
     ],
@@ -520,7 +520,7 @@
       "output_type": "display_data",
       "png": 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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556aaac88>"
+        "<matplotlib.figure.Figure at 0x7f3e637a9c88>"
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@@ -576,7 +576,7 @@
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d543c50>"
+        "<matplotlib.figure.Figure at 0x7f3e5ba31c18>"
       ]
      }
     ],
@@ -759,7 +759,7 @@
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556b16ba8>"
+        "<matplotlib.figure.Figure at 0x7f3e637e8588>"
       ]
      }
     ],
@@ -907,7 +907,7 @@
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       "png": 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       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d4b9a90>"
+        "<matplotlib.figure.Figure at 0x7f3e5a100b38>"
       ]
      }
     ],
@@ -1005,7 +1005,7 @@
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d4e26a0>"
+        "<matplotlib.figure.Figure at 0x7f3e5a179048>"
       ]
      },
      {
@@ -1070,7 +1070,7 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d4b2208>"
+        "<matplotlib.figure.Figure at 0x7f3e5bca7b38>"
       ]
      }
     ],
@@ -1302,7 +1302,7 @@
       "output_type": "display_data",
       "png": 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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556aaee80>"
+        "<matplotlib.figure.Figure at 0x7f3e5a0b2898>"
       ]
      },
      {
@@ -1310,7 +1310,7 @@
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a366a0>"
+        "<matplotlib.figure.Figure at 0x7f3e637325f8>"
       ]
      },
      {
@@ -1318,7 +1318,7 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f355290dc88>"
+        "<matplotlib.figure.Figure at 0x7f3e637851d0>"
       ]
      }
     ],
@@ -1381,7 +1381,7 @@
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAA0YAAAGkCAYAAAAVNzW4AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a56080>"
+        "<matplotlib.figure.Figure at 0x7f3e63836400>"
       ]
      },
      {
@@ -1430,7 +1430,7 @@
      "\n",
      "ac = ACSim(ac_pos, (100,0), 0.02)\n",
      "\n",
-      "kf = UKF(4, 2, dt, fx=f, hx=h, kappa=-1)\n",
+      "kf = UKF(4, 2, dt, fx=f, hx=h, kappa=0)\n",
      "\n",
      "kf.Q[0:2,0:2] = Q_discrete_white_noise(2, dt=dt, var=0.1)\n",
      "kf.Q[2:4,2:4] = Q_discrete_white_noise(2, dt=dt, var=0.1)\n",
@@ -1464,9 +1464,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a96860>"
+        "<matplotlib.figure.Figure at 0x7f3e637ae400>"
       ]
      },
      {
@@ -1553,7 +1553,7 @@
       "output_type": "display_data",
       "png": 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       "text": [
-        "<matplotlib.figure.Figure at 0x7f35569d8b38>"
+        "<matplotlib.figure.Figure at 0x7f3e636a6128>"
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@@ -1620,7 +1620,7 @@
       "output_type": "display_data",
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556ac2828>"
+        "<matplotlib.figure.Figure at 0x7f3e5a0335c0>"
       ]
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@@ -1672,7 +1672,7 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a8ec50>"
+        "<matplotlib.figure.Figure at 0x7f3e6381f2b0>"
       ]
      }
     ],
@@ -1738,12 +1738,12 @@
      "    return measurement(sa_pos, sb_pos, pos)\n",
      "\n",
      "\n",
-      "def moving_target_filter(std_noise):\n",
+      "def moving_target_filter(std_noise, Q_variance):\n",
      "    dt = 0.1\n",
      "    f = UKF(dim_x=4, dim_z=2, dt=dt, hx=hx, fx=fx, kappa=0.)\n",
