Bragg grating-based all-optical continuous two-dimensional force perceptron

The main component of this force perceptron consists of a cylinder with three embedded FBGs and a sphere located at the top of the cylinder. Additionally, an open-cone-shaped strong strain isolator is included. To simulate the signal generated by the strain applied to the perceptron, we simplify the entire perceptron to a cantilever beam loaded with a weight at the top. The cantilever beam has a length of h, a circular cross-section, and a bending stiffness of $EI$ (where $E$ is the Young's modulus and $I$ is the second moment of area) is calculated based on the Bernoulli beam equation.

The reflection wavelength of a FBGs primarily changes with the grating spacing $\Lambda$, and temperature and strain are the fundamental factors determining the variation. The thermal effect, which varies with temperature, is effective for basic FBGs Flockhart et al [20]. A differential strain configuration can be employed to compensate for the thermal effect, ensuring that the reflection wavelength $\lambda$ of the FBG depends solely on the strain magnitude $\varepsilon$. The relationship between strain and reflection wavelength is given by equation (1)

$$\begin{equation}\varepsilon = \frac{{\Delta \Lambda }}{\Lambda } = \frac{{\Delta \lambda }}{\lambda }.\end{equation} \tag{ 1 }$$

For determining the strain magnitude and direction of the neutral plane, we first express the wavelength shift as signal strength $\Delta {\lambda _1}$, $\Delta {\lambda _2}$, $\Delta {\lambda _3}$. Three standard deviations ${D_1}$, ${D_2}$, ${D_3}$ are defined as the difference between the received signal strength and the mean signal strength

$$\begin{equation}{D_1} = \Delta {\lambda _1} - \frac{{\Delta {\lambda _1} + \Delta {\lambda _2} + \Delta {\lambda _3}}}{3}\end{equation} \tag{ 2 }$$

$$\begin{equation}{D_2} = \Delta {\lambda _2} - \frac{{\Delta {\lambda _1} + \Delta {\lambda _2} + \Delta {\lambda _3}}}{3}\end{equation} \tag{ 3 }$$

$$\begin{equation}{D_3} = \Delta {\lambda _3} - \frac{{\Delta {\lambda _1} + \Delta {\lambda _2} + \Delta {\lambda _3}}}{3}.\end{equation} \tag{ 4 }$$

The total strain magnitude is calculated using the Pythagorean theorem. We square each component, add them together, and then take the square root of the sum. This is equivalent to finding the length of a vector representing strain in three-dimensional space, where each component corresponds to a dimension. When Bragg grating wavelength is ${\lambda _a}$, The amplitude of the strain received by the column is given by:

$$\begin{equation}\varepsilon = \frac{{\sqrt {{D_1}^2 + {D_2}^2 + {D_3}^2} }}{{{\lambda _a}}}.\end{equation} \tag{ 5 }$$

The direction of strain is determined by the relative intensities of the signals received from the three fiber channels. In this case, we can use an improved algorithm to calculate the direction based on the wavelength shifts with positive and negative intensity signals. Figure 2(b) illustrates the partitioning of the direction of stimulation. After receiving an external stimulus, we can classify it to any partition based on the different signal intensities from the three FBGs. Angle recognition calculations for force directions involve taking the wavelength shifts generated by the three FBGs and projecting them onto the xy-plane established in figure 3(a). The wavelength shifts are projected separately in the $XY$-directions, thus we can get: $y = - [\Delta {\lambda _1} - (\Delta {\lambda _2} + \Delta {\lambda _3})\cos {60^\circ }]$, $x = (\Delta {\lambda _2} - \Delta {\lambda _3})\cos {30^\circ }$. By utilizing the differences in wavelength along the $XY$-directions, the angle values can be reconstructed using the arctangent function. It is important to note that the goal is to achieve 360° angle recognition, which is typically computed using the $a\tan 2(y,x)$ function borrowed from computer programming languages

$$\begin{align} {\text{direction}} &= a\tan 2\left({ - \left( {\Delta {\lambda _1} - \frac{{\Delta {\lambda _2} + \Delta {\lambda _3}}}{2}} \right),} \right. \nonumber\\ & \left.\quad \ {\frac{{\sqrt 3 }}{2}\left( {\Delta {\lambda _2} - \Delta {\lambda _3}} \right)} \right). \end{align} \tag{ 6 }$$

