Optimization Design and Parameter Analysis of a Wheel with Array Magnets

Abstract

At present, a large number of magnetic wall-climbing robots are applied to various magnetically conductive metal facades for detection and anti-corrosion work. Limited by the wall-climbing mechanism and adsorption device, most wall-climbing robots can only climb on smooth walls, and it is difficult to adapt to complex walls. Therefore, by studying the multi-media magnetic circuit conduction mechanism, a permanent magnetic adsorption wheel with a magnet array arrangement was designed in this study and applied to a hinge-type wall-climbing robot. By analyzing the influence of structural parameters on the adsorption performance and optimization design, a magnetic wheel structure with a symmetric structure that can meet a variety of adsorption requirements was obtained. To analyze the mechanical characteristics of the wall-climbing robot under complex facade conditions, we researched the adsorption performance of the designed magnetic wheel in different wall structures. Finally, the adhesion force of the magnetic wheel was verified through experimental measurements, and it was found that the hinged wall-climbing robot could adapt to different structural features and complete wall-transition and obstacle-crossing movements.

1. Introduction

With the development of the petrochemical and shipping industries, more and more wall-climbing detection robots are used in high-altitude operations instead of manual labor, improving operational safety and detection efficiency [1,2]. Permanent magnet adsorption technology is pursued by a large number of scholars for stable and reliable performance and spaced adsorption methods and is also a research hotspot in the field of robotics [3,4].

H. L. Dong [5] designed a permanent magnet adsorption mechanism that can adjust the distance between the magnetic pole and the wall and developed a tracked wall-climbing robot to realize the automatic non-destructive detection of large walls. In addition, a large number of scholars at home and abroad installed the magnet directly on the tracked belt and designed a series of crawler wall-climbing robots, which allowed for stable adsorption, but its movement flexibility was poor [6,7,8,9]. Compared with the high load and high stability of crawler wall-climbing robots, wheeled robots have relatively flexible motion and a better ability to cross obstacles and transition between walls. J. C. Fan [10] installed a Halbach permanent magnet on a wall-climbing robot to provide a strong and stable adhesion force to realize robot crawling on an underwater catheter. A four-wheel wall-climbing robot was designed, and its permanent magnet was placed under the front and rear body modules for ship weld detection [11]. M. L. Zhang [12] proposed a method for calculating the magnetic adhesion force of the Halbach array magnetic circuit by using the equivalent magnetic flux density and determined a magnetic circuit structure that produces the largest magnetic adhesion force. The PMAD magnetic circuit was optimized and analyzed, and a new permanent magnet adsorption device was proposed to improve the utilization rate of the magnet [13]. To meet the demand for adjustable adhesion force under different working conditions, J. C. Fan [14] used the principle of internal force compensation to fuse electromagnetism and permanent magnetism and designed an adjustable adsorption mechanism to improve the adaptability of the wall-climbing mechanism. Y. L. Hu [15] carried out hybrid ring-symmetric magnetic circuit optimization research and developed a new permanent magnet wheel with a high magnet utilization rate by adjusting the structural parameters. Y. Tavakoli [16] combined an omnidirectional wheel with spacing permanent magnet adsorption to design an Omni-Climber wall-climbing robot with excellent flexibility to achieve safe and stable movement on different surfaces. M. Kawamura [17] put the ring magnet in the wheel and uses steel plates on both sides to conduct magnets to generate an adhesion force. M. L. Zhang [18,19] optimized the layout of the Helbach array, designed an arc magnet and put it in the rubber wheel, and used the lever mechanism to achieve rapid demagnetization. Many new wall-climbing robots were developed to enable wall transitions. M. Tavakoli [20] utilized gimbals as driving wheels, aided by a magnetically controllable adhesion module and a two-DOF simple manipulator for the wall transition. S. T. Nguyen [21] and R. Bisht [22] used a multi-DOF arm to connect two permanent magnet adhesion wheeled moving mechanisms, and by controlling the movement of the arm, the robot could achieve a multi-angle wall transition. Y. Zhang [23] presented a class of inchworm-inspired multimodal soft crawling–climbing robots that could achieve crawling, climbing, and transitioning between horizontal and vertical planes. M. M. Khalil [24] proposed an insect-scale climbing robot with a low-cost dry adhesive technology to address surface-to-surface transitions.

