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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.