The chain drive as a motion mechanism has good adaptability, expandability, and high efficiency of motion. As shown in Fig. 1, a chain with lugs mounts the rope-climbing foot as the crawler mechanism that moves to the rope. Guide wheels are installed on the frame at both ends to form a stable rope-climbing interval. The rope-climbing interval is installed with upper and lower crawler rope-climbing mechanisms, while the rope passes through the middle. The crawler rope-climbing mechanism is equipped with a chain-tensioning device, which reduces the vibration of the chain drive and serves as the chain-guiding crawler between rope-climbing intervals. The upper chain-tensioning device converts the self-weight of the machine into a compression force to make the rope-climbing foot press against the rope. The lower chain-tensioning device can be adjusted with bolts. Therefore, a good friction transmission interface is formed between the rope-climbing foot and the rope in the rope-climbing interval. The movement of the RCR along the rope direction is realized by synchronously driving the chains in the upper and lower crawler rope-climbing mechanisms.

The load capacity of the robot is closely related to the friction force generated at the friction interface between the rope-climbing foot and the rope. From classical friction mechanics, it is known that the friction force is related to the friction surface pressure and the coefficient of friction: 1f=μG+Fpress, where μ is the coefficient of friction between the rope-climbing foot and the rope, G is the sum of the gravity of the robot and the load, and Fpress is the clamping force provided by the pressing mechanism.

The driving force of the rope-climbing robot is the reaction force of the static friction between the rope-climbing foot and the rope, so the maximum static friction determines the maximum driving force of the rope-climbing robot. The rope-climbing foot uses rubber material whose contact surface with the wire rope is the Hertzian contact, so the maximum static friction can be increased in the form of increasing the contact area. The rope is made of a steel cable with a core, and the contoured cross section can be regarded as circular. The contact area between the planar-shaped crawler and the rope is small, so the circular rope-climbing foot structure is designed as shown in Fig. 2. Expand the contact area between the rope-climbing foot and the rope to improve the overall load capacity of the robot.

The majority of the existing RCRs adopt the form of a single rope plus a protection rope, which is prone to tilt and derailment during operation. This study designs a dual rope-crawling climbing robot with double guide ropes based on the crawling rope-climbing mechanism, which ensures the balance and stability of the robot while climbing. The mechanism principle of the robot is shown in Fig. 3, and the detailed structure of the robot is shown in Fig. 4. The output shaft of motor 1 is connected to the dual-output reducer 2 for speed reduction. The output of the reducer, the active sprockets 4 and 5, and the gear 3 are located on shaft 7. The active sprockets 4' and 5' and the gear 3' are located on shaft 7'. The transmission between shaft 7 and shaft 7' is carried out through the gears 3 and 3', which realizes the synchronous reverse rotation of the active sprockets 4 and 5 and the active sprockets 4' and 5'. Thus, the rope-climbing mechanism of the robot crawler is synchronized.

The RCR moves repeatedly along the overhead ropes. The overhead rope is divided into two forms, equal height pivot fixed and unequal height pivot fixed; however, their system mechanics analysis principle is the same. The double ropes in the system are considered a single rope for analysis. Suppose the load of the rope is uniformly distributed along the horizontal direction and the uniform load in the horizontal direction is q,q=wl0. Then, the parabolic equation of the rope without load with the left pivot point as the origin can be obtained as follows: 2y=xtgα-4s(l0-x)xl0, where w is the mass of the rope, l0 is the horizontal span distance of two fixed pivot points, q is the rope horizontal direction uniform load, α is the tilt angle of the two fixed pivot points, f0 is the disturbance of the rope's midpoint, and s=f0l0.

Then the length of the rope between the two fixed pivot points can be obtained as 3L=∫0l01+dydx2dx=∫0l01+tan⁡α-4sl02,dx≅l0secα1+83s2cos4α. When the robot crawls on the rope, the robot is considered to act as a concentrated load on the overhead rope, and its motion trajectory is related to the ratio of robot load P to rope self-weight W.