Conventional wall-climbing robots usually have the adsorption device fixed to the robot chassis or placed on the wheels near the wall, which results in the robot wheels not having circumferential adsorption capability. In this study, a permanent magnetic adhesion wheel with a magnetic array arrangement was designed by investigating the multi-media magnetic circuit conduction mechanism. Through simulation experiments, various factors that affect the magnetic adhesion force were analyzed to provide a method for optimizing the magnetic wheel structure to meet different adhesion requirements. A hinge-type robot with strong motion capability while enabling wall transition was obtained by combining the magnetic wheel with a hinge-type robot construction.

The main sections of this paper are arranged as follows. The Section 2 introduces the overall structure of the hinge-type wall climbing robot and the specific structure of the array magnetic wheel. The Section 3 gives the analysis results of various structural parameters and changing trends that affect the adsorption performance of the magnetic wheel in detail. The Section 4 describes the performance of the magnetic wheel under a complex wall surface. The Section 5 describes the experiments that were carried out to verify the designed magnetic wheel and wall-climbing robot.

2. Overall Structure

With reference to the physiological structure and movement patterns of arthropods and mollusks, we fused the permanent magnet adsorption principle and mobile robot technology to obtain a hinge-type wall-climbing robot. The hinge-type wall-climbing robot is mainly composed of vehicle modules, magnetic wheels, and a flexible connecting rod. We set up three flexible hinges on the connecting rod to simulate the joints and muscles of the creature, the vehicle module to simulate the body, and the magnetic wheels to simulate the feet. The vehicle modules mainly consist of driving and control units, and the wheels provide adhesion and power. Because of the flexible hinges, the rod has three degrees of freedom (DOFs) in different directions, which can produce active or passive deformation when the robot actively adjusts its posture or when the wall features change, achieving flexible movements, as well as passive adaptive wall surface to ensure stable adsorption. The vehicle modules and flexible hinge are relatively independent, and we can flexibly configure them to expand into a multi-body flexible robot according to different needs and scenarios. In this study, we only took the two vehicle modules as an example for analysis and research and verified the proposed new methods and ideas.

2.1. The Hinge-Type Wall-Climbing Robot

The moving mechanism has a symmetrical four-wheel drive structure, in which the magnetic wheels are fixedly connected to the vehicle module, and each wheel control unit is integrated inside the vehicle modules. The magnetic wheel is based on the principle of array magnetic adsorption, placing multiple cylindrical magnets in the aluminum wheel hub, using the high permeability of the yoke to transfer magnetic energy, reducing magnetic loss to increase the adhesion force. The rubber layer of the wheel hub improves the friction between the magnetic wheel and the wall to ensure stable movement.

In Figure 1, the two vehicle modules are connected using a passive adaptive flexible hinge mechanism, which has three DOFs in the pitch, yaw, and roll directions. The roll joint ensures that the four wheels fit closely to the curvature-changing wall while improving the ability of a single wheel to cross obstacles. By actively adjusting the wheel speed, the yaw joint enables the robot to adjust its attitude and move flexibly with a very small turning radius. By adjusting the front and rear vehicle wheel module rotational speed actively and adjusting passively after encountering obstacles, the pitch joint can be used to change the direction of the rear vehicle drive force to facilitate obstacle crossing. Through the active adjustment of three flexible joints and passive adaptive deformation, the hinge-type wall climbing robot has strong curved surface adaptability, movement flexibility, wall transition, and obstacle-crossing ability.

2.2. Structural Design of the Magnetic Wheel

The robot works on a magnetically conductive elevation, whose adsorption performance directly affects the motion reliability. Therefore, the adsorption mechanism design optimization is the key to ensuring movement reliability, where the greater the adhesion force, the better the robot’s movement reliability. At present, the improvement in the adhesion force mainly depends on increasing the weight of the magnet, resulting in technical conflicts between the improvement in adhesion force and lightweight design. Therefore, it is very important to design a lightweight adsorption mechanism. This study innovated and designed an array-type adsorption structure with high magnetic energy product utilization. Through magnetic circuit optimization research, a lightweight and strong adsorption permanent magnet adsorption wheel was obtained.

