Velocity Obstacle for Polytopic Collision Avoidance for Distributed Multi-robot Systems

TL;DR

This work addresses real-time collision avoidance in distributed multi-robot systems with polytopic shapes by extending the velocity obstacle (VO) concept to polytopes, constructed in an optimization-free manner from vertex coordinates and relative states. The authors introduce a vertex-based VO_p for polytopes, along with combined VO (and RVO_p/HRVO_p variants) to enable distributed navigation where each robot independently selects velocities that avoid collisions while progressing toward its goal, using a cost function when necessary. Key contributions include (i) an optimization-free VO construction for polytopes, (ii) a VO-based distributed navigation framework with RVO_p and HRVO_p, and (iii) extensive simulations comparing against circular baselines that show improvements in completion rate, deadlock rate, and travel distance, especially in large-scale and obstacle-rich scenarios. The approach offers real-time scalability and less conservative behavior for polytopic robots, with practical implications for applications like warehouses and search-and-rescue; future work aims to extend the method to 3D space and address uncertainties and integration with global planning.

Abstract

Obstacle avoidance for multi-robot navigation with polytopic shapes is challenging. Existing works simplify the system dynamics or consider it as a convex or non-convex optimization problem with positive distance constraints between robots, which limits real-time performance and scalability. Additionally, generating collision-free behavior for polytopic-shaped robots is harder due to implicit and non-differentiable distance functions between polytopes. In this paper, we extend the concept of velocity obstacle (VO) principle for polytopic-shaped robots and propose a novel approach to construct the VO in the function of vertex coordinates and other robot's states. Compared with existing work about obstacle avoidance between polytopic-shaped robots, our approach is much more computationally efficient as the proposed approach for construction of VO between polytopes is optimization-free. Based on VO representation for polytopic shapes, we later propose a navigation approach for distributed multi-robot systems. We validate our proposed VO representation and navigation approach in multiple challenging scenarios including large-scale randomized tests, and our approach outperforms the state of art in many evaluation metrics, including completion rate, deadlock rate, and the average travel distance.

Figures (7)

• Figure 1: Snapshot of the distributed multi-robot navigation with polytopic shapes using our proposed approach. Each robot $\text{R}_i$ has its own destination $\mathbf{x}_{i}^{\text{goal}}$, and each robot could decide independently to move towards the destination while avoiding collisions. In this figure, we demonstrate each robot position at four different ticks with lighter shades of a color indicating robot's position in the past.
• Figure 2: Circular-shaped based velocity obstacle $\text{VO}_{\text{R}_i|\text{O}_j}(\mathbf{v}_{\text{O}_j})$ (blue conic region) of robot $\text{R}_i$ (blue circular region) induced by the obstacle $\text{O}_j$ (yellow circular region) with velocity $\mathbf{v}_{\text{O}_j}$. $\text{CC}_{\text{R}_i|\text{O}_j}$ (gray conic region) represents the collision cone between $\text{R}_i$ and $\text{O}_j$. If the relative velocity $\mathbf{v}_{\text{R}_i}-\mathbf{v}_{\text{O}_j} \in \text{CC}_{\text{R}_i|\text{O}_j}$ or the absolute robot velocity $\mathbf{v}_{\text{R}_i} \in \text{VO}_{\text{R}_i|\text{O}_j}(\mathbf{v}_{\text{O}_j})$, a collision will occur between $\text{R}_i$ and $\text{O}_j$. The direction vectors $\mathbf{vl}$ and $\mathbf{vr}$ (bold solid lines) is same for $\text{CC}_{\text{R}_i|\text{O}_j}$ and $\text{VO}_{\text{R}_i|\text{O}_j}(\mathbf{v}_{\text{O}_j})$ of robot $\text{R}_i$.
• Figure 3: Velocity Obstacle $\text{VO}_{\text{R}_i|\text{O}_j}(\mathbf{v}_{\text{O}_j})$ of robot $\text{R}_i$ induced by the obstacle $\text{O}_j$ for polytopic-shaped robots. First, we need to obtain the direction vectors $\mathbf{vl}$ and $\mathbf{vr}$ (bold black solid lines) on both sides of the collision cone $\text{CC}_{\text{R}_i|\text{O}_j}$ by connecting the vertices of $\text{R}_i$ and $\text{O}_j$ in any two pairs as done in \ref{['eq:vl_and_vr']}. Then we construct the $\text{VO}_{\text{R}_i|\text{O}_j}(\mathbf{v}_{\text{O}_j})$ with the direction vectors $\mathbf{vl}$ and $\mathbf{vr}$, and $\text{VO}_{\text{R}_i|\text{O}_j}(\mathbf{v}_{\text{O}_j})$ is a cone with its apex at $\mathbf{v}_{\text{O}_j}$.
• Figure 4: Illustration for the distributed navigation for multi-robot system. Robot $\text{R}_2$ only considers the robots and obstacles within a certain range ($l$), so $\text{R}_2$ only considers the VO induced by $\text{R}_1$.
• Figure 5: Simulation results of eight polytopic-shaped robots in a circle scenario using our proposed approach under (b) polytopic-shaped based velocity obstacle ($\text{VO}_\text{p}$), (c) polytopic-shaped based reciprocal velocity obstacle ($\text{RVO}_\text{p}$), (d) polytopic-shaped based hybrid reciprocal velocity obstacle ($\text{HRVO}_\text{p}$) representations. In (a), we show the initial and goal position of each robot in a circle scenario, where the goal position of each robot is shown as a circle with different colors.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3