From Underactuated, the manipulator equation is of the form:
$$
\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}} = \tau_{g} + \mathbf{Bu}
$$
Aside from differing notation, this form differs significantly from the current state-space form $\dot{\mathbf{x}} + \mathbf{Ax} + \mathbf{Bu} + g$ with the introduction of configuration-dependent matrices $\mathbf{M}(\mathbf{q})$ and $\mathbf{C}(\mathbf{q}, \dot{\mathbf{q})}$. For state $\mathbf{x} = (\mathbf{q}, \dot{\mathbf{q}})$, the variable state matrix $\mathbf{A}(\mathbf{x})$ can be partitioned as follows, yielding $\dot{\mathbf{x}} = (\dot{\mathbf{q}}, \ddot{\mathbf{q}}) = \mathbf{A}(\mathbf{x})\mathbf{x}$, which relates the manipulator equation form to the state-space form.
$$
\mathbf{A}(\mathbf{x}) =
\begin{pmatrix}
\mathbf{0} & \mathbf{I} \\
\mathbf{0} & -\mathbf{M}(\mathbf{q})^{-1}\mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})
\end{pmatrix}
$$
From Underactuated, the manipulator equation is of the form:
Aside from differing notation, this form differs significantly from the current state-space form$\dot{\mathbf{x}} + \mathbf{Ax} + \mathbf{Bu} + g$ with the introduction of configuration-dependent matrices $\mathbf{M}(\mathbf{q})$ and $\mathbf{C}(\mathbf{q}, \dot{\mathbf{q})}$ . For state $\mathbf{x} = (\mathbf{q}, \dot{\mathbf{q}})$ , the variable state matrix $\mathbf{A}(\mathbf{x})$ can be partitioned as follows, yielding $\dot{\mathbf{x}} = (\dot{\mathbf{q}}, \ddot{\mathbf{q}}) = \mathbf{A}(\mathbf{x})\mathbf{x}$ , which relates the manipulator equation form to the state-space form.