  Astrometry is one of the main science case which drives the requirements of
the next multiconjugate adaptive optics (MCAO) systems for future extremely
large telescopes. The small diffraction limited point-spread function (PSF) and
the high Signal-to-Noise Ratio (SNR) of these instruments, promise astrometric
precision at the level of micro-arcseconds. However, optical distortions have
to be as low as possible to achieve the high demanding astrometry requirements.
In addition to static distortions, the opto-mechanical instabilities cause
astrometric errors that can be major contributors to the astrometry error
budget. The present article describes the analysis, at design level, of the
effects of opto-mechanical instabilities when coupled with optical surface
irregularities due to the manufacturing process. We analyse the notable example
of the Multi-conjugate Adaptive Optics RelaY (MAORY) for the extremely large
telescope (ELT). Ray-tracing simulations combined with a Monte Carlo approach
are used to estimate the geometrical structure and magnitude of field
distortion resulting from the optical design. We consider the effects of
distortion on the MCAO correction showing that it is possible achieve the
micro-arcseconds astrometric precision once corresponding accuracy is obtained
by both optical design and manufacturing. We predict that for single-epoch
observations, an astrometric error below 50$\mu$as can be achieved for exposure
times up to 2 min, provided about 100 stars are available to remove fifth-order
distortions. Such performance could be reproducible for multi-epoch
observations despite the time-variable distortion induced by instrument
instabilities.
