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Comparative Analysis of Control Barrier Functions and Artificial Potential Fields for Obstacle Avoidance
Andrew Singletary, Karl Klingebiel, Joseph Bourne, Andrew Browning, Phil Tokumaru, Aaron Ames
TL;DR
The paper asks how APFs and CBFs compare for real-time obstacle avoidance. It proves that APFs are a special case of CBFs, then compares both methods in examples and quadrotor simulations and experiments. Across these comparisons, CBFs outperform APFs.
Problem
The paper examines how established artificial potential fields compare with newer control barrier functions for real-time obstacle avoidance.
Method
The paper combines a theoretical APF-to-CBF construction with obstacle-avoidance comparisons in simple examples and quadrotor simulations and hardware experiments.
Results
CBFs outperform APFs in obstacle avoidance, including hardware tests where APFs show more oscillations and fail to reach the goal through a narrow door.
Takeaways & Limitations
APFs can be formulated as CBFs, while the resulting CBF framework provides smoother behavior and supports safety-oriented control beyond the original potential-field formulation.
Abstract
from arXiv · showhide
Artificial potential fields (APFs) and their variants have been a staple for collision avoidance of mobile robots and manipulators for almost 40 years. Its model-independent nature, ease of implementation, and real-time performance have played a large role in its continued success over the years. Control barrier functions (CBFs), on the other hand, are a more recent development, commonly used to guarantee safety for nonlinear systems in real-time in the form of a filter on a nominal controller. In this paper, we address the connections between APFs and CBFs. At a theoretic level, we prove that APFs are a special case of CBFs: given a APF one obtains a CBFs, while the converse is not true. Additionally, we prove that CBFs obtained from APFs have additional beneficial properties and can be applied to nonlinear systems. Practically, we compare the performance of APFs and CBFs in the context of obstacle avoidance on simple illustrative examples and for a quadrotor, both in simulation and on hardware using onboard sensing. These comparisons demonstrate that CBFs outperform APFs.
I. INTRODUCTION
The paper compares established artificial potential fields with newer control barrier functions for real-time robot obstacle avoidance. It combines theoretical analysis with simulation and quadrotor experiments, concluding that CBFs provide smoother, minimally invasive behavior while guaranteeing avoidance.
- Real-time obstacle avoidance must preserve safety while minimally altering robot behavior away from potential collisions.
- APFs are established, simple, and easy to implement, with extensive use in real-time obstacle avoidance.
- CBFs provide safety guarantees for nonlinear systems through optimization-based controllers and support real-time operation.
- The paper asks how control barrier functions compare with artificial potential fields.
- The study combines theoretical analysis, simple examples, quadrotor simulations, and onboard-sensing experiments.
- CBFs outperform APFs by providing smooth, minimally invasive behavior while guaranteeing obstacle avoidance.
II. BACKGROUND & MOTIVATION
The paper introduces APFs and CBFs through a single-integrator obstacle-avoidance example, motivating CBFs as a generalization of potential fields.
- The section compares APFs and CBFs for obstacle avoidance with a single integrator in the plane.
- The comparison motivates treating control barrier functions as a generalization of potential fields.
A. Artificial Potential Fields
The APF controller combines attraction toward a goal with repulsion from obstacles and applies the resulting gradient as the velocity input. Its behavior depends strongly on parameter tuning, with poor choices causing local minima or oscillations.
- The single-integrator model uses position x and velocity v, with velocity serving as the control input.
- APFs combine an attractive goal potential with repulsive obstacle potentials, then use the gradient to synthesize control.
- The repulsive potential is active within a region of influence and is zero outside it, allowing attraction to dominate at larger distances.
- The example models two obstacles and requires the robot to remain at least 0.5 meters away from each.
- ρ0 = 0.5 causes a local minimum, while ρ0 = 0.1 produces oscillations; ρ0 values of 1 and 0.25 work well.
B. Motivating Control Barrier Functions
CBFs encode obstacle-free regions as safe sets and enforce safety through an instantaneous derivative condition. A quadratic-program controller minimizes deviation from a desired velocity while satisfying that condition, avoiding the APF example’s oscillations.
- A safety function h defines a safe set, interpreted here as the complement of the obstacles.
- Because positivity of h alone does not ensure safety, CBFs impose a derivative condition that can be checked instantaneously.
- Satisfying the CBF inequality makes the safe set forward invariant for the closed-loop system.
- The controller is obtained through a quadratic program that minimizes deviation from the desired velocity while enforcing safety.
- Unlike APFs, CBF synthesis optimizes directly over controllers satisfying the safety constraint rather than using gradients.
- For all tested α values, the CBF robot completes the mission without oscillations and modifies nominal goal trajectories less than potential fields.
III. POTENTIAL FIELDS AS CONTROL BARRIER FUNCTIONS
The paper shows that artificial potential fields are a specific instance of control barrier functions, and that this transformation adds useful properties while generalizing potential-field methods.
- Potential fields are a specific instance of control barrier functions.
A. Control Barrier Functions
Control barrier functions provide conditions for safety of nonlinear systems by defining a safe set and enforcing a derivative constraint through a controller or safety filter.
- CBFs apply to general nonlinear control systems with state x, input u, and dynamics defined by f and g.The dynamics functions f and g are assumed to be Lipschitz continuous.
- A continuously differentiable function h defines the safe set S through positive values in its interior.The interior is Int(S) = {x ∈ R^n : h(x) > 0}.
