  The Extremely Large Telescope and the Thirty Meter Telescope will use state
of the art multiconjugate adaptive optics (MCAO) systems to obtain the full D4
advantage that their apertures can provide. However, to reach the full
astrometric potential of these facilities for on-sky science requires
understanding any residual astrometric distortions from these systems and find
ways to measure and eliminate them. In this work, we use Gemini multiconjugate
adaptive optic system (GeMS) observations of the core of NGC 6723 to better
understand the on-sky astrometric performance of MCAO. We develop new methods
to measure the astrometric distortion fields of the observing system, which
probe the distortion at the highest possible spatial resolution. We also
describe methods for examining the time-variable and static components of the
astrometric distortion. When applied to the GeMS Gemini South Adaptive Optics
Imager (GSAOI) data, we are able to see the effect of the field rotator at the
subpixel level, and we are able to empirically derive the distortion due to the
optical design of GeMS-GSAOI. We argue that the resulting distortion maps are a
valuable tool to measure and monitor the on-sky astrometric performance of
future instrumentation. Our overall astrometry pipeline produces high-quality
proper motions with an uncertainty floor of 45 uas per year. We measure the
proper motion dispersion profile of NGC 6723 from a radius of 10 arcsec out to
1 arcmin based on 12000 stars. We also produce a high-quality optical-near
infrared color-magnitude diagram, which clearly shows the extreme horizontal
branch and main-sequence knee of this cluster.
