Although autonomous vehicle technology has evolved significantly in recent years, the navigation of self-driving vehicles in complex scenarios is still an open issue. One of the major challenges in these conditions is safe navigation on roads open to public traffic. The main issue is the interaction of the autonomous vehicle with regular traffic, as the behaviors and intentions of human-driven vehicles are hard to predict and understand. In this paper we propose a strategy to allow an autonomous vehicle to safely cross a multi-lane roundabout. Our approach uses a High-Definition (HD) map to predict at lane level the future situation, harnessing the concept of virtual instances of road users, which is a key concept in anticipating the situation in a roundabout that can be represented by a navigation graph with loops. This paper presents a methodology that uses intervals representing road occupancy by vehicles, with the road being widened to reflect uncertainties in localization. Our method safely avoids collisions and guarantees that no priority constraints are violated during the insertion maneuver. Moreover, the method does not provide an overly cautious insertion policy, i.e., an autonomous vehicle does not wait for a long time before the insertion. The performance of our strategy was evaluated using the SUMO simulation framework. To better evaluate the complexity of the simulation scenario, a highly interactive vehicle flow was generated using real dynamic traffic data from the INTERACTION dataset. We report real tests carried out with an experimental self-driving vehicle on a test circuit. Our results show that this approach is easy to integrate into an embedded system and that it allows roundabouts to be crossed with a high level of safety.