daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Added air drag to kalman filter ball tracking.

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Roger Labbe committed 2014-06-02 23:08:47 -07:00
1 parent 12c67ec3e5
commit 70f34843c7
2 files changed
+361 -249
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Designing_Kalman_Filters.ipynb
+278 -249
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@@ -1,7 +1,7 @@
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"nbformat": 3,
"nbformat_minor": 0,
@@ -244,7 +244,7 @@
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@@ -336,9 +336,9 @@
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],
@@ -672,9 +672,9 @@
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truncated
"text": [
"<matplotlib.figure.Figure at 0x3b35190>"
"<matplotlib.figure.Figure at 0x31aa610>"
]
}
],
@@ -751,9 +751,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x3f866d0>"
"<matplotlib.figure.Figure at 0x35fa750>"
]
},
{
@@ -838,7 +838,7 @@
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x3cdf7d0>"
"<matplotlib.figure.Figure at 0x334e110>"
]
}
],
@@ -903,7 +903,7 @@
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truncated
"text": [
"<matplotlib.figure.Figure at 0x3b6d390>"
"<matplotlib.figure.Figure at 0x3350250>"
]
}
],
@@ -1175,9 +1175,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x3b57cd0>"
"<matplotlib.figure.Figure at 0x31c6550>"
]
}
],
@@ -1285,12 +1285,13 @@
" 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"
" self.y += randn() * self.noise_scale\n",
" return (self.x, self.y)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 43
"prompt_number": 27
},
{
"cell_type": "markdown",
@@ -1324,11 +1325,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x4243690>"
"<matplotlib.figure.Figure at 0x3bb37d0>"
]
}
],
"prompt_number": 46
"prompt_number": 28
},
{
"cell_type": "markdown",
@@ -1378,13 +1379,13 @@
"cell_type": "code",
"collapsed": false,
"input": [
"def ball_filter(dt):\n",
"def ball_filter(dt,R=1., Q = 0.1):\n",
" f1 = KalmanFilter(dim=5)\n",
" g = 9.8\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, 1., dt, dt*dt*g],\n",
" [0, 0, 0, 1., dt],\n",
" [0, 0, 0, 0, 1.]])\n",
"\n",
@@ -1395,10 +1396,11 @@
" [0,0,0,0,0]])\n",
"\n",
"\n",
" f1.R = np.eye(5) * 1.\n",
" f1.R = np.mat(np.eye(5)) * R \n",
" \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.Q[4,4] = Q\n",
" f1.x = np.mat([0, 0 , 0, 0, 0]).T\n",
" f1.P = np.eye(5) * 5.\n",
" f1.B = 0.\n",
" f1.u = 0\n",
@@ -1408,277 +1410,304 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 170
"prompt_number": 29
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def plot_ball_filter (traj):\n",
" xs, ys = [],[]\n",
" pxs, pys = [],[]\n",
" \n",
" for z in zs:\n",
" m = np.mat([z[0], 0, z[1], 0, 0]).T\n",
"\n",
" f1.predict ()\n",
" f1.update (m)\n",
"\n",
" xs.append (f1.x[0,0])\n",
" ys.append (f1.x[2,0])\n",
" pxs.append (z[0])\n",
" pys.append(z[1])\n",
" \n",
" if z[1] < 0:\n",
" break;\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": [],
"prompt_number": 41
},
{
"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",
"f1 = ball_filter(dt,R=0.1, Q=0.1)\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",
"traj = BallTrajectory2D (20., angle_degrees=45., noise_scale=0.0)\n",
"zs = [traj.update(t) for t in np.arange(dt,100,dt)]\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()"
"plot_ball_filter (zs)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"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",
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"[[ 8.68840872 21.11463517 8.5903118 15.46920598 -8.16329089]]\n",
"10.7741338917 9.0855524088\n",
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"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",
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"18.5595673308 15.3828782525\n",
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"[[ 18.94991144 20.7210924 16.07943823 9.23089798 -6.38486712]]\n",
"22.0627856661 16.4447654453\n",
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"23.6303586575 17.1945292524\n",
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"25.5961104482 18.1753812749\n",
