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Abstract
A fiber Bragg grating (FBG) based three-dimensional (3D) force sensor for a humanoid prosthetic hand is designed, which can precisely detect 3D force and compensate for ambient temperature. FBG was encapsulated in polydimethylsiloxane (PDMS) for force sensitization and immobilization, and the structural parameters of the sensor were optimized by using finite element simulation, so that its sensitivity to 3D force is enhanced. In the meantime, the calibration experiments for normal force fZ, shear force fX/fY, and temperature were conducted, and the 3D force data were decoupled using the least square (LS) and backpropagation (BP) neural networks decoupling methods, so that an overall decoupling error is 0.038. The results show that the sensor has a simple structure, high sensitivity, high linearity, good creep resistance, and rapid decoupling, providing a successful design for the 3D force detection of a humanoid prosthetic hand.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
To obtain accurate force vector information on the contact with the environment of the prosthetic hand, the robot force sensor must simultaneously measure the force information in several dimensions. Single-dimensional force sensors cannot accurately sense the hand-grasping state to meet precise operating requirements [1]. The evolution of multidimensional force sensors for prosthetic hands is geared toward flexibility, miniaturization, and intelligence [2]. The majority of extant sensors are piezoresistive [3,4], magnetosensitive [5], capacitive [6–8], piezoelectric [9,10], photoelectric [11], etc. These sensors have the advantages of large dynamic range and high accuracy, but there are disadvantages such as large unit size, complex manufacturing processes, susceptibility to electromagnetic interference, high hysteresis, the inability to measure statically, and complex processing circuits.
The fiber grating-based multidimensional force sensor has the advantages of small size, simple structure, a separate processing circuit from the sensor, rapid response, high sensitivity, and excellent electrical insulation performance [12,13]. It has been utilized extensively in robotic surgical equipment [14], wearable devices [15], robotic manipulators [16], etc.
Guo et al. investigated a three-axis FBG force sensor for robotic fingers [17], which created a distinct elastic structure, and embedding six FBGs with varying wavelengths enables efficient sensing of the 3D force. Li et al. designed a small 3D force sensor with FBGs encased in a tube [18], in which four FBGs suspended in a biofluid-filled environment were employed for mutual decoupling and temperature correction to increase the sensor's durability. Long et al. proposed an ultrathin, three-axis force sensor for the wrist of a robot [19], whose results indicate that the sensor's maximum sensitivity may reach 34.07 pm/N, which applies to the wrist of the collaborative robot UR5. The above research has demonstrated the feasibility of 3D force detection for FBG in robots, but these optical fiber tactile sensors face problems such as laborious manufacturing processes and complicated decoupling techniques. In addition, another challenge in detecting multidimensional forces is the decoupling of forces on sensors. The LS decoupling method was popular due to its its simple calculation process and high accuracy in linear decoupling [20]. However, machine learning techniques such as BP neural networks [21], artificial neural networks (ANN) [22], radical basis function (RBF) neural networks [23], etc. They can efficiently decouple large nonlinear data sets. Multidimensional force sensors require a simple, fast and highly accurate decoupling method.
Currently, some progress has been made in the research on FBG tactile sensors, which encapsulate FBG into polymer materials to detect tactile force. Luca Massari et al. made a tactile sensor for robotic hands by putting optical fibers inside flexible polymer materials and putting them on robotic hands [24]. Armitage L. et al. devised a fiber Bragg grating sensor that can simultaneously detect normal and shear strain and was inserted in a foam pad at the interface of a prosthesis [25], whose results indicate that the measured normal strain and shear strain are in good agreement for loads less than 20 N, but the variances become large above this load. Qian et al. suggested a sliding sensor based on an FBG-based two-dimensional distributed sensing array [26], whose sensor can estimate the sliding direction based on the changing features of distinct grating sections, hence enhancing the mechanical finger's capacity to detect sliding. Sun et al. proposed a double-layer distributed fiber grating sensor array that can detect pressure, sliding, and ambient temperature information [27], whose results demonstrate that the sensor has a pressure sensitivity of 7.287 nm/Mpa, a temperature sensitivity of 13 pm/$^\circ $C, and can accurately determine the sliding direction and velocity of an item.
Due to the performance of the packaging materials and the sensor structural design, the current research has problems such as low linearity of the signal measured by the sensor, an inability to accurately detect multi-directional shear forces, poor creep resistance, high coupling error, and inaccurate temperature compensation.
