# What's the result of multiplying a rotation matrix having only ψ components by another which has (−φ,θ,−ψ) components?

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?

• 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. – Felix Crazzolara Mar 17 '20 at 4:10
• I ported it directly from CoppeliaSim, and the multiplication is part of a working control flow for a quadcopter. – Aly Shmahell Mar 17 '20 at 16:31
• $R(-φ, θ, -ψ)$ 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. – Aly Shmahell Mar 17 '20 at 16:34