Control Barrier Function Quadratic Programs (CBF-QPs) provide formal safety guarantees for general control-affine systems and accommodate actuation constraints, but do not inherently guarantee liveness, i.e., sustained progress toward and convergence to a goal. They may admit undesirable equilibria or dysfunctional circulation that impede goal convergence. In contrast, Modulation of Dynamical Systems (Mod-DS) promotes liveness through geometric reshaping of nominal vector fields, but is primarily suited to fully actuated systems and does not directly accommodate general actuation constraints. We establish theoretical connections between these paradigms, showing that normal Mod-DS is a special case of CBF-QP and that reference Mod-DS is linked to CBF-QP through a shared formulation. These connections motivate the Modulated CBF-QP (MCBF-QP) framework, which incorporates geometric guidance from Mod-DS into CBF-QPs to enhance liveness while retaining formal safety and control-affine generality. We develop reference MCBF-QP, which reduces undesirable equilibria and dysfunctional circulation for star-shaped obstacles, and on-manifold MCBF-QP, which eliminates undesirable equilibria and reduces dysfunctional circulation for arbitrarily shaped concave obstacles. Although no Lyapunov function is explicitly constructed, we prove that the tangent guidance constraint introduced into the CBF-QP formulation implicitly enforces the decrease condition of a hybrid Lyapunov function, yielding asymptotic goal convergence and thus liveness under the stated assumptions. Extensive simulation and hardware experiments demonstrate safe and live obstacle avoidance with the proposed MCBF-QP framework in static and dynamic environments.

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