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Safe Navigation in Cluttered Environments Via Spline-Based Harmonic Potential Fields

Theodor-Gabriel Nicu, Florin Stoican, Daniel-Mihail Ioan, Ionela Prodan

TL;DR

This work introduces a spline-based harmonic potential framework for safe navigation in cluttered environments by decomposing the free space into polyhedral cells and constraining motion to a prescribed cell sequence. Each cell carries a harmonic potential with Dirichlet boundary conditions induced by a cardinal B-spline boundary, enabling smooth funneling between adjacent cells. The method provides explicit harmonic surface expressions via Fourier and polylogarithm tools, along with principled control-point selection to enforce desired boundary behavior, and a switched control law for a double-integrator model that follows a graph-guided path. Practically, the approach yields obstacle-avoiding trajectories with guaranteed exit through designated facets, and is demonstrated with illustrative examples and downloadable code. The framework offers a flexible, mathematically grounded alternative to traditional APF methods, with potential extensions to richer spline configurations and optimized inter-cell performance.

Abstract

We provide a complete motion-planning mechanism that ensures target tracking and obstacle avoidance in a cluttered environment. For a given polyhedral decomposition of the feasible space, we adopt a novel procedure that constrains the agent to move only through a prescribed sequence of cells via a suitable control policy. For each cell, we construct a harmonic potential surface induced by a Dirichlet boundary condition given as a cardinal B-spline curve. A detailed analysis of the curve behavior (periodicity, support) and of the associated control point selection allows us to explicitly compute these harmonic potential surfaces, from which we subsequently derive the corresponding control policy. We illustrate that the resulting construction funnels the agent safely along the chain of cells from the starting point to the target.

Safe Navigation in Cluttered Environments Via Spline-Based Harmonic Potential Fields

TL;DR

This work introduces a spline-based harmonic potential framework for safe navigation in cluttered environments by decomposing the free space into polyhedral cells and constraining motion to a prescribed cell sequence. Each cell carries a harmonic potential with Dirichlet boundary conditions induced by a cardinal B-spline boundary, enabling smooth funneling between adjacent cells. The method provides explicit harmonic surface expressions via Fourier and polylogarithm tools, along with principled control-point selection to enforce desired boundary behavior, and a switched control law for a double-integrator model that follows a graph-guided path. Practically, the approach yields obstacle-avoiding trajectories with guaranteed exit through designated facets, and is demonstrated with illustrative examples and downloadable code. The framework offers a flexible, mathematically grounded alternative to traditional APF methods, with potential extensions to richer spline configurations and optimized inter-cell performance.

Abstract

We provide a complete motion-planning mechanism that ensures target tracking and obstacle avoidance in a cluttered environment. For a given polyhedral decomposition of the feasible space, we adopt a novel procedure that constrains the agent to move only through a prescribed sequence of cells via a suitable control policy. For each cell, we construct a harmonic potential surface induced by a Dirichlet boundary condition given as a cardinal B-spline curve. A detailed analysis of the curve behavior (periodicity, support) and of the associated control point selection allows us to explicitly compute these harmonic potential surfaces, from which we subsequently derive the corresponding control policy. We illustrate that the resulting construction funnels the agent safely along the chain of cells from the starting point to the target.
Paper Structure (8 sections, 6 theorems, 71 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 8 sections, 6 theorems, 71 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Preliminaries on cardinal B-splines
  4. Computing the potential surface with cardinal B-splines
  5. Computation of the harmonic function
  6. Choosing the control points
  7. Control policy
  8. Conclusions

Key Result

Lemma 1

For a given $T \in \mathbb{Z}_{> 0}$, let $\lambda = 2\pi / T$ and $P_{k} = P_{k\pm T}$ for all $k \in \mathbb{Z}$. Then, the function $h(\theta)$ defined in eq:dirichlet_spline$\,$ satisfies

Figures (7)

  • Figure 1: Proof of concept illustration for the connectivity graph construction
  • Figure 2: Cardinal B-splines
  • Figure 3: Mapping of the induced potential surface from the unit disk to the polyhedral case.
  • Figure 4: A cluttered environment, partitioned into polyhedral obstacles and free cells as in \ref{['eq:decomposition']}, overlaid with the facet connectivity graph \ref{['eq:graph-nodes']}--\ref{['eq:graph-edges']}. Graph nodes are shown at the centroids of the corresponding facets.
  • Figure 5: "Waterfall" effect of the surfaces along a path segment (surfaces and trajectory are vertically shifted for clarity).
  • ...and 2 more figures

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Lemma 3
  • ...and 4 more