To address the aforementioned issues with current fiber grating sensors, a fiber grating 3D force sensor for prosthetic hands will be presented in this paper, which can detect the 3D force with precision and compensate for ambient temperature. In order for the sensor to detect the greatest range of 3D force, it is significant to design an appropriate FBG embedding position in Section 2. To improve the sensor's sensitivity to 3D force, it will also be proposed to optimize the sensor's structural parameters by finite element analysis in Section 3. A whole sensing system will be constructed, 3D force calibration and temperature experiments conducted, and two decoupling methods—LS and BP neural networks utilized for decoupling in Section 4. In conclusion, some achieved results, and their discussions will be shown in Section 5.
2. Sensing principle and structural design
2.1 Design requirements
The humanoid prosthetic hand is a functional substitute for the human hand, and perception of the surrounding environment is a crucial component of the prosthesis’ closed-loop control. The key to achieving steady grasping with prosthetic limbs is obtaining accurate contact force throughout the gripping phase [1]. However, traditional tactile sensors lack accurate detection of 3D force, or the detection signal has significant nonlinearity and high coupling errors. To design a sensor that can accurately detect 3D contact force, this study combines the advantages of the optical fiber sensor, such as its small size, resistance to environmental disturbances, and quick response. According to the usage scenario of the humanoid prosthetic hand, the optical fiber sensor replaces the fingers of the distal knuckle of the original humanoid prosthetic hand, and its structural size must meet the design position requirements of the finger. In the meantime, the sensor range must correspond to the maximum grasping force of the humanoid prosthetic hand to ensure a stable grip. The average size (length*width*height) of the far knuckle of the adult finger is 32*15*13 mm. Referring to the grasping requirements and average size of the humanoid prosthetic hand [28], we set the maximum size for the humanoid prosthetic hand's distal knuckle at 30*13*15 mm, with the sensor's maximum size at 25*11*12 mm, as depicted in Fig. 1. The purpose is to achieve 3D force detection without interfering with the prosthetic hand's natural grasping capability. The range of the sensor was chosen according to the maximum grip force of the humanoid prosthetic hand during regular grasping operations, and the requirement for stability of the grasping [29]. The selected normal force range for the sensor is 0–20 N, and its shear force range is −10–10 N (refer to Fig. 2 for the direction of the shear forces).
In this paper, three optical fibers (four gratings) are designed for 3D force detection and temperature compensation. As shown in Fig. 1, the 3D force measurement module of the sensor is fixed on the distal knuckle of the prosthetic hand, and the temperature compensation module of the sensor is fixed on the middle knuckle of the prosthetic hand. This is to prevent the optical fiber from breaking due to excessive pulling during the operation of the prosthetic hand. In this study, the maximum bending angle of the optical fiber in the inner groove of the humanoid finger shell is 70–80°. It could be adjusted based on the installation context. Once bent, fixed in the inner groove of the humanoid finger shell, and then connected to the outside along with other leads inside the humanoid hand.
2.2 Sensor structure design
This study uses PDMS material to encase the optical fiber grating because the bare FBG is easily broken and cannot simultaneously detect 3D force [30]. On the one hand, it can prevent damage to the fiber grating, and on the other, it can boost the optical fiber sensor's sensitivity to 3D force and temperature [31]. When the normal or shear forces are applied to the top surface of the packaging material, the external force can be converted into the interaxial force of the optical fiber.
The three fiber gratings (FBG1, FBG2, and FBG3) are used to detect 3D forces, and the reference grating (FBG4) compensates for the ambient temperature. The objective is to enable the sensor to obtain accurate external information when the dexterous hand is grasping. The sensor construction is depicted in Fig. 2. When the upper surface of the sensor is subjected to an external force, the three optical fiber FBGs can detect the external force of the same position. Thus, the design places FBG1, FBG2, and FBG3 in the same space line. FBG is sensitive to axial force but insensitive to lateral force. By utilizing the varying responses of three optical fibers to the same external force, the coupling error in detecting 3D force is diminished, so as to better decouple the 3D force. To measure the maximum shear force of the X-axis and Y-axis, FBG1 and FBG2 are designed parallel to the X-axis and Y-axis, respectively. FBG1 and FBG2 are embedded in the packaging material parallel to the XOY plane and are positioned at 90° to each other. The FBG3 was embedded in the packaging material with perpendicular orientation to the Y-axis, and embedded obliquely on the X-axis plane (XOZ plane). This method of embedding increases FBG3's sensitivity towards X shear forces while decreasing its sensitivity towards Y shear force. For the detection of shear forces in different directions, the sensor utilizes an asymmetric form, and FBG3 is obliquely embedded into the packaging material. Moreover, the accurate embedding sites of the three optical fibers and the size of the packaging material were established following optimization by simulation using finite elements in Section 3.1.
