Abstract
This paper presents the analysis of ropes’ bending on threedimension edges by ascending robots. A rope ascending robot (RAR) is a type of exterior wallworking robot that utilizes a synthetic rope to traverse the outer surface of a building. Ropebased façade cleaning robots demonstrate effective performance in wellstructured buildings. However, in unstructured buildings, the rope used by these robots may become entangled or caught on various structures, presenting a significant challenge for their operation. If the rope becomes caught on a structure, the robot will be unable to move to its intended position. In more severe cases, the rope may become damaged, leading to potential failure or even a fall of the robot. Therefore, solving this problem is crucial for safe and efficient robot operation. Consequently, this study defines the issue of the rope becoming caught on a structure as a ropelocking problem and analyzes it by categorizing it based on the dimensions of contact between the rope and the edge. To address the varying tension experienced in different areas, the rope was divided into micro units and subjected to a threedimensional analysis to resolve the ropelocking problem. Additionally, the analysis was verified by experiments.
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Introduction
Modern buildings are becoming increasingly diverse with the development of architectural technologies. Modern buildings are designed to be more aesthetically pleasing than existing buildings. The use of ropes for building maintenance and repair can pose a significant risk of accidents for workers. This is due to the fact that the method used for maintenance and repair typically involves workers climbing the building directly using ropes.
This study aimed to address the safety concerns associated with building maintenance and repair by developing a ropeascending robot (RAR) (Fig. 1). The RAR is equipped with robotembedded ascenders, which allows it to move freely on structured buildings using two fiber ropes. This is different from other robots used for building maintenance and repair, such as dualrope winch robots^{1}, tworopedriven mobile robots^{2}, and dualascender robots^{3,4}, which rely on an external winch. However, the use of ropes in the RAR presents a limitation for its application in unstructured buildings, where the lack of anchor points and support structures for the ropes makes it challenging to use the robot.
Specifically, support structure can makes RAR critical problems. RAR hangs from a building by tying end of the rope to a fixed anchor and winding other end to robot respectively. RAR’s operation characteristic can cause the rope to become lock onto its support structure that existed in rope path. This phenomenon is usually caused by the frictional force generated when the rope comes in contact with the support structure due to rope’s bending. Rope may cut by accumulated damage in the rope and rope’s cut can cause robot’s falling under the building in severe cases. However, using a separate control technique for RAR could potentially enable its application in unstructured buildings.
A separate control technique for RAR’s application in unstructured buildings was named Rope Impact Control (RIC). When rope is locked on support structure of unstructured buildings, the rope reaches force equilibrium and also forms a specific angle according to the rope path. So, if rope’s contact angle with structure can be change to reach force disequilibrium, rope can be unlock from lock state. In short, RIC is a control that take advantage of force disequilibrium uses impact to change rope’s contact angle with support structure. However, in order to implement RIC to RAR, not only required impact to rope by RAR’s movement also analysis of the angle and frictional force of ropes and support structures is required. Therefore, rope analysis on structures preceded the development of a control technique for the use of RAR in unstructured buildings.
Therefore, to address the ropelocking problem of the rope, this study analyzed the force generated when the rope comes into contact with the edge of a structure. In general, force analysis research on ropelocking problems has existed before and is still actively ongoing^{5,6,7,8}. Among them, commonly used methods include analysis rope as group of nodes^{5,6} and finite analysis^{7,8}. However, those analysis methods are about force analysis in the rope axial direction regarding the frictional force that occurs when a rope is wrapped around an object different with direction of analysis. In short, there was a requirement of novel analysis that conducted to the direction perpendicular to the direction of rope movement for implement of RIC. Therefore, the ropelocking problem on shear direction according to the robot’s movement was solved through dynamic analysis in this study.
