Euler angles derivation

Question

I have been trying to grasp the idea of Euler angles for a while. Can anyone point out if my understanding is correct or not.

Situation: We have 3 axes known as principal axes of inertia which define rotation axes of a body, the body is rotationally symmetric about each of these axes, and If we start from a Cartesian $i,j,k$ system, we need to orientate such that they are in position or match the principal axes,each of the orientation the angles rotates defines a unique change in orientation i.e. rotation, so we can construct any rotation by combining these Euler angles vectors?

I don't think I fully understand how the Eulers angles were defined, my intuition to these derivation should be defined like this, 1. rotate about $z$ axis 2. rotate about $y$ axis, 3 rotate about $x$ axis. But I can't convince myself why I'm wrong. I hope any experts can point me out.

Answer 1

I don't think I fully understand how the Eulers angles were defined, my intuition to these derivation should be defined like this, 1. rotate about z axis 2. rotate about y axis, 3 rotate about x axis. But I can't convince myself why I'm wrong. I hope any experts can point me out.

The sequence depicted in the original post involved

• A rotation by angle $\phi$ about the z axis, forming the x',y',z' system (with the z' axis being the same as the z axis) followed by
• A rotation by angle $\theta$ about the x' axis, forming the x'', y'', z'' system (with the x'' axis being the same as the x' axis), followed by
• A rotation by angle $\psi$ about the z'' axis, forming the desired coordinate system.

That intrinsic "ZXZ" sequence is the canonical Euler sequence, canonical not because it's simpler but just because that is how Euler initially defined this rotation. There are in fact 23 other Euler-like sequences. For example, instead of rotating about the rotated axes, one could use a rotation about the original z axis, then by a rotation about the original x axis, and finally about the original z axis. That's the extrinsic ZXZ rotation sequence.

Similarly, that second rotation could have been about the y' axis (intrinsic ZYZ) or about the y axis (extrinsic ZYZ). There are twelve sequences of the form ABA, six intrinsic and six extrinsic.

The other twelve Euler-like sequences involve rotations about three distinct axes. You intrinsic ZYX sequence represents one of these twelve Tait-Bryan sequences. There are six intrinsic and six extrinsic Tait-Bryan sequences.

Any one of these 24 Euler-like rotation sequences will obviously yield some proper orientation in three dimensional space. What about the reverse: Can any one of these 24 Euler-like rotation sequences be used to describe an arbitrary proper orientation in three dimensional space. The answer is yes, by construction. There are algorithms for converting any arbitrary (proper) orientation to each of those Euler-like rotation sequences.

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Note that I used "proper" above. There is no way to express the XYZ frame where $\hat X = \hat z$, $\hat Y = \hat y$, $\hat Z = \hat x$ as an Euler sequence. This is a reflection in the plane $x=z$. Transformations involving a reflection are "improper" orientations. Such improper orientations cannot be represented via an Euler sequence of any form.

Answer 2

The rigid body that's being rotated is arbitrary; it need not be rotationally symmetric.

To use Euler angles, we assume that rotation is executed by the body being rotated (say, an airplane with flaps, or spacecraft with thrustors) instead of by some external manipulator. So for all intermediate rotations, we have to consider the perspective of the body.

The body has its own notion of "left", "right", "front", "back", "ceiling", and "floor". This is represented by the body's coordinate frames, defined by $3$ direction vectors (considered exist independently as abstract entities). Let the initial frame be $(x,y,z)$ and final be $(X,Y,Z)$. Euler rotation decomposes rotation into $3$ elementary rotations with 2 intermediate steps:

$$(x,y,z) \to (x',y',z') \to (x'',y'',z'') \to (X,Y,Z).$$

Since rotation is actuated by the body, so for $(x',y',z') \to (x'',y'',z'')$, we "forget" about the original frame $(x,y,z)$, and can only use $(x',y',z')$. Same goes with $(x'',y'',z'') \to (X,Y,Z)$.

For Euler rotation, each intermediate rotation is applied about an existing coordinate axis of the immediately previous frame. In the Wikipedia article, the "Classical Definition" lists the choice as $z$-$x'$-$z''$. There are other possibilities, but this is not arbitrary because there are "dumb" choices like $z$-$z'$-$z''$, which cannot represent all rotations.

It's rather strange that in $z$-$x'$-$z''$, we don't see any $\{y,y',y''\}$. My guess is that the convention is useful because it allows you to only use 2 independent rotating devices in a vehicle. The 3rd can be omitted altogether, or be used as backup.