A Fiber-Optic Six-Axis Force Sensor Based on a 3-UPU-Compliant Parallel Mechanism

Abstract

Traditional six-axis force sensors are mostly based on resistance strain, piezoelectricity and capacitors, which have poor resistance to electromagnetic interference. In this paper, a six-axis force sensor based on bending-sensitive optical fibers is proposed. A 3-UPU-(universal joint–prismatic joint–universal joint) compliant parallel mechanism is adopted in the sensor. The bending-sensitive optical fiber is encapsulated to form a fiber encapsulation module (FEM). The configuration of the FEMs within the six-axis force sensor is investigated. Static and stiffness analyses of the sensor are conducted and a force mapping matrix for the sensor is established. Simulation experiments are performed to verify the correctness of the established force mapping matrix. The detection system of the sensor is fabricated and the experiments are carried out to evaluate the performance of the sensor. The experiment results show that the maximum values of type-I errors and type-II errors are 4.52%FS and 3.26%FS, respectively. The maximum hysteresis and repeatability errors are 2.78% and 3.27%. These results verify the effectiveness of the proposed sensor.

1. Introduction

Six-axis force sensors are widely used in industrial automation, robotics technology, automotive engineering, aerospace and other fields [1]. The elastic body of a force sensor has a significant impact on its performance. According to the different structural forms, the elastic body of a six-axis force sensor can be divided into two types: integral structure and assembled structure.

Integrated processing technology is adopted in elastic bodies with integral structure. Cross-beam structure [2] and column structure [3] are two typical structures of an integral elastic body. The advantage of the integral elastic body lies in its good mechanical decoupling performance and small measurement error. However, its measurement accuracy is greatly affected by the processing accuracy of the elastic body, which usually requires precision machining. In the assembled elastic body, modular components are assembled to transmit multidimensional mechanical information. The Stewart structure is one of its typical examples [4]. Compared to the integral elastic body, the component processing of the assembled elastic body is simpler and the cost is lower.

The force sensors based on resistance strain [5], piezoelectricity [6] and capacitors [7] are well-developed and widely used nowadays. The force sensors based on resistance strain convert force signals into electrical signals by resistance strain gauges fixed on elastic bodies. And then, each component of force can be obtained by using the force–electric coupling mathematical model [8]. Piezoelectric material (such as quartz) is used as the sensitive element in piezoelectric force sensors. When an external load is applied to the piezoelectric material, the internal charge of the material is changed and an electrical signal proportional to the applied force is generated. By solving the electrical signal, the size and direction of the external load can be obtained [9]. For the force sensors based on capacitors, force signal detection is achieved by detecting changes in capacitance. The capacitor is connected with the elastic body. When an external load is applied to the elastic body, the parameters (such as electrode spacing, effective area and dielectric constant) of the capacitor change, which results in a change in capacitance. By detecting the change in capacitance, external load information can be inferred [10]. However, the types of sensors mentioned above demonstrate the poor resistance to electromagnetic and cross-sensitivity to temperature. Optical fiber sensors offer advantages such as compact size, immunity to electromagnetic interference, corrosion resistance, and high-temperature tolerance, making them more suitable for application in complex environments [11]. Optical fibers have been used to measure parameters such as temperature, pressure, current, displacement, and gas concentration. In recent years, optical fibers have gained attention from researchers and have been increasingly applied in force sensors [12].

A six-axis force sensor based on bending-sensitive optical fibers is proposed in this paper. The bending-sensitive optical fiber is produced by introducing a sensitive region on the fiber surface. It can directly detect the curvature of structure. The authors of this paper have conducted relevant studies on it in previous studies. The operation principles of the bending-sensitive optical fiber are shown in [13]. In [14], the Monte Carlo simulation by ray tracing and an orthogonal matrix are used to optimize the configuration of the sensitive region. This kind of fiber has many advantages, such as high sensitivity, bending direction identification and linear output. It has been used to measure many parameters, such as wind speed, hand joint angle and water waves [15,16]. In this paper, the parameters of the sensitive region are the same as the optimized configuration in [14]. A 3-UPU-(universal joint–prismatic joint–universal joint) compliant parallel mechanism is designed to accommodate the bending-sensitive optical fibers. The static and stiffness analyses of the mechanism are conducted, and relevant experimental studies are carried out.

