Here is my take on the hurdle. I wish that I had some pictures to express what I am talking about, but I do not unfortunately.
The hurdle by definition is simply a transition from a run such that the athlete can maintain as much of the horizontal momentum as possible that was generated from the run into their tumbling skill(s) or vault.
When walking/running, the center of mass is constantly moving upwards and downwards. This can be seen graphically and is the theoretical construct under the idea of the "inverted pendulum" theory of walking. With that said, the goal is to keep the center of mass as constant as possible as any major displacement will affect the horizontal momentum that has been generated.
Despite this, I believe that there should be a bit of a vertical component to the hurdle. I like to see the hurdle go both upwards and forwards. Will you lose a little horizontal momentum? Yes. The same happens in vaulting. But, I believe it's minimal. The reason that I feel that you must go slightly upwards is because of the back leg and where it contacts the ground.
David Kenwright, Olympic coach in Canada, and someone that I respect greatly, teaches his athletes to try and "scoot" (I believe is his terminology) their initial contact leg (which is the "kicking" leg of a round-off/cartwheel) underneath of them as it initially contacts the ground. When watching the top tumblers, this definitely seems to be a trend as all of the best do it. Whether they were taught or they do it naturally, it happens.
The question has perplexed me greatly even with a mechanical background. After much thought, I think it's no different than why landing the feet in front of the body for forward vaulting is necessary. In vaulting, as the feet contact the board and are essentially "stopped," the rest of the body continues to travel forward due to its horizontal momentum. How far the feet should be in front obviously depends on the velocity of the athlete. Athletes who are very fast will need their feet further in front because they tend to generate much greater momentum. If the feet are not far enough in front, the center of mass is too far forward, too great of a torque (i.e. rotation) is created, and the athlete struggles to get the center of mass moving vertically enough to effectively contact and/or clear the vaulting table at the appropriate height and trajectory.
In the floor hurdle, I believe that the initial foot (leg) must contact directly underneath or even slightly in front of the hip to allow a more optimal position of the center of mass. Whether its underneath or slightly in front will again be dependent upon the velocity and the momentum generated by a particular athlete. If the athlete does not pull the initial leg under upon contact, they must compensate in some way so that the center of mass remains within their base of support and they remain "stable" as they are transitioning into the round-off. How do they compensate? From my observation, they often push the hips backwards, which is completely opposite of what you want to happen! This allows a better position of the center of mass so that they remain stable and do not lose their balance. Another compensation is kicking the leg around the side in some fashion. (There are other issues that cause this, of course...head position, etc. so if your athlete kicks their leg around the side of a round-off, don't assume it's purely a hurdle issue. I am just citing another possible compensatory strategy)
(Also, as a side note, if you watch kids who do not bend the front leg knee in the lunge as they land out of the hurdle, their hips move backwards as well. That's why lunging is so important! Teach your kids to lunge when they first walk in the gym!)
Another thing to consider when we're talking about the hurdle is the position of the body. The body should be upright with no closure at the hips. Many gymnasts "reach" to the floor and do so by bending (piking) at the hips. The body should act like a lever and pivot over the lead ("lunge") leg with the back leg initiating the "kick" over the top. The gymnasts should think about "kicking" the back leg to turnover - not reaching to the floor. The weight of the head, arms, and trunk (HAT) actually creates an additional torque as the body is pivoting over that lead leg and should be helpful to the overall turnover in my opinion in addition to the active "kicking" of the back leg.
However, if the upper body bends down at the waist too soon, the gymnast has lost this additional torque and may also create stability problems because their center of mass will now move too far forward of the lead leg over which they are pivoting. I believe that this, too, can also cause a gymnast to innately move the hips backwards as well as to kick around the side.
So, I promote a hurdle that moves up and forward and stress getting the initial contact leg in underneath or even slightly in front of the contact leg's hip. I also promote an upright body position and any leaning is a total body lean that is caused by the bending of the lead knee (i.e. passing through a lunge) just prior to the back leg "kick" and the body pivoting over the lead leg. Another thing about the lunge is that when the knee is extended, that also contributes to the angular momentum that is being created.
Hopefully, all of that makes some sense and gives you something to think about.
Lastly, the hurdle is like any other skill. Spend some time teaching it. Don't assume that gymnasts know how to hurdle.
