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this is my first post on this forum

I have $n$ IMUs which outputs its rotation their matrices in the $XYZ$ world coordinate system $w$. I would like to use multiple IMUs $n$, such as $n_{1}, n_{2}, n_{3} ...$. IMUs are connected by spherical joints with a fixed arm length $d$ such as $d_{1}, d_{2}, d_{3} ...$.

The relative rotation represented in $w$ between two IMUs $n_{1}$ and $n_{2}$ is:

$R_{n_{1}\leftarrow n_{2}} = (R_{n_{1}\leftarrow w})(R_{n_{2}\leftarrow w})^{T}$

but instead of the $w$ coordinate system, I would like to read the relative rotation of the $R_{n_{1}\leftarrow n_{2}}$ in the $n_{1}$ coordinate system as $X'Y'Z'$.

How do I calculate it?

reference frame

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I've found the solution.

Instead of using intrinsic matrix to angle conversions such as:

$yaw = atan2(R_{n_{1}\leftarrow n_{2}}(2,1),R_{n_{1}\leftarrow n_{2}}(1,1))$

$pitch = asin(−R_{n_{1}\leftarrow n_{2}}(3,1))$

$roll = atan2(R_{n_{1}\leftarrow n_{2}}(3,2),R_{n_{1}\leftarrow n_{2}}(3,3))$

I should use extrinsic matrix to angle conversions such as:

$yaw = atan2(−R_{n_{1}\leftarrow n_{2}}(1,2),R_{n_{1}\leftarrow n_{2}}(1,1))$

$pitch = asin(R_{n_{1}\leftarrow n_{2}}(1,3))$

$roll = atan2(−R_{n_{1}\leftarrow n_{2}}(2,3),R_{n_{1}\leftarrow n_{2}}(3,3))$

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