A necessary and sufficient condition for uniquely registering a system of point clouds (whose positions are known in different coordinate systems) in a global coordinate system  has been obtained. This was done by relating the registration problem to the rigidity of bar-and-joint frameworks. In particular, a particularly simple connectivity-based test for point clouds in two dimensions. Was obtained. Testing uniqueness however remains open in three and higher dimensions.
A. V. Singh and K. N. Chaudhury, “On uniquely registrable networks,” IEEE Transactions on Network Science and Engineering, 2019.