The general formulation for equation of motion is shown in the following:
$$\begin{equation*}
\mathbf{M}(\mathbf{q}) \ddot{\mathbf{q}} +\mathbf{C}(\mathbf{q}, \dot{\mathbf{q}}) = \mathbf{\tau}_g (\mathbf{q}) + \mathbf{B} u
\end{equation*}$$
The mass-inertia matrix $\mathbf{M}$ for AcroMonk is
$$\begin{equation*}
\mathbf{M}(\mathbf{q}) =
\begin{bmatrix} I_1 + I_2 + m_2 l_1^2 +
2m_2 l_1 l_{c2} c_2 & I_2 + m_2 l_1 l_{c2} c_2 \\ I_2 + m_2 l_1 l_{c2} c_2
& I_2
\end{bmatrix}
\end{equation*}$$
with the shorthand notation $s_1 = \sin(q_1), c_1 = \cos(q_1)$.
Accordingly, Coriolis matrix is:
$$\begin{equation*}
\mathbf{C}(\mathbf{q},\dot{\mathbf{q}}) =
\begin{bmatrix}
-2 m_2 l_1 l_{c2} s_2 \dot{q}_2 & -m_2 l_1 l_{c2} s_2
\dot{q}_2 \\
m_2 l_1 l_{c2} s_2 \dot{q}_1 & 0
\end{bmatrix}
\end{equation*}$$
with the gravity vector as:
$$\begin{equation*}
\mathbf{\tau}_g(\mathbf{q}) =
\begin{bmatrix}
-m_1 g l_{c1}s_1 - m_2 g (l_1 s_1 + l_{c2}s_{1+2}) \\
-m_2 g l_{c2} s_{1+2}
\end{bmatrix}
\end{equation*}$$
and the actuation matrix is $\mathbf{B} = [0 \quad 1]^T$, which captures the underactuation of the system.
