daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Merge branch 'master' of https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

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Roger Labbe committed 2014-06-02 14:36:18 -07:00
2 parents 896103f1ad 394f0a2570
commit 76101fbb0c
2 files changed
+486 -49
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Designing_Kalman_Filters.ipynb
+458 -23
View File
@@ -1,7 +1,7 @@
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@@ -244,7 +244,7 @@
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"text": [
"<matplotlib.figure.Figure at 0x2a773d0>"
"<matplotlib.figure.Figure at 0x3ab7990>"
]
}
],
@@ -672,9 +672,9 @@
{
"metadata": {},
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truncated
"text": [
"<matplotlib.figure.Figure at 0x2a772d0>"
"<matplotlib.figure.Figure at 0x3b35190>"
]
}
],
@@ -751,9 +751,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x3ed5e90>"
"<matplotlib.figure.Figure at 0x3f866d0>"
]
},
{
@@ -836,9 +836,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x3ae2a10>"
"<matplotlib.figure.Figure at 0x3cdf7d0>"
]
}
],
@@ -901,9 +901,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x3b5ffd0>"
"<matplotlib.figure.Figure at 0x3b6d390>"
]
}
],
@@ -1175,9 +1175,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x430aa50>"
"<matplotlib.figure.Figure at 0x3b57cd0>"
]
}
],
@@ -1209,14 +1209,436 @@
"outputs": [],
"prompt_number": 18
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Tracking a Ball"
]
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Physics of a Ball in a Vacuum"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's turn our attention to a situation where the physics of the object that we are tracking is constrained. A ball thrown in a vacuum must obey Newtonian laws. In a constant gravitational field it will travel in a parabola. I will assume you are familiar with the derivation of the formula:\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"y &= \\frac{g}{2}t^2 + v_{y0} t + y_0 \\\\\n",
"x &= v_{x0} t + x_0\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"where $g$ is the gravational constant, $t$ is time, $v_{x0}$ and $v_{y0}$ are the initial velocities in the x and y plane. If the ball is thrown with an initial velocity of $v$ at angle $\\theta$ above the horizon, we can compute $v_{x0}$ and $v_{y0}$ as\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"v_{x0} = v \\cos{\\theta} \\\\\n",
"v_{y0} = v \\sin{\\theta}\n",
"\\end{aligned}\n",
"$$\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can simulate this with the following code. I added a way to specify the initial angle in either radians or degrees, and a noise scaling factor which will be useful when we test our Kalman filter. "
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"input": [
"from math import cos, tan, radians\n",
"from numpy.random import randn\n",
"\n",
"def sec (x):\n",
" return 1. / cos(x)\n",
"\n",
"class BallTrajectory2D(object):\n",
" def __init__(self, v0, angle_radians=0., x0=0., angle_degrees=None, noise_scale=0.0):\n",
" if angle_degrees is not None:\n",
" self.theta = radians(angle_degrees)\n",
" else:\n",
" self.theta = angle_radians\n",
" \n",
" self.x = self.x0 = x0\n",
" self.y = self.y0 = 0\n",
" \n",
" self.v0_y = v0 * math.sin(self.theta)\n",
" self.v0_x = v0*math.cos(self.theta)\n",
" \n",
" self.noise_scale = noise_scale\n",
" \n",
" def update(self, t):\n",
" self.x = self.v0_x*t + self.x0\n",
" self.y = -0.5*9.8*t**2 + self.v0_y*t + self.y0\n",
" \n",
" self.x += randn() * self.noise_scale\n",
" self.y += randn() * self.noise_scale"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 18
"prompt_number": 43
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So to create a trajectory with a velocity of $60 \\frac{m}{s}$ and an angle of $65^\\circ$ we would write:\n",
