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A Closed-Form CLF-CBF Controller for Whole-Body Continuum Soft Robot Collision Avoidance

Kiwan Wong, Maximillian Stölzle, Wei Xiao, Daniela Rus

Abstract

Safe operation is essential for deploying robots in human-centered 3D environments. Soft continuum manipulators provide passive safety through mechanical compliance, but still require active control to achieve reliable collision avoidance. Existing approaches, such as sampling-based planning, are often computationally expensive and lack formal safety guarantees, which limits their use for real-time whole-body avoidance. This paper presents a closed-form Control Lyapunov Function--Control Barrier Function (CLF--CBF) controller for real-time 3D obstacle avoidance in soft continuum manipulators without online optimization. By analytically embedding safety constraints into the control input, the proposed method ensures stability and safety under the stated modeling assumptions, while avoiding feasibility issues commonly encountered in online optimization-based methods. The resulting controller is up to $10\times$ faster than standard CLF--CBF quadratic-programming approaches and up to $100\times$ faster than traditional sampling-based planners. Simulation and hardware experiments on a tendon-driven soft manipulator demonstrate accurate 3D trajectory tracking and robust obstacle avoidance in cluttered environments. These results show that the proposed framework provides a scalable and provably safe control strategy for soft robots operating in dynamic, safety-critical settings.

A Closed-Form CLF-CBF Controller for Whole-Body Continuum Soft Robot Collision Avoidance

Abstract

Safe operation is essential for deploying robots in human-centered 3D environments. Soft continuum manipulators provide passive safety through mechanical compliance, but still require active control to achieve reliable collision avoidance. Existing approaches, such as sampling-based planning, are often computationally expensive and lack formal safety guarantees, which limits their use for real-time whole-body avoidance. This paper presents a closed-form Control Lyapunov Function--Control Barrier Function (CLF--CBF) controller for real-time 3D obstacle avoidance in soft continuum manipulators without online optimization. By analytically embedding safety constraints into the control input, the proposed method ensures stability and safety under the stated modeling assumptions, while avoiding feasibility issues commonly encountered in online optimization-based methods. The resulting controller is up to faster than standard CLF--CBF quadratic-programming approaches and up to faster than traditional sampling-based planners. Simulation and hardware experiments on a tendon-driven soft manipulator demonstrate accurate 3D trajectory tracking and robust obstacle avoidance in cluttered environments. These results show that the proposed framework provides a scalable and provably safe control strategy for soft robots operating in dynamic, safety-critical settings.
Paper Structure (29 sections, 1 theorem, 31 equations, 8 figures, 1 table)

This paper contains 29 sections, 1 theorem, 31 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Kinematics of a Tendon-Driven Soft Robotic Arm
  4. Tendon Length Model
  5. Differential Actuation Kinematics
  6. Control Barrier Functions and Control Lyapunov Functions
  7. Closed-Form Solution for CLF-CBF Controller
  8. Problem Formulation
  9. Control Barrier Functions
  10. Control Lyapunov Function
  11. Quadratic Program
  12. Aggregation of CBFs
  13. Closed-form Solution Derivation
  14. Lagrangian and KKT conditions boyd2004convex
  15. Role of the slack variable
  16. ...and 14 more sections

Key Result

Lemma 1

Let $\{b_{i,j}(\bm{x})\}_{i \in \mathbb{N}_{N_{\mathrm{res}}}; j \in \mathbb{N}_{N_{\mathrm{obs}}}}$ be real-valued functions and define the Log-Sum-Exp aggregation If $b_{\mathrm{LSE}}(\bm{x}) \ge 0$, then $b_{i,j}(\bm{x}) \ge 0$ for all $i,j$.

Figures (8)

  • Figure 1: Experimental setup of the tendon-driven soft robotic manipulator used to validate the proposed closed-form CLF–CBF controller. The two-segment robot is mounted to a ceiling fixture and operates in an environment with spherical obstacles. Reflective markers are attached to each segment for real-time motion capture and feedback control during safe trajectory tracking experiments.
  • Figure 2: Simulation results. Circular trajectory tracking pattern in 2D. The trajectories represent the motion path of the soft robot tip center, while the red-shaded region indicates the collision area. The RMSE of each case is shown in the legend, consistently demonstrating that the closed-form controller achieves more accurate and smoother trajectories than the baseline QP controller.
  • Figure 3: Quantitative evaluation of body discretization and controller efficiency. (a) Increasing $N_{\mathrm{res}}$ improves geometric fidelity, with rapidly diminishing Hausdorff error after $\sim 400$ spheres. (b) The closed-form CLF–CBF solver maintains low computation time even at high discretization resolutions, in contrast to the generic QP solver.
  • Figure 4: Setpoint regulation simulation results for the distance to the target (CLF objective) and to the obstacles (CBF constraints). Left. Behavior of a low-level controller tracking a sequence of configuration setpoints provided by an RRT* obstacle avoidance planner. Right. Behavior of the closed-loop system under the proposed closed-form CLF-CBF controller. The distance to the target does not reach absolute zero due to the presence of nearby obstacles.
  • Figure 5: Sequence of stills of the setpoint regulation simulations visualizing the motion of a soft robot consisting of two segments (blue and green) under the closed-form CLF–CBF controller, with time progressing from left to right. The controller steers the robot toward the target denoted with a red dot while preserving a safe distance from surrounding obstacles (grey).
  • ...and 3 more figures

Theorems & Definitions (4)

  • Definition 1: Control Barrier Function ames2019controlxiao2021high
  • Definition 2: Control Lyapunov Function ames2019control
  • Lemma 1: LSE barrier implies all pairwise barriers
  • proof