2. Bending-Sensitive Optical Fiber

2.1. Characteristics of Bending-Sensitive Optical Fiber

As is well known, the optical loss of an untreated plastic multimode fiber caused by bending is minimal, which is insufficient to detect the bending deformation [17]. To enhance the sensitivity of the fiber to bending curvature, a special sensitive region is introduced on the fiber, which increases optical leakage during fiber bending. The sensitive region is produced by removing a part of the fiber core, as shown in Figure 1. Compared to untreated optical fiber, this kind of bending-sensitive optical fiber has significantly increased sensitivity to bending.

The static characteristic analysis has been conducted on bending-sensitive optical fiber. It exhibits excellent linear output during curvature measurement when the bending radius is greater than 60 mm [18]. The relationship between the electrical signal output of the fiber and the bending curvature of sensitive region can be expressed as:

$$U=aC+b$$

(1)

where U is the output signal of the fiber (the optical signal is converted into a voltage signal by a phototransistor), C is the curvature of the sensitive region, a and b are constants related to the sensor. The optical fiber used in this study is a plastic multimode fiber with a diameter of 0.5 mm, which is produced by Mitsubishi Company (Tokyo, Japan). LED IF-E96-R made by Industrial Fiber Optics Company (Tempe, AZ, USA, the wavelength is 660 nm) is used as a light source and the corresponding receiver is phototransistor IF-D92 (made by Industrial Fiber Optics Company).

2.2. Encapsulation of Bending-Sensitive Optical Fiber

In order to measure the force applied to the force sensor, the fiber needs to be arranged on the elastic body of the force sensor to detect its deformation. Due to the presence of the sensitive region, the fiber becomes more fragile in this area and needs to be protected by encapsulation. Additionally, to ensure the fiber bends in a predetermined direction, it is pre-bent during encapsulation. According to the above requirements, the material used for encapsulating optical fibers must be flexible [19]. In this study, the fiber encapsulation module (FEM) is introduced to facilitate the arrangement of optical fibers.

Liquid silicone (Ecoflex00-30) made by Smooth-On Company (Macungie, PA, USA) is used as the base material to encapsulate the fiber. The molding of FEMs is achieved by pouring liquid silicone into a pre-made mold and curing it. The pre-made mold consists of upper and lower parts with identical curved arcs. There is a semicircular groove with the same diameter as the optical fiber on both upper and lower mold parts. The semicircular grooves are used to fix the optical fiber. When fixing optical fibers, the surface of the fiber sensitive region should be perpendicular to the bending direction of the curved arc as much as possible. The photograph of FEM is shown in Figure 2. In Figure 2, the optical fiber has the same bending curvature as the FEM, which ensures that the optical fiber can measure the deformation curvature of FEM.

To verify the protective effect and strength enhancement effect of FEM on optical fiber, the bending experiments are carried out. In the experiments, the fiber breakage occurs in the sensitive region when the bending radius of the fiber is around 30 mm, while the breakage of fiber in FEM occurs when the bending radius of FEM is around 10 mm. This result indicates that FEM effectively protects the fiber and improves its strength.

The calibration experiments were conducted on the FEM. The experimental results are linearly fitted by using the least squares method. The calibration curve between the output of fiber optic and the deformation curvature of FEM is shown in Figure 3, and the corresponding curve-fitting equation can be expressed as:

$$U=0.0795C-5.35375$$

(2)

In Figure 3, linearity L_o of calibration curve can be expressed as:

$$L_o={\displaystyle \frac{\Delta U_{se}}{U_{FS}}}\times 100\%$$

(3)

where ΔU_se is the maximum deviation between the actual output and the fitted line, U_FS is the full output range of fiber. According to Figure 3, the linearity of FEM is 2.3%.

3. Structure Design and Detection Principle of Six-Axis Force Sensor

3.1. Structural Design

A 3-UPU parallel mechanism is adopted in the six-axis force sensor. The designed structure is shown in Figure 4. It consists of a base platform, three elastic force-measuring branches and a loading platform. The three force-measuring branches are identical and uniformly distributed between the loading platform and the base.

Each force-measuring branch employs a hybrid rigid-flexible structure, as shown in Figure 5. It consists of a rectangular connecting block, two connecting rods, two semi-arc-shaped elastic bodies (featuring a single-sided flexible hinge structure), and two FEMs fixed between the semi-arc-shaped elastic bodies. The two elastic bodies are symmetrically distributed on both sides of the rectangular connecting block by using connecting rods. The material of the force-measuring branches is 7075 aluminum alloy.

To ensure that the FEMs can effectively detect the deformation of the elastic bodies, soft silicone adhesive is used to fix the FEMs onto the elastic bodies. The adhesive offers high strength, high-temperature resistance, and excellent waterproof properties.

