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A nonholonomic under-actuated robot with a bounded control range travelsinside an unknown and unstructured 3D tunnel bounded by a generic 2D surface.The robot has access to the nearest point of the surface and measures thedistance to the surface along any ray from a narrow beam emitted from the robotaround the direction to the nearest point. The paper presents some mathematicalfacts underlying justification of the convergence and performance of a newnavigation law that ensures constant advancement of the robot through thetunnel, along with respecting a given safety margin to its surface, driving therobot to the desired distance to it, and subsequently maintaining thisdistance.

Proofs of Technical Results Justifying an Algorithm of Reactive 3D Navigation of a Mobile Robot through an Unknown Tunnel