In this paper, an efficient motion planning approach with grid-based generalized Voronoi diagrams (G$ \mathbf{^2} $VD) is newly proposed for mobile robots. Different from existing approaches, the novelty of this work is twofold: 1) a new state lattice-based path searching approach is proposed, in which the search space is reduced to a novel Voronoi corridor to further improve the search efficiency; 2) an efficient quadratic programming-based path smoothing approach is presented, wherein the clearance to obstacles is considered to improve the path clearance of hard-constrained path smoothing approaches. We validate the efficiency and smoothness of our approach in various challenging simulation scenarios and outdoor environments. It is shown that the computational efficiency is improved by 17.1% in the path searching stage, and path smoothing with the proposed approach is 6.6 times faster than an advanced sparse-banded structure-based path smoothing approach and 53.3 times faster than the popular timed-elastic-band planner. A video showing outdoor navigation on our campus is available at https://youtu.be/iMXGthgvp58.

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