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"[[ 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",
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"[[ 29.85422175 21.22365422 20.87132005 2.06340731 -7.36298674]]\n",
"31.9311062718 20.6213988152\n",
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"33.5827618634 21.4409160537\n",
"[[ 34.08758237 21.21331765 22.2279603 -0.61107519 -7.82639727]]\n",
"[[ 33.97841219 21.11074164 21.98563339 -1.43037209 -8.16354081]]\n",
"35.7254947678 21.7649117661\n",
"[[ 36.08948635 21.11074164 22.64262318 -2.24672617 -8.16354081]]\n",
"[[ 36.01447654 21.04430138 22.36752568 -3.19257556 -8.55671087]]\n",
"38.3247380297 22.8875572193\n",
"[[ 38.11890668 21.04430138 22.88682579 -4.04824665 -8.55671087]]\n",
"[[ 38.15938964 21.07820678 22.88705811 -4.04744762 -8.5563786 ]]\n",
"40.2030744561 22.9801889478\n",
"[[ 40.26721032 21.07820678 23.32083845 -4.90308548 -8.5563786 ]]\n",
"[[ 40.25515225 21.06863132 23.21176316 -5.27407846 -8.70978083]]\n",
"42.565696679 22.3666619174\n",
"[[ 42.36201538 21.06863132 23.53791383 -6.14505654 -8.70978083]]\n",
"[[ 42.39867745 21.09630619 23.16178094 -7.40098292 -9.22422372]]\n",
"44.557567354 22.7491277508\n",
"[[ 44.50830807 21.09630619 23.32565657 -8.32340529 -9.22422372]]\n",
"[[ 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]]"
"<matplotlib.figure.Figure at 0x31d73d0>"
]
},
}
],
"prompt_number": 42
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here, we can see that the filter tracks the trajectory quite closely, but not perfectly. Let's look at the effect of a slower and faster initial velocity."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"dt = 0.1\n",
"f1 = ball_filter(dt,R=.1, Q=.1)\n",
"traj = BallTrajectory2D (10., angle_degrees=60., noise_scale=0.0)\n",
"zs = [traj.update(t) for t in np.arange(dt,100,dt)]\n",
"plot_ball_filter (zs)\n",
"\n",
"f1 = ball_filter(dt, R=0.1, Q=0.1)\n",
"traj = BallTrajectory2D (60., angle_degrees=60., noise_scale=0.0)\n",
"zs = [traj.update(t) for t in np.arange(dt,100,dt)]\n",
"plot_ball_filter (zs)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x46b2a50>"
"<matplotlib.figure.Figure at 0x31c9c10>"
]
}
],
"prompt_number": 177
"prompt_number": 43
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"cell_type": "markdown",
"metadata": {},
"outputs": [],
"prompt_number": 46
"source": [
"Can you explain what is happening here?\n",
"\n",
"When the initial velocity is low, the change in velocity per time period is large compared to the velocity. In the physical world the ball's velocity is changing continuously, but we are imposing a step function on it because we are stepping by $0.1s$. This means our model of the ball does not exactly match the true behavior of the ball, and so the Kalman filter will not exactly predict the ball's behavior. A slow initial velcocity exaberates that \n",
"\n",
"**author's note** complete this"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So we seem to be doing a good job of tracking the ball. What happens if we inject noise?"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"%run?"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 166
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"math.cos(math.pi/2.)"
"dt = 0.1\n",
"f1 = ball_filter(dt, R=.16, Q=0.0)\n",
"\n",
"traj = BallTrajectory2D (40., angle_degrees=45., noise_scale=0.4)\n",
"zs = [traj.update(t) for t in np.arange(dt,100,dt)]\n",
"plot_ball_filter (zs)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 21,
"output_type": "display_data",
"png": 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truncated
"text": [
"6.123233995736766e-17"
"<matplotlib.figure.Figure at 0x31d7d10>"
]
}
],
"prompt_number": 21
"prompt_number": 44
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"You can see that the filter follows an almost perfect parabola despite the noise in the signal. Because I know the noise characteristic of my filter, $\\sigma=0.4$, I was able to specify the variance as $\\mathbf{R} = \\sigma^2=0.16$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's make this more realistic. We don't have much call to track balls thrown on airless planets. On Earth, balls are subject to the effects of the atmosphere. \n",
