This paper presents a conceptual design and proof-of-concept to control magnetic force using a wedge mechanism. The concept starts with the general knowledge that magnetic force is inversely proportional to distance. A magnetic model is generated consisting of a permanent magnet (PM), two magnetic flux paths, two wheels, and a steel plate. This model is subjected to formulate a mathematical relationship between the movement of the PM and a magnetic force exerted onto the steel plate. A 3D simulation has been conducted to verify the effectiveness of the developed mathematical model. The comparison between the mathematical and simulation models is likely to be fair. The next is to design a wedge mechanism which is composed of two wedges, a trenching plate, and a stepping motor. Finally, the proof-of-concept is placed on a magnetic force measurement and subjected to experiments. The result shows that the changes in the magnetic force according to the PM movements are 84.5% with simulation and 80.1% with the experiments, respectively. It can be concluded that the proposed concept is effective in reasonably controlling the magnetic force.

## I. INTRODUCTION

MAGNETIC force control (MFC) can significantly improve the mobility or efficiency of robots or devices. Compared with other adhesion methods such as the vacuum adhesion method, the magnetic adhesion method in a ferromagnetic environment has the advantages of superior surface adhesion and simple design structure. In modern ferromagnetic environments such as large plumbing facilities, large structures, bridges, and ships, etc.,^{1,2} it is often difficult to access and inspect the environments due to space constraints or safety issues. As an alternative, the mobile robots can be used to perform tasks in such environments.^{3–7} In such cases, there are vertical walls and inclined walls that require corridor adhesion, which can be considered the most important design element of the wall or floor. This type of adhesion is mainly achieved through magnetic force or vacuum adsorption. Especially, when the attachment surface is made of ferromagnetic materials or the surface is not smooth, the robot uses the magnetic adhesion method. Compared with the vacuum adhesion method, the driving efficiency of the magnetic wheel is relatively high. To enhance work efficiency on complex or inclined walking paths, the degree of freedom can be increased by controlling the adhesion of the magnetic adhesion method.

There are three types of magnetic adhesion methods for MFC: the permanent magnet (PM), electromagnet, and hybrid adhesion methods. An electromagnet consists of a magnetic core and a surrounding energized coil.^{8} The polarity of the electromagnet can be controlled by the external current and the strength of the electromagnet magnetism can be controlled by varying the amplitude of the current. This method is desirable considering the characteristics of MFC, but it may be difficult to get a desired level of magnetic field intensity. The PM adhesion method for MFC is usually in the form of magnetic wheel. To realize MFC, the magnetic wheels are designed by employing the interaction between two PMs.^{9} In most cases, one PM is assembled within the wheel and the other PM is placed at a distance location. The servo motor is used to rotate the outer PM for the MFC. This method is suitable for gaining the desired level of magnetic field intensity but less suitable for the MFC. The hybrid adhesion method uses both the PM and electromagnet. This method is acceptable in terms of both attaining the desired level of magnetic field intensity and MFC, but results in a bulky size. As such, each adhesion method has pros and cons.

Therefore, this paper proposes the wedge mechanism as a new way to control magnetism by adjusting the PM movement and verifies its validity through modeling, simulating and experimenting a prototype. The proposed MFC uses a single PM as the source of magnetic energy.

## II. CONCEPT AND MATHEMATICAL MODEL

### A. Concept and mathematical modeling

Fig. 1 shows the concept of the MFC system which consists of a single PM, flux paths, wheels, and a (invisible) steel plate. Only half the schematic is shown due to its symmetry. It is assumed that no gaps exist between the components. The concept starts with the general knowledge that magnetic force is inversely proportional to distance. In this design, the distance in 2D space can be equivalent to the effective area in 3D space, A_{PM}, as shown in Fig. 1. It is defined as the area in which the PM and flux path overlap, and can be expressed as

where, the respective *w*_{PM}, *h*_{PM}, and *x* are the width, the height, and the PM movement.

Using the Maxwell Stress Tensor,^{10} the magnetic force, **F**_{mag}, can be derived as

Here, B is the density of the magnetic flux between the wheel and the steel plate, and the surface integral, ∮_{s}*ds*, is the same as the defined effective area. The PM is the only source, so *B* can be replaced by *B*_{r} of the PM.

Equation (2) can be rewritten as

### B. Simulation and comparison

After obtaining the mathematical model, the magnetic force can be estimated according to the change of the position of the PM. For the estimation, rectangular NdFeB N52 and AISI 1008 steel have been selected as the PM and the materials for other components, respectively. The PM has a relative permeability of 1.05 with a residual flux density of 1.45 T. The AISI 1008 steel has an initial magnetic relative permeability of approximately 400 and a saturation magnetization of roughly 1,650 kA/m. The PM movement ranges from 0 to 25 mm.

When the PM movement is set to 0 mm, i.e., maximum effective area, the magnetic force between the wheel and the steel plate reaches its maximum value of 523 N. A simulation with FEMM has been carried out and compared with the mathematical model. The results indicate that the magnetic force variation is consistent between the simulation and the mathematical model. Therefore, the proposed MFC system is likely to be feasible for detailed design.

## III. DESIGN OF THE WEDGE MECHANISM AND SIMULATION

Figure 2 shows a general wedge mechanism. When Wedge 1 is pushed forward or backward along the y-axis, a vertical movement along the x-axis appears in the Wedge 2. The resolution along the x-axis depends on the angle of Wedge 1 as well as the accuracy of the stepper motor.

