A Closed-Form CLF-CBF Controller for Whole-Body Continuum Soft Robot Collision Avoidance
Authors: Kiwan Wong, Maximillian Stölzle, Wei Xiao, Daniela Rus
Organizations: Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Robotics Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA
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× faster than standard CLF--CBF quadratic-programming approaches and up to 100× 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.
Figures & tables
Fig. 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.
Fig. 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.
Methods
Runtime [ μ s/call]
Rel. Speedup
QP solver ( qpax , JIT)
45.53
1 x (baseline)
Closed-form
4.22
10.8 x faster
TABLE I : Runtime and numerical accuracy for the two-constraint QP. The Closed-form method denotes the closed-form solution derived from KKT conditions, and the QP solver refers to the JIT-compiled numerical optimizer ( qpax ). Metrics are defined as follows: ∥uc−uqp∥∞ and ∥uc−uqp∥2 measure the maximum and Euclidean norm of the solution difference, ∣f(uc)−f(uqp)∣ is the objective gap, and max(Au−b,0) quantifies constraint violation. All results show machine-precision agreement between the closed-form and numerical solutions, with the closed-form approach achieving over an order-of-magnitude faster runtime.
Fig. 3 : Quantitative evaluation of body discretization and controller efficiency. (a) Increasing Nres improves geometric fidelity, with rapidly diminishing Hausdorff error after ∼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.
Fig. 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.
Fig. 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).
Fig. 6 : Setpoint regulation simulation results for the time evolution of the end-effector (EE) position under the RRT* baseline collision avoidance strategy. The dotted lines represent the desired EE targets (i.e., the task encoded in the Control Lyapunov Function), the dashed lines the sequence of the setpoints provided by the RRT* planner that serve as intermediate goals for the low-level kinematic controller, and the solid lines the actual motion of the closed-loop system.
Fig. 7 : Experimental results. Sequence of stills showing the system evolution under: a) the closed-form CLF–CBF controller, b) a QP CLF controller. The target position is highlighted in red, and the obstacle is outlined in green. Consistently, the closed-form CLF–CBF controller avoids the obstacle entirely, whereas the pure CLF controller reaches the target but violates the safety constraints.
Fig. 8 : Experimental results. Comparison of target and obstacle distances for closed-form CLF-CBF controller, and a pure QP CLF controller during the hardware experiment.
Collision avoidance for robotic manipulators requires enforcing full-body safety constraints in high-dimensional configuration spaces. Control Barrier Function (CBF) based safety filters have proven effective in enabling safe behaviors, but enforcing the high number of constraints needed for safe manipulation leads to theoretic and computational challenges. This work presents a framework for full-body collision avoidance for manipulators in dynamic environments by leveraging 3D Poisson Safety Functions (PSFs). In particular, given environmental occupancy data, we sample the manipulator surface at a prescribed resolution and shrink free space via a Pontryagin difference according to this resolution. On this buffered domain, we synthesize a globally smooth CBF by solving Poisson's equation, yielding a single safety function for the entire environment. This safety function, evaluated at each sampled point, yields task-space CBF constraints enforced by a real-time safety filter via a multi-constraint quadratic program. We prove that keeping the sample points safe in the buffered region guarantees collision avoidance for the entire continuous robot surface. The framework is validated on a 7-degree-of-freedom manipulator in dynamic environments.
Meg Wilkinson, Gilbert Bahati, Ryan M. Bena +3
Computing and Mathematical Sciences, California Institute of Technology. · Mechanical and Civil Engineering, California Institute of Technology.
Safe navigation often relies on well-defined conditions based on the shape of robots and obstacles, and can be challenging when they have irregular geometries. While Control Barrier Functions (CBFs) offer an efficient mechanism to enforce safe set forward invariance, common shape surrogates (e.g., spheres or super-ellipsoids) either are overly conservative in unstructured scenes or require many local primitives, which inflates constraint counts and degrades real-time performance. In this paper, we introduce a novel geometry-aware Control Barrier Function (CBF) based on Bernstein-Polynomial Signed Distance Fields (BP-SDFs). It provides a unified way to represent the obstacles and robots, so as to represent the barrier function with a unified minimum distance. Benefiting from the differentiability of the Bernstein polynomials, one can easily enforce the control constraints in a closed loop. We validate the method's efficiency and performance to guarantee safety in single-robot navigation and heterogeneous multi-robot collision avoidance via simulations under different environments.
Siwon Jo, Yanze Zhang, Yupeng Yang +1
GRASP Laboratory, University of Pennsylvania, Philadelphia, PA, USA · Department of Computer Science, University of Illinois Chicago, Chicago, IL, USA · Department of Computer Science, University of North Carolina at Charlotte, Charlotte, NC, USA
Real-time motion generation for tendon-driven continuum robots requires accurate modeling of nonuniform bending and whole-body collision avoidance. This paper presents a unified actuation-space framework for planar multi-segment tendon-driven continuum robots. An energy-based variable-curvature model captures spatially varying tendon spacing and bending stiffness and provides analytical Jacobians for differential inverse kinematics and safety monitoring. A multipoint CBF-QP enforces backbone clearance under obstacle motion and actuation-velocity bounds, while its decision dimension depends only on the number of independently actuated segments. The model closely agrees with GVS references, with a maximum curvature error of 5.223×10−2m−1. Over 100 MuJoCo trials, the proposed method achieves collision-free success rates of 96% and 100% in static and dynamic scenarios, respectively, compared with approximately 60% and 80% without CBF constraints. Hole-traversal tests further demonstrate safe motion in constrained environments. With 600 backbone monitoring points, the mean control-step time is 6.66 ms, demonstrating real-time whole-body safe motion generation.