2.3 Measurement principle
According to coupled-mode theory, when broadband light is transmitted in FBG, light that satisfies the fiber Bragg grating's criteria is reflected [32]. The center wavelength of the light that's reflected is given by
where λB is the central wavelength of the fiber grating, neff is the effective refractive index of the fiber grating, and Λ is the intrinsic period.The central wavelength λB of a fiber grating varies with the effective refractive index neff and the grating's intrinsic period Λ. The effective refractive index of the fiber grating will change when the external environment is altered.
When the temperature does not change, only the stress changes, the change of the central wavelength is Eq. (3).
When the fiber grating is axially deformed, the center wavelength of the reflected light will shift, and the above equation can be rewritten as follows:
where pe is the effective elastic-optic coefficient and pe is a fixed value for a particular type of fiber. The conventional quartz fiber utilized in this investigation, where pe is 0.216 and ɛz is the axial strain of the fiber grating.Under the influence of the 3D force $F = ({{f_x},{f_y},{f_z}} )$, the central wavelengths of FBG1, FBG2, and FBG3 change in relation to the force in three axes. Assuming that there is a good linear relationship between the change in central wavelength and the 3D force $F = ({{f_x},{f_y},{f_z}} )$. Then, the 3D force is solved by the central wavelength shift. Equation (5) is then utilized to determine the 3D force.
To facilitate uniform calculations, Eq. (5) is simplified as follows.
Due to the sensitivity of FBG to both temperature and strain, it is required to develop a temperature compensation structure to eliminate the effect of temperature on the central wavelength when detecting external forces. In this study, the reference grating FBG4, which was not affected by external pressure, was added to the same fiber for temperature compensation and encapsulated with PDMS. The main objective is to increase its temperature sensitivity and protect the fiber grating. The principle of compensation can be stated in the following Eq. (7).
3. Sensor structure optimization and fabrication
3.1 Structure parameter selection/design
To maximize the sensitivity of the sensor to external forces, its structural properties were tuned by finite elements. The basic processes of sensor structure optimization are presented in Fig. 3.
Since optical fibers were embedded in packaging materials during experiments, it was challenging to directly find the changes between optical fibers and packaging materials when applying force. PDMS was used as the packaging material due to its low modulus of elasticity, high elasticity, softness, corrosion resistance, and good resistance to deformation. The sensor was comprised of bare optical fiber, acrylate coating, and packaging material. The bare optical fiber and coating layer, as well as the acrylate coating and packaging material, were tightly bound without relative slippage. Table 1 lists the parameters associated with the material settings of finite element models.
Table 1. Material parameters table
As a result, finite element simulation was used to construct the fiber grating 3D force sensor model encased in PDMS material. On the upper surface of the PDMS packaging material, a normal force was applied, while fixed constraints were imposed on the lower surface. The simulation calculated the maximum displacement of the position of each fiber grating region along the axis for FBG1, FBG2, and FBG3. The structural design parameters of the sensor were determined by finite element simulation: the optimal embedding depths H1, H2, and H3 of FBG1, FBG2, and FBG3, the embedding angle α of FBG3, and the length L, width W, and height T of the packaging material. The structural design parameters are shown in Fig. 2.
The method of Spearman's rank correlation was used for correlation coefficient calculation. The objective was to determine which structural design parameters have the greatest effect on the sensor‘s performance. Figure 4 shows the overall sensitivity table between each structural design parameter and the maximum displacement of each FBG along the axis.
Fig. 4. The overall sensitivity table between sensor structure design parameters and the maximum displacement of each FBG along the axis.
The overall sensitivity scale measures the degree to which each independent variable influences the target variable. The sensitivity of each structural parameter was calculated using the Spearman's’ rank order correlation coefficients method. This method can take multiple influencing factors into the sensitivity calculation at the same time. H1 and H3 are the structural parameters that have the most effect on the maximum displacement sensitivity of FBG1. H1, H2, and H3 are the structural parameters that have the most effect on the maximum displacement sensitivity of FBG2. H3, T, and α are the structural parameters that have the most effect on the maximum displacement sensitivity of FBG3. Among these parameters, L and W seem to have little influence on the maximum displacement of each FBG. Consequently, the number of parameters to be optimized decreased from 7 to 5. This can eliminate structural parameters with negligible optimization impact, enhancing efficiency and accuracy when optimizing the optimal structure. In summary, the most important structural parameters for the subsequent optimization of the sensor sensitivity are H1, H2, H3, T, and α, all of which are shown in Fig. 4.
In this work, the Nonlinear Programming by Quadratic Lagrangian (NLPQL) algorithm was used for optimizing the main structural parameters of the sensor, through a direct optimization design approach. Through this method, the maximum displacement along the axial direction of three FBGs under different structural parameters was calculated, thereby determining the optimal structural parameters for the sensor. Table 2 shows the original and optimized structural design parameters of the sensor.