In the study of permanent magnet adsorption wheels, it was found that circular magnets are generally used as permanent magnet excitation sources to provide a magnetic adhesion force for wheels. Figure 2 shows the evolution and design process of the magnetic wheel. Figure 2a,b show the magnetic simulation of radial magnetic ring magnets and axial magnetic ring magnets as adsorption sources. Both magnetic charging methods can provide a better adhesion force, but radial magnetization causes more difficulty in the production process. We prefer magnets with axial magnetization. After observing the magnetic simulation results of the axially magnetized arc magnet, it was found that only the magnetic induction lines near the wall formed a circuit through the conductive wall to generate adhesion force, and the magnetic induction lines in other regions formed a circuit with the air medium, resulting in a relatively low magnetic energy utilization rate of the magnet. After studying the large collection ability of a yoke on magnetic induction lines, we added a yoke on both sides of the magnet. Through the simulation results, we found that a yoke has a good cohesive ability and the adhesion force was greatly improved. Since the adhesion force is related to the magnetic properties, the adhesion force after adding the yoke was about 5–10 times higher than before. On this basis, considering the lightweight design and wheel structure, we used an array of cylindrical magnets instead of ring magnets, which still have good performance.

Based on the above analysis, we carried out the multi-media magnetic circuit optimization design of the magnetic wheel. By increasing the yoke with high magnetic permeability, we could adjust the magnetic circuit, reduce the loss of magnetic energy, improve the utilization rate of magnetic energy, and maximize the performance of the magnet. Combined with the mechanical design method, a permanent magnet array adsorption wheel was obtained. Its specific structure is shown in Figure 3.

The magnetic wheel is arranged symmetrically and consists of a wheel hub with a rubber layer, high permeability yokes, and cylindrical permanent magnets. The permanent magnets are installed in an aluminum hub wrapped with rubber according to the same polarity array. The two sides of the magnets are closely attached to the ring yoke made of electrician pure iron, forming a coaxial matching with the wheel hub through the shoulder on the yokes. At the same time, there are various sizes of threaded holes on the yoke for axial fastening, and the yoke can be easily disassembled by using screws to lift it. The outer diameter of the rubber is greater than the outer diameter of the yoke, which ensures that the rubber is in contact with the wall when the magnetic wheel works. Rubber deformation is used to improve the friction coefficient between the magnetic wheel and the wall while protecting the wall surface and solving the problem of stable movement. Through the strength curve of the magnetic induction line in Figure 4b, it can be clearly found that the yokes on both sides of the wheel can converge the magnetic induction lines generated by each permanent magnet and conduct them to the contact position with the wall with small loss to form a magnetic circuit, which can greatly improve the magnetic energy volume utilization rate of the magnets, and then generate constant and strong adsorption.

3. Optimization of the Magnetic Wheel Parameters

Aiming at the above magnetic wheel structure, simulation experiments were systematically carried out to optimize the structural parameters and finally obtain a lightweight magnetic wheel with a strong adhesion force. It was found that the adhesion force of the magnetic wheel was directly related to the performance and number of magnets, the thickness of the yoke, the thickness of the yoke shoulder, and the distance between the wheel and the wall. We carried out simulations to optimize the above influencing parameters.

To analyze the influence of the magnetic material, number of magnets, yoke thickness, yoke shoulder thickness, air gap height, and other factors on the adhesion force, we conducted several comparative tests with 3–4 types of magnetic wheels as prototypes for each variable. Therefore, a large number of simulation experiments were carried out to analyze the influence of different parameters of the magnetic wheel on the adsorption performance. Furthermore, a naming rule was developed to facilitate a clear understanding of each structural parameter in the magnetic wheel, as shown in

Table 1. Naming rules for different configurations of magnetic wheels.

Parameter	Value
Magnetic material	N42
Number of magnets	20
Thickness of yoke	2
Thickness of yoke shoulder	2

.