"\n",
"blah blah\n"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import baseball\n",
"dt = 0.01\n",
"f1 = ball_filter(dt, R=1.16, Q=5.5)\n",
"\n",
"x,y = baseball.compute_trajectory(v_0_mph = 90., theta=65., dt=dt)\n",
"zs = zip (x,y)\n",
"plot_ball_filter (zs)\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x3bee8d0>"
]
}
],
"prompt_number": 61
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def ball_filter6(dt,R=1., Q = 0.1):\n",
" f1 = KalmanFilter(dim=6)\n",
" g = -9.8\n",
" g=1\n",
"\n",
" f1.F = np.mat ([[1., dt, dt**2, 0, 0, 0],\n",
" [0, 1., dt, 0, 0, 0],\n",
" [0, 0, 1., 0, 0, 0],\n",
" [0, 0, 0., 1., dt, dt*dt*g],\n",
" [0, 0, 0, 0, 1., dt],\n",
" [0, 0, 0, 0, 0., 1.]])\n",
"\n",
" f1.H = np.mat([[1,0,0,0,0,0],\n",
" [0,0,0,0,0,0],\n",
" [0,0,0,0,0,0],\n",
" [0,0,0,1,0,0],\n",
" [0,0,0,0,0,0],\n",
" [0,0,0,0,0,0]])\n",
"\n",
"\n",
" f1.R = np.mat(np.eye(6)) * R \n",
" \n",
" f1.Q = np.zeros((6,6))\n",
" f1.Q[4,4] = Q\n",
" f1.x = np.mat([0, 0 , 0, 0, 0, 0]).T\n",
" f1.P = np.eye(6) * 5.\n",
" f1.B = 0.\n",
" f1.u = 0\n",
" \n",
" return f1"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 67
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def plot_ball_filter6 (zs):\n",
" xs, ys = [],[]\n",
" pxs, pys = [],[]\n",
" \n",
" for i,z in enumerate(zs):\n",
" m = np.mat([z[0], 0, 0, z[1], 0, 0]).T\n",
"\n",
" f1.predict ()\n",
" f1.update (m)\n",
"\n",
" xs.append (f1.x[0,0])\n",
" ys.append (f1.x[3,0])\n",
" pxs.append (z[0])\n",
" pys.append(z[1])\n",
" \n",
" if i > 0 and z[1] < 0:\n",
" break;\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": [],
"prompt_number": 85
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"dt = 0.1\n",
"noise = 0.3\n",
"f1 = ball_filter6(dt, R=.16, Q=0.1)\n",
"\n",
"x,y = baseball.compute_trajectory(v_0_mph = 90., theta=65., dt=dt)\n",
"zs = zip (x,y)\n",
"znoise = [(i+randn()*noise,j+randn()*noise) for (i,j) in zs]\n",
"plot_ball_filter6 (znoise)\n",
"print (zs[0:5])\n",
"print (znoise[0:5])"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x3e35190>"
]
},
{
"output_type": "stream",
"stream": "stdout",
"text": [
"[(0.0, 0.0), (1.700193266982834, 3.646076227248443), (3.363396716383785, 7.114827534704153), (4.989282707845675, 10.407750130907692), (6.577522827472383, 13.526428179417952)]\n",
"[(0.0077428794280200615, -0.24993790836595048), (1.6319949358597867, 3.906624591863025), (2.927767750429539, 7.282837525906626), (4.889897158515106, 10.321582956313172), (6.55031695691016, 13.756792084374545)]\n"
]
}
],
"prompt_number": 88
}
],
"metadata": {}
baseball.py
+83
View File
@@ -0,0 +1,83 @@
# -*- coding: utf-8 -*-
"""
Spyder Editor
This temporary script file is located here:
/home/rlabbe/.spyder2/.temp.py
"""
"""
Computes the trajectory of a stitched baseball with air drag.
Takes altitude into account (higher altitude means further travel) and the
stitching on the baseball influencing drag.
This is based on the book Computational Physics by N. Giordano.
The takeaway point is that the drag coefficient on a stitched baseball is
LOWER the higher its velocity, which is somewhat counterintuitive.
"""
import math
import matplotlib.pyplot as plt
def a_drag (vel, altitude):
""" returns the drag coefficient of a baseball at a given velocity (m/s)
and altitude (m).
"""
v_d = 35
delta = 5
y_0 = 1.0e4
val = 0.0039 + 0.0058 / (1 + math.exp((vel - v_d)/delta))
val *= math.exp(-altitude / y_0)
return val
def compute_trajectory(v_0_mph, theta, v_wind_mph=0, alt_ft = 0.0, dt = 0.01):
### comput
theta = math.radians(theta)
v_0 = v_0_mph * 0.447 # mph to m/s
v_x = v_0 * math.cos(theta)
v_y = v_0 * math.sin(theta)
v_wind = v_wind_mph * 0.447 # mph to m/s
altitude = alt_ft / 3.28 # to m/s
ground_level = altitude
x = [0.0]
y = [altitude]
i = 0
while y[i] >= ground_level:
g = 9.8
v = math.sqrt((v_x - v_wind) **2+ v_y**2)
x.append(x[i] + v_x * dt)
y.append(y[i] + v_y * dt)
v_x = v_x - a_drag(v, altitude) * v * (v_x - v_wind) * dt
v_y = v_y - a_drag(v, altitude) * v * v_y * dt - g * dt
i += 1
return (x,y)
if __name__ == "__main__":
x,y = compute_trajectory(v_0_mph = 110., theta=35., v_wind_mph = 0., alt_ft=5000.)
plt.plot (x, y)
plt.show()