Table 2. Sensor structure parameters
Figure 5(a) shows the structure of the original sensor. When 20 N normal force was applied to the upper surface of the original sensor, the maximum displacement of each grating region was along the axis for FBG1, FBG2, and FBG3. Figure 5(b) shows the structure of the optimized sensor. When 20 N normal force was applied to the upper surface of the optimized sensor, the maximum displacement of each grating region was along the axis for FBG1, FBG2, and FBG3. To be consistent with the actual use of the sensor, the simulations of the shear forces fX and fY must be carried out under the conditions of the preload of the fZ force, and the applied normal force is more than the shear force. When a force of 20 N was applied in the fZ direction, vary the Y/X shear forces to determine the sensitivity of FBG to shear force.
Fig. 5. Finite element simulation results of the maximum displacement of each grating region along the axis (a) The structure of the original sensor (b) The structure of the optimized sensor.
The original and optimized sensitivities of each optical fiber of the sensor are shown in Fig. 6 and Table 3. The maximum displacement of each fiber grating region along the axis is positively associated with the normal force, as shown by the optimization findings. FBG2's normal force sensitivity was the highest among them. The FBG2 was embedded in more packaging materials than other optical fibers. And the position of the fiber was closer to the upper surface of the packaging material. In addiction, FBG1 was buried deeper than the other fibers, which caused it to be less sensitive to shear forces in the X/ Y direction than to its normal force. Table 3 shows that the FBG sensitivity of the optimized sensor has increased for the normal force fZ and most of the shear forces. The sensitivity of FBG2 to fX and fY shear forces decreased by 9.24% and 9.66% respectively. The reason is that the normal force was used for structural optimization of the sensor, but the shear force in a certain direction is not applied separately for structural optimization. The result is a slight reduction in the sensitivity of some FBGs to shear forces in certain directions. There are many cases of shear force, and the optimization of the sensor's structure cannot be based on a specific shear force. Finally, we optimize the sensor structure when subjected to normal force. The optimized sensor increased the maximum displacement of FBG1, FBG2, and FBG3 along the axis, thus increasing the amount of change in the center wavelength.
Table 3. The normal force sensitivity of the sensor's original and optimized structures
3.2 Sensor fabrication
After determining the optimal parameters of the sensor structure, the next step was to fabricate the sensor. FBG1, FBG2, FBG3, and FBG4 have the following central wavelengths: 1549.981 nm, 1554.857 nm, 1550.087 nm, and 1550.021 nm. All FBGs had a reflectivity better than 70%, a reflectivity bandwidth of less than 3 dB, and a grating length of 3 mm. FBG2 and FBG4 are on the same fiber, 6 mm apart. Figure 7 shows the essential fabrication procedures as well as the fiber grating 3D force sensor.
Fig. 7. Sensor fabrication process (a) Front view of sensor pre-tightening (b) Top view of sensor pre-tightening (c) The mold to fix FBG (d) Fiber grating 3D force sensor.
The main fabrication steps were the following. First, a mold was designed and fabricated for solidifying the packaging material and securing the three optical fibers in a determined place. Second, it was necessary to fix the three optical fibers with the mold. Next, a fiber pre-tightening device was designed to pretension the three optical fibers as shown in Fig. 7(a), (b). The purpose of pre-tensioning the optical fiber was to prevent inaccurate sensor detection due to fabrication errors during the fabrication process. And the fiber pre-tightening device consisted of a fiber holder, clamps, and a standard weight, as shown in Fig. 7(a), (b). The main steps in pre-tightening were to attach optical fibers to the groove of the bracket, use a clamp to clamp the end of the optical fiber, and hang the standard weights to pre-tighten the optical fiber. Then, the polydimethylsiloxane and auxiliary curing agent were combined and defoamed at a mass ratio of 10:1 before being poured into the mold is shown in Fig. 7(c). Finally, after the packaging material had hardened, the sensor was removed. Package the reference grating FBG4 in the same way. The completed fiber grating 3D force sensor is shown in Fig. 7(d). The size of the 3D force detection module of the sensor was 24*10*10.88 mm (L*W*T), and the size of the temperature compensation module was 10*2.5*2.5 mm (a*b*c) as shown in Fig. 7(d).
4. Experiment and data processing
4.1 Normal force calibration
After the fiber grating 3D force sensor had been fabricated, its static performance was evaluated, as depicted in Fig. 8. The experimental equipment consisted of the FBG interrogator (FBGScan 904), the 3D force sensor, a normal force loading platform (with an accuracy of 0.01 N), a load block, and PC.