### A. Design of wedges

Fig. 3 illustrates how to estimate a wedge angle. Fig. 3(a) shows the ideal wedge geometry. In the figure, y is the length of the PM, and x is the wedge height to be estimated. Fig. 3(b) shows the geometry of the wedge that has been moved. In the figure, y’ indicates the moved distance of Wedge 1 by F in Fig. 2, and x’ is the moved distance of Wedge 2 accordingly. In the design, y and x’ are set to be 60mm and 5mm, respectively. First, take tangent of an angle θ from both geometries, then, put the two tangents in equality. Finally, x can be estimated by substituting various numbers into y’ considering the geometric limitation. When y’ is around 8mm, x becomes 42mm. The minimum wedge angle is estimated to be 31.47° when the respective x’ and y’ are 5mm and 8mm. The final wedge angle is determined to be 35° due to manufacturing convenience, etc. Wedge 1 needs a 15mm raise in height for additional space required for the ball screw. The height of Wedge 2 includes the housing for the PM. The dimensions of Wedge 1 and Wedge 2 are determined to be 60 x 57 x 35mm^{3} and 60 x 77 x 35mm^{3}, respectively.

### B. Motor selection

Based on statics, the minimum force for motor driving is estimated to be 250N. The selected motor is a stepper motor with a maximum force of 280 N and with a step angle of 1.8°. The ball screw has a pitch of 2 mm. The motor and ball screw are connected through coupling.

### C. 3D simulation

Due to the availability of manufactured components, a cylindrical NdFeB N52 has been selected. The dimensions of the PM are determined to be 25mm in diameter and 60mm in length. Simulation results confirm that the selected PM can provide sufficient magnetic force. Once the parameters of the PM have been determined, the 3D model of the MFC system is established in a commercial solver, Maxwell3D. Compared with the 2D basic model, the changes in the magnetic force in the 3D model are closer to the actual experimental results. Figs. 4(a) and 4(b) show the distribution of magnetic flux when the PM movement is 0mm and 25mm, respectively. It has been observed that when the PM movement is 0mm, most of the magnetic flux flows through the path and the flux density at the path is estimated to be 1.2 T. When the PM movement is 25mm, most of the magnetic flux flows into the air and the flux density at the path is estimated to be 0.32T. The simulation results show that the magnetic force decreases with the lessening of the effective area. It agrees well with the known knowledge mentioned in the concept.

## IV. FABRICATION

Two flux paths, two wheels, two wedges, and a trenching plate has to be made. The trenching plate, made of Al6061-T6, has a milled pocket to place and guide the wedges. The two wedges are also made of Al6060-T6. Wedge 1 has a male T-shaped protrusion along the center, and Wedge 2 has a female structure for the same shape. Such T-shape structures on the wedges and the pocket of the trenching plate allow the two wedges to securely move together, as designed. A hole with a diameter of 4mm is added to the design of the PM which is to be assembled onto Wedge2 with a screw. The stepper motor is connected to Wedge 1 through coupling and a ball screw. Power is provided through a DC power supply. The schematic and photo of the MFC system is shown in Figure 5.

## V. EXPERIMENTAL PROCEDURE

The experiments for the MFC system are carried out through the in-house magnetic force testing rig which consists of a stepping motor, a linear motion guide, force gauge (IMADA DS2-1000N), connection component, fixture, 20-mm thick steel plate, power supply, and controller. The operating range of the testing rig is up to 300mm, and the maximum load capacity is up to 300 N. First, the MFC system is placed on the steel plate. Then, the pulse input is sent to the stepper motor to raise the MFC system. Six experiments have been conducted at an interval of 5mm in PM movement. Each experiment is repeated five times to reduce the errors of the equipment and measurement process. Finally, the average value has been obtained.

## VI. RESULTS AND DISCUSSION

Figure 6 is a comparison of the mathematical modeling, simulation and experimental results. Overall, all the results show similar trends. According to the observation, the magnetic force decreases as the PM movement increases. At the beginning, the magnetic force of the simulation is higher than that of the experiment. This may be due to the assumptions made for the simulation. At midpoint, the result is reversed which may be due to the experimental errors. It is also observed that the change in the magnetic force according to the PM movements is 100% for mathematical model, 84.5% for simulation and 80.1% for the experiments, respectively. Such difference may be due to assumptions made for the modeling and simulation, and due to the PM cross section change from rectangle in mathematical modeling to round in simulation and experiments. Therefore, it can be concluded that the proposed concept is effective to reasonably control the magnetic force.

## VII. CONCLUSIONS

This paper proposes a new MFC system using the wedge mechanism for optimal operation of mobile robots. To validate the concept, the study created a magnetic model, and carried out mathematical modeling using the created magnetic model and simulation using FEMM, a 2D tool. A pair of wedges have been designed to achieve the concept, i.e., physical mechanism to securely move the PM. Once obtaining all the dimensions and material properties of the components for the MFC system, a 3D CAD model has been created by using the ANSYS Maxwell program, then was subjected for simulation. All components for the MFC system have been made and properly assembled. Experiments to measure the magnetic force have been carried out by mounting the MFC system onto a magnetic force testing system. According to observations, the magnetic force decreases as the PM movement increases. Observations also confirm that the changes in magnetic force according to the PM movements is 84.5% for simulation and 80.1% for the experiments, respectively. It can be concluded that the proposed concept is effective in reasonably controlling magnetic force.

## ACKNOWLEDGMENTS

This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (NO. 20171520101780).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.