On the static structural design of climbing robots: part 2
 Ausama Hadi Ahmed^{1} and
 Carlo Menon^{1}Email author
Received: 5 August 2015
Accepted: 10 November 2015
Published: 25 November 2015
Abstract
This manuscript is the second of two parts of a work investigating optimal configurations of legged climbing robots while loitering on vertical surfaces. In this Part 2, a structural analysis based on the finite element method, specifically the stiffness method, is performed to address the problem. Parameters that are investigated in this Part 2 include the inclination of both the body and the legs of the robot. Outcomes of the performed study are validated by analyzing the posture of 150 ants when loitering on vertical surfaces. The obtained validation ensures the predictions of the developed structural model are correct and can be used to identify optimal configurations of legged robots when loitering on vertical surfaces.
Keywords
Background
Climbing robots have a wide range of potential applications such as inspecting airplane wings, bridges and wind turbine blades, cleaning sky scrapers, welding and painting ships and tanks, maintaining nuclear plants, and in agriculture, surveillance and security. The structural design of some climbing robots developed by the scientific community was inspired by nature’s living organisms including worms [1], spiders [2], geckos [3], cockroaches [4] or a combination of different species [5].
Climbing robots have been designed to adhere to climbing surfaces using different mechanisms such as magnets [6, 7], electrostatics [8, 9] and air vortex [10, 11]. A number of the attaching mechanisms also mimic the nature. For instance, Sky Cleaner IV [12] and Robicen III [13] use suction to adhere to the climbing surfaces mimicking the suction cups of the octopus; Stikybot [14], MiniWhegs [15], Waalbot [16] and Abigaille [2, 17] use dry adhesives inspired by geckos; and Spinybot [18], and RiSE [19] use hooks inspired by cockroaches to adhere to nonsmooth surfaces.
In this work, the effect of different geometrical parameters on the maximum attachment force required by a robot to stay attached to a vertical surface is investigated. Specifically, the optimal configuration is considered to be the one that minimizes the adhesion requirements for the robot, that is minimizes the minimum of the maximum adhesion force of any foot.
In the first part (Part 1 [20]) of this twopart work, the body of the robot is assumed to be perpendicular to the legs and parallel to the climbing surface. Dimensions are normalized, that is, they are divided by the distance between the front and hind feet of the robot. Main findings of Part 1 are: the body height to body length ratio should be as small as possible; the normalized middle leg position should to be between 0.24 and 0.41 where the middle leg is positioned at 0 when it overlaps the hind leg and at 1 when it overlaps the front leg; and the stiffness of the body should be much larger than the one of the legs (e.g., the thickness of the body should be larger than that of the legs).
In this second part (Part 2) of the work, the role of both the inclination of the body and legs is investigated using the finite element method (FEM) [21]. The investigated parameters provide guidelines to design robots that loiter on vertical surfaces with minimum normal adhesion force. In addition, an example from nature is investigated to validate the assumptions and simplifications used in the developed FEM model. Specifically, the posture of ants when they are on vertical surfaces is investigated. An experiment is carried out to measure the geometrical parameters of the ants’ postures while on vertical surfaces. Many researchers have studied the stepping patterns in ants under the influence of speed [22, 23], curvature [23], different body geometries [24], under the influence of different loads [25] and on different slopes [26]. To the best of the authors ‘knowledge, little work was, however, performed on ants’ postures while loitering on vertical surfaces.
This paper is organized as follows. “Investigated parameters” section completes the theoretical investigation of Part 1. Specifically, the inclination of both the body and the legs are investigated. “Model verification: structural analysis for ants’ stance on vertical surfaces” section presents a verification of the performed twodimensional FEM analysis. Specifically, predicted results are compared to postures ants have on vertical surfaces. Conclusions and recommendations for the design of legged robots operating on vertical surfaces are presented in “Discussion” and “Conclusion” sections.
Investigated parameters
Body inclination and middle leg’s position
Similar to Part 1, the robot in this paragraph is assumed to have a height to body length ratio of 1:2. Results drawn from this specific geometry are generalized in a subsequent section.
Using a robot with height to body length ratio of 1:2 implies that collision occurs when the body inclination is either over 45° or less than −45°, where 0° inclination is defined when the robot’s body is parallel to the climbing surface (see angle \(\theta_{\text{B}}\) in Fig. 1). The height and the distance between the front and the hind legs are considered to be fixed to keep the height to length ratio fixed at every inclination, see Fig. 1. As discussed in Part 1, no units are used, as the results can be scaled up/down.
The inclination of the body by a positive angle \(\theta_{\text{B}}\) (counterclockwise angle in Fig. 2) causes the beams of the front half body of the robot to become longer (see Fig. 2) and therefore more flexible, whereas the beams on the hind half body (see Fig. 2) to become shorter and therefore stiffer. A negative body inclination angle causes an opposite effect. In other words, the body inclination affects the stiffness of the different parts of the robot. This general behavior can be generalized to robots having different height to length ratios. It should be noted that a robotic structure with a smaller height to length aspect ratio would have a smaller inclination range, thus limiting the choice of optimal body inclinations.
Leg’s inclination angle and middle leg’s position
The effect of having the legs inclined instead of perpendicular (see Part 1) to the climbing surface is investigated in this section. Similar to the previous sections, the adhesion force required to keep the robot on a perpendicular surface is assumed to be minimized.
The range of the front leg’s angle is from −90° to −45°, and the hind leg’s angle to range from −135° to −90°. At the maximum inclination, i.e., θ _{f} = θ _{m} = −45° and θ _{h} = −135°, all of the three joints are located at the center of mass. A wider inclination range at this height is not feasible without increasing the distance between the tips of the front and the hind legs.
From Fig. 8, the backward inclination of the legs improves the adhesion requirement when the middle leg’s position is below 0.38 (see red circle in Fig. 8b). The backward legs’ inclination for any other position will cause an increase in the required force. The effect of legs’ inclination for different height to length ratios has the same effect as the investigated structure with 1:2 height to length ratio; with the exception that the point that improves with the backward legs inclination is varied to be between 0.38 and 0.41 for the range of heights considered in Part 1.
The optimal configuration found is when the inclination of the front and the middle legs is at the maximum front, at −45°, and the hind leg is at the maximum from the perpendicular, at −135°. This result is similar to that found by Yasong et al. [2] with the assumption that the structure has infinite stiffness.
Optimal geometry
Model verification: structural analysis for ants’ stance on vertical surfaces

