Performance Analysis of a Lower Limb Multi Joint Angle Sensor Using CYTOP Fiber: Influence of Light Source Wavelength and Angular Velocity Compensation

This paper presents the analysis of an intensity variation polymer optical fiber (POF)-based angle sensor performance, i.e., sensitivity, hysteresis and determination coefficient (R2), using cyclic transparent optical polymer (CYTOP) fiber. The analysis consisted of two approaches: influence of different light source central wavelengths (430 nm, 530 nm, 660 nm, 870 nm and 950 nm) and influence of different angular velocities (0.70 rad/s, 0.87 rad/s, 1.16 rad/s, 1.75 rad/s and 3.49 rad/s). The first approach aimed to select the source which resulted in the most suitable performance regarding highest sensitivity and linearity while maintaining lowest hysteresis, through the figure of merit. Thereafter, the analysis of different angular velocities was performed to evaluate the influence of velocity in the curvature sensor performance. Then, a discrete angular velocity compensation was proposed in order to reduce the root-mean-square error (RMSE) of responses for different angular velocities. Ten tests for each analysis were performed with angular range of 0∘ to 50∘, based on knee and ankle angle range during the gait. The curvature sensor was applied in patterns simulating the knee and ankle during the gait. Results show repeatability and the best sensor performance for λ=950 nm in the first analysis and show high errors for high angular velocities (w=3.49 rad/s) in the second analysis, which presented up to 50% angular error. The uncompensated RMSE was high for all velocities (6.45∘ to 12.41∘), whereas the compensated RMSE decreased up to 74% (1.67∘ to 3.62∘). The compensated responses of application tests showed maximum error of 5.52∘ and minimum of 1.06∘, presenting a decrease of mean angular error up to 30∘ when compared with uncompensated responses.


Introduction
The biomechanics of human movement are defined as the study of the human movement using methods of mechanical engineering [1]. The human movement analysis includes gait analysis, which comprises the systematic study of human walking, performed by collecting kinematic and kinect data, among them the joint angles [2,3]. In the clinical field, changes of the joint angles in the normal gait pattern may reveal key information about person's quality of life [4]. The traditional scales used to analyze joint angles in clinical field are semi-subjective, based on specialists observation of in previously defined parameters between velocities groups. Finally, the proposed compensation technique was applied on the POF angle sensor in two tests: simulating the knee and ankle angle patterns during the gait using the light source central wavelength which results in the best response.
The remainder of this paper is organized as follows. Section 2 presents the materials and methods used for the fabrication and the analysis of the sensor, including the experimental procedures, the statistical analysis and the proposal of angular velocity compensation. The results and discussion of each analysis, as well as the comparison of the responses with and without compensation and the application in movement analysis are presented in Section 3. Finally, the final considerations and future works are discussed in Section 4.

Materials and Methods
The POF was made of a commercial gradient index multimode CYTOP fiber (Chromis Fiberoptics Inc, USA) with a core diameter of 120 µm, a cladding thickness of 20 µm and a polycarbonate overcladding. The light sources were five light emitting diodes (LEDs) with different central wavelengths (λ): 430 nm, 530 nm, 660 nm, 870 nm and 950 nm (IF-E92A, IF-E93, IF-E97, IF-E91D and IF-E91A, respectively, Industrial Fiber Optics, USA). The selected LEDs correspond to different central wavelength of visible light region (blue -430 nm, green -530 nm and red -660 nm) and different central wavelength of infrared region (870 nm and 950 nm), and are commonly commercially available. The optical power was converted into an electrical signal using a phototransistor IF-D92 (Industrial Fiber Optics, USA) and the acquisition was made through a microcontroller FRMD-KL25Z (NXP Semiconductors, Netherlands), at a sampling rate of 100 Hz. The data was filtered through the moving average filter with span of 5%, in order to eliminate the outliers observed in the measurements. The angle and the angular velocity were controlled using a servo motor.
The sensor operation principle is based on intensity variation, in which the attenuation of the transmitted optical signal power is related to the bending angle of the fiber, i.e., the transmission loss occur due to radiation losses caused by macrobending and is proportional to the variance of optical fiber's bending angle.

