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In ROS I've recorded a bag file from a custom robot (in real world) that does not provide covariance matrix and I want to use /odom to feed an EKF, but covariance matrix is 0. How can I calculate it?

Note: Covariance matrix is needed by EKF to estimate position.

It's a sample of /odom:

pose: 
  pose: 
    position: 
      x: 0.082
      y: 0.507
      z: 0.0
    orientation: 
      x: 0.0
      y: 0.0
      z: -0.789272088731
      w: 0.614043622188
  covariance: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
twist: 
  twist: 
    linear: 
      x: 0.104
      y: 0.0
      z: 0.0
    angular: 
      x: 0.0
      y: 0.0
      z: 0.0663225115758
  covariance: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
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Well, you can't estimate the covariance from the state. You need the equations used to find the state.

This is because the covariance (along with the markov assumption) represents how the robot got to the state. A robot at location (10,5,1) does not always have the same covariance matrix, right? You need to know what actions brought it to that state.

Barring any ros-based solution (are you sure the SLAM / EKF nodes can't do this for you?) you'll have to do this:

  1. Find the kinematics model for the robot (differential drive, ackerman, whatever)
  2. Read in the odometery and control command, and use the kinematics equations and control inputs as the u and f() for the EKF (using this) as a reference.
  3. Find the Jacobian of the kinematics equations with respect to each state element
  4. Feed the Jacobian and control / odometery into the EKF as the F term (using this) as a reference.

In case you are wondering, yes this effectively reproduces the work done by the /odom node, since the output of the kinematics equations is precisely what /odom reports.

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