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I have radar mounted on a car. For each detection, the radar returns these variables.

  • relative distance between the object/host vehicle in forward direction
  • standard deviation of the relative forward distance
  • relative distance between the object/host vehicle in left/right direction
  • standard deviation of the relative left/right distance

I'm trying to do coordinate transform of the above data, so that I get relative distance/standard deviation in global coordinates (North, South, East, West) Distance is easy since it only requires to rotate the axis by the amount of angle between the vehicle body-fixed axis and the global axis.

How about standard deviation? How do I transform the standard deviation from the vehicle body-fixed axis to global axis?

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Standard deviation is used to represent probability distribution - in your case if you have $\sigma = 1m$ for forward distance it means that there is ~68% chance of true forward distance to the obstacle be less than 1m away from your measurement. You have two variables, so you should project your probability distribution on a 2D surface - you will get something like that: enter image description here

Each line represents points of equal probability of being a true position. Center of ellipsis is your measurement and has a highest probability.

Now, when you project these values to global coordinate system you will likely get something like:

enter image description here

You cannot represent such a probability distribution using only two variables - you have to use 2x2 covariance matrix.

Good tutorial on graphical interpretations of covariance matrix can be found here: http://www.visiondummy.com/2014/04/geometric-interpretation-covariance-matrix/

In general you can find your covariance matrix for global coordinate system with: $\Sigma = R S S R^{-1}$ where $R$ is rotation matrix from local to global coordinates and $S$ is diagonal matrix with your variances as diagonal elements.

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