Egg cartons works as a poor mans sound proofing for buildings and is cheap.
Egg cartons is very poor in dampening the sound intensity eg. outside a rehearsal room, but it works for getting the reverb down, which is unfortunately called dampening as well. To measure the effect of this dampening, you usually use a ball shaped array of speakers and a sound output that is a bit like a heavy machine gun firing short bursts and then measuring the delay and strength of the return signal.
I think it has a lot to do with the shape. [...]
My office is over-dampened by flat
baffles mounted on the wall and ceiling (and the floor is covered in grounded carpet with carbon fibers - glued on to a grounded copper net with carbon loaded glue (expensive beyond imagination).
Take a sonar sensor and test it against some different materials. also I think you need to use the foam style cartons not the cardboard ones.
I agree on the testing with actual sonar and since one is probably used on the 'bot, it won't mean an extra outlay
I'd go for something like whatever_I_could_find and make lots of test to acquire some knowledge and then go buy what works best.
2 layers might be beneficial if different densities are chosen - light stuff on the outside and something like a thin layer of neoprene on the inside against the 'bot.