**2. Principle**

The structure of the piezoelectric tactile feedback device is shown in Figure 1, mainly including set screws, piezoelectric ceramics, ciliary body touch beams, bracket, control system, and power supply, of which the control system includes an analog-to-digital conversion module, power amplifier modules, and Bluetooth modules.

When a sinusoidal signal with a frequency close to the natural frequency of the touch beam is supplied to the piezoelectric sheets, the touch beam resonates and produces a bending vibration. The ciliary body at di fferent positions of the touch beam will vibrate in di fferent directions. As shown in Figure 2, the ciliary bodies under the finger are distributed on the right and the left side of the vibration peak. The ciliary bodies of 1, 2, and 3 indicate the ciliary body distributed on the right side of the peak, and 4 and 5 indicate the ciliary body distributed on the left side of the peak. At this time, the ciliary bodies distributed on the right side of the peak will give the finger upward inertial pressure and rightward thrust. The ciliary bodies distributed on the left side of the peak will give the finger upward inertial force and leftward thrust. Therefore, when one finger moves to the right, the vibration direction of the ciliary body on the right side of the peak is the same as the direction of movement of the finger. The vibration direction of the ciliary body on the left side of the peak is opposite to the direction of movement of the finger. When the finger touches the beam, the coe fficient of friction of the di fferent parts is constantly changing, which makes the subject feel that the ciliary body on the right side of the peak is smoother than the left side. When the finger moves to the left, the result is reversed. The change in tactile sensation is due to the fact that the equivalent friction coe fficient between the finger and the plane is modulated by the vibrating ciliary body beam, and the sliding friction coe fficient between the finger and the contact surface depends on the ratio of the ciliary body in both directions covered by the finger. The change in the frequency of the excitation signal causes the vibration mode of the touch beam to change. When the excitation frequency is close to the natural frequency, the acceleration increases, and decreases away from the natural frequency. When the excitation frequency approaches the next natural frequency, the vibration mode of the beam is also changed, and the distribution of the vibration direction of the ciliary bodies are changed as the changing of the vibration mode. The tactile effect is di fferent compared to the previous vibration mode.

**Figure 1.** The schematic diagram of the piezoelectric tactile feedback device.

**Figure 2.** Working principle.

The structure of the ciliary body touch beam is shown in Figure 3. The ciliary bodies' density and number can set multiple sets of data. It can produce a variety of tactile sensations. Piezoelectric sheets are pasted on the upper and lower surfaces of the touch beam, and the position of the paste is one peak of the mode function of the touch beam.

**Figure 3.** Structure of the ciliary body touch beam.

Since the ciliary body touch beam accomplishes different surface roughness by the anisotropic vibration of ciliary bodies, the number of the ciliary bodies in the two vibrating directions is the main factor in controlling the overall equivalent friction coefficient under the skin-covering length. As shown in Figure 4, the ciliary bodies' anisotropic vibration dynamic model has a vibration of a wavelength, λ. When the ciliary touch beam vibrates between the solid line and the dashed line, the ciliary bodies will squeeze the skin from the bottom to the top. In the process of squeezing the finger skin upwards, the ciliary bodies, *a*, will move from left to right in the horizontal direction, and the ciliary bodies, *b*, will move from right to left in the horizontal direction.

**Figure 4.** Anisotropic vibration dynamic model of ciliary bodies.

Let the skin move from left to right at a certain speed on the touch beam slowly. When the skin moves above the ciliary body, *a*, the horizontal component of the vibrational direction of the ciliary body is the same as the direction of movement of the skin, and gives the skin a certain thrust along the direction of motion and reduces the total sliding friction between the skin and the touch beam. Therefore, the equivalent friction coefficient of the skin and the touch beam will be reduced by the total sliding friction of the ciliary bodies. When the skin moves above the ciliary body, *b*, the horizontal component of the vibration direction of the ciliary body is opposite to the direction of skin movement, which gives the skin the opposite directional resistance. Therefore, the equivalent friction coefficient of the skin and the touch beam affected by the total sliding friction of the ciliary bodies will increase. In the process of moving the skin, the friction coefficient decreases when the ciliary bodies with the same vibration direction as the skin movement direction are touched, and the tactility of the touch beam becomes smooth. The friction coefficient increases when the ciliary bodies with the opposite vibration direction as the skin movement direction are touched and the tactility of the touch beam becomes rough. Additionally, the vibrating direction of ciliary bodies is distributed according to the following Equation (1):

$$\begin{cases} \phi^{(i)}(\mathbf{x}) \cdot \phi^{(i)'}(\mathbf{x}) < 0, \text{ Right vibration area} \\ \phi^{(i)}(\mathbf{x}) \cdot \phi^{(i)'}(\mathbf{x}) > 0, \text{ Left vibration area} \end{cases} \tag{1}$$

where φ(*i*)(*x*) is the *i*-th order mode function of the touch beam [19].

