Almost all known robots fear impacts. Unlike humans , robots keep guarded motions to near zero-velocity prior to establishing contacts with their surroundings. This significantly slows down robotic tasks involving physical interaction. Two main ingredients are necessary to remedy this limitation: impact-friendly hardware design, and impact-friendly controllers. Our work focuses on the controller aspect. Task-space controllers formulated as quadratic programming (QP) are widely used in robotics to generate modular and reactive motion for a large range of task specifications under various constraints. We explicitly introduce discrete impact dynamics model into the QP-based controllers to generate robot motions that are robust to impact-induced state jumps in the joint velocities and joint torques. Our simulations, validate that our proposed impact-friendly QP controller is robust to contact impacts, shall they be expected or not. Therefore, we can exploit it for establishing contacts with high velocities, and explicitly generate task-purpose impulsive forces.