# On the static structural design of climbing robots: part 1

- Ausama Hadi Ahmed
^{1}and - Carlo Menon
^{1}Email author

**Received: **5 August 2015

**Accepted: **10 November 2015

**Published: **1 December 2015

## Abstract

This manuscript is the first of two parts of a work investigating optimal configurations of legged climbing robots while loitering on vertical surfaces. In this part 1, a mathematical model of a climbing robot based on the finite element method (FEM), specifically the stiffness method, is generated. A number of parameters, namely the height of the robot, the length of its body and the position of its legs, are investigated to assess their effect on the adhesion requirements needed for the robot to stay attached to a wall. Predictions of the developed mathematical model are validated using FEM commercial software. The body and the legs are assumed to be perpendicular to each other in this part 1. The effect of their inclination is investigated in the subsequent part 2 of our work. In part 2, the model is also used to predict postures that ants have while standing on vertical surfaces. The model is validated by comparing the predicted results to images of loitering ants. The parameters investigated provide guidelines to design legged climbing robots.

## Keywords

## Background

A variety of legged robots having different size and design have been proposed in the literature to climb vertical surfaces. Such robots have different locomotion systems including tracks [1–3], wheels [4, 5], and legs [6, 7]. Legged robots, which are of primary interest for this work, use a variety of mechanisms to adhere to climbing surfaces, including magnets [8], dry adhesion, in the form of single layer [9, 10] or hierarchical [11], spines/claws [12–15], and negative pressure [16].

The design of some climbing robots was inspired by nature’s living organisms including geckos [7], spiders [6], cockroaches [12] or a combination of different species such as geckos and cockroaches [17]. The arrangement and inclination of the legs are not the same among the different designs of the robots. In fact, some of the robots, including Spinybot II [15], had their legs inclined forward; others, including Abigaille III [11] and ROBUG II [8], have some of their legs inclined forward and some backward; some others, including the RiSE and DIGbot [12, 18], have their legs on the sides of their bodies, and others, including Abigaille II [6], have legs symmetrically distributed around their bodies.

In this work, the size and the configuration of legged climbing robots are analyzed by investigating the effect that different design parameters have on the maximum attachment force required by the robot to stay attached to a vertical surface. This work investigates the design of six-legged robots. They are analyzed in a 2-D space by taking into account the geometrical symmetries hexapods generally have. The finite element method (FEM) is used to analyze the force distribution on the tips of the robotic legs; FEM is selected as it provides more accurate results than approximated quasi-static methods generally used to investigate this problem [1, 6, 9–11, 19–21]. In this work, dimensionless parameters are selected as the beams used in the structures investigated in this work are scalable as long as the second moment of inertia is fixed.

The amount of force available to the robot to adhere to a vertical surface is critical regardless of the type of attachment mechanism (e.g., dry adhesion, suction, magnets) used to climb. A successful climbing robot should, in fact, always have enough force to be able to adhere to the wall. In this work, the optimal structure of the robot is considered to be the one that requires the least attachment force to adhere to a vertical wall. This optimal structure would maximize the safety factor to avoid detachment or equivalently minimize the adhesion strength required by the adhesive pads or grasping mechanisms used by the robot to adhere to a vertical surface. The formulation developed in this work is, therefore, general and is suitable to model legged robots relying on a large variety of different sources of adhesion.

This manuscript is organized as follows: “Kinematics” section presents the model and the kinematics of the examined multi-legged structure; “Structural analysis” section describes the proposed method to analyze the force distribution of the robot; “Investigated parameters” section presents results obtained by changing the different geometrical parameters of the considered structure on the force distribution on the tips of the robot’s legs. Conclusions and recommendations for the design of climbing legged robots are drawn at the end of the manuscript.

## Kinematics

It should be noted that while this article specifically addresses robots in a static configuration, results of this work could be generalized to a certain extent to dynamic systems, as inertial forces resulting from accelerations of the robot would simply add to the weight of the robot, without affecting the optimal geometries investigated in this work. Variation of posture during walking is not addressed in this work as it resides outside the scope of this study.

## Structural analysis

*K*is the structural stiffness matrix, \(F\) is a vector representing both the known forces applied to the structure and the unknown reaction forces of the nodes and \(D\) is a vector comprising the known and the unknown displacements of the nodes. Damping is not included as a static analysis is considered in this work.

