More thoughts on TPS. The two factors that affect TPS are
1. brain speed
2. finger speed
Brain speed - You can't do anything before you know what you're supposed to do. So knowing what you need to do is the very first step to making any move at all (this should be obvious). That's why look-ahead is so important. But, most people aren't talking about look-ahead when they talk about improving TPS. Usually it's simply turning faster when you already know what to do. Finger speed - This I think basically comes down to practice. The more you practice, the more accustomed your fingers will be to making turns. You're just exercising muscles at first and then it eventually becomes reflex action. This is related to "muscle memory". Finger speed can be broken down into two sections. 1) How fast you can make a turn and 2) how fast you can transition into the next turn.
But both of these are just starting points. Knowing what to do and how to do them and getting comfortable with it may not be enough. I believe you have to practice speed itself. First, take a simple move, let's say it's U. How fast are you able to execute it? One move, not U many times in a row. Now, this is just the smallest part. One single move. You know what to do before you execute it. You're going to make a U move. Then you move your finger as fast as possible. Easy, right? And I don't think it would take you a second to do it... not even half a second. 1/10th of a second, perhaps? And the brain is nearly always faster than the finger. So next is the trick of getting from one move to another without even the slightest pause. So you try a longer series of moves like U R'. Back to brain speed and finger speed, your brain has to think faster. Don't think U then R'. Think U R' as one single move. Basically, skip right to the R' as soon as you think of starting the U. Or even more specifically, aim to complete your R' at the same time you start your U. Otherwise it becomes Start U/Finish U/Start R'/Finish R' which is much too slow. Our cubes are quite sophisticated nowadays and the corner cutting ability allows you to start the second move before the first one is completed. You can practice U R'... pause... R U'... pause... (repeat). R U' is simply the reverse movement. Eventually you can train your mind and your fingers in similar ways to execute longer triggers (say, 4 moves) or entire algs (say, T-perm:14 moves). Triggers are good (R U R' U' is a common one). If you can get those so it's thought of and executed as one "move" then you know you are quickly getting to the U' once you start the R. Sometimes you can work on a specific section of an algorithm to improve that part (maybe it includes a regrip, for example).
Anyway, those are my thoughts about that. Nobody, when running, waits for their weight to roll from the heel to the toe before moving their other leg.
