Feedback-Based Mobile Robot Navigation in 3-D Environments Using Artificial Potential Functions Technical Report

TL;DR

This work develops a polynomial, 3-D navigation-function framework for motion planning in environments with spherical and cylindrical obstacles, directly in 3-D space without global coordinate transformations. By constructing a base function $\hat{\varphi}$ and two composite functions $\varphi$ and $\psi$ via normalization and sharpening, the approach achieves polarity and admissibility, guaranteeing a unique minimum at the target and maximal boundary values. The authors address pairwise intersecting obstacles using smooth p-Rvachev compositions and provide gradient/Hessian analyses to prove no interior local minima for large $k$, supported by simulations in obstacle-rich workspaces. They further extend to spherical robots through a transformation that expands obstacles by the robot radius and contract the workspace, and validate the framework on complex configurations, including a composite truss, with insights on obstacle-merging to reduce required $k$. The results offer a rigorous, topology-aware planning paradigm bridging motion planning, control, and formal guarantees for 3-D navigation in nonconvex environments.

Abstract

This technical report presents the construction and analysis of polynomial navigation functions for motion planning in 3-D workspaces populated by spherical and cylindrical obstacles. The workspace is modeled as a bounded spherical region, and obstacles are encoded using smooth polynomial implicit functions. We establish conditions under which the proposed navigation functions admit a unique non-degenerate minimum at the target while avoiding local minima, including in the presence of pairwise intersecting obstacles. Gradient and Hessian analyses are provided, and the theoretical results are validated through numerical simulations in obstacle rich 3-D environments.

Figures (29)

• Figure 1: Simulation of a quadrotor navigating in a two-story parking garage. The environment consists of a non-convex polyhedral outer boundary together with cylindrical and spherical internal obstacles.
• Figure 2: Different types of obstacles in a spherical room: (a) Spherical internal obstacle. (b) Full cylinder. (c) Half cylinder capped by a hemisphere. (d) Finite cylinder capped by two hemispheres.
• Figure 3: Composite obstacle consisting of pairwise unions between a spherical obstacle and three half-cylinder obstacles. The black shell represents the outer walls of the spherical room.
• Figure 4: 2-D illustration of the five regions into which the free space is divided
• Figure 5: Spatial relationships between two 2-D disc obstacles. (a) Disjoint obstacles. (b) Tangent obstacles with a single shared point. (c) Intersecting obstacles with two intersection points.

Theorems & Definitions (12)

• Claim 3.1
• proof
• Claim 3.2
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• Claim 3.3
• proof
• Claim 3.4
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• Claim 3.5
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