# A Fiber-Optic Six-Axis Force Sensor Based on a 3-UPU-Compliant Parallel Mechanism
## 5 Experiments and Results Analysis
### 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 \(1\times 6\) 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:  
\[H=\frac{2}{C}\sin\left(\frac{SC}{2}\right) \tag{27}\]  
From Equations (1) and (27), we obtain:  
\[H=\frac{2a}{U-b}\sin\left(S\cdot\frac{U-b}{2a}\right) \tag{28}\]  
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).  
Figure 15: Detection system of the six-axis force sensor.  
Figure 16: The experimental setup for the sensor loading.  
Figure 17: Photograph of the detection system.  
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:  
\[\begin{bmatrix}F\\ T\end{bmatrix}=D_{0}\begin{bmatrix}H_{1}\\ H_{2}\\ H_{3}\\ H_{4}\\ H_{5}\\ H_{6}\end{bmatrix} \tag{29}\]  
where \(H_{1}\), \(H_{2}\), \  
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 \(D_{0}\) is reasonable. Due to a deviation of 1.046 times in slope, we compensate for matrix \(D_{0}\) by multiplying by 1.046. After compensation, the calibration matrix of the sensor can be expressed as:  
\[\begin{bmatrix}F\\ T\end{bmatrix}=\begin{bmatrix}47.547&-47.692&95.332&-95.348&47.589&-47.554\\ 82.566&-82.582&0.004&0.001&-82.576&82.583\\ -95.352&-95.364&-95.387&-95.346&-95.385&-95.437\\ 1.893&1.923&-3.841&-3.838&1.907&1.836\\ 3.364&3.353&-0.001&-0.001&-3.303&-3.324\\ 3.823&-3.785&-3.887&3.845&3.803&-3.849\\ \end{bmatrix}\begin{bmatrix}H_{1}\\ H_{2}\\ H_{3}\\ H_{4}\\ H_{5}\\ H_{6}\end{bmatrix} \tag{31}\]  
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.  
Figure 19: The measurement results obtained using matrix \(D_{0}\).  
Figure 20: Comparison of measurement results before and after compensation.