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Fast Shortest Path Polyline Smoothing With $G^1$ Continuity and Bounded Curvature

Patrick Pastorelli, Simone Dagnino, Enrico Saccon, Marco Frego, Luigi Palopoli

TL;DR

The paper addresses the problem of fast, physically realizable smoothing of a collision-free polyline for nonholonomic robots with a minimum turning radius. It introduces DPS, which decomposes the interpolation into 3-point subproblems and constructs tangent points on a circle of radius $r$ to produce a sequence of CSC Dubins paths, ensuring $G^1$ continuity and bounded curvature. Under explicit existence and feasibility conditions, the concatenation of subpaths is proven to be the shortest possible path within the admissible class, with linear-time complexity and both sequential and parallel implementations. Extensive experiments on interpolation, road-map planning, sampling-based methods, and real hardware validate improved speed and path quality, including deployment on embedded systems. The work provides practical guarantees for collision-free, optimal smoothing suitable for real-time robotic motion planning and CNC-like applications.

Abstract

In this work, we propose a novel and efficient method for smoothing polylines in motion planning tasks. The algorithm applies to motion planning of vehicles with bounded curvature. In the paper, we show that the generated path: 1) has minimal length, 2) is $G^1$ continuous, and 3) is collision-free by construction, if the hypotheses are respected. We compare our solution with the state-of.the-art and show its convenience both in terms of computation time and of length of the compute path.

Fast Shortest Path Polyline Smoothing With $G^1$ Continuity and Bounded Curvature

TL;DR

The paper addresses the problem of fast, physically realizable smoothing of a collision-free polyline for nonholonomic robots with a minimum turning radius. It introduces DPS, which decomposes the interpolation into 3-point subproblems and constructs tangent points on a circle of radius to produce a sequence of CSC Dubins paths, ensuring continuity and bounded curvature. Under explicit existence and feasibility conditions, the concatenation of subpaths is proven to be the shortest possible path within the admissible class, with linear-time complexity and both sequential and parallel implementations. Extensive experiments on interpolation, road-map planning, sampling-based methods, and real hardware validate improved speed and path quality, including deployment on embedded systems. The work provides practical guarantees for collision-free, optimal smoothing suitable for real-time robotic motion planning and CNC-like applications.

Abstract

In this work, we propose a novel and efficient method for smoothing polylines in motion planning tasks. The algorithm applies to motion planning of vehicles with bounded curvature. In the paper, we show that the generated path: 1) has minimal length, 2) is continuous, and 3) is collision-free by construction, if the hypotheses are respected. We compare our solution with the state-of.the-art and show its convenience both in terms of computation time and of length of the compute path.
Paper Structure (10 sections, 8 theorems, 23 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 8 theorems, 23 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Statement
  4. Solution formalization
  5. Experimental Results
  6. Path Interpolation
  7. Road Maps
  8. Path Sampling Interpolation
  9. Physical Robot Experiment
  10. Conclusion

Key Result

Lemma 2

Let $P=(p_i,p_m,p_f)$ be a polyline of 3 non aligned points, and $\alpha_m$ the angle in $p_m$, as in Figure fig:p3. The following constraint must be satisfied to ensure the existence of a $G^1$ approximation of the polyline:

Figures (9)

  • Figure 1: In blue the original 4-points polyline $P=\{p_0,\hdots,p_3\}$ and in red the smoothed path from DPS.
  • Figure 2: Visual representation of the standard placement for the 3-points sub-problem.
  • Figure 3: In red, an example of the J Dubins path type in the standard setting used in Theorem \ref{['th:DubinsPath']}.
  • Figure 4: An example showing three obstacles (in gray), their bounding boxes (in red) computed with \ref{['eq:minDistance']} and two robots in green and blue. The green one uses DPS and is guaranteed not to hit any obstacle. The blue one uses a Dubins interpolation algorithm 8989830 and may still collide.
  • Figure 5: A scenario in which the mitered offset $O$, in purple, is larger than the radius of the robot $h$ in orange. In blue, the turning radius of the robot and in green the path.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Definition 1: Polyline
  • Definition 2: Problem
  • Lemma 2: Existence constraint
  • proof
  • Corollary 3: Global existence condition
  • Corollary 4: Tangent points
  • Corollary 5: Distance from the polyline points
  • Theorem 7: Optimality of $\mathbf{Q}$
  • proof
  • Corollary 8: Optimality of $\mathcal{P}$
  • ...and 4 more