Aerospace toolbox rotations: right-handed or left-handed?

I'm getting myself all confused over this. Supposedly, the toolbox uses a right-handed coordinate system ( ref ). But this is producing a left-handed rotation matrix:

>> which angle2dcm
...../Matlab/toolbox/aero/aero/angle2dcm.m
>> angle2dcm(pi/2,0,0,'zyx')

ans =

     0.000000000000000   1.000000000000000                   0
    -1.000000000000000   0.000000000000000                   0
                     0                   0   1.000000000000000

This behavior seems relatively consistent throughout the toolbox. Is the documentation just wrong?

Answer 1

[Mischa Kim]

It is proper right-handed. The rotation matrix for a right-handed rotation about the z-axis with a rotation angle psi is given by

Rz(psi)   = [ cos(psi) sin(psi) 0; 
             -sin(psi) cos(psi) 0; 
                     0        0 1]

For a rotation angle of 90 deg (pi/2) this results in

angle2dcm(pi/2,0,0,'zyx')
ans =

      0.0000    1.0000         0
     -1.0000    0.0000         0
           0         0    1.0000

As an example, the vetor v in the graph below has coordinates [1; 0; 0] in the un-primed reference frame (left). When rotating the reference frame about 90 deg (right-handed) about the z-axis the vector points in negative y direction, [0; -1: 0], which corresponds to the result from the matrix-vector multiplication:

angle2dcm(pi/2,0,0,'zyx')*[1; 0; 0]
ans =

      0.0000
     -1.0000
           0

[Tyler]

That is wrong, the equation you give is for a left-handed rotation matrix. (see wikipedia)

In your example you have rotated the coordinate system, but a rotation matrix times a vector rotates the vector. A 90deg right-handed, i.e. counter-clockwise, rotation of [1; 0; 0] about the z-axis points in the positive y direction.

[Mischa Kim]

I appreciate your feedback because it might help clarify the difference between rotating a vector and describing the attitude of two frames of reference relative to each other. It also illustrates the fact that there is a lot of good information on wikipedia.

Rotation matrices are defined in the standard literature as one of many ways to describe the attitude of one reference frame relative to another, which is consistent with the documentation. In other words, given the components of a vector v in one reference frame one can use the rotation matrix to compute the components of that same vector in the other frame. Note, that this is not about rotating the vector itself.

For rotating vectors you would use MATLAB's rotx command and equivalent.

[Tamas Sarvary]

@Mischa from the behavior of quatrotate I can see your observation is right but in the documentation that isn't mentioned at all.

"n = quatrotate(q,r) calculates the rotated vector, n, for a quaternion, q, and a vector, r."

[James Tursa]

See also this post. The quatrotate function should probably use the phrase "coordinate system transformation" instead of the phrase "rotated vector".

https://www.mathworks.com/matlabcentral/answers/465053-rotation-order-of-quatrotate