Differential equation of motion of combined rigid body I am reading the following paper:
http://tinyurl.com/jv2pu2m
I have a question about (1) and (2):

To (1), I think the following link provides good explanation.
Equation of Motion for Rigid Body Motion
However, I do not know how to do the similar steps to (2), particularly for $\omega$.
Specifically, we can write $v = v_1 +\omega \otimes r$ and take derivative to it w.r.t $t$; how to do this step to $\omega$?
 A: I think you are asking about how to transform various vector quantities between points attached to a rigid body. Here are a rundown of the rules of transformation between an arbitrary point A (located at $r_A$) and the center of mass C (located at $r_C$).
$$\begin{align}
 v_C & = v_A + \omega \times (r_C - r_A) & & \text{linear velocity at C} \\
 \dot{v}_C & = \dot{v}_A + \dot{\omega} \times (r_C - r_A) + \omega \times (v_C - v_A) & & \text{linear acceleration at C} \\
  p & = m v_C & & \text{linear momentum}\\
  L_C & = \mathtt{J}_C\, \omega & & \text{angular momentum at C} \\
  L_A & = L_C + (r_C-r_A) \times p & & \text{angular momentum at A} \\
  F & = \frac{{\rm d}}{{\rm d}t} p = m \dot{v}_C & & \text{net force on body} \\
  M_C & = \frac{{\rm d}}{{\rm d}t} L_C = \mathtt{J}_C\,\dot{\omega}+ \omega \times \mathtt{J}_C\, \omega & & \text{net torque on body at C} \\
  M_A & = M_C + (r_C-r_A) \times F & & \text{net torque at A}
\end{align}$$
Edit 1
If you combine the above you have
$$\begin{aligned} \begin{pmatrix}v_{C}\\
\omega
\end{pmatrix}&=\begin{bmatrix}1 & -[c\times]\\
0 & 1
\end{bmatrix}\begin{pmatrix}v_{A}\\
\omega
\end{pmatrix} \\ \begin{pmatrix}\dot{v}_{C}\\
\dot{\omega}
\end{pmatrix}&=\begin{bmatrix}1 & -[c\times]\\
0 & 1
\end{bmatrix}\begin{pmatrix}\dot{v}_{A}\\
\dot{\omega}
\end{pmatrix}+\begin{pmatrix}\omega\times\left(\omega\times c\right)\\
0
\end{pmatrix} \\
\begin{pmatrix}p\\
L_{A}
\end{pmatrix}&=\begin{bmatrix}m [1] & -m[c\times]\\
m[c\times] & \mathtt{J}_{C}-m[c\times] [c\times]
\end{bmatrix}\begin{pmatrix}v_{A}\\
\omega
\end{pmatrix} \\ \begin{pmatrix}F\\
M_{A}
\end{pmatrix}&=\begin{bmatrix}m[1] & -m[c\times]\\
m[c\times] & \mathtt{J}_{C}-m[c\times] [c\times]
\end{bmatrix}\begin{pmatrix}\dot{v}_{A}\\
\dot{\omega}
\end{pmatrix}+\begin{bmatrix}[1] & [0]\\ 
[c\times] & [1]
\end{bmatrix}\begin{pmatrix}m\left(\omega\times\left(\omega\times c\right)\right)\\
\omega\times\mathtt{J}_{C}\omega
\end{pmatrix}
\end{aligned}$$
where $c = r_C-r_A$ is the relative CM location, and $[c\times] = \begin{vmatrix} 0 & -z & y \\ z & 0 & -x \\  -y & x & 0 \end{vmatrix}$ is the 3×3 cross product operator, and $[0]$, $[1]$ are the 3×3 zero and identity matrices.