The force analysis of the rope on the edge sets a constraint similar to the situation in which the robot is hung on the rope for application in robot control. When the robot was suspended from the rope, the weight of the robot remained constant, and the tension in the rope did not change. Therefore, the bending angle was used to unlock or move the rope. Force analysis was conducted in a relatively simple twodimensional and threedimensional order, and the induction equation of the capstan equation, which is commonly used for rope tension analysis, was used.
The remainder of this paper is organized as follows. In “Mathematical analysis” Section, the analysis is conducted by dividing the definition of the rope contact that occurs when the RAR moves on an unstructured building into two or three dimensions. In “Experimental verification” Section, the planning of the verification experiment and the actual experiment are presented. “Analysis of Experimental results” Section presents an analysis of the experimental results obtained using a fiber rope. Finally, in “Conclusion” Section, the conclusions are drawn.
Mathematical analysis
Problem definition
The definition of the “ropelocking problem” that occurs when a robot moves on unstructured building should be preceded to analyze the rope characteristics for robot movement. This problem is closely related to the rope characteristics. For the RAR to move on a building, both ends of the rope must be fixed to the building and ascender. When both ends of the rope were fixed, the rope always attempted to be located at the shortest distance between connected endpoints. This rope characteristic is generally not a problem in structured buildings where there are few obstacles between the endpoints when the robot is roped on the building to move to the top of the building. However, in unstructured buildings, a rope is inevitably caught at the edge of an obstacle (Fig. 2). Therefore, in this section, a relatively simple mathematical twodimensional analysis of the ropelocking problem of the rope was performed. Accordingly, a threedimensional analysis was then conducted to predict the rope movement in the ropelocking problem for robot control.
Rope contact for twodimensional edge
In this section, we analyze the ropelocking problem that occurs when the rope contacts the edge on a plane (Fig. 3a). The ropelocking problem is typically caused by friction on the edge and occurs when the rope is in contact with the edge. Friction is defined as: Friction Coefficient \((\mu )\) X Normal force \((F_N)\). If \(\mu \) is constant, then friction is affected only by \(F_N\)^{9,10}. \(F_N\) is caused by tension at both ends of the rope on ropelocking problem. Both ends of the tension vary according to the capstan equation (\(T_{load} = T_{hold} * e^{\mu \theta }\)). The rope bending angle \(\theta \) in this equation significantly affects the ropelocking problem^{11,12,13}. To analyze the relationship between the bending angle \(\theta \) and \(F_N\), a differential analysis of the force according to the bending angle of the rope on the xy surface is conducted considering the situation shown in Fig. 3a. The force equation for the y axis is as follows:
where \(d\theta \) is the degree to which the rope is wound around the edge and \(dF_N\) is the normal force that the rope receives from the edge.
Further, the rope was hung at the edge at a bending angle of \(\theta \), as shown in Fig. 3b. If the rope in the suspended state is pulled at a certain angle, no force is applied by the upper rope tension based on the edge, and only the force due to the lower rope tension exists. Therefore, the force equation for the z axis is as follows:
where \(\mu \) is the friction coefficient between the rope and the edge and \(\phi \) is the pulling angle. Finally, the integration of Equations using by Eqs. (1) and (2) is as follows:
Equation (3) implies that even if the rope is caught at the edge, the rope locking at the edge can be moved using only the angle at which the rope is pulled, provided that Please check if this part should be expressed as the friction coefficient \((\mu )\) and the bending angle \((\theta )\).
Rope contact for n threedimensional edge
The threedimensional ropelocking problem is illustrated in Fig. 4a. The appearance of the rope on the edge is different from that in the twodimensional analysis. Nonetheless, the force is generated in a similar manner. Therefore, the analysis in this section starts similarly to the interpretation in the previous section. However, to simplify the analysis complexity according to the parameters that occur when the rope makes contact, this study assumed two analysis methods for the rope.

Assumptions

1.
The shape of rope winding is assumed to be helical.

2.
Rope contacts the threedimensional edge by surrounding it in a circular manner.

1.