As shown in Fig. 5, when the ratio of robot load P to rope self-weight W is infinitesimal, its trajectory can be regarded as a parabola; when the ratio of robot load P to rope self-weight W is infinite, its trajectory can be regarded as an ellipse with A and B as the focal points. However, under the actual working condition, the value of the ratio of robot load P to rope self-weight W is between 0.1 and 3.0, where the above two cases do not exist, so the trajectory is between the ellipse and parabola.

When the RCR is operating, it is equivalent to a concentrated load acting on the overhead rope. The linearity of the overhead rope is shown in Fig. 5a, and the AC section and the BC section are two parabolic curves. T is the rope tension, H is the rope horizontal tension, and V is the rope vertical tension. As shown in Fig. 5b, the x–A–y coordinate system is established with the fixed pivot point A as the origin, and the separation is intercepted from a point near the A end of the AC section for analysis. Similarly, the separation is intercepted from a point near the B end of the BC section for analysis in the X–B–Y coordinate system, as shown in Fig. 5c.

The equation of the curve of the AC section is 4y=xtanα-ql01+2n1-kx-qx22H. The equation of the curve of the BC section is 5Y=Xtanα-ql01+2nkX-qX22H, where x=kl0 is the AC section and X=(1-kl0) is the BC section.

Assume that the rope does not elongate before and after the robot operation. The length of the rope where there is self-weight is L, the length of the AC rope where the robot works is L1, and the length of the BC rope where the robot works is L2. We obtain the following equation. 6L=L1+L2=l0secα1+83s2cos4α7L1=∫0kl01+dydx2dx8L2=∫01-kl01+dYdX2dx Let x=kl0 be the horizontal coordinate of the robot at a certain position, and the trajectory equation of the robot can be obtained from Eqs. (5), (6), (7), (8), and (9). 9y=xtanα-4sl01+2nxl0-x2l021+12n+n2xl0-x2l02

As shown in Fig. 6, the RCR is suspended from the rope by the guide wheel, and the O1O2 linear rope-climbing interval is formed between the guide wheels. Considering that the guide wheel is the slave and its friction is minimal compared to the whole machine drive, it can be neglected. The drape curve of the rope under the action of its weight q and travelling load G consists of AO2, O1B, two parabolas, and an O1O2 straight line. Furthermore, in the RCR and the horizontal direction, there is an inclination angle β, an O1B section of rope, O1 rope tension T1, the horizontal direction inclination angle γ, AO2, a section of rope O2, rope tension T2, and the horizontal direction inclination angle ω.

The coordinate system X–O–Y is established with the horizontal direction X and the vertical direction Y at the center O of the RCR. The rope force on the robot in the O1O2 linear rope-climbing area is along the I–I direction, and the rope tension T1 on both sides of the guide wheel O1 is equal. The combined force is N1, the tension T2 on both sides of the guide wheel O2 is equal, and the combined force is N2.

When the RCR is in equilibrium, T1=T2, θ=δ, the combined force of N1 and N2 is along the I–I direction, and the RCR tends to move along the ∥-∥ direction. This is obtained according to the parabolic theory of the overhead ropes.

The angle γ of the RCR and rope O1B is 10tanγ=q2Hl0-2x+PHl0l0-x-tanα. The angle ω of the RCR and rope AO2 is 11tanω=q2H2x-l0+PHl0x+tanα. Because the inclination angle β of the RCR is small, it can be approximated by the equation 12β=arctantanγ-tanω2=arctantanα+2x-l02HPl0+q. The RCR is based on the frictional drive motion of the rope-climbing foot and the rope. The frictional force is related to the robot load and the robot tilt angle β, where the RCR drive force F should satisfy 13Psinβ<F<µPcosβ+Fpress. When F<Psinβ, the RCR cannot move; when F>μPcosβ+Fpress, the RCR will slip.

