## 6 Conclusions  
This paper presents a novel two-step path planning method for smart HDS systems in warehousing logistics. The proposed method first topologically represents the HDS layouts into mesh-tree grid maps and then introduces the Lattice (mesh-grid)-Hopping (jumping) algorithm for path planning. During representation, our method accurately captures the spatial and physical information, as well as the traffic constraints of the HDS layout, through a hierarchical 'backbone-branch' tree structure. The main track, the aisle where cargo is shipped along to the storage area from the entrance, is the backbone, and subtracks, where the shipping cargos are guided to the specific storage position, are branches. We designed a main track priority strategy and jump mechanism in our algorithm based on this 'backbone-branch' tree structure. By implementing these two designs, strong feasibility and optimality with low computational costs can be obtained without obstacles. Given obstacles in both the main track and the sub-tracks, the proposed algorithm still displays rigid stability and robustness. Both scenarios are theoretically proven. Practically, numerical and case studies were undertaken to check the usability of our method. Three renowned benchmark algorithms, A-star (\(l_{1}\)), BFS (\(b_{w}^{*}=0.7\)), and RRT-star (\(\eta=3\)), after being properly modified according to the HDS system, were adopted for performance comparison. Two scenarios, without obstacles and with randomly settled obstacles, were considered. In the numerical study, three map scales were simulated to compare different types of HDS systems in modern warehousing logistics. The case study was conducted in a real warehousing environment from Huade Co., Ltd., our cooperator. A time window of 120 s was assigned to check the operational feasibility for industrial production. In both the numerical and case studies, our proposed algorithm performs best, with the lowest computational cost and the highest feasibility when obstacles and time windows are given. These results exhibit the algorithm's exceptional efficiency, stability, and robustness. Although the feasibility of the proposed algorithm is not sufficient to guarantee direct production deployment yet, our method still significantly enhances the likelihood of planning viable shipping routes under production settings when operational challenges cannot be neglected. In a word, our method displays outstanding performance and solid practical usage in modern warehousing logistics, and it could be a potential solution for difficulties in modern HDS systems. For future work, our method can serve as a foundation for developing more efficient and robust algorithms in advanced HDS systems. Future algorithm development could \begin{table}
\begin{tabular}{c c c c c} \hline
**HUADE Co.,** & & & & \\
**Ltd.** & & & & \\ \hline std\({}_{\text{ship}}\) (s) & 0.4 & 0.9 & 1.3 & 3.1 \\ \hline CI & \((15.20,15.40)\) & \((16.17,16.63)\) & \((17.46,18.14)\) & \((18.40,20.00)\) \\ \hline \end{tabular}
\end{table} 
Table 3: _Cont._integrate machine learning algorithms, e.g., reinforcement learning, to optimize time window allocation and enhance robustness when larger HDS systems are considered. With multiple shipping equipment in consideration, our method can be extended to design smart multi-vehicle cooperative transportation systems. Because lower computational cost usually implies low energy consumption, the proposed method can also be applied for sustainable equipment development for modern warehousing transportation. Conceptualization, S.Z. and Z.G. ; methodology, S.Z. and Z.G. ; software, S.Z. ; validation, Y.S. and Z.C. ; formal analysis, S.Z. and Z.G. ; investigation, S.Z. ; resources, G.C. and Y.C. ; data curation, X.X. ; writing--original draft preparation, S.Z. ; writing--review and editing, Z.G. and X.X. ; visualization, X.X. ; supervision, Z.G. and X.X. ; project administration, Y.S. ; funding acquisition, Z.G. All authors have read and agreed to the published version of the manuscript.