Foundations.

I have been pondering the feet lately. What I find is that there are a lot of Budoka who only think of their hands and their upper body, with the feet being even less than an afterthought.
Pondering the feet in conjunction with the whole body as a single unit, say, a brick, can be a useful exercise; going through Kata with this mindset one comes to realize that there really is very little movement other than shifting position and the body does this as a whole as opposed to first moving the arms, or first moving the legs, it all happens at once.

Now there are some instances where the hands do move first, but only slightly, and even this means there really is very little movement, thus, very little energy expended.
That is the very essence of economy of motion and such an exercise helps to shave off unnecessary movement such as 'tells' where the shoulder moves before the punch is initiated.

Pondering the feet also helps to realize that we need foundation, instead of bouncing around like crazy monkeys on a dance floor.
One thing I have been doing lately is imagining a triangle on the ground, inside a square, inside a circle, and playing around with footwork inside that design, which covers a very small space.
The design itself covers all 'stances' within Karate and shows how to apply them from a single point, with the circle acting as the boundary; if you go outside the boundary you have lost your ground and, therefor, your foundation.
Naturally it is not an 'end all' exercise, there is no such thing, but it does cover rooting and unrooting while limiting the range of motion.

What I find this helps to do is get the practitioner out of the 'static stances' mindset and actually engage in the practice of fluid footwork in conjunction with strategy.
How do these strange foot positions apply when they take on an active role in the execution of Sabaki? Apply that with the brick exercise, then top it off by playing around with axis of the body, moving the whole body as one unit within a sphere.
Then apply the Yin/Yang idea of completing a movement on one side with its' simultaneous opposite on the other.
You have a winner that covers all fronts of proper Tai Sabaki. At least in my own opinion.