## 6 Conclusions  
In this study, the tire-road contact mechanism was studied theoretically, and numerical analyses were conducted to develop the tire contact force equations. The methodology for identifying the tire-road contact force by combining the derived equations and computer vision techniques was verified with field experiments on passenger cars and trucks. The conclusions can be summarized as follows:  
1. The theoretical equation for the tire contact forces was developed based on the tire-road contact mechanics. The relationship between the contact force and the tire inflation pressure, tire vertical deflection, and other physical parameters was established for the vehicle weight identification.
2. Numerical 3D models for different types of tires were established to analyze the deformation process of the tire. Simulation results show that the tire footprint approximates different shapes at different deformation stages. For passenger cars, the tire footprint is not a constant oval shape as assumed. It is actually a rhombus when the contact width is smaller than the tread width, and a hexagon when the contact width reaches the tread width. As for the tires of heavy trucks, the footprint is an oval when the contact width is smaller than the tread width, and a rectangle with two half circles on each side afterward.
3. The vision-based vehicle weight identification method was established by combining the derived theoretical equations and computer vision techniques. The tire vertical deflection, inflation pressure, and tire specification can be extracted from the tire image using computer vision techniques, which are then input into the theoretical equation to calculate the contact force of each tire, and finally, estimate the vehicle weight by summing the contact forces of all the tires.
4. Two passenger cars and two trucks with tires of different specifications were used in field tests to verify the accuracy of the proposed method. The results showed that the identified weights were in good agreement with the actual measured weights, which proves the performance of the proposed method.  
The tire force-deformation equation established in this study is based on the proposed shape of the tire footprint. In the simulation analysis and field test, the established equation has better accuracy compared with previous methods that assume the footprint as simple shapes (oval, rectangular, and combination shape). The proposed vision-based vehicle weight identification method has the advantages of low cost, convenience, and high efficiency compared with the static weighing and WIM methods. Further studies are still needed to improve the practicality and accuracy of the method, including the influence of low inflation pressure of tires, tire loss, load eccentricity in the DWS, and other factors that impact on the mechanical properties of tires. Additionally, computer vision may be affected by various outdoor conditions in the field, such as poor lighting, shadowing, and camera vibration; these will be addressed in the next work.  
Conceptualization, X.K., T.W. and J.Z. ([PERSON]); methodology, X.K. and T.W.; investigation and result analysis, [PERSON] ([PERSON]) and T.W.; writing--original draft, T.W.; supervision and writing--review and editing, X.K., [PERSON] ([PERSON]) and L.D.; project administration, L.D., [PERSON] ([PERSON]), Y.C. and S.X. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research work was fully supported by the National Natural Science Foundation of China (Project No. 52008160), Excellent Young Scholars Fund of Hunan Province (Project No.: 2021 JJ20015), the Postgraduate Scientific Research Innovation Project of Hunan Province (Project No.: CX20200405).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.