Here's a reply, not an answer. Just some information, but not enough information. You've applied an acceleration. That's fine, but acceleration doesn't come from thin air, and for the human body it most certainly is not constant. Acceleration comes from force (Newton's second law of motion), and the force that pushes a human body forward is the difference beween internal muscular forces and the friction force between the foot and the ground. The net muscular forces exceed the frictional force, so that the net force applied is nonzero, allowing the center-of-mass to accelerate forward. The body deforms in such a way that the foot remains fixed on the ground for a period of time while the pelvis moves foward. Then, the next foot gets placed on the ground, as the muscles pull the pelvis foward again. The way the body interacts with the floor is pretty complex, and the deformation involves translational and rotational dynamics within the skeleton...caused by muscular impulses. The human body cannot walk without bone rotation...without bone rotation, the body would be rigid, and in that case an external energy source...expelling some kind of gas (laugh!) or an object pushing from behind...would be the only way to achieve locomotion, there simply would be no way for muscle actions to even do anything if the skeleton was rigid. So, in real life, a lot of very complex dynamics are going on.

The equation you posted, as written really is a form that is built around acceleration being constant for all time, never changing. That is simply not the case with human motion, and you have to vary acceleration to do what you want to do. See This Thread for a discussion of how to extend that formula to deal with non-constant acceleration.

I don't have any immediate advice as to how you can model acceleration for very simple physics-based walking. I'm absolutely certain there's a way to do it, though it probably won't be any better than applying some simple sinusoidal speed function rather than using numerical physics. But, I can say that physics-based walking simulations have been done by other people many many many times in the past, for things like insect simulation, simple robot simulation, and even extremely complex human-like motion. See Natural Motion's Morpheme product for example, which is partially physics-based. Do a google search on physically-based walking...lots of stuff...some super complex, some easier to understand. Just today, I saw a nice web app demonstrating physics-based walking of a 2D biped robot....let me see if I can find that link....well, I can't find it quickly and I have to go to sleep. I'll see if I can find the link at the office tomorrow..I was looking at that demo on my office computer.

There was a nice article in the Journal of Game Development (JOGD), first issue, that presented an elegant way to do realistic procedural walks, on flat terrain at least. It does involved sinusoidal animation of limbs, and may be manageable for games...it might not be much more expensive than interpolated quaternion keyframes. Here's a link to the journal:

Journal of Game Development

The journal is pricey, but if you aren't inclined to subscribe, look in the abstracts link for the article "Versatile Walk Engine". Then google the authors. You may very well find some freely available publications from there.

I have a feeling the Natural Motion stuff isn't purely physics-based. They do have a real-time SDK, Morpheme... I believe what they are doing is using mo-cap or hand key-framed animation that is then re-targeted---tweaked---using physics and/or IK. That is, physics-based response superimposed on a more traditional animation technique. Er....that "traditional" animation technique may be sophisticated and not directly artist or actor-created, e.g., they may have some collection of representative animation data or keyframes or gesture information, then mimic that to eliminate the need to actually do mo-cap or artistic key-framing.

The idea of example-based or sketch-based animation that can generate reasonable-looking motion from prescribed samples or simplistic gesture data.....

Motion Doodles


A Sketching Interface for Articulated Figure Animation

Skeleton-free Example-aased Animation

Combining some of those ideas with physics or IK can be quite powerful.

But, I'm not sure. I don't think its impossible to produce good-looking physics-based animation of characters, and especially if you focus on certain very specific motions---not trying to look right for every kind of situation.