# PDDL Task Planning for Tunnel Spraying Based on a Multivariate Coating Accumulation Model
## 2 Methods
### Hierarchical Planning of Spraying Tasks Based on over- and Under-Excavation Volume
#### 2.1.2 Spraying Task Hierarchical Planning  
Based on the analysis of the over-excavation and under-excavation, the maximum blast face profile, blast face profile, and average blast face profile of the actual tunnel contour are determined. Combined with the tunnel design contour and the arch profile, the tunnel spraying layers are divided into the pit-filling layer, filling layer, and sweeping layer, with their respective thicknesses denoted as \(\delta_{1}\), \(\delta_{2}\), and \(\delta_{3}\), \(R_{g}\), \(R_{b}\), \(R_{j}\), and \(R_{d}\) represent the arch profile radius, blast face profile radius, average blast face profile radius, and design profile radius, respectively. \(\Delta\) is the distance between the blast face profile and the design profile, as shown in Figure 2.  
The pit-filling layer is the region between the maximum blast surface contour and the average blast surface contour. The pit-filling task involves determining the locations of the pits and their opening radii, which are then used to plan the positions of the stopping points and the spraying distance \(H_{1}\). The filling layer is the region between the average blast surface contour and the arch contour. In this layer, the grid stopping points are planned. The grid spacing is determined using the optimal trajectory spacing model, and the spraying distance \(H_{2}\) is determined based on manual experience. The sweeping  
Figure 1: Estimation of the vehicle body pose of the wet spraying trolley.  
layer is the region between the arch contour and the design contour. For this layer, the endpoints of the spraying path are planned, the spraying distance \(H_{3}\) is determined based on manual experience, and the spray gun movement speed is calculated using the dynamic spraying model.  
To determine the spraying thickness of each task layer, it is necessary to adaptively select a reasonable average blast surface contour based on the actual blasting conditions. The radius \(R_{j}\) of the average blast surface contour is defined as the mean value of the distances from various points on the tunnel blast surface contour to the center of the design contour, which can be expressed as:  
\[R_{j}=\frac{1}{\pi}\int\frac{\pi}{\frac{2}{2}}\ R_{\text{b}}d\omega \tag{9}\]  
Thus, the intersection points \(A_{2i-1}\) and \(A_{2i}\) between the average blast face profile and the blast face profile can be obtained, and the distance between the two points is the opening radius \(S_{A_{2i-1},A_{2i}}\) of the deep pit., where \(i\geq 1\).  
According to the adaptively selected average blast face profile radius \(R_{j}\), let the thickness to be sprayed for the pit-filling layer be \(T_{1}\), for the filling layer be \(T_{2}\), and for the sweeping layer be \(T_{3}\). The thickness to be sprayed for each layer can be calculated as follows:  
\[T_{1}=\begin{cases}\Delta-(\delta_{2}+\delta_{3}),&\Delta>\delta_{2}+\delta_{ 3}\\ 0,&\Delta\leq\delta_{2}+\delta_{3}\end{cases} \tag{10}\]  
\[T_{2}=\begin{cases}\delta_{2},&\delta_{2}+\delta_{3}<\Delta\\ \Delta-\delta_{3},&\delta_{3}<\Delta<\delta_{2}+\delta_{3}\\ 0,&\Delta<\delta_{3}\end{cases} \tag{11}\]  
\[T_{3}=\begin{cases}\delta_{3},&\Delta>\delta_{3}\\ \Delta,&0<\Delta<\delta_{3}\\ 0,&\Delta<0\end{cases} \tag{12}\]  
Figure 2: Schematic diagram of the tunnel profile.