Experimental Procedures
The experimental protocol was divided into two approaches in order to evaluate the performance of the CYTOP fiber under curvature: influence of different light source central wavelengths and influence of different angular velocities. Both approaches consisted of curvature tests with angular range from 0 • to 50 • and steps of 10 • . This range was defined according to the knee and ankle angle patterns and limited at 50 • to ensure the reliability of the tests, due to servo motor limitation for high variation at high velocities. The analysis of the opposite bending (−50 • to 0 • ) is not necessary, since the knee and ankle movements cannot be performed for the another direction due to the body limitation. However, it would be solved through lateral sections made in the fiber to create a sensitive zone. In the case of concave bending, there is an increase in reflections on the convex side of the curvature and a decrease on the concave side. Thus, if the lateral section is made in the concave side, the output power is higher for concave bending and lower for convex bending, and it is possible to identify the bending direction. Figure 1 shows the experimental setup for evaluation of the POF-based curvature sensing performance using CYTOP fiber.  In the first approach, curvature tests with angular velocity of 0.87 rad/s were performed. In addition, LEDs with different central wavelengths were employed to compare the responses related to wavelength. Sensitivity, hysteresis and R 2 were analyzed, and through these factors, the figure of merit (FoM) was calculated (as shown in Equation (1)) in order to characterize the wavelength which results in the higher FoM, consequently the best sensor performance. In Equation (1), S is the sensitivity, h is the hysteresis, R 2 is the determination coefficient, and α, β, θ are coefficients which define the weight of each performance factor.

Light source
Thereafter, tests with different angular velocities (0.70 rad/s, 0.87 rad/s, 1.16 rad/s, 1.75 rad/s and 3.49 rad/s) and the light source central wavelength previously defined by FoM were conducted to evaluate the influence of velocity in the curvature sensor performance (sensitivity, hysteresis and R 2 ).
Based on these tests, an angular velocity compensation technique was proposed in order to reduce the root-mean-square error (RMSE) in different angular velocities (see Section 2.2). In both approaches, 10 tests were performed. Finally, the proposed compensation technique was applied on the POF angle sensor in two tests: simulating the knee and ankle patterns during the gait.

Statistical Analysis and Angular Velocity Compensation
A Shapiro-Wilk test was used to verify the data normality. Since the data are normal, one-way ANOVA (analysis of variance) was applied to determine if significant differences in sensitivity, hysteresis and angular error existed among different angular velocities with a significance level of 0.05, and the significant angular velocity groups were defined through this analysis.
To reduce the RMSE of angle measurements, compensation models were developed for each angular velocity group. Figure 2 shows the state machine diagram describing the proposed angular velocity compensation, in which "QS" block comprises the model obtained on a quasi-static test, i.e., calibration curve with the lowest influence of the sensor angular velocity, and C n comprises the compensation models for respective angular velocity range.

C n-1 Compensation model n-1
ω n-1 < ω ≤ ω n Since the CYTOP is a viscoelastic material, its strain response depends on the time and the polymer relaxation can lead to the sensor hysteresis and possible variations on its linearity. To obtain a response close to the ideal (servo motor input), i.e., linear response, the compensation models were based on a sum of exponential with order 2, as shown in Equation (2), where a n are the model coefficients, α is the compensated angle ( • ) and P is the sensor power variation (ADC unit).

