Design of 3D Magnetic Tactile Sensors with High Sensing Accuracy Guided by the Theoretical Model

The past decade has witnessed a surging interest in the study of magnetic tactile sensors that can detect subtle changes in both normal and shear forces. However, due to the lack of guidance by appropriate theoretical models, the development of previous magnetic tactile sensors relies either on a trial‐and‐error manner or tedious point‐by‐point experimental calibrations, which are costly and time‐inefficient. Here, a theoretical model integrating magnetics, artificial neural networks, and nonlinear solid mechanics is proposed for the first time to guide the design of 3D magnetic tactile sensors. Then, a button‐shaped magnetic tactile sensor prototype that can detect subtle triaxial force changes is fabricated, which relates the nonlinear magnetic flux density to the external force, without burdensome calibration procedures. The sensor can achieve an axial measurement error of less than 1% and an in‐plane error of less than 3.7% with excellent durability. This study provides a comprehensive understanding of magnetic tactile sensors and sheds light on their applications in soft robotics, intelligent manipulation, and human–robot interaction (HRI).


Introduction
Touch is a fundamental approach for humans to sense and interact with their surroundings. [1] Temperature changes, pain, irritation, vibration, or other stimulations can be transmitted to the brain through the tactile receptors and connected nerves. [2] Besides humans, the interpretation of touch in robotics also plays a crucial role in the further development of humanrobot interaction (HRI), where a fundamental concept is that a robot can perceive objects, understand processed signals, and respond to stimuli.
As the interactions between humans and robots become increasingly complex, there is a growing need to develop tactile sensors with the advantages such as smaller size, low manufacturing cost, durability, and high precision. [3] Recently, magnetic tactile sensors have garnered considerable academic attention in next-generation HRI technologies. [3,4] For clarity, we will briefly introduce some representative works in this field: In 1988, Clark first proposed a prototype of the magnetic tactile sensor composed of magnets and sensors. [5] The sensor can detect the magnetic flux density changes due to the magnet displacement under the action of external forces. To date, the operation of almost all magnetic tactile sensors still relies on this principle. At about the same time, Lee et al. proposed several designs of magnetic tactile sensors. [6] After that, restricted by the development of magnetic sensing technology, the research on magnetic tactile sensors has almost stagnated for over 20 years. [3] In 2010, Goka et al. developed a sensor that can measure triaxial force vector and slip using giant magnetoresistances and inductances array. [7] Still, the sensor volume was quite large, with a characteristic size close to 50 mm. In 2013, Ledermann et al. used a 3D Hall sensor to fabricate a novel magnetic tactile sensor . [8] Powered by advanced technology in magnetic sensing, the size of the whole sensor structure is reduced to less than 10 mm. In the past decade, a wealth of research has been conducted to integrate 3D magnetic sensor technologies [3,9,10] with state-ofthe-art machine learning (ML) models such as artificial neural networks. [11][12][13][14] Despite the progress, the field of magnetic tactile sensors still faces significant challenges. First, a number of previous works that adopt the magnetic localization approach ignored the magnet's size. The magnet is simplified as a magnetic dipole to calculate the magnetic field. Nevertheless, this hypothesis is invalid for cases where the magnet is close to the sensor. As a result, the dipole model [15,16] may lead to inaccurate predictions if adopted improperly. Second, a DOI: 10.1002/aisy.202200291 The past decade has witnessed a surging interest in the study of magnetic tactile sensors that can detect subtle changes in both normal and shear forces. However, due to the lack of guidance by appropriate theoretical models, the development of previous magnetic tactile sensors relies either on a trial-and-error manner or tedious point-by-point experimental calibrations, which are costly and time-inefficient. Here, a theoretical model integrating magnetics, artificial neural networks, and nonlinear solid mechanics is proposed for the first time to guide the design of 3D magnetic tactile sensors. Then, a button-shaped magnetic tactile sensor prototype that can detect subtle triaxial force changes is fabricated, which relates the nonlinear magnetic flux density to the external force, without burdensome calibration procedures. The sensor can achieve an axial measurement error of less than 1% and an in-plane error of less than 3.7% with excellent durability. This study provides a comprehensive understanding of magnetic tactile sensors and sheds light on their applications in soft robotics, intelligent manipulation, and human-robot interaction (HRI).
