Simulating mechanical properties of human tissues or organs based on magnetorheological fluid for tactile display

Robot-assisted minimally invasive surgery enables surgeons to tele-perform elaborate surgical operations to patients with less damage and pain. Besides force feedback provided by the surgical robot to the surgeon, touching sensations are also important for the surgeon to acquire the complete conditions of the patient. Thus, tactile display devices are crucial elements in surgical robots. Meanwhile, various sensations of magnetorheological (MR) fluid can be provided to human fingers because its stiffness, elasticity, and viscosity can be controlled by applied magnetic field. Therefore, in this paper, a new tactile display device based on MR fluid is proposed. This device has high magnetic conduction efficiency, less magnetic leakage, no MR fluid leakage, and overcomes the major drawbacks of the existing tactile display devices based on MR fluid in literatures. Firstly, the design of the tactile display device is described in detail, followed by its fabrication and assembling methods. Secondly, the working current range of the tactile display device is determined by using electromagnetic finite element method (FEM) simulation. Thirdly, the mathematical model to characterize the compression and shear behaviors of the tactile display device is developed. Then, the tactile display device is tested in terms of normal and shear contact forces, followed by its elastic and shear moduli analysis. Finally, the unknown parameters in the mathematical model are figured out, and the model is validated by using structural FEM simulation. The experimental results show that the elastic and shear modulus range of the proposed tactile display device are respectively 3–7.5 kPa and 1.4–5.0 kPa, which can cover the mechanical properties of various human viscera.

Robot-assisted minimally invasive surgery enables surgeons to tele-perform elaborate surgical operations to patients with less damage and pain. Besides force feedback provided by the surgical robot to the surgeon, touching sensations are also important for the surgeon to acquire the complete conditions of the patient. Thus, tactile display devices are crucial elements in surgical robots. Meanwhile, various sensations of magnetorheological (MR) fluid can be provided to human fingers because its stiffness, elasticity, and viscosity can be controlled by applied magnetic field. Therefore, in this paper, a new tactile display device based on MR fluid is proposed. This device has high magnetic conduction efficiency, less magnetic leakage, no MR fluid leakage, and overcomes the major drawbacks of the existing tactile display devices based on MR fluid in literatures. Firstly, the design of the tactile display device is described in detail, followed by its fabrication and assembling methods. Secondly, the working current range of the tactile display device is determined by using electromagnetic finite element method (FEM) simulation. Thirdly, the mathematical model to characterize the compression and shear behaviors of the tactile display device is developed. Then, the tactile display device is tested in terms of normal and shear contact forces, followed by its elastic and shear moduli analysis. Finally, the unknown parameters in the mathematical model are figured out, and the model is validated by using structural FEM simulation. The experimental results show that the elastic and shear modulus range of the proposed tactile display device are respectively 3-7.5 kPa and 1.4-5.0 kPa, which can cover the mechanical properties of various human viscera. * Author to whom any correspondence should be addressed.
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Introduction
In medical field, surgery is often applied as a specialty to help improve bodily function, appearance, or to repair unwanted ruptured areas. Generally, a surgeon needs to hold a scalpel to manipulate the tissues or organs in a surgery. However, no matter how proficient and experienced the surgeon is, some elaborate and exquisite operations cannot be finished by human hands. Moreover, manual manipulations sometimes may lead to unnecessary and unwanted damage to patients due to carelessness of the surgeon. To tackle this problem, robotassisted minimally invasive surgery (RMIS) arises in recent years. RMIS is often used for surgery requiring elaborate work to minimize physiological damage and less pain to speed up recovery [1,2].
