From forces to angles

So far, we have converted our electromyography signal, which is the electrical signal generated by a muscle when it contracts, to force. That involved quite a bit of mathematical modelling: first what we call activation dynamics, then muscle contraction dynamics.

Give yourselves a pat on the back, we’ve come a long way.

What’s left is pure mechanics: when we have a number of forces pulling on bodies, how will the bodies move? In our case, the forces come from muscles and the bodies are the bones of the hand, but the principles are the same as, say, pulling on a handle to open a door.

Step 3: From forces to moments

As you may have noticed, handles are placed as far from the door hinges as possible. Why is that? To produce a turning effect, known as a moment, a force needs to be applied at a distance from the axis of rotation.

You can test this easily if you find yourself in front of a door you have to push to open. If you place your hand very close to the hinges, you will probably not be able to open it, no matter how hard you push. If you place it at the opposite end (where, helpfully, you will often find a push plate), all you need is a small nudge.

This very important distance (technically the perpendicular distance between the line of action of a force and the axis of rotation) is called moment arm.

If you have a large enough moment arm, you can convert even a tiny force into a huge moment. As Archimedes said, “Give me a long enough lever and a place to rest it and I will move the earth”.

I’m not sure where Archimedes is resting his lever, is it Mars? — oh, and here’s the XKCD version.

Based on how muscles insert and wrap around bones, our model knows all the muscle moment arms. So it can easily convert forces to moments.

Step 4: From moments to angles

The final step! We’ll start by using Newton’s 2nd law for rotational motion: if you know the moment acting on a body, and how its mass is distributed, you can calculate how much it is accelerating (as it rotates).

That’s easy, but what we really want to know are the angles of the hand joints, not their acceleration. If I imagine myself going for a run, I will accelerate and slow down quite often as I am running, but what I care about is how far I’ve gone. If it’s more than 100 metres I call that a success.

Converting acceleration to angle is difficult. We cannot solve the required mathematical equations exactly, we need to solve them approximately. Don’t raise your eyebrows, I’m not being lazy: an exact solution is simply not possible.

That in itself is not a problem, as we can use numerical methods to solve our equations approximately. The problem is that our equations are stiff*. This means that numerical methods take tiny steps to solve them. Simulating one second of movement could take hours. Or days. But if we want to use our model for prosthesis control, simulating one second of movement needs to take less than one second.

Luckily, we’ve managed to crack this problem. Here I’d like to mention Professor Ton van den Bogert, my mentor and friend, amazing biomechanist and Academy Award winner. He developed new methods for musculoskeletal simulation for leg movement, which we then expanded and applied to arms and hands. Here is the scientific paper explaining it all, and here is the application to hands.

The final graph shows the angles of the three joints of the middle finger. The angles are zero when the finger is fully extended. So what you see is that the finger starts at a relaxed flexed position, then when the Extensor Digitorum Communis muscle activates the finger extends, and when the muscle relaxes it flexes again.

When we include the action of all muscles on all fingers, the hand movement looks like this:

From closed to open to closed again. Our volunteer was not moving only the middle finger :)

We can now compare the model movement with the movement of our volunteer, that we recorded with our motion capture system.

But first: the Royal Society Summer Science Exhibition, the reason we started this blog! It’s starting in just a couple of days. We have put together a learning pack with information and things to consider when visiting our exhibit.

I am hoping that all this modelling talk has inspired you to come and see us, so we can have a more in-depth discussion of stiff systems :)

*I don’t know why these problems are called stiff. Does anyone out there know? A synonym for stiff is punishing, which these equations certainly are.