- #1
Bullwinckle
- 10
- 0
I understand that a gyroscope undergoes precession, nutation, spin.
And that the order of the rotations are such that the precession and spin share a common "local axis."
I also understand there are, for totally different purposes, Euler angles to model rotations.
In this case, the order of the rotations are ALSO such that the first and third share a common local axis.
For the Tait-Bryan angles, it so happens that each rotation is about a different local axis (corresponding to pitch, Yaw, roll).
I understand that I can cover all possible rotational configurations of the body with each of these. I can see it by example.
But is there a mathematical statement that "ensures" that I have indeed "covered" all possible rotational configurations with these selections?
Maybe it is so obvious, it escapes me.
But I have this very naive (ignorant?) understanding that just like in translation space, where one must cover all three orthogonal directions, that here, too, one must cover each local angle separately. I know this is silly. But could someone explain how they KNOW that the choice of rotation angles for the three I mentioned -- gyroscope, Euler, Tait -- do, indeed, cover all configurations.
For in my ignorant understanding, the Tait makes the most sense because the three local axes are different -- and yes, I know that is silly, but I am using it as a springboard to solicit guidance.
(I do "see" how they all work. But how can I "know" it?)
(And I am aware of Gimbal Lock and this is not about that.)
And that the order of the rotations are such that the precession and spin share a common "local axis."
I also understand there are, for totally different purposes, Euler angles to model rotations.
In this case, the order of the rotations are ALSO such that the first and third share a common local axis.
For the Tait-Bryan angles, it so happens that each rotation is about a different local axis (corresponding to pitch, Yaw, roll).
I understand that I can cover all possible rotational configurations of the body with each of these. I can see it by example.
But is there a mathematical statement that "ensures" that I have indeed "covered" all possible rotational configurations with these selections?
Maybe it is so obvious, it escapes me.
But I have this very naive (ignorant?) understanding that just like in translation space, where one must cover all three orthogonal directions, that here, too, one must cover each local angle separately. I know this is silly. But could someone explain how they KNOW that the choice of rotation angles for the three I mentioned -- gyroscope, Euler, Tait -- do, indeed, cover all configurations.
For in my ignorant understanding, the Tait makes the most sense because the three local axes are different -- and yes, I know that is silly, but I am using it as a springboard to solicit guidance.
(I do "see" how they all work. But how can I "know" it?)
(And I am aware of Gimbal Lock and this is not about that.)