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A single nonholonomic under-actuated mobile robot with a bounded controlrange travels in a 3D workspace. This workspace also hosts a compact unknowndomain with a smooth boundary. The robot has access to the horizontal distanceto this domain and to a certain direction (typically, vertical) in the space,along with its coordinate (typically, altitude) in this direction. We present anew navigation law that drives the robot to the desired distance to the domainand ensures subsequent full scan of its surface within a desired range of"altitudes". The proposed strategy is computationally inexpensive and achievesfull scan at the lowest control level via generating the current control as asimple, reflex-like reaction to the current observation. The paper presents andproves key technical facts underlying mathematically rigorous analysis andjustification of the proposed guidance approach.

Proofs of the Technical Results Justifying an Algorithm of Reactive 3D Navigation for a Surface Scan by a Nonholonomic Mobile Robot