"\n",
" traj = BallTrajectory2D (60, angle_degrees=65)\n",
" \n",
"and then call $\\verb,update(t),$ for each time step. For example:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"traj = BallTrajectory2D (60, angle_degrees=65)\n",
"t = 0.0\n",
"while t >= 0 and traj.y >= 0:\n",
" traj.update(t)\n",
" plt.scatter(traj.x, traj.y)\n",
" t += 0.1\n",
"plt.axis('equal')\n",
"plt.show()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
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truncated
"text": [
"<matplotlib.figure.Figure at 0x4243690>"
]
}
],
"prompt_number": 46
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If you are not completely familiar with this math I suggest playing with the velocity and angle parameters to see how changing each affects the trajectory. "
]
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Designing the Kalman Filter"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Our assumption is that the ball is travelling in a vacuum, so the velocity in x is a constant, and the acceleration in y is solely due to the gravitational constant $g$. The Newtonian equations for $x$ and $y$ in terms of $\\Delta t$ are:\n",
"\n",
"$$\\begin{aligned}\n",
"y_t &= \\frac{g}{2} {\\Delta t}^2 + v_{y(t-1)} {\\Delta t} + y_{t-1} \\\\\n",
"v_{yt} &= g {\\Delta t} + v_{y(t-1)} \\\\\n",
"\\\\\n",
"x_t &= v_{x(t-1)} {\\Delta t} \\\\\n",
"v_{xt} &= v_{x(t-1)}\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"We need to incorporate acceleration for $y$ into the Kalman filter, but not for $x$. This suggests the following state variables.\n",
"\n",
"$$\n",
"\\mathbf{x} = \n",
"\\begin{bmatrix}\n",
"x \\\\\n",
"\\dot{x} \\\\\n",
"y \\\\\n",
"\\dot{y} \\\\\n",
"\\ddot{y}\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def ball_filter(dt):\n",
" f1 = KalmanFilter(dim=5)\n",
" g = 9.8\n",
"\n",
" f1.F = np.mat ([[1., dt, 0, 0, 0],\n",
" [0, 1., 0, 0, 0],\n",
" [0, 0, 1., dt, -dt*dt*g],\n",
" [0, 0, 0, 1., dt],\n",
" [0, 0, 0, 0, 1.]])\n",
"\n",
" f1.H = np.mat([[1,0,0,0,0],\n",
" [0,0,0,0,0],\n",
" [0,0,1,0,0],\n",
" [0,0,0,0,0],\n",
" [0,0,0,0,0]])\n",
"\n",
"\n",
" f1.R = np.eye(5) * 1.\n",
" f1.Q = np.zeros((5,5))\n",
" f1.Q[4,4] = 0.1\n",
" f1.x = np.mat([0, 21.2 , 0, 20.7, -9.8]).T\n",
" f1.P = np.eye(5) * 5.\n",
" f1.B = 0.\n",
" f1.u = 0\n",
" \n",
" return f1"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 170
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"dt = 0.1\n",
"f1 = ball_filter(dt)\n",
"# initialize storage and other variables for the run\n",
"count = 30\n",
"xs, ys = [],[]\n",
"pxs, pys = [],[]\n",
"\n",
"traj = BallTrajectory2D (30., angle_degrees=45., noise_scale=0.3)\n",
"traj.update(0)\n",
"x,y= (traj.x,traj.y)\n",
"traj.update(dt)\n",
"print (f1.x)\n",
"print (traj.x - x, traj.y - y)\n",
"log = True\n",
"#print (f1.F*f1.x)\n",
"for t in np.arange(dt,5,dt):\n",
" traj.update(t)\n",
" z = np.mat([traj.x, 0, traj.y, 0,0]).T\n",
" if log: print (traj.x, traj.y)\n",
"\n",
" f1.predict ()\n",
" if log: print (f1.x.T)\n",
" \n",
" #f1.x[4]=-9.8\n",
" \n",
" f1.update (z)\n",
" if log: print (f1.x.T)\n",
"\n",
"\n",
" xs.append (f1.x[0,0])\n",
" ys.append (f1.x[2,0])\n",
" pxs.append (traj.x)\n",
" pys.append(traj.y)\n",
"\n",
"p1, = plt.plot (xs, ys, 'r--')\n",
"p2, = plt.plot (pxs, pys)\n",
"plt.legend([p1,p2], ['filter', 'measurement'], 2)\n",
"plt.show()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"[[ 0. ]\n",
" [ 21.2]\n",
" [ 0. ]\n",
" [ 20.7]\n",
" [ -9.8]]\n",
"2.26334154798 2.2701914807\n",
"2.7486356737 1.99227097822\n",
"[[ 2.12 21.2 3.0304 19.72 -9.8 ]]\n",
"[[ 2.64472895 21.25195336 2.16251132 19.6432216 -9.71658223]]\n",
"4.00471898834 3.77548778907\n",
"[[ 4.76992429 21.25195336 5.07905854 18.67156338 -9.71658223]]\n",
