On theoretical mechanics essay at the university, I have stumbled in the issue below:
Given $\textbf{u}^T \textbf{D}(\mathbf{\theta}, \alpha)\textbf{v}(\textbf{d}) = \textbf{m}(\textbf{d})$, find out if it is possible to find a funciont $f: \textbf{p} \rightarrow \textbf{q}$, for $\textbf{q}^T = [\theta_1, \theta_2, \theta_3]$ and $\textbf{p}^T = [\textbf{d}^T, \alpha]$, $\textbf{d}^T = [x, y]$ and find it, if so. The equations below give the structure of $\textbf{D}(\mathbf{\theta}, \alpha)$ and $\textbf{v}(d)$.
\begin{equation} v(d) = \begin{bmatrix} \textbf{v}_1(\textbf{d}) \\ \textbf{v}_2(\textbf{d}) \\ \textbf{v}_3(\textbf{d})\end{bmatrix} \end{equation}
\begin{equation} \textbf{v}_i(d) = \begin{bmatrix} \textbf{d} - d_0^1 \\ \textbf{d} - d_0^2 \\ \textbf{d} - d_0^3 \\ \end{bmatrix} \end{equation}
\begin{equation} \label{eq:D} \textbf{D}(\mathbf{\theta}, \alpha) = \begin{bmatrix} \mathbf{D_1}(\mathbf{\theta_1}, \alpha) & 0 & 0 \\ 0 & \mathbf{D_2}(\mathbf{\theta_2}, \alpha) & 0 \\ 0 & 0 & \mathbf{D_3}(\mathbf{\theta_3}, \alpha) \\ \end{bmatrix} \end{equation}
\begin{equation} \label{eq:Di} \mathbf{D_i}(\mathbf{\theta_i}, \alpha) = \begin{bmatrix} -\mathbf{R}(\theta_i + \beta_i) & -\mathbf{R}(\theta_i + \alpha + \beta_i)\\ \mathbf{R}(\alpha) & 0 \\ \end{bmatrix} \end{equation}
\begin{equation} \label{eq:R} \mathbf{R}(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta) \\ \end{bmatrix} \end{equation}
A priori, the equation has 6 possible solutions for given feasible and reachable $\mathbf{p}$. For the sake of curiosity, the solutions comes for the position of this mechanism.
If the problem is ill-formed or lacks information, make me aware of it. Furthermore, if a numerical is better suited for this case, explain the manners to reach $\textbf{ALL}$ solutions of the problem.
I thank in advance for the attention and given effort.
Best regards, Bruno.