3.2. Force Detection Principle

During force analysis, each force-measuring branch can be equivalently modeled as a six-bar linkage mechanism composed of four binary links and two ternary links. The schematic diagram of the six-bar linkage mechanism is shown in Figure 6.

As shown in Figure 6, the ternary link 2 is connected to two binary links by using single-sided flexible hinges (R1 and R2 in Figure 6). The connections between the remaining links are achieved by pin joints. The two ternary links (ternary link 1 and ternary link 2 in Figure 6) are connected to the loading platform and the base through rigid hinges, respectively.

The axes of the revolute pairs (R1–R6) on the binary links are all parallel. The revolute pairs R7 and R8, where the ternary links are connected to the loading platform and the base, have axes perpendicular to those of the revolute pairs (R1–R6) on the binary links. Therefore, the three revolute pairs on every ternary link can be equivalent to a universal joint (U-joint). When an external load is transmitted to the connecting rods and elastic bodies through the ternary links, the combined deformation effect can be equivalent to a prismatic joint (P-joint, as shown in Figure 6). Thus, a force-measuring branch can be equivalent to an UPU structure.

4. Analysis of the Force Mapping Matrix for the Six-Axis Force Sensor

4.1. Static Analysis of the Sensor

As analyzed above, the sensor can be equivalent to a 3-UPU configuration. The equivalent diagram is shown in Figure 7. The model consists of a fixed platform M, a moving platform m, and three UPU-structured branches. The two U-joints at the top and bottom of each force-measuring branch ensure that the branch is only subjected to tension or compression forces in the direction perpendicular to the axis of the flexible hinge. The realistic physical factors (such as friction, mechanical clearances, adhesive joint deformation and internal material damping) have a certain impact on the sensor model. However, these realistic physical factors are complex and some of them cannot be quantified in the sensor model. For the convenience of establishing the sensor model, we ignore these factors in static and stiffness analysis.

The 6 × 6 velocity Jacobian matrix J for the 3-UPU parallel structure can be expressed as:

$$J={\displaystyle \frac{1}{2}}\left(\begin{array}{cccccc}0& 0& 2& -e& -\sqrt{3}e& 0\\ 0& 0& 2& 2e& 0& 0\\ 0& 0& 2& -e& \sqrt{3}e& 0\\ 1& \sqrt{3}& 0& 0& 0& 2e\\ 2& 0& 0& 0& 0& -2e\\ 1& -\sqrt{3}& 0& 0& 0& 2e\end{array}\right)$$

(4)

where e (e = 40 mm) is the distance from the U-joint on the force-measuring branch to the center of the platform, as shown in Figure 7.

Based on the principle of virtual work, the static model of the 3-UPU structure can be expressed as [20]:

$$\left[\begin{array}{c}{F}_{a1}\\ F_{a2}\\ F_{a3}\\ F_{c1}\\ F_{c2}\\ F_{c3}\end{array}\right]=-{\left(J^T\right)}^{-1}\left[\begin{array}{c}F\\ T\end{array}\right]$$

(5)

where F is the external force acting on the sensor; T is the external torque acting on the sensor; F_a1, F_a2 and F_a3 are the forces along the direction of the three force-measuring branches; F_c1, F_c2 and F_c3 are the forces perpendicular to the direction of three force-measuring branches.

For the force F_a1 along the direction of the force-measuring branch, the equation is as follows:

$$F_{a1}=\left(F_{a11}+F_{a12}\right)\cos \alpha$$

(6)

where F_a11 and F_a12 are the forces acting on the connecting rods under the action of force F_a1 and α is the angle between the connecting rod and the vertical direction, as shown in Figure 8a.

Since the 3-UPU parallel mechanism has only three translational degrees of freedom, the force-measuring branch only experiences translational motion. For the force F_c1 perpendicular to the direction of the force-measuring branch, the equation is as follows:

$$F_{c1}=\left(F_{c11}+F_{c12}\right)\cos \alpha$$

(7)

where F_c11 and F_c12 are the forces acting on the connecting rods under the action of force F_c1, as shown in Figure 8b.

Based on the above analysis, Equation (5) can be rewritten as:

$$\left[\begin{array}{c}F\\ T\end{array}\right]=-J^T\cos \alpha \left[\begin{array}{c}{F}_{a11}+F_{a12}\\ F_{a21}+F_{a22}\\ F_{a31}+F_{a32}\\ F_{c11}+F_{c12}\\ F_{c21}+F_{c22}\\ F_{c31}+F_{c32}\end{array}\right]$$

(8)

where F_a21 and F_a22 are the forces acting on the connecting rods under the action of force F_a2, F_c21 and F_c22 are the forces acting on the connecting rods under the action of force F_c2, F_a31 and F_a32 are the forces acting on the connecting rods under the action of force F_a3, and F_c31 and F_c32 are the forces acting on the connecting rods under the action of force F_c3.

