For safe and effective grasping in a dynamic environment, planning algorithms need real-time to deal with changing the target’s movement and obstacles. This paper proposes a new sequential Sense-Plan-Act (SeqSPA) dynamic grasping framework to generate a robot’s real-time and smooth grasping trajectory. Specifically, we cluster all stable grasps of the target, transform the clustering centers into pregrasps, and predict the future motion of the moving objects by using the observed values. The trajectory optimization algorithm constructing the approximative joint space gradient field can generate a smooth trajectory for a 6-DOF industrial robot arm within 2 ms. Our method generates trajectories for multiple pregrasps and selects the time-optimal trajectory for execution. Simulation comparison and actual experiments verify that our framework can immediately respond to environmental changes and efficiently find a grasping trajectory of the near-optimal time. The trajectory optimization algorithm in the framework can also be used alone to generate a real-time grasping trajectory when the prediction module cannot accurately predict the target motion.