For example, N42-20-2-2 indicates a magnetic wheel consisting of 20 magnets (N42), a 2 mm thick yoke, and a 2 mm thick yoke shoulder, while N52-28-3-1 indicates a magnetic wheel consisting of 28 magnets (N52), a 3 mm thick yoke, and 1 mm thick yoke shoulder.

3.1. Influence of the Material and Number of Magnets on the Adsorption Performance of the Magnetic Wheel

Restricted by the structure of the magnetic wheel, the shape and size of the magnet were fixed. However, we could change the magnetic material to adjust the adhesion force, as well as increase or decrease the number of magnets according to different adsorption needs. The following figure shows the simulation results.

Through the simulation results shown in Figure 5, we can see that in the same structure and environment, the better the magnet performance, the greater the adhesion force of the magnetic wheel, and the number of magnets was positively correlated with the adhesion force. At point “20”, the magnetic circuit formed by each magnet in the yoke was affected by the polarity of the neighboring magnets, which made the magnetic induction near the wall increase slowly and lead to a change in the magnetic adhesion force. Compared with the magnetic material, the magnetic field strength on both sides of the magnetic wheel could be quickly improved by adding the magnet. Then, the yokes on both sides were converged and transmitted so that the magnetic field strength in the contact area between the wheel and the wall was continuously improved, leading to the adsorption performance being simply and directly improved. Therefore, to enhance the adsorption performance of the wheel on a large scale, the preferred method was to increase the number of magnets and subsequently consider improving the magnetic material performance.

3.2. Influence of the Yoke Thickness on the Magnetic Wheel Adsorption Performance

With the optimization of multi-dielectric magnetic circuit for the magnetic wheel, we found that the yokes on both sides had a good converging conduction effect on the magnets. To explore the effect of the yoke on the adhesion force, we selected four combinations, namely, N = 8, N = 16, N = 24, and N = 32, for the simulation analysis, and the results are shown in Figure 6.

As shown in Figure 6, increasing the thickness of the yoke could improve the adsorption performance of the wheel to the best condition regardless of the magnet configuration for this wheel structure. However, as the number of magnets increased, the thickness of the yoke required to reach the maximum adsorption state became larger, mainly owing to the following reasons. As the number of magnets increased, the wheel generated a greater magnetic field strength and energy, which required a thicker yoke to gather and conduct. As the thickness of the yoke increased, it had a stronger ability to collect and conduct magnetic induction lines, which could increase the local magnetic field strength and improve the adhesion force of the magnetic wheel.

3.3. Influence of the Yoke Shoulder Thickness on the Magnetic Wheel Adsorption Performance

We also selected four combinations of N = 8, N = 16, N = 24, and N = 32 magnetic wheels for the simulation analysis to study the influence on the magnetic wheel adhesion force by the yoke shoulder, and the results are shown in Figure 7.

As shown in Figure 7, as the thickness of the yoke shoulder increased, the contact area between the magnetic wheel and the wall was enlarged, which improved the efficiency of the magnetic circuit transfer and increased the magnetic wheel adhesion force. When the shoulder thickness continued to increase, the spacing between the two yokes gradually decreased, and more and more magnetic induction lines formed a circuit directly between the two yokes through the air, resulting in a reduction in the adhesion force of the magnetic wheel. By observing Figure 7, it was found that relatively good adsorption performance could be achieved for each magnetic wheel at 2 mm. To simplify the optimization process, we considered that the magnetic wheels could excite the best performance under the existing structural conditions with the yoke shoulder thickness of 2 mm.

3.4. Influence of the Distance between the Magnetic Wheel and the Wall on the Adsorption Performance

Regarding permanent magnet adsorption, the air gap between the magnet and the wall directly affected the adsorption performance. In order to choose a more suitable rubber thickness, we used the following five configurations of magnetic wheels to simulate for verification: “N42-16-2-2”, “N42-20-2-2”, “N42-24-2-2”, “N42-28-2-2”, and “N42-32-2-2”. The specific simulation results are shown in the figure below.

