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I'm tracking the position of a vehicle along a certain trajectory using the Kalman filter and the idea is to check for improvements in position estimation through fusion of data from multiple sensors (We consider data from 2 inputs: GPS and Odometry)

GPS data is recorded through an app on the iPhone and gives information about the Latitude, Longitude, Elevation and Timestamp Information. This GPS data is in 3D spherical coordinates and is converted to 2D Cartesian using UTM (UTM is projected cartesian System). Thus, we have the position of the vehicle through GPS in world coordinates (in m).

Odometry data is recorded through measurement sensors on vehicle and gives the x-position of the vehicle, y-position of the vehicle and orientation of the vehicle. This is calculated based on the wheelticks and is relative to the inertial origin of coordinates. Thus, we have the position of the vehicle through Odometry in vehicle coordinates (in mm). The GPS data updates once every 1000ms and Odometry data is given once every 20ms.

  1. How do we relate the position data from vehicle coordinates system with the data from world coordinates?
    (Eg: If I check the data for some point, I see GPS data for the position (x,y) to be (661272.59, 1.517914e+06) and Odometry data for some point to be (414,6) and, both are measurements given to the Kalman. How do we relate the position given by odometry with the corresponding GPS point?)
  2. What is meant by inertial origin of coordinates and inertial coordinate system? (I read about dead reckoning systems but, in this case, we do not have information about the accelerometer, gyroscopes, etc and only get information about positioning of vehicle from wheel ticks)
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The GPS gives you the absolute position (in world coordinate system), while the encoders the relative position of the vehicle, according to the initial assumption that you made. To simplify from GPS you get the actual position, while from the encoders you get the path traveled between 2 measurements. In your position I would get the as initial position the one from the GPS and then when I would get measurement from the encoders I would just update the position, finally when I would get again both the measurements feed them to the filter.

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As nionios said, the GPS gives you latitude and longitude in global coordinates, while odometry gives the difference between two coordinates, normally using encoders in the motors and/or IMU.

You can use your initial position from the GPS data as your initial position for odometry.

However, as stated here, GPS data has a precision in meters and you won't necessarily increase your accurate position if you use it alongside IMU and encoder information in a Kalman Filter.

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