Aims: Over the past decade, discoveries of multiple and binary asteroid systems have played a significant role in our general understanding of small solar system bodies. Direct observations of satellites of asteroids are rare and difficult since they require the use of already over-subscribed facilities such as adaptive optics (AO) on large 8-10 m class telescopes and the Hubble Space Telescope (HST). The scarcity of data and the long temporal baseline of observations (up to 10 years) significantly complicate the determination of the mutual orbits of these systems. Methods: We implemented a new approach for determining the mutual orbits of directly-imaged multiple asteroids using a genetic-based algorithm. This approach was applied to several known binary asteroid systems (22 Kalliope, 3749 Balam, and 50 000 Quaoar) observed with AO systems and HST. This statistical method is fast enough to permit the search for an orbital solution across a large parameter space and without a priori information about the mutual orbit. Results: From 10 years of observation, we derived an orbital solution for Linus, companion of (22) Kalliope, with an accuracy close to the astrometric limit provided by the AO observations, assuming a purely Keplerian orbit. A search for non-Keplerian orbit confirmed that a J2 ~ 0 is the best-fitting solution. We show that the precession of the nodes could be detected without ambiguity, implying that Kalliope's primary may have an inhomogeneous internal structure. HST astrometric observations of Weywot, companion of the trans-Neptunian object (50 000) Quaoar, were used to derive its mass and its bulk density, which appears to be higher than the density of other TNOs. Finally, we derived a bundle of orbital solutions for (3749) Balam, with equally good fits, from the limited set of astrometric positions. They provide a realistic density between 1.3 and 3.7 g/cm3 for this S-type asteroid.