Given a set of robots with arbitrary initial location and no agreement on a global coordinate system, convergence requires that all robots asymptotically approach the exact same, but unknown beforehand, location. Robots are oblivious— they do not recall the past computations — and are allowed to move in a one-dimensional space. Additionally, robots cannot communicate directly, instead they obtain system related information only via visual sensors. We prove ([4]) necessary and sufficient conditions for the convergence of mobile robots despite a subset of them being Byzantine (i.e. they can exhibit arbitrary behavior). Additionally, we propose a deterministic convergence algorithm for robot networks and analyze its correctness and complexity in various synchrony settings. The proposed algorithm tolerates f Byzantine robots for (2f + 1)-sized robot networks in fully synchronous networks, (3f + 1)-sized in semi-synchronous networks and (4f + 1)-sized in asynchronous networks. The bounds obtained for the ATOM model are optimal for the class of cautious algorithms, which guarantee that correct robots always move inside the range of positions of the correct robots.