Robotics Stack Exchange is a question and answer site for professional robotic engineers, hobbyists, researchers and students. It only takes a minute to sign up.
Sign up to join this community
For my master thesis in robotics I have to compute the orientation error between two coordinate frames, called E and H. Their orientation is expressed through rotation matrices (3x3) with respect to a "world frame", W. Remark: writing $R_A^B$ I'm indicating the orientation of frame A with respect to frame B.
What I have understood is that I have to do the following operation:
$R_{error} = R_E^W * R_W^H$
is it correct? If yes, that error in which frame is expressed? Thanks a lot Ale
The rotation error between two frames can be viewed in two ways:
The orientation of one frame as seen from the other, calculated by multiplying the inverse of the observing frame by the observed frame. For frames $E$ and $H$, this error in your notation would be $$ \tag{1} {R_{1}}^{E}_{H} = (R^{W}_{E})^{-1}R^{W}_{H} = R^{E}_{W}R^{W}_{H} $$ for $H$ as seen from $E$, or $$ \tag{2} {R_{1}}^{H}_{E} = (R^{W}_{H})^{-1}R^{W}_{E} = R^{H}_{W}R^{W}_{E} $$ for $E$ as seen from $H$. These matrices encode the rotations around local axes (the axes of the superscript frame) that map the superscript frame to the subscript frame, $$ \tag{3} R^{W}_{E}{R_{1}}^{E}_{H} = R^{W}_{H} $$ and $$ \tag{4} R^{W}_{H}{R_{1}}^{H}_{E} = R^{W}_{E} $$
The orientation that would be reached by starting at one frame and moving by the inverse of the second frame's orientation relative to the world, $$ \tag{5} {R_{2}}^{E}_{H} = R_{H}^{W} (R_{E}^{W})^{-1} = R_{H}^{W} R_{W}^{E} $$ or $$ \tag{6} {R_{2}}^{H}_{E} = R_{E}^{W} (R_{H}^{W})^{-1} = R_{E}^{W} R_{W}^{H} $$ These matrices encode the rotations around the world axes that map the frames to each other, $$ \tag{7} {R_{2}}^{E}_{H} R^{W}_{E} = R^{W}_{H} $$ and $$ \tag{8} {R_{2}}^{H}_{E}R^{W}_{H} = R^{W}_{E} $$
The fundamental source of there being two interpretations of rotational error is that rotations are not commutative, and so our intuition that the differences $A + (-B)$ and $(-B) + A$ should be the same does not hold in this case.
The equations have a left-right duality. When we take the difference by putting the inverse rotation on the left (globally rotating the second frame by the inverse of the first), we find the (local) rotation error that maps between the frames when placed to the right of the starting frame, and when we put the inverse rotation on the right (locally rotating the first frame by the inverse of the second), we find the (global) rotation error that maps between the frames when placed to the left of the starting frame.
A rotation matrix represents the rotation between two frames. Therefore, it does not make sense to talk about in which "one" frame the error rotation is expressed. Namely, the rotation matrix $R^B_A$ represents the rotation from frame $B$ to frame$A$. Therefore, the error rotation matrix you defined as
$$ R_{error} = R^W_E R^H_W $$
can also be seen as the rotation from frame $H$ to frame $W$ and then from frame $W$ to frame $E$, which is equivalent to $R^H_E$.
