## 5 Conclusions  
In this study, the theory of the APF-Dijkstra fusion algorithm with volume and steering model constraints is proposed to improve the algorithm in response to the local oscillation phenomenon of the APF algorithm in the face of static U-shaped obstacles, the phenomenon of sharp corners of the path when avoiding obstacles, and the lack of consideration of corner constraints, curvature limitations, and the volume of carriers in the planning of the path. This fusion algorithm theory not only avoids static and dynamic obstacles on the basis of ensuring the safety of the path but also smoothly avoids U-shaped obstacles compared to the local oscillation problem that occurs in the A* + DWA algorithm and the traditional APF algorithm. In response to most of the literature, local planning is carried out first, and then smoothing (post-processing) is performed on it. This paper adopts the three-point arc smoothing method on the basis of the steering model, and the smoothing process is carried out in real time to ensure the smoothness of the paths, and the paths planned by using this theory can be used as the reference paths for the subsequent tracking and control, but the algorithm also suffers from the following defects:  
1. Although the steering angles of the fusion algorithms all comply with the limitations, there are cases where the turning angles are too large.
2. Although dynamic obstacles are considered, the speed and future position of dynamic obstacles are not taken into account.  
This theory can be considered for experiments in warehouse logistics vehicles. Since there are usually many goods (static obstacles), irregular obstacles (U-shaped obstacles), narrow driving spaces, and randomly appearing people (dynamic obstacles) in warehouse depots, traditional logistics vehicles may be difficult to cope with flexibly in such environments. The improvement in this paper can ensure the safety and smoothness of the path by real-time calculation of the obstacle avoidance path and optimization of the moving trajectory so that the logistics vehicle can navigate smoothly in the complex environment. In future research, we can consider how to plan the optimal path without affecting the local obstacle avoidance on the basis of maximizing the step size, and at the same time, the speed factor needs to be taken into consideration, and we should focus on how to introduce the speed information of the obstacles and then optimize the dynamic path planning algorithm.  
Conceptualization, X.W. and G.L.; Methodology, X.W. and Z.B.; Software, X.W.; Validation, X.W. and G.L.; Formal analysis, X.W.; Investigation, X.W. and Z.B.; Resources, G.L. and X.W.; Data curation, X.W.; Writing--original draft preparation, X.W.; Writing--review and editing, X.W. and G.L.; Visualization, X.W. and G.L.; Supervision, X.W. and Z.B.; Project administration, G.L.; Funding acquisition, G.L. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Innovation Fund Project of Industry, University and Research of Chinese Universities (2024 HT007), the Key Research Project of Liaoning Provincial Department of Education (JYTZD2023081) and Overseas Cultivation Project for Higher Education Institutions in Liaoning Province(2018 LNCXGJWPY-YB014).  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.