## 2 Extended Pick Failure Probability Assessment Approach
### Pick Tip Failure Probability during the Sump-in Stage  
In the sump-in stage, the drum normally cuts into the face along the roof for a sump-in depth of about the cutting radius (i.e., the half of the tip-to-tip diameter) of the drum. To implement the required sump-in depth, the drum needs to rotate multiple revolutions. The pick tip failure probability in this stage is a combination of the tip failure probability over individual drum revolutions. If the roof rock thickness is less than the tip-to-tip diameter of the drum (which is quite common during real production), the drum will partially cut roof hard rock and partially cut the coal seam over a single revolution. As the coal is generally much softer and weaker than the rock, the pick failure probability due to cutting coal can be ignored [11]. To estimate the pick tip failure probability over each drum revolution, the thickness of the roof rock in each individual drum revolution needs to be determined. In reality, the roof rock thickness generally fluctuates randomly, and it is normally difficult to obtain the distribution of roof rock thickness. For simplicity, instead of using a continuous distribution, the roof rock thickness can be approximated by several mean values with their corresponding occurrence likelihoods in industrial applications. In this case, based on the total probability theorem, the failure probability of a pick tip with a given allowable bending force \(x_{a}^{t}\) in a drum revolution can be estimated by  
\[F_{ri}=\sum\
olimits_{j=1}^{n_{br}}F_{ri}\Big{(}h_{rj},\ x_{a}^{t}\Big{)}p\Big{(} h_{rj}\Big{)} \tag{1}\]  
where \(F_{ri}\) is the pick tip failure probability at the \(i\)-th drum revolution in the sump-in stage; \(n_{br}\) is the number of mean roof rock thickness values in consideration; \(F_{ri}\Big{(}h_{rj},\ x_{a}^{t}\Big{)}\) is the pick tip failure probability at the \(i\)-th drum revolution in the sump-in stage when the mean roof rock thickness is \(h_{rj}\) (mm) and tip allowable bending force is \(x_{a}^{t}\); and \(p\Big{(}h_{rj}\Big{)}\) is the likelihood (%) that the mean roof rock thickness is \(h_{rj}\). Obviously,  
\[\sum\
olimits_{j=1}^{n_{br}}p\Big{(}h_{rj}\Big{)}=100\% \tag{2}\]The pick failure probability \(F_{rl}\!\left(h_{rj},\ x_{d}^{t}\right)\) can be estimated using the models given in [11]. Nevertheless, when using the models given in [11], one needs to consider the following points:  
The first issue is the determination of the cutting sector of the drum, \(\alpha_{rc}\) (degree) (see Figure 3). In [11], since the impact of cutting soft coal is ignored, the cutting sector indicates the hard rock cutting sector and is given by  
\[\alpha_{rrj}=\arccos\left(1-\frac{2h_{rj}}{D_{d}}\right)*\frac{180}{\pi} \tag{3}\]  
where, \(\alpha_{rrj}\) is the hard rock cutting sector of the drum (in degrees) when the mean roof rock thickness is \(h_{rj}\) (mm). \(D_{d}\) is the tip-to-tip diameter of the drum (mm). However, determination of the cutting sector in this way would be inconvenient when multiple drum revolutions are involved, because of the uncertainty of the rock thickness in each individual drum revolution. To resolve this problem, the cutting sector of a drum in one revolution is defined as the central angle of the drum engaging coal and/or rock cutting. With this definition, the cutting sector is between 90 degrees and 180 degrees in the sump-in stage (see Figure 3). The cutting sector of the drum at the \(i\)-th revolution can be obtained by solving the following equation:  
\[\frac{1000v_{a}}{60}\left[\frac{60(i-1)}{n}+\frac{\alpha_{rci}}{6n}\right]+ \frac{D_{d}}{2}\sin\frac{\pi\alpha_{rci}}{180}-\frac{D_{d}}{2}=0 \tag{4}\]  
where, \(\alpha_{rci}\) is the cutting sector of the drum at the \(i\)-th revolution (in degrees), \(n\) is the drum rotational speed (rpm), and \(v_{a}\) is the drum advance speed (m/min).