I have very simple multi-body dynamic system from which I have to solve following DAE:

$ \textbf{q}(t) - 3 \times1 \text{ vector of known state variables} $

$ \phi(\textbf{q}(t))=0 - 2 \times 1 - \text{ vector function of multibody constraints } $

$ A(t,\textbf{q}(t)), M - 3 \times 3 \text{ known matrices} $

$ \textbf{p}(t), \textbf{w}(t)- 3 \times 1 \text{ unknown vector functions} $

$ \textbf{m}(t)-2 \times 1 \text{ unknown vector function} $

\begin{equation} \dot{\textbf{p}}=A \cdot \textbf{w} + \phi_q^T \cdot \textbf{m} \end{equation} \begin{equation} M \cdot \dot{\textbf{w}} = \textbf{p} \end{equation} \begin{equation} 0 = \phi_q \cdot \textbf{w} \end{equation} \begin{equation} \textbf{p}(0)=[1,0,0]^T \end{equation} \begin{equation} \textbf{w}(0)=[0,0,0]^T \end{equation}

The system contains algebraic equation for $\textbf{w}$ which by itself gives index equal to 1. However, equation ($1$) has one algebraic function ($\textbf{m}$) that confuses me.

- Should it also be taken into account while evaluating index? If so, could you show how to calculate it?
- Moreover, if the index is higher than 1, I will have to transformate this system into one equal to it. If this is simple task, can I ask you to show how to perform it? If not, could you navigate me to useful sites or books?
- I need to solve this system in Matlab, preferably using
`ode15s`

. Any help regarding how to map this DAE into Matlab environment would be appreciated.