Previous works on parallel robots have shown that their visual servoing using the observation of their leg directions was possible. There were however found two main results for which no answer was given. These results were that (i) the observed robot which is composed of n legs can be controlled using the observation of only m leg directions (m < n) arbitrarily chosen among its n legs, and that (ii) in some cases, the robot does not converge to the desired end-effector pose, even if the observed leg directions did. Recently, it has been shown that the visual servoing of the leg directions of the Gough-Stewart platform and the Adept Quattro with 3 translational degrees of freedom was equivalent to controlling other virtual hidden robots that have assembly modes and singular configurations different from those of the real ones. In this paper, the concept of hidden robot model is generalized for any type of parallel robots controlled using visual servos based on the observation of the leg directions. It is shown that the concept of hidden robot model is a powerful tool that gives useful insights about the visual servoing of robots using leg direction observation. With the concept of hidden robot model, the singularity problem of the mapping between the space of the observed robot links and the Cartesian space (including the analysis of the local minima and of the diffeomorphism between the observation space and the robot space) can be addressed. And above all, it is possible to give and certify the information about the controllability of the observed robots using the proposed controller. All these results are validated through experiments on a Quattro robot.