In RMIS, instead of directly performing surgery on the patient, a surgeon usually operates a surgical robot to perform the surgical operations. A typical surgical robot generally contains two parts: operation device and manipulator. The operation device functions as a console where the surgeon directly operates. The operations of the surgeon are transferred as the position and force data that are sent to the central computer. The position and force data are then sent to the manipulator after being processed by the central computer. The manipulator performs the operations on the patient according to the position and force data. Meanwhile, the manipulator also obtains the force data from the patient and sends them back to the central computer and the operation device. The surgeon can then feel the force feedback from the operation device which is conducive to knowing the conditions of the patient in real time.
However, only force feedback provided for the surgeon is not enough to obtain the complete information of the tissues or organs that the surgeon is operating on. Besides force feedback, other factors such as compliance, shape, sliding resistance are also contributing to human sensations. Different tissues or organs may deliver the same force feedback, but their sensations to human fingers are totally different. In fact, the word 'haptic' refers to 'force (kinesthetic) feedback and tactile (cutaneous) display' [3,4]. Therefore, tactile display is also a crucial element for a surgical robot.
Tactile display can be applied in RMIS in two forms [22]. The first form is that the tactile display device installed in the operation device reproduces the tactile feeling of the tissues or organs in the manipulator side. The second form is that the tactile display device itself mimics the tissues or organs in the manipulator side for surgical training and simulation.
Magnetorheological (MR) fluid, as a new kind of material, is applied in tactile display devices in recent years. The working mechanism of MR fluid is that its yield stress can be controlled by applied magnetic field, which further leads to stiffness, elasticity, and viscosity change. This change can be sensed by human touching. When no magnetic field is applied, it behaves like normal viscous fluid; when magnetic field is applied, it becomes semi-solid. Therefore, its touching sensation can be adjusted by the magnitude of the applied magnetic field intensity. Some researchers have developed tactile display devices based on MR fluid. Researchers at University of Pisa developed a series of tactile display devices (Pinch Grasp, Haptic Black Box I, Haptic Black Box II) to mimic the compressional compliance of biological tissues for surgical applications [23][24][25]. Liu et al [26] designed a single cell MR fluid-based tactile display device equipped with two kinds of electromagnets to display different surface profiles. Lee et al [27] further designed a multi-cell tactile display device based on MR fluid to display different shapes, compliances, sliding resistances, and tribological perceptions. Tsujita et al [28] proposed a tactile display device based on MR fluid to mimic soft tissues being cut by scalpels, functioning as surgical simulation and cutting force display. Ishizuka et al [29,30] encapsulated MR fluid in chambers to reproduce the spatial stiffness distribution of normal tissues and tumors. Han et al [31] used pin array mechanism to design a tactile display device for RMIS based on MR fluid and used fuzzy logic control algorithm to obtain the desired palpation force. Researchers at Inha University conducted a series study to develop tactile display devices based on MR fluid and sponge to mimic different animal organs or tissues [32][33][34][35]. Researchers at Korea Advanced Institute of Science and Technology conducted a series of studies to develop miniature haptic buttons based on MR fluid to display the resistive force sensed by human fingers for mobile devices [36][37][38][39][40][41].
The existing tactile display devices based on MR fluid can be divided into two types according to their application methods: (1) human fingers directly touch MR fluid [23][24][25][26][27][28]; (2) MR fluid is packaged in containers or mixed with solids [29][30][31][32][33][34][35][36][37][38][39][40][41]. Some drawbacks of these devices are listed as follows: (a) Fluid has its viscosity, so its touch sensation is quite different from that of solids; (b) Fluid is amorphous. When nonuniform magnetic field is applied, the fluid shape would be irregular and uncontrollable; (c) There is MR fluid leakage; (d) External magnet (usually at the bottom) is used. There is a non-magnetizable container between the magnet and MR fluid, which significantly reduces the magnetization efficiency. Additionally, magnetic flux also flows outwards, so the magnetic flux applied to MR fluid is less.
