This paper proposes a collision avoidance control strategy for constrained differential-drive robots moving in static but unknown obstacle scenarios. We assume that the robot is equipped with an on-board path planner providing a sequence of obstacle-free waypoints, and we design an ad-hoc constrained control strategy for ensuring absence of collisions and velocity constraints fulfillment. To this end, the nonlinear robot kinematics is redefined via a dynamic feedback linearization procedure, while a receding horizon control strategy is tailored to deal with time-varying state and input constraints. First, by considering the worst-case constraints realization, a conservative solution is offline determined to guarantee stability, recursive feasibility, and absence of collisions. Then, online, the tracking performance is significantly improved leveraging a non-conservative representation of the input constraints and set-theoretical containment conditions. Simulation results involving a differential-drive robot operating in a maze-like obstacle scenario are presented to show the effectiveness of the proposed solution.