[EricH]

You'd have to get the ramp just about below your bumpers for that to work properly (below the bumpers, you can just have an angle down towards the drivetrain). Let's do some rules/field work here:

The bridge settles at 12" off the floor. Bumpers, by rule, must have their bottom between 2" and 5" off the floor (due to the bumper zone). So, whatever angle you use must go beyond the bumpers far enough to lower the bridge by 7" or more.

Per [R02-E], unless the appendage exemption ([R02-D]) is used, nothing can go beyond the vertical projection of the frame perimeter, which is defined in the Bumper Zone. Therefore, you must have an appendage. If a part of the robot, specifically an angle like you want, is stationary, I doubt that you could classify it as an appendage (particularly because [G01] forces all appendages to be within the frame perimeter to start the match). Translation: It needs to be able to move from inside the frame perimeter to outside the frame perimeter.

Now, to some math. You will need to extend out beyond the frame perimeter a minimum of 3" to get the appendage clear of the bumpers. You'll need to be able to drop the bridge 7-10", and of your 14" extension, you have 11" left (due to the 3" clearance of bumpers). This is very doable with a 45* angle. But, you'll need to keep the appendage extended while putting the bridge down. That's the hard part, and ways to do it are left as an exercise to the reader.

Short version: What you want to do (drop the bridge without deploying anything) is not possible without being in violation of [R02]. There are ways to have a simple deployment that would then act as part of the robot once deployed; technically, those would be very difficult to find illegal. However, you'd need to be careful about contacting other robots with the appendage, as there is a risk that some of the gameplay rules could be violated.