Working A Star implementation on a grid, however, using a shape for our 'robot' along with polygonal obstacles. Seems to work okay, still needs work. Since I need a hardcopy of the source I've put up a LaTeX document and PDF.

The proper solution when faced with polygonal obstacles and whatnot is to use a navigation mesh with offsets for the shape and size of the robot itself. However, this is pretty difficult and I had only two weeks to learn most of the relevant bits for JavaFX and make it work so it's good enough. Acceptable is the key word here.

The other thing to note is the path A* will generate on a grid might not always be the most straightforward. This is because some F values may tie, that is where the tie breaking modification comes into play. It helps reduce the number of tying F values so you A* does not explore much when it has a direct LOS with the goal. However, it'll still do some funky stuff that seems off. Unfortunately there's not too much you can do about that.