Did Frodo complete his quest?

[Ancalime]

I just finished reading LotR for the first time and I have a nagging question: did Frodo really complete his quest to destroy the Ring? Sure, the Ring ends up in the fires of Mount Doom, but I got the impression that if it had been up to Frodo, he wouldn't have thrown it in by choice. And that's another thing-I thought that if the Ring was taken by force, the former bearer would go 'mad' or something-so why didn't Frodo? But that's an entirely different topic. [img]smilies/wink.gif[/img]
Basically, I guess my point is that in a way, Frodo failed in his mission because he claimed the Ring as his and refused to destroy it. The only reason it ended up in the fires of Mount Doom was because of Gollum's interference.