## 2 Problem Description
### Workspace Analysis of FPTLIR  
The workspace analysis of the FPTLIR focuses on evaluating its reachable range and safe zone during the landing process. During the landing process, it is crucial to monitor the robot's attitude and displacement changes to ensure it does not collide with the PTL, maintains optimal control performance, and avoids compromising the insulation clearance of the PTL. Therefore, quantifying the robot's workspace provides a foundation for better evaluating its dynamics during and after the landing process, ensuring safe and efficient operation.  
Assuming that the local deformation of the PTL is negligible, the analysis of the roll workspace is as follows. When the PTL is horizontal, the roll workspace is as shown in Figure 3a. The width of the robot is \(D_{Ry}\), and the distance between the contact point of the lower side of the travelling wheel with the PTL and the rotor plane of the robot is \(L_{c}\)  
Figure 2: Analysis framework for the research.  
Therefore, when the PTL is horizontal, the safe range of the roll angle is \((-\Phi_{\rm c},\Phi_{\rm c}\) ), where \(\Phi_{\rm c}\) is given by:  
\[\Phi_{\rm c}=\arctan\frac{2L_{c}}{D_{Ry}} \tag{1}\]  
The robot's roll workspace varies with the PTL slope. For a 200 m span PTL, its shape and slope distribution are shown in Figure 4a,b The roll workspace at \(x\) can be expressed as \((-\Phi_{\rm c}+\alpha(x),\Phi_{\rm c}+\alpha(x)\) ), where the PTL shape is fitted by a quadratic function \(z=Ax^{2}+Bx+C\) and its slope is calculated by \(\alpha(x)=\arctan(2 Ax+B)\), \(\alpha(x)\in(-5.1,5.1)\).  
In the yaw motion, the flying mechanism avoids collision with the PTL, but excessive yaw may cause the travelling wheel to detach. With a travelling wheel diameter \(D\), inner width \(B\), and PTL diameter \(d\) (Figure 3d), the safe yaw range \((-\Psi_{c},\Psi_{\rm c})\) is calculated by:  
Figure 4: Line shape and slope changes of a 200 m PTL: (**a**) PTL shape; (**b**) PTL slope.  
Figure 3: Schematic diagram of the operating zones of attitude for the FPTLIR: (**a**) roll workspace; (**b**) yaw workspace; (**c**) pitch workspace; (**d**) zoomed-in view of yaw workplace.  
\[\Psi c=\tan^{-1}\frac{B-d/\cos\Psi c}{D} \tag{2}\]  
The formula simplifies to:  
\[\Psi c\approx\tan^{-1}\frac{B-2d}{D} \tag{3}\]  
For pitch motion, there are no collision constraints as no obstacles exist in this direction. However, excessive pitch can cause significant power loss. Considering the robot's weight and maximum rotor thrust \(F_{\mathrm{zmax}}\), the maximum pitch angle is calculated as:  
\[\Theta_{c}=\cos^{-1}\frac{F_{\mathrm{zmax}}}{m_{B}g} \tag{4}\]  
In power systems, transmission lines consist of phase conductors and ground wires. The robot and PTL1 are safe from flashover when inside the insulation zone (Figure 5). Sufficient spacing between conductors is required to prevent flashover. When the FPTLIR lands on the PTL, its displacement should comply with the following constraints in:  
\[Z_{c}=D_{12}-D_{Rz}-D_{E} \tag{5}\]  
where \(D_{12}\) represents the distance between PTL1 and PTL2, \(D_{Rz}\) denotes the geometric height of the FPTLIR, and \(D_{E}\) indicates the safe clearance between PTL1 and PTL2.  
By substituting the corresponding values into the derived equations and performing the necessary calculations, the workspace of the FPTLIR is determined. The results, including key parameters and their ranges, are presented in Table 2. This quantified workspace provides a comprehensive understanding of the operational boundaries and serves as a foundation for further analysis of the robot's dynamics and safety during the landing process.