systematic mechanical model linking the sensor deformation to the measured force is still lacking. In previous works, neural network models were adopted to directly map the magnetic field to the force information, ignoring the intermediate mechanical variables such as the mechanical strain, which could result in increased measurement errors. Moreover, training data acquisition in previous approaches heavily depends on experimentation. If the sensor needs to change its operation parameters, it will take enormous time and effort to recalibrate within its workspace.
We systematically design, fabricate, and characterize magnetic tactile sensors to address the above challenges. From the magnetics perspective, the integral model developed by Norman [17] is used to precisely determine the axial and radial magnetic flux density in the air domain around a cylindrical magnet. From the mechanics perspective, a theoretical model based on continuum mechanics has been constructed to link the sensor deformation to the measured force. To this end, a back-propagation (BP) artificial neural network is adopted to map the magnetic field to position information. Finally, to characterize the performance of the proposed magnetic tactile sensor guided by the theoretical model, experiments are performed and the sensitivity is discussed in detail by finite element analysis (FEA). Both the axial and radial predicting errors are acceptable (less than 5%), and the detected values are repeatable, indicating that the theoretical model can effectively guide the design of magnetic tactile sensors.
The rest of the manuscript is organized as follows: Section 2 introduces the basic concept of the magnetic tactile sensor. Equations governing the magnetic fields are formulated, and the spatial localization algorithm and the mechanical model are established in this section. Section 3 presents experimental results on the characterization of the sensor's performance. Section 4 discusses the influence of the constitutive model adopted on the measurement performance. Concluding remarks and experimental methods are presented in Section 5 and 6, respectively.

Structure of the Sensor
The basic prototype of the magnetic tactile sensor is proposed in Wang and his fellows' work. [3,18] We develop a new version of the magnetic tactile sensor based on the previous works, as illustrated in Figure 1a. The sensor comprises four main parts: A cylindrical magnet that can generate the magnetic field in the surrounding air domain. A magnetic chip detects the magnetic flux density due to the magnet's movement. A polymeric layer between the magnet and the chip mainly sustains the applied force. A specially designed PCB board plays a vital role in the communication between the sensor chip and the computer. The tactile sensor possesses a dimension of 26 Â 26 Â 10 mm (Figure 1b). It can be easily integrated into other devices for further usage, such as walking robots [12] and grippers. [19] The polymeric layer underneath will deform when an external force (normal or shear) is applied to the magnet, as shown in Figure 1c. The change of the magnetic flux density detected by the chip is transmitted to the computer and processed by the theoretical mode. Next, we will discuss the formulation of the theoretical model.

Magnetic Flux Density Around a Cylindrical Magnet
The applied external force changes the magnet's position. As a result, the magnet's axis will shift relative to the sensor's axis. Theoretically, the magnetic field around a cylindrical magnet can only be expressed in elementary functions along its axis. At off-axis points, the calculation is much more complicated. [17] Either way, exact expressions for the field strength can be expressed using special functions such as elliptic integrals or various Bessel functions. In 2010, Norman et al. presented an exact solution for the magnetic flux density of a cylindrical magnet in an easy-to-use form. [17] Before calculating the global magnetic flux density, an elliptic integral is defined as  ðc cos 2 φ þ s sin 2 φÞdφ ðcos 2 φ þ p sin 2 φÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi The radial and axial magnetic flux densities defined in Figure 2a are In Equation (1), (2), and (3), k c , p, c, s are the parameters related to magnet size; B 0 is the magnet's remanence; a is the magnet's radius; l is its thickness B ρ and B z can be determined in MATLAB once the size of the magnet is given. The calculated magnetic field is compared with that obtained from the experiment and COMOSL simulation. Detailed calculation results can be found in Section 3.2.