      "    f.x = np.array([target_pos[0], 1., target_pos[1], 1.])\n",
      "\n",
-      "    Q = Q_discrete_white_noise(2, dt, 1.1)\n",
+      "    Q = Q_discrete_white_noise(2, dt, Q_variance)\n",
      "    f.Q[0:2,0:2] = Q\n",
      "    f.Q[2:4, 2:4] = Q\n",
      "\n",
@@ -1781,7 +1781,7 @@
      "target_pos = [100, 200]\n",
      "\n",
      "std_noise = math.radians(0.5)\n",
-      "f = moving_target_filter(std_noise)\n",
+      "f = moving_target_filter(std_noise, Q_variance=1.1)\n",
      "plot_straight_line_target(f, std_noise)"
     ],
     "language": "python",
@@ -1792,7 +1792,7 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d2f2390>"
+        "<matplotlib.figure.Figure at 0x7f3e59fa3550>"
       ]
      }
     ],
@@ -1812,7 +1812,7 @@
     "collapsed": false,
     "input": [
      "target_pos = [0, 0]\n",
-      "f = moving_target_filter(std_noise)\n",
+      "f = moving_target_filter(std_noise, Q_variance=1.1)\n",
      "plot_straight_line_target(f, std_noise)"
     ],
     "language": "python",
@@ -1823,7 +1823,7 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d3814a8>"
+        "<matplotlib.figure.Figure at 0x7f3e5a09bef0>"
       ]
      }
     ],
@@ -1839,6 +1839,226 @@
      "**author's note: this is a fairly naive approach to overdetermined systems. For example, suppose you had 6 sensors, not the minimum of 2 which is needed for 2D positions. While you can use the approach above, systems like GPS use something like an iterative Least Squares approach to estimate the 3D position for each epoch based on the pseudorange it receives from each satellite. This estimate is then passed into the GPS as the measurement. I will be writing a section on this very soon. In the meantime if you google for 'gnss iterative least squares' tons of relevant links are returned.**"
     ]
    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##Iterative Least Squares for Sensor Fusion"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The previous section took a somewhat naive approach to integrating several sensor measurements into the filter. Let's work with a real world scenerio, albeit somewhat simplified. The GPS system is designed so that at least 6 satellites are in view at any time, assuming no line of sight blockage, at any point on the globe. The GPS receiver knows the location of the satellites in the sky relative to the Earth. At each epoch (instant in time) the receiver gets a signal from each satellite from which it can derive the *pseudorange* to the satellite. In more detail, the GPS receiver gets a signal identifying the satellite along with the time stamp of when the signal was transmitted. The GPS satellite has an atomic clock onboard so this timestamp is extremely accurate. The signal travels at the speed of light, which is constant in a vacuum, so in theory the GPS should be able to produce an extremely accurate distance measurement to the measurement by measuring how long the signal took to reach the receiver. There are several problems with that. First, the signal is not travelling through a vacuum, but through the atmosphere. The atmosphere causes the signal to bend, so it is not travelling in a straight line. This causes the signal to take longer to reach the receiver than theory suggests. Second, the on board clock on the GPS *receiver* is not very accurate, so deriving an exact time duration is nontrivial. Third, in many environments the signal can bounce off of buildings, trees, and other objects, causing either a longer path or *multipaths*, in which case the receive receives both the original signal from space and the reflected signals. \n",
