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Safe Stabilizing Control for Polygonal Robots in Dynamic Elliptical Environments

Kehan Long, Khoa Tran, Melvin Leok, Nikolay Atanasov

TL;DR

This work tackles safe navigation for polygon-shaped robots in dynamic 2D environments populated by moving ellipses. It introduces an analytic distance function in SE(2) to compute the closest approach between an ellipse and a polygon and builds a time-varying control barrier function (TV-CBF) to enforce safety while stabilizing to a goal. The control problem is solved online via a quadratic program that enforces CLF-based stabilization and CBC-based safety, enabling real-time obstacle avoidance for both ground robots and multi-link robot arms. Comparative simulations show that SE(2)-aware distance-based CBFs outperform simpler R^2 or circular models, particularly in narrow passages, and demonstrate robust safety during dynamic obstacle motion. The approach generalizes to unicycle-like systems and planar robot arms, with planned extensions to 3D manipulation and onboard perception for environment geometry estimation.

Abstract

This paper addresses the challenge of safe navigation for rigid-body mobile robots in dynamic environments. We introduce an analytic approach to compute the distance between a polygon and an ellipse, and employ it to construct a control barrier function (CBF) for safe control synthesis. Existing CBF design methods for mobile robot obstacle avoidance usually assume point or circular robots, preventing their applicability to more realistic robot body geometries. Our work enables CBF designs that capture complex robot and obstacle shapes. We demonstrate the effectiveness of our approach in simulations highlighting real-time obstacle avoidance in constrained and dynamic environments for both mobile robots and multi-joint robot arms.

Safe Stabilizing Control for Polygonal Robots in Dynamic Elliptical Environments

TL;DR

This work tackles safe navigation for polygon-shaped robots in dynamic 2D environments populated by moving ellipses. It introduces an analytic distance function in SE(2) to compute the closest approach between an ellipse and a polygon and builds a time-varying control barrier function (TV-CBF) to enforce safety while stabilizing to a goal. The control problem is solved online via a quadratic program that enforces CLF-based stabilization and CBC-based safety, enabling real-time obstacle avoidance for both ground robots and multi-link robot arms. Comparative simulations show that SE(2)-aware distance-based CBFs outperform simpler R^2 or circular models, particularly in narrow passages, and demonstrate robust safety during dynamic obstacle motion. The approach generalizes to unicycle-like systems and planar robot arms, with planned extensions to 3D manipulation and onboard perception for environment geometry estimation.

Abstract

This paper addresses the challenge of safe navigation for rigid-body mobile robots in dynamic environments. We introduce an analytic approach to compute the distance between a polygon and an ellipse, and employ it to construct a control barrier function (CBF) for safe control synthesis. Existing CBF design methods for mobile robot obstacle avoidance usually assume point or circular robots, preventing their applicability to more realistic robot body geometries. Our work enables CBF designs that capture complex robot and obstacle shapes. We demonstrate the effectiveness of our approach in simulations highlighting real-time obstacle avoidance in constrained and dynamic environments for both mobile robots and multi-joint robot arms.
Paper Structure (12 sections, 2 theorems, 33 equations, 5 figures)

This paper contains 12 sections, 2 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Preliminaries
  4. Control Lyapunov Function
  5. Control Barrier Function
  6. Analytic distance between ellipse and polygon
  7. Polygon-Shaped Robot Safe Navigation in Dynamic Ellipse Environments
  8. Time-Varying Control Barrier Function Constraints
  9. Ground-Robot Navigation
  10. K-joint Robot Arm Safe Stabilizing Control
  11. Evaluation
  12. Conclusion

Key Result

Proposition IV.1

Let ${\@fontswitch\mathcal{E}}'$ be an ellipse and $l_i'$ be a line segment in the frame of the ellipse. Denote $\tau^*$ as the argument of the minimum in eq: polygon_line_seg_1. Then, the distance where $\underline{\mathbf{p}i'}$ and $\underline{\mathbf{p}{[i+1]{M}}'}$ are the points on the ellipse closest to $\mathbf{p}i'$ and $\mathbf{p}{[i+1]{M}}'$, respectively. These points are determined u

Figures (5)

  • Figure 1: Comparative analysis of the $SE(2)$ and $\mathbb{R}^2$ signed distance functions for elliptical obstacles. The cyan triangle represents the rigid-body robot, with its orientation varying across the sequence. The importance of considering robot orientation in distance computations becomes evident: while the $SE(2)$ function accounts for this orientation, the $\mathbb{R}^2$ approximation treats the robot as an encapsulating circle with radius $1$. Level sets at distances $0.2$ and $2$ are depicted for both functions.
  • Figure 2: Safe navigation in a dynamical elliptical environment. (a) shows the initial pose of the triangular robot and the environment. (b) shows the triangular robot passing through the narrow space between two moving ellipses. (c) shows the robot adjusts its pose to avoid the moving obstacle. (d) shows the final pose of the robot that reaches the goal region. In (e), we plot the trajectory of navigating a circular robot in the same environment.
  • Figure 3: Safe stabilization of a 3-joint robot arm. The green circle denotes the goal region, and the gray box denotes the base of the arm. The arm is shown in blue and the trajectory of its end-effector is shown in red. The trajectories of the moving elliptical obstacles are shown in purple.
  • Figure 4: Control input of the 3-joint robot arm.
  • Figure 5: Lyapunov function and barrier function values over time.

Theorems & Definitions (4)

  • Definition III.1
  • Definition III.2
  • Proposition IV.1
  • Proposition IV.2