## 2 Extended Pick Failure Probability Assessment Approach
### Pick Tip Failure Probability during the Sump-in Stage
As the hard rock cutting sector can be calculated using Equation (3) and the failure probability for cutting coal is ignored, the cutting sector in the sump-in stage can be set to 180 degrees in the calculation if the roof rock thickness is not greater than \(\frac{D_{d}}{2}\). In shearing-down stage, the cutting sector is 90 degrees when the distance between the center of the drum and the floor is not less than \(D_{d}\). The estimation of the cutting sector when the distance between the center of the drum and the floor is shorter than \(D_{d}\) will be discussed in the next section. Obviously, the hard rock cutting sector for a given revolution cannot be greater than the full cutting sector of this revolution. The second issue for consideration is the calculation of the DOC of the pick at the first revolution. According to [11; 18], the maximum DOC (nominal DOC) can be calculated by  
\[D_{max}=\frac{v_{n}}{m} \tag{5}\]  
where \(m\) is the number of starts of the drum and \(v_{n}\) is the drum advance distance per revolution (mm/r):  
\[v_{n}=\frac{1000v_{a}}{n} \tag{6}\]  
As the cutting starts from the initial position on the first revolution, the maximum DOC of the pick in consideration in this revolution is only a quarter of that in other revolutions because the pick rotates only a quarter of a revolution to reach the maximum advance distance in this revolution \(v_{n}/4\) (see Figures 2 and 3). For example, with the parameters in Figure 2 and the number of the starts being 2, the maximum DOC of this pick in the first revolution is only 7.5 mm, while it is 30 mm in the other revolutions. The final point is the estimation of the pick tip failure probability of the last revolution. On this revolution, the pick may not complete a full cut of the roof rock before the drum moves to shearing-down stage. However, as the drum will continue cutting when its cutting mode changes from sum-in to shearing-down, and the variations of the DOC during the transition period can be ignored, the calculation of the pick failure probability of the last revolution can be treated as the same as that in the other revolutions (see Figure 2). Once the failure probability of the pick tip in each drum revolution in the sum-in stage has been estimated, its failure probability in the sum-in stage, \(F_{1}^{t}\left(x_{a}^{t}\right)\), is  
\[F_{1}^{t}\left(x_{a}^{t}\right)\ =1-\prod\
olimits_{i=1}^{n_{dr}}\left[1-\sum \
olimits_{j=1}^{n_{dr}}F_{ri}\left(h_{rj},\ x_{a}^{t}\right)p\left(h_{rj} \right)\right] \tag{7}\]  
where \(n_{dr}\) is the number of drum revolutions for implementing the sum-in cutting process, which is given by  
\[n_{dr}\ =\ \text{ceil}\left(\frac{D_{s}}{v_{n}}\right) \tag{8}\]  
In Equation (8), \(D_{s}\) is the sum-in distance (mm). In reality, the sum-in depth cannot be accurately controlled due to limitations of the equipment control systems, resulting in the number of the drum revolutions for sum-in cutting fluctuating randomly. This issue will be studied in due course. In this paper, only fixed sum-in depth is considered. Figure 3: Cutting sector in sum-in stage.