+      "\n",
+      "In the figure below I have shown how three pseudoranges might intersect in a 2D space. The width of the circle is meant to illustrate the measurement error for each pseudorange."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "with book_format.figsize(10,6):\n",
+      "    ukf_internal.show_three_gps()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f3e5a11ad30>"
+       ]
+      }
+     ],
+     "prompt_number": 76
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "So we need to solve for the 3 dimensions of space, and 1 dimension of time. That is 4 unknowns, so in theory with 4 satellites we have all the information we need. However, we normally have at least 6 satellites in view, and often more than 6. This means the system is *overdetermined*. Finally, because of the noise in the measurements none of pseudoranges intersect exactly.\n",
+      "\n",
+      "Probably the most common approach used by GPS receivers to find the position is the *iterative least squares* algorithm, commonly abbreviated ILS. As you know, if the errors are Gaussian then the least squares algorithm finds the optimal solution. In other words, we want to minimize the square of the residuals. We minimize by taking the derivative of the squares and setting it to 0, like so."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$ \\frac{\\partial}{\\partial \\mathbf{x}}({\\delta \\mathbf{z}}^\\mathsf{T}\\delta \\mathbf{z}) = 0,$$\n",
+      "\n",
+      "where\n",
+      "\n",
+      "$$\\delta \\mathbf{z}=\\delta \\mathbf{z}^- - \\mathbf{H}\\delta \\mathbf{x}.$$\n",
+      "\n",
+      "\n",
+      "For various reasons you may want to weigh some measurement more than others. We can do that with the equation\n",
+      "\n",
+      "$$ \\frac{\\partial}{\\partial \\mathbf{x}}({\\delta \\mathbf{z}}^\\mathsf{T}\\mathbf{W}\\delta \\mathbf{z}) = 0$$\n",
+      "\n",
+      "If we solve the first equation for ${\\delta \\mathbf{x}}$ (derivation shown below) we get\n",
+      "\n",
+      "$${\\delta \\mathbf{x}} = {{(\\mathbf{H}^\\mathsf{T}\\mathbf{H})^{-1}}\\mathbf{H}^\\mathsf{T} \\delta \\mathbf{z}}\n",
+      "$$\n",
+      "\n",
+      "And the second equation yields\n",
+      "\n",
+      "$${\\delta \\mathbf{x}} = {{(\\mathbf{H}^\\mathsf{T}\\mathbf{WH})^{-1}}\\mathbf{H}^\\mathsf{T}\\mathbf{W} \\delta \\mathbf{z}}\n",
+      "$$\n",
+      "\n",
+      "Since the equations are overdetermined we cannot solve these equations exactly. There are several techniques to find an approximate answer, but GPS receivers typically use the *iterative least squares* technique. An initial guess for the position is made, and then one of the equations above are iterated upon. Each iteration produces an updated estimate which is fed back into the equation for the next iteration. The iterations continue until the residuals are acceptably small. I will do that in code, but first let's derive these equations. You can skip the derivation if you want, but it is somewhat instructive and not too hard if you know basic linear algebra and partial differential equations."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## Derivation of ILS Equations (Optional)"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Substituting $\\delta \\mathbf{z}=\\delta \\mathbf{z}^- - \\mathbf{H}\\delta \\mathbf{x}$ we get\n",
+      "\n",
+      "$$ \\frac{\\partial}{\\partial \\mathbf{x}}(\\delta \\mathbf{z}^- -\\mathbf{H} \\delta \\mathbf{x})^\\mathsf{T}(\\delta \\mathbf{z}^- - \\mathbf{H} \\delta \\mathbf{x})=0$$\n",
