Let's assume I have $N$ 3d bodies and I want to register them with a rigid transformation (Rotation and translation). To simplify the problem, let's assume that we are only looking for a rotation. In this case we are looking for an orthogonal matrix for each body $R_{i}$.
Between each pair of bodies ${i,j}$, We have a set of point matches. The matches between two bodies i and j are $X_{i}^{j}$ (points in body i that match j) and $X_{j}^{i}$ (corresponding points in body j).
For the case N = 2, there is an analytical solution to this problem: https://igl.ethz.ch/projects/ARAP/svd_rot.pdf . I tried to follow the same approach as and I found that the is minimizing the following quantity:
$$ - \sum_{i=1}^{n} \sum_{j=1}^{n} <R_{i} X_{i}^{j}, R_{j} X_{j}^{i}> $$
But I don't see where to go from here. I tried to use the lagrange multiplier but I didn't get anything. How would you proceed ?