Fig. 8. Normal force calibration experiment system (a) Experimental setup (b) Partial enlarged view of sensor loading.
During the normal force fZ static calibration experiment, the temperature of the experimental environment was maintained at 23 °C. The upper surface of the sensor was applied to normal force (Z-axis) by means of the fZ force loading platform, with the force increasing in increments of 1 N from 0 N to 20 N over the surface area of load block. In order to match the actual usage scenario of the sensor and to ensure that the loading position of each experiment was consistent, we used a load block the size of the top surface of the sensor to load the sensor. The load block was placed between the normal force load platform and the sensor, as shown in Fig. 8(b). Each experiment was sampled at a sampling frequency of 100 Hz for 30 s, and the average value of the center wavelength shift of the FBG during the experiment was calculated. The loading and unloading experiments were repeated 10 times. The experimental system is shown in Fig. 8(a). The purpose was to ensure the accuracy and rationality of the experimental results. Figure 9 plots the relationship between the central wavelength shift of the four FBGs and the normal force when loading and unloading different normal forces in fZ.
The results of Fig. 9 show that the central wavelengths of FBG1, FBG2, and FBG3 vary linearly with fZ, and the reference grating FBG4 is not affected by fZ. The average sensitivity of FBG1∼FBG3 when loading are: KZ1 = 0.03118 nm/N, KZ2 = 0.08704 nm/N, KZ3 = 0.01822nm/N, and the average sensitivities when unloading are $K_{Z1}^{\prime}$ = 0.03141 nm/N, $K_{Z2}^{\prime}$=0.0874 nm/N, $K_{Z3}^{\prime}$=0.01832nm/N. FBG2 has the highest sensitivity to normal force fZ. The hysteresis inaccuracy of the loading and unloading processes is 1.48%, suggesting that the sensor has low hysteresis. The highest error of 10 repetitions during the loading phase is 2.57%, and the maximum error of 10 repetitions during the unloading process is 4.03%. The sensor has a low repetition error. According to the above results, the fiber grating 3D force sensor has an average linearity R2 of 0.9998 during loading and unloading, which has high linearity and low repeatability error. Comparing the sensitivity of the simulation in Fig. 6, the maximum difference in fZ sensitivity when the sensor is actually loaded is 18.6%, 14.6%, and 14%. The rationale for the drop in sensitivity is that the conditions of the finite element simulation are more ideal, and the outcomes of the sensitivity experiment are generally congruent with those of the simulation.
4.2 Shear force calibration
The shear force calibration experimental system is shown in Fig. 10. The experimental setup consisted of the FBG interrogator (FBGScan 904), the 3D force sensor, a normal force loading platform (with an accuracy of 0.01 N), M1 standard weights, fixed pulleys, a load block, strings, and PC.
Fig. 10. Shear force calibration experiment system (a) Experimental setup (b) Partial enlarged view of sensor loading.
The sensor could accurately detect the shear force of the X and Y axes during experiments. There was no error due to sliding, and it was consistent with the actual use of the sensor. Therefore, the calibration experiments of the shear forces fX and fY must be carried out under the conditions of the preload of the fZ force, and the applied normal force is more than the shear force. In order to match the actual usage scenario of the sensor and to ensure that the loading position of each experiment was consistent, we used a load block to load the sensor. The load block was placed between the normal force load platform and the sensor, and the standard weights of different masses were loaded through the string thread to connect the load block to load the shear force as shown in Fig. 10(b). In addition, it is to prevent errors caused by friction between the normal force loading platform and the top surface of the load block. The lubricant was added between the loading platform and the load block to ensure the accuracy of the shear force on the top surface of the sensor.
The measuring range of shear forces along the X and Y axes is −10–10 N (refer to Fig. 2 for the direction of the shear force). The two fixed pulleys were fixed at the X and Y axis positions facing the center of the sensor. The standard weights of varying masses were suspended by fixed pulleys, and loaded at a fixed location on the upper surface of the sensor. Then shear forces in the X / Y axes were applied at intervals of 1 N. Each experiment was sampled at a sampling frequency of 100 Hz for 30 s. And the changes of the central wavelength of each fiber gratings were recorded during the whole process. Shear force experiments were conducted using normal forces of 8 N, 11 N, 14 N, 17 N and 20 N respectively. The experimental system diagram of shear force fX is shown in Fig. 10, and the experimental procedure of shear force fY is the same as it. Figures 11 and 12 plot the central wavelength shift of the four FBGs versus the shear force when loading with different fZ.
