## 2 Theoretical Background  
Figure 1 illustrates the kinematic diagram of a track-wheeled system in contact with the surface of a PV panel (1). Let \(j\) be the index of the suspension units \(j=\{-2,-1,0,1,2\}\), with the middle suspension unit corresponding to \(j=0\). The track-wheeled system comprises two timing pulleys (2), a multi-suspension unit (3), and a Polyurethane (PU) endless timing belt (4). The motion is powered by a brushless direct current motor, which drives one timing pulley through the PU timing belt, resulting in the rotation of the other. The contact area between the track-wheeled system and PV panels is defined as the surface area of the PU timing belt limited by the two timing pulleys. The multi-suspension unit is integrated into the track-wheeled system to ensure proper contact between the PU timing belt and the PV panels. Additionally, the multi-suspension unit serves to suppress vibrations induced by the brushing action, as well as vibrations caused by the movement of the SPCRs on the panel support during operation. The multi-suspension unit proposed in this paper integrates five suspension units, each with its typical kinematic diagram, as shown in Figure 2a. Each suspension unit comprises an elbow-shaped bogie arm (5) with an angle \(\gamma\), a freewheel (6), and a shock absorber. The shock absorber is constructed using a slide tube (1), a bolt nut (2), a compressed spring (3), and a piston rod (4). All five suspension units are mounted to the whole frame at point A, serving as their rotary center. The bogie arm is connected to the shock absorber at point B and the freewheel at point D, allowing it to leverage point C and apply pressure to the freewheels, ensuring proper contact with the PU timing pulley. Considering the potential variations in the altitude of freewheels due to PV panel deformation, the compressed springs can be adjusted using a spring adjuster mechanism inspired by [15]. By sliding the bolt nut along the slide tube, the displacement of the compressed springs can be changed, enabling adjustable spring stiffness. Figure 2b illustrates the 3D CAD design of the suspension, utilizing two freewheels to coincide and spread the entire width of the PU timing belt. Generally, four situations might occur as SPCRs move on PV panels, as shown in Figure 3. First, the SPCR moves on PV panels with minimal deformation, resulting in the freewheels' altitude being roughly the same. The second situation defines the largest deformation of PV panels, where the altitude of these wheels is noticeably different. The third situation illustrates the SPCR moving across the mounting clamps or encountering PV panel misalignment, defining the highest travel of the freewheel. In the final situation, the SPCR moves across the gap between two adjacent PV panels, defining the lowest travel of the freewheel. Figure 1: Kinematic diagram of track-wheeled system. According to [12], situation 2 commonly occurs when the SPCR passes over PV panels along its long edge. For the sake of simplicity, the design process for the multi-suspension unit is implemented by defining multiple contact points between the deformed PV panel and the multi-suspension unit, taking into account the two timing pulleys with the assumption of negligible timing belt thickness, as shown in Figure 4. It is assumed that the track-wheeled system is in contact with the PV panel along the center line. Let \(G_{i}\) (where \(i\) is the index ranging from \(-3\) to \(3\)) represent the contact points of the freewheels and the two timing pulleys with the deformed PV panel. Point \(G_{0}\) is the contact point of the middle freewheel, while the other contact points are located on both sides of \(G_{0}\) and spread throughout the contact area. Ideally, these contact points should align on the same line, referred to as basis contact points, when the track-wheeled system is in contact with an undeformed PV panel, and the PU timing belt thickness is uniform.