Therefore, to tackle the problems in the existing tactile display devices based on MR fluid, a new tactile display device based on MR fluid for RMIS is proposed in this paper. This device adopts the internal electromagnet design to increase magnetic conduction efficiency. The electromagnet is covered by a polyurethane (PU) sponge and immersed in MR fluid. A polydimethylsiloxane (PDMS) membrane is used for sealing. The idea of immersing the PU sponge into MR fluid originates from [32][33][34]. The rest of this paper is organized as follows. First, the detailed design, fabrication, and assembly of the proposed tactile display device are described in section 2. The current applied to the coil is then determined by using electromagnetic finite element method (FEM) simulation. Second, the tactile display device is modeled mathematically in terms of normal and shear contact forces in section 3. Third, in section 4, the normal and shear contact forces of the tactile display device are measured, and the elastic and shear moduli range of the tactile display device are analyzed. Then, in section 5, the unknown parameters in the mathematical model are determined according to the testing data, and the mathematical model is validated by using structural FEM simulation. Finally, conclusions are drawn in section 6.

3D model of the tactile display device
To tackle the drawbacks of the existing tactile display devices based on MR fluid, a new tactile display device is designed, fabricated, and assembled. Figure 1 shows the 3D model of the proposed tactile display device and each component. A piston wound by a coil is installed in a container by using bolts. A box-shape PU sponge is used to cover the piston. The PU sponge and the piston are immersed in MR fluid in the container. The PU sponge is used as a substrate to absorb and trap the amorphous MR fluid into its porous structure. A PDMS membrane (Hangzhou Westru Technology Co., Ltd) is fixed on the top of the container. A top cover can be put on this tactile display device when it is not used. Materials and key parameters of each component are listed in table 1. The piston is made of silicon steel with high relative magnetic permeability. The relative magnetic permeabilities of the piston [42] and MR fluid [43] are expressed by magnetic flux density versus magnetic field intensity (B-H) curves. The container and the top cover are made of acrylic for transparency. The relative magnetic permeabilities of the container, the top cover, the coil, the PU sponge, and the PDMS membrane are one, which are hardly magnetic conductive. Copper wire with diameter of 0.2 mm and turns of 2100 is used for the coil. The PU sponge is open celled with porosity of 45 pores per inch (ppi) and density of 2.9 g cm −3 . The thickness of the top part of the PU sponge is chosen as 5 mm to achieve the balance of relatively large average magnetic flux density at the top plane of MR fluid and the experimental requirements. The thickness of the PDMS membrane is 125 µm. Other mechanical properties of the PDMS membrane are also given in table 1. MR fluid MRF-132-DG from LORD Corporation is used in this device.
The magnetic circuit of the tactile display device is shown in figure 2 as the orange dashed curve. The magnetic field generated from the coil flows around the piston in MR fluid. Because the relative permeabilities of the container and the PDMS membrane are one, most of the magnetic field are locked in the container and are fully applied to MR fluid with little magnetic leakage. Therefore, there is less magnetic loss in this device compared with the existing devices in literatures with external coils. With the same current input, larger magnetic field can be applied to MR fluid, the efficiency of this device significantly increases as a result.