The structure of the robot is divided into six separate beams, see Fig. 2. Specifically, each of the three legs, the connection between the hind leg and the center of mass, the connection between the center of mass and the middle leg and the connection between the middle leg and the front leg are all considered to be separate beams. The case when the middle leg is located between the hind leg and the center of mass is also formed using six beams: specifically, the three legs, the connection between the hind leg and the middle leg, the connection between the middle leg and the center of mass and the connection between the center of mass and the front leg.

*x*and

*y*axes (see Fig. 2), are equal to zero. The unknown degrees of freedom are the distance the unconstrained nodes moved after applying the known forces on the structure; from Fig. 2, the unknown degrees of freedom are the linear movement of nodes 1, 2, 3 and 4, and the rotation movement of all of the nodes, namely nodes 1–7. The known forces are the weight of the robot at the center of mass, and the linear force components of all of the unconstrained nodes, namely nodes 1–4 along with the moment on all of the nodes, namely nodes 1–7, are equal to zero. The unknown forces are the reaction forces at the hinges, namely \(F_{\text{hx}} , F_{\text{hy}} , F_{\text{mx}} , F_{\text{my}} , F_{\text{fx}}\) and \(F_{\text{fy}}\), which are shown in Fig. 2. Equation (1) can, therefore, be written as:

Commercially available finite element method (FEM) software, i.e., ANSYS (V14.0), is used to verify the correct implementation of the stiffness method. Specifically, the beam element BEAM188 based on Timoshenko beam theory is used to analyze 2D structures [29]. The MATLAB code is tested against the ANSYS software by comparing randomly selected cases solved using MATLAB with the same cases solved using ANSYS.

## Investigated parameters

The FEM presented in “Structural analysis” is used in this section to minimize the normal adhesion required by the robot to stay attached to a vertical surface, i.e., \(F_{\text{hy}} , F_{\text{my}}\) and \(F_{\text{fy}}\) in Fig. 2. The normal force is investigated by examining different parameters of the structure with the assumption that the legs are always perpendicular and the body is parallel to the climbing surface. The investigated parameters are: (1) the position of the middle leg; (2) the body height to body length aspect ratio; (3) the cross-sectional area of the beams forming the body and the legs. These three parameters are investigated in the following sections in pairs. Specifically, first the parameters 1 and 2 are investigated while parameter 3 is fixed; subsequently, the parameters 1 and 3 are investigated while parameter 2 is fixed. The combined effect of the three parameters is generalized in a later section.

### Effect of height to length ratio and middle leg’s position

The *x* axis in Fig. 3 represents the position of the middle leg, where 0 means that the middle leg is positioned at the back of the robot. In this configuration, the middle leg has the same position as the hind leg. The value of the *x* axis increases as the middle leg gets closer to the front leg and the value equals 1 when the middle and front legs have the same position. The *y* axis represents the height to length aspect ratio and the *z* axis represents the normal force per body weight. From Fig. 3, increasing the body height to body length ratio, on the *x* axis, requires higher force to keep the robot attached to the vertical surface because, a robot with higher height and fixed weight causes an increase in the torque applied to the robot’s structure due to gravity which needs higher forces on the tips of the legs to keep the robot in equilibrium than that required by a robot with lower height.

The normal force in Fig. 3 has two peaks located at middle leg positions of 0.07 and 0.93; the first peak is located at the hind leg with a maximum of 64.16 and the second is located at the middle leg with a maximum of 56.66. The peaks can be explained by analyzing the shear and normal forces distributions for a specific robotic structure with fixed height to length aspect ratio. The shear force distribution due to changing the middle leg’s position is explained first and the normal force distribution resulting from changing the middle leg’s position is explained next.

#### Shear force distribution due to middle leg’s position

Figure 5, which is obtained through an ANSYS simulation, shows the deflections in the structure. In Fig. 5, \(B_{\text{m}}\) is the beam connecting \(J_{\text{Hm}}\) to \({\text{cg}}\). \(B_{\text{f}}\) and \(B_{\text{h}}\) are instead the beams connecting \(J_{\text{Hf}}\) to \(J_{\text{Hm}}\) and \(J_{\text{Hh}}\) to \(cg\), respectively, when the middle leg is located between \({\text{cg}}\) and \(J_{\text{Hf}}\). These two parameters, that is \(B_{\text{f}}\) and \(B_{\text{h}}\), are the beams connecting \(J_{\text{Hf}}\) to \(cg\) and \(J_{\text{Hh}}\) to \(J_{\text{Hm}}\), respectively, when the middle leg is located between \(cg\) and \(J_{\text{Hh}}\).

