Model Predictive Control (MPC) has shown the great performance of target optimization and constraint satisfaction. However, the heavy computation of the Optimal Control Problem (OCP) at each triggering instant brings the serious delay from state sampling to the control signals, which limits the applications of MPC in resource-limited robot manipulator systems over complicated tasks. In this paper, we propose a novel robust tube-based smooth-MPC strategy for nonlinear robot manipulator planning systems with disturbances and constraints. Based on piecewise linearization and state prediction, our control strategy improves the smoothness and optimizes the delay of the control process. By deducing the deviation of the real system states and the nominal system states, we can predict the next real state set at the current instant. And by using this state set as the initial condition, we can solve the next OCP ahead and store the optimal controls based on the nominal system states, which eliminates the delay. Furthermore, we linearize the nonlinear system with a given upper bound of error, reducing the complexity of the OCP and improving the response speed. Based on the theoretical framework of tube MPC, we prove that the control strategy is recursively feasible and closed-loop stable with the constraints and disturbances. Numerical simulations have verified the efficacy of the designed approach compared with the conventional MPC.