howto Rotate Rotation matrix from relative frame back to reference frame [ subsequent rotation ]

As we know that the angle from 'pitch, roll, yaw' is measured from first reference frame.

Even though, we have subsequent frame.

For example, if we rotate 90 degree around x axis, then rotate 90 degree around y axis.

we noticed from the last couple that it is rotate 90 around y axis of the first frame ( not -90 degree around z axis in the secound frame )

We can use rotation formula, like R = RyRx

However, in some case, if we want to use the the rotated frame as a reference frame, ( for example, -90 degree around z axis in the second frame, in this case ), we can do that

and the formula to convert from rotated frame to the first reference frame is

BRB^-1 or BRB^T ( because the rotation matrix is orthonormal )

for example, from the example above

we rotate 90 degree around x axis and then 90 degree around y axis, so

R = RyRx

octave:1> [0 0 1;0 1 0;-1 0 0]*[1 0 0;0 0 -1;0 1 0]
ans =

0   1   0
0   0  -1
-1   0   0

if we want to rotate 90 around x axis and then -90 degree around z axis, our R will be

R = (RxRx,zRx^-1)Rx

octave:2> ([1 0 0;0 0 -1;0 1 0]*[0 1 0;-1 0 0; 0 0 1]*[1 0 0;0 0 -1;0 1 0]')*[1 0 0;0 0 -1;0 1 0]
ans =

0   1   0
0   0  -1
-1   0   0

more info : wiki