Ideas for Engineering Course

[IKE]

Sounds like you are looking for a challenge. As I am from the area and know a little about your school/team, why don't you do a complete design study for a system that your team may someday want to use.

I.E. do a complete parametric crab drive design using FIRST components. If you are doing AP physics you should have the background to complete spec out a system from component design, speed, wheel torque, and robot pushing power.
Personally I prefer 6WD bots, but having a well thought out crabdrive in your back pocket would be a great Ace in the hole. The big thing on this would be keeping ego in check should your design be great, but your team does not want to build it.

In college I did a design for a CVT transmission as an independent study. This included doing the stress calculations and tractive power transmission. Doing independent study is tough. As you have already realized you can get as much or as little out of it as you want. What area of engineering do you want to go into? Hope this helps.