The inertial reference frame is denoted by $I$, the position, velocity, acceleration, angular velocity, and angular acceleration of reference frame $S$ is measured in the inertial frame, reference frame $R$ is fixed relative to $S$ and has the homogeneous transformation $T_{S}^{R}$ relating the two, point $P$ is measured in the inertial frame (velocitie, acceleration, etc). With this, I would like to know the velocity and acceleration of point $P$ as measured in the $R$ reference/coordinate frame.
It has been a few years since I did this type of work, and I feel a bit uncomfortable, so here is my attempt:
I recognize that point $P$ can measured in $R$ via homogeneous coordinate transformations: $${}^{R}P={}^{R}_{S}T{}^{S}_{I}T{}^{I}P,$$ where ${}^{R}_{S}T$ is constant, and ${}^{S}_{I}T$ changes in time according to the dynamics of $S$. Now, taking the 2nd time derivative of this is gross, and I would prefer not to do it, so my thought was that I could use the transport theorem to solve for $\frac{d{}^{R}P}{dt}$:
$${}^{I}V_{P}={}^{I}V_{S}+{}^{S}V_{P/S}+{}^I\omega^{S}\times r_{P/S}$$ Then solving for the velocity of point $P$ relative to $S$, as measured in the $S$ reference frame: $${}^{S}V_{P/S}={}^{I}V_{P}-{}^{I}V_{S}-{}^I\omega^{S}\times r_{P/S}$$ Next, I recognize that ${}^{S}V_{P/S}={}^{S}V_{P/R}$ since $R$ is not changing relative to $S$, so I can write $${}^{S}V_{P/R}={}^{I}V_{P}-{}^{I}V_{S}-{}^I\omega^{S}\times r_{P/S}$$ Next, to get ${}^{R}V_{P/R}$, I believe that I could use the homogeneous coordinate transformation ${}^{R}_{S}T$: $${}^{R}V_{P/R}=\textbf{I}_{0}^{T}[{}^{R}_{S}T \textbf{I}_{0} ({}^{I}V_{P}-{}^{I}V_{S}-{}^I\omega^{S}\times r_{P/S})+[0,0,0,1]^{T}],$$ where $\textbf{I}_{0}$ is constructed by taking the $3\times 3$ identity matrix and adding an additional row of zeros to the bottom, so to convert to to homogeneous coordinates. Then the transpose converts back.
But, is this really the velocity of point $P$ as measured in the $R$ coordinate frame? Something is making me feel uncomfortable and I can't place my finger on what it is.
Any help in relieving me of this uncertainty is greatly appreciated, thanks!