Fabrication and assembly of the tactile display device
Each component of the tactile display device is fabricated based on its 3D model. The assembling procedures of the tactile display device are shown in figure 3 and are expounded as follows: (a) Figure 3(A). All the components including MR fluid MRF-132-DG, container, top cover, piston, copper wires, PU sponge, and PDMS membrane are prepared. (b) Figure 3(B). The copper wires are wound around the piston by using FZ-180 coil winding machine (Ningbo Feizhi Electric Tools Co., Ltd, China). The piston is then installed in the container by using bolts. (c) Figure 3(C). The PU sponge is immersed in MR fluid to fully absorb the fluid. (d) Figure 3(D). The PU sponge is put into the container to cover the piston. A syringe is used to inject more MR fluid into the container. Therefore, the piston and the PU sponge are fully immersed in MR fluid to facilitate magnetic conduction. (e) Figure 3(E). The PDMS membrane is covered on the PU sponge. Kafuter K-5905L organosilicon sealants (Guangdong Hengda New Materials Technology Co., Ltd, China) is used to fix and seal the PDMS membrane. (f) Figure 3(F). The finished assembly. (g) Figure 3(G). The finished assembly with the top cover and extended wires.   [43] PDMS is a kind of silicone elastomer often used in microfluidic or lab-on-a-chip applications to form devices with defined microstructures. Functions and advantages of using the PDMS membrane in this tactile display device are listed as follows: (a) The PDMS membrane has soft and skin-like touch sense, making it appropriate to be used in this device to simulate human tissues or organs; (b) As mentioned before, the PDMS membrane can ensure little magnetic leakage; (c) With the help of the organosilicon sealants, the PDMS membrane can be used for MR fluid sealing without MR fluid leakage; (d) Because PDMS is oil resistant, the silicon oil of MR fluid cannot permeate and penetrate the PDMS membrane. When touching this device, human finger will not be polluted by MR fluid; (e) According to the mechanical properties given in table 1, this PDMS membrane is highly resistant to local force loading, thus it will not break when poked by tip-shape objects. This feature is quite crucial in the following experiments. In summary, to solve the four problems of the existing tactile display devices based on MR fluid, the proposed tactile display device has several advantages: (a) The PDMS membrane provides skin-like touch sense to human fingers without pollution when directly touching MR fluid; (b) The PU sponge can be used to fix the shape of MR fluid.
The stiffness and elasticity of the PU sponge is similar to that of human tissues or organs and can be adjusted by changing the applied current; (c) Thanks to the PDMS membrane and the organosilicon sealants, there is no MR fluid leakage; (d) The electromagnet (the piston and the coil) is in the container and fully immersed in the MR fluid, so that the magnetic field is fully applied to MR fluid. Since the relative magnetic permeabilities of the container and the PDMS membrane are small, most of the magnetic field is locked in this device with little magnetic leakage. Therefore, the proposed tactile display device is highly efficient in magnetic conduction.

Electromagnetic FEM simulation of the tactile display device
To investigate the electromagnetic performance of the tactile display device, electromagnetic FEM simulation is performed in ANSYS Electronics Desktop. After the 3D model is built, relative magnetic permeabilities and other parameters in table 1 are respectively assigned to each material. Currents I ranging from 0 to 1.5 A at a step of 0.1 A are respectively input to the coil. The whole 3D model is surrounded by an air sphere with radius of 1.0 m. Zero tangential field is set at the outer surface of the air sphere as boundary condition. Figure 4 shows the magnetic flux density distribution when I = 0.4 A at the top and vertical cross-sectional plane of MR fluid. The maximum magnetic flux density at the top plane can reach 0.04 T. The magnetic flux density distribution around the piston is relatively smooth and large, which effectively contributes to the stiffness change of MR fluid. According to the B-H curve of MR fluid MRF-132DG in [43], MR fluid will be magnetic saturated when the applied magnetic field reaches a certain magnitude. Average magnetic flux density B ave can be an index to characterize whether MR fluid is magnetic saturated under the applied current. B ave at the top plane of MR fluid under every current is calculated and plotted in figure 5. When I ⩽ 0.7 A, B ave significantly increases with the increment of I. When I > 0.7 A, B ave becomes almost saturated and its increment is relatively slow. If this device works under I > 0.7 A, it is not energy efficient.
In addition, large current input will also induce considerable temperature rise. Therefore, in the following experiments in section 4, currents ranging from 0 to 0.7 A will be applied in order to increase energy efficiency and reduce temperature rise.