The tilt angle of the RCR is related to the location of the RCR and the ratio of robot load to rope self-weight, which impacts the motion performance of the rope-climbing robot. Based on the prototype rope-climbing robot and platform parameters in Table 2, and according to the parabolic theory of the overhead rope, the tangent value of the theoretical inclination angle of the RCR under no load, 40 kg load, and 80 kg load can be obtained as shown in Fig. 7. From the robot tilt angle a change can be seen when the rope-climbing robot is close to the fixed pivot point at both ends of the rope, the value of robot tilt angle β is the largest, and the required robot drive power is the largest. Comparing the angle of change in robot inclination for different loads in Fig. 8, the change values of robot inclination under a 40 kg load and an 80 kg load and robot inclination in the robot self-weight case can be obtained. The rope-climbing robot load case has less influence on the rope-climbing robot inclination in the middle part of the rope and has more influence when it is near the fixed pivot points at both ends of the rope. Comparing the robot inclination changes at span distances of 15, 30, and 45 m in Fig. 9, with increasing span distance, the inclination maximum decreases and the inclination changes become slower. Through the tilt angle change process and the change in tilt angle under load, the RCR in engineering applications should be considered a reasonable safe working space that should be avoided near the rope fixed pivot point area. Establish a tilt angle of less than 10∘ for the rope-climbing robot safe working space.

During RCR motion, the tension of the rope is an important indicator of the stability and safety for the system. The dynamics of the rope-climbing robot system is carried out to analyze the dynamic tension of the rope. The double ropes in the system are mainly for maintaining the balance of the robot. To emphasize the influence on the rope tension during the robot motion, the simplified system treats the double ropes as a single rope for analysis. As shown in Fig. 10, the RCR moves along the rope according to the law of “acceleration (a1) (0-t1) – uniform speed (v0) (t1-t2) – deceleration (a2) (t2-t3)”, only considers the influence of the RCR movement on the rope tension, and makes the following assumptions.

Ignore the vibration of the rope under the external disturbance.

The mass of one-third of the rope is accounted for in the weight of the robot by the Rayleigh method, thus saving the rope as a massless elastomer. Taking the ropes on the left- and right-hand sides of the RCR as the object of study, the time-varying model of the dynamic tension at the fixed pivot points A and B is established.

14T¨A=EAm+ρL3m+ρl1t3gcosωsin⁡ω+γ+EAa-EAm+ρl1t3cosωsin⁡ω+γ+aTA-2∫0tatdtT˙A/l1t15T¨B=EAm+ρL3m+ρl2t3gcosγsin⁡ω+γ-EAa-EAm+ρl2t3cosγsin⁡ω+γ-aTB+2∫0tatdtT˙B/l2t TA and TB are the tension at two points of A and B, E is the modulus of elasticity of the rope, A is the cross-sectional area of the rope, m is the mass of the robot and the load, g is the acceleration of gravity, a is the acceleration, v is the running velocity of the robot, l1t,l2t is the length of the rope at the left and right ends of the robot, L is the length of the rope, and t is the time.

From Eqs. (15) and (16), rope tensions at points A and B of the rope are both significant time-varying systems, and the time-varying models at points A and B are established using MATLAB/Simulink, as shown in Fig. 11 (take the example of point A).

Example analysis was performed based on the prototype rope-climbing robot and platform parameters in Table 2. The analysis is carried out with the rope-climbing robot under self-weight, an acceleration of 10 mm s-2, a maximum robot speed of 100 mm s-2, an acceleration and deceleration time of 10 s, and a moving displacement of 1500 mm along the rope. From Fig. 12, at A of the rope, the maximum static tension value is 1586 N, the maximum dynamic tension value is 2115 N, and the maximum value of dynamic tension fluctuation is 1366 N. At B of the rope, the maximum static tension value is 1579 N, the maximum dynamic tension value is 2144 N, and the maximum value of dynamic tension fluctuation is 1340 N. The results show that the maximum tension of the rope reaches its maximum at the middle position, and the rope tension does not produce significant changes due to the process of acceleration change.

In order to research the stability of the rope-climbing robot system under different speeds of the rope-climbing robot, the rope tension changes under the speeds of 75, 80, and 100 mm s-1 are compared. From Fig. 13, it can be seen that the maximum value of dynamic tension and the maximum value of dynamic tension fluctuation at fixed pivot A of rope increases with the increase in speed.

In order to research the influence of the load condition of the rope-climbing robot system on the stability of the system, we compare the rope tension variation when the robot is running at 100 mm s-1 under self-weight, 40 kg load, and 80 kg load conditions. From Fig. 14, it can be seen that the rope tension increases with the increase in load, and the maximum value of the dynamic tension fluctuation increases with the increase in load.