+      "\n",
+      "which expands to\n",
+      "\n",
+      "$$ \\frac{\\partial}{\\partial \\mathbf{x}}({\\delta \\mathbf{x}}^\\mathsf{T}\\mathbf{H}^\\mathsf{T}\\mathbf{H}\\delta \\mathbf{x} - \n",
+      "{\\delta \\mathbf{x}}^\\mathsf{T}\\mathbf{H}^\\mathsf{T}\\delta \\mathbf{z}^- - \n",
+      "{\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H}\\delta \\mathbf{x} +\n",
+      "{\\delta \\mathbf{z}^-}^\\mathsf{T}\\delta \\mathbf{z}^-)=0$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We know that \n",
+      "\n",
+      "$$\\frac{\\partial \\mathbf{A}^\\mathsf{T}\\mathbf{B}}{\\partial \\mathbf{B}} = \\frac{\\partial \\mathbf{B}^\\mathsf{T}\\mathbf{A}}{\\partial \\mathbf{B}} = \\mathbf{A}^\\mathsf{T}$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Therefore the term\n",
+      "\n",
+      "$$\\frac{\\partial}{\\partial \\mathbf{x}}{\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H}\\delta \\mathbf{x} = {\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H}$$\n",
+      "\n",
+      "and the term\n",
+      "\n",
+      "$$\\frac{\\partial}{\\partial \\mathbf{x}}{\\delta \\mathbf{x}}^\\mathsf{T}\\mathbf{H}^\\mathsf{T}\\delta \\mathbf{z}^-={\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H}$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We also know that\n",
+      "$$\\frac{\\partial \\mathbf{B}^\\mathsf{T}\\mathbf{AB}}{\\partial \\mathbf{B}} = \\mathbf{B}^\\mathsf{T}(\\mathbf{A} + \\mathbf{A}^\\mathsf{T})$$\n",
+      "\n",
+      "\n",
+      "Therefore\n",
+      "\n",
+      "$$\n",
+      "\\begin{aligned}\n",
+      "\\frac{\\partial}{\\partial \\mathbf{x}}{\\delta \\mathbf{x}}^\\mathsf{T}\\mathbf{H}^\\mathsf{T}\\mathbf{H}\\delta \\mathbf{x} &= {\\delta \\mathbf{x}}^\\mathsf{T}(\\mathbf{H}^\\mathsf{T}\\mathbf{H} + {\\mathbf{H}^\\mathsf{T}\\mathbf{H}}^\\mathsf{T})\\\\\n",
+      "&= {\\delta \\mathbf{x}}^\\mathsf{T}(\\mathbf{H}^\\mathsf{T}\\mathbf{H} + \\mathbf{H}^\\mathsf{T}\\mathbf{H}) \\\\\n",
+      "&= 2{\\delta \\mathbf{x}}^\\mathsf{T}\\mathbf{H}^\\mathsf{T}\\mathbf{H}\n",
+      "\\end{aligned}$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Finally,\n",
+      "\n",
+      "$$ \\frac{\\partial}{\\partial \\mathbf{x}}\n",
+      "{\\delta \\mathbf{z}^-}^\\mathsf{T}\\delta \\mathbf{z}^-=0$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Replacing the terms in the expanded partial differential equation gives us\n",
+      "\n",
+      "$$\n",
+      " 2{\\delta \\mathbf{x}}^\\mathsf{T}\\mathbf{H}^\\mathsf{T}\\mathbf{H} -\n",
+      " {\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H} - {\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H}\n",
+      " =0\n",
+      "$$\n",
+      "\n",
+      "$${\\delta \\mathbf{x}}^\\mathsf{T}\\mathbf{H}^\\mathsf{T}\\mathbf{H} -\n",
+      " {\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H} = 0$$\n",
+      " \n",
+      " $${\\delta \\mathbf{x}}^\\mathsf{T}\\mathbf{H}^\\mathsf{T}\\mathbf{H} =\n",
+      " {\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H}$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Multiplying each side by $(\\mathbf{H}^\\mathsf{T}\\mathbf{H})^{-1}$ yields\n",
+      "\n",
+      "$${\\delta \\mathbf{x}}^\\mathsf{T} =\n",
+      "{\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H}(\\mathbf{H}^\\mathsf{T}\\mathbf{H})^{-1}$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Taking the transpose of each side gives\n",
+      "\n",
+      "$${\\delta \\mathbf{x}} = ({{\\delta \\mathbf{z}^-}^\\mathsf{T}\\mathbf{H}(\\mathbf{H}^\\mathsf{T}\\mathbf{H})^{-1}})^\\mathsf{T} \\\\\n",
