Robotic manipulators are essential for precise industrial pick-and-place operations, yet planning collision-free trajectories in dynamic environments remains challenging due to uncertainties such as sensor noise and time-varying delays. Conventional control methods often fail under these conditions, motivating the development of Robust MPC (RMPC) strategies with constraint tightening. In this paper, we propose a novel RMPC framework that integrates phase-based nominal control with a robust safety mode, allowing smooth transitions between safe and nominal operations. Our approach dynamically adjusts constraints based on real-time predictions of moving obstacles\textemdash whether human, robot, or other dynamic objects\textemdash thus ensuring continuous, collision-free operation. Simulation studies demonstrate that our controller improves both motion naturalness and safety, achieving faster task completion than conventional methods.