In order to test the performance of the rope-climbing robot and to carry out further research, we developed a RCR prototype. The prototype uses 86 hybrid stepper motors, the prototype weighs 43 kg and uses 14 mm diameter wire rope to build a platform with a horizontal span of 1500 mm as shown in Fig. 15, and the main technical parameters of the system are shown in Table 1.

The RCR reciprocates along the rope, with the RCR moving from a fixed pivot point of the rope to another fixed pivot point with the side of the output shaft of the reducer in the positive direction. The test results show that the maximum no-load speed of the RCR is 115.8 mm s-1 in the forward direction and 104.7 mm s-1 in the reverse direction of the RCR.

The load capacity of the RCR is an important index for evaluating the RCR. This study tested the movement effect of the RCR under the weights of 20, 40, 60, and 80 kg, respectively. It is found that the RCR runs stably under an 80 kg load; however, after exceeding 80 kg, it runs stably in the position of a gentle inclination, and when the RCR is near the position of the fixed pivot points at both ends of the rope, the robot will slip when the inclination is large.

The rope will form certain obstacles on the surface of the rope due to rust and external factors. Obstacle segments of different diameters are set on the rope to test the ability of the robot to cross obstacles. The RCR prototype designed in this study grasps the rope foot inner-circle diameter of 14 mm. The test results found that the obstacle diameter is less than 17 mm and that the RCR can cross normally; however, when the obstacle diameter is 17–17.5 mm, the RCR does not pass smoothly, and when the obstacle diameter is greater than 17.5 mm, the RCR cannot pass.

The YOLOv5 algorithm was used to obtain the trajectory of the rope-climbing robot based on the principle of monocular recognition to verify the accuracy of the parabolic theory to calculate the trajectory. The motion trajectory was recorded using a Sony NEX-5N micro-single camera with a Sony E 18–55 mm f/3.5–5.6 OSS (SEL1855) lens. During the test, the calibration of the camera internal reference and the camera external reference was firstly completed based on Zhang's calibration method. The YOLOv5 algorithm is a single-stage target detection algorithm. As shown in Fig. 18, the center point of the recognition frame of the rope-climbing robot is regarded as the center position of the robot, the lower-left corner of the recognition screen is the origin of the Cartesian coordinate system, and the Euclidean distance between the two points of the algorithm recognition center and the manual anchor frame center is taken as the recognition accuracy of the algorithm. The test results show that the average Euclidean distance is 1.2962 mm and the maximum distance is 2.3212 mm during the forward and reverse motions of the robot with an 80 kg load.

The repeated motion trajectory of the RCR with self-weight, a 40 kg load, and an 80 kg load is shown in Fig. 19. Furthermore, bringing Table 1 parameters into Eq. (10), we calculated the theoretical motion trajectory of the RCR with self-weight, a 40 kg load, and an 80 kg load. Using the difference between the actual trajectory and the theoretical trajectory within 10 mm as the reference basis, analyze the motion trajectory identified by the YOLOv5 algorithm and the theoretical motion trajectory. From Fig. 20, the maximum error of the rope-climbing robot trajectory in the case of self-weight is 31.39 mm in the forward direction and 56.33 mm in the reverse direction, and the difference between the actual trajectory and the theoretical trajectory within 10 mm is 73.39 % in the forward direction and 52.54 % in the reverse direction. The maximum error in the case of 40 kg is 31.10 mm in the forward direction and 30.70 mm in the reverse direction, and the difference between the actual trajectory and the theoretical trajectory within 10 mm is 48.29 % in the forward direction and 64.22 % in the reverse direction. The maximum error in the case of 80 kg is 31.46 mm in the forward direction and 42.45 mm in the reverse direction. The difference between the actual trajectory and the theoretical trajectory within 10 mm is 50 % in the forward direction and 58.53 % in the reverse direction. So, the maximum error between the trajectory and the overhead rope parabolic theory trajectory under different loads is 56.33 mm, and the difference between the actual trajectory and the theoretical trajectory within 10 mm is more than 48.29 %, so the motion trajectory of the RCR is in line with the motion trajectory calculated based on the overhead rope parabolic theory.