Solid body rotation around 2-axes

Question

I am trying to understand how to describe the rotation of a solid body flying in 3D space. From physics forums, I understand that the rotation of any solid object in space, is around 2 axes simultaneously.

Can someone help me understand what it means and how can I describe this kind of rotation by Euler angles, axis-angle or with quaternions?

Look here for my previous question in this area, which includes videos of an actual asteroid movement: Aren't asteroids contradicting Euler's rotation theorem?

I would like to describe that movement using one of the above mentioned representations. Is it possible in an easy way? What I had in mind, is that I can write an algorithm like

1. rotates an object around axis A by n degrees
2. rotates axis B around axis A by n degrees
3. rotates the object around axis B by n degrees
4. rotates axis A around axis B by n degrees

And run it in a loop with small n. But I don't know if it would give the above described rotation movement. Is there any literature in mathematics which would describe this "2 axis" rotation?

I asked the previous question in Physics forum here, and I got the following response, what I don't know what to do with.

You might also want to google the phrase "The polhode rolls without slipping on the herpolhode lying in the invariable plane."

together with 2 youtube videos showing objects producing this rotation:

an animation: http://www.youtube.com/watch?v=zmXpSL91fzk

a book on space station: http://www.youtube.com/watch?v=GgVpOorcKqc

Can you tell me where should I start reading and where can I find a method or an algorithm to describe that movement with a common rotation representation? Also, can someone shed some light on what does that polhode statement mean?

Answer 1

To me, the statement about rotation about two axes simultaneously doesn't really make sense.

To begin with, as clarified by the answers and comments to your previous question, the way mathematicians use the word "rotation", it's only about the initial and final state, not about the actual motion that takes place in between. When we say things like "rotation by an angle of $\phi$ about the $z$ axis" it is easy to be fooled into thinking that this must take place by really spinning something continuously around the $z$ axis. But really, all it means is the transformation from an initial to a final state that's described by the rotation matrix $$ \begin{pmatrix} \cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1 \end{pmatrix}, $$ regardless of what happened along the way. If you really want to describe the process of spinning around the $z$ axis, you need to use a time-dependent rotation matrix (or a time-dependent quaternion, or however you represent your rotations). For example, we can say that at time $t$ the body is rotated (compared to the initial position at time zero) by the rotation $$ \begin{pmatrix} \cos t & -\sin t & 0 \\ \sin t & \cos t & 0 \\ 0 & 0 & 1 \end{pmatrix}. $$ Any type of spinning or tumbling motion of a rigid body (with the center of mass fixed, for simplicity) can be described by a time-dependent rotation matrix, but in general the time-dependence will of course be much more complicated than this simple "spinning around a single axis" case.

The type of motion actually performed by a rigid body in empty space is described by something called Euler's equations. They are expressed in terms of the angular velocity vector $\omega$, which is related to the time derivative of the time-dependent rotation matrix, so even if one manages to solve the Euler equations for $\omega$ there is still some integration left to do in order to really get the rotation.

Anyway, Euler's equations simplify when no torques act on the body, because the right-hand side becomes zero, but they are still not completely trivial. In that case there are some simple solutions which just consist of pure spinning around a single axis (namely one of the principal axes of inertia of the body), but most solutions are more complicated, like the tumbling motion of your asteroid, or the book in that video when the astronaut tries to spin it around the "middle" axis. (Spinning about that axis is possible in principle, but unstable, so that the slightest deviation will make the rigid body tumble erratically instead.) I have no idea what those people mean who say that these tumbling solutions are described by "rotating around two axes simultaneously". The most concrete geometric description of the tumbling is given by Poinsot's ellipsoid, which is what that "polhode" stuff refers to. (The quotation is from a footnote in the section about this in Goldstein's book Classical Mechanics.) In short, for any rigid body, one can define an abstract "inertia ellipsoid" which is imagined to be attached to the body, and the body tumbles in just the same way that this ellipsoid would tumble if it were rolling on a certain abstract "invariable plane" which is imagined to be fixed in space (in just the right way for this whole thing to work, of course).

I suggest you try to find some mechanics textbook that you like, and read about rigid body dynamics there; it's a bit too involved to try to explain in detail here, I think.