Model mismatches can cause multi-dimensional uncertainties for the receding horizon control strategies of automated vehicles (AVs). The uncertainties may lead to potentially hazardous behaviors when the AV tracks ideal trajectories that are individually optimized by the AV's planning layer. To address this issue, this study proposes a safe motion planning and control (SMPAC) framework for AVs. For the control layer, a dynamic model including multi-dimensional uncertainties is established. A zonotopic tube-based robust model predictive control scheme is proposed to constrain the uncertain system in a bounded minimum robust positive invariant set. A flexible tube with varying cross-sections is constructed to reduce the controller conservatism. For the planning layer, a concept of safety sets, representing the geometric boundaries of the ego vehicle and obstacles under uncertainties, is proposed. The safety sets provide the basis for the subsequent evaluation and ranking of the generated trajectories. An efficient collision avoidance algorithm decides the desired trajectory through the intersection detection of the safety sets between the ego vehicle and obstacles. A numerical simulation and hardware-in-the-loop experiment validate the effectiveness and real-time performance of the SMPAC. The result of two driving scenarios indicates that the SMPAC can guarantee the safety of automated driving under multi-dimensional uncertainties.