1. A computer-implemented method for performing computer-assisted procedures involving a first set of N three-dimensional points representing locations in a first space of first fiducials and a second set of N three-dimensional points representing locations in a second space of second fiducials corresponding to the first fiducials, where N is an integer greater than 2, the method comprising: employing a current rigid transformation to the first set to obtain a current transformed set of N three-dimensional points; determining a linearized transformation that minimizes a weighted fiducial registration error between the current transformed set and the second set, the linearized transformation comprising a translation vector and a linearized matrix; obtaining an updated rigid transformation based on the current rigid transformation, the translation vector of the linearized transformation, and a rotation matrix that is closest to the linearized matrix; applying the updated rigid transformation to the first set to obtain an updated transformed set of N three-dimensional points; and if a motion associated with the updated transformed set fails to meet at least one criterion, repeating the steps of determining, obtaining, and applying using the updated rigid transformation and the updated transformed set.