Tube RRT*: Efficient Homotopic Path Planning for Swarm Robotics Passing-Through Large-Scale Obstacle Environments

TL;DR

Tube RRT* introduces an efficient homotopic path planning framework for swarm robotics by extending RRT* to optimize both path length and gap volume within a virtual-tube formulation. It generates multiple boundary-boundary homotopic paths and interpolates between them to yield infinite homotopic options, while maintaining probabilistic completeness and asymptotic optimality. The method achieves higher gap-volume metrics and reduced cross-section variability, enabling smoother, congestion-resistant swarm navigation in large-scale obstacle environments. Empirical results across simulations and real-world drone experiments demonstrate practical gains in traversal efficiency and swarm safety, with a tunable tradeoff between path length and gap openness via a velocity-gap weight parameter.

Abstract

Recently, the concept of homotopic trajectory planning has emerged as a novel solution to navigation in large-scale obstacle environments for swarm robotics, offering a wide ranging of applications. However, it lacks an efficient homotopic path planning method in large-scale obstacle environments. This paper introduces Tube RRT*, an innovative homotopic path planning method that builds upon and improves the Rapidly-exploring Random Tree (RRT) algorithm. Tube RRT* is specifically designed to generate homotopic paths, strategically considering gap volume and path length to mitigate swarm congestion and ensure agile navigation. Through comprehensive simulations and experiments, the effectiveness of Tube RRT* is validated.

Figures (8)

• Figure 1: Virtual tube in obstacle environments. (a) The purple and blue polyhedrons are terminals. The black curve is the trajectory from ${\bf q}_0\in{\mathcal{C}_0}$ to ${\bf q}_m \in \mathcal{C}_1$. (b) The colorful points denote path points of the homotopic paths and the gray lines represent homotopic paths $\boldsymbol{\sigma}\left(\left({\bf q}_0,{\bf q}_m\right),t\right)$.
• Figure 2: The scheme of Tube RRT* algorithm. Gray circles are obstacles and blue circles are centered in the red and black points. (a) Step 1: Sample a sphere ${\bf x}_\text{rand}$ which is denoted by the blue circle centered in ${\bf q}_\text{rand}$ with radius $r_\text{rand}$ in free space. (b) Step 2: Find the nearest sphere ${\bf x}_\text{nearest}$ in the tree. (c) Step 3: Steer the sample sphere ${\bf x}_\text{rand}$ towards the nearest sphere ${\bf x}_\text{nearest}$ to obtain ${\bf x}_\text{new}$ which has an intersection with ${\bf x}_\text{nearest}$. (d) Step 4: Find all the spheres in the tree which have intersections with the ${\bf x}_\text{new}$ to obtain the set $X_\text{near}$. (e) Step 5: Rewire the sphere ${\bf x}_\text{new}$ to a sphere in the tree to approach the minimum cost. And the spheres in $X_\text{near}$ also are rewired.
• Figure 3: Examples of homotopic paths generation. (a) Red line and blue line represent terminal $\mathcal{C}_0$ and $\mathcal{C}_1$ respectively. (b) Tube RRT* algorithm generates a path $\boldsymbol{{\sigma}}_c$ from ${\bf q}_0$ to ${\bf q}_4$ represented by the black line. (c) Boundary paths $\boldsymbol{\sigma}_1$ and $\boldsymbol{\sigma}_2$ are denoted by green lines. (d) Homotopic paths $\boldsymbol{\sigma}_{\boldsymbol{\theta}}$ are represented by the gray lines.
• Figure 4: Comparisons of metrics for algorithms in environments with different number of obstacles.
• Figure 5: Comparisons of paths generated by Tube RRT*. The red and green points represent the start area and goal area respectively. And the green curves and red curves are generated by Tube RRT* with $\rho_v = 0.15$ and $\rho_v = 0$ respectively. The colorful curves represent trajectories for robots and the colors from blue to red represent the speed from zero to maximum.

Theorems & Definitions (12)

• Definition 1: Virtual Tube mao2023optimal
• Definition 2: Path
• Definition 3: Homotopic Paths
• Definition 4: Path Length
• Theorem 1: Probabilistic completeness of Tube RRT*
• proof
• Theorem 2: Asymptotic optimality of Tube RRT*
• proof
• Lemma 1
• proof