# PDDL Task Planning for Tunnel Spraying Based on a Multivariate Coating Accumulation Model
## 2 Methods
### Operation Parameter Planning Based on Multi-Variable Spraying Layer Accumulation Model
#### 2.2.2 Task Layer Parameter Planning  
The spray distance \(H\) for the pit-filling layer is the distance from the nozzle of the spray gun to the tunnel surface. To ensure the density and quality of the sprayed concrete, engineering experience suggests selecting \(H\approx 1.5\) m.  
For the pit-filling layer, the positions of the pits are determined based on an even distribution of explosion-fragmented rock surfaces, treating individual pits or connected pits as independent. The selection of operating points for spraying is related to the size of the pit openings. When the distance \(H_{1}\) between the spray gun and the pit meets the static spray conditions, the approximate spray coverage radius is \(R=H_{1}KQ^{a_{1}}P^{a_{2}}\). For pits with an opening radius \(S_{A_{1},A_{2}}/2\) smaller than the spray coverage radius \(R\), the operating point is selected as the position directly above the center of the largest superimposed pit. This is marked as \(U_{1}\) (\(X\),\(Y\),\(Z\)), and its coordinates are determined as shown in the figure. If the opening radius of a pit \(S_{A_{1},A_{2}}/2\) is greater than the spray coverage radius \(R\), the connected region \(E\) of the pit will require multiple operating points (\(U_{1}\), \(U_{2}\),..., \(U_{i}\)) within \(E\) to ensure full coverage of the spray area, as shown in Figure 5. If any operating point \(U_{i+1}\
otin E\), the selection of operating points for the pit-filling layer will stop. The station point is the position of the spray gun during the spraying task, aligned with the normal direction of the operating point \(V_{i}\). Its coordinates \(V(X\),\(Y\),\(Z)\) are determined based on the spray distance \(H_{1}\), the over-excavation amount \(\Delta\) at the operating point, the spray angle \(\theta\), and the coordinates of the operating point, as shown in Figure 6. Thus, the spray angle \(\theta\) is the angle between the normal direction of the station point and the \(Y\)-axis of the tunnel coordinate system, where:  
\[\cos\theta=\frac{U_{y}}{\sqrt{U_{y}^{2}+U_{z}^{2}}} \tag{22}\]  
\[\sin\theta=\frac{U_{z}}{\sqrt{U_{y}^{2}+U_{z}^{2}}} \tag{23}\]  
The station point and the operating point lie in the \(YOZ\) plane, so the \(X\)-coordinate of the station point is the same as the \(X\)-coordinate of the operating point. The coordinates of the station point \(V(X\),\(Y\),\(Z)\) are calculated as follows:  
\[V_{i}=U_{i}-H_{1}\left[ \begin{array}{c}0\\ \cos\theta\\ \sin\theta\end{array} \right] \tag{24}\]  
Figure 5: Schematic diagram of pit-filling operation point planning. Based on the static spraying model, the station spraying time \(t\) for the pit-filling layer with the planned spraying thickness \(T_{1}\) is calculated as:  
\[t=\frac{\pi K^{2}T_{1}H_{1}{}^{2}P^{2a_{2}}}{Q^{1-2a_{1}}\beta} \tag{25}\]  
The spray distance \(H_{2}\) for the filling layer refers to the distance from the nozzle of the spray gun to the designed contour. To ensure the density and quality of the sprayed concrete, the spray distance \(H_{2}\) is selected as 1.5 m based on manual experience. To ensure the surface flatness of the coating, it is necessary to control the grid spacing \(L\) between adjacent grids such that the difference between the maximum accumulated thickness \(d_{L/2}\) in the overlapping region of two adjacent trajectory cross-sections and the maximum thickness \(d_{\text{max}}\) in a single trajectory cross-section is minimized., as shown in Figure 7.  
The accumulated thickness in the overlapping region of adjacent trajectories can be modeled as the superposition of the accumulated thickness contributions from both individual trajectories within the overlapping area.