Additionally, the analysis of the ropelocking problem involves the rope coming into contact with the edge. Figure 2b depicts the RAR suspended from the end of the rope, and while locking can occur in the robot, this problem is mitigated owing to the constraints of the rope. Because the RAR moves using two ropes, the two ropes are always in a stable state (assumption 1), and the same solution method as the Capstan equation can be applied (assumption 2) by assuming that the rope comes into contact with the edge by encircling it in a circular manner. Thus, the two assumptions were transformed into two conditions.

Conditions

1.
A certain angle is maintained as long as the rope is hanging from the edge.

2.
Rope bending can be solved similar to the problem of a pulley contacting a threedimensional edge.

1.
The force generated by the ropelocking problem between the rope and the edge in three dimensions is expressed in (Fig. 4b). For ease of expression, global and rope coordinates are expressed as X–Y–Z and x–y–z, respectively. v represents the speed of the rope on the edge, and because one end of the rope is fixed as an anchor, the direction is expressed as the yaxis in the rope coordinates (circular motion). \(F_r\) denotes the friction between the rope and edge and is expressed as a vector in the opposite direction of motion. \(\alpha \) denotes the angle between the edge and rope and \(d\phi \) indicates the angle of the rope movement. The problem is difficult to solve because of the variation in the direction of the force. Therefore, to lower the order of the force analysis, the force was analyzed using its projection on the x–z plane and the force equation was solved (Fig. 5a).
Sum of forces on X–Z plane
where \({d\theta }_1\) denotes the ropebending angle in the X–Z plane.
However, there are several variables that can be used to solve the force equation using only Eq. (4). Therefore, the force equation can be solved by simplifying it, which can be achieved by projecting the force onto the Y–Z plane to reduce the number of variables, as illustrated in Fig. 5b.
Sum of forces on Y–Z plane
where \({d\theta }_2\) denotes the ropebending angle in the Y–Z plane. By combining Xaxis equation in Eq. 4 X and Yaxis equation in Eq. 5, we obtain Eq. (6) as follows:
\(F_r\) follows the Coulomb friction. Furthermore, substituting Zaxis equation in Eq. (4), the final equation can be obtained as
where \(\mu \) denotes the total friction coefficient between the edge and rope. Finally, integration must be performed, and an independent judgment between the variables is inevitably required to solve 7. However, \(\phi \), \(\alpha \) and \({\theta }_1\) are independent of the changes in each variable integrated over the variables individually, we obtain:
Equation (8) shows that regardless of the amount of tension applied to the rope, rope locking can occur on a threedimensional edge. This result is similar to that expressed in Eq. (3) in the twodimensional analysis. However, the rope is influenced by gravity and exists only below the edge. Therefore, the ropelocking problem can be solved under the constraint \(\alpha + \phi \le 180^{\circ }\).
Experimental verification
Experimental verification was conducted to determine if the force analysis of ropelocking problem performed in Section 2 on two and threedimensional edges is accurate. To create a situation similar to that in which the equations were solved, a verification experiment was performed for each equation. A testbench was used to perform verification experiments. The testbench was largely composed of an anchor and an anglemoving device (Fig. 6a). The anchor was installed at the end of the testbench such that the rope gets locked during its installation. The anglemoving device consisted of a rope pulley, a motor, and a ball screw, which allowed the rope pulley to move in both directions(Fig. 6a).
In addition, a unique method was used to set variables used in the equations as this could not be achieved solely using the testbench. Experiments were conducted by changing the rope type to vary the friction coefficient \(\mu \). Two types of ropes were used in the experiment: red rope (\(\mu = 0.081\)) and black rope (\(\mu = 0.11\)) (Fig. 6a). Except for different surface roughness values, the two ropes had identical structures with a radius of \(6\pi \). To set the bending angle \(\theta \), experiments were conducted by varying the height of the anchor to determine \(\theta \). For this experiment, the values of \(\theta \) used were \(90^{\circ }\) and \(127^{\circ }\) that selected as smallest and largest, which assumed the contact scenario between the rope and edge. This was accomplished by adjusting the height of the anchor, as shown in Fig. 6b.