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To further investigate the magnitude of the forces that cause strain, this columnar structure can be equivalent to a cantilever beam. The mechanical characteristics of the cantilever beam Benham et al [21] are characterized by equations (7) and (8), where the strain at the cantilever height of $z$, the neutral axis bending distance of $d$ show in figure 3(b), and the cantilever top displacement of $\Delta \tau$ after receiving the force $F$ at the top of the cantilever are determined together. Among them, when the force magnitude is fixed, $\Delta \tau$ mainly depends on the cantilever height and the material's flexural rigidity

$$\begin{equation}\Delta \tau = \frac{{F{h^3}}}{{3EI}}\end{equation} \tag{ 7 }$$

$$\begin{equation}\varepsilon = - \Delta \tau \frac{{3d\left(h - z\right)}}{{{h^3}}}.\end{equation} \tag{ 8 }$$

The cylindrical part of the strain sensor is equipped with three Bragg gratings arranged in an equilateral triangle pattern show in figure 2. The silica outer layer of the optical fiber is not removed. In order to achieve compatibility and integration with multi-level strain sensing in the future, the Bragg grating wavelength is mainly selected within the range of ${\lambda _a} = 1550 \pm 50$ ${\text{nm}}$ in the communication band.

The bending stiffness $K$ plays a crucial role in the mechanical performance of the fiber-supported structure of the strain sensor and is a key parameter affecting its sensing performance. It particularly depends on the elastic properties of the support cross-section and fibers and is strongly influenced by their geometry and composition. The composite structure of the support matrix and fibers implies that the bending stiffness $EI$ is a function of the mechanical properties of both materials and is influenced by their relative proportions and arrangement. Additionally, the bending stiffness of each individual fiber depends on its distance to the neutral bending axis, which is defined as the axis around which the fiber can rotate without any additional bending. Importantly, the bending axis is not fixed and can rotate as the sensor is deflected in different directions. However, we have shown that the key mechanical parameter of the fiber-supported structure, the bending stiffness $K$, does not need to be related to the variable bending axis offset $\alpha$.

When the strain sensor receives a strain force impact, and the sensor body is deflected by an angle $\alpha$ relative to its initial state, the relative positions of the three fibers in the $XY$ direction change show in figure 3(b). We characterize the strain generated by the fibers themselves when the strain sensor receives an impact by the perpendicular distance ${d_{x1}}$, ${d_{x2}}$, ${d_{x3}}$ and ${d_{y1}}$, ${d_{y2}}$, ${d_{y3}}$ between the fibers and the bending axis show in figure 3(b). The overall structure of the cylinder has no preferred bending direction and is mechanically isotropic in bending stiffness $K$. Therefore, we can determine a direction-independent or isotropic bending stiffness at the top of the cylinder structure.

$$\begin{equation}K = \frac{{3\pi }}{{4 \cdot {h^3}}}\left[{E_s} \cdot {r_s}^4 + 4\left({E_c} - {E_s}\right)\left({r_c}^2 \cdot {p^2} + {r_c}^4\right)\right]\end{equation} \tag{ 9 }$$

where ${r_c}$ and ${r_s}$ denote the cladding and support radii and ${E_c}$ and ${E_s}$ their respective Young's moduli, $p$ represents the distance between the fiber cores.

Next, we provide the relationship between the main body deflection angle and the top displacement $\Delta \tau$ of the cylinder when it receives an impact, as well as the differential strain vector of the fiber. We define the fiber spacing in the $XY$ direction after the bending axis of the cylinder is offset by angle $\alpha$ as follows:

