This study addresses an argument on the disconnection between the computational side of the robot navigation problem with the control problem including concerns on stability. We aim to constitute a framework that includes a novel approach of using quantizers for occupancy grids and vehicle control systems concurrently. This representation allows stability concerned with the navigation structure through input and output quantizers in the framework. We have given the theoretical proofs of qLQR in the sense of Lyapunov stability alongside with the implementation details. The experimental results demonstrate the effectiveness of the qLQR controller and quantizers in the framework with real-time data and offline simulations.