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

## TL;DR

This paper provides mathematical proofs justifying a new reactive 3D navigation algorithm for a nonholonomic robot operating in unknown tunnels, ensuring safe, continuous movement towards a target distance from the tunnel surface.

## Contribution

It introduces a novel navigation law with rigorous proofs of convergence and safety guarantees for a robot in unstructured 3D tunnels.

## Key findings

- Proves convergence of the navigation law.
- Ensures safe distance maintenance from tunnel surfaces.
- Guarantees continuous progress through the tunnel.

## Abstract

A nonholonomic under-actuated robot with a bounded control range travels inside 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 the distance to the surface along any ray from a narrow beam emitted from the robot around the direction to the nearest point. The paper presents some mathematical facts underlying justification of the convergence and performance of a new navigation law that ensures constant advancement of the robot through the tunnel, along with respecting a given safety margin to its surface, driving the robot to the desired distance to it, and subsequently maintaining this distance.