Re: rotation matrices-

bob@xxxxxxxxxxxxxx wrote:

Please take a look at this:

http://msdn.microsoft.com/library/default.asp?url=/library/en-us/directx9_c/D3DXMatrixRotationYawPitchRoll.asp

It says:

"The order of transformations is roll first, then pitch, then yaw."

Why did it choose that order? What are the implications?



It is a convention in the mathematics of the theory of flight (and also in flight simulation) to depart from the standard attitude (horizontal roll axis, nose to the North, level wings and upside up) and first to apply the yaw rotation, secondly the pitch, and lastly the roll.

Now the matrix algebra:
When one wants to know coordinates with respect to the Earth-fixed coordinate system one needs to left-multiply by M-yaw, M-pitch and M-roll in the conventional order Yaw, Pitch, Roll.
The resulting matrix formula reads

(X, Y, Z)^T (Earth-fixed) = (M'-roll).(M'-pitch).(M'-yaw). (X, Y, Z)^T,

where the M' matrices are of course referred to the Earth-fixed system.

When one wants body-fixed coordinates, i.e. coordinates with respect to the coordinates system fixed to the aircraft, one needs to =right-multiply= in the conventional order Yaw, Pitch, Roll.
The resulting matrix formula reads

(x, y, z)^T (Body-fixed) = (M''-yaw).(M''-pitch).(M''-roll). (X, Y, Z)^T,

where the M'' matrices are referred to the aircraft-fixed system. The M' and M'' matrices are each other's inverses.

In practice one always needs the body-fixed coordinate system.

A complete account of all this is at my website at URL http://www.xs4all.nl/~jemebius/Jemkk.htm .

Implications for the student:
(1) review the general relations of matrices to the linear transformations which they represent.
(2) when you also study linear algebra and matrix algebra as applied in theoretical physics, please be keen on the similarities and differences of "Earth-fixed (laboratory-fixed) versus body-fixed (particle-fixed)" at the one side and "active versus passive coordinate transformations" (found in the older literature) at the other side.

Happy rolling and flying: Johan E. Mebius

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