I am trying to calculate the Euler angles of a rigid body given 4 force vectors and a location relative to the center of mass.
I know the distance from the center of mass to the four force vectors, $r_i = [x_i,y_i,z_i]$, and I know the four force vectors, $F_i = [f_{xi},f_{yi},f_{zi}]$
I've been trying to work out how to plug the rotation matrices into the moment equilibrium equation to get three equations, three unknowns and solve that.
\begin{align} R_x &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & cos(\alpha) & -sin(\alpha) \\ 0 & sin(\alpha) & cos(\alpha) \end{bmatrix} \\ R_y &= \begin{bmatrix} cos(\beta) & 0 & sin(\beta) \\ 0 & 1 & 0 \\ -sin(\beta) & 0 & cos(\beta) \end{bmatrix} \\ R_z &= \begin{bmatrix} cos(\gamma) & -sin(\gamma) & 0 \\ sin(\gamma) & cos(\gamma) & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ R_i &= R_z \cdot R_y \cdot R_x \cdot R_i^* \label{eq:EndEffectorRotation} \end{align}
\begin{equation} \sum M = 0 = \sum_{i=1}^{4} (r_i*R) \times F_{i} \label{eq:momentEquilibrium} \end{equation}
After expanding this moment equilibrium equation out into three nasty equations for rotation in x, y, and z, I'm lost on what to do.
I'd prefer to not solve this system numerically. If numerically is the best way, I'd like to understand the approach to do so.