i have 3 translations and 3 rotations known in global coordinate system.
I need to know these 6 movements in a new CS, that is rotated and translated from the global coordinate system: in which I know only the direction of the X axis of the new CS, plus its origin.
How can I convert the known translations and rotations from global CS to the new CS ?
If you know a transformation between the old and the new systems, say $\hat{A}$ such that $$ \hat{A}\, \vec{v}_{\text{old system}} = \vec{v}_{\text{new system}} $$ and it has an inverse $$ \hat{A}^{-1} \vec{v}_{\text{new system}} = \vec{v}_{\text{old system}}, $$ than any transformation $\hat{T}_{\text{old system}}$ in the new system can be presented as $$ \hat{T}_{\text{new system}} = \hat{A} \hat{T}_{\text{old system}} \hat{A}^{-1}. $$ If these are given in terms of matrices, you can simply multiply them to convert everything into one matrix.
Intuition behind this is as follows. Say $$ \hat{T}_{\text{old system}} \vec{v}_{\text{old system}} = \vec{u}_{\text{old system}}, $$ and we want certain transformation $\hat{T}_{\text{new system}}$ such that $$ \hat{T}_{\text{new system}} \vec{v}_{\text{new system}} = \vec{u}_{\text{new system}}. $$ Consider that $$ \hat{A} \hat{T}_{\text{old system}} \hat{A}^{-1} \vec{v}_{\text{new system}} = \hat{A} \hat{T}_{\text{old system}} \vec{v}_{\text{old system}} = \hat{A} \vec{u}_{\text{old system}} = \vec{u}_{\text{new system}}, $$ thus $\hat{A} \hat{T}_{\text{old system}} \hat{A}^{-1} = \hat{T}_{\text{new system}}$.