Results and Discussion
To verify the stability of the sensor material related to temperature and humidity variation, two tests were performed. One test consisted of the sensor response analysis with the temperature increase and the other test consisted of the sensor response analysis with the increase of relative humidity. Figure 3a,b present the results of these tests, which showed that the temperature and the humidity do not significantly influence the sensor response. The pH was not evaluated in this work since this application is not influenced by this factor, because there is no pH variation in the environment where the sensor is positioned. The analysis of POF curvature sensor in CYTOP fiber is based on three performance factors: sensitivity, hysteresis and R 2 . Figure 4a shows the concept of sensitivity and hysteresis applied in one result, where R 2 corresponds to the determination coefficient with an exponential regression, as shown in Figure 4b.   Figure 5a presents the sensors' responses to the curvature applied to the CYTOP fiber (angular range from 0 • to 50 • and constant angular velocity of 0.87 rad/s) for different central wavelengths (λ), in which the markers represent the measured output and the dashed lines represent the sensors fit. The sensors' responses presented exponential behavior with R 2 higher than 0.99 in all characterization tests. It is possible to observe the normal distribution of sensitivities in Figure 5b, in which the light source central wavelength of 950 nm provide the higher sensitivity (32.63 ADC unit/deg) of the sensor, whereas the light source central wavelength of 870 nm provide the lower sensitivity (0.94 ADC unit/deg). This may occur due to the some reasons, such as the decreasing optical attenuation curve of CYTOP fiber which presents higher attenuation for λ = 430 nm and lower optical attenuation for λ = 950 nm [8]. In addition, the photodetector as a function of the wavelength presents higher responsivity for 800 nm < λ < 950 nm.

Analysis of Different Light Source Central Wavelengths
Thus, the sensor using LED with central wavelength of λ = 950 nm presented highest sensitivity. The λ = 870 nm should present the second highest sensitivity; however, the current of the tests is lower than the forward current presented in the light source datasheet [25]. Since the currents used in all tests has the same value and the λ = 870 nm need a higher current value, the optical power is relatively lower, as the sensitivity when compared with others light source central wavelengths. In addition, the λ = 660 nm presented the lower sensitivity among the visible spectrum (λ = 430 nm, 530 nm, 660 nm) due to the lowest full-spectral bandwidth of the light source, resulting in a lower output power, since the photodetector acquires the integral of the light source spectrum, and, consequently, lower sensitivity. Based on these performance factors, a FoM was applied to calculate the weighted sum of the factors and to select the central wavelength that results the higher FoM. The weighted coefficients for FoM calculation were defined as α = 0.5, β = 0.2, θ = 0.3, since the sensitivity is a important factor which significantly decreases as the angular velocity increases, and hysteresis can be smoothed through some compensation technique [23]. Table 1 presents the results of each FoM, in which the light source central wavelength of 950 nm provides the higher FoM, being the best option for following applications.