As shown in Figure 5, we assumed that the ciliary bodies' density on the touch beam is sufficiently large. It can be seen that the directions of vibration of the ciliary bodies change at the peak or trough of a wave. Hence, when one finger's skin moves from left to right, the increase and decrease of the friction coefficient shows a periodic variation pattern.

However, if the direction of the vibration of the ciliary body covered by the finger skin is positive and opposite when the finger touches the beam, the sense of touch perceived by the receptor is determined by the ratio of the two vibration directions covered by the skin. In addition, the length of the beam covered by the receptor with a single finger and multi-fingers is not the same. Additionally, the single finger covers less than half of the wavelength of the vibration form. This is called local coverage here. Multi-fingers cover the range of more than one wavelength of the vibration form. This

is called full-coverage. Therefore, the tactile model under both full-coverage and local-coverage needs to be analyzed and calculated separately.

**Figure 5.** The distribution of the different directional ciliary bodies.

#### **3. Analysis for Anisotropic Vibrating Tactile Models**

#### *3.1. Analysis for Full-Coverage Anisotropic Vibration Tactile Model*

The full-coverage anisotropic vibration tactile model is shown in Figure 6. The skin covers the full touch beam and moves to the right at a uniform velocity, *v*. The skin gives pressure to the touch beam, and the pressure between the skin and the touch beam includes the static pressure and the inertia pressure, which are given to the skin during the vibration of the ciliary bodies. Therefore, the total pressure between the skin and the touch beam can be written as:

$$F = (a+b)f + f\_{p\*} \tag{2}$$

where *a* is the number of vibrating ciliary bodies in the same direction, *b* is the number of vibrating ciliary bodies in the opposite direction, *f* is the static pressure of the skin on each ciliary body, and *fp* is the ciliary bodies' inertial pressure on the skin of the hand.

**Figure 6.** Anisotropic vibration tactile model of full-coverage.

According to the previous analysis, we know the total sliding friction includes the static pressure, the inertial pressure, and the sliding friction caused by lateral vibration of the ciliary bodies. From Equation (2), the total sliding friction force of the hand skin in full-coverage is:

$$\begin{split} F\_n^{(f)} &= \mu(fa + fb) + \eta(\mu f \cdot b - \mu f \cdot a) + \mu f\_p \\ &= \frac{3}{2}\mu fb + \frac{1}{2}\mu fa + \mu f\_p \end{split} \tag{3}$$

where μ is the general sliding friction coefficient between the finger and the touch beam. η is the effective coefficient of the anisotropic vibration of the ciliary bodies. The effect coefficient of the ciliary bodies' anisotropic vibration is the degree of the effect of the skin of the ciliary bodies in the process of touching the beam. The ciliary bodies have a more obvious effect on the skin when moving upwards. However, when the ciliary bodies move away from the skin, the effect is faint. So, here, the effect on the fingers is negligible. Therefore, the vibrating inertial force acting time of the ciliary bodies can be approximated as 1/2 of the total time, so η = 1/2 is taken.

Let the total number of ciliary bodies remain unchanged on the touch beam. In order to simplify the ciliary bodies' tactile model, it was assumed that the ciliary bodies are dense enough. From Equation (3), the total sliding friction force of the hand skin in full-coverage can be written as:

$$F\_n^{(f)} = \mu(\frac{3}{2} - \frac{a}{u} + \frac{f\_p}{uf})F. \tag{4}$$

Then, the equivalent friction coefficient can be written as:

$$
\mu' = \mu(\frac{3}{2} - \frac{a}{u} + \frac{f\_p}{uf})\_\prime \tag{5}
$$

where *u* is the total number of ciliary bodies on the touch beam. μ*'* is the equivalent friction coefficient under the vibration state.

