In my Lindy Hop lessons in March 2011, the question arose of what is the low-level mechanism that implements the "connection" that allows the lead to control the follow. Most people (including me) manage to learn to use the connection properly by trial and error. The experience goes straight into their motor memory, and the principles are never consciously understood. While this is a perfectly satisfactory way of learning to dance, it is no comfort to the enquiring mind, which inevitably develops a further ambition to understand it properly.
It is a difficult problem. This is partly because dancing is a very complicated motion. To make any progress at all it is first necessary to separate out its many features, and then to distinguish the many irrelevant ones from the few relevant ones. Then, to have any hope of finding a complete answer, it is necessary to limit the scope of the question quite brutally.
I made my choices after much discussion with David Laban and Gérald Bianchi (and via Gérald also with Maria Andersson) for which I am very grateful. I think that we ended up broadly in agreement, but to avoid any risk of mis-representing them I won't claim any authority but my own.
My choice of scope is summarised in the following quotation from those discussions (in which the question is Gérald's and the answer is mine):
First of all, I'd like to ensure that we are talking about the same problem. As far as I understand, you guys are trying to figure out what makes the follow move. Is that correct?
Very sensible. :-)
Yes. In particular, the motivating example was the 1 & 2 of a swing out, and the distinction between the variant where the follow moves on 1 and the variant where she counterbalances on 1 and moves on 2.
I am also looking for a broader understanding, especially after Caroline and Will's Wednesday block. In particular we did an exercise in which the follow pushes or pulls the lead across the floor. This rather unnatural exercise (which I found I could do!) is an excellent counter-example to some of the theories of how leading and following works.
I think if you boil the above swing-out distinction down to its simplest terms you end up studying the distinction between a sugar push (in which lead and follow travel) and a counterbalance (in which they don't). Both of them are periodic oscillations, but in the counterbalance the lead and follow are in antiphase, while in the sugar push they are almost in phase (the lead is slightly ahead). I can understand Caroline and Will's exercise (in which the follow is slightly ahead) as an unusual linear combination of the two.
I think it is helpful for all of that to be in scope, but I don't want to widen it further than that. We should stick to movement along a rail, for example, because I expect the generalisation to other degrees of freedom (e.g. turns) will be straightforward. Similarly, I don't want to get into "virtual mass", "visual leads", "cheating" or similar unless it is necessary to explain the above cases, which I don't believe it is (feel free to persuade me otherwise).
I think this scope is the same as you are describing.
My choice of how to simplify the model is illustrated by this quotation (mine again):
I think what makes this question so hard is that there are a large number of obfuscating factors. For example:
- There appears to be a special case when the force is zero, in which it is impossible for the follow to match a sudden change in force. Idiomatic dancing uses a "prep" to work around this case.
- More generally, a follow takes time to react to match a change in force. The lead can exploit this to control her arm independently of the rest of her frame. A gradual motion will move her entire frame, while a sudden jerk will only move her arm.
- David pointed out that by raising and lowering your centre of mass, it is possible to pretend to have a different height. This allows you to vary the ratio of force to position needed to maintain a counterbalance.
- David pointed out that by moving your bum more than your shoulders, it is possible to pretend to have a larger mass. This too allows you to vary the force to position ratio needed to maintain a counterbalance. It can also be felt in dynamic forces, e.g. centrifugal force, or the force needed to achieve a rapid acceleration.
- If we model the follow's contact with the floor as a point force, then the follow has two different ways of moving that point around. First, she can move her weight from heel to toe, and second, she can take a step.
- Human muscles do not behave like springs. Two opposed muscles (e.g. bicep and tricep) do not just cancel each other out. They can be controlled independently, and when they are both tense it feels different from when they are both relaxed. This "feeling" is manifested on timescales that are faster than the human's reflexes or conscious movements.
- Even a good follow cannot maintain a perfect frame. She requires time to react.
- As you say, a couple can use Bayesian inference. In particular, a good follow can compensate remarkably well for bad leading - a practice we call "cheating" - and similarly a good lead can learn to place a follow correctly even if she systematically throws her own weight around.
I think we will not have answered the question unless we understand the case where none of the above apply.
In summary, each person is modelled as a point mass supported on a massless rigid rod: an inverted pendulum. In reality a person has many other degrees of freedom, but they can all be simplified away. The base of the rod represents the point at which the person places their weight on the floor. In reality, the person moves their weight around using the muscles in their ankle, and when that is insufficient, by taking steps. However, these details too can be simplified away. The feet are instead modelled as a massless wheel, equipped with a motor that allows the person to move their weight around at will.
After the first few salvos in the discussion, it became apparent that our various theories were not going to be distinguishable from the truth by abstract thought alone. I therefore decided to build a robot, which David christened Sugar Push-Button. After exploring the parts list, and working out the costs (about £50 for a 30cm tall version, plus lots of work), I back-tracked and wrote a computer simulation instead. It immediately disproved one of my theories, which was highly satisfactory. I have subsequently written more simulations. They are all collected below, with accompanying explanations.