- A function h is a CBF when an admissible input exists that satisfies the derivative-based safety condition for every state in S.The condition uses an extended class K function α and requires a nonzero gradient on the boundary.
- Controllers satisfying the CBF condition make S forward invariant, providing a sufficient and necessary condition for safety relative to S.
- A quadratic program can enforce the affine CBF constraint as a real-time safety filter.The explicit solution adds a safety correction only when the nominal controller would violate the condition.
B. Potential Fields as a Special Case of CBFs
The paper constructs CBFs from repulsive potential fields, using attraction as the nominal motion and barrier enforcement for safety. This yields safe, stable, minimally intervening behavior and extends potential fields to nonlinear systems.
- B. Potential Fields as a Special Case of CBFs: An artificial potential field combines attractive and repulsive potentials into one controller based on the negative gradient of their sum.
- B. Potential Fields as a Special Case of CBFs: The CBF paradigm includes potential fields as a special case by using attraction as the desired velocity and repulsion as the barrier function.
- B. Potential Fields as a Special Case of CBFs: The constructed CBF yields a forward-invariant and asymptotically stable safe set for the single-integrator system.
- B. Potential Fields as a Special Case of CBFs: The safe set approaches the complement of the obstacles as δ decreases, and the CBF remains defined while stabilizing trajectories back to the safe set from outside it.
- B. Potential Fields as a Special Case of CBFs: CBF synthesis combines attraction and repulsion pointwise optimally, minimally altering the desired velocity only when safety requires it.
- B. Potential Fields as a Special Case of CBFs: Compared with the original APF, the APF-CBF path gets closer to obstacles with fewer oscillations and converges faster toward the desired path after passing them.The example uses K_rep = K_att = α = 1.
- B. Potential Fields as a Special Case of CBFs: The APF-to-CBF connection extends potential fields to nonlinear control systems when the repulsive potential satisfies the CBF condition.The resulting controller renders the set outside the minimum obstacle distance forward invariant.
IV. APPLICATION TO QUADROTORS
The quadrotor evaluation compares APF and CBF velocity-based controllers on systems with nontrivial dynamics, using simulation scenarios designed to test safety, obstacle passage, and oscillations. CBFs maintained smoother, safer behavior and approached obstacles more closely in the reported tests.
- Application to Quadrotors: The study applies APF and CBF velocity-based controllers to quadrotors with nontrivial dynamics.The comparison uses high-fidelity simulation with ArduPilot SITL, ROS velocity commands, and simulated LIDAR data.
- Controller Design: The CBF controller used D_obs = 0.3 as the minimum obstacle distance and α = 1 without tuning to improve the results.The barrier function uses simulated laser-scan points as obstacle locations.
- Simulation Scenarios: Five scenarios test obstacle avoidance, including offset obstacles, a strongly obstructing obstacle, narrow doorways, and a wall blocking the goal.The tests increase in difficulty and include a wall that the drone cannot pass through.
- Simulation Results: CBF velocity control produced no oscillations, whereas APF oscillations occurred when the drone could not reach the goal, including the wall and 0.7 m doorway cases.The comparison is based on the five simulation tests and their recorded paths and velocities.
- Simulation Results: CBFs approached obstacles more closely while remaining safe because of their pointwise optimality.The reported comparison attributes closer obstacle passage to the CBF controller’s pointwise optimization.
V. EXPERIMENTAL RESULTS
The hardware experiments reproduce the five quadrotor scenarios using onboard localization and obstacle sensing. CBF results remained similar to simulation, while APFs showed more oscillations and failed in the narrow-door case.
- Experimental Setup: The hardware comparison uses onboard localization and obstacle detection with the APF and CBF controllers implemented as in simulation.The quadrotor uses a RealSense camera, Hokuyo LIDAR, Cartographer SLAM, and an onboard computer.
- Experimental Setup: The hardware tests reproduce the same five scenarios used in simulation, with the drone additionally yawing 180° and returning along the path.The altered return path increases the amount of data collected for analysis.
- Experimental Results: CBF hardware results were similar to simulation, whereas APFs exhibited significantly more oscillations in hardware.The reported difference suggests greater robustness of the CBF implementation to model uncertainty and noise.
- Experimental Results: APFs failed to reach the goal through the narrow door, while CBFs succeeded and outperformed APFs on hardware.The paper links the APF failure to the need for retuning under differences between simulation and reality.
- Manual Flight: During manual flight toward obstacles, the CBF prevented crashes and maintained safety near a wall.The manual test used sporadic user inputs and a more complicated obstacle course.
VI. CONCLUSIONS
The paper concludes that CBFs are a viable and arguably improved alternative to APFs for real-time obstacle avoidance. It reports smoother behavior, untuned quadrotor implementation, and better performance than an expertly tuned APF system, while identifying model-free guarantees as future work.
- Conclusion: CBFs provide a viable and arguably improved alternative to APFs for real-time obstacle avoidance.The conclusion frames this as the paper’s overall assessment.
- Conclusion: APFs can be formulated as CBFs, and the resulting behavior is smoother than the APF alone.The conclusion summarizes the theoretical translation and its observed behavioral benefit.
- Conclusion: CBFs were implemented on a quadrotor in simulation and hardware with no tuning, outperforming an expertly tuned APF-based collision-avoidance system.The comparison covers both simulated and physical quadrotor experiments.
- Future Work: Future work extends the model-based formal guarantees to the velocity-based model-free controller used in practice.The conclusion also calls for further hardware demonstrations on dynamic robotic systems.