"[[ 4.40728351 21.01906479 4.44234009 18.35760646 -9.36803761]]\n",
"6.67939142944 5.71113377806\n",
"[[ 6.50918999 21.01906479 7.19616842 17.4208027 -9.36803761]]\n",
"[[ 6.57184657 21.10323034 6.60307241 16.91399233 -8.79487404]]\n",
"8.7006186025 7.65399749306\n",
"[[ 8.6821696 21.10323034 9.1563693 16.03450493 -8.79487404]]\n",
"[[ 8.68840872 21.11463517 8.5903118 15.46920598 -8.16329089]]\n",
"10.7741338917 9.0855524088\n",
"[[ 10.79987223 21.11463517 10.93723491 14.6528769 -8.16329089]]\n",
"[[ 10.79129279 21.09747628 10.25646109 13.95990014 -7.44098872]]\n",
"12.5014218437 10.9726682202\n",
"[[ 12.90104042 21.09747628 12.381668 13.21580126 -7.44098872]]\n",
"[[ 12.76815754 20.83559032 11.87913651 12.70112583 -6.98521569]]\n",
"14.7386837611 12.3742790314\n",
"[[ 14.85171657 20.83559032 13.83380023 12.00260426 -6.98521569]]\n",
"[[ 14.81461232 20.76678774 13.33419469 11.46474553 -6.63369164]]\n",
"17.3062844439 13.911438362\n",
"[[ 16.8912911 20.76678774 15.13077103 10.80137637 -6.63369164]]\n",
"[[ 17.02389252 20.99392907 14.73222341 10.32239088 -6.4478821 ]]\n",
"18.5595673308 15.3828782525\n",
"[[ 19.12328543 20.99392907 16.39635494 9.67760267 -6.4478821 ]]\n",
"[[ 18.94991144 20.7210924 16.07943823 9.23089798 -6.38486712]]\n",
"22.0627856661 16.4447654453\n",
"[[ 21.02202068 20.7210924 17.628245 8.59241127 -6.38486712]]\n",
"[[ 21.32785126 21.16337047 17.2713798 7.98750161 -6.41802682]]\n",
"23.6303586575 17.1945292524\n",
"[[ 23.4441883 21.16337047 18.69909659 7.34569893 -6.41802682]]\n",
"[[ 23.49623645 21.23273566 18.2562726 6.44318999 -6.59337121]]\n",
"25.5961104482 18.1753812749\n",
"[[ 25.61951002 21.23273566 19.54674198 5.78385287 -6.59337121]]\n",
"[[ 25.6132985 21.22507762 19.14723094 4.82358101 -6.8688087 ]]\n",
"27.4769545972 19.4389444698\n",
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"[[ 27.67059765 21.15039996 20.05032567 3.44265248 -7.10848858]]\n",
"30.0724069863 20.348544138\n",
"[[ 29.78563765 21.15039996 21.0912228 2.73180362 -7.10848858]]\n",
"[[ 29.85422175 21.22365422 20.87132005 2.06340731 -7.36298674]]\n",
"31.9311062718 20.6213988152\n",
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"[[ 36.01447654 21.04430138 22.36752568 -3.19257556 -8.55671087]]\n",
"38.3247380297 22.8875572193\n",
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"[[ 38.15938964 21.07820678 22.88705811 -4.04744762 -8.5563786 ]]\n",
"40.2030744561 22.9801889478\n",
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"42.565696679 22.3666619174\n",
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"[[ 42.39867745 21.09630619 23.16178094 -7.40098292 -9.22422372]]\n",
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"[[ 44.51680959 21.10242071 23.14078376 -8.9268641 -9.46854847]]\n",
"46.3895725412 22.6008802881\n",
"[[ 46.62705166 21.10242071 23.1760151 -9.87371895 -9.46854847]]"
]
},
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
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"48.5150711204 22.4991528062\n",
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"[[ 48.66652608 21.05653143 22.77132118 -11.82039523 -9.85681741]]\n",
"50.4225553907 22.1793888942\n",
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"[[ 50.71853752 21.02275275 22.43726881 -13.1639266 -9.99616068]]\n",
"53.0049706805 21.9264343895\n",
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Multidimensional_Kalman_Filters.ipynb
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View File
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truncated
"text": [
"<matplotlib.figure.Figure at 0x350fb90>"
"<matplotlib.figure.Figure at 0x2734110>"
]
}
],
@@ -568,7 +568,7 @@
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"text": [
"<matplotlib.figure.Figure at 0x32ab190>"
