Kalman filtering for movement in long coordinates

I am trying to implement a simple kalman filter that will be used to filter / predict vehicle movement in long / local coordinates.

There are no measurements from the vehicle sensors, this is just a new update of the observed long / lat position, so basically the condition that I will try to predict and correct is the longitude and latitude of the vehicle at any given time.

As far as I understand, the model is not linear as direction changes and so on can occur, but I think this can be largely ignored as long as I keep track of the bearing state in my state as well. My problem is that I don't know how to model this system in terms of the state and prediction matrix, and furthermore, it seems that it is necessary to transform / project the long / lat coordinates into some cartesian xy system so that the two become independent, but I'm not really sure how to do this.

It seems that converting back to wgs84 from xy is not trivial and potentially a bit computationally intensive. Can anyone shed some light on this?