The results of Fig. 11 show that the central wavelength changes of FBG1, FBG2, and FBG3 are linearly related to fX, and the reference grating FBG4 is not affected by external forces. In the shear force calibration experiment, the maximum fX shear force is less than the maximum fZ normal force of 2 N, in order to prevent the load block from sliding due to excessive shear force, as shown in Fig. 11. When the fZ was loaded with the same force, different shear forces were applied on the negative axis of X. The maximum sensitivities of the three FBGs of the X—shear force are $K_{X1}^ - $ = 0.00879 nm/N, $K_{X2}^ - $=0.02109 nm/N, and $K_{X3}^ - $=0.04844 nm/N, respectively. FBG3 has the highest sensitivity. Both FBG2 and FBG3 are linearly positively correlated with fX, and FBG1 is linearly negatively correlated. When fZ was loaded with the same force, the different shear forces were applied on the positive axis of X. The maximum sensitivities of the three FBGs of the X+ are $K_{X1}^ + $ = 0.0042 nm/N, $K_{X2}^ + $=0.0046 nm/N, and $K_{X3}^ + $=0.03915 nm/N, respectively. FBG3 has the highest sensitivity and is linearly negatively connected with fX. FBG1 and FBG2 are positively correlated with fX. When the preload fZ force changes, the X-axis shear force sensitivity of FBG1, FBG2, and FBG3 changes little as the fZ force increases. This is because of the sensor's shear force sensitivity variation when subjected to a combined force in both axes. The linearity of each fiber changes with different fZ. The average R2 of FBG1, FBG2, and FBG3 are 0.9441, 0.9667, and 0.9982, respectively.
Figure 12 shows that the central wavelength changes of FBG1, FBG2, and FBG3 are linearly related to fY, while the reference grating FBG4 is not changed by external forces. In the shear force calibration experiment, the maximum fY shear force is less than the maximum fZ normal force of 2 N, in order to prevent the load block from sliding due to excessive shear force, as shown in Fig. 12. When the fZ was loaded with the same force, the different shear forces on the negative axis of Y were applied. The Y—shear force maximum sensitivities of the three FBGs are: $K_{Y1}^ - $ = 0.00528 nm/N, $K_{Y2}^ - $=0.02206 nm/N, and $K_{Y3}^ - $=0.000571 nm/N, respectively. Both FBG1 and FBG2 have negative and linear correlations to fY, while FBG3 is positively and linearly correlated. When fZ was loaded with the same force, the different shear forces on the positive axis of Y were applied. The Y+ shear force maximum sensitivities of the three FBGs are as follows: $K_{Y1}^ + $ = 0.00623 nm/N, $K_{Y2}^ + $=0.0243 nm/N, and $K_{Y3}^ + $=0.0034 nm/N, respectively. FBG1, FBG2, and FBG3 are all linearly and positively related to the fY. When the preloading fZ force changes, the Y-axis shear force sensitivity of FBG1, FBG2, and FBG3 changes little as the fZ force increases. The average R2 of FBG1, FBG2, and FBG3 are 0.9857, 0.9958, and 0.5293, respectively. FBG2 exhibits the highest sensitivity to shear force fY. FBG3 has low linearity of shear force in the Y-axis. The reason is that FBG3 has the lowest sensitivity to the Y shear forces. The FBG3 was embedded in the packaging material with perpendicular orientation to the Y-axis, and embedded obliquely on the X-axis plane. This method of embedding increases FBG3's sensitivity towards X shear forces while decreasing its sensitivity towards Y shear force.
4.3 Creep experiment
The experiment system shown in Fig. 8 was used, and the external temperature was kept constant. The sensor was fixed to the normal force loading platform, applied to the top of the sensor with a normal force of 5 N, and remained there for 20 minutes. The sampling frequency of the FBG interrogator was set to 100 Hz, and the central wavelength changes of FBG1, FBG2, and FBG3 during this period were recorded. The creep characteristic curve of the sensor is shown in Fig. 13.
Fig. 13. Experimental analysis of sensor creep resistance (a) Experiment results for creep resistance. (b) FBG1 (c) FBG2 (d) FBG3.
It can be seen from the experimental results in Fig. 13(a) that after the sensor was loaded with external force, the central wavelength of each FBG remained stable after a rapid rise during loading. After the external force was stabilized, the central wavelength change of each FBG remained stable and showed little change. And a central wavelength of 1 minute was selected for analysis, as shown in Fig. 13. The maximum shift in the central wavelengths of FBG1, FBG2, and FBG3 are 0.5 pm, 1.2 pm, and 0.7 pm, respectively. According to the maximum center wavelength shift of the three optical fibers, the maximum creep errors are calculated to be 0.378%, 0.224%, and 0.98% respectively. Since the maximum creep error of the FBG is less than 1%, the sensor has good creep resistance characteristics that can meet the needs of humanoid prosthetic hand experiments.
