Osteoarthritis, insomnia, and litterbox issues

I've had issues with my cat peeing on the rug before, but now there's a weird pattern becoming obvious to me, and I know that this isn't related to a uti or something similar, as she just had a BUNCH of tests done. So:
She doesn't pee in the same place on the rug every time and I clean it very well with enzymatic cleaners every time, pretty much immediately after the occurrence. She's just been diagnosed with osteoarthritis, which I know could make using the litterbox painful, but: she only pees on the carpet late at night, like between 2 and 4 am, and only when I'm awake that late. I've got fairly bad insomnia; this happens twice a week at least. But I check the rug constantly, and when no one is in the room, she only uses the box. When my roommate and I are present, awake or asleep, she uses the box. But when I'm in bed and I've been trying to sleep for about three hours, sometimes she pees on the rug. When I sleep through the night, she only uses the box. I guess my only thought is that maybe she picks up on my stress when I can't fall asleep?? I don't know what to think, but it is definitely only when my insomnia is really bad.
Thoughts? Advice?

To clarify before anything else, you use both 'rug' and 'carpet' in your original post. I know some people use them to mean the same thing, but a rug is usually something you can roll up and move somewhere else, while a carpet is usually wall to wall covering that is firmly attached to the floor.  Which one are we discussing? If it is just a rug, can't you just remove it at night or maybe entirely?

We have a very comprehensive series of articles on litter box problems, which I am linking for you here.  Please read through all of these  and see whether you think any of the advice applies to your situation.

http://www.thecatsite.com/a/how-to-solve-litterbox-problems-in-cats-the-ultimate-guide
http://www.thecatsite.com/a/how-to-choose-the-right-litterbox
http://www.thecatsite.com/a/how-to-choose-the-right-cat-litter
http://www.thecatsite.com/a/how-often-should-i-clean-the-litter-box
http://www.thecatsite.com/a/how-many-litterboxes-should-you-have
http://www.thecatsite.com/a/litter-box-maintenance
http://www.thecatsite.com/a/litter-box-location-secrets

http://www.thecatsite.com/a/combat-cat-urine
http://www.thecatsite.com/a/how-to-remove-cat-urine

http://www.thecatsite.com/a/is-your-cat-stressed-out
http://www.thecatsite.com/a/six-surefire-strategies-to-reduce-stress-in-cats

After you read through some of these, let us know if you have any questions.

Good luck resolving this issue- I know how frustrating it can be.

It is a rug, sorry for the ambiguity. We live in a dorm room, so there's really nowhere to put the rug at night, and my roommate and I absolutely hate stepping on the tile. I've read every article on litterbox issues on this site and pretty much anywhere else.

What's stumping me here is that it literally only happens when my insomnia is really bad and I'm up past 2 am but trying to sleep. When I am asleep, absent from the room, or purposefully awake, she always uses the box.