The foot accelerates when the head feels an acceleration. The foot accelerates twice as much as the head, so as to "get in front" and correct the acceleration. The foot acceleration also includes a damping term to prevent oscillations; specifically, if the head's acceleration is increasing, the foot accelerates more, and if the foot's acceleration is decreasing it accelerates less. View the source for details.
These rules implement a self-righting inverted pendulum. This is a reasonable model of how a person stands upright without falling over. Of course, a real person is a many-jointed thing, with numerous degrees of freedom. Also, a real person cannot accelerate their feet arbitrarily - they have to take steps. On the other hand, a person can move their weight from heel to toe without moving their foot. For simplicity, these features of a real person are not modelled. Instead we pretend that all of a person's mass is at their head, and we give the person a wheel for a foot, whose position represents the place that the person has placed their weight.
The hypothesis was that by applying a force to the person's head (or, more precisely, to their centre of mass) would cause the person to move in the way that a follow does in Lindy Hop. The hypothesis was rejected. The observed behaviour is that the person is unwilling to travel, and instead adopts a stubborn counterbalance.
It is interesting that it is not possible to induce the simulated person to travel at a uniform velocity. The person is unaware of their velocity, and is perfectly capable of maintaining a constant velocity, if only they started with one. However, no amount or timing of force seems to set up such a velocity.
In this version, the mouse directly controls both dancers, and the arm is redundant. The vertical position of the mouse determines the separation of the dancers, while the horizontal position of the mouse moves the couple as a whole. This simulates "cheating", i.e. the situation where the dancers do not communicate during the dance. Moreover, there is no counterbalancing.
Each person places their feet so as to accelerate their head towards the desired position. The head acceleration includes a damping term to prevent oscillations; specifically, if the head is moving towards the desired position the head accelerates less, and if it is moving away then it accelerates more. View the source for details.
The simulation is deliberately slowed down to allow you to study it, so please use slow, smooth mouse movements if you want a realistic effect. Here are some things to try (in this and all later simulations):
In this version, the mouse still directly controls both dancers, but the arm is used to change the separation of the dancers. The thickness of the arm represents the force; compression makes it thicker and tension makes it thinner. Cheating is still needed for travelling; the follow's feet respond directly to the mouse. There is still no counterbalancing.
In this version, the follow obeys a simple rule: the foot is always underneath the body. This means the follow behaves like a "frictionless particle". The lead is completely in control of the follow (the follow ignores the mouse) so there is no cheating. The communication between the lead and the follow is mediated only by the force in the arm; it is one-dimensional.
In order to start or stop travelling, it is necessary for the couple as a whole to go off-balance. It is clear from watching the simulation that the lead has the unique privilege of taking the couple as a whole off balance. This is correct according to the conventions of leading and following in Lindy Hop.
Although this version gets a lot of things right, it doesn't look like proper Lindy Hop because there is still no counterbalancing. This is unsatisfactory for two reasons:
In this version, an extra arm force is added that is proportional to the arm length (minus the natural arm length). The size of the force is chosen to be exactly that required to maintain a static counterbalance.
The follow obeys a slightly more complicated rule: she places her feet in front or behind her body as necessary to compensate for this extra arm force (only). The lead is again completely in control of the follow. However, the communication is now mediated by both the force in the arm and the length of the arm; it is two-dimensional.
This version is my best answer, and represents my complete understanding (as of April 2011) of what actually happens in real dancing (a full-speed version might be more convincing). It is notable that the connection is two-dimensional. I have not so far found a version that works with only a one-dimensional connection, but nor have I proved that it is impossible. This is still an open question.
Apart from the original simulation (2011-03-31), all the above simulations assume that the lead and follow both know exactly where they are (relative to the mouse, at least) and which way is up. In reality, this ability is forbidden by the laws of Physics - to be precise it is Galilean relativity that spoils the party. No physical being can measure their own position or velocity, except relative to other objects. Only acceleration can be measured.
In practice humans have an excellent sense of balance which gives them an intuitive feel for which way is up, even on a sloping treadmill, with a wind blowing, and with their eyes closed. Humans can dance even under such adverse circumstances. I therefore no longer feel that hard-wiring an understanding of "up" into the simulations is a significant failing. It is certainly a useful simplification.
However, for the very first simulation above, written when I was still considering building a robot, I did go to the trouble of making the physics pedantically correct (I think). All the behaviour of the robot is determined by the accelerations it feels, as required. It is a useful illustration of how it can be done. The basic technique is to differentiate the equations of motion until all positions and velocities have been turned into accelerations, and then add extra feedback and damping terms as necessary to keep the robot stable.
Any attempt to build a robotic follow based on the other simulations should start by doing the maths needed to make the equations relativistically invariant. In other words, you have to give the robot a sense of balance that it can use to work out which way is up, even in adverse circumstances. Otherwise, the robot will not work. I have not done this. You have been warned!