Modeling of the tactile display device
Since this tactile display device is used to simulate the mechanical properties of human tissues of organs, it is quite necessary to develop a mathematical model to accurately predict its mechanical properties, such as contact force, elastic modulus, shear modulus, etc. Human tissues or organs are kinds of soft biomedical materials that is compliant and can withstand large amount of deformation before failure. Their most conspicuous feature is that they usually absorb lots of surrounding fluid, constituting multi-phase systems. There are numerous modes of interaction between different phases. Thus, their mechanical behaviors are complex due to multiple length and time scale phenomena and their hierarchical structures. Meanwhile, due to the fluid-solid interaction, they usually show both viscosity and elasticity when undergoing deformation, i.e. viscoelasticity. In addition, their mechanical responses are highly dependent on strain and time. Under different strain rates, they may show linear or nonlinear mechanical responses. Under different loading rates, their mechanical responses may be totally different.
When modeling viscoelastic materials, standard linear solid (SLS) method is often used. There are two basic SLS models: Maxwell and Kelvin-Voigt models. In the Maxwell model, a linear spring is in parallel with two elements in series (a linear spring and a dashpot); while in the Kelvin-Voigt model, a linear spring is in series with two elements in parallel (a linear spring and a dashpot). These two models show high accuracy when modeling simple viscoelastic materials, such as sponge, foam, rubber, etc. However, they cannot accurately characterize the material nonlinearity and time-dependent behaviors that is often seen in biomedical materials.  Therefore, in this section, inspired by [44], the classical Maxwell model is modified as the nonlinear fractal order Maxwell model to characterize the nonlinear and timedependent behaviors caused by the interaction of MR fluid and the PU sponge. This model includes two parts: normal force model and shear force model. A linear spring is used to characterize the linear mechanical response under low strain, while a nonlinear spring is used to characterize the nonlinear mechanical response under high strain. Fractal derivative is used to capture the rate-dependent effects under different loading rates. Figure 6 shows the schematic diagram of the nonlinear fractal order Maxwell model. E 1 and G 1 are respectively the linear elastic and linear shear moduli of the linear spring. η and λ are respectively the normal and shear viscous coefficients of the dashpot. p is the order of the fractal derivative.

Normal force model
In the normal force model, σ 1 is the normal stress of the linear spring and can be expressed as where ε 1 is the normal strain of the linear spring and t is time. The exponent p in the fractal derivative is used to characterize the time scale effect. Meanwhile, σ 1 can also be expressed by the normal strain of the dashpot ε 2 as σ 2 is the normal stress of the nonlinear spring and can be expressed by a fifth order polynomial as where ε is the normal strain of the nonlinear spring (total normal strain); α 1 , α 2 , α 3 , α 4 , and α 5 are the polynomial coefficients. The reason why fifth order polynomial is used here is that its universality and accuracy are relatively higher than those of the traditional lower order polynomial curve fitting with acceptable calculation burden. In addition, fractal derivative is more general because it covers more possibilities than those of normal integer derivative. According to the strain relationship of the Maxwell model, the following equation should be satisfied: Substituting equations (1) and (2) into equation (4), the following equation is obtained: According to the stress relationship of the Maxwell model, the total normal stress σ can be obtained as Substituting equations (3) and (6) into equation (5), the following equation is obtained: Reorder equation (7) as Integrate equation (8) on the both sides, the following equation is obtained: Reorder equation (9) as Equation (10) is the normal stress-normal strain relationship of the tactile display device.