ML-Based Spatial Localization Algorithm
Magnetic localization is a technique that is widely used in medical applications. [20][21][22] The magnetic tracking system uses a small enclosed magnet to serve as an excitation source without connecting wires and an external power supply. The movementinduced field change can be analyzed and transferred into position information by an appropriate algorithm. The traditional algorithm relies on a good initial guess of the parameters and is thus limited by low time-efficiency and high complexity, [16] which greatly hinder their practical applications. Because of the nonlinearity and crosstalk effect, the correlation between the displacement D and the magnetic field B is nontrivial and challenging to solve analytically. [3] Notably, it is also timeinefficient to solve the problem numerically. Real-time feedback is one of the basic requirements for sensor design, while artificial neural network fitting can satisfy all of the above requirements. The network can train itself by learning an adequate number of samples and obtaining a nonlinear mapping from the input space to the output space. The off-line training process often takes minutes, but the trained neural network can predict the position of the magnet in milliseconds, well satisfying the stringent requirement of real-time feedback for the tactile sensor. The integral model in Section 2.2 ensures that the position and the magnetic field are in one-to-one correspondence. The approach provides sufficient training samples in the software aspect without measuring the magnetic field point by point. [11] As shown in Figure 2a, the norm of the magnetic flux density B norm and the angel φ between axial and radial magnetic flux density constitute the input vector, and the components are In Equation (5) and (6), B x ,B y , and B z are magnetic flux density components in Cartesian coordinate, θ ¼ arc tanðB y =B x Þ, In Equation (7), f À1 is determined by the network training. As shown in Figure 2b, the output vector ðρ, zÞ is obtained from the linear transformation of weight matrices V and W , together with a nonlinear transformation by bipolar Sigmoid function (Detailed information can be found in the classic textbook [23] ). The magnet position in the xy-plane can be hence obtained as In Figure 2c, the possible trajectory is limited in a cylindrical zone, and feature plane is investigated. The plane is divided into 100 Â 60 equal-sized squares with a side length of 100 μm. The coordinates of the rectangle vertices ðρ, zÞ and the magnetic flux density ðB ρ , B z Þ corresponding are the variables we care about. These points are sequentially numbered in the scan order as shown (red lines with arrow). The coordinates and magnetic flux density of all the points are obtained. This one-to-one mapping for 4312 points (70% of the 6161 points) is used as training sets input into the algorithm, then the nonlinear inverse correspondence can be obtained by the training process. That is, in the cylindrical coordinate system, the point's magnetic induction intensity is known, then the point's position information can be obtained.

Mechanical Model
The mechanical behavior of the tactile sensor ( Figure 3a) is investigated. The polymeric layer mainly sustains the mechanical loads and deforms accordingly. The resin layer and PCB board can be regarded as rigid bodies because their modulus is much higher than the soft polymeric material.
First, the uniaxial compressive deformation state is considered in Figure 3b. The coordinates of the deformed polymer can be expressed as where λ is the uniaxial stretch ratio along the z-direction, X, Y, and Z are the coordinates in the undeformed state. Hence, the deformation gradient tensor can be obtained as It is assumed that the polymeric material obeys the neo-Hookean material law Here Ψ is the strain energy function, c 1 is half of the shear modulus, I 1 is the first invariant of the right Cauchy-Green deformation tensor, p is the hydrostatic pressure, and J is the determinant of the deformation tensor. Then the nominal stress tensor can be obtained.