The fiber spacing in the $XY$ direction is:

$$\begin{equation}\begin{gathered} {d_{x1}} = \frac{{\sqrt 3 }}{3}p\sin \left(\frac{\pi }{2} - \alpha \right) \hfill \\ {d_{x2}} = \frac{{\sqrt 3 }}{3}p\sin \left(\frac{\pi }{6} - \alpha \right) \hfill \\ {d_{x3}} = \frac{{\sqrt 3 }}{3}p\sin \left(\frac{\pi }{6} + \alpha \right) \hfill \\ \end{gathered} \end{equation} \tag{ 10 }$$

$$\begin{align} {d_{y1}} & = \frac{{\sqrt 3 }}{3}p\sin \alpha \nonumber\\ {d_{y2}} & = \frac{{\sqrt 3 }}{3}p\sin \left(\frac{\pi }{3} - \alpha \right) \nonumber\\ {d_{y3}} & = \frac{{\sqrt 3 }}{3}p\sin \left(\frac{\pi }{3} + \alpha \right).\end{align} \tag{ 11 }$$

According to equation (8), we can obtain the differential strain in the $XY$ direction as follows:

$$\begin{align} \Delta {\varepsilon _x} & = {\varepsilon _{x2}} - {\varepsilon _{x1}} - {\varepsilon _{x3}} \nonumber\\ & = a \cdot - 2\cos \left(\alpha - \frac{\pi }{3}\right) \end{align} \tag{ 12 }$$

$$\begin{align} \Delta {\varepsilon _y} & = \,{\varepsilon _{y2}} + {\varepsilon _{y3}} - {\varepsilon _{y1}} \nonumber\\ & = a \cdot - 2\sin \left( {\alpha - \frac{\pi }{3}} \right) .\end{align} \tag{ 13 }$$

With

$$\begin{equation}a = - \Delta \tau \cdot \frac{{\sqrt 3 \left(h - z\right)p}}{{{h^3}}}.\end{equation} \tag{ 14 }$$

The size of the absolute strain difference is calculated using polar coordinates, and we can obtain the size ${R_{\Delta \varepsilon }}$ and angle ${\theta _{\Delta \varepsilon }}$ of the differential strain

$$\begin{equation}{R_{\Delta \varepsilon }} = \Delta \tau \cdot \frac{{3\left(h - z\right)p}}{{{h^3}}}\end{equation} \tag{ 15 }$$

$$\begin{equation}{\theta _{\Delta \varepsilon }} = \alpha - \frac{\pi }{3}.\end{equation} \tag{ 16 }$$

Similarly, the wavelength shift difference can be used to obtain the wavelength difference in the $XY$ direction, and the relationship between the wavelength drift difference $\overrightarrow {\Delta \lambda }$ and the deflection $\overrightarrow {\Delta \tau }$ is given by:

$$\begin{equation}\Delta {\lambda _x} = \Delta {\lambda _2} - \Delta {\lambda _1} - \Delta {\lambda _3} = \Delta {\tau _x} \cdot \frac{{3\left(h - z\right)p}}{{{h^3}}}\lambda \end{equation} \tag{ 17 }$$

$$\begin{equation}\Delta {\lambda _y} = \Delta {\lambda _2} - \Delta {\lambda _1} + \Delta {\lambda _3} = \Delta {\tau _y} \cdot \frac{{3\left(h - z\right)p}}{{{h^3}}}\lambda .\end{equation} \tag{ 18 }$$

For the overall force applied to the cylindrical body, we have the relationship is as follows:

$$\begin{equation}\Delta \lambda = \frac{{F\left(h - z\right)p}}{{EI}}\lambda .\end{equation} \tag{ 19 }$$

The processing and fabrication of the main body of the force perceptron involve critical factors such as the spacing between the fiber cores, the fixed position of the grating region, the length of the column, and the size and material of the stimulus-receiving sphere. The optimal dynamic range of the sensor is achieved with a spacing of $313.4$ ${{\mu m}}$ between the fiber cores and a distance of $5$ ${\text{mm}}$ from the grating region to the bottom of the cantilever Wolf et al [11]. To approach the optimal solution, we fabricated the column as shown in figure 3(a), with a length of $7$ ${\text{cm}}$ and a fiber core spacing of $700$ ${{\mu m}}$. To enhance the sensitivity of the sensor, we focused on the structural design of the column. For this purpose, we made the column hollow, as reducing its flexural stiffness can increase the sensitivity of the sensor, as shown in equation (7). Another important consideration is the necessity of the stimulus-receiving sphere at the top. The spherical shape of the stimulus receiver ensures the consistency of stimulus transmission to the grating, as it eliminates any preferred direction of stimulus perception. The diameter and material of the spherical stimulus receiver are determined based on the main objective of the mechanical sensor design, which is to enable perception of various complex force stimulus. To amplify the perception threshold of the sensor for force stimuli in different complex fluids, we can control the boundary thickness and diameter of the stimulus-receiving sphere to amplify the stimulus. To test the perception threshold of the sensor itself, we set the sphere diameter to $15$ ${\text{mm}}$, and the column material is made of silicone gel (SI587,LOCTITE, Germany) (van Netten & MacKinnon, 2013) [22].