Through Figure 8, we clearly found that the air gap between the magnetic wheel and the wall had a relatively large impact on the adsorption performance, and as the air gap became larger, the magnetic field loss in the air medium increased, causing the magnetic force to decay. The adsorption performance of the magnetic wheel decayed by about 50% when the wheel was about 1 mm away from the wall. Therefore, when the rubber was too thick, this caused great attenuation of adhesion force, and when the rubber layer was too thin, there was scratching of the wall surface. Taking this into account, the thickness of the rubber layer was chosen to be 0.25 mm above the outer diameter of the yoke.

The parameters that affected the adsorption performance of the magnetic wheel were the magnetic wheel structure parameters (magnet material and number, yoke thickness, yoke shoulder width, etc.), air gap, and other factors. Through software simulation, we analyzed the impact of the above factors on the adsorption performance of the designed wheel systematically to provide certain basic support for obtaining a lightweight, high-performance, and demand-compliant magnetic wheel. Since there is a strong nonlinear coupling between each wheel structure parameter, we selected and optimized the magnetic wheels in the order of magnet material, number of magnets, yoke thickness, and shoulder thickness.

So far, we have described the analysis of the transfer mechanism of the magnetic circuit in multi-media and obtained the influence of each parameter on the adhesion force by simulating and analyzing the adsorption performance of the magnetic wheel. Under the same size structure, by adjusting the magnet material, number of magnets, yoke thickness, and increasing the yoke shoulder width, we could cause the magnet wheel mechanism to have a large adhesion force range (0–200 N), which could basically adapt to most of the detection wall-climbing robots without redesigning the adsorption mechanism, which greatly improved the design cost and efficiency.

4. Influence of the Wall Characteristics on the Magnetic Wheel Adhesion Force

Through the above analysis, we obtained the influence of the key parameters of the magnetic wheel on the adsorption performance by using simulations and result analysis, which provided a reference to carry out reasonable magnetic wheel and parameter configuration according to different adsorption requirements. Since this study focused on the movement of the robot on a complex facade, we needed to study the performance of the magnetic wheel under different wall characteristics.

4.1. Magnetic Homogeneity of the Magnetic Wheels

In this study, the adsorption principle of the magnetic wheel is that the array of magnets relies on the yokes to converge the magnetic energy to the area near the wall, which generates a superposition of magnetic fields and increases the magnetic field strength. As shown in the Figure 9. to further verify the permeability of the yoke and the stability of the magnetic wheel’s adsorption performance in each circumferential position, we conducted simulations with three configurations of magnetic wheels: “N42-20-2-2”, “N42-24-2-2”, and “N42-28-2-2”. We set a magnet directly above the wall as the initial state, and then gradually changed the angle θ to change the relative position of the wheel and the wall to simulate the wheel rotation by observing the adhesion force to verify the adsorption performance under the circumferential position of the magnetic wheel. Since the magnet wheel structure has a relatively regular symmetry, we only needed to analyze the position between two adjacent magnets.

The simulation data is shown in

Table 2. Simulation data sheet for different configurations.

	0°	3°	6°	9°	12°	15°	18°	21°	24°
N42-20-2-2	69.6	70.2	69.8	71.1	68.9	70.7	70.3	69.7	69.9
N42-24-2-2	76.6	78.1	77.0	76.7	78.5	76.7	78.0	77.9	77.5
N42-28-2-2	105.3	104.2	104.7	105.5	104	104.1	105	105.7	104.8

.

We could calculate the mean and variance of each measured value, as well as the magnitude of the maximum amount of error with respect to the mean, and the constancy of the magnetic wheel adsorption could be measured using the above parameters. For the “N42-20-2-2” magnetic wheel:

$$\left\{\begin{array}{c}\overline{F}=\frac{1}{n}{\displaystyle \sum _{i=1}^m}{F}_i=70.0N\\ {\delta }^{\prime }=\frac{F_{max}-F_{min}}{\overline{F}}=0.016\end{array}\right.$$

(1)

