The tricycles ride on a circular track (see video or pictures below), whose surface forms what I might call a fanned catenary. Note that the inner wheels of the tricycles are smaller squares than the outer wheels, in order that they fit the smaller track length there. Note also that the two rear wheels each have their own axle, and the smaller inner wheel axle is correspondingly lower, although they are geared so that they turn together at the same angular rate. Thus, the two wheels turn in unison, with their corners sinking into the crevices between the catenary pieces at the same time. Another detail is that the turn angle of the tricycle is fixed — the rider cannot steer.
The point of the exhibit, of course, is that despite the square wheels and bumpy road, the tricycles offer a smooth ride around the course. Each axle stays the same distance above the floor as it turns, because the uneven track exactly makes up for the change in the radius of the square as it turns.
My question has to do with the observation that each tricycle operates properly only at a certain fixed radius from the center of the track, namely, the radius at which the circumference of the wheels matches that of the track surface. If you look in the photos, you can see that this radius is marked on the track with a solid red center line for the front wheel and dashed lines for the rear wheels. The wheels have a rubber surface that does not slip on the surface of the track.
Question. Is the geometry of this arrangement stable under small perturbations?
In other words, it seems inevitable that the tricycles will get bumped in some way or pushed a little off their line, and then the wheels will no longer exactly fit the track. Will the precise way that they don't fit tend to force them back onto the track properly? And what if the fixed angle of the steering is a little off? Will the operation of the tricycle simply force it back into place?
Looking at the operation of the exhibit in practice, it seems to be fairly stable, and I have never seen a rider simply get stuck. I asked the guy running the exhibit at MoMath, but he made only a non-committal answer, although he did suggest that it might be at least a little stable.
I am hoping that someone with a strong geometrical sense will be able to explain to me why the tricycles seem to operate smoothly even when subject to random forces of kids jostling them.