+      "={{(\\mathbf{H}^\\mathsf{T}\\mathbf{H})^{-1}}^T\\mathbf{H}^\\mathsf{T} \\delta \\mathbf{z}^-} \\\\\n",
+      "={{(\\mathbf{H}^\\mathsf{T}\\mathbf{H})^{-1}}\\mathbf{H}^\\mathsf{T} \\delta \\mathbf{z}^-}\n",
+      "$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "For various reasons you may want to weigh some measurement more than others. We can do that with the equation\n",
+      "\n",
+      "$$ \\frac{\\partial}{\\partial \\mathbf{x}}({\\delta \\mathbf{z}}^\\mathsf{T}\\mathbf{W}\\delta \\mathbf{z}) = 0$$\n",
+      "\n",
+      "Replicating the math above with the added $\\mathbf{W}$ term results in\n",
+      "\n",
+      "$${\\delta \\mathbf{x}} = {{(\\mathbf{H}^\\mathsf{T}\\mathbf{WH})^{-1}}\\mathbf{H}^\\mathsf{T}\\mathbf{W} \\delta \\mathbf{z}^-}\n",
+      "$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## Implmenting ILS"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Our goal is to implement an iterative solution to \n",
+      "$${\\delta \\mathbf{x}} = {{(\\mathbf{H}^\\mathsf{T}\\mathbf{H})^{-1}}\\mathbf{H}^\\mathsf{T} \\delta \\mathbf{z}^-}\n",
+      "$$\n",
+      "\n",
+      "\n"
+     ]
+    },
    {
     "cell_type": "heading",
     "level": 2,
@@ -1915,7 +2135,7 @@
      "np.random.seed(12283)\n",
      "std_noise = math.radians(0.5)\n",
      "target_pos = [0, 0]\n",
-      "f = moving_target_filter(std_noise)\n",
+      "f = moving_target_filter(std_noise, Q_variance=1.1)\n",
      "plot_circular_target(f, std_noise, target_pos)"
     ],
     "language": "python",
@@ -1926,7 +2146,7 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d353b38>"
+        "<matplotlib.figure.Figure at 0x7f3e5a00b278>"
       ]
      }
     ],
@@ -1944,28 +2164,16 @@
     "cell_type": "markdown",
     "metadata": {},
     "source": [
-      "The filter tracks the movement of the target, but never really converges on the track.  This is because the filter is modeling a constant velocity target, but the target is anything but constant velocity. As mentioned above we could model the circular behavior by defining the `fx()` function, but then we would have problems when the target is not moving in a circle. Instead, lets tell the filter we are are less sure about our process model by making $\\mathbf{Q}$ much larger."
+      "The filter tracks the movement of the target, but never really converges on the track.  This is because the filter is modeling a constant velocity target, but the target is anything but constant velocity. As mentioned above we could model the circular behavior by defining the `fx()` function, but then we would have problems when the target is not moving in a circle. Instead, lets tell the filter we are are less sure about our process model by making $\\mathbf{Q}$ larger. Here I have increased the variance from 0.1 to 1.0"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
-      "def circular_target_filter(std_noise):\n",
-      "    f = UKF(dim_x=4, dim_z=2, dt=0.1, hx=hx, fx=fx, kappa=2.)\n",
-      "    f.x = np.array([target_pos[0], 1., target_pos[1], 1.])\n",
-      "\n",
-      "    Q = Q_discrete_white_noise(2, dt, 10.1)\n",
-      "    f.Q[0:2,0:2] = Q\n",
-      "    f.Q[2:4, 2:4] = Q\n",
-      "\n",
-      "    f.R *= std_noise**2\n",
-      "    f.P *= 1000\n",
-      "    \n",
-      "    return f\n",
-      "\n",
+      "np.random.seed(12283)\n",
      "std_noise = math.radians(0.5)\n",
-      "cf = circular_target_filter(std_noise)\n",
+      "cf = moving_target_filter(std_noise, Q_variance=10.)\n",
      "target_pos = [0, 0]\n",
      "plot_circular_target(cf, std_noise, target_pos)"
     ],