For the “N42-24-2-2” magnetic wheel:

$$\left\{\begin{array}{c}\overline{F}=\frac{1}{n}{\displaystyle {\displaystyle \sum }_{i=1}^m}{F}_i=77.4N\\ {\delta }^{\prime }=\frac{F_{max}-F_{min}}{\overline{F}}=0.024\end{array}\right.$$

(2)

For the “N42-28-2-2” magnetic wheel:

$$\left\{\begin{array}{c}\overline{F}=\frac{1}{n}{\displaystyle {\displaystyle \sum }_{i=1}^m}{F}_i=104.8N\\ {\delta }^{\prime }=\frac{F_{max}-F_{min}}{\overline{F}}=0.009\end{array}\right.$$

(3)

Through the above simulation and analysis, we could intuitively find that the adhesion force generated by the magnetic wheel designed in this study was basically constant when it was in contact with the wall at different positions. In turn, it could be confirmed that the circumferential array arrangement of permanent magnets could adjust the magnetic circuit after passing through the yoke and concentrate most of the magnetic energy to the contact area with the wall, thus ensuring a relatively stable and uniform magnetic output of the magnetic wheel in the circumferential direction.

4.2. Influence of a Convex Corner of Two Intersecting Walls on the Performance of the Magnetic Wheel

The robot needs to achieve a convex transition when climbing on a facade; therefore, we studied the change in magnetic force for the magnetic wheel during the convex transition. Since the two walls are at convex angles and the magnetic wheel was farther away relative to the second wall at the position where the walls intersected, we only needed to analyze the adsorption on the first wall, which referred to the influence of the overlapping area between the magnetic wheel and the wall on the adsorption performance. Three configurations of magnetic wheels, namely, “N42-20-2-2”, “N42-24-2-2”, and “N42-28-2-2”, were used for the simulation verification. The schematic diagram and simulation results are shown in Figure 10, where x represents the distance between the contact point and the intersecting line.

From the above figure, we can see that when x = 0, that is, the center of the wheel is above the intersecting line, the effective overlap of the wall that the magnetic wheel could touch was the smallest and the minimum adhesion force was about 50%. With the increase in x, the effective overlap area between the magnetic wheel and the wall also increased, and after x > 20 mm, the adhesion force did not change and achieved the best performance. It can be seen that during the transition across the convex-angled wall, the closer the magnetic wheel was to the wall intersection line, the smaller the adhesion force until it was reduced to 50%, which also provided data support for subsequent mechanical analysis.

4.3. The Influence of a Concave Corner of Two Intersecting Walls on the Performance of the Magnetic Wheel

The robot also needs to realize concave corner transitions; therefore, we studied the change in magnetic force during the transition. As shown in Figure 11, before the magnetic wheel touches the second wall, the influence of the second wall on the wheel is small, and thus, we only need to analyze the adsorption of the wheel on the first wall; when the wheel touches the second wall, both walls and the wheel generate magnetic field loops, forming a vertical wall adhesion force. Because the environment is the same, we assumed that the adhesion force generated by the two walls was the same, and the total adhesion force direction could be calculated using an angled parallelepiped.

From the geometric relationship, the following was obtained:

$$F_{mag}=2{F^{\prime }}_{mag}sin\left(\frac{\theta }{2}\right)$$

(4)

Two configurations of magnetic wheels, namely, “N42-24-2-2” and “N42-28-2-2”, were used for the simulation. Moreover, by using the above equation, we processed the data at different wall transition angles to obtain the adhesion force on the two wall surfaces, as shown in Figure 12.

The adhesion force generated by the magnetic wheel at different pinch angles of the wall varied with θ, but the single-sided adhesion force calculated according to Equation (4) was within a constant range, basically between 80% and 90%. The reason for this was that when there were multiple wall contacts on the magnetic wheel, the magnetic energy generated by the magnets was dispersed to the area of the given contact position and still provided a constant magnetic field strength to produce magnetic adhesion force on the wall. Based on the above analysis, when the magnetic wheel touched two walls at the same time, the adhesion force of the magnetic wheel to each wall was vertical to the wall, and the adhesion force was the same.

