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Explained how covariance would look if we only had 1 radar station, not 2.
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Roger Labbe 2015-02-16 07:09:07 -08:00
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06_Multivariate_Kalman_Filters.ipynb
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"You can see how the multivariate Gaussian's shape reflects the geometry of the problem. The first radar system was at a 45 degree angle to the aircraft, and its error in the bearing measurement was much smaller than the error in the range. This resulted in a long and narrow covariance ellipse whose major axis was aligned with the angle to the radar system. The next radar system was also at a 45 degree angle, but to the right, so the two measurements were orthogonal to each other. This allowed us to *triangulate* on the aircraft, which resulted in a very accurate final estimate. We didn't explicitly write any code to perform triangulation; it was a natural outcome of multiplying the Gaussians of each measurement together.\n",
"\n",
"To make sure you understand this, what would the Gaussian look like if we only had one radar station, and we receive several measurements from it over a short period of time? Clearly the Gaussian would remain elongated in the axis of the bearing angle. Without a second radar station no information would be provided to reduce the error on that axis, so it would remain quite large. As the aircraft moves the bearing will typically change by a small amount, so over time some of the error will be reduced, but it will never be reduced as much as a second radar station would provide.\n",
"\n",
"To round this out lets quickly redo this example but with the first radar system in a different position. I will position it directly to the left of the aircraft. The only change I need to make is to the Gaussian for the measurement from the radar. I used\n",
"\n",
"$$\\Sigma = \\begin{bmatrix}2&1.9\\\\1.9&2\\end{bmatrix}$$\n",