4.4 Temperature sensitivity calibration
This study encapsulated fiber optic grating with PDMS, and FBG temperature sensitivity was influenced by the packaging material. The 3D force sensor was placed in the water bath thermostat with a precision of 0.1 °C for the temperature test, and the experimental setup is depicted in Fig. 14. During the experiments, the temperature surrounding the sensor was altered using a thermostat. The experimental calibration temperature was between 10-55 °C, the sampling interval was 2 °C, and the duration of each time was 2 minutes. To compare the temperature characteristics of packaged FBGs, temperature experiments were also conducted on a bare FBG with a central wavelength of 1549.997 nm. The temperature sensitivity of each FBG is shown in Fig. 15.
As shown in Fig. 15, the changes of in the central wavelength of each fiber grating are positively and linearly correlated with temperature. In the temperature range of 10-55 °C, the sensor temperature fitting linearity is high. The temperature sensitivity of the encapsulated FBG grating is greater than that of the bare FBG. The temperature sensitivity of FBG1, FBG3, and FBG4 is about the same, KT1 = 0.01353 nm/°C, KT3 = 0.01439 nm/°C, and KT4 = 0.0127 nm/°C, respectively. The temperature sensitivity of FBG2 is greater than that of other fibers, KT2 = 0.026 nm/°C. The center wavelength of FBG2 is 1554.857 nm, and the center wavelengths of other three FBGs are around 1550 nm. Meanwhile, it has the most fiber encapsulated in the packaging material, and the highest temperature sensitivity. Figure 15 shows that when the ambient temperature changes greatly, the temperature has a significant effect on the central wavelength of the FBG. According to Eq. (7) and the temperature sensitivity coefficient measured in the experiment, the 3D force can be accurately measured through the reference grating FBG4. The purpose is to eliminate the effect of ambient temperature on the central wavelength of other fiber gratings during 3D force detection.
4.4 3D force decoupling and validation
4.4.1 3D force decoupling
In the 3D force calibration experiment, the integrated structure of the sensor results in structural coupling. The key to decoupling the 3D force sensor is determining the mapping relationship between the input and output signals of the sensor. Then decoupling the 3D force using various decoupling methods [33].
The measured FBG central wavelength values in three axes exhibit high linearity to the applied normal and shear forces. The LS decoupling method is effective for the decoupling of multi-dimensional forces with high linearity. As a result, the LS decoupling method was selected to decouple the measured signals. Multiple regression analysis was done on the sensor data taken on the three axes [24], and the calibration coefficient in Eq. (6) was produced, allowing Eq. (6) to be transformed into the following Eq. (8).
Then the sensitivity of the sensor in each axis are: KX = 0.044 nm/N, KY = 0.012 nm/N, and KZ = 0.032 nm/N. The accuracy of the FBG interrogator used in the experiment is 0.3 pm, and the minimum precision of the sensor in each axis are: fX-min = 0.0068 N, fY-min =0.025 N, and fZ-min =0.0094 N.
Some of the sensor's measured data exhibits nonlinear characteristics, which resulted in inaccurate decoupling results of some data using the LS decoupling method. In terms of decoupling nonlinear data, the BP neural networks decoupling method was more effective [25].
The decoupling method of BP neural networks was chosen to compare with the LS decoupling method. The 3D force sensor was a three-input and three-output system, and BP neural networks were built in MATLAB. The network model included three layers: an input layer, a hidden layer, and an output layer. According to the empirical equations [34], we calculated that the selection range of the number of hidden layer nodes of the neural network was 3–36. To prevent overfitting caused by increasing the number of nodes, we used the K-fold cross-validation method to improve the network and avoid falling into local optimality [35]. It was finally determined that the hidden layer of the network has three layers, the number of hidden layer nodes are: 22, 15, 33. As the activation function of the hidden layer unit nodes, the S-type (Tansig) function were employed. And the linear (Purelin) function was used for the output layer unit nodes. Figure 16 shows the mean-square error (MSE) iteration curve of BP neural network training. When the number of iterations (n) reaches 225, the mean square error decreases to 7.3556e-4. After the training was over, the neural network model was exported and used as the decoupling program parameters of the 3D force sensor.