Maintaining the proportion of the same direction of ciliary bodies and the opposite direction of ciliary bodies, let:

$$\frac{a}{b} = \chi, \chi \in \mathcal{N}.\tag{6}$$

Substituting Equation (6) into (5), we can obtain the relationship between the equivalent friction coefficient and the total number of the ciliary bodies as:

$$
\mu' = \mu \left[ (\frac{3}{2} - \frac{\chi}{\chi + 1}) + \frac{f\_P}{\mu f} \right]. \tag{7}
$$

During the contact of the skin with the touch beam, the inertial pressure also affects the sliding friction between the skin and the touch beam. As shown in Figure 7, the touch beam vibrates from the solid line to the dotted line. One vibrating ciliary body produces a normal pressure on the skin that is the inertia force, *fp*. The inertial pressure is related to the forced response of the touch beam. The effect coefficient of the inertial pressure, η*p*, and the effect coefficient, η, of the anisotropic vibration are the same, taking η*p* = 1/2, so the inertial force of the ciliary bodies can be expressed as:

$$f\_p = \eta\_p a\_c^{(f)} m = \frac{m}{T} \cdot \sum\_{j=1}^{u} \left| \int\_0^{\frac{T}{2}} \left| w^{(i)"} \left( x\_j, t \right) \right| \text{d}t \right|, \tag{8}$$

where *w*(*i*) is the acceleration response of the touch beam, *m* is the mass of one ciliary body, and *ac(f)* is the average acceleration over one vibration period of all ciliary bodies. *xj* is the positional coordinate of the *j*-th ciliary body. *T* is the vibrational cycle time of the touch beam.

In the case where the density of the ciliary bodies is sufficient, one ciliary body can be approximated as a micro-element, and then the inertial pressure of the full-coverage ciliary bodies can be expressed as:

$$f\_p^{(f)} = \frac{1}{2} a\_\mathcal{L} m = \frac{m}{T} \cdot \int\_0^\mathcal{L} \int\_0^{\frac{T}{2}} |w^{(i)\prime\prime}(x\_j, t)| \mathrm{d}t \mathrm{d}x\_j. \tag{9}$$

Substituting Equation (9) into (5), the equivalent friction coefficient in full-coverage can be shown by:

$$\mu' = \mu \left( \frac{3}{2} - \frac{a}{u} + \frac{m}{Tuf} \cdot \int\_0^L \int\_0^{\frac{L}{2}} \left| w^{(i)\prime\prime} \left( \mathbf{x}\_{j\prime}, t \right) \right| \mathbf{d}t d\mathbf{x}\_j \right). \tag{10}$$

**Figure 7.** The inertial pressure of the ciliary body.

The cantilever touch beam was chosen as the research object, and the relevant parameters of the piezoelectric sheet and touch beam are shown in Tables 1 and 2. The sliding friction coe fficient of the skin and the copper was μ = 0.4, and the parameters related to ciliary bodies are shown in Table 3.

**Table 1.** Related parameters of the touch beam.


**Table 2.** Related parameters of a piezoelectric sheet.


**Table 3.** Related parameters of the ciliary bodies.


The relationship between the equivalent friction coe fficient of the cantilever touch beam and the number of the vibration direction, *a*, of the ciliary bodies in full-coverage calculated from Equation (10) is shown in Figure 8.

**Figure 8.** The relationship between μ' and *a*.

Figure 8 shows that if the total number of ciliary bodies on the touch beam is 20, the equivalent friction coe fficient between the touch beam and the skin gradually decreases with the increase in the number of vibrating ciliary bodies in the same direction, and the two change linearly. When the number of the same directional ciliary bodies is less than half, the sliding friction force of the touch beam is greater than the general sliding friction force, and it is in the rougher state. When the number of the same directional ciliary bodies is more than half, the sliding friction force of the touch beam is smaller than the general sliding friction force and is in the smoother state. The equivalent friction coe fficient of the cantilever touch beam varies from about 0.2 to 0.6.

#### *3.2. Analysis for Anisotropic Vibration Tactile Model in Local-Coverage*

The local-coverage anisotropic vibration tactile model is shown in Figure 9. The skin of the finger covers the local length, *ls*, of the touch beam and moves to the right at a uniform velocity, *v*. The finger gives pressure to the touch beam. There are a certain number of the same directional ciliary bodies and the opposite directional ciliary bodies under the skin. The pressure between the skin of the finger and the touch beam also includes the static pressure and the inertia pressure, which is given to the finger during the vibration of the ciliary bodies. Due to the changes of the ciliary bodies' vibration distribution state during the constant movement of the finger, the changeable law of the equivalent friction coe fficient with two parts of the pressure at di fferent positions, *x*, must be analyzed.

