## 2 Theoretical Background  
Figure 1 depicts a PV cleaning robot equipped with two belt-based track mechanisms operating on PV panels. The belt-based track comprises two timing pulleys interconnected by a polyurethane (PU) endless timing belt coated with slip-resistant material. The coating layer serves a crucial role in the preventing slippage of the PV cleaning robot on tilt PV panels and facilitating its movement across module-supporting brackets that are significantly higher than the PV panels' aluminum edge. To enable the application of slip-resistant material, such as vulcanized rubber, on the PU belt, grooves are made on this layer, allowing for the easy bending of the belt-based track. The area, stiffness, pattern, and thickness of the track grooves formed by the adjacent grooves directly influence the slippage of PV cleaning robots. This research investigates these factors to provide valuable insights into the properties of track grooves reflected by the coefficient of friction. To compute the coefficient of friction, we consider a displacement mass (\(m\)) placed on an adjustable tilt angle (\(\alpha\)) floor surface, as depicted in Figure 2. The motion of the displacement mass on the floor surface is analyzed using the free-body diagram method. The force of gravity acting on the displacement mass is directly proportional to its weight, allowing for further analysis of the coefficient of friction. \(P_{x}=P\sin(\alpha),P_{y}=P\cos(\alpha)\) are, respectively, the components of the force of gravity projected onto the \(x\) and \(y\) axis. Figure 1: PV cleaning robot equipped with two belt-based track mechanisms. Figure 2: Free body diagram of displacement mass on tilt surface. When the tilt angle is zero, it should be noted that the displacement mass is placed on the level surface, resulting in no frictional force. Increasing the tilt angle, the force of static friction \(F_{s}\) is increased from zero to counteract the component of the force of gravity \(P_{x}\). According to [PERSON]'s second law, while the displacement mass rests on the tilt surface, the forces that act on the displacement mass along the \(x\)-axis are in equilibrium \(\sum F_{\rm x}(=P_{\rm x}-F_{s})=0\). Recall that the \(x\)-component of the force of gravity \(P_{x}=mgsin(\alpha)\) and the force of static friction \(F_{s}=\mu_{s}N\) can be obtained as follows:  
\[mgsin(\alpha)=\mu_{s}N \tag{1}\]  
where \(\mu_{s}\) is the coefficient of static friction (COSF) and \(N\) is the normal force. Applying [PERSON]'s second law and using the \(y\)-component of the force of gravity \(P_{y}=mg\cos(\alpha)\), the normal force \(N\) can be determined by  
\[N=mg\cos(\alpha) \tag{2}\]  
By substituting Equation (2) into Equation (1) and solving for \(\mu_{s}\), this yields  
\[\mu_{s}=tan(\alpha) \tag{3}\]  
Equation (3) shows that the COSF can be obtained by analyzing the tilt angle of the floor surface. The tilt angle is incrementally increased until it reaches a specific value at which the displacement mass starts to slide off. This indicates that the \(x\)-component of the force of gravity surpasses the maximum static friction force. An alternative method for measuring the COSF involves not incrementally increasing the tilt angle. That is, the displacement mass, which corresponds to the weight-bearing frame, is connected to an electronic load cell through an inextensible strut bar. When the \(x\)-component of the force of gravity is lower than the static friction (\(P_{x}<F_{s}\)), the weight-bearing frame remains stationary on the PV panel surface. Consequently, there is no tensile force exerted by the displacement mass, as indicated by an ideally constant value of the electronic load cell. Conversely, if the \(x\)-component of the force of gravity exceeds the static friction, the electronic load cell registers a tensile force, indicating that the weight-bearing frame has started to slide off the PV panel surface. This occurs when the tilt angle reaches a certain threshold.