This paper presents a unified framework for trajectory tracking and static obstacle avoidance of differential-drive mobile robots. A nominal exact-decoupling controller achieves pure exponential longitudinal convergence and globally avoids angular singularities via a stereographic projection; its cascade structure is proved globally asymptotically stable, and a time-scale separation analysis yields an explicit lateral settling-time constant, enabling prescribed convergence design. To handle obstacles, an artificial potential field (APF) correction is superimposed in a structure-preserving manner: the cancellation term responsible for longitudinal decoupling is retained, so the longitudinal error dynamics remain decoupled and are only additively perturbed during avoidance. The perturbed closed loop is modelled as a cascade driven by bounded inputs, and a bounded-input bounded-state (BIBS) analysis guarantees that tracking errors stay bounded during avoidance and recover exponentially with the nominal time constants once the influence zone is left. Quantitative error bounds and a recovery-time estimate enable an explicit safety–performance trade-off. Simulations on circular and figure-eight trajectories with single and multiple static obstacles confirm the predicted recovery behaviour and reduce the total integral squared error (ISE) by factors of 7.4 and 20 relative to a CBF-QP safety filter, with lateral recovery about twice as fast as that of a Kanayama controller augmented by APF.