Could it be Alien: Extinction by The Asylum?
The film has two separate storylines that eventually intertwine, both centered on typical all-American young white men who gather their diverse teams to fight an alien menace that has sent robots to Earth to capture humans and take them back to their masters’ planet for harvesting. One of our heroes is Captain James Wheeler (Anthony Marks), who, like too many protagonists in this kind of thing, had quit the Air Force after a mission gone wrong, and has now been convinced by a trusting General (Robert Picardo) to come out of retirement for one last mission. Why the heck these films always seem to think a single person is the difference between winning and losing, I don’t know, but there you have it.
On the other side of the spectrum we have the good-hearted family man Chris Meher (Lane Townsend), a firefighter who gets caught up in the alien invasion when his daughter Lindsay (Taylor Coliee), who he, of course, isn’t close enough with, is abducted by the alien robots and taken to their world.
Government physicist Dr. Gordon (Kelly Hu) assists Wheeler’s team in flying onto the aliens’ asteroid so they can blow it up, eventually figuring out that if they get shot by one of their floating spherical robots they will be teleported to the aliens’ home world, where all the human hostages have been taken. Soon enough, Wheeler and Chris have both found themselves fighting the aliens (or, should I say, people in very minimalistic costumes) – the former with military gear, the latter with an axe – in truly trashy B-movie fashion.
It's from 2014, which is the only part that doesn't match.
Trailer
I found it by searching for science fiction film spherical flying robots "with an axe"
