  The VLTI instrument GRAVITY will provide very powerful astrometry by
combining the light from four telescopes for two objects simultaneously. It
will measure the angular separation between the two astronomical objects to a
precision of 10 microarcseconds. This corresponds to a differential optical
path difference (dOPD) between the targets of few nanometers and the paths
within the interferometer have to be maintained stable to that level. For this
purpose, the novel metrology system of GRAVITY will monitor the internal dOPDs
by means of phase-shifting interferometry. We present the four-step
phase-shifting concept of the metrology with emphasis on the method used for
calibrating the phase shifts. The latter is based on a phase-step insensitive
algorithm which unambiguously extracts phases in contrast to other methods that
are strongly limited by non-linearities of the phase-shifting device. The main
constraint of this algorithm is to introduce a robust ellipse fitting routine.
Via this approach we are able to measure phase shifts in the laboratory with a
typical accuracy of lambda/2000 or 1 nanometer of the metrology wavelength.