Due to the presence of two connecting rods on each force-measuring branch, there are, in total, six connecting rods. Through the analysis of Figure 8, the forces (F_l1, F_l2, F_l3, F_l4, F_l5, F_l6) of the six connecting rods can be derived as:

$$\left[\begin{array}{c}{F}_{l1}\\ F_{l2}\\ F_{l3}\\ F_{l4}\\ F_{l5}\\ F_{l6}\end{array}\right]=\left[\begin{array}{c}{F}_{a11}-F_{c11}\\ F_{a11}+F_{c11}\\ F_{a31}-F_{c32}\\ F_{a31}+F_{c32}\\ F_{a51}-F_{c52}\\ F_{a51}+F_{c52}\end{array}\right]$$

(9)

From Equations (8) and (9),

$$\left[\begin{array}{c}F\\ T\end{array}\right]=-J^T\cos \alpha A\left[\begin{array}{c}{F}_{l1}\\ F_{l2}\\ F_{l3}\\ F_{l4}\\ F_{l5}\\ F_{l6}\end{array}\right]$$

(10)

where

$$A=\left[\begin{array}{cccccc}1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 1\\ -1& 1& 0& 0& 0& 0\\ 0& 0& -1& 1& 0& 0\\ 0& 0& 0& 0& -1& 1\end{array}\right]$$

4.2. Stiffness Analysis of the Force-Measuring Branch

The deformation of the elastic body comes from the flexible hinges. The morphology and size of the flexible hinges are very important parameters. In this paper, a single-sided circular flexible hinge structure is adopted in the flexible hinge, as shown in Figure 9. Next, we will derive the relationship between structural parameters and sensor output.

The rotational stiffness G_Z of single-sided circular flexible hinge can be expressed as [21]:

$$G_Z={\displaystyle \frac{Eh}{12}}{\displaystyle \frac{R^2}{D}}$$

(11)

where

$$D={\int }_{-\pi /2}^{\pi /2}{\displaystyle \frac{\cos \theta }{{\left(1-\cos \theta +t/R\right)}^3}}d\theta$$

E is the elastic modulus of the material; h (h = 10 mm) is the width of the flexible hinge; R (R = 2.5 mm) is the radius of the flexible hinge; t (t = 2 mm) is the minimum thickness of the flexible hinge; θ is the central angle; dθ is the central angle corresponding to the infinitesimal deformation of the flexible hinge.

Here, taking the force F_l1 as an example to analyze the force on the elastic body. When the force F_l1 is applied to the elastic body at an angle Ψ (Ψ = 25°), as shown in Figure 10, it can be decomposed into horizontal force F_x and vertical force F_y.

$$\left.\begin{array}{c}{F}_y=F_{l1}\cos \psi \\ F_x=F_{l1}\sin \psi \end{array}\right\}$$

(12)

When subjected to the force F_y, the semi-arc-shaped elastic body structure can be equivalent to a cantilever beam, as shown in Figure 10.

The moment M acting on the elastic body by the force F_y can be expressed as:

$$M=F_y\cdot l$$

(13)

where l (l = 19.5 mm) is the arm length of the equivalent cantilever beam in the direction of the force F_y.

The work done by the force F_y acting on the elastic body is equal to the work done by the equivalent moment. Thus,

$$F_y\cdot {\delta }_v=M\cdot \phi$$

(14)

where δ_v is the displacement of the equivalent p-joint in the vertical direction; $\phi$ is the bending angle of the flexible hinge under the force F_y, as shown in Figure 10.

The deformation trajectory of the equivalent cantilever beam is an arc with a radius of r₀. δ_v can be represented as:

$${\delta }_v=r_0\cos \beta d\beta$$

(15)

where β is the central angle corresponding to the equivalent cantilever beam. Since dβ represents the infinitesimal angular deformation of the equivalent cantilever beam, it is equal to $\phi$. δ_v can be rewritten as:

$${\delta }_v=r_0\cos \beta \cdot \phi$$

(16)

From Figure 10, we can obtain:

$$r_0\cos \beta =l$$

(17)

Combining the above analysis, the equivalent stiffness K_y of the equivalent cantilever beam in the direction of the force F_y can be expressed as:

$$K_y={\displaystyle \frac{F_y}{{\delta }_v}}={\displaystyle \frac{G_Z}{l^2}}$$

(18)

Similarly, the equivalent stiffness $K_x$ in the direction of force F_x can be expressed as:

$$K_x={\displaystyle \frac{F_x}{{\delta }_v}}={\displaystyle \frac{G_Z}{L\cdot l}}$$

(19)

where L (L = 39 mm) is the arm length of equivalent cantilever beam in the direction of the force F_x, as shown in Figure 10.