**Figure 9.** Anisotropic vibrational tactile model of local-coverage.

In the case where the ciliary bodies' density is su fficient, every ciliary body is approximated as a micro-element. One-half vibration cycle of each ciliary body is its e ffective period on the finger. So, from Equation (9), the inertial pressure in the local-coverage can yield:

$$f\_p^{(l)} = \frac{1}{2} a\_c^{(l)} m = \frac{m}{T} \cdot \int\_{x\_j - l\_s}^{x\_j} \int\_0^{\frac{T}{2}} |w^{(i)\prime\prime}(x\_{j\prime}, t)| dt dx\_{j\prime} \tag{11}$$

where *a* (*l*) *c* is the average acceleration within a half vibration period of the locally covering ciliary bodies; ls is the local-coverage length of the finger. Substituting Equation (11) to (5), the equivalent friction coe fficient in local-coverage can be shown as:

$$
\mu' = \mu \left( \frac{3}{2} - \frac{a}{u} + \frac{m}{Tuf} \cdot \int\_{x\_j - l\_s}^{x\_j} \int\_0^{\frac{T}{2}} \|w^{(i)"} (x\_{j'}, t)\| dt dx\_j \right). \tag{12}
$$

From Equation (12), it is apparent that the equivalent friction coe fficient under local-coverage is mainly a ffected by two parts. One is the inertial force, *fp*, of the ciliary bodies, and the other is the proportion, *<sup>a</sup>*/*<sup>u</sup>*, of the total number of ciliary bodies in the same direction of vibrating ciliary bodies. The functional relationship between *fp* and the positional coordinate, *x*, was obtained from Equation (11), and the functional relationship between *a*/*u* and the positional coordinate, *x*, needs to be further analyzed.

One cantilever touch beam was selected as the research object, and the relevant parameters were the same as in Tables 1–3. One operating mode of the touch beam is shown in Figure 10, and the covered length of the finger was considered as *l*s = 0.01 m. In the course of the uniform movement of the finger on the touch beam, according to the theory of the ciliary body's anisotropic vibration, the ratio of the ciliary bodies of the first whole wave segment, *<sup>a</sup>*/*<sup>u</sup>*, exists in six stages. Then, seven key positions of the finger exist on the touch beam. The change rules were as follows:


$$\frac{a}{u} = \frac{\lambda}{4l\_s}.\tag{13}$$


$$\frac{a}{\mu} = \left(l\_s - \frac{\lambda}{4}\right) / l\_s = 1 - \frac{\lambda}{4l\_s}.\tag{14}$$

• When the finger moves from position VI to position VII, the same directional vibration area of the finger-covering part gradually increases. The ratio of the same-directional ciliary bodies increases from 1 − λ/(4*ls*), and the increased ratio is (*x* − 0.0331 − λ/2)/*ls*.

**Figure 10.** *Cont.*

**Figure 10.** Key positions of the finger on the cantilever beam. (**a**) Position I; (**b**) position II; (**c**) position III; (**d**) position IV; (**e**) position V; (**f**) position VI; (**g**) position VII.

After that, all the whole wave segments met the above change rules 3 to 6. So, the coefficient, γ, was introduced to indicate the number of the whole wave segments after the first. In summary, according to the change rules of the ratios of the same directional ciliary bodies in the five stages, the relationship between *a*/*u* and the positional coordinates are as follows:

$$\frac{a}{u} = \begin{cases} 0 & \mathbf{x} \in [0, 0.0131) \\ \frac{\mathbf{x} - 0.0131}{l\_s} & \mathbf{x} \in [0.0131, 0.0218) \\ \lambda / 4l\_s & \mathbf{x} \in [0.0218 + \frac{\lambda \mathcal{V}}{2}, 0.0231 + \frac{\lambda \mathcal{V}}{2}) \\ \frac{\lambda}{dl\_s} - \frac{\mathbf{x} - 0.0231 - \frac{\lambda \mathcal{V}}{2}}{l\_s} & \mathbf{x} \in [0.0231 + \frac{\lambda \mathcal{V}}{2}, 0.0318 + \frac{\lambda \mathcal{V}}{2}) \\ 1 - \lambda / 4l\_s & \mathbf{x} \in [0.0318 + \frac{\lambda \mathcal{V}}{2}, 0.0331 + \frac{\lambda \mathcal{V}}{2}) \\ 1 - \frac{\lambda}{4l\_s} + \frac{\mathbf{x} - 0.0331 - \frac{\lambda \mathcal{V}}{2}}{l\_s} & \mathbf{x} \in [0.0331 + \frac{\lambda \mathcal{V}}{2}, 0.0418 + \frac{\lambda \mathcal{V}}{2}) \end{cases} \tag{15}$$