T _{B} is the distance between the front and hind legs’ tips.

S _{B} and T _{cm} are the body’s length and position, respectively.

S _{h} is the position of the middle leg’s coxa.

T _{h} is the position of the middle leg’s tip.

h is the height.

r, not shown in Fig. 11, is the body and legs’ thickness.
The ants are photographed when they are standing still to feed off honey drops on a surface of vertically mounted plexiglass. In total 150 ants are used, where 91 ants are photographed from either the top or the bottom, similar to that shown in Fig. 11b, and 59 ants are photographed from the side, similar to that shown in Fig. 11a.
In both of the photographed positions, i.e., from above and the side, the following parameters are measured: T _{B}, S _{B}, T _{cm}, S _{h}, and T _{h} (see Fig. 11). The height is measured only using the photos captured from the side. The photos captured from above are used to make sure that the ants are standing up vertically with an angle range of ±20° from vertical, since a change in the orientation of 20° would decrease the force pulling an ant to the back, due to gravity, by only ±6 %.
All of the measurement data are presented as ratios to overcome the variation in ants’ sizes. T _{B} is considered as a measuring unit for all of the measurements, i.e., T _{B} = 1, except for S _{h} which is measured with S _{B} as the measuring unit.
Parameters measured experimentally and the equivalent values used in calculations
Parameter  Experimentally  Value 

