# How to change between these two coordinate systems?

I have a problem with the coordinate system change between two 3D rotation sensors A and B. The coordinate systems have the same origin and are (mostly) perpendicular. I tested a somewhat pure rotation around each axis, while measuring for both sensors at the same time.

These are the angle axis rotations over time (x = red, y = green, z = blue), A on the left chart and B on the right chart:

I have now two questions:

1. Does it make sense to compare two angle axis representations or is it completely nuts? Is there a better way of comparing rotations?
2. What is the transformation between the two coordinate systems?

Looking at the chart I came up with this:

$z_{A} = -x_{B}$

$x_{A} = y_{B}$

$y_{A} = z_{B}$

Hence the rotation matrix (coordinate system change) from B to A is:

$R_{BA} = \begin{bmatrix} 0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{bmatrix}$

Supposedly by then doing $R_{BA} * R_{B}$ I should get $R_A$, but it's not what happens, I get completely different values.

• When you say "angle axis rotations", do you mean some kind of Euler angles? For example, "rotate by $\alpha$ around the $x$ axis, then by $\beta$ around the $y$ axis, then by $\gamma$ around the $z$ axis"? – David K Jan 22 '16 at 18:52
• Nope, I mean the logarithm of an SO3 group, maybe calling it angle-axis is not right. – aledalgrande Jan 22 '16 at 18:52
• Or the result of applying the Rodrigues theorem to a rotation matrix, if you will. – aledalgrande Jan 22 '16 at 18:54