According to the principle of force superposition, the displacement change dy₁ of the equivalent p-joint can be derived as:

$$d_{y1}={\displaystyle \frac{F_y}{K_y}}+{\displaystyle \frac{F_x}{K_x}}=F\left({\displaystyle \frac{\cos \psi }{K_y}}+{\displaystyle \frac{\sin \psi }{K_x}}\right)$$

(20)

The equivalent stiffness K of the equivalent p-joint can be expressed as:

$$K={\displaystyle \frac{F_{l1}}{d_{y1}}}={\left({\displaystyle \frac{\cos \psi }{K_y}}+{\displaystyle \frac{\sin \psi }{K_x}}\right)}^{-1}$$

(21)

As shown in Figure 11, the deformation of the equivalent p-joint is dy₁, and the change in the chord length of FEM is δ₁. The equivalent stiffness K_s of the chord deformation at the FEM can be expressed as:

$$K_S={\displaystyle \frac{F_{l1}}{{\delta }_1}}={\displaystyle \frac{F_{l1}}{d{y}_1}}\cdot {\displaystyle \frac{d{y}_1}{{\delta }_1}}=K\cdot {\displaystyle \frac{d{y}_1}{{\delta }_1}}$$

(22)

In Figure 11, the following equations can be obtained:

$$\left.\begin{array}{r}d{y}_1=r_1\sin \left({\theta }_1+d\theta \right)-r\sin {\theta }_1=\cos {\theta }_1\cdot r_1\cdot d\theta \\ {\delta }_1=r_2\sin \left({\theta }_2+d\theta \right)-r\sin {\theta }_2=\cos {\theta }_2\cdot r_2\cdot d\theta \end{array}\right\}$$

(23)

where θ₁ (θ₁ = 15°) and θ₂ (θ₁ = 45°) are the central angles corresponding to the equivalent p-joint and the FEM, respectively; r₁ (r₁ = 20.15 mm) and r₂ (r₂ = 49.85 mm) are the radius of the deformation arcs corresponding to the equivalent p-joint and the FEM, respectively.

From Equations (22) and (23), we can obtain:

$$K_S=K\cdot {\displaystyle \frac{\cos {\theta }_1\cdot r_1}{\cos {\theta }_2\cdot r_2}}$$

(24)

From Equations (10) and (22),

$$\left[\begin{array}{c}F\\ T\end{array}\right]=-J^T\cos \alpha A{K}_S\left[\begin{array}{c}{\delta }_1\\ {\delta }_2\\ {\delta }_3\\ {\delta }_4\\ {\delta }_5\\ {\delta }_6\end{array}\right]=D\left[\begin{array}{c}{\delta }_1\\ {\delta }_2\\ {\delta }_3\\ {\delta }_4\\ {\delta }_5\\ {\delta }_6\end{array}\right]$$

(25)

where δ₁, δ₂, δ₃, δ₄, δ₅ and δ₆ are the changes in the chord length of FEMs. Matrix D is the mapping matrix from the external load to the changes in the chord length of the FEMs. $D=-J^T\cos \alpha A{K}_S$. The J and K_s can be obtained from Equations (4) and (24). cosα and A are all constant.

5. Experiments and Results Analysis

5.1. Simulation

To verify the correctness of the force mapping matrix D, a virtual simulation of the six-axis force sensor was conducted.

The 3D model was imported into the ANSYS simulation software (version number: 2023). Before running the simulation, it was necessary to complete some settings, including setting material properties, adding constraints, and dividing mesh division, as shown in Figure 12. Here, the force loading in the Z-direction is taken to verify the correctness of the force mapping matrix D.

Within the range of −60 N to 60 N, forces F_Z were applied with a step size of 10 N. The changes in the chord length at the FEM were recorded separately, as shown in Figure 13.

And then the changes of the chord length were substituted in Equation (25) and the loading force were calculated. The comparison between the resolved loading force and applied loading force is presented in Figure 14.

The errors of simulation are analyzed. The error E_F can be expressed as:

$$E_F={\displaystyle \frac{\max \left|F-F^{\prime }\right|}{F_{mr}}}$$

(26)

where F is the applied loading force, F′ is the resolved load, and F_mr is the full measurement range. According to Figure 14, the maximum error of the simulation is 0.89%, which shows the correctness of the established force mapping matrix.