4.4. Influence of a Step Obstacle Crossing on the Performance of the Magnetic Wheel

For the obstacle-crossing process as shown in Figure 13, when encountering small obstacles, such as welds, we can use a small height step obstacle for the analysis, and when encountering large obstacles that exist at an angle, we can refer to the convex transition form for consideration; therefore, our main analysis here was the effect of step obstacles on the performance of the magnetic wheel. By analyzing the adhesion performance of the magnetic wheel at the Section 4.2 convex angle transition, we assumed that when the magnetic wheel encountered a step-type obstacle, the first wall adsorption characteristics basically did not change and the direction was perpendicular to the wall, and thus, we could obtain the force between the obstacle and the magnetic wheel using the simulation results. Two configurations of magnetic wheels, namely, “N42-24-2-2” and “N42-28-2-2”, were simulated to investigate the adsorption performance of the magnetic wheels in this scenario.

Through the geometric relations, we could obtain the following:

$$\left\{\begin{array}{c}{F}_{2}cos\sigma =F_{mag-x}\\ F_1+F_{2}sin\sigma =F_{mag-y}\end{array}\right.$$

(5)

According to Equation (5), we could calculate the adhesion force of the wall and the adhesion force provided by the obstacle, as shown in the Figure 14:

Similar to the situation of the concave corner transition, the adhesion force of the wheel varied similarly with the height of the obstacle when crossing a step-type obstacle. However, the adhesion force on the first wall was more constant and was essentially 90% of the normal value of the magnetic wheel. The adhesion force generated by the obstacle became larger but grew more slowly with the height of the obstacle and had a maximum of 15%.

4.5. Influence of Wheel Tilt on the Adsorption Performance

As shown in Figure 15, when the robot is climbing on a circular wall or the road surface undulates, the left and right wheels cannot fit the wall completely and there is a certain angle between the magnetic wheels and the wall; therefore, we needed to consider the influence of this angle on the adhesion force of the magnetic wheels. Two configurations of magnetic wheels, namely, “N42-24-2-2” and “N42-28-2-2”, were used for the simulation verification.

When the wheel is tilted at an angle to the wall, the yoke on one side directly touched the wall, and the distance between the other yoke and the wall became larger, resulting in the adhesion force becoming smaller. The rubber layer was in contact with the wall in the initial state, and when the angle started to appear, the compression on one side of the rubber layer became larger, which directly led to the yoke touching the wall and the other yoke was relatively close to the wall at this time; therefore, there was a sharp increase in the adhesion force when θ = 3°. As the angle became larger, the magnetic wheel adhesion force dropped sharply. At the initial design of the magnetic wheel, a safety factor k = 1.2 is often reserved to improve the reliability of the robot adsorption; therefore, we set “s = 0.8” as the safety threshold, meaning that when the adhesion force decays to less than “s”, the magnetic wheel is in a dangerous state and it was hard to ensure stable adsorption. When the robot is in this condition, there may be safety hazards, such as dropping and overturning. Analyzing the simulation data, we could find that the adhesion force dropped to 80% at a tilt angle of about 8°, and thus, we assumed that the maximum tilt angle at which the magnetic wheel could meet the effective adsorption was 8°.

When the robot moves along the axis of the circular wall, the wheels on both sides have the same tilt angle with respect to the wall where they are located, as shown in Figure 16.

Through geometric relations, we could obtain that the minimum arc radius that the hinged robot can crawl safely and steadily is

$$R_{wall}·sin{\theta }_{wall-min}=\frac{B}{2}$$

(6)

When B = 300 mm, the minimum curvature radius that the robot can adapt to is about 1 m, which can basically cover most of the structural areas of large metal structure equipment under the premise of meeting the magnetic wheel adhesion force.

5. Experiment

By analyzing the mechanical characteristics of the robot in the space elevation, we obtained the required adhesion force of the magnetic wheel as 100 N. Combined with the above analysis of the influence of the magnetic wheel structure, under the premise of meeting the design requirements, we took the adhesion force and its weight as the comprehensive optimization target, and finally obtained the parameters of the magnetic wheel as shown in

Table 3. Magnetic wheel parameters for hinge-type wall climbing robots.