Before proceeding with the experiment, an initial setup was required. In both experiments, the rope was hung in the order of an anchoredge rope pulley to create a bending angle, and a weight was attached to the end of the rope. The weight was set to 20 kg (10 kg \(\times \) 2), which was the weight of the RAR. However, the twodimension and threedimension experiments require different angles when a rope is caught on an edge. In the twodimensional edge scenario, the initial setting of the angle of anchor edge weight should be \(\phi = 0^{\circ }\) (Fig. 7a,b), whereas for the threedimensional edge scenario, it should be \(\phi = \alpha \) (Fig. 7c,d). After this initial setup, in each experiment, the ball screw was moved using a motor in the direction in which the angle of the anchoredge weight increased, and \(\phi \) was measured precisely at the moment when the rope was unlocked at the edge. Each same experiment was repeated five times to reduce uncertainty.
Experimental result
The resulting graph was divided into two graphs according to the bending angle \(\theta \). The estimated value was denoted using a bar graph, and the error between the estimated and experimental values is denoted by an error graph above the bar (Fig. 8 and Table 1). Because it is difficult to measure the friction coefficient \(\mu \) according to the rope type, and to compare the estimated and experimental values to verify the force analysis, we assumed friction coefficient values of 0.081 and 0.11 for the red and black ropes, respectively^{9,10}.
Case 1: Bending angle \(\theta = 90^{\circ }\)
First, the experimental verification results of the proposed force analysis method in the case of “locking problems” with a bending angle of \(\theta = 90^{\circ }\) show similar results for the estimated values (Eqs. 3 and 8) and experimental values (Fig. 8a). Examining the results from the error perspective of the estimated and experimental values, the “locking problem” on the bending angle \(\theta = 90^{\circ }\) tends to result in higher error values in twodimensional edge scenarios compared to those in threedimensional edge scenarios (\(0.15^{\circ }\)–\(0.19^{\circ }\)). This error is caused by the difference in the constraint conditions, and it is obvious that the threedimensional edge scenario is more vulnerable to the collapse of the force equilibrium than its twodimensional counterpart.
Additionally, analyzing the results according to the rope type, a lower error was observed in the case of black rope compared to that obtained for the red rope, which showed a maximum error of \(0.15^{\circ }\) between the results in the threedimensional edge scenario. However, the tendency of the result according to the friction coefficient \(\mu \) (rope type) is similar, and the error between the estimated and experimental values is in acceptable range. Therefore, it can be considered as an uncertainty caused by the rope characteristics.
Case 2: Bending angle \(\theta = 127^{\circ }\)
Further, the experimental verification results obtained in the case of ”locking problems” with a bending angle of \(\theta = 127^{\circ }\) are shown in Fig. 8b. The overall results were similar to that observed in the case of \(\theta = 90^{\circ }\). Under all conditions, the estimated and experimental values were similar, and fewer errors occurred in the case of black rope compared to those observed when using the red rope. However, a singularity of the result occurred in the twodimensional edge case using a black rope. The average of the experimental values was only \(9.65^{\circ }\), whereas the estimated value was \(13.92^{\circ }\). This result indicates that Eq. 3 was incorrect, owing to which the error rate exceeded \(30.6\%\).
Singularity analysis
Various factors are responsible for the error rates of the estimated and experimental values for bending angle \(\theta = 127^{\circ }\), in the case of twodimensional edge using black rope. However, considering that the range of the error between the estimated and experimental values was \(9.25^{\circ }\) to \(10.06^{\circ }\) and that the equation yielded accurate results when the bending angle \(\theta =90^{\circ }\), it can be concluded that the issue lies with the friction coefficient \(\mu \). Therefore, further experiments were conducted by fixing the experimental conditions in two dimensions: the black rope condition and changing the bending angle \(\theta \) between \( 90^{\circ } \& 127^{\circ }\).
