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JohnnyCode

3d rotation speeds

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hi,

 

I wonder wheather any rotaional velocity of a mass object can be expressed by three speeds around three axises.

 

I cannot somehow believe it. Consider object that rotates around x, around y, around z, and yet around some arbitrary direction vector.

So is it possible to express any rotational velocity by three quaternions?

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yes, but I wonder wheather absolutely any rotating tendence can be expressed by three of them.


Physicists figured this out probably more than three centuries ago. There is no need for you to wonder. All you need to do is learn it.

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I have actualy proved it to myself.

 

Lets break 3 quaternion characteristics to 3 base axises and 3 angular speeds scalars.

 

This composes a single rotation matrix against a base matrix (the identity rotation matrix I).

 

We compose a rotation matrix H by s degress around y.

 

We now perform I*H=D ; D*H=C ; C*H=E  ;......

 

thus we are rotating by s degrees in a time step (second), rotating by quaternion that costructed H matrix, - the quaternion (0,1,0,s);

 

We now have to extend rotating by this quaternion along with one other quternion , quaternion (1,0,0,2s).

 

We construct H matrix from previous quaternion (0,1,0,s) and construct G matrix by the second quaternion (1,0,0,2s).

 

We now perform I*H*G=D ; D*H*G=C ; C*H*G=E  ;..... . and we can reduce to J=H*G thus i*J=D ; D*J=C ; C*J=E ; ....

 

This means that whatever amount of descibing queternions adds up to describe tendence, they cen be incomposed to a single rotation matrix that transforms after itself the object timestep by timestep. A strange J/s unit.

 

But this yields problem of order of composited quaternion rotations. But I gess this is the responsibility of expressing last added quaternion corectly, rather than physical dilema.

 

 

 

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But this yields problem of order of composited quaternion rotations. But I gess this is the responsibility of expressing last added quaternion corectly, rather than physical dilema.

 

 

And this is what gives so many permutations of Euler or Tait-Bryan angles.

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The group of 3D rotations is called SO(3). This is a Lie group, and the tangent space at the identity is R^3. That is another way of saying that if you have a quaternion changing over time, its derivative lives in R^3. Since the angular velocity is the time derivative of the attitude, that's what we are talking about here. That's the way I understand it, although perhaps there are more elementary descriptions.

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