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Differentiable Optimization Based Time-Varying Control Barrier Functions for Dynamic Obstacle Avoidance

Bolun Dai, Rooholla Khorrambakht, Prashanth Krishnamurthy, Farshad Khorrami

TL;DR

This work solves dynamic obstacle avoidance by extending differentiable-optimization-based control barrier functions (diffOpt CBFs) to time-varying safe sets, enabling safe interaction with moving obstacles. The proposed TVCBFQP minimizes deviation from a reference control while enforcing a set of time-varying CBF constraints across convex primitives, and it incorporates measurement noise via a Mahalanobis-distance based worst-case obstacle configuration and actuation limits via velocity inflation with a tunable factor. The approach is validated through simulations against MPC and spherical-CBF baselines and is demonstrated on a 7-DOF Franka Research 3 manipulator, showing safety robust to noise and improved conservatism due to geometric modeling with multiple primitives. The results indicate real-time capability (sub-millisecond to a few milliseconds per step) and practical relevance for dynamic obstacle scenarios in robotic manipulation and safe autonomous operation.

Abstract

Control barrier functions (CBFs) provide a simple yet effective way for safe control synthesis. Recently, work has been done using differentiable optimization (diffOpt) based methods to systematically construct CBFs for static obstacle avoidance tasks between geometric shapes. In this work, we extend the application of diffOpt CBFs to perform dynamic obstacle avoidance tasks. We show that by using the time-varying CBF (TVCBF) formulation, we can perform obstacle avoidance for dynamic geometric obstacles. Additionally, we show how to extend the TVCBF constraint to consider measurement noise and actuation limits. To demonstrate the efficacy of our proposed approach, we first compare its performance with a model predictive control based method and a circular CBF based method on a simulated dynamic obstacle avoidance task. Then, we demonstrate the performance of our proposed approach in experimental studies using a 7-degree-of-freedom Franka Research 3 robotic manipulator.

Differentiable Optimization Based Time-Varying Control Barrier Functions for Dynamic Obstacle Avoidance

TL;DR

This work solves dynamic obstacle avoidance by extending differentiable-optimization-based control barrier functions (diffOpt CBFs) to time-varying safe sets, enabling safe interaction with moving obstacles. The proposed TVCBFQP minimizes deviation from a reference control while enforcing a set of time-varying CBF constraints across convex primitives, and it incorporates measurement noise via a Mahalanobis-distance based worst-case obstacle configuration and actuation limits via velocity inflation with a tunable factor. The approach is validated through simulations against MPC and spherical-CBF baselines and is demonstrated on a 7-DOF Franka Research 3 manipulator, showing safety robust to noise and improved conservatism due to geometric modeling with multiple primitives. The results indicate real-time capability (sub-millisecond to a few milliseconds per step) and practical relevance for dynamic obstacle scenarios in robotic manipulation and safe autonomous operation.

Abstract

Control barrier functions (CBFs) provide a simple yet effective way for safe control synthesis. Recently, work has been done using differentiable optimization (diffOpt) based methods to systematically construct CBFs for static obstacle avoidance tasks between geometric shapes. In this work, we extend the application of diffOpt CBFs to perform dynamic obstacle avoidance tasks. We show that by using the time-varying CBF (TVCBF) formulation, we can perform obstacle avoidance for dynamic geometric obstacles. Additionally, we show how to extend the TVCBF constraint to consider measurement noise and actuation limits. To demonstrate the efficacy of our proposed approach, we first compare its performance with a model predictive control based method and a circular CBF based method on a simulated dynamic obstacle avoidance task. Then, we demonstrate the performance of our proposed approach in experimental studies using a 7-degree-of-freedom Franka Research 3 robotic manipulator.
Paper Structure (17 sections, 1 theorem, 21 equations, 9 figures)

This paper contains 17 sections, 1 theorem, 21 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Barrier Function
  4. Differentiable Optimization Based Control Barrier Function Formulation
  5. Problem Formulation
  6. Method
  7. Motivating Example -- Moving Circles
  8. Time-Varying Control Barrier Function
  9. Measurement Noise
  10. Actuation Limits
  11. Experiments
  12. Dynamic Rectangular Obstacle Avoidance
  13. FR3 Robotic Manipulator
  14. Avoiding a Moving Board
  15. Avoiding Two Moving Boxes
  16. ...and 2 more sections

Key Result

Theorem IV.1

For the CBF defined in eq:diff_opt_cbf, let Then, the solution to the TVCBFQP with $h(x, \Tilde{\psi})$ in the TVCBF constraint also guarantees safety for all $\psi \in \mathcal{S}_{k}$.

Figures (9)

  • Figure 1: Control pipeline for using time-varying diffOpt CBFs for dynamic obstacle avoidance.
  • Figure 2: This figure illustrates the motion generated by the TVCBFQP controller. The robot we control is in blue. The obstacle is in orange. The light blue and orange dashed lines represent the path the robot and the obstacle traveled, respectively. The timestamp of each snapshot is given in the lower right corner of each figure. Each small grid is $2~m\times2~m$.
  • Figure 3: This figure illustrates the motion generated by the TVCBFQP controller under sensor noise. The top two rows show motion snapshots generated by the controller that considers sensor noise. For the motion snapshots, the color scheme follows Fig. \ref{['fig:moving_circle_visualization']} with the only difference being the light orange dashed lines representing the estimated obstacle trajectory. The bottom plot compares the CBF value for two controllers, one considering sensor noise using the proposed method and the other not considering sensor noise. The bottom plot only shows from $[1.5, 2.5]~s$ since the CBF values are positive for the rest of the time.
  • Figure 4: This figure illustrates the motion generated by the TVCBFQP controller under actuation limits. The top two rows show snapshots of the motion generated when actuation limits are considered. The color scheme follows Fig. \ref{['fig:moving_circle_visualization']}. The bottom plots compare the CBF value for two controllers, one considering actuation limits using the proposed method in Section \ref{['sec:method-actuation']}, and the other not considering actuation limits. The bottom right plot zooms in on the bottom left plot between time $[1.5, 2.5]$s to show that the controller not considering actuation limits leads to unsafe behavior.
  • Figure 5: This figure illustrates the motion generated by our proposed controller and the controller in WhiteJWH22 on the moving rectangle task described in Section \ref{['sec:experiments-rectangle']}. The color scheme follows Fig. \ref{['fig:moving_circle_visualization']}. Comparing the figures in the 3rd - 6th columns shows that the MPC-based method (top row) generates a much larger collision avoidance maneuver than our proposed approach (bottom row).
  • ...and 4 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Theorem IV.1
  • proof
  • Remark 3
  • Remark 4