# 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  
#### 2.2.1 Multi-Variable Spraying Layer Accumulation Model  
In spraying operations, the process typically involves fixed-point spraying or continuous linearly moving spraying, requiring the establishment of a spraying process parameter model based on the static and dynamic spraying layer accumulation models. When the spray nozzle is stationary relative to the work surface, maintaining a steady flow rate \(Q\) and air pressure \(P\), the concrete particles attach to the wall surface, forming a circular area with a spraying radius \(R\). The maximum accumulation thickness \(d_{max}\) occurs in the circular area, as shown in Figure 3.  
The coating thickness on the conical diameter section follows a beta distribution model, meaning that the cumulative thickness \(D(r)\) at a distance rrr from the circle center \(O\) satisfies:  
\[D(r)=d_{\max}(1-\frac{r^{2}}{R^{2}})^{\beta-1} \tag{13}\]  
where \(\beta\) is an empirical value determined by the operating parameters during spraying (which can be obtained through empirical formulas or experimental fitting), and the volume of the cone is:  
\[V=\int_{0}^{R}D(r)2\pi rdr \tag{14}\]  
If the sprayed volume after continuously spraying for time \(t\) at a spraying flow rate \(Q\) (the volume of concrete sprayed) is:  
\[V=Qt \tag{15}\]  
By combining Equations (13)-(15), the maximum cumulative thickness \(d_{max}\) after continuous spraying is:  
\[d_{\max}=\frac{Qt\beta}{\pi R^{2}} \tag{16}\]  
From the geometric relationship in Figure (3), the relationship between the spray radius \(R\), the spraying distance \(H\), and the spray angle \(\varphi\) is:  
\[R=H\tan\varphi \tag{17}\]  
According to reference [11], the relationship between the spray angle \(\varphi\), spray flow rate \(Q\), and air pressure \(P\) is:  
\[\tan\varphi=KQ^{\theta_{1}}P^{\theta_{2}} \tag{18}\]  
Figure 3: Static coating accumulation distribution. The parameters \(K\), \(a_{1}\), and \(a_{2}\) are experimental fitting parameters. Thus, by solving the system of Equations (13), (16)-(18), a static spraying model related to the spray flow rate \(Q\), injection pressure \(P\), injection distance \(H\), and spraying time \(t\) can be established as follows:  
\[D(r)=\frac{Q^{1-2a_{1}}t\beta}{\pi K^{2}H^{2}P^{2a_{2}}}(1-\frac{r^{2}}{K^{2}H^ {2}Q^{2a_{1}}P^{2a_{2}}})^{\beta-1} \tag{19}\]  
When the spray gun moves perpendicularly to the work surface at distance \(H\), maintaining a linear velocity \(v\) with flow rate \(Q\) and air pressure \(P\), concrete particles adhere to the rock surface, forming a rectangular area with a thicker middle and thinner edges. The spray width radius (half-width) is \(R\), and the maximum accumulated thickness is \(d_{max}\), as shown in Figure 4.  
Assuming that the accumulated thickness distribution in the direction perpendicular to the spray direction still follows the beta distribution described by Equation (13), the accumulated coating thickness \(D(x_{0})\) at a point \(A\), located at a distance \(x_{0}\) from the centerline of the spray operation, satisfies:  
\[D(r)=d_{\max}(1-\frac{r^{2}}{R^{2}})^{\beta-1} \tag{13}\]0  
The equation \(t_{0}=2\sqrt{R^{2}-x_{0}^{2}}/v\) represents the duration during which point \(A\) is covered by the spray during the operation, i.e., the time it takes for point \(A\) to enter and leave the spray area (assuming point \(A\) enters the spray area at \(t=0\)s). \(r=\sqrt{\left(\sqrt{R^{2}-x_{0}^{2}}-vt\right)^{2}+x_{0}^{2}}\) represents the distance between point \(A\) and the spray center at time \(t\). Equation (16) expresses the accumulated thickness at point \(A\) as the result of integration over time during the spray operation.