5.2. Experiment

The detection system of the six-axis force sensor based on bending-sensitive optical fibers is illustrated in Figure 15. The light source is generated by a modulation circuit, which is then split into six optical paths by a 1 × 6 fiber optic splitter to provide light for the six FEMs. When an external load is applied to the loading platform, the force is transmitted to the elastic bodies through the connecting blocks and rods, causing deformation of the elastic bodies. The deformation changes the bending curvature of the FEMs fixed on the elastic bodies, which changes the output signals of the optical fibers. The phototransistors IF-D92 capture the optical signals and convert them into voltage signals. These signals are then collected by the PCL818L data acquisition card (Advantech, Taipei, Taiwan) and the computer obtains the external load information through calculation.

The loading experiment was carried out using weights. The loading method refers to [22], and the separate loadings of the X, Y, Z directions, as well as the torque for each dimension, were carried out. The output signals of the optical fibers were recorded. Figure 16 illustrates the experimental setup for the sensor loading. The photograph of the detection system is shown in Figure 17.

During the loading, the relationship between the chord length H, arc length S, and curvature C of the FEM can be expressed as:

$$H={\displaystyle \frac{2}{C}}\sin \left({\displaystyle \frac{SC}{2}}\right)$$

(27)

From Equations (1) and (27), we obtain:

$$H={\displaystyle \frac{2a}{U-b}}\sin \left(S\cdot {\displaystyle \frac{U-b}{2a}}\right)$$

(28)

Based on Equation (28), the chord length H of FEM can be deduced from the optical fiber output signal U. And then, the loading force information can be calculated using Equation (25).

In order to realize the calibration of the sensor, F_x, F_y, F_z, T_x, T_y and T_z are loaded separately by ANSYS simulation. The calibration relationship of the sensor can be expressed as:

$$\left[\begin{array}{c}F\\ T\end{array}\right]=D_0\left[\begin{array}{c}{H}_1\\ H_2\\ H_3\\ H_4\\ H_5\\ H_6\end{array}\right]$$

(29)

where H₁, H₂, H₃, H₄, H₅ and H₆ are the chord length of FEMs, which can be obtained from Equation (28). Matrix D₀ is the matrix from the external load to the chord length of the FEM.

$$D_0=\left[\begin{array}{cccccc}43.562& -43.562& 87.124& -87.124& 43.562& -43.562\\ 78.935& -78.950& 0.038& 0.001& -78.944& 78.951\\ -91.185& -91.170& -91.192& -91.153& -91.190& -91.240\\ 1.810& 1.838& -3.672& -3.669& 1.823& 1.755\\ 3.216& 3.205& -0.001& -0.001& -3.158& -3.178\\ 3.655& -3.618& -3.716& 3.676& 3.636& -3.680\end{array}\right]$$

Here, the influence of the mechanical effects on the calibration matrix D₀ is investigated. According to Figure 5 and Figure 6, it can be seen that the mechanical effects are from the mechanical friction of revolute pairs (R3–R6), which introduce frictional resistance during force transmission. As is known to all, the mechanical friction is proportional to external force and the corresponding frictional resistance torque M_f can be expressed as:

$$M_f=G\cdot f\cdot r$$

(30)

where G is the external force, f is the coefficient of friction and r is the radius of the rotation axis. There are revolute pairs at both ends of the connecting rod, as shown in Figure 5. If the friction is not considered, the force is directly transmitted in the direction of the connecting rod. Because of the presence of friction, there is an angle Ψ₁ between the direction of force transmission and connecting rod, as shown in Figure 18. In this case, the Matrix D in Equation (25) is changed to be $D=-J^T\cos (\alpha -{\psi }_1)A{K}_S$. That is to say, the relationship between theoretical matrix and the actual calibration matrix is regarded as an approximate proportional relationship and the proportion is $\cos (\alpha -{\psi }_1)/\cos \alpha$. To determine the size of this proportional relationship, compensation analysis is conducted using a loading experiment. The force loading method is shown in Figure 16. Figure 19 shows the measurement results.