In the following experiments in section 4, the total normal strain ε can be expressed as where y is the displacement in the vertical direction, v N is the normal moving speed of the sensor probe, and l 0 is the initial thickness of the top part of the PU sponge. The normal force F N can be obtained via the following equation: where A is the cross-sectional area of the sensor probe. Substituting equations (11) and (12) into equation (10), the normal force-time relationship of the tactile display device can be obtained as

Shear force model
In the shear force model, similar mathematical deduction can be adopted. For the linear spring, τ 1 and γ 1 are respectively the shear stress and the shear strain. For the dashpot, γ 2 is the shear strain. For the nonlinear spring, τ 2 and γ are respectively the shear stress and shear strain (total shear strain). τ is the total shear stress. The following equations can be obtained: where β 1 , β 2 , β 3 , β 4 , and β 5 are the polynomial coefficients. Following the similar procedures, the shear stress-shear strain relationship can be obtained as In the following experiments in section 4, the total shear strain γ can be expressed as where x is the displacement in the shear direction, and v S is the shear moving speed of the sensor probe. The shear force F S can be obtained via the following equation: Substituting equations (19) and (20) into equation (21), the shear force-time relationship of the tactile display device can be obtained as The proposed model is an empirical one, in which the unknown parameters need to be determined by experiments. These parameters are divided into two categories: (1) parameters with physical meanings; (2) parameters without physical meanings. E 1 , η, G 1 , and λ belong to category (1), while α 1 , α 2 , α 3 , α 4 , α 5 , β 1 , β 2 , β 3 , β 4 , β 5 , and p belong to category (2). In traditional polynomial curve fitting for material properties, parameters in category (2) are used as nonlinear scale parameters for time (displacement), whose values do not directly reflect physical meanings. However, the proposed model combines these two categories of parameters in consideration of both physical meanings and nonlinearity of time (displacement) scale. Parameters with physical meanings characterize the basic properties of the materials, while parameters without physical meanings are used to account for the nonlinear effect induced by strain (α 1 , α 2 , α 3 , α 4 , α 5 , β 1 , β 2 , β 3 , β 4 , β 5 ) and time (p). The unknown parameters in equations (13) and (22) will be determined in section 5 based on the experimental results in section 4. The normal force model and the shear force model will then be validated via structural FEM simulation in section 5.

Experimental setup
After the tactile display device is assembled, a test bench is built to measure its normal and shear contact forces. A sensor probe is installed on a force sensor. The sensor probe is used to contact the tactile display device and the contact force can be measured by the force sensor in real time. A ball screw linear guide rail with motor is used to move the force sensor and the sensor probe which are fixed on the rail via a sensor fixture. The motor is controlled by a motor controller via a motor driver. The force signal is transmitted to the computer in real time via a transducer and an adapter. A switching power supply is used to provide 24 V DC electrical power to the motor driver and the transducer (sensor). The adapter is powered by the computer via a USB cord. A DC electrical power source is used to provide electrical current to the tactile display device to change its stiffness. 220 V AC power supply is used to power the motor controller, the switching power supply, the computer, and the DC electrical power source. Figure 7 shows the experimental setup and figure 8 shows the schematic diagram of the experiments.
During the experiments, the diameter of the sensor probe is d probe = 4 mm, and the current applied to the tactile display device ranges from 0 to 0.7 A as suggested in the electromagnetic FEM simulation in section 3. The resolution of the force sensor is 0.0001 N, and its sampling frequency is 10 Hz. In the normal force measurement, the sensor probe moves vertically downward at a speed of v N = 0.1 mm s −1 for 30 s and moves vertically upward at the same speed for the same time. Thus, the maximum vertical displacement is 3 mm, and the maximum normal strain is 60%, which will not damage the top  part of the PU sponge and the PDMS membrane. In the shear force measurement, the sensor probe moves tangentially forward at a speed of v S = 0.1 mm s −1 for 50 s and moves backward at the same speed for the same time. Thus, the maximum shear displacement is 5 mm, and the maximum shear strain is 100%. The moving speed of the sensor probe is determined based on the low-speed touching motion of human fingers in RMIS to investigate the quasi-static properties of the proposed tactile display device. There is a pre-compression of 0.5 mm in the vertical direction in the shear force measurement. The contact force measured by the force sensor can be displayed in real time on the computer screen.

