dcm2angle

Create rotation angles from direction cosine matrix

Syntax

[r1 r2 r3] = dcm2angle(n)
[r1 r2 r3] = dcm2angle(n,s)
[r1 r2 r3] = dcm2angle(n,s,lim)
[r1 r2 r3] = dcm2angle(n,s,lim,action)
[r1 r2 r3] = dcm2angle(n,s,lim,action,tolerance)


Description

[r1 r2 r3] = dcm2angle(n) calculates the set of rotation angles, r1, r2, r3, for a given direction cosine matrix, n. n is a 3-by-3-by-m matrix containing m direction cosine matrices. r1 returns an m array of first rotation angles. r2 returns an m array of second rotation angles. r3 returns an m array of third rotation angles. Rotation angles are output in radians. This function applies only to direction cosine matrices that are orthogonal with determinant +1. The rotation used in this function is a passive transformation between two coordinate systems.

[r1 r2 r3] = dcm2angle(n,s) calculates the set of rotation angles, r1, r2, r3, for a given direction cosine matrix, n, and a specified rotation sequence, s.

The default rotation sequence is 'ZYX', where r1 is z-axis rotation, r2 is y-axis rotation, and r3 is x-axis rotation.

Supported rotation sequences are 'ZYX', 'ZYZ', 'ZXY', 'ZXZ', 'YXZ', 'YXY', 'YZX', 'YZY', 'XYZ', 'XYX', 'XZY', and 'XZX'.

[r1 r2 r3] = dcm2angle(n,s,lim) calculates the set of rotation angles, r1, r2, r3, for a given direction cosine matrix, n, a specified rotation sequence, s, and a specified angle constraint, lim. lim specifies either 'Default' or 'ZeroR3'. See Assumptions and Limitations for full definitions of angle constraints.

[r1 r2 r3] = dcm2angle(n,s,lim,action) performs action if the direction cosine matrix is invalid (not orthogonal).

• Warning — Displays warning and indicates that the direction cosine matrix is invalid.

• Error — Displays error and indicates that the direction cosine matrix is invalid.

• None — Does not display warning or error (default).

[r1 r2 r3] = dcm2angle(n,s,lim,action,tolerance) uses a tolerance level to evaluate if the direction cosine matrix, n, is valid (orthogonal). tolerance is a scalar whose default is eps(2) (4.4409e-16). The function considers the direction cosine matrix valid if these conditions are true:

• The transpose of the direction cosine matrix times itself equals 1 within the specified tolerance tolerance (transpose(n)*n == 1±tolerance)

• The determinant of the direction cosine matrix equals 1 within the specified tolerance (det(n) == 1±tolerance).

Examples

Determine the rotation angles from direction cosine matrix:

dcm = [1 0 0; 0 1 0; 0 0 1];
[yaw, pitch, roll] = dcm2angle(dcm)
yaw =

     0

pitch =

     0

roll =

     0

Determine the rotation angles from multiple direction cosine matrices:

dcm        = [ 1 0 0; 0 1 0; 0 0 1];  
dcm(:,:,2) = [ 0.85253103550038   0.47703040785184  -0.21361840626067; ...
              -0.43212157513194   0.87319830445628   0.22537893734811; ...
               0.29404383655186  -0.09983341664683   0.95056378592206];
[pitch, roll, yaw] = dcm2angle(dcm,'YXZ')

pitch =

         0
    0.3000

roll =

         0
    0.1000

yaw =

         0
    0.5000

Determine the rotation angles from direction matrices validated within tolerance:

dcm        = [ 1 0 0; 0 1 0; 0 0 1];  
dcm(:,:,2) = [ 0.85253103550038   0.47703040785184  -0.21361840626067; ...
              -0.43212157513194   0.87319830445628   0.22537893734811; ...
               0.29404383655186  -0.09983341664683   0.95056378592206];
[pitch, roll, yaw] = dcm2angle(dcm,'YXZ','Default','None',0.1)

pitch =
         0
    0.3000

roll =
         0
    0.1000

yaw =
         0
    0.5000

Assumptions and Limitations

The 'Default' limitations for the 'ZYX', 'ZXY', 'YXZ', 'YZX', 'XYZ', and 'XZY' implementations generate an r2 angle that lies between ±90 degrees, and r1 and r3 angles that lie between ±180 degrees.

The 'Default' limitations for the 'ZYZ', 'ZXZ', 'YXY', 'YZY', 'XYX', and 'XZX' implementations generate an r2 angle that lies 0–180 degrees, and r1 and r3 angles that lie between ±180 degrees.

The 'ZeroR3' limitations for the 'ZYX', 'ZXY', 'YXZ', 'YZX', 'XYZ', and 'XZY' implementations generate an r2 angle that lies between ±90 degrees, and r1 and r3 angles that lie between ±180 degrees. However, when r2 is ±90 degrees, r3 is set to 0 degrees.

The 'ZeroR3' limitations for the 'ZYZ', 'ZXZ', 'YXY', 'YZY', 'XYX', and 'XZX' implementations generate an r2 angle that lies 0–180 degrees, and r1 and r3 angles that lie between ±180 degrees. However, when r2 is 0 or ±180 degrees, r3 is set to 0 degrees.