This is a throw that I've spent a lot of time investigating and exploring over the past couple of years, and I'm always finding something new about it.
First off, we tried a simple little drill to emphasize the basic body relationship. We had uke stand still, feet square. Tori just stepped forward, letting his right arm collapse (hand to shoulder and his elbow in uke's armpit). By the time tori is turned, facing the same way as uke, he's radically pitched forward. For this, we didn't even have a grip with the left elbow. All the action came from the center stepping to uke's side, and the turning of our center transmitting through our right hand, which pitches uke forward. Uke, then can just do a front roll out of it, much like you see here at about the 1:45 mark:
This whole approach has a very aikido feel to it, almost like tenchi nage ("heaven and earth") or maybe the guruma throws from the big ten. If my center was even slightly in front of uke, he could brace and kill it. My center had to be at least even with his, side by side, or even slightly behind (which is a little different than some of the rest of the judo world; they tend to do it right in front, uke still rolling over tori's back, which is a little hip throw-ish to me). It's almost like you're stepping in (a little off to the side) and scooping under him while turning, if that makes any sense. Done right, you virtually don't even need the foot, of course.
I also liked this turning drill that happens at the 4:05 mark, so we played with that this morning (another version of it appears at the 4:50 mark, where uke steps over tori's leg so tori can get a lot of repetitions in).
It's actually quite nice. Basically, as you keep turning, uke's center will gradually drift a little until it's no longer pointed at yours, but a little off to the side. Once you sense that this is happening, the turn and throw are as smooth as butter, no speeding up or anything.
The drawback was, however, as is often the case with new guys, uke would starting turning so his back went into the throw. Which is fine for tori; he can just do osoto gari. But, that's not the drill, and besides, this is a good time to start teaching the new guys how to keep their centers (and their weapons) pointed at tori (we glanced over the fact that a very nice tani otoshi lives there for uke if tori isn't doing his job right, but you've got to keep your center "active" and pointed at tori).
Now, the interesting thing about tai otoshi is that it can be thrown two ways. Most of what I see is done perpendicular to the line of uke's feet, but you can also do it down the line of uke's feet. Chuck Caldwell Sensei liked to throw it that way, and I remember his teaching a subtle balance tweek set-up to it that's pretty nice. You can see a bit of it at the 1:50 mark.
Incidentally, at the 3:10 mark, you'll see the other guy use his foot to brush back uke's far leg to set it up, which is interesting (you get a similar effect if you try an advancing ouchi gari that fails, which he also does).
For Wednesday, I think we'll stick with the same drills and see if we can refine it. We also may talk a bit about Zdenek Matl Sensei's verion, just for fun.