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

The paper proves that, for the single-integrator with V=U_att and B=U_rep, choosing σ(x)=||b(x)||^2 and Γ(x)=||d(x)||^2+α(h(x))−d(x)u_nom yields an RCBF-QP controller identical to the classic APF law: −Fatt when ρ≥ρ0 and −Fatt−Frep when ρ<ρ0 (Lemma 4, Lemma 5, Theorem 1) . The candidate solution reproduces exactly this construction: it derives u_CLF=−Fatt from the CLF-QP and then, using the one-constraint QP closed form with the stated Γ, gets φ=||d||^2 and u_CBF=u_nom−d^T, i.e., −Fatt−Frep in the repulsive region, matching the APF controller. The only notable gap is that the model omits the paper’s explicit requirement that Γ be positive semidefinite (ensured by a sufficiently large α), but this does not change the computed controller. Aside from a minor constant-factor typo in the paper’s α(h) expression, both arguments align and are correct in substance .