Parameter	Value
Dimensions	$\varnothing 154\ast 24$
Wight	650 g
Adhesion force	105 N
Magnet material	N42
Number and size of magnets	$24\times \varnothing 8\ast 20$
Yoke size	$\varnothing 153.5\ast 2$
Yoke shoulder size	2
Rubber layer size	$\varnothing 154\ast 16$

.

As shown in Figure 17,to verify the adhesion force of the designed wheel, we tested the adhesion force of the wheel on a 5 mm thickness plane with different numbers of magnets for yoke thicknesses t_yoke = 2 mm and t_yoke = 2.5 mm. For different working conditions, namely, horizontal and vertical wheel positions, several sets of measurements were carried out by means of a dynamometer. Meanwhile, different positions of the magnetic wheel were selected for each measurement and the data were recorded in

Table 4. Actual magnetic wheel measurement data.

	1	2	3	4	5	6	7	8	Average	Simulation Data
N42-20-2-2	59.6	60.2	59.8	61.1	60.7	60.3	59.7	59.9	60.16	70.8
N42-24-2-2	70.6	71.1	71.0	70.7	71.2	71.8	70.9	70.5	70.98	78.47
N42-28-2-2	95.3	94.2	89.7	91.5	88.1	93	88.7	90.8	91.4	105.25
N42-20-2.5-2	81.3	80.2	80.1	80.1	79.8	79.5	80.1	80.9	80.3	90.54
N42-24-2.5-2	101.4	100.3	101	99.8	101.5	98.1	101	99.2	100.3	106.8
N42-28-2.5-2	106.8	108	100.5	103.0	109.4	99.5	98.9	109.5	104.5	116.2

.

By analyzing the above experimental measurement data, we found that the adhesion force did not fluctuate significantly after changing the measurement position for the same wheel configuration, proving the uniformity of the adhesion force of the wheel. We then compared the measured and simulated values of each wheel and found that the measured values were about 15–20% smaller than the simulated values, with the reason being that the simulated data were conducted under an ideal condition and some influencing parameters were ignored; at the same time, the wheel material properties, processing accuracy, and measurement errors would lead to small measured values of the wheel.

After completing the optimization of the magnetic wheel, we carried out experiments using the hinge-type robot as shown in Figure 18. We performed movement experiments on vertical surfaces, circular arc surfaces, and intersecting walls and found that the robot could adsorb to the walls stably and reliably without dropping or slipping. By using the deformation of the flexible connecting rod, the robot can perform flexible movements, wall transitions, and obstacle transitions up to 70 mm in height.

6. Conclusions

Aimed at satisfying the demand for the automatic detection of steel structures, such as petrochemical tanks, ships, and bridges, we designed an array-type permanent magnetic wheel that can provide a large range of adhesion force and applied it to a hinge-type wall-climbing robot. Through theoretical analysis, simulations, and experiments, we obtained the following conclusions.

By analyzing the conduction mechanism of magnetic energy in the magnetic wheel and studying the influences of the parameters on the adsorption performance, a lightweight and high-performance magnetic wheel was obtained. The adhesion force requirements were met by changing the structural parameters of the magnetic wheel for different environmental and task requirements. For example, the magnetic wheel structure used in this study could reach a maximum adhesion force of 2000% of its weight. The robot used in this study, for example, provided 105 N of adsorption force with a 650 g wheel, which was 1.6 times its weight.

Subsequently, we studied the performance of the magnetic wheel in various complex and special environments, including intersecting walls, obstacles, and inclined walls. We analyzed and explored the adsorption performance of the magnetic wheel in these situations, and provided a theoretical basis for analyzing the robot’s mechanical properties. We analyzed the influence of each structural parameter on the adhesion force and then optimized the magnetic wheel performance to achieve an increase in adhesion force with a small increase in mass. Indirectly, the described technical conflict between weight and adsorption force was solved.

Finally, a hinge-type wall-climbing robot with an array of magnetic wheels was designed. The experiments showed that the measured values of the adhesion force of the designed magnetic wheel were basically in accordance with the theoretical simulation analysis, and the robot could stably adsorb on the wall while completing complex motions, such as a wall transition and obstacle crossing.