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We present a mostly vision-based controller for mapping and completely covering a rectangular area by meandering cleaning lanes. The robot is guided along a parallel course by controlling the current distance to its previous lane. In order to frequently compute and-if necessary-correct the robot's distance to the previous lane, a dense topological map of the robot's workspace is built. The map stores snapshots, i.e. panoramic images, taken at regular distances while moving along a cleaning lane. For estimating the distance, we combine bearing information obtained by local visual homing with distance information derived from the robot's odometry. In contrast to traditional mapping applications, we do not compute the robot's full pose w.r.t. an external reference frame. We rather rely on partial pose estimation and only compute the sufficient and necessary information to solve the task. For our specific application this includes estimates of (i) the robot's distance to the previous lane and of (ii) the robot's orientation w.r.t. world coordinates. The results show that the proposed method achieves good results with only a small portion of overlap or gaps between the lanes. The dense topological representation of space and the proposed controller will be used as building blocks for more complex cleaning strategies making the robot capable of covering complex-shaped workspaces such as rooms or apartments.