The external force components T x and T y are 0 around the cylindrical free boundary of the polymeric layer, then p can be obtained as p ¼ 2c 1 λ À1 . Equation (11) can be transformed into So, the force applied normal to the xy-plane is If the polymeric layer is under the simple shear state (x-direction, Figure 3c), the displacement can be expressed as The deformation gradient tensor is defined as follows According to Equation (10), the nominal stress tensor is In Equation (15), the vanishing stress s xx and s yy at the boundary of the polymeric layer gives that p ¼ 2c 1 , and thus Equation (15) can be simplified into So, the force applied to the xz-plane is Accordingly, the force T zy can be obtained (Figure 3d) as From Equation (13), (17), and (18), the components of the traction TðT zx , T zy , T z Þ on the top surface are calculated independently of each other. It is worth noting that T zx and T zy are often applied in the form of friction and T z is applied as the force normal to the top surface. The surface friction coefficient may influence the measurement of the forces, which has been investigated by Liu et al. [19] It is worth noting that the conclusion above is only valid for neo-Hookean material. For other materials that obey Mooney-Rivlin law (which contains I 2 ) or Yeoh law (which incorporates I 2 and I 3 ), the material's stress distribution is more complicated than those expressed in Equation (12) and (16), as discussed in Section 4.

Mechanical Properties of the Polymeric Layer
In the present work, the polymeric layer is made of a highly elastic polydimethylsiloxane (PDMS) network swollen by silicon oil. PDMS is a silicone rubber with excellent transparency, desirable gas permeability, and high dielectric constant. It has been widely exploited in sensor fabrication. [24] Compared with other polymers, PDMS deforms more reversibly. [25] However, it cannot be regarded as a purely elastic material. As shown in Figure 4a left, after being stretched for one cycle with an elongation ratio of 1.5, the loading-unloading curve forms a hysteresis loop, which indicates the occurrence of energy dissipation. Hysteresis mainly results from the viscoelastic behavior of the elastomer and should be eliminated in sensor designs. Otherwise, it will lead to measurement errors. [3] Bosnjak's work suggests that the viscoelasticity of swollen gel disappears due to solvent absorption. [26] Inspired by his work, silicon oil is chosen as the solvent to reduce the hysteresis of PDMS. As the mix ratio of PDMS and silicon oil increases to 4:6, the hysteresis loop almost vanishes (yellow curve in the left panel of Figure 4a), which indicates a highly elastic mechanical behavior. One possible explanation is that when solvent molecules swell the polymer network, the friction between polymer chains becomes negligible because the distance between individual chains increases, significantly reducing the viscoelasticity. [26] Silicon oil functions as a solvent to eliminate hysteresis and also acts as a softener: the tensile modulus of the material can be adjusted by one order of magnitude, from 1.33 MPa to 162 kPa, which provides a route for tuning the sensitivity of the sensor. The mixing ratio of 4:6 is adopted in the subsequent experiments.
In most cases, tactile sensors are used to detect pressure. Therefore, the uniaxial compressive test is performed on samples of polymeric layers with the same dimensions. The middle panel of Figure 4a indicates that the stretch-stress curve is nonlinear, and the compressive modulus is fitted to be 252 kPa using the neo-Hookean model (fitted from Equation (10)). To a certain extent, the tensile and compressive moduli inequality may result from the difference in the stress state near the constrained boundary. A parameter (shape factor) that can represent the effect is defined as α ¼ c compression =c tensile % 1.5(1.55 is obtained from the experiment, and 1.53 is obtained by COMSOL simulation). As shown in the right panel of Figure 4a, the area of the hysteresis loop is negligible even after 200 loading-unloading cycles, indicating that the polymeric material is almost purely elastic, which is essential for enabling durable magnetic tactile sensors.

