I am not a materials engineer. I'm not a mechanical engineer. What I am is a hobbyist who has built several boats out of composite materials and a few robots. Here are some things you might want to think about in addition to "flexibility."
- Titanium is a pain to work with. It's hard to cut, nearly impossible to bend, and has to be welded in an inert gas environment. It is springy, though, which meets one of your criteria. The chance of your robotics team being able to work with titanium is low, however, unless you have a mentor with real experience with the metal. I wouldn't consider it.
- The reason people like carbon fiber composites is that they are incredibly stiff, and light for the strength and stiffness. The thing that carbon fiber composites are best at -- stiffness -- is exactly what you aren't looking for. It's probably not your best choice.
- I see a lot of stuff on this board where "fiberglass" is used in addition to "composites." What is usually known as "fiberglass" is glass fiber matt, cloth, or biax held together in a polyester resin matrix. Generically, fiberglass is just one type of composite material. The way you normally experience it, fiberglass is not very stiff, not particularly strong, and heavy for the strength and stiffness. The advantage to fiberglass in boat building, for example, is that it can be laid up by fairly unskilled labor, and doesn't cost much.
More advanced composites use epoxy (or sometimes, vinylester resins, but let's keep it simple) to bind a material together. The materials used include glass fiber in a variety of orientations and compositions, kevlar, and carbon fiber. One of the best things about glass fibers is that they are available in a variety of pre-made types -- woven glass (in the US, measured in ounces per square yard of fabric -- 6 ounce is lightweight, 10 or 12 ounces is heavy), bundles of glass fiber held together in crossing orientations of different weights (called "biaxial"), and other things like mat or chopped fibers which are generally not used with epoxy. Matt and chopped fibers are used to bulk up thicker laminations used with cheap polyester resins.
Since epoxy is expensive, very strong, and impervious to moisture, it is usually laid up with a fiber over a core material. We won't go over cores in detail here, but for amateur use it's very popular to use lightweight, high-quality plywood as a core for structural use. Something like BS1088 meranti or okoume plywood is stiff, strong, and lightweight. Some very expensive, very impressive sail- and motor-boats are built up with multiple layers of biaxial fiberglass laid up in epoxy over a core of 1088 plywood.
For really, truly expensive boats (see "America's Cup") they will lay up carbon fibers over a foam core with a thermo-setting epoxy to hold the whole thing together. If you want a mind-blowing experience, start pricing things like carbon-fiber spinnaker poles for 80-foot racing boats. For us mortals, carbon fiber is not a great choice for large assemblies -- it's picky, brutally hard to cut, messy, and demanding.
With one exception, if you want to lay up a composites structure for a robot, I would stick to either woven or biaxial glass fibers laid up in epoxy. If you need to make a long, strong pole, you might follow the plans like this:
http://boatbuildercentral.com/products.php?cat=36 to build a pole, which can then be bonded to attachment pads for assembly to your robot. This is what I wanted to do with the tetra game a few years ago, but the students didn't want to mess with composites. By the way, it's the availability of carbon fiber "socks" that makes this possible for simple CF composite poles. (That link is to a commercial Website to which I have no affiliation, other than as a happy customer.)
Probably more than you wanted to know...