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Safe Multi-Robotic Arm Interaction via 3D Convex Shapes

Ali Umut Kaypak, Shiqing Wei, Prashanth Krishnamurthy, Farshad Khorrami

TL;DR

The paper addresses safe coordination of multiple robotic arms operating in a shared workspace by extending high-order control barrier functions (HOCBFs) to 3D convex collision bodies (ellipsoids) and implementing centralized and decentralized safety filters. It constructs pairwise HOCBFs for inter-arm link collisions and tackles computational overhead with Savitzky-Golay-based numerical Hessian contribution estimation, enabling real-time performance. The authors validate the approach through extensive simulations with four Franka arms and real-world experiments with two arms, showing robust safety guarantees and significantly higher control update rates when Hessian contributions are estimated. The work demonstrates a practical, real-time reactive safety layer that complements offline planning for multi-robot arm systems and highlights areas for improvement in decentralized feasibility and scalability.

Abstract

Inter-robot collisions pose a significant safety risk when multiple robotic arms operate in close proximity. We present an online collision avoidance methodology leveraging High-Order Control Barrier Functions (HOCBFs) constructed for safe interactions among 3D convex shapes to address this issue. While prior works focused on using Control Barrier Functions (CBFs) for human-robotic arm and single-arm collision avoidance, we explore the problem of collision avoidance between multiple robotic arms operating in a shared space. In our methodology, we utilize the proposed HOCBFs as centralized and decentralized safety filters. These safety filters are compatible with many nominal controllers and ensure safety without significantly restricting the robots' workspace. A key challenge in implementing these filters is the computational overhead caused by the large number of safety constraints and the computation of a Hessian matrix per constraint. We address this challenge by employing numerical differentiation methods to approximate computationally intensive terms. The effectiveness of our method is demonstrated through extensive simulation studies and real-world experiments with Franka Research 3 robotic arms. The project video is available at this link.

Safe Multi-Robotic Arm Interaction via 3D Convex Shapes

TL;DR

The paper addresses safe coordination of multiple robotic arms operating in a shared workspace by extending high-order control barrier functions (HOCBFs) to 3D convex collision bodies (ellipsoids) and implementing centralized and decentralized safety filters. It constructs pairwise HOCBFs for inter-arm link collisions and tackles computational overhead with Savitzky-Golay-based numerical Hessian contribution estimation, enabling real-time performance. The authors validate the approach through extensive simulations with four Franka arms and real-world experiments with two arms, showing robust safety guarantees and significantly higher control update rates when Hessian contributions are estimated. The work demonstrates a practical, real-time reactive safety layer that complements offline planning for multi-robot arm systems and highlights areas for improvement in decentralized feasibility and scalability.

Abstract

Inter-robot collisions pose a significant safety risk when multiple robotic arms operate in close proximity. We present an online collision avoidance methodology leveraging High-Order Control Barrier Functions (HOCBFs) constructed for safe interactions among 3D convex shapes to address this issue. While prior works focused on using Control Barrier Functions (CBFs) for human-robotic arm and single-arm collision avoidance, we explore the problem of collision avoidance between multiple robotic arms operating in a shared space. In our methodology, we utilize the proposed HOCBFs as centralized and decentralized safety filters. These safety filters are compatible with many nominal controllers and ensure safety without significantly restricting the robots' workspace. A key challenge in implementing these filters is the computational overhead caused by the large number of safety constraints and the computation of a Hessian matrix per constraint. We address this challenge by employing numerical differentiation methods to approximate computationally intensive terms. The effectiveness of our method is demonstrated through extensive simulation studies and real-world experiments with Franka Research 3 robotic arms. The project video is available at this link.

Paper Structure

This paper contains 14 sections, 3 theorems, 33 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Notations
  4. Background
  5. High order control barrier functions
  6. Convex shapes for defining safety in robotic arms
  7. Safe multi-robotic arm interaction
  8. Construction of HOCBFs for multi-robotic arm setting
  9. Centralized-decentralized HOCBF filters
  10. Computational challenges in computing HOCBFs for multi-robotic arm safety
  11. Experiments
  12. Simulation studies
  13. Real-world experiments
  14. Conclusion

Key Result

Theorem 1

Given an HOCBF $h$ from Definition def:hocbf, and with the associated sets $C_k$, $k \in \{1, \hdots, r\}$ defined by (eq:psi_definitions), if $x(t_0) \in \mathcal{C}:= C_1 \cap \cdots \cap C_{r}$, then any Lipschitz continuous controller $\mathbf{u}(t) \in K_{\text{HOCBF}}(\mathbf{x}) := \{ \mathbf

Figures (8)

  • Figure 1: Three examples of Hessian contribution estimation utilizing Savitzky-Golay filter in a real multi-robot environment. The top plots compare the estimated signals to the ground truth, and the bottom plots present the corresponding estimation errors. The root-mean-square error (RMSE) for each case is reported below. The data and ground truth signals were collected from real experiments that use analytically computed Hessian contributions. The estimation was performed offline, simulating an online setting. The window length and polynomial order in the filter are set to 5 and 2, respectively.
  • Figure 2: Trajectory following examples in the MuJoCo environment. Ellipses covering the robot parts represent the 3D shapes used to define HOCBFs. Dark blue line segments in front of the end-effectors illustrate examples of intersecting trajectories.
  • Figure 3: Subplots (a)–(c) show the deviations $\Delta x$, $\Delta y$, and $\Delta z$ between executed and planned trajectories for robots 1–4 over the course of a simulation experiment using centralized filter. Subplot (d) shows the time evolution of the HOCBF values $\psi_{0}$ defined between each pair of end-effectors. Subplot (e) shows the distance between the origins of the end-effector frames for each robot pair. In subplots (a)–(c), the legend identifies individual robots; in subplots (d)–(e), it identifies the six inter-robot pairs.
  • Figure 4: Comparison of the same experiment under the nominal controller (top row), where collisions occur, and with the centralized HOCBF filter (bottom row), which prevents collisions. Red ellipsoids indicate robot collisions.
  • Figure 5: Real-world experimental environment.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Definition 1: Time-invariant HOCBFs xiao2021high
  • Theorem 1: Theorem 4 in xiao2021high for time-invariant HOCBFs
  • Definition 2: Scaling functions wei2024diffocclusion
  • Theorem 2: Theorem 2 in wei2024collision
  • Definition 3
  • Theorem 3
  • proof
  • Remark 1: QP feasibility in the safety filters