## 2 Problem Description  
A self-developed FPLIR mainly consists of two parts, a flight mechanism and a walking mechanism, as shown in Figure 1, and the system parameters are listed in Table 1. The flight mechanism adopts an X-type structural design, and six rotor motors with serial numbers M1-M6 are distributed around the center bin. The walking mechanism includes a main traveling wheel, an auxiliary traveling wheel, a main compression wheel, an auxiliary compression wheel, a compression motor, a screw, an encoder, etc. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Types** & **Description** & **Value** \\ \hline FPLIR & Mass, m/kg & 38 \\ design parameters & Dimensions, L \(\times\) W \(\times\) H/m & 1.76 \(\times\) 1.76 \(\times\) 1.1 \\  & Distance from rotor center to fuselage center, r/m & 0.88 \\ \hline  & Body \(x\)-axis moment of inertia, \(l_{x}\)/kg\(\cdot\)m\({}^{2}\) & 5.8 \\  & Body \(y\)-axis moment of inertia, \(l_{y}\)/kg\(\cdot\)m\({}^{2}\) & 5.8 \\  & Body \(z\)-axis moment of inertia, \(l_{z}\)/kg\(\cdot\)m\({}^{2}\) & 8.5 \\ model parameters & Atmospheric drag coefficient, \(k_{f}\) & 1.6865 \(\times\) 10\({}^{-6}\) \\  & Rotor torque coefficient, \(k_{c}\) & 2.9250 \(\times\) 10\({}^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of FPLIR. Figure 2 shows the working principle of the robot, where it lifts off the ground to approach the power line and then precisely lands on the line and switches to walking mode to begin the line inspection. When the task is completed, the robot lifts off the line in flying mode. Table 1 shows that the FPLIR has a large mass, volume, and rotational inertia, which require a large minimum turning radius. Since the flight path of the robot is obstructed by the power line during ascent and descent, the reference path inevitably contains multiple corners and line shapes. Power lines are densely distributed, and in order to maintain safety when approaching them, higher requirements are placed on the path-following accuracy of the landing. When the robot has finished inspecting the entire section of the power line, it switches from the walking mode to the flying mode to fly over obstacles. It then repeats the above process. Given the flexible cable structure of the power line, if the robot takes off from a power line with a large following error or slow error convergence, the robot may collide with the power line and cause an accident. Therefore, it is important to carry out path-following research that allows the robot to work under such conditions. To address the mentioned issues concerning the robot's large turning radius, substantial following error, and slow convergence during path following, an adaptive acceptance circle strategy is proposed. This strategy considers the performance the constraints of the UAV and the reference path information. When the path angle is small, a larger acceptance circle is used, and reference path switching is performed in advance to avoid problems with slow convergence and large overshoot after path switching. If the path angle is large, a smaller acceptance circle is used to make the robot better fit the following path. During the process of following the 3D path, a heading-scaling factor is introduced to regulate the control volume. The spatial distribution of power lines is taken into account to adjust the robot's priority for its plane path and altitude. This ensures that higher-priority path following is completed more quickly and increases the convergence speed of the UAVs. Figure 1: Schematic of FPLIR. Figure 2: Schematic diagram of workflow. Figure 3 shows the position of the robot relative to the reference path in the inertial coordinate system, xyz. The variable \(V\) is the flight velocity; \(V_{P}\) is the projected velocity in the horizontal plane; \(\gamma\) and \(\chi\) are the heading angle and the path inclination, respectively; \(P\) is the position of the robot; and \(P^{{}^{\prime}}\) is the projected point of the robot in the path segment.