In Figure 19, the slope of the fitted curve is 0.956. The slope of the theoretical curve is 1. As mentioned above, the reason for the error lies in the influence of mechanical effects. According to the above analysis, a proportional compensation for matrix D₀ is reasonable. Due to a deviation of 1.046 times in slope, we compensate for matrix D₀ by multiplying by 1.046. After compensation, the calibration matrix of the sensor can be expressed as:

$$\left[\begin{array}{c}F\\ T\end{array}\right]=\left[\begin{array}{cccccc}47.547& -47.692& 95.332& -95.348& 47.589& -47.554\\ 82.566& -82.582& 0.004& 0.001& -82.576& 82.583\\ -95.352& -95.364& -95.387& -95.346& -95.385& -95.437\\ 1.893& 1.923& -3.841& -3.838& 1.907& 1.836\\ 3.364& 3.353& -0.001& -0.001& -3.303& -3.324\\ 3.823& -3.785& -3.887& 3.845& 3.803& -3.849\end{array}\right]\left[\begin{array}{c}{H}_1\\ H_2\\ H_3\\ H_4\\ H_5\\ H_6\end{array}\right]$$

(31)

Here, the force loading experiment in Z-direction is carried out to demonstrate the effectiveness of compensation. The experiment results are shown in Figure 20. Equation (26) is used to calculate the errors of Figure 20. The maximum errors before and after compensation are 5.83%FS and 2.38%FS, respectively, which shows the effectiveness of compensation.

5.3. Results and Analysis

5.3.1. Accuracy and Hysteresis

In order to study the measurement accuracy of the sensor, the above calibration Equation (31) is used to measure the loading force in the individual direction. Forces F_X, F_y and F_z were applied with a step size of 6 N within the range of −60 N to 60 N, and the torques T_X, T_y and T_z were applied with a step size of 0.25 N·m within the range of −2.5 N·m to 2.5 N·m. And then, the same load was applied in the opposite direction for all measurement directions. The loading methods for all forces and torques are shown in Figure 16. Figure 21 illustrates the measurement results. To study the hysteresis of the sensor, the measured values of the corresponding directional force are also shown in Figure 19 when the load is applied in the opposite direction. The hysteresis H_S can be expressed as:

$$H_S={\displaystyle \frac{\max \left|F_F-F_R\right|}{F_{mr}}}\times 100\%$$

(32)

where F_F is the measured force of forward loading (increasing loading force), F_R is the measured force of reverse loading (decreasing loading force).

According to Figure 21, the maximum hysteresis occurs when the torque is applied in the X-direction, with a value of 2.78%.

The corresponding residual error analysis result is shown in Figure 22. In Figure 22, all values of Pearson’ r and determination coefficient R² are greater than 0.99, and all p-values are smaller than 0.01, which shows the validation of the experiment.

The error matrix E_r computed by substituting the data in Figure 21 into Equation (26) is as follows:

$$E_r=\left[\begin{array}{cccccc}{E}_{Fx}& E_{Fy}& E_{Fz}& E_{Tx}& E_{Ty}& E_{Tz}\end{array}\right]=\left[\begin{array}{cccccc}0.0452& 0.0012& 0.0125& 0.0069& 0.0148& 0.0122\\ 0.0141& 0.0365& 0.0085& 0.0295& 0.0106& 0.0054\\ 0.0117& 0.0035& 0.0263& 0.0251& 0.0011& 0.0121\\ 0.0012& 0.0326& 0.0159& 0.0253& 0.0294& 0.0116\\ 0.0166& 0.0023& 0.0072& 0.0094& 0.0264& 0.0075\\ 0.0235& 0.0152& 0.0325& 0.0154& 0.0035& 0.0183\end{array}\right]$$

(33)

The elements on the diagonals constitute the type-I errors and are equivalent to the maximum relative errors between the real forces acting in the corresponding directions and the measured force. The remaining elements constitute the type-II errors and are equivalent to the coupled outputs in other directions resulting from forces in individual dimensions. According to the matrix E_r, type-I errors are 4.52%FS, 3.65%FS, 2.63%FS, 2.53%FS, 2.64%FS and 1.83%FS in the F_x, F_y, F_z, T_x, T_y, and T_z principal measurement directions, respectively. The maximum type-II error is 3.26%FS, corresponding to the coupled torque output in the M_x direction when the load is applied in the F_y direction. The maximum values of type-I errors and type-II errors are both smaller than that of [8].

5.3.2. Repeatability

Repeatability refers to the difference between the results of multiple measurements under the same conditions. Repeatability R_e can be expressed as:

$$R_e={\displaystyle \frac{\Delta F}{F_{mr}}}\times 100\%$$

(34)

where ΔF is the maximum difference between multiple measured results.

To study the repeatability of the sensor, the loading experiments for all forces and torques are carried out three times. The force loading method is shown in Figure 16, and the measurement results are shown in Figure 23. It is shown that the maximum repeatability error R_e = 3.27% when the force is applied in the Z-direction. In order to better demonstrate the repeatability and stability of the sensor, the three experimental results are processed and plotted into error bands, as shown in Figure 24.