@@ -1975,9 +2183,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d353828>"
+        "<matplotlib.figure.Figure at 0x7f3e59f8dcf8>"
       ]
      }
     ],
@@ -2040,7 +2248,7 @@
      "sb_pos = [200,-200]\n",
      "plt.scatter(*sa_pos, s=200)\n",
      "plt.scatter(*sb_pos, s=200)\n",
-      "cf = circular_target_filter(std_noise)\n",
+      "cf = moving_target_filter(std_noise, Q_variance=10.)\n",
      "target_pos = [0, 0]\n",
      "plot_circular_target(cf, std_noise, target_pos)"
     ],
@@ -2050,9 +2258,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d2afda0>"
+        "<matplotlib.figure.Figure at 0x7f3e64a57400>"
       ]
      }
     ],
@@ -2171,9 +2379,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a18b00>"
+        "<matplotlib.figure.Figure at 0x7f3e5ba312e8>"
       ]
      }
     ],
@@ -2242,9 +2450,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d53d160>"
+        "<matplotlib.figure.Figure at 0x7f3e63732358>"
       ]
      }
     ],
@@ -2342,9 +2550,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d54e0b8>"
+        "<matplotlib.figure.Figure at 0x7f3e59f88390>"
       ]
      }
     ],
@@ -2377,9 +2585,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a798d0>"
+        "<matplotlib.figure.Figure at 0x7f3e637dd710>"
       ]
      }
     ],
@@ -2491,7 +2699,7 @@
       "output_type": "display_data",
       "png": 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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a017b8>"
+        "<matplotlib.figure.Figure at 0x7f3e59f6f780>"
       ]
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      {
@@ -3558,7 +3766,7 @@
       "output_type": "display_data",
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f35569d8eb8>"
+        "<matplotlib.figure.Figure at 0x7f3e6376f668>"
       ]
      },
      {
@@ -3566,7 +3774,7 @@
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556accb00>"
+        "<matplotlib.figure.Figure at 0x7f3e5a1a1668>"
       ]
      },
      {
@@ -3574,7 +3782,7 @@
       "output_type": "display_data",
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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a89ef0>"
+        "<matplotlib.figure.Figure at 0x7f3e5a151828>"
       ]
      }
     ],
@@ -3607,7 +3815,7 @@
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d2decf8>"
+        "<matplotlib.figure.Figure at 0x7f3e5a03fc18>"
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@@ -3677,7 +3885,7 @@
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       "text": [
-        "<matplotlib.figure.Figure at 0x7f3556a367b8>"
+        "<matplotlib.figure.Figure at 0x7f3e63722518>"
       ]
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@@ -3685,7 +3893,7 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d401e48>"
+        "<matplotlib.figure.Figure at 0x7f3e5a0035f8>"
       ]
      },
      {
@@ -3693,7 +3901,7 @@
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAA0YAAAGkCAYAAAAVNzW4AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7f354d2c1f98>"
+        "<matplotlib.figure.Figure at 0x7f3e59faf898>"
       ]
      },
      {
@@ -42,6 +42,33 @@ def show_two_sensor_bearing():
    ax.add_artist(circle2)
    plt.show()

+
+def show_three_gps():
+    circle1=plt.Circle((-4,0),5,color='#004080',fill=False,linewidth=20, alpha=.7)
+    circle2=plt.Circle((4,0),5,color='#E24A33', fill=False, linewidth=8, alpha=.7)
+    circle3=plt.Circle((0,-3),6,color='#534543',fill=False, linewidth=13, alpha=.7)
+
+    fig = plt.gcf()
+    ax = fig.gca()
+
+#    plt.ylim((-10,10))
+
+    #plt.plot ([-4,0], [0,3], c='#004080')
