In Gollum's final scene, Frodo gets to the top of Mt. Doom but can't bring himself to toss the ring into the fire. He puts on the ring and becomes invisible to Sam. At that moment Gollum comes out from nowhere, wrestles with invisible Frodo, bites his finger off, wins the ring, and stumbles into the fire.
Gollum never stumbles:
First time we actually meet Gollum is in the Two Towers. After Frodo and Sam spend two days climbing down a cliff using magic Elf-rope, Gollum scales down bare-handed in record time. He is very dexterous.
Gollum escaped the elves in both Mirkwood and Lorien, and snuck into Gondor as well as sneaking in and out of Mordor.
He lived in the deep dark places for eons.
Consequently, he didn't fall. So he either jumped or was pushed.
I was surprised to see it stated in such clear terms. At this point Frodo was not wearing the ring but he was in it's possession and he was using it's will to command Gollum. The passage is entirely prophetic: ‘In the last need, I should put on the precious... And such would be my command.’
There isn't evidence that Frodo deliberately commanded Gollum on Mt. Hood, but here he is earlier quite clearly commanding him in advance. He clearly says, ‘When I put the ring on, you are ordered to take it from me and cast yourself in the fire.’
Gollum was only following Frodo's final command as Master. The command comes just after Frodo warns him to beware, the Precious will twist his words and use him.