  Context. The GRAVITY+ upgrade implies a complete renewal of its adaptive
optics (AO) systems. Its complex design, featuring moving components between
the deformable mirrors and the wavefront sensors, requires the monitoring and
auto-calibrating of the lateral mis-registrations of the system while in
operation. Aims. For preset and target acquisition, large lateral registration
errors must be assessed in open loop to bring the system to a state where the
AO loop closes. In closed loop, these errors must be monitored and corrected,
without impacting the science. Methods. With respect to the first requirement,
our method is perturbative, with two-dimensional modes intentionally applied to
the system and correlated to a reference interaction matrix. For the second
requirement, we applied a non-perturbative approach that searches for specific
patterns in temporal correlations in the closed loop telemetry. This signal is
produced by the noise propagation through the AO loop. Results. Our methods
were validated through simulations and on the GRAVITY+ development bench. The
first method robustly estimates the lateral mis-registrations, in a single fit
and with a sub-subaperture resolution while in an open loop. The second method
is not absolute, but it does successfully bring the system towards a negligible
mis-registration error, with a limited turbulence bias. Both methods proved to
robustly work on a system still under development and not fully characterised.
Conclusions. Tested with Shack-Hartmann wavefront sensors, the proposed methods
are versatile and easily adaptable to other AO instruments, such as the
pyramid, which stands as a baseline for all future AO systems. The
non-perturbative method, not relying on an interaction matrix model and being
sparse in the Fourier domain, is particularly suitable to the next generation
of AO systems for extremely large telescopes that will present an unprecedented
level of complexity and numbers of actuators.