+    #plt.plot ([4,0], [0,3], c='#E24A33')
+    #plt.text(-4, -.5, "A", fontsize=16, horizontalalignment='center')
+    #plt.text(4, -.5, "B", fontsize=16, horizontalalignment='center')
+
+    ax.add_patch(circle1)
+    ax.add_patch(circle2)
+    ax.add_patch(circle3)
+
+    plt.axis('equal')
+    #ax.set_xlim(-15,15)
+    #ax.set_ylim(-15,15)
+
+    plt.show()
+
+
 def show_sigma_transform():
    fig = plt.figure()
    ax=fig.gca()
@@ -157,6 +184,7 @@ def show_sigmas_for_2_kappas():


 if __name__ == '__main__':
-    show_sigma_transform()
+    show_three_gps()
+    #show_sigma_transform()
    #show_sigma_selections()

@@ -0,0 +1,141 @@
+# -*- coding: utf-8 -*-
+
+
+import numpy as np
+from numpy.linalg import norm, inv
+from numpy.random import randn
+from numpy import dot
+
+
+numpy.random.seed(1234)
+user_pos = np.array([1000, 100])  # d5, D6
+
+pred_user_pos = np.array([100, 0]) #d7, d8
+
+
+t_pos = np.asarray([[0, 1000],
+                    [0, -1000],
+                    [500, 500]], dtype=float)
+
+
+def transmitter_range(pos, transmitter_pos):
+    """ Compute distance between position 'pos' and the list of positions
+    in transmitter_pos"""
+
+    N = len(transmitter_pos)
+    rng = np.zeros(N)
+
+    diff = np.asarray(pos) - transmitter_pos
+
+    for i in range(N):
+        rng[i] = norm(diff[i])
+
+    return norm(diff, axis=1)
+
+
+
+
+# compute measurement of where you are with respect to seach sensor
+
+
+rz= transmitter_range(user_pos, t_pos) # $B21,22
+
+# add some noise
+for i in range(len(rz)):
+    rz[i] += randn()
+
+
+# now iterate on the predicted position
+pos = pred_user_pos
+
+
+def hx_range(pos, t_pos, r_est):
+    N = len(t_pos)
+    H = np.zeros((N, 2))
+    for j in range(N):
+        H[j,0] = -(t_pos[j,0] - pos[0]) / r_est[j]
+        H[j,1] = -(t_pos[j,1] - pos[1]) / r_est[j]
+    return H
+
+
+def lop_ils(zs, t_pos, pos_est, hx, eps=1.e-6):
+    """ iteratively estimates the solution to a set of measurement, given
+    known transmitter locations"""
+    pos = np.array(pos_est)
+
+    converged = False
+    for i in range(20):
+        r_est = transmitter_range(pos, t_pos) #B32-B33
+        print('iteration:', i)
+        #print ('ra1, ra2', ra1, ra2)
+        print()
+
+        H=hx(pos, t_pos, r_est)
+        
+        Hinv = inv(dot(H.T, H)).dot(H.T)
+
+        #update position estimate
+        y = zs - r_est
+        print('residual', y)
+
+        Hy = np.dot(Hinv, y)
+        print('Hy', Hy)
+
+        pos = pos + Hy
+        print('pos', pos)
+
+        print()
+        print()
+
+        if max(abs(Hy)) < eps:
+            converged = True
+            break
+
+    return pos, converged
+
+
+
+print(lop_ils(rz, t_pos, (900,90), hx=hx_range))
+
+
+
+#####################
+"""
+# compute measurement (simulation)
+rza1, rza2 = transmitter_range(user_pos) # $B21,22
+
+rza1 += randn()
+rza2 += randn()
+
+# now iterate on the predicted position
+pos = pred_user_pos
+
+
+for i in range(10):
+    ra1, ra2 = transmitter_range(pos) #B32-B33
+    print('iteration:', i)
+    print ('ra1, ra2', ra1, ra2)
+    print()
+
+    H = np.array([[-(t1_pos[0] - pos[0]) / ra1, -(t1_pos[1] - pos[1]) / ra1],
+                  [-(t2_pos[0] - pos[0]) / ra2, -(t2_pos[1] - pos[1]) / ra2]])
+    Hinv = inv(H)
+
+    #update position estimate
+    residual_t1 = rza1 - ra1
+    residual_t2 = rza2 - ra2
+    y = np.array([[residual_t1], [residual_t2]])
+    print('residual', y.T)
+
+
+    Hy = np.dot(Hinv, y)
+
+    pos = pos + Hy[:,0]
+    print('pos', pos)
+
+    print()
+    print()
+
+    if (max(abs(y)) < 1.e-6):
+        break
+"""