# A Force Measurement Method Based on Flexible PDMS Grating
## 4 Results
### Force Measurement  
Two PDMS grating samples with mixing ratios of 10:1 and 15:1 are tested respectively. For the 10:1 samples, the processed photos of light spots are shown in Figure 9. The accurate position of each light spot is calculated and illustrated. It is obvious that the spots move closer with the increase of force.  
Figure 8: The data distribution of elastic modulus along with the mixing raito. (Curing temperature = 80 \({}^{\circ}\)C)  
Figure 7: The average data of elastic modulus under different mixing ratios and curing temperatures.  
The data series of the light spot position and the corresponding tensile force are analyzed using linear regression method. The fitting curves are shown in Figure 10.  
The discussion on Figure 10 are listed as below,  
(1) Linearity. The coefficient of determination (R\({}^{2}\)) for the 10:1 sample is 0.9999, and that for 15:1 sample is 0.9985. Therefore, the relationship between light spot position and tensile force is linear, which coincides with Equation (9).  
(2) Sensitivity. The slope for the red line (10:1 sample) is -0.57 mm/N, while the slope for the blue line (15:1 sample) is -0.67 mm/N. The 15:1 sample has larger sensitivity than the 10:1 one. As illustrated in Figure 10, the elastic modulus is \(\sim\)1.5 Mpa for 10:1 mixing ratio and \(\sim\)1.1 Mpa for 15:1 mixing ratio. Therefore, the 15:1 sample has larger deformation than the 10:1 sample under the same external force, leading to a larger shift of the light spot. Based on this, the sensitivity of the PDMS grating force sensor can be adjusted by changing the mixing ratio, or even the cured temperature.  
(3) Accuracy. The residual error of linear regression is used to evaluate the accuracy of force measurement. The mean residual error of the data points in Figure 10 is 0.05N and the maximum error is 0.09N, which is sufficiently precise for force measurement. The error mainly comes from the uncertainty of spot position measurement, which is \(\sim\)0.05 mm. According to the sensitivity analysis above, 0.05 mm error in spot position leads to \(\sim\)0.03N error in force measurement.  
Figure 10: The experimental data and fitting curve of the light spot position and tensile force.  
Figure 9: Images of light spot position under different forces: (**a**) F=5N, (**b**) F=7N, (**c**) F=10N.  
(4) Range. The force data in Figure 10 is ranging from 5N to 12N. However, the maximum range is related with the fracture force of the sample. The fracture force is \(\sim\)30N by a series of experiments. It can be estimated that linear and reliable results can be obtained using this method when the force is less than 20N.  
The force sensor based on PDMS waveguide in [11] has linearity (R2) of 0.96, error of \(\sim\)0.3N, and range of 5N. The proposed force sensor based on PDMS grating has a larger footprint than [11], and possesses better performance. Also using a PDMS grating, the relative error of strain measurement in [12] is less than 0.6%. In this paper, the average relative error is estimated as 0.59% (0.05N/8.5N, 8.5N is the average force in the test), which is comparable with this key factor in [12]. However, the proposed soft casting method in this paper does not require any special equipment, and it is even cheaper and easier than the oxygen treatment method in [12]. Moreover, the PDMS grating pattern can be faithfully replicated from the master grating, which is favorable for mass production.