Analysis of Different Angular Velocities
After the first set of tests to select the light source central wavelength, tests with the angular range from 0 • to 50 • and different angular velocities are performed, as shown in Figure 1. Figure 6 shows the results of measurements without data treatment for the different angular velocities. All graphs show curvature cycles performed during the same time variation at different angular velocities. It is noticeable that the sensor responses present low noise; however, the moving average filter is used to eliminate the outliers observed in the measurements. The three performance factors (sensitivity, hysteresis and R 2 ) are analyzed, and Table 2 shows the mean and the standard deviation (SD) of sensitivity (ADC unit/deg), hysteresis (%) and angular error ( • ) of tests for each angular velocity.   According to Table 2, it is noticeable that w = 3.49 rad/s presented worst performance compared with others velocities, with lower sensitivity, equivalent to approximately 40% of the others, error of 25.25 • (2.00 • ), corresponding to 50% of error and highest hysteresis of 3.12%(2.82%). Excluding this angular velocity (w = 3.49 rad/s), all data are normal and the one-way ANOVA test showed that the sensitivities did not showed significant difference (p = 0.1831), as well the hysteresis (p = 0.1042) and the angular error (p = 0.1841). In addition, the maximum angular error was of 1.31 • (0.77 • ), which corresponds to 2.62% of error, and the minimum angular error was of 0.78 • (0.58 • ), which corresponds to 1.56% of error. These results show that the sensor is repeatable and presents similar responses for this angular velocity range (0.70 to 1.75 rad/s) and as the velocity increases, the sensor performance decline. All angular velocities presented R 2 higher than 0.99. The mean of the cycles at each angle for all tested angular velocities presented low hysteresis, as shown in Figure 7, in which h is the hysteresis and ω is the angular velocity.
In addition to angular errors, the sensor responses presented high RMSE with maximum mean of 12.41 • (1.49 • ) and minimum of 6.45 • (0.43 • ). For this reason, the angular velocity compensation technique was applied in order to decrease the RMSE of angle measurements according to the angular velocity. Since the angular velocity range from 0.70 to 1.75 rad/s did not present significant differences, the angular velocity compensation technique was fitted for two groups: first group (angular velocities lower than or equal to 1.75 rad/s) and second group (angular velocities higher than 1.75 rad/s). The C 1 and C 2 blocks comprise the compensation models for two angular velocity groups. Equations (3)- (5) show the compensation models QS, C1 and C2 (see Figure 2) relating the optical power variation (P) with angle, respectively. QS(P) = 16.43 · e −0.0007·P − 16.40 · e −0.0061·P (3) C 1 (P) = 8.93 · e 0.0010·P − 9.75 · e −0.0080·P (4) C 2 (P) = 12.53 · e 0.0025·P − 12.84 · e −0.0091·P (5) Figure 8 shows the angle curves of characterized sensors responses in a quasi-static tests (uncompensated) and with the angular velocity compensation (compensated) compared to the servo motor input (reference) in one test, and Table 3 shows the RMSE of angle measurements for uncompensated and compensated response in each angular velocity.  The compensated responses presented a RMSE decrease up to 74% (w = 1.16 rad/s), with minimum error of 1.67 • (0.27 • ) for angular velocity of w = 1.16 rad/s. The higher angular velocity (w = 3.49 rad/s) presented two cycles with major differences from the others, resulting in RMSE 6.85 • (5.23 • ). However, excluding these two cycles the RMSE was 3.62 • (1.32 • ). Although the responses demonstrate worst performance in this angular velocity, the two different cycles may be related to servo motor performance at high velocities, presenting errors up to three times higher than the mean, which can be considered outliers.  Table 4 shows the results of uncompensated and compensated responses for the two gait applications.  The high errors of uncompensated responses are due to the variation of angular velocity, resulting in a different response from the characterization. When the angular velocity compensation was applied, according to the angular velocity of the curves, the response was more similar to the reference, with a decrease of 30 • angular error for p 2 , approximately.

Conclusions
This paper presented the performance analysis of a POF curvature sensor using CYTOP fiber under two conditions: different light source central wavelengths and different angular velocities. In addition, the sensor was applied on two approaches in the gait analysis: simulating knee and ankle patterns during the gait. The curvature sensor comprises of POF based on intensity variation principle, which present the lowest cost and highest simplicity in the fabrication and the signal processing. Results showed differences in sensitivity and hysteresis between light source central wavelengths, where best performance of λ = 950 nm was obtained through FoM evaluation. In addition, results showed high RMSE for different velocities, especially under w = 3.49 rad/s. The angular velocity compensation decreased the RMSE up to 74% and presented an improvement of responses in the gait simulating application, decreasing the error up to 86.37% (32.95 • ). Future works include the application of the sensor in a multiplexing technique to assess all joints simultaneously.
Author Contributions: L.A. and A.L.-J. designed and fabricated the proposed sensor and the experimental setup. L.M.A. was responsible for the sensors characterization, validation tests and data analysis. C.M. and A.F. analyzed the data and revised the paper. All authors were involved in the paper writing and revision of the paper. All authors have read and agreed to the published version of the manuscript.
Funding: This research is financed by CAPES (88887.095626/2015-01), FAPES (80605893, 85426300 and 84336650), CNPq (304192/2016-3) and Petrobras (2017/00702-6). C. Marques acknowledges FCT through the program UID/CTM/50025/2019 and by the National Funds through the Fundação para a Ciência e a Tecnologia/Ministério da Educação e Ciência, and the European Regional Development Fund under the PT2020 Partnership Agreement. This work is also funded by national funds (OE), through FCT-Fundação para a Ciência e a Tecnologia, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of 29 August, changed by Law 57/2017, of 19 July.

Conflicts of Interest:
The authors declare no conflict of interest.