The results of these experiments are shown in Fig. 9a. The graphs obtained have different shapes despite using the same rope for the experimental values. The slopes of the three graphs are similar and it was confirmed that the results had a tendency to vary depending on the bending angle \(\theta \). The tendency checks confirmed that it behaved differently depending on the bending angle \(\theta \). Therefore, the friction coefficient \(\mu \) at the bending angle \(\theta = 127^{\circ }\) was modified to 0.077, which is equivalent to the experimental value provided in Fig. 9b and Table 2. After modification, the estimated value was \(9.74^{\circ }\), and the error from the experimental value was in the range of \(\pm 4\%\).
Analysis of experimental results
To more accurately examine the difference between the estimated and experimental values, the values were compared using the rootmeansquare error (RMSE). The RMSE of the angle of rope movement \(\phi \) was calculated based on Eq. (9) and is plotted in Fig. 10. Overall, a slight difference was observed between the experimental and estimated values obtained by force analysis. However, the synthetic rope used in the experiment was a timevarying system, that is, its characteristics often changed over time^{14,15,16}. Considering these properties, an error of less than \(1.5^{\circ }\) between the estimated and experiment values increases the reliability of the estimation accuracy. Therefore, the force analysis relationship between rope and edge (Eqs. 3 and 8) was confirmed to be accurate, and the ropelocking problem can be solved if angle of rope \(\phi \) can be changed whatever their condition is.
Conclusion
This study presents a method for creating an external wall working robot that utilizes rope by conducting a force analysis on the ropelocking problem that may arise during its operation. The term ropelocking problem refers to a situation where the tensioned rope becomes trapped by a structure. Through force analysis, it was determined that the rope could be moved at an angle \(\phi \), regardless of the applied tension. The calculated equation was verified by experiment on the testbench, which was created assuming that the rope is hung on the robot. When comparing the estimated and experimental values with RMSE after the verification experiment, it was confirmed that errors of less than \(15\%\) (red rope) and less than \(10\%\) (black rope) occurred according to the rope type. However, because the RMSE size is not large (\(<1.3^{\circ }\)), it was judged to be uncertain due to the characteristics of the synthetic fiber rope used, and the ropelocking problem was concluded to be solved.
Therefore, as a followup to this study, a motion control algorithm will be developed and implemented to address the ropelocking problem, and to ensure the smooth operation of the RAR exterior wall working robot. By applying the corresponding motion control algorithm, obstacles in the application of an exterior wallworking robot, such as RAR, which utilizes ropes for climbing unstructured buildings, can be eliminated. Moreover, if the constraints utilized in this study are generally applied to analyze the forces required to resolve the ropelocking problem, its application may not be limited to robots that utilize ropes, and can assist in addressing movement disturbance in studies related to cranes and winches.
Data availibility
All data generated or analysed during this study are included in this published article.
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Acknowledgements
This research was supported by a National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT for Bridge Convergence R &D Program (NRF2021M3C1C3096807, NRF2021M3C1C3096808) and partly supported by the MOTIE (Ministry of Trade, Industry, and Energy) in Korea, under the Human Resource Development Program for Industrial Innovation(Global) (P0017306, Global Human Resource Development for Innovative Design in Robot and Engineering) supervised by the Korea Institute for Advancement of Technology (KIAT).
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M.C. wrote the main manuscript text and studied the dynamics of the rope. S.A. assisted conducting experiment. H.S.K. and T.S. supervised the research and developed the project. M.C. and T.S. made idea of the research, all authors read the manuscript and contributed to its final form.
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Choi, M., Ahn, S., Kim, H.S. et al. Examining the mechanics of rope bending over a threedimensional edge in ascending robots. Sci Rep 13, 20663 (2023). https://doi.org/10.1038/s41598023480785
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DOI: https://doi.org/10.1038/s41598023480785
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