## 2 Structural Model of the Climbing Robot
### The Safety Recovery Mechanism Model of the Climbing Robot  
When electrical failure occurs, the robot freely falls to the ground from high altitude and the fast rate of descent and impact velocity damages the equipment and even threatens the safety of workers. In order to avoid fast gliding speed of the robot after electrical failure, a centrifugal-friction recovery mechanism was designed. The braking force was dynamically adjusted in accordance with different inclination angles of cables as shown in Figure 3a. The recovery mechanism consisted of mass blocks, a brake disc, a friction plate, a gearbox and a rotary shaft (Figure 3b-d) and each part was set on the rotary shaft. The chute was processed on one side of the brake disc and mass blocks slid in the chute to push the brake disc to overcome the spring force and then move along the rotary shaft. The brake disc contacted with the friction plate to generate braking torque, while the gearbox generated an acceleration to increase the braking torque of the friction plate on the robot. When the descent speed of the robot is low, the brake disc is separated from the friction plate under the effects of the spring (Figure 4a). If the robot drops quickly after a power outage, the mass blocks slide outside under the effects of centripetal forces and the axial component thereof overcomes  
Figure 3: Safety recovery structure of the robot. 1—Main case; 2—Brake disc; 3—Mass blocks; 4—Base; 5—Rotary shaft; 6—Spring mechanism; 7—Gearbox; 8—Ball bearing; 9—slideway. (**a**) Schematic diagram of recovery mechanism; (**b**) The inner structure of the recovery device; (**c**) Breakdown drawing of recovery mechanism; (**d**) Centrifugal–friction recovery mechanism. the spring forces, so that the brake disc makes complete contact with the friction plate and braking forces are dynamically transferred to the robot (Figure 4b). With the changes of descent speed of the robot, contact forces between the friction plate and brake disc changes constantly. In theory, the braking torque can automatically reach a stable value according to the inclination angle of cables and loads on the robot. Both sides of mass blocks are designed for a certain inclination angle \(\varphi\) so as to transform the centripetal force into an axial force. The component of this force in the axial direction of the sum of centripetal forces of six mass blocks can be expressed as:  
\[F=3\cdot m\cdot w^{2}\cdot l_{1}\cdot\cot\varphi \tag{1}\]  
where \(m\), \(w\) and \(l_{1}\) represent the mass of mass blocks, the angular velocity of the rotary shaft and the distance from the center (Point P) of gravity of the mass blocks to the axis of the rotary shaft, respectively. If small changes in the spring forces of the brake in working are ignored, axial force can be expressed as:  
\[\left\{ \begin{array}{ll}F^{\prime}=0&(F_{t1}>F)\\ F^{\prime}=F-F_{t1}=3\cdot m\cdot w^{2}\cdot l_{1}\cdot\cot\varphi-F_{t1}&(F_ {t1}<F)\end{array} \right. \tag{2}\]  
where \(F_{t1}\) denote the spring force. The braking torque thus generated is:  
\[M^{\prime}=F^{\prime}\cdot\mu_{f}\cdot l_{2}=(3\cdot m\cdot w^{2}\cdot l_{1} \cdot\cot\varphi-F_{t1})\cdot\mu_{f}\cdot l_{2} \tag{3}\]  
where \(\mu_{f}\) denote the coefficient of friction. \(l_{2}\) represents the distance from the geometrical center of the mass blocks to the axis of the rotary shaft. When the robot runs at a high speed, it needs to satisfy the following equation in order to limit the speed and move more uniformly in a certain range:  
\[i\cdot M^{\prime}=M \tag{4}\]  
where \(i\) is the gear ratio and \(M\) is the driving torque during the descent of the robot, thus giving the angular speed of the rotary shaft:  
\[w=\sqrt{\left(\frac{M}{\mu_{f}l_{2}\cdot i}+F_{t1}\right)\bigg{/}\left(3\cdot m \cdot l_{1}\cdot\cot\varphi\right)} \tag{5}\]  
The descent speed of the robot is:  
\[v=\frac{1}{i}w\cdot d \tag{6}\]  
Figure 4: Movement state of the recovery mechanism. (**a**) Low-speed movement of the robot; (**b**) High-speed movement of the robot.