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

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.