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

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Roger Labbe 2016-09-23 12:49:37 -07:00
commit 91e603c592
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2
07-Kalman-Filter-Math.ipynb
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@ -754,7 +754,7 @@
"source": [
"## Design of the Process Noise Matrix\n",
"\n",
"In general the design of the $\\mathbf Q$ matrix is among the most difficult aspects of Kalman filter design. This is due to several factors. First, the math requires a good foundation in signal theory. Second, we are trying to model the noise in something for which we have little information. Consider trying to model the process noise for a thrown baseball. We can model it as a sphere moving through the air, but that leave many unknown factors - the wind, ball rotation and spin decay, the coefficient of drag of a ball with stitches, the effects of wind and air density, and so on. We develop the equations for an exact mathematical solution for a given process model, but since the process model is incomplete the result for $\\mathbf Q$ will also be incomplete. This has a lot of ramifications for the behavior of the Kalman filter. If $\\mathbf Q$ is too small then the filter will be overconfident in its prediction model and will diverge from the actual solution. If $\\mathbf Q$ is too large than the filter will be unduly influenced by the noise in the measurements and perform sub-optimally. In practice we spend a lot of time running simulations and evaluating collected data to try to select an appropriate value for $\\mathbf Q$. But let's start by looking at the math.\n",
"In general the design of the $\\mathbf Q$ matrix is among the most difficult aspects of Kalman filter design. This is due to several factors. First, the math requires a good foundation in signal theory. Second, we are trying to model the noise in something for which we have little information. Consider trying to model the process noise for a thrown baseball. We can model it as a sphere moving through the air, but that leaves many unknown factors - the wind, ball rotation and spin decay, the coefficient of drag of a ball with stitches, the effects of wind and air density, and so on. We develop the equations for an exact mathematical solution for a given process model, but since the process model is incomplete the result for $\\mathbf Q$ will also be incomplete. This has a lot of ramifications for the behavior of the Kalman filter. If $\\mathbf Q$ is too small then the filter will be overconfident in its prediction model and will diverge from the actual solution. If $\\mathbf Q$ is too large than the filter will be unduly influenced by the noise in the measurements and perform sub-optimally. In practice we spend a lot of time running simulations and evaluating collected data to try to select an appropriate value for $\\mathbf Q$. But let's start by looking at the math.\n",
"\n",
"\n",
"Let's assume a kinematic system - some system that can be modeled using Newton's equations of motion. We can make a few different assumptions about this process. \n",

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code/mkf_internal.py
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@ -144,7 +144,7 @@ def show_position_prediction_chart():
plt.xticks(np.arange(1,5,1))
plt.yticks(np.arange(1,5,1))
plt.scatter ([4], [4], c='g',s=128, color='#8EBA42')
plt.scatter ([4], [4], s=128, color='#8EBA42')
ax = plt.axes()
ax.annotate('', xy=(4,4), xytext=(3,3),
arrowprops=dict(arrowstyle='->',