[Paper Review] Proofs of the Technical Results Justifying a Biologically Inspired Algorithm for Reactive Navigation of Nonholonomic Robots in Maze-Like Environments

Alexey S. Matveev, Michael Hoy+1 more|arXiv (Cornell University)|Nov 21, 2011

Control and Dynamics of Mobile Robots7 references3 citations

TL;DR

This paper provides rigorous mathematical proofs for a biologically inspired reactive navigation algorithm that enables nonholonomic Dubins cars to reach a target in maze-like environments while avoiding obstacles using only bearing and relative distance-to-obstacle data. The key contribution is proving global convergence under minimal sensing and nonholonomic constraints, ensuring safe, obstacle-avoiding path convergence despite discontinuous control and limited curvature.

ABSTRACT

We present technical results justifying a method for guidance of a Dubins-like vehicle with saturated control towards a target in a steady simply connected maze-like environment. The vehicle always has access to to the target relative bearing angle and the distance to the nearest point of the maze if it is within the given sensor range. The proposed control law is composed by biologically inspired reflex-level rules. Mathematically rigorous analysis of this law is provided; its convergence and performance are confirmed by computer simulations and experiments with real robots.

Motivation & Objective

• To establish mathematical justification for a reactive navigation strategy inspired by animal behavior in maze-like environments.
• To address the lack of formal convergence proofs for existing bearing-only navigation algorithms under nonholonomic constraints.
• To prove that a Dubins car-type robot can reliably reach a target while maintaining a safe distance from obstacles using only local, minimal sensing.
• To validate the robustness of a discontinuous control law based on relative bearing and sign of distance rate, even under control saturation and limited curvature.
• To resolve inconsistencies in prior algorithms by proving convergence under realistic robotic constraints, such as no reversing and bounded turning radius.

Proposed method

• The navigation strategy uses two distinct modes: mode 𝔸 for long-range target pursuit using relative bearing, and mode 𝔹 for short-range obstacle avoidance via sharp turns.
• Control input is defined by a discontinuous law: u = ±ū × sgn(β) in mode 𝔸, and u = sgn(β) or −σū in mode 𝔹 depending on whether the distance to obstacle is increasing or decreasing.
• The system is analyzed in the Filippov sense to handle discontinuities in the control law.
• The proof relies on geometric arguments involving the projection of the robot’s position onto the obstacle boundary and the use of angular measures (spherical angles) to track cumulative turning.
• Key concepts include exit arcs on the obstacle boundary, the notion of a 'cave' (a region containing all past paths), and the use of angular deviation from target line-of-sight to enforce convergence.
• Theoretical tools include lemmas on path monotonicity, angular evolution, and contradiction-based proofs to rule out infinite loops or non-convergence.

Experimental results

Research questions

• RQ1 Can a reactive navigation algorithm based on minimal sensory input (bearing and sign of distance rate) guarantee global convergence to a target in a maze-like environment?
• RQ2 How can convergence be proven for a nonholonomic Dubins car under control saturation and limited curvature, when prior algorithms assumed unbounded curvature or absolute orientation?
• RQ3 What geometric and topological conditions ensure that the robot does not get trapped in loops or fail to escape from obstacle proximity?
• RQ4 How does the discontinuous control law, which switches between pursuit and avoidance behaviors, maintain stability and avoid oscillation?
• RQ5 Under what conditions does the robot’s path remain within a bounded region and eventually reach the target despite the absence of absolute orientation or full obstacle boundary tracking?

Key findings

• The proposed control law ensures global convergence to the target under the given nonholonomic constraints and minimal sensing, even with discontinuous control.
• The proof establishes that the robot cannot enter an infinite loop around the obstacle, as shown by contradiction using the concept of 'caves' and angular deviation bounds.
• The cumulative turning angle of the robot’s line-of-sight to the target is strictly bounded, preventing unbounded rotation and ensuring progress toward the target.
• The algorithm remains valid even when the robot performs full loops around the obstacle, as long as the relative bearing and distance rate are correctly interpreted.
• The convergence is robust to the choice of the turn direction parameter σ, with the proof holding for both σ = + and σ = −, though the paper focuses on the right-turn case.
• The result holds even when the robot is not able to trace the obstacle boundary exactly, as long as it can detect the sign of the distance derivative and the relative bearing.