The section starts off with one of the most famous arcade machines Sonic ever had. It was famous despite it's very limited release to certain arcades. This could be seen/played in Japan, and also in England.at Segaworld London. It was unique for two reasons, one being (as seen) it had to be controlled using the trackball, and the second it had it's own unique character. This game featured Sonic, Mighty (Armadillo) and Ray the Flying squirrel. Ray never appeared again, but Mighty was in Knuckles Chaotix. (Mighty is essentially a Sonic re-color without spikes)
It's called SegaSonic Arcade, and appeared in 1993. A LOT of people wanted to play this, but dissapointment reigned! It hasn't yet appeared on any compilation (IE Sonic Gems etc) due to the trackball control. Supposedly they can't make it work right for home consoles.

However...a board (the thing that makes it run) was supposedly leaked, and so there are videos of the actual gameplay on Youtube sometimes. It had a 3/4 overhead perspective (new at the time) so it's an interesting look. Screen shot with can be found in the gallary. 

It's seldom-seen because there were few machines made, and it was never ported to anything. This game was the first with Mighty & Ray, of which Mighty only ever appeared again in Chaotix. The gameplay often (or maybe always) was 'chase' style, where some hazard (in this case lava) was rapidly going after the characters, pushing them to run faster through the level. Look where the character is on the screen: about mid-way, and that's on purpose so that any obstacle coming ahead would be difficult to miss. So you're rushed down a set path making snap decisions ahead of an insta-kill hazard.

Having to spin a trackball at an odd angle plus buttons to control the character/s would likely make it difficult. Of course...this was probably specifically geared to mess you up and make you waste more coins trying to see what was next.