  High-precision astrometry requires accurate point-spread function modeling
and accurate geometric-distortion corrections. This paper demonstrates that it
is possible to achieve both requirements with data collected at the high acuity
wide-field K-band imager (HAWK-I), a wide-field imager installed at the Nasmyth
focus of UT4/VLT ESO 8m telescope. Our final astrometric precision reaches ~3
mas per coordinate for a well-exposed star in a single image with a systematic
error less than 0.1 mas. We constructed calibrated astro-photometric catalogs
and atlases of seven fields: the Baade's Window, NGC 6656, NGC 6121, NGC 6822,
NGC 6388, NGC 104, and the James Webb Space Telescope calibration field in the
Large Magellanic Cloud. We make these catalogs and images electronically
available to the community. Furthermore, as a demonstration of the efficacy of
our approach, we combined archival material taken with the optical wide-field
imager at the MPI/ESO 2.2m with HAWK-I observations. We showed that we are able
to achieve an excellent separation between cluster members and field objects
for NGC 6656 and NGC 6121 with a time base-line of about 8 years. Using both
HST and HAWK-I data, we also study the radial distribution of the SGB
populations in NGC 6656 and conclude that the radial trend is flat within our
uncertainty. We also provide membership probabilities for most of the stars in
NGC 6656 and NGC 6121 catalogs and estimate membership for the published
variable stars in these two fields.