When the middle leg is located between the center of mass and the front leg, the body’s deflection creates a compression in \(B_{\text{m}}\) and \(B_{\text{f}}\) and an expansion in \(B_{\text{h}}\), thus causing the distances \(\delta_{\text{f}} ,\) \(\delta_{\text{m}}\) and \(\delta_{\text{h}}\) to be less than \(\delta_{cg}\). The distance \(\delta_{\text{h}}\) is equal to the compression in \(B_{\text{h}}\) subtracted from \(\delta_{cg}\); also, \(\delta_{\text{m}}\) is equal to the elongation in \(B_{\text{m}}\) subtracted from \(\delta_{cg}\), and \(\delta_{\text{f}}\) is equal to the compression in \(B_{\text{f}}\) subtracted from \(\delta_{\text{m}}\).

The maximum distance that \(J_{\text{Hm}}\) travels is when it is located at the center of mass \(cg,\) which corresponds to the maximum force it experiences. The expansion in \(B_{\text{f}}\) and the compression in \(B_{\text{h}}\) cause \(\delta_{\text{f}}\) and \(\delta_{\text{h}}\) to be less than \(\delta_{\text{m}}\); these expansion and compression generate less shear force in the hind and the front legs than that in the middle one (see Fig. 4 when the middle leg’s position is at 0.5). The front and middle legs have the same shear force when the middle leg’s position is at 1, because \(\delta_{\text{m}}\) and \(\delta_{\text{f}}\) are equal. The amount of expansion in \(B_{\text{f}}\) increases with the length, causing \(\delta_{\text{f}}\) to be smaller than \(\delta_{\text{m}}\) and thus generating less shear force in the front leg than that in the middle one (see Fig. 4 for a middle leg’s position ranging between 0.5 and 1).

The case when the middle leg is located between the center of mass and the hind leg could be analyzed as done previously. The main difference is that the beam \(B_{\text{m}}\) undergoes compression instead of expansion.

#### Normal force distribution due to middle leg’s position

*x*axis, a shear \(F_{\text{P}}\) towards the positive

*y*axis, and a torque \(\tau\) applied at \(J_{\text{Hm}}\). The effect of the wrench can be analyzed by considering the structure that contains the middle leg, the front leg and \(B_{\text{f}}\) in Fig. 7. The effect of the individual components of the wrench is explained as follows:

First, the effect of \(\tau_{\text{L}}\): applying a torque \(\tau_{\text{L}}\) at \(J_{\text{Hm}}\) causes a tension force to be generated in the front leg and a compression force in the middle leg. A decrease in the length of \(B_{\text{f}}\), due to a change in the middle leg’s position towards the front, causes an increase in the tension and the compression magnitude in the front and middle legs, and vice versa.

Second, the effect of \(F_{\text{L}}\): applying force \(F_{\text{L}}\) at \(J_{\text{Hm}}\) causes a tension force to be generated in the middle leg and a compression force in the front leg. A decrease in the length of \(B_{\text{f}}\), due to a change in the middle leg’s position, causes an increase in the amount of the tension and compression, and vice versa. Third, the effect of \(F_{\text{P}}\): applying force \(F_{\text{P}}\) in the positive direction of the *y* axis at \(J_{\text{Hm}}\) causes a tension force in only the middle leg, i.e., a decrease in the normal force of the middle leg.

Each of the first two wrench components, namely \(\tau_{\text{L}}\) and \(F_{\text{L}}\), causes opposite forces along the middle and the front legs, thus causing the big difference in the normal forces of the middle and front legs, which are colored green and blue in Fig. 6. The difference in the forces is a result of the opposite forces generated in the middle and front legs by \(\tau_{\text{L}}\) and \(F_{\text{L}}\).

#### Optimal middle leg position and height to length ratio

The optimizer is configured to search for the optimal middle leg’s position within the range of 0–0.95 to prevent the optimizer from converging to the global optimum at 1. In fact, the global optimum is not considered to be the most desirable value as a small variation from the minimum causes a dramatic increase in the adhesion requirement. In fact, in Fig. 6, a small variation of the leg from its optimal position causes the maximum adhesion requirement to increase dramatically. For example, a 0.01 variation in position causes a more than 250-fold increase in the required adhesion.

It should be noted that the 2.1 is the smallest height that is considered in this study. In fact, the radius of the structure is assumed to be two and a minimum gap between the robot and the surface is assumed to be 0.1. As expected, it can be concluded that the height should be as low as possible.