Experimental results
The normal force-time and the shear force-time curves under different currents are respectively shown in figures 9 and 10. It is clear that the normal and shear contact forces gradually increase with the increasing of the applied current. The maximum normal force is 0.057 N, and the maximum shear force is 0.043 N. Meanwhile, there exists magnetic hysteresis in the unloading period. The main reason is that the elastic force and the viscous force are in the same direction in the loading period, while they are in the opposite direction in the unloading period. In addition, there exists force-time lag when the sensor probe starts to reverse its direction. Specifically, the normal and shear forces reach the maximum value at 32 s and 51 s, respectively. The main reason is that the PDMS membrane, the PU sponge, and MR fluid are separated for a short time when the sensor probe approaches the maximum stroke. The PU sponge continues to be compressed for a short time due to inertia and reverses its direction afterwards to follow the probe. Moreover, in the shear force measurement, there exists force in the opposite direction when the sensor probe approaches the initial point in the unloading period. The main reason is that in the unloading period, the elastic force from other parts of the material is applied to the sensor probe. This force is in the opposite direction of the probe motion and increases as the probe approaches the initial point. It finally surpasses the original elastic force in the loading period.
According to table 1, the initial thickness of the top part of the PU sponge is l 0 = 5 mm. Then, y and x can be respectively transformed to ε and γ using equations (11) and (20). The elastic modulus E and shear modulus G of the tactile display device can be respectively calculated as The elastic modulus-normal strain and the shear modulusshear strain curves are respectively plotted in figures 11 and 12. For better plotting, the normal and shear strain values are sampled at a step of 4% without influencing their trends. It is clear that the elastic modulus and the shear modulus gradually increase with the increasing of the applied currents. Because there is space between the piston and the PU sponge, the semisolid MR fluid in the space (initially not in the PU sponge) will be gradually absorbed into the PU sponge with the increasing of the normal strain, generating more resistance to the vertical motion of the sensor probe. Thus, the elastic modulus increases with the increasing of the normal strain. Because the direction of the link structures of the magnetizable particles of MR fluid (the direction of magnetic field) is the same as the shear direction of the sensor probe, the shear motion of the sensor probe will damage the link structures. With the increasing of the shear strain, the link structures break to a larger   The elastic modulus of the tactile display device ranges from 3 kPa to 7.5 kPa, and the shear modulus ranges from 1.4 kPa to 5.0 kPa. By controlling the applied current and the normal or shear strain, this tactile display device can simulate the mechanical properties of various human tissues or organs [45] as listed in tables 2 and 3. In real applications of the proposed tactile display device, the desired normal or shear strain can be obtained by using a two-axis probe equipped with a strain gauge. With specific control strategies, the desired normal or shear strain can be obtained by moving the two-axis probe to the designated place. The strain gauge is used to measure the real-time strain and send feedback to the control board for calibration. The proposed tactile display device can simulate the mechanical properties of many human viscera that is often in surgery, such as lung, liver, kidney, etc. In RMIS, the surgeon can use the operation device to control the manipulator to perform the surgery on human tissues or organs, and the mechanical properties of the human tissues or organs can be transmitted to the proposed tactile display device in real time by adjusting and calibrating the applied current and the strain with specific control strategies embedded in a control board. The surgeon can feel the sensation of the human tissues or organs by touching this device, which provides more information to the surgeon other than force feedback.
In real applications, this device can be equipped with a moving bracket or a moving supporter to change the normal or shear strain by changing its locations or moving directions. Cooling apparatus is also preferred for this device because the applied current leads to temperature rise. Control strategies should be investigated to adjust the applied current and strain according to the feedback signal from the manipulator. The geometric dimensions of the components of this device can also be changed and optimized when the external attachments are equipped.