Determination of the Sensor Size
Significant progress has been made in magnetic sensing technology to decrease the dimension of magnetic chips. The size of the magnetic tactile sensor mainly depends on the chosen magnet and the sensitivity of the magnetic chip. [18] We choose the MLX-90 393 chip, of which the sensitivity along the z-axis and x/y axis are 0.242 and 0.150 uT LSB À1 , respectively, at 25°C. The measuring range along the z-axis is À7.93 to 7.93 mT, and that along the x/y axis is À4.92 to 4.92 mT. According to the previous work, [27,28] the magnetic flux density near the surface of the magnet can reach hundreds of mT, which is far beyond the measurement range of the magnetic chip. The magnetic flux density reaches the maximum value near the surface and decays rapidly with increasing distance from the measurement point to the magnet's surface. [3] Figure 4b illustrates the magnet's radial and axial magnetic flux density (with a diameter of 3 mm and thickness of 1 mm) calculated from Equations (2), (3) and simulated in COMSOL Multiphysics. The result indicates that Equation (2) and (3) can perfectly predict the magnetic flux density in the air domain around the magnet and can be further used in developing the spatial localization algorithm. The magnetic saturation intensity of the magnetic chip (B sat ) determines the lower limit of the magnet position (Z min ). Meanwhile, the upper limit of the position (Z max ) is www.advancedsciencenews.com www.advintellsyst.com determined by the sensitivity of the magnetic chip (S). Let Δ min be the minimum displacement that the tactile sensor can recognize. Then we have the following requirements 8 > > > > > > > < > > > > > > > : Here, Δ min is set to be 10 um. The position of the magnet along the z-axis is limited by max fZ min 1 , Z min 2 g ≤ Z ≤ min fZ max 1 , Z max 2 g In the present work, Z min is 4.13 mm and Z max is 17.5 mm, as shown in Figure 4c (the region with magnetic flux density outside the sensor range is white). The distance from the magnetic sensor to the magnet is 8 mm. The side length of the polymeric layer is set to be 11 mm, which is much larger than the diameter of the magnet, to ensure that the magnet remains horizontal during the deformation process (a prerequisite for adopting the integral model). These dimensions are proven appropriate by experiments and simulations, as discussed in the following.

Algorithm Accuracy Analysis
In Section 2.3, the feature plane has been divided into 100 Â 60 equal sized squares with a side length of 100 μm. The coordinates of these vertices fitted by the spatial localization algorithm are compared with the original given value, as shown in Figure 5a. The fitted points completely cover the original point, indicating the validation of the algorithm. To furtherly investigate the positioning accuracy, errors are calculated. Radial error is defined as Error ρ ¼ ðρ Predicted À ρ Original Þ=ρ Original , and axial error is defined as Error z ¼ ðz Predicted À z Given Þ=z Given . It can be obtained that the radial errors of nearly 90% points are less than 0.1%, and errors of 80% points are less than 0.01% for the axial position fitting (Figure 5b,c).

Installation Accuracy
The fixed magnetic chip can detect the change of the magnetic flux density, which is generated by the magnet's position change. Figure 6a shows the test platform to perform the uniaxial tension and compression alongz-axis. Cyclic and step motions are examined ( Figure 6b top and bottom), and the magnet is set to move within 5-10 mm relative to the chip. For cyclic motion, the detected magnetic flux density (B z ) decreases from 4.5 to 0.7 mT nonlinearly while B x and B y remains zero. The magnetic flux density is processed to obtain the input vector for the spatial localization algorithm, as discussed in Section 3.3.1. The upper and lower limits of the predicted cyclic motion are 5 and 9.8 mm, respectively, and the error within the stroke range is 4%. For step motion, the predicted position change perfectly follows the trend of magnetic flux density change, and the error within the stroke range is 3%. As shown in Figure 6c, the magnet moves along the radial direction and is also examined (for convenience, only the x-axis is considered). Both B x and B z vary with the distance (Figure 6d), which is different from the situation of axial magnet movement. The radial fitting error is 4% after data processing. The radial and axial position fitting methods are suitable for simple linear motions and valid for complex spatial 3D motions. Figure 6e demonstrates the test platform, and the magnet can perform three-axis movement relative to the platform surface. Spiral motion of the magnet is performed, and magnetic flux density is monitored (Figure 6f left). Three components of the spatial magnetic flux vector ðB x , B y , B z Þ change nonlinearly with the magnet position. The predicted magnet motion is plotted after the magnetic information is processed (Figure 6f, right). Compared with the original trajectory, the fidelity of the reproduced trajectory is acceptable.

