The bubble chair was a start but not quite the solution I needed. There was also an issue of engineering to be considered, how do you rotate a sphere 3 ways without the one set of wheels stopping the other.

Happily this is a problem also worked out long ago :

Omni wheels are one solution :

or perhaps the weirder looking mecanum wheel

This was the core of my idea now. I could build a large ball balancing robot, flip it upside down, put a bubble chair on it instead of a wheel and a small child would have a great time. But I would need a better chair.