Get derivation of symbolic matrix equations

I want to use Mathematica to infer kinematics equations (derivate position equations to speed equations).

\begin{equation} \vec{c}=\vec{t}_{m}+R_{b m} \vec{c^{m}} \end{equation} \begin{equation} \vec{d}=\vec{t}_{m}+R_{b m} \vec{t_{s}}+R_{b m} R_{m s} \vec{d^{s}} \end{equation}

where $ R_i$ means the rotation matrix, the $ \vec{c^m},\vec{d^s}$ are constant, and others are relative to time $ t$

I want to use Mathematica to get the symbolic form of $ \frac{\text{d}\vec{c}}{\text{d}t}\frac{\text{d}\vec{d}}{\text{d}t}$ . The answer I get by hand is this (for convenience, I don’t write the vector hat $ \vec{}$ in the equations below):

$ $ \begin{array}{c} v_{c}=\frac{d c}{d t}=R_{b m}^\top \dot{t_{m}}+R_{b m}^\top S\left(w_{m}\right) R_{b m} c^{m}=R_{b m}^\top \dot{t_{m}}+S\left(R_{b m}^\top w_{m}\right) c^{m} \ =R_{b m}^\top \dot{t_{m}}-S\left(c^{m}\right) R_{b m}^\top w_{m} \ v_{d}=\frac{d d}{d t}=\dot{t_{s}}+R_{b m}^\top \dot{t_{m}}+R_{b m}^\top S\left(w_{m}\right) R_{b m} t_{s}+R_{b m}^\top S\left(w_{m}\right) R_{b m} R_{m s} d^{s} \ \quad+R_{b m}^\top R_{b m}\left(S\left(w_{s m}\right) R_{m s} d^{s}\right) \ =\dot{t_{s}}+R_{b m}^\top \dot{t_{m}}-S\left(t_{s}\right) R_{b m}^\top w_{m}-S\left(R_{m s} d^{s}\right) R_{b m}^\top w_{m}+\left(S\left(w_{s m}\right) R_{m s} d^{s}\right) \ \left.=\dot{t_{s}}+R_{b m}^\top \dot{t_{m}}-S\left(t_{s}\right)\right) R_{b m}^\top w_{m}-S\left(R_{m s} d^{s}\right) R_{b m}^\top w_{m}-\left(S\left(R_{m s} d^{s}\right) w_{s m}\right) \end{array} $ $ where $ S(\vec{a})$ means the (antisymmetric) skew matrix form of vector $ \vec{a}$ . How could I get the result?

By the way, what bothers me most is how could I represent a symbolic matrix with a number?

The problem source: [1] Y. Cai, S. Zheng, W. Liu, Z. Qu和J. Han, 《Model Analysis and Modified Control Method of Ship-Mounted Stewart Platforms for Wave Compensation》, IEEE Access, 卷 9, 页 4505-4517, 2021, doi: 10.1109/ACCESS.2020.3047063.