Performance Evaluation for Force Sensing
As discussed in Section 3.3, the change of spatial magnetic flux density can be transformed into the change of position by the spatial localization algorithm. The force applied to the tactile sensor can be calculated by the position information through Equation (13), (17), and (18). As evident from Figure 7a, during the compression test shown in Figure 6a, the axial force T z by the magnetic tactile sensor system increases in a stepwise manner, while the in-plane forces T zx and T zy remain trivial. The results well match the force value applied by the universal testing machine, with the average error being %1.0%. According to the material modulus and sensor size, the allowed maximum measuring force is 10 N. Otherwise, mechanical failure or magnetic saturation of the chip may occur. Figure 7b shows that negligible hysteresis occurs during the 1-cycle compressive test, and the excellent performance is owed to the nearly pure elastic polymeric material. The sensor returns to its initial configuration after the external force returns to zero, without any plastic deformation and internal damage. Thus, the parameters obtained after the first calibration do not require further modification in subsequent uses. The stability of the sensor is verified by a 50-cycle compression test, as shown in Figure 7c. The average peak-to-peak value of the force measured by the tactile sensor is 97% of that applied by the tensile machine and remains constant.
The experiment also verified the in-plane force measured by the tactile sensor during the shear test, as shown in Figure 6c (Figure 7d). Without losing generality, assume the shear force is applied in the x-direction, so only T zx is involved and verified. The average error is calculated to be %3.7%, which is larger than that along the axial direction. The error may arise from the boundary effect, i.e., the cylinder is not in homogeneous pure shear deformation. It also may come from imperfect hardware installation. For example, the magnet surface and the sensor surface are not aligned, or the track of the tensile machine is not perfectly parallel to the axis of the sensor, etc.
Two demonstrations are presented to show the sensing capability of the magnetic tactile sensor: 1) As shown in Figure 7e, the sensor can measure the weight of a business card (1.37 g). The average weight measured by the magnetic tactile sensor is 1.12 g www.advancedsciencenews.com www.advintellsyst.com (In-plane force T zx and T zy is not plotted because their variation is negligible).
2) The sensor can detect finger thrusts in different directions (Figure 7f ). The proposed results and demonstrations are the most straightforward cases, but they are also the most basic and essential concepts for measuring the actual forces.

Sensitivity of the Tactile Sensor
1D sensitivity of the magnetic tactile sensor is deduced and calculated by Wang et al. [3] However, the magnet is not limited to performing 1D motion along a straight line. Herein, we derive the 2D sensitivity for magnetic tactile sensors undergoing combined compression and shear, which is defined as Here,T is the tri-axis force as mentioned in Section 2.4. Equation (21) can be further expanded as The deformation is assumed to be located in the xz-plane and Equation (22) can be transformed into From Equation (23), the sensitivity can be improved by decreasing the contact area (a 2 ) or selecting a softer polymeric Figure 6. The test platform and results (Hardware). a) The uniaxial displacement test along the z-direction. b) The top figures show the magnetic flux density detected and the corresponding cyclic motion measured by the magnetic tactile sensor. The bottom figures present the magnetic flux density detected and the related step motion measured by the magnetic tactile sensor (Black, red and blue lines correspond to x-, y-, z-directions, respectively). c) The uniaxial displacement test along the x-direction. d) Magnetic flux density detected and the corresponding motion measured. e) The homemade test platform for 3D motion. f ) Magnetic flux density is detected, and the related spiral motion is measured.
