## 3 Research Methods
### Tracking and Speed Measurement Central-ByteTrack Module
#### 3.2.1 Transformation Between the World Coordinate System and the Image Coordinate System  
In the task of detecting the velocity of drilling debris, it is essential to accurately map the motion trajectory of the debris presented in the image to the physical space in order to achieve precise estimation of its true velocity. To address this, this paper establishes a mapping relationship between the image coordinate system and the real-world (world) coordinate system. The image coordinate system represents the two-dimensional pixel position of the target on the camera's sensing plane, while the world coordinate system describes the physical position of the target in the actual three-dimensional space. Considering the stability of the unmanned drilling machine operation environment and the controllability of camera parameters, this paper utilizes fixed industrial cameras, along with parameters such as field of view, camera height, lens focal length, and image resolution, to map pixel points in the image to physical space. Specifically, by calibrating the size and position of a reference object, a conversion relationship between pixels and real-world distance is established, allowing for accurate conversion of debris displacement in the image to real-world physical displacement. Figure 3 illustrates the process of transforming between different coordinate systems, visually demonstrating how a three-dimensional spatial point is mapped to a two-dimensional image plane through various coordinate systems. Figure 3: Visual representation of coordinate transformation from 3D world space to 2D image space, supporting pixel-to-physical mapping. As shown in Figure 4, during the actual camera calibration process, the range of the drilling machine's working surface is relatively limited, and the vertical height difference can be considered negligible. As a result, the drilling debris ejection surface can be approximated as a plane, with minimal error introduced by this approximation. Therefore, the core task primarily focuses on the two-dimensional coordinate transformation between the image pixel coordinate system and the working surface coordinate system. As shown in Figure 5, this paper systematically outlines the transformation relationships between the four coordinate systems and rigorously derives the transformation process using mathematical equations, ultimately establishing the mapping relationship between the image pixel coordinate system and the world coordinate system. The specific transformation equations are as follows:  
(1) The translation relationship between the physical image coordinate system and the image pixel coordinate system. \[u=\frac{X}{d_{x}}+u_{0} \tag{1}\]  
\[v=\frac{X}{d_{y}}+v_{0} \tag{2}\]  
In the equations, \((u_{0},v_{0})\) denotes the origin of the image pixel coordinate system, while \(d_{x}\) and \(d_{y}\) represent the size of each pixel along the \(x\) and \(y\) axes in the pixel plane, and \(X\) and \(Y\) correspond to the values of the point in the image physical coordinate system. (2) The relationship between the image physical coordinate system, the camera coordinate system, and the world coordinate system. \[Z_{p} \begin{bmatrix}u\\ v\\ 1\end{bmatrix} = \begin{bmatrix}\frac{f}{dx}&0&u_{0}\\ 0&\frac{f}{dy}&v_{0}\\ 0&0&1\end{bmatrix}\begin{bmatrix}X_{p}\\ Y_{p}\\ Z_{p}\end{bmatrix} =K \begin{bmatrix}X_{p}\\ Y_{p}\\ Z_{p}\end{bmatrix}  \tag{3}\]  
\[ \begin{bmatrix}X_{p}\\ Y_{p}\\ Z_{p}\end{bmatrix} =R \begin{bmatrix}X_{w}\\ Y_{w}\\ Z_{w}\end{bmatrix} +t \tag{4}\]  
In the equations, the coordinates of point \(p\) in the camera coordinate system are \((X_{p},Y_{p},Z_{p})^{T}\), in the image physical coordinate system \((X,Y)^{T}\), with the focal length being  
Figure 4: Schematic diagram of the conversion method between image pixel coordinate system and work surface coordinate system. (The red frame is the image plane, and T1, T2, and T3 are the position simulations of the same drill cuttings at different times.) \(f\).