A Comparative Study of Artificial Potential Fields and Reciprocal Control Barrier Function-based Safety Filters

TL;DR

The paper establishes a formal equivalence between artificial potential field (APF) controllers and reciprocal control barrier function quadratic program (RCBF-QP) safety filters by introducing tightened versions of CLFs and RCBFs (T-CLFs and T-RCBFs). By using an attractive potential as a tightened CLF and a repulsive potential as a tightened RCBF, APF-based controllers can be recovered within the RCBF-QP framework, and with suitable auxiliary-function choices, this equivalence generalizes beyond fixed formulations. The authors prove the equivalence and show how to construct a generalized APF-based controller that combines APF and RCBF-QP safety-filter ideas. A collision-avoidance example demonstrates the practical connection and highlights how different choices of the auxiliary functions affect stability, safety, and trajectory smoothness. This work provides a unifying perspective for safe, stabilizing motion planning in robotic systems by bridging classic APF methods and modern RCBF-QP safety filters, with implications for robust, efficient autonomous navigation.

Abstract

In this paper, we demonstrate that controllers designed by artificial potential fields (APFs) can be derived from reciprocal control barrier function quadratic program (RCBF-QP) safety filters. By integrating APFs within the RCBF-QP framework, we explicitly establish the relationship between these two approaches. Specifically, we first introduce the concepts of tightened control Lyapunov functions (T-CLFs) and tightened reciprocal control barrier functions (T-RCBFs), each of which incorporates a flexible auxiliary function. We then utilize an attractive potential field as a T-CLF to guide the nominal controller design, and a repulsive potential field as a T-RCBF to formulate an RCBF-QP safety filter. With appropriately chosen auxiliary functions, we show that controllers designed by APFs and those derived by RCBF-QP safety filters are equivalent. Based on this insight, we further generalize the APF-based controllers (equivalently, RCBF-QP safety filter-based controllers) to more general scenarios without restricting the choice of auxiliary functions. Finally, we present a collision avoidance example to clearly illustrate the connection and equivalence between the two methods.

Figures (2)

• Figure 1: APF Operational Mechanism: The shaded red area represents an obstacle with radius $r$, where the solid red circle denotes its boundary. The light blue shaded area illustrates regions where $\rho(\mathbf{x})<\rho_0$, with the solid blue circle indicating the boundary $\rho(\mathbf{x}) =\rho_0$. The green star denotes the goal position of a navigation task, while $\Omega_{\mathrm{c}}$ (black dashed circle) denotes the super level set of the attractive potential field $\mathrm{U}_{\mathrm{att}}(\mathbf{x})$. The red solid point signifies the robot, showcasing the force exerted on it when within the region where $\rho(\mathbf{x})<\rho_0$. Conversely, the magenta solid point depicts the attractive force exerted on the robot when it is situated within the region where $\rho(\mathbf{x})\geq \rho_0$.
• Figure 2: Different generalized APF-based controllers with different $\Gamma(\mathbf{x})$ in three cases.

Theorems & Definitions (21)

• Definition 1
• Proposition 1
• Definition 2
• Definition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof