# Geometry: why yaw, pitch, roll in that order?

1. Apr 11, 2014

### Avichal

It is basically a question in computer graphics but I guess math sub-forum will suit this question.

While rotating a point, why we first apply yaw, then pitch and then roll in that order? Of course if we change the order, final rotation changes but why this specific order works? Some proof or logical explanation will help.

Thanks!

2. Apr 11, 2014

### SteamKing

Staff Emeritus
Who or what applies yaw, pitch, and roll in that order?

As long as you specify a certain order of rotations, derive the transformations for that order, and stick to it, the order doesn't matter.

3. Apr 11, 2014

### D H

Staff Emeritus
Any permutation of yaw/pitch/roll will "work". There are six such permutations. All six "work" in the sense that any one these rotation sequences can represent any possible orientation of a body in three dimensional space.

In fact, there are six more sequences that "work" in the sense of being able to represent any possible orientation in three dimensional space. There's a yaw/roll/yaw sequence (that's the canonical Euler sequence), but also yaw/pitch/yaw, etc. Even better (even worse?), there are twelve more such Euler-like sequences that "work" in this sense. Typically a yaw/pitch/roll sequence involves a rotation about the z axis, then about the y' axis, then about the x'' axis. Well, what if you rotate about the original z axis, then the original y axis, and then the original x axis? That "works", too, in the sense of being able to represent any possible orientation in three dimensional space. This makes for a total of 24 such Euler-like rotation sequences.

I intentionally put "works" in quotes. Not one of these 24 Euler-like sequences is very useful mathematically. They are useful only because we humans have a hard time grasping the mathematics of the group SO(3).

4. Apr 12, 2014

### Stephen Tashi

Given that "apply" means to perform a linear transformation relative to the current coordinate frame, I find it easier to visualize applying yaw first. It gives the general "heading". Likewise, I think its easier to visualize the effect of applying pitch first and then roll. Thinking of an airplane, suppose you apply roll first. If you then apply yaw, the direction of the plane is turned about the current vertical axis, not the original vertical axis.

5. Feb 2, 2017

### infernus

The order of operations actually does matter here. Basically when multiplying non-diagonal matrices, you must do so in order. The best way to prove this is hold a paper airplane in your hand. Now rotate in yaw (about body z axis) to the left by 45deg, then pitch up (around body y axis) by 45 deg. Now restart the aircraft back to the original orientation and reverse the order. Pitch up by 45deg (in body y axis) then rotate in yaw (in body z axis) to the left by 45 deg and you'll see that the aircraft ends up in a different attitude.

The reason for the order that we use is just a matter of standard convention as far as I know. If I were to give you a set of Euler angles you would know the aircraft attitude relative to the earth if you follow this process. First rotate in yaw from North by the heading angle, then rotate in pitch, then rotate in roll. When you're trying to go from sensors onboard, which measure in body axes, back to earth fixed coordinates, you reverse the order and rotate in roll, then pitch, then yaw.

Hope this helps.