Three SMF-28 optical fibers, with the jacket removed and a core radius (${r_c}$) of $62.5$ ${{\mu m}}$ and Young's modulus of $75$ ${\text{GPa}}$, were arranged in a precise equilateral triangle configuration. The positioning of the fiber cores was carefully controlled to ensure a maximum deviation of $0.1$ ${\text{mm}}$ from the grating region to the fixed point of the cantilever. These fibers were encapsulated in the selected material for the column structure. To achieve this, we designed a specialized processing apparatus that enabled precise fiber fixation, installation, and column formation. The cross-section of the fabricated column is shown in figure 3. The base of the sensor was created using 3D printing and included a specific mounting component. The column was partially inserted into the mounting component and secured with adhesive, serving as the fixed cantilever point of the sensor. Additionally, a shielding cover was added to the external part of the sensor to effectively shield stray stimuli. Moreover, it prevented damage to the sensor in the event of excessive stimulation when reaching the sensing limit.

The mechanical sensor testing procedure is illustrated in figure 4. A precision microbalance (ESE, ES1035A, D&T, China) with a design accuracy of $0.01$ ${\text{mg}}$ was used to precisely control the applied force stimulus on the sensing sphere. The wavelength shift of the FBG was recorded using an optical demodulator (BJTW, TV-1600, TONG WEI SENSING, China) with a resolution of $0.1$ ${\text{pm}}$. The applied force on the sensing sphere was adjusted by controlling the linear stage's (WN28VM, WN28VM25M, Winner Optical Instruments, China) movement distance. The microbalance was used to display the magnitude of the applied force, allowing for precise control of force stimulus of any desired magnitude to be conveyed to the sensing sphere.

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We used a linear stage and a precision balance to control and measure the force applied to the receiving sphere. First, we examined whether the dynamic response of the sensor under fixed deflection force was consistent. It was essential to ensure that the three FBGs in the sensor were mechanically symmetric and had no preferred bending direction. This was crucial for determining the independent or isotropic bending stiffness $K$ of the column in any direction. Figure 5 illustrates the wavelength shift of the column when subjected to a positive force stimulus. We used the linear stage to apply a fixed force stimulus to the column incrementally by a value of $0.1$ ${\text{mN}}$, ranging from $0$ ${\text{mN}}$ to $1$ ${\text{mN}}$. We measured the wavelength shifts of fiber core1, fiber core2, and fiber core3. The experimental results showed that the sensor was capable of detecting forces as low as $0.06$ ${\text{mN}}$, with a resolution of $0.1$ ${\text{mN}}$. Figure 6(a) demonstrates that the overall wavelength shift of fiber core1, fiber core2, and fiber core3 reached $0.04$ ${\text{nm}}$ when the force ranged from $0$ ${\text{mN}}$ to $1$ ${\text{mN}}$. Under the increment of $0.1$ ${\text{mN}}$, the wavelength shifts of fiber core1, fiber core2, and fiber core3 were in the range of $0.003$ ${\text{nm}}$ to $0.004$ ${\text{nm}}$, proving that the force resolution of the sensor reached $0.1$ ${\text{mN}}$. Furthermore, in figure 6(a), it can be observed that when the initial stimulus increased from $0$ ${\text{mN}}$ to $0.1$ ${\text{mN}}$, the wavelength shift was $0.003$ ${\text{nm}}$. As the force stimulus continued to increase to $0.9$ ${\text{mN}}$ and received an additional $0.1$ ${\text{mN}}$ stimulus, the wavelength shift of the sensor remained stable at $0.003$ ${\text{nm}}$ to $0.004$ ${\text{nm}}$.Therefore, when the same incremental force stimulus was added to the sensor under different states, the dynamic characteristics of the sensor remained stable. However, when the applied stimulus approached the bending limit of the FBG, the column material's deformation started to affect the sensitivity of the sensor. Thus, in figure 6(b), we continued to increase the mechanical stimulus while ensuring that the FBG did not reach its bending limit. Based on a baseline of $1$ ${\text{mN}}$, we incrementally added a force stimulus of $0.1$ ${\text{mN}}$. The total increase of force stimulation was from $1$ ${\text{mN}}$ to $2$ ${\text{mN}}$.The experimental results in figure 6(b) showed that as the force stimulus steadily increased, the wavelength shifts of FBG1, FBG2, and FBG3 exhibited unstable increments, ranging from $0.002$ ${\text{nm}}$ to $0.005$ ${\text{nm}}$ for each $0.1$ ${\text{mN}}$ increment. As the force stimulus approached the bending limit of the internal grating in the sensor, the reflection peaks of the FBG also became unstab [23].When the sensor is subjected to forces nearby $2$ ${\text{mN}}$, the wavelength shift per $0.1$ ${\text{mN}}$ increment starts to decrease noticeably. This suggests that as the applied force linearly increases, the wavelength shift provided by the linear within its sensing threshold, Ensure the consistency of the force perception output signal in the inner gate area of the sensor, and support the accuracy of perception direction.