www.advancedsciencenews.com www.advintellsyst.com layer. The influence of thickness (h) on sensitivity is more complicated since a larger thickness indicates a larger displacement but a smaller magnetic flux density gradient at the detector. A conclusion can be drawn that both the mechanical and magnetic properties determine the sensitivity of the tactile sensor. Typically, when the sensitivity of the sensor needs to be changed (the size of the magnet is fixed), only the modulus or size of the polymer layer needs to be changed to achieve satisfactory results. The magnet is located at the center of the interface, as shown in Figure 8a. Two motion trajectories for uniaxial compression and simple shear are considered. From point 1-3, the position along the z-axis changes, and the contribution to the sensitivity is mainly derived from Γ xz and Γ zz according to Equation (23). From Figure 8c,e, the value of Γ zz is two orders of magnitude larger than Γ xz . Specifically, Γ zz is the scalar sensitivity for the uniaxial compressive test. From point 1-2, the position along the x-axis changes, and the contribution to the sensitivity is mainly derived from Γ xx and Γ zx . As shown in Figure 8b,d, during the deformation process, Γ xx decreases while Γ zx increases. Therefore, when the sensor performs a relatively small shear deformation, Γ xx is dominant and otherwise Γ zx is dominant. The sensor is generally more sensitive in the radial direction (0.268 mT N À1 ) than in the axial direction (0.120 mT N À1 ).

Discussions
We next discuss the limitation of the neo-Hookean model as mentioned in Section 2.4. To give a reasonable correction, the Mooney-Rivlin model is adopted as the constitutive model of the polymeric material Ψ ¼ c 1 ðI 1 À 3Þ þ c 2 ðI 2 À 3Þ À pðJ À 1Þ For the sample under the uniaxial deformation state, the Cauchy stress tensor is expressed as The stress at the free boundary is 0 and p can be obtained as p ¼ 2c 1 λ À1 À 2c 2 λ. The nominal stress tensor is expressed as So, the force applied normal to the xy-plane is The external force applied on the boundary is along the z-axis, if c 2 ¼ 0, Equation (26) degenerates into Equation (12). When the sample is under the simple shear state, the Cauchy stress tensor is expressed as www.advancedsciencenews.com www.advintellsyst.com deformation is beyond the scope of this work and will be investigated in future studies.

Conclusion
A theoretical model consisting of an ML-based spatial localization algorithm and a finite-deformation mechanical model is derived, which relates the highly nonlinear spatial magnetic field to the external force applied on the button-like magnetic tactile sensor. Guided by the theoretical model, we fabricate a durable tactile sensor prototype that can detect triaxial force. The predictions on displacement and force are validated by the experiments, in which all errors are reasonable and acceptable. This research is expected to provide a promising strategy for designing magnetic tactile sensors with potential applications in many fields, such as soft robotics, smart manipulation, and HRI.
Fabrication of the Magnetic Tactile Sensor: Polymeric Material Synthesis: 4 g PDMS was weighed and poured into the centrifugal mixing bottle, then 4 μL Pt-catalyst and 6 g silicon oil were added. After that, they were mixed by a planetary mixer (Thinky mixer ARE-310, USA) for 30 s (2000 rpm). 0.4 g curing agent SYLGARD was immersed into the mixture, followed by the same mixing procedure. The precursor was then injected into the mold of a sandwich-like structure (a square with a side length of 11 mm, and a height is 8 mm), and the mold was placed in the oven for 1 h at 65°C.
Fabrication of the Magnetic Tactile Sensor: Installation of the Magnetic Tactile Sensor: A magnet with the size of 3 mm diameter and 1 mm height was selected. The magnet was embedded into a resin sheet, which is further bonded on the surface of the polymeric layer with silicone glue (5562 silicone glue). A resin sheet with a square hole in the middle was bonded on the other surface of the polymeric layer and then placed on the PCB board. The primary circuit diagram and component layout of the PCB circuit board were designed by Altium Designer, and electronic components were purchased from Lichuang Mall. The magnetic tactile sensor was installed on a tilt angle adjustment platform (X-Z plane and Y-Z plane) and a three-axis displacement adjustment platform. The subsequent tests were based on this experimental platform.