Parameter identification
Generally, the force-displacement relationship of a kind of material should be monotonic (cyclic loading is not considered here), so the experimental results in the unloading period are not considered in the modeling. The normal force-time curves and the shear force-time curves are truncated. The first 32 s data (loading period) are left for the normal force, and the first 51 s (loading period) are left for the shear force. Curve fitting for equations (13) and (22) are performed based on the data in the loading periods of figures 9 and 10. It should be mentioned that the cross-sectional area of the sensor probe is A = 1 4 π d 2 probe = 1.2566 × 10 −5 mm 2 . The curve fitting results for the normal force model and the shear force model are respectively plotted in figures 13 and 14. The parameters of the normal force model and the shear force model via curve fitting are respectively listed in tables 4 and 5. Meanwhile, the goodness of fit R 2 values are also included for each curve.
The linear elastic modulus E 1 , the normal viscous coefficient η, the linear shear modulus G 1 , and the shear viscous coefficient λ increase with the increasing of the applied current. There are physical meanings for these four parameters. They characterize the linear elastic and linear shear properties of the tactile display device. The nonlinearity of the tactile display device is characterized by the polynomial coefficients α 1 , α 2 , α 3 , α 4 , α 5 , β 1 , β 2 , β 3 , β 4 , and β 5 . In addition, the combination of MR fluid and the PU sponge is also more rate-dependent (|p| increases) when current increases. Since R 2 value for each fitted curve is close to 1, the proposed models are suitable for predicting the compression and the shear behaviors of the tactile display device. The accuracy of the models will be validated in the next part.

Model validation
To validate the proposed normal force and shear force models, structural FEM simulation is performed. The top part of the PU sponge with MR fluid and the sensor probe are respectively simplified as a rectangular plate and a cylinder with the same geometric dimensions in ANSYS Workbench, as shown in figure 15. Because MR fluid is immersed in the PU sponge and is in semi-solid state when magnetic field is applied, MR fluid and the PU sponge can be treated as one solid entity (rectangular plate) during the simulation. Therefore, all the linear elastic and linear shear parameters in tables 4 and 5 are used to characterize the solid entity made of MR fluid and the PU sponge.
According to the densities of the PU sponge and MR fluid given in table 1, the density of the rectangular plate is set as 3.0 g cm −3 . Structural steel built in ANSYS Workbench is assigned to the cylinder (Density: 7.85 g cm −3 , Young's modulus: 2 × 10 11 Pa, Poisson's ratio: 0.3). The motion of the sensor probe is the same as that in the experiments. Transient analysis is performed with the time step of 0.01 s. The original testing data in the loading period are also plotted for comparison. It can be seen that the results of the structural FEM simulation based on the normal and shear force models match well with the testing data, which validates the accuracy and effectiveness of the proposed models. The normal stress distribution in the normal force simulation and the shear

Conclusion
In this paper, in view of the drawbacks in the existing tactile display devices based on MR fluid, a new tactile display device is designed, simulated, fabricated, assembled, and tested. Firstly, the new tactile display device is designed by incorporating a piston immersed in the PU sponge and MR fluid into a container, and then by covering a PDMS membrane on the PU sponge. Secondly, electromagnetic FEM simulation is performed to determine the maximum current applied to the coil. Thirdly, mathematical models to predict the normal and shear forces of the tactile display device are developed. Then, the tactile display device is tested under normal and shear contact forces. The experimental results are used to determine the unknown parameters in the proposed models. Finally, the proposed models are validated via structural FEM simulation. The main conclusions of this paper are summarized as follows: (1) The proposed tactile display device can overcome the drawbacks of the existing devices in literatures, such as low magnetic conduction efficiency, magnetic leakage, MR fluid leakage, pollution to human skin, etc; (2) The proposed tactile display device can simulate various human tissues or organs in terms of elastic modulus and shear modulus; (3) The proposed nonlinear fractal order Maxwell model accurately predicts the normal and shear forces of the tactile display device.
The proposed tactile display device is still a preliminary prototype, and more work needs to be finished. (1) This device should be made smaller and more compact for better use in RMIS. (2) Suitable control strategies also need to be investigated to adjust the properties of this device. (3) Human touching tests on different biological tissues or organs will be performed as well.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.