Aqwa defines the orientation of a structure using Euler
angles. These are the angles which are printed in the .LIS file and passed in the argument list to the subroutine user_force. They are the angles by which the structure
has to be rotated about the FRA axes to reach its given position,
applied in the order rx, ry, rz (or roll, pitch, yaw). This worksheet
shows how to obtain the direction cosines of the LSA axes from these
angles.
Assume the Euler angles are 
,
, and 
for
rotations about the x- (roll), y- (pitch), and z-axes (yaw).
|
Euler angles |
| Unit vectors in LSA |
|
| Roll about FRA x-axis |
|
| |
| Pitch about FRA y-axis |
|
| |
| Yaw about FRA z-axis |
|
| |
| Overall motion |
| ||
The columns of 
give
the direction cosines of the LSA x-, y-, z-axes in the FRA:
Note that the inverse of 
is
the same as its transpose, so use 
to convert from FRA to LSA.
If the position of the COG in the FRA is CG, then the position of x in the FRA is given by:
|
|
|
|
Expanding multiplication, using 
and 
:
|
Rotations applied in the order roll, pitch, yaw |
|
|
Equivalent single transformation matrix |
|
