# 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
Therefore, the maximum accumulated thickness in the overlapping region can be calculated from Equation (13) as follows:  
\[d_{L/2}=2 td_{\text{max}}(1-\frac{L^{2}}{4R^{2}})^{\beta-1} \tag{26}\]  
Similarly, according to Equation (13), the maximum accumulated thickness at the center of a single trajectory is calculated as follows:  
\[D_{(r=0)}=td_{\text{max}} \tag{27}\]  
By combining Equations (17), (18), (26) and (27), the calculation of the optimal spacing \(L\) between adjacent trajectories is derived as follows:  
\[L=2R(1-2^{\frac{1}{1-\beta}})^{\frac{1}{2}}=2 HKQ^{a_{1}}P^{a_{2}}(1-2^{\frac{ 1}{1-\beta}})^{\frac{1}{2}} \tag{28}\]  
Figure 6: Schematic diagram of pit-filling layer stance point calculation. Figure 7: Schematic diagram of optimal spraying distance. The first operating point of the filling layer is selected at the sidewall positions on both sides of the tunnel, with its coordinates \(U(X,Y,Z)\) obtained through vehicle-mounted radar scanning. Using Equations (24)-(26), the coordinates of the first grid station point in the tunnel coordinate system \(\{\)G\(\}\) are calculated as \(V_{1I}(X_{1},Y_{1},Z_{1})\). The positions of the adjacent grid station points are distributed as shown in Figure 8.  
Calculation of the coordinates for adjacent stopping points \(V_{1,j}\) in the filling layer:  
\[V_{i,j}=V_{1,1}+\left[ \begin{array}{c}(i-1)L\\ \sqrt{\gamma_{1}^{2}+Z_{1}^{2}}\cos(\theta_{1}+(j-1)\Delta\theta)\\ \sqrt{\gamma_{1}^{2}+Z_{1}^{2}}\sin(\theta_{1}+(j-1)\Delta\theta)\end{array} \right] \tag{29}\]  
\(\Delta\theta\) is the angle between adjacent stopping points in the tunnel's circumferential direction. \[\Delta\theta=\frac{L}{\sqrt{\gamma_{1}^{2}+Z_{1}^{2}}} \tag{30}\]  
According to the static spraying model, the spraying time \(t\) at a stopping point is determined by the different spraying thickness \(T_{2}\) at that point. \[t(Q,P,H_{2},T_{2})=\frac{\pi K^{2}T_{2}H_{2}^{2}P^{2a_{2}}}{Q^{1-2a_{1}}\beta} \tag{31}\]  
The spraying distance \(H_{3}\) for the sweeping layer refers to the distance from the nozzle tip to the design contour. To ensure the compaction and spraying quality of the sprayed concrete, \(H_{3}\) is selected as 2 m based on manual experience. The first operational endpoint of the sweeping layer is selected in the region with the extreme mileage of the tunnel to be sprayed. The height from the ground is determined by the safe distance maintained between the spray gun and the tunnel floor. Its coordinates \(U(X,Y,Z)\) are obtained via vehicle-mounted radar scanning. The first spray gun motion path endpoint \(V_{1I}(X,Y,Z)\) is calculated using Formula (25). Let the difference in the extreme mileage of the tunnel's spraying region be \(B\), as shown in Figure 9. The coordinates of the spray gun motion path endpoint are calculated as follows:  
\[V_{i,j}=V_{1,1}+\left[ \begin{array}{c}(i-1)B\\ \sqrt{\gamma_{1}^{2}+Z_{1}^{2}}\cos(\theta_{1}+(j-1)\Delta\theta)\\ \sqrt{\gamma_{1}^{2}+Z_{1}^{2}}\sin(\theta_{1}+(j-1)\Delta\theta)\end{array} \right] \tag{32}\]  
Figure 8: Distribution of grid stance point positions in the filling layer. According to the dynamic spraying model, during sweeping operations, the spraying speed \(v\) along the sweeping path is calculated based on the spraying thickness \(T_{3}\) of the sweeping layer:  
\[v(Q,P,H_{3},T_{3})=\frac{Q^{1-a_{1}}\beta}{2K\pi T_{3}H_{3}P^{a_{2}}} \tag{33}\]