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Given the following rotation matrix:

$ R(φ, θ, ψ) = \left( \begin{smallmatrix} cos(θ)*cos(ψ) & -sin(φ)*sin(θ)*cos(ψ)-cos(φ)*sin(ψ) & -cos(φ)*sin(θ)*cos(ψ)+sin(φ)*sin(ψ) \\ cos(θ)*sin(ψ) & -sin(φ)*sin(θ)*sin(ψ)+cos(φ)*cos(ψ) & -cos(φ)*sin(θ)*sin(ψ)-sin(φ)*cos(ψ) \\ sin(θ) & sin(φ)*cos(θ) & cos(φ)*cos(θ) \end{smallmatrix} \right) $

The multiplication $ R(0, 0, ψ)^T * \begin{pmatrix}ẋ& ẏ& ż\end{pmatrix}^T $ returns new linear velocities: \begin{pmatrix}ẋ_n& ẏ_n& ż\end{pmatrix} Which account for the effect of yaw on $ ẋ $ and $ ẏ $, because only the components of $ ψ $ remain in $ R $ and since $ ψ $ is around $ Z $, its components are in $ X $ and $ Y $.

Now my question is, what does the following multiplication return: $ R(0, 0, ψ)^T * R(-φ, θ, -ψ)^T $ ? and what does it account for?

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  • $\begingroup$ Are you sure that you got the signs correct in the entries (1,2), (2,1), (2,3) and (3,2)? The rotation matrix R looks very similar to a Euler XYZ angles, but it additionally flips the y axis before and after the rotation. $\endgroup$ – Felix Crazzolara Mar 17 at 4:10
  • $\begingroup$ I ported it directly from CoppeliaSim, and the multiplication is part of a working control flow for a quadcopter. $\endgroup$ – Aly Shmahell Mar 17 at 16:31
  • $\begingroup$ $ R(-φ, θ, -ψ) $ is basically like multiplying entries (1,2), (2,1), (2,3) and (3,2) by -1. However, what I don't understand is what the final multiplication represents, what it accounts for. $\endgroup$ – Aly Shmahell Mar 17 at 16:34

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