T _{B}  1  200 
T _{h}  0.62  124 
h  0.1439  28.78 
S _{B}  0.169  33.8 
S _{h}  0.358  12.1 
T _{cm}  0.616  123.25 
r  0.022  4.4 
The developed mathematical model presented earlier is used to compute the maximum required adhesion force for S _{h} and T _{h} while keeping all the other parameters fixed. The value of S _{h} is varied within the full range of 0–1, with the middle leg’s coxa coinciding with the hind leg’s coxa when S _{h} = 0, and coinciding with the front leg’s coxa when S _{h} = 1. Similarly, the value of the middle leg’s tip position T _{h} is considered to be 0 when the tip is aligned with the hind leg’s tip, and 1 when the middle leg’s tip is aligned with the front leg’s tip, i.e., the distance to the hind leg’s tip is 200.
Curves of maximum normal force for a range of heights for the robot are shown in Fig. 14 and are colored with shades varying from blue to green. The different heights are added to this figure to analyze the effect of changing the height; the authors in fact noticed that ants change their body height while loitering. The magenta line in Fig. 14 is obtained by intersecting all the curves in this figure and considering their lower values for each middle leg’s tip position. It, therefore, represents the maximum normal adhesion requirement for each position of the middle leg’s tip. It should be noted that the maximum value of this line is close to the averaged middle leg position of the ants (vertical red line in Fig. 14 at 0.62 normalized middle leg’s position). The maximum of that curve represents the position of the middle leg’s tip that experiences the minimum adhesion force requirement for the different heights. That point is different by only 2.6 % from the middle leg position used by the ants. It is interesting to note that the position of the middle leg used by ants to minimize the maximum adhesion required to adhere to vertical surfaces is 0.62, which is very close to the golden ratio often found in nature [27–29].
Discussion

For a fixed body length, all the legs of the robot should be inclined outwards, that is they should be extended as far as possible to increase the distance between the front and the hind tips of the legs.

The optimal position for the middle leg is to be as close as possible to the front leg and the inclination should be forward for a middle leg’s position between approximately 0.4 to 1, and backward otherwise.

The optimal result is found when the hip joints for the legs coincide and the middle leg’s tip position is at the same position as the front leg’s; this finding is confirmed though an optimization performed using GA [see Eq. (3)].

Depending on the height to length ratio, tilting the body mostly improves the adhesion requirement for the robot.

The optimal body and legs inclination for any middle leg’s position could be chosen from Fig. 9.
The steps used to investigate the effect of the different parameters on the structure of the robot are used to analyze the stance the ants use on vertical surfaces. Analytically, it is found that the closer the middle leg’s coxa is to the hind leg’s coxa the less the adhesion force is over the entire range of the middle leg’s tip position. Interestingly, the coxas of the middle and the hind legs are touching each other in ants, that is they are as close as their size allows. The distance from the center of the coxa of the middle leg to the center of the coxa of the hind leg is 33 % of the body length on average on the collected ants. The optimal middle leg’s tip position is at approximately 61 % of the distance between the tips of the hind and front legs pointing forward.
Conclusion
In this work, an investigation of the effect of the inclination of the body and legs on the minimal adhesion requirement for a climbing robot to adhere to a vertical surface is investigated. It is found that the optimal configuration to minimize the force required to adhere to a vertical surface is when the front and the middle legs are inclined forward and their tips overlap. Also, tilting the front of the body reduces the required adhesion force. The structural model used to investigate the effect of the different parameters on the adhesion requirements is used also to explain the positioning of the tip of the middle leg the ants use to stand on vertical surfaces; the stance that ants use minimizes the maximum adhesion force over the full range of the middle leg’s tip position. The model developed in this work is applicable to sixlegged robots. Similar to the analysis performed to investigate the ants’ stance, different robotic structures can be investigated following the same procedure used in this article.
Declarations
Authors’ contributions
Both authors were equally involved in the study and preparation of the manuscript. Both authors read and approved the final manuscript.
Acknowledgements
This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Libyan Ministry of Higher Education. The authors thank the members of the Menrva Lab for their support.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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