"<matplotlib.figure.Figure at 0x28ceed0>"
]
}
],
@@ -658,7 +658,7 @@
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"text": [
"<matplotlib.figure.Figure at 0x31b2290>"
"<matplotlib.figure.Figure at 0x28e3650>"
]
}
],
@@ -712,7 +712,7 @@
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truncated
"text": [
"<matplotlib.figure.Figure at 0x31b2290>"
"<matplotlib.figure.Figure at 0x2f9a390>"
]
}
],
@@ -743,7 +743,7 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x32a74d0>"
"<matplotlib.figure.Figure at 0x27c7950>"
]
}
],
@@ -796,7 +796,7 @@
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truncated
"text": [
"<matplotlib.figure.Figure at 0x1f04890>"
"<matplotlib.figure.Figure at 0x27c7dd0>"
]
}
],
@@ -1315,9 +1315,9 @@
{
"metadata": {},
"output_type": "display_data",
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truncated
"text": [
"<matplotlib.figure.Figure at 0x3bb2cd0>"
"<matplotlib.figure.Figure at 0x112c750>"
]
},
{
@@ -1325,7 +1325,7 @@
"output_type": "display_data",
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truncated
"text": [
"<matplotlib.figure.Figure at 0x32a38d0>"
"<matplotlib.figure.Figure at 0x28d06d0>"
]
}
],
@@ -1494,17 +1494,17 @@
{
"metadata": {},
"output_type": "display_data",
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"text": [
"<matplotlib.figure.Figure at 0x378d7d0>"
"<matplotlib.figure.Figure at 0x29b6b10>"
]
},
{
"metadata": {},
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x35b3490>"
"<matplotlib.figure.Figure at 0x29b5c90>"
]
}
],
@@ -1585,7 +1585,9 @@
"\n",
"So white noise has the variance $U_n$ and a mean of 0, which we notate as $w \\sim \\mathcal{N}(0,Q)$. \n",
"\n",
"Finding an analytic value for $\\textbf{Q}$ in a simple problem like this is not difficult, but it quickly becomes difficult to impossible as the number of state variables increase. So in this chapter we will use the simplification that only the variance of the last term is important. In the Kalman filter math chapter we will discuss finding an analytic solution ofr $\\textbf{Q}$, and then present C. F. van Loan's extremely useful numerical technique for finding $\\textbf{Q}$, which is what you will typically use in practice. \n"
"Finding an analytic value for $\\textbf{Q}$ in a simple problem like this is not difficult, but it quickly becomes difficult to impossible as the number of state variables increase. So in this chapter we will use the simplification that only the variance of the last term is important. In the Kalman filter math chapter we will discuss finding an analytic solution ofr $\\textbf{Q}$, and then present C. F. van Loan's extremely useful numerical technique for finding $\\textbf{Q}$, which is what you will typically use in practice. \n",
"\n",
"**author's note: text needs to move to kalman math chapter. leaving here for now **"
]
},
{
@@ -1615,9 +1617,9 @@
{
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truncated
"text": [
"<matplotlib.figure.Figure at 0x3771310>"
"<matplotlib.figure.Figure at 0x24673d0>"
]
}
],
@@ -1687,7 +1689,7 @@
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x3285850>"
"<matplotlib.figure.Figure at 0x27c2a50>"
]
}
],
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truncated
"text": [
"<matplotlib.figure.Figure at 0x31b8510>"
"<matplotlib.figure.Figure at 0x28cec50>"
]
}
],
@@ -1805,17 +1807,17 @@
{
"metadata": {},
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truncated
"text": [
"<matplotlib.figure.Figure at 0x32bad10>"
"<matplotlib.figure.Figure at 0x29a62d0>"
]
},
{
"metadata": {},
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x378dc10>"
"<matplotlib.figure.Figure at 0x29a1990>"
]
}
],
@@ -1882,7 +1884,7 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x32ab190>"
"<matplotlib.figure.Figure at 0x24c7e10>"
]
}
],