From Figure 23, it can be seen that the maximum repeatability error occurs when the force loading in the Z-direction is −12 N. To further investigate the repeatability of the sensor, a force of −12 N is repeatedly applied in the Z-direction 10 times. The results are shown in Figure 25. Here, the standard deviation of repeatability S_r is used to reflect the stability of the sensor, and it can be expressed as:

$$S_r=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^{10}{(x_i-x_a)}^2}}{n-1}}}$$

(35)

where x_i is the measured force, x_a is the average of the measurement results and n is the measurement times (n = 10). According to Figure 25, S_r = 0.718, which indicates the stability of the sensor.

5.3.3. Linear Range and Strength Limits

Experimental analysis is conducted to investigate nonlinear effects within a large load range. Forces F_x, F_y and F_z are applied with a step size of 6 N within the range of −96 N to 96 N, and the torques T_x, T_y and T_z were applied with a step size of 0.25 N·m within the range of −4.25 N·m to 4.25 N·m. Figure 26 illustrates the measurement results. The results show that the linear ranges of force loading F_x, F_y and F_z are ±60 N, ±72 N and ±90 N, and the linear ranges of torque loading T_x, T_y and T_z are ±2.5 N·m, ±3.25 N·m and ±3.75 N·m, respectively.

Here, ANSYS Workbench is used to simulate the strength limits and material lifespan. The simulation results show that the maximum stress occurs at single-sided circular flexible hinge. The material of the force-measuring branches is 7075 aluminum alloy and its allowable stress [σ] = 419 MPa. Therefore, the maximum stress at the single-sided circular flexible hinge should be less than this allowable stress. The results of the strength limits of sensor in all directions are shown in

Table 1. Strength limits of sensor in all directions.

Load Limits	Fx = 130 N	Fy = 160 N	Fz = 200 N	Tx = 4.9 N·m	Ty = 6.5 N·m	Tz = 7.6 N·m
corresponding stress (MPa)	395.7	405.8	400.2	402.6	408.3	415.2

. Although the maximum loads are significantly greater than the loads of the linear range, it is generally not allowed to exceed the load of the linear range when using sensors. The material lifespan simulation result shows that the fatigue life is greater than 1.2 × 10⁶ under the maximum linear load. To investigate whether nonlinear effects occur under material fatigue aging conditions, the maximum linear load (90 N) is applied in the Z-direction repeatedly. The experiment results show that the sensor still maintains a good linear output after 5000 loading cycles.

5.3.4. Thermal Effects

In order to study the instability of the sensing system caused by temperature, a relevant experiment was carried out. Without any loading, we recorded the output of fiber every 30 min for 7 h (the temperature is from 15 °C to 32 °C) in a day. The results are shown in Figure 27. It shows that maximal variation is within 0.7%, which implies that the temperature has a minor effect on the sensor. The variation of the fiber output mainly comes from the instability of LED light sources caused by temperature variations.

6. Conclusions

This study presents a six-dimensional force sensor based on bend-sensitive optical fiber. A theoretical mapping matrix was established through static analysis and stiffness analysis. Finite element simulation was carried out to obtain the chord length changes of the FEMs. The loading forces were further calculated by using the established mapping matrix. The results show that the maximum error of the simulation is 0.98%, which verifies the correctness of the established mapping matrix. The measurement experiments were carried out, and the experimental results reveal that the type-I errors are 4.52%FS, 3.65%FS, 2.63%FS, 2.53%FS, 2.64%FS and 1.83%FS in the F_x, F_y, F_z, T_x, T_y, and T_z principal measurement directions, respectively. The maximum type-II error is 3.26%FS. The relative experiments are carried out to investigate the repeatability and linear range of sensor. The maximum repeatability error is 3.27% when the force is applied in the Z-direction. The linear ranges of force loading F_x, F_y and F_z are ±60 N, ±72 N and ±90 N, and the linear ranges of torque loading T_x, T_y and T_z are ±2.5 N·m, ±3.25 N·m and ±3.75 N·m, respectively.

Due to the many advantages of optical fibers, the proposed fiber-based six-axis force sensor has the characteristics of resistance to electromagnetic interference and adaptability to high-temperature and humid environments. This provides a novel approach for measuring force information in harsh environments.

In the experiment, the temperature variations can lead to the instability of LED light sources, which cannot be ignored. In the future, we will introduce reference fiber in the sensing system to reduce the influence of the temperature. Improving the performance of the sensor is what we need to further consider. This may involve consideration of doping elements [23] and optimizing structural parameters.