### Effect of cross-sectional area and middle leg position on force distribution

In this section, the effect of the stiffness of the legs is considered. Specifically, we investigate whether a robot should have stiff or compliant legs to minimize the adhesion force required to adhere to a vertical surface. To change the stiffness of the legs, their cross-sectional area was varied.

At first, a structure with body length of 200 and a height of 100 is arbitrarily chosen to explore the effect of changing the cross-sectional area on the normal force distribution of a robot. Results drawn from this specific geometry are generalized in a subsequent section.

The cross-sectional area and the area moment of inertia are varied, while the weight of the robot is considered to be fixed at one and applied at the center of mass. The area moment of inertia is calculated to be equivalent to that of a circle; the radius is calculated from the cross-sectional area, and the area moment of inertia is then calculated accordingly. The normal force distribution is calculated for different cross-sectional area values for the legs, between \(0.0001\;{\text{and}}\;3.16 \times 10^{4}\), while keeping the cross-sectional area of the body, formed by the horizontal beams in Fig. 1, fixed at one. This analysis, therefore, explores if the legs should be more or less stiff than the body to minimize the maximum required adhesion.

The best position for the middle leg, in the range between 0 and 0.99, is located between 0.3 and 0.42 for the range of legs’ cross-sectional area from 0.045 to 100, while the best range for smaller cross-sectional area, less than 0.045, jumps to be at 0.99, see Fig. 11b. For any cross-sectional area, the best position of the middle leg is when it overlaps the front leg, i.e., the middle leg has a position equal to one for any cross-sectional area value.

In summary, the optimal configuration when the body is parallel and the legs are perpendicular to the vertical surface is when the structure has a minimum legs’ cross-sectional area of \(0.0001\) and a middle leg’s position of 0.99. Changing the body’s cross-sectional area and fixing the legs’ cross-sectional area have an opposite adhesion force requirement behavior to that shown in Fig. 10; the lowest point of the graph is when the body cross-sectional area is at minimum, which equals \(10^{ - 4}\), and the maximum point is when the radius at maximum, which equals \(3.16 \times 10^{4}\).

### Effect of middle leg position, height and legs’ cross-sectional area

The best middle leg’s position for a range of height to length ratios, chosen arbitrarily between 0.01 and 10, and different cross-sectional area between \(0.0001\) and \(3.16 \times 10^{4}\) is bounded between 0.24 and 0.41. Figure 12 allows the designer to identify the optimal middle leg’s position for different legs’ cross-sectional areas at different height to length ratios.

In Fig. 12, the best configurations are the ones with the lowest height to length ratio and it is found that the best configuration at a specific height is the one with the highest legs cross-sectional area value. In fact, the improvement in the adhesion for the entire cross-sectional areas investigated range, and the height to length ratios is calculated to be between 22 and 71 %. For example, at the specific height to length ratio of 1, arbitrarily chosen, the best adhesion requirement at the considered cross-sectional areas sorted from best to worse is: [3.14 × 10^{4} 4.07 × 10^{3} 530.9 66.48 12.57 1.54 0.283 0.0314], where the first four cross-sectional areas have almost the same adhesion requirement value and the improvement in the adhesion force requirement between the cross-sectional area of \(3.14 \times 10^{4}\) and \(0.0314\) is calculated to be 29 %.

In part 2 of the paper, the posture of living ants loitering on a vertical surface is used to confirm the validity of the assumptions used in this paper. The authors simplified the ants’ structure in the same manner as in this part 1. The model proposed in this part 1 paper was used to predict the configuration of the ants’ posture and the effect of both their middle leg’s position and body’s cross-sectional area.

## Conclusion

In this work, the effect of different geometrical parameters on the structure of a legged robot is investigated and analyzed using the stiffness method. To improve the efficiency of vertical climbing, the height to length ratio of the robot should be kept as low as possible, the cross-sectional area of the legs should be as big as possible, and the middle leg should be positioned between 0.24 and 0.41. The presented parametric study is utilized as an outline to assist designing the structure of climbing robots when loitering on a vertical surface. The presented results are applicable to any robot size as they are unit-less. Equation (6), which is at the foundation of the developed code, can potentially be implemented in a microcontroller to optimize in real time the posture of the robot. Results presented in this work yield the following guidelines to design climbing robots: (1) height to length ratio should be as small as possible; (2) the cross-sectional area of the structure should be as big as possible; and (3) the position of the middle leg can be selected from Fig. 12, which provides the optimal middle leg’s position as a function of the height to length ratio and the cross-sectional area of the structure.

## 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 was 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 Access**This 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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