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We are developing robots with socially appropriate spatial skills not only to travel around or near people, but also to accompany people side-by-side. As a step toward this goal, we are investigating the social perceptions of a robot's movement as it follows behind a person. This paper discusses our laser-based person-tracking method and two different approaches to person-following: direction-following and path-following. While both algorithms have similar characteristics in terms of tracking performance and following distances, participants in a pilot study rated the direction-following behavior as significantly more human-like and natural than the path-following behavior. We argue that the path-following method may still be more appropriate in some situations, and we propose that the ideal person-following behavior may be a hybrid approach, with the robot automatically selecting which method to use.