  Adaptive optics systems require a calibration procedure to operate, whether
in closed loop or even more importantly in forward control. This calibration
usually takes the form of an interaction matrix and is a measure of the
response on the wavefront sensor to wavefront corrector stimulus. If this
matrix is sufficiently well conditioned, it can be inverted to produce a
control matrix, which allows to compute the optimal commands to apply to the
wavefront corrector for a given wavefront sensor measurement vector.
Interaction matrices are usually measured by means of an artificial source at
the entrance focus of the adaptive optics system; however, adaptive secondary
mirrors on Cassegrain telescopes offer no such focus and the measurement of
their interaction matrices becomes more challenging and needs to be done on-sky
using a natural star. The most common method is to generate a theoretical or
simulated interaction matrix and adjust it parametrically (for example,
decenter, magnification, rotation) using on-sky measurements. We propose a
novel method of measuring on-sky interaction matrices ab initio from the
telemetry stream of the AO system using random patterns on the deformable
mirror with diagonal commands covariance matrices. The approach, being
developed for the adaptive secondary mirror upgrade for the imaka wide-field AO
system on the UH2.2m telescope project, is shown to work on-sky using the
current imaka testbed.