4.4.2 Verification
During prosthetic manipulation, the sensor is typically subjected to simultaneous forces in three axes. In order to validate the sensor's decoupling accuracy after being subjected to 3D forces, the sensor is simultaneously subjected to external forces in three axes using the same experimental setup (Figs. 8, 10). To verify the reliability of the experimental results, the orthogonal experiment method was used, and the three-factor, five-level orthogonal tables were selected [36]. Using two different decoupling methods, the decoupling errors eX, eY, and eZ of the X, Y, and Z axes and the overall decoupling error eT were analyzed, respectively. eT is calculated as in Eq. (9).
As shown in Fig. 17 and Table 4, the decoupling error of the LS decoupling method is typically greater than that of the BP neural networks. The maximum errors of eX, eY, eZ, and eT are 0.27, 1.43, 0.19, and 1.45, respectively, when decoupling is performed using the LS decoupling method. Both the decoupling accuracy of the X and Z axes is superior to that of the Y axis demonstrates. This demonstrates that the nonlinearity of the Y-axis sensor’s data is greater than that of the other axes, leading to inaccurate decoupling. When using the BP neural networks decoupling method for decoupling, eX, eY, eZ, and eT the maximum errors are 0.033, 0.035, 0.02, and 0.038. Overall decoupling accuracy has improved. Therefore, it is necessary to use the BP neural networks decoupling method for decoupling. Analysis showed that the LS decoupling effect is lower than the BP neural networks as a whole. The main reason is that some FBGs have a nonlinearity in the experimental process, which interferes with the three-dimensional force decoupling process.
Fig. 17. Error comparison of two decoupling methods (a) LS decoupling maximum error (b) BP neural networks decoupling maximum error.
Table 4. Accuracy comparison of decoupling algorithms
5. Discussion and conclusion
In this work, a novel fiber grating sensor capable of detecting 3D force precisely for a humanoid prosthetic hand has been developed. The force and temperature sensitization and fixation were accomplished by embedding FBG into the PDMS packaging material. Using the FBG sensing principle, it is possible to translate the applied external force into the axial force carried by the optical fiber. Then, the sensor have been designed according to the size of the prosthetic finger. Three optical fibers were used for three-dimensional force detection so that the sensor could detect the maximum range of 3D forces. And the ambient temperature was compensated by the reference grating. In addition, the structural parameters were determined by finite element optimization to increase the sensor's sensitivity to external forces. We designed a novel pre-tightening structure for optical fibers and fabricated the sensor to build a complete sensing system. Next, we calibrated the normal and shear forces in three axes, and conducted creep and temperature experiments. Before decoupling, the experimental data analysis showed that the signal measured by the sensor had good linearity for 3D forces. The 3D force data were decoupled using the decoupling methods of LS and BP neural networks, with an overall decoupling error of no more than 0.038.
In comparison with other researchers, this research has great advances in sensor structural design, multi-directional shear force measurement, and decoupling. For instance, Massari et al. utilized Pe-Lite foam as the packaging material for FBG in order to detect the normal direction and shear stress [24]. The results demonstrated that the measured signal was not as stable as when PDMS was used as the packaging material and that the multi-directional shear force could not be detected precisely. Qian et al. used a symmetrical structure to embed FBG for pressure and sliding detection [26]. Sun et al. developed a two-layer distributed array structure for touch and sliding detection [27]. K. Dey et al. designed a semi-etched FBG sensor for axial force measurement with high sensitivity [37]. The asymmetric structure used in this study can identify the multi-directional shear force while utilizing less FBG, and the oblique embedding can boost the shear force's sensitivity. In addition, the decoupling method is simple and high accuracy. Two decoupling methods of LS and BP neural networks were compared. And the force decoupling method of the BP neural networks can provide sufficient decoupling precision to the sensor.
The measurement ranges of the sensor's X, Y, and Z axes are: −10–10 N, −10–10 N, and 0–20 N. The sensitivity of the sensor in each axis are: KX = 0.044 nm/N, KY = 0.012 nm/N, and KZ = 0.032 nm/N. The minimum precision of the sensor in each axis are: fX-min = 0.0068 N, fY-min =0.025 N, and fZ-min =0.0094 N. And the maximum temperature sensitivity is KT = 0.026 nm/°C. The sensor structure of this study is flexible. Depending on the robot's actual requirements, different numbers of sensing units can be embedded in the packaging material. The sensor has great linearity, high sensitivity, good creep resistance, and high adaptability. Therefore, a new design strategy for 3D force sensing of humanoid prosthetic hands is provided.
Future research will concentrate on assembling sensors on the humanoid prosthetic hand for contact force detection. Then, the sensor structure will be further optimized to detect sliding and texture recognition. And a more comprehensive prosthetic hand detection system will be developed.
Funding
National Key Research and Development Program of China (2017YFB1300303); Science and Technology Project of Xi’an (21RGZN0007).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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