Please remember, I am not japanese and my knowledge of japanese language is limited. (And sometimes plain wrong!!)
Only remember two diffent forms of osu.
The first is iirc the short form for ohayo gozaimasu and means something like ggodmorning.
The second osu is iirc the same as shinobu and means something like to push oneself forward to endure.
I only heard people of Yoshinkan Aikido use it and assumed it was the second one, but I am not really certain.
Maybe one of the people with better knowledge of japanese can explain it all much better.