I solved my quest for a cool looking platform base which was a requirement from my wife.
Just ordered this two wheeled self-balancing scooter to modify for a laundry / grocery cart.
$425 with shipping.
I plan to add a triangular platform with a 3rd swivel wheel for increased stability.
I already have the LIDAR Lite for room recognition and object avoidance:
I'll use 3 ping sensors to prevent the added 3rd wheel from banging things on turns.
I found a video of one being disassembled:
I'm guessing they use two accelerometers that control the speed and direction of each wheel independently. They probably use pulse width modulation to control the speed.