where γ ∈ Z, 0 ≤ γ ≤ 5, *x* ≤ 0.1.

From Equation (15), the relationship between the proportion of the same directional ciliary bodies and the positional coordinates is shown in Figure 11. From the image, we know that the ratio, *<sup>a</sup>*/*<sup>u</sup>*, increases from zero and then decreases when the finger moves from left to right, and periodically alternates in the interval of [0,0.87]. The value of *a*/*u* maintains a shorter distance when it is at a maximum or minimum position.

**Figure 11.** The relationship between the ratio, *<sup>a</sup>*/*<sup>u</sup>*, and the position coordinates, *x*.

Substituting Equation (15) into (12), the relationship between the equivalent friction coe fficient and the position of the touch beam at one operating mode of the cantilever touch beam can be obtained, as shown in Figure 12. The length of the dotted line, *l*b, is the piezoelectric sheet pasted length. From Figure 12, the following observations were worth noting:

The equivalent friction coe fficient of the touch beam also alternates periodically and fluctuates around 0.2 to 0.6 and centered at μ' = 0.4. There are four troughs of the equivalent friction coe fficient on the left side of the length, *lb*, of the piezoelectric sheet, which is the place where the equivalent friction coe fficient is relatively low and is also a relatively smooth position. However, the distance between the fourth valley (from left to right) and the segmen<sup>t</sup> where the piezoelectric sheet is located is short, and it is di fficult to move in the length of this segmen<sup>t</sup> during the actual touch process. Therefore, the first three valleys, *l*I, *l*II, and *l*III, are the three easily perceived smoother positions on the left side of the piezoelectric sheet. Taking the position of the trough at each position as the base point, they are respectively located at 0.0137, 0.0275, and 0.0422 m on the abscissa, *x*.

**Figure 12.** The relationship of μ'-*x* at the operating frequency of 24,221 Hz.

#### **4. E** ff**ect of System Parameters on Tactile Changes**

#### *4.1. E*ff*ect of System Parameters on Tactile Changes in Full-Coverage*

Four parameters were selected to analyze the e ffect on the equivalent friction coe fficient, including the number of the same directional ciliary bodies, *a*; the total number of ciliary bodies, *u*; the operating frequency, ω; and the excitation voltage amplitude, *V*. Figure 13 shows the variation of the equivalent friction coe fficient with the parameters changed under the conditions of 2, 10, and 20 N finger pressures. From

 Figure 13, the following observations were worth noting:


**Figure 13.** The effect of key parameters on the equivalent friction coefficient in full-coverage. (**a**) μ*'*-*<sup>a</sup>* relational figure; (**b**) μ*'-u* relational figure; (**c**) μ*'*-frequency relational figure; (**d**) μ*'*-*<sup>V</sup>* relational figure.

#### *4.2. E*ff*ect of System Parameters on Tactile Changes in Local-Coverage*

The parameters of the operating voltage, *V*, and frequency, ω, of the local-coverage cantilever touch beam were changed. The relationship between the equivalent friction coefficient and the position of the touch beam under the effect of the operating parameters was obtained, as shown in Figure 14.

**Figure 14.** The changes of μ*'-x* under the effect of operating parameters in local-coverage. (**a**) Operating voltage changed; (**b**) operating frequency changed.

From Figure 14, the following observations were worth noting:

As the operating voltage amplitude, *V*, increases, the μ*'* variation amplitude of the touch beam gradually increases with 0.4 as the center, and the corresponding position becomes smoother or rougher. With the decrease of the operating frequency, ω, the μ' variation amplitude of the touch beam slightly increases with 0.4 as the center. However, when the frequency decreases, the number of peaks and troughs gradually decreases.