Mechanical Properties Measuring of the Polymeric Layer: Samples for the stretch test were cured into a dog-bone shape, with the narrow strip part having a length of 20 mm, a width of 4 mm, and a thickness of 2 mm. The sample was pulled uniaxially using the tensile machine with a 10 N load cell at 50 mm min À1 stroke speed. The stretch was terminated and the gripper returned at the same speed when the stretch ratio was 1.5. Three specimens were tested for each data point we presented, and the average value was reported.
Compression tests were carried out using the testing machine above (10 N load cell) at a 2 mm min À1 loading speed. The sample size was the same as that of the polymeric layer (11 Â 11 Â 8.5 mm). The termination condition was the force exceeds 10 N. For each data point, three specimens were tested, and the average values were calculated.
Cyclic compressive test was also performed using the parameters mentioned above and the number of cycles was 200.
Magnetic Field-Position Conversion: Verification of the Axial Magnetic Flux Density Measuring: The magnet was fixed on the gripper of the tensile machine, and the test platform was then placed beneath the magnet. All of the position was verified by the naked eyes for rough adjustment.
The tensile machine was adjusted until the magnet just touched the magnetic sensor (without a polymeric layer), and the force displayed on the screen was zero at this time. The magnet was then moved up to 5 mm from the current position, and the new position was defined as the zero-displacement position. The program was set up so that the stretching machine performed a cyclic motion with a speed of 100 mm min À1 and a stroke of 5 mm. The test platform was connected to the computer through an Arduino UNO board, and the magnetic flux density vector ðB x , B y , B z Þ could be monitored. By adjusting the three-axis displacement platform and the angle tilt platform, the values of B x and B y were close to 0. This step ensured the sensor was only sensitive to the z-direction (axial direction) when the magnet position changed.
The magnet was then set to return to the initial position (5 mm from the sensor), and the program was set up so that the stretching machine performed a movement of 10 mm min À1 , each time it traveled for 1 mm and stopped for three seconds, and the total stroke is also 5 mm. During the upward movement of the magnet, the change in the magnetic flux density will be detected by the magnetic sensor and transmitted to the computer in real time. The measured magnetic flux density was compared with that simulated from COMSOL Multiphysics to verify the parameter settings.
Magnetic Field-Position Conversion: Verification of the Spatial Localization Algorithm: The magnetic flux density obtained in Section 6.4.1 was analyzed in MATLAB for linear motion verification. For 3D motion in space, the magnet was fixed on the needle of a homemade glue dispenser. A program was written to make the magnet perform an upward spiral movement with a radius of 2 mm and a pitch of 1 mm. The magnetic flux density was analyzed in MATLAB using the spatial localization algorithm mentioned in Section 2.3.
Verification of the Force Sensing: Measurement of Normal Force T z : The test platform (Figure 5a) was placed beneath the compressive gripper, and the gripper's position was adjusted until the magnet was just in contact with it. This new position was defined as the start point (displacement is zero and force is zero). Then the tensile machine was programmed to perform a step movement (moving at a speed of 1 mm min À1 to add 2 N in each step, followed by stopping for 3 s; repeating the procedure until the force reaches to 8 N).
Verification of the Force Sensing: Measurement of Tangential Force: Without losing generality, only T zx was verified in the test. The test platform (Figure 5c) was placed next to the gripper, and the resin layer was bonded to the gripper. The tensile machine was programmed to perform a cyclic movement (a speed of 5 mm min À1 and a stroke of 2 mm). The adjustment platforms were tuned for tilt angle and the three-axis displacement so that T zx was detected. After that, let the tensile machine perform a 1-cycle movement (at a speed of 2 mm min À1 , until the force reaches 1 N).