footwork, strock production, technique, legs and abs are both very important for moving fast and keeping balance and power, i find alot coming from my abs. so i would do ab rotations, crunchs, leg raises, sit-ups, etc.
Definately the legs - as long as you're using complex multi-joint movements like the squat, deadlift or clean. Think about it this way: you have a large load on your back, over your head or on the front of your shoulders (back squat, overhead squat, front squat), or at hip level (deadlift). Your legs exert a huge force on the ground to displace both yourself and the big weight you're carrying. Now, what is there between your legs and your arms/shoulders to transfer that force from your feet to the weight? Hopefully not just your spine! Instead, your core muscles are there to do that. Now, tell me what kind of training would help you more in badminton. Is it doing endless repetitions using repeated very low intensity forces across a dubious range of motion (what is the effective range of motion for trunk movements anyways? not much. if you really try to maximize it, think about how far you're bending your spine around - think your lower back likes being all flexible, loose and unstable like that?). Instead you could train the core muscles to protect the spine by keeping the trunk rigid, as well as economizing the transfer of force from the powerful leg muscles to the shoulders. Hmmm... decrease in injury potential with an increase in power transfer. Sounds like an easy decision to me!
One concept that I think people need to learn to accept is that some muscle groups have an "energy transfer" role, especially under high load or high speed situations. Think about how much power the the puny abdominal muscles can generate (tension*range of motion/contraction length) compared to the mighty hip muscles. In such a case, if the abdominal muscles can simply minimize the power wasted when transferring force across, then it is doing a good job. As a result, don't think about it as a generator of movement, but rather as a resistor of movement. In addition to the squat and deadlift variations I mentioned above, there are a few additional exercises that are needed for completeness. Doing any 1 arm standing exercise (e.g. 1 arm deadlift, 1 arm snatch, 1 arm press, etc) will train your core to resist lateral flexion. 1-arm horizontal exercises (1 arm bench press, 1 arm row) will train your core in an anti-rotational manner. As for the ways to train, the two I like to use are isometrics and oscillatory isometrics. For example, all the above exercises train the core isometrically. An oscillatory isometric exercise would be like this: sit in a lat pulldown machine or roman chair. Straighten yourself out so that your torso is horizontal to the ground. If you were to simply hold this position statically, this would be an isometric exercise. In the oscillatory isometric version, maximize the tension in your abs, then release as much tension as possible momentarily before tightening again. The range of motion here should be very small, and upon retightening you should bounce right up effortlessly. Have fun trying this out
