# Photogrammetric Measurement and Analysis of the Shape Profile of Pneumatic Artificial Muscles
## 5 Analysis
### Shape Profile Analysis
#### 5.1.2 Profile Fourier Series Fit  
Previous assumed shape profiles have failed to accurately capture the complex curvature of the PAM's shape profile. Piece-wise profile definitions used for attempted corrections on the cylindrical profile approximation--which typically use separate equations to define the profile of the midsection and the region near each end-fitting--increase modeling complexity with unvalidated improvements in accuracy. A single equation representation of the measured shape profiles is desired for simplicity of analysis.  
Curve fit equations are used to fit the measured shape profiles. With multiple forms of fit functions to choose from (e.g., polynomial, exponential, etc.), it is important to select a function with a characteristic shape that resembles the shape of the observed profiles to enable an accurate and low-order fit equation. Since the observed profiles draw a resemblance to a half-sine pulse with higher-order harmonics, a Fourier series curve fit equation was selected. The form of the fit equation is given as:  
Figure 14: Comparison of assumed profiles to the measured shape profiles of the PAM at 0, 8.75, 17.5, 26.25, and 35 (maximum) percent contraction at 689 kPa (100 psi): (**a**) Three-term Fourier series fit equation profiles. (**b**) Assumed cylindrical profiles that have diameters equal to the average diameter of the respective measured shape profiles.  
Figure 15: Initial analysis of the measured profiles for the full contraction ranges of 139–689 kPa (20–100 psi) tests: (**a**) Average diameter. (**b**) Internal air volume.  
\[y=a_{0}+\sum_{n=1}^{3}a_{n}cos(nxw)+b_{n}sin(nxw) \tag{7}\]  
where \(x\) represents the position along the length of the shape profile, \(a_{0}\) provides a horizontal shift, \(n\) indicates the order of the harmonic, and \(a_{n}\) and \(b_{n}\) are terms that scale the magnitude of each harmonic. Values of \(a_{0}\), \(a_{n}\), \(b_{n}\), and \(w\) are found for each measured profile by performing a nonlinear least-squares fit (using MATLAB's fit() function). The number of terms used for the fit is adjusted to achieve a good fit while also keeping the number of terms to a minimum for simplicity. A third-order Fourier series fit was found to achieve a very good fit of the shape profile data (Figure 14a). The resulting fit has a low combined average error for all tested states of contraction. The accuracy of the fit of this PAM begins to plateau with the use of three or more terms. Table 2 provides a comparison of the accuracy of the fit depending on the number of terms used. Values for each term of the three-term Fourier series fit performed in Figure 14a are provided in Table 3. Some of the coefficient values found for the fit functions, as shown in Table 3, can appear to be erratic, and may require additional constraints on the fit function for smoother variations of parameters between measured profiles.  
For a complete characterization, a fit function can be found for all of the profiles measured during tested states of pressures and contraction. As noted from previous observation of Figure 15a, the profile is predominantly a function of the contraction state with little variation with changes in pressure. This enables a reduction in the number of required fits to fully define the shape profile of the PAM by only needing to characterize the shape profiles of the maximum pressure tested.