User Manual

© Xsens Technologies B.V.
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Here, the arctangent (tan
-1
) is the four quadrant inverse tangent function.
NOTE: that the output is in degrees and not radians.
11.7.3 Rotation Matrix Orientation Output Mode
The rotation matrix (also known as Direction Cosine Matrix, DCM) is a well-known,
redundant and complete representation of orientation. The rotation matrix can be
interpreted as the unit-vector components of the sensor coordinate system S expressed in G.
For R
GS
the unit vectors of S are found in the columns of the matrix, so column 1 is X
S
expressed in G etc. A rotation matrix norm is always equal to one (1) and a rotation R
GS
followed by the inverse rotation R
SG
naturally yields the identity matrix I
3
.
||R|| = 1
R
GS
R
SG
= I
3
The rotation matrix, R
GS
, can be interpreted in terms of quaternions;
2 2 2 2
0 1 2 3 1 2 0 3 0 2 1 3
2 2 2 2
0 3 1 2 0 1 2 3 2 3 0 1
2 2 2 2
1 3 0 2 2 3 0 1 0 1 2 3
22
0 1 1 2 0 3 1 3 0 2
22
1 2 0 3 0 2 2 3 0 1
13
2 2 2 2
2 2 2 2
2 2 2 2
2 2 1 2 2 2 2
2 2 2 2 1 2 2
22
GS
q q q q q q q q q q q q
R q q q q q q q q q q q q
q q q q q q q q q q q q
q q q q q q q q q q
q q q q q q q q q q
q q q

    

     


    

   
    

22
0 2 2 3 0 1 0 3
2 2 2 2 1q q q q q q q




  

or in terms of Euler-angles (according to the XYZ Euler sequence);
cos sin 0 cos 0 sin 1 0 0
sin cos 0 0 1 0 0 cos sin
0 0 1 sin 0 cos 0 sin cos
cos cos sin sin cos cos sin cos sin cos sin sin
cos sin sin sin sin cos cos cos sin si
Z Y X
GS
R R R R
  
   
   
   
           
        


     
     

     
     

     

n sin cos
sin sin cos cos cos
  
    







As defined here R
GS
, rotates a vector in the sensor co-ordinate system (S) to the global
reference system (G):
()
T
GS SG
RR
G S S
x x x
It follows naturally that, R
SG
rotates a vector in the global reference co-ordinate system (G)
to the sensor co-ordinate system (S).
For the rotation matrix (DCM) output mode it is defined that: