Proofs of the Technical Results Justifying an Algorithm of Reactive 3D   Navigation for a Surface Scan by a Nonholonomic Mobile Robot
  A single nonholonomic under-actuated mobile robot with a bounded control range travels in a 3D workspace. This workspace also hosts a compact unknown domain with a smooth boundary. The robot has access to the horizontal distance to this domain and to a certain direction (typically, vertical) in the space, along with its coordinate (typically, altitude) in this direction. We present a new navigation law that drives the robot to the desired distance to the domain and ensures subsequent full scan of its surface within a desired range of "altitudes". The proposed strategy is computationally inexpensive and achieves full scan at the lowest control level via generating the current control as a simple, reflex-like reaction to the current observation. The paper presents and proves key technical facts underlying mathematically rigorous analysis and justification of the proposed guidance approach. 
Alexey S. Matveev, Kirill S. Ovchinnikov, and Andrey V.Savkin
math.OC cs.RO