- 2016 IEEE. There are several practical reasons to endow a mobile robot with a tether, but doing so adds considerable complexity to the problem of moving the robot. The feasibility of a particular motion in such systems depends on topological constraints imposed by the interplay of the robot's tether and its environment. The physical properties of the tether may also rule out configurations that would be possible otherwise. Little work has addressed these latter constraints, despite the considerable interest in motion planning for tethered robots recently. We examine the problem of planning motions of a planar robot connected via a cable of limited length to a fixed point in R2 when the tether has a constraint on its curvature, which adds appreciably to the realism of the cable model over existing work. We incorporate Dubins's theory of curves with work on planning with topological constraints to concisely represent the configuration space manifold, leading to an atlas of the manifold consisting of locally continuous charts that represent the cable's curvature limits conveniently. Any configuration of the tether and the robot is described in our representation with two elements: (1) a discrete structure that characterizes the cable's position and (2) an element within a single continuous chart. We provide an algorithm that explores the necessary parts of this atlas on-the-fly to locate paths efficiently.