# Photogrammetric Measurement and Analysis of the Shape Profile of Pneumatic Artificial Muscles
## 5 Analysis
### Shape Profile Analysis
#### 5.1.3 Comparison to Previous Cylindrical Profile Estimates  
The shape profile of PAMs has traditionally been modeled as a cylindrical profile. The diameter of the PAM can be estimated for all states of contraction by using this cylindrical approximation and an assumed initial geometry. The estimated profile diameter has long been assumed to be nominally accurate enough for use in almost all modeling efforts of PAMs. However, with no previous method of measuring the actual shape profile of PAMs, there has not been any quantifiable data to check the accuracy of these estimated cylindrical profiles until now. With the shape profile measurements acquired using photogrammetry, the accuracy of the cylindrical shape profile approximation can now be tested. The cylindrical approximation has served as the basis of most PAM modeling since the inception of the PAM back in 1958 [28]. The cylindrical approximation, prior to the addition of any applied correction factors, makes the following assumptions: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number of Terms** & **1** & **2** & **3** & **4** & **5** \\ \hline Avg. Max Error, mm & 12.70 & 1.49 & 0.69 & 0.68 & 0.39 \\ Avg. RMSE, mm & 2.74 & 0.39 & 0.12 & 0.12 & 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The effect of the number of Fourier series fit terms used for fitting the profile data on the maximum and Root-Mean-Square Error (RMSE) averaged between all tested pressures. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**\%L\({}_{0}\)** & \(a_{0}\) & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(b_{1}\) & \(b_{2}\) & \(b_{3}\) & \(w\) \\ \hline
0\% & 1.03 & \(-\)0.08 & \(-\)0.07 & \(-\)0.02 & 0.02 & 0.04 & 0.02 & 5.72 \\
8.75\% & 1.36 & \(-\)0.36 & \(-\)0.13 & \(-\)0.00 & \(-\)0.19 & \(-\)0.21 & \(-\)0.06 & \(-\)5.78 \\
17.5\% & 1.51 & \(-\)0.57 & \(-\)0.10 & 0.04 & 0.44 & 0.42 & 0.10 & 6.02 \\
26.25\% & 1.16 & \(-\)0.87 & 0.40 & 0.20 & 1.40 & 0.81 & 0.02 & 5.78 \\
35\% & \(-\)15.39 & 9.24 & 9.06 & \(-\)1.82 & 25.68 & \(-\)7.50 & \(-\)1.08 & 3.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coefficient values for the three-term Fourier Series fit performed in Figure 14a. The shape profile is cylindrical (constant diameter with respect to length)
* The braid angle is constant with respect to length
* There is no strain in the braid fibers  
With these qualifying assumptions, the geometry of the PAM can be defined using the _triangle relationship_ which defines a set trigonometric relationship between the active length, \(L\), diameter, \(D\), and braid angle, \(\theta\), of the PAM through fixed values that are the length of each braid fiber, \(B\), and the number of turns of each braid fiber around the circumference of the PAM, \(N\)[4]. The fixed values of \(N\) and \(B\), can then be found using the initial geometry of the PAM defined by the initial braid angle, \(\theta_{0}\), the initial diameter, \(D_{0}\), and the initial active length, \(L_{0}\), through the following relations:  
\[N=\frac{L_{0}}{D_{0}\pi tan(\theta_{0})} \tag{8}\]  
\[B=\frac{L_{0}}{sin(\theta_{0})} \tag{9}\]  
Finally, the diameter of the PAM can be estimated as a function of active length through the following relationship:  
\[D_{est}(L)=\frac{\sqrt{B^{2}-L^{2}}}{\pi N} \tag{10}\]  
The established direct relationship between \(D_{est}\) and \(L\) produces estimated diameter values--commonly used as an input parameter to models--through relatively simple geometric relationships. However, it is also relatively sensitive to the assumed initial geometry of the PAM. The value of \(D_{0}\) is commonly assumed to be equal to the resting outer diameter of the bladder (assuming a thin braid), or the braid diameter at its point of insertion into the end-fitting. The value of \(\theta_{0}\) can then be determined using the method covered in [PERSON], (2015) [29] which uses the linear relationship between blocked force and pressure to solve for \(\theta_{0}\).