does the frame perimeter have to be a complete rectangle? or could we make a U shaped robot?
You can make whatever shape robot you want. As long as it follows the bumper rules set out in 4.3.2 of the game manual.
Take note of the definition of FRAME PERIMETER
FRAME PERIMETER – the polygon defined by the outer-most set of exterior vertices on the HOSTBOT (without the BUMPERS attached) that are within the BUMPER ZONE.
To determine the FRAME PERIMETER, wrap a piece of string around the HOSTBOT at the level of the BUMPER ZONE - the string describes this polygon.
Note: to permit a simplified definition of the FRAME PERIMETER and encourage a tight, robust connection between the BUMPERS and the FRAME PERIMETER, minor protrusions such as bolt heads, fastener ends, rivets, etc are excluded from the determination of the FRAME PERIMETER.
Taken from the definitions, page 8 of the competition manual.
Be aware that MOST, but not all, ‘U’ shaped robots will violate the bumper rules.
The frame perimeter is the convex polygon that is formed by the external vertices of the robot. So, you can have a u-shaped robot, in so far as it meets the bumper requirements, but its frame perimeter will still be a that polygon formed by the external vertices.
There are rules in the glossary in section 1.6! Frame perimeter and Bumper Zone are two important examples… The blue box explanation for the Frame Perimeter has a good way of understanding it.
By definition, a U shaped robot violates the bumper rules because it is concave.
Is it OK to reminisce about the years before bumpers? :o
This year you can have an indentation, but it’s limited in size and depth. (in the frame perimeter not the bumper itself)
At which point having a U-shaped robot is pretty silly since you have that big bumper in the way.
No one said anything about ‘silly’, just if it was legal … Hence my ‘most’ 
.
FRAME PERIMETER – the polygon defined by the outer-most set of exterior vertices on the HOSTBOT (without the BUMPERS attached) that are within the BUMPER ZONE.
and <R07>, applicable sections:
A. BUMPERS must provide complete protection of the entire FRAME PERIMETER of the ROBOT (i.e. BUMPERS must wrap entirely around the ROBOT). As part of the 100% coverage, BUMPERS must protect all exterior corners of the FRAME PERIMETER. For adequate protection, a full segment of BUMPER must be placed on each side of the corner (see Figure 3-2).
K. BUMPERS must attach to the FRAME PERIMETER of the ROBOT with a rigid fastening system to form a tight, robust connection to the main structure/frame (e.g. not attached with Velcro). The attachment system must be designed to withstand vigorous game play. All removable fasteners (e.g. bolts, locking pins, pip-pins, etc.) will be considered part of the BUMPERS.
The BUMPER backing must be supported by the structure/frame of the ROBOT (i.e. the gap between the backing material and the frame must not be greater than 1/4” and no section of BUMPER greater than 8” may be unsupported). See Figure 3-3.
The important item here is to understand how the FRAME PERIMETER interacts with the bumper rules. The FRAME PERIMETER is convex - there are no concave (or U shaped) openings - If you built a U shaped robot, the FRAME PERIMETER would include the outside edge of the U, and a straight line across the top of the U, turning it into a sideways D. That means that you need a bumper going straight across the top of the U, and it needs to be supported every 8 inches.