© 2020 IEEE.Robust and safe feedback motion planning and navigation is a critical task for autonomous mobile robotic systems considering the highly dynamic and uncertain nature scenarios of modern applications. For these reasons motion planning and navigation algorithms that have deep roots in feedback control theory has been at the center stage of this domain recently. However, the vast majority of such policies still rely on the idea that a motion planner first generates a set of open-loop possibly time-dependent trajectories, and then a set of feedback control policies track these trajectories in closed-loop while providing some error bounds and guarantees around these trajectories. In contrast to trajectory-based approaches, some researchers developed feedback motion planning strategies based on connected obstacle-free regions, where the task of the local control policies is to drive the robot(s) in between these particular connected regions. In this paper, we propose a feedback motion planning algorithm based on sparse random neighborhood graphs and constrained nonlinear Model Predictive Control (MPC). The algorithm first generates a sparse neighborhood graph as a set of connected simple rectangular regions. After that, during navigation, an MPC based online feedback control policy funnels the robot with nonlinear dynamics from one rectangle to the other in the network, ensuring no constraint violation on state and input variables occurs with guaranteed stability. In this framework, we can drive the robot to any goal location provided that the connected region network covers both the initial condition and the goal position. We demonstrate the effectiveness and validity of the algorithm on simulation studies.