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Next, we further validated the directional recognition capability of the all-optical two-dimensional sensor. To achieve directional recognition in a two-dimensional space, it is necessary to have contrasting differences in both horizontal and vertical directions. Therefore, we designed the directional recognition based on the centroid of the equilateral triangle formed by the three FBGs, establishing the Cartesian coordinate system shown in figure 3(a). The two-dimensional plane is divided into three major directions as shown in figure 2(b): region 1 represented by FBG1, region 2 represented by FBG2, and region 3 represented by FBG3. The goal was to identify which region the stimulus came from when the receiving sphere received stimulus from any direction, enabling directional recognition.

To test the effectiveness of directional recognition for microforce stimulus, we used a step rotation stage to rotate the sensor at angles of $- 90^\circ$, $30^\circ$ and $150^\circ$, while applying fixed force deviations of $0.1$ ${\text{mN}}$ to $1$ ${\text{mN}}$ at the top of the sensor. The important reason for choosing these three angles is that the current ideal direction recognition target is the general direction recognition of three regions, and the stimulus under these three angles can cover the direction perception of the whole two-dimensional plane. Next recorded the wavelength shifts of FBG1, FBG2, and FBG3 under the same force deviation in the three directions, as shown in figure 7. In table 1, we have presented the wavelength shifts of FBG1, FBG2, and FBG3 under three different force angles. When conducting large-area recognition, one can directly determine the source of the force direction (i.e. which quadrant it belongs to) by examining the signs of the three FBG signals. The quadrant associated with the positive signal corresponds to the direction of the applied force.

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For further angle recognition, the received signals can be input into equation (6) for calculation. Table 1 also demonstrates the experimental results of three repeated tests at each angle. The results indicate that under the same force stimulus, the sensor maintains stability in recognizing the force stimulus. The maximum error angle in the experimental results is 2.6° . We standardize the error angle, which is defined as error rate=|Identify angle-Force angle|/360°, and the error rate is less than 1%. The source of the perceptual error of the perceptron is mainly due to the inconsistency in the production of the fiber Bragg grating, and there will be a certain difference in wavelength offset when receiving the same force stimulus, and at the same time, in the actual production, the fiber Bragg grating arrangement can not be arranged in a perfect regular triangle, resulting in a difference between the force stimulus and the applied force stimulation angle when the force stimulus is transmitted to the internal fiber Bragg grating. In order to reduce the error, we can choose the fiber Bragg grating with better consistency and ensure that the fiber is arranged in a perfect triangle as much as possible.

The results demonstrate that, despite these challenges, the system achieves a relatively low angle recognition error of $2.6^\circ$ with an error rate of $1\%$, indicating its effectiveness in recognizing force directions and angles. As a result, we achieved directional recognition of force stimulus within the entire two-dimensional plane.