# Photogrammetric Measurement and Analysis of the Shape Profile of Pneumatic Artificial Muscles
## 5 Analysis
### Volume Approximation  
Measurement of a PAM's internal volume is often required for actuation system design and modeling efforts. Estimates of volume are necessary to determine requirements of the system such as the working fluid flow rate and storage requirements. Modeling efforts also use values of internal volume for force-response models (e.g., virtual work method), as well as for providing estimates of actuation efficiency. The acquisition of accurate volume measurements has often proved difficult for researchers due to the specialized equipment that is often required to do so. As a consequence, researchers have often settled for using dimensions provided by using the estimated cylindrical approximation diameter. The photogrammetric data acquired in this research provides the opportunity to easily obtain an accurate estimation of the internal volume of the PAM for all tested pressure and contraction states. Since the profiles only provide a two-dimensional depiction of the PAM, a three-dimensional measurement of volume requires the assumption that the PAM is axisymmetric about its longitudinal axis. The symmetry of many previously tested PAMs usually makes this a reasonable assumption. Appreciable deviations from symmetry have only been observed for PAMs with flaws in their fabrication. Using the shape profile data, the internal volume of the PAM, \(V_{int}\), can be found using the following basic equation:  
\[V_{int}=V_{prof}-V_{bld}-V_{brd} \tag{11}\]where,  
\[V_{prof}= \begin{cases}w_{p}\sum_{i=1}^{L_{p}}\frac{\pi}{4}D_{i}^{2},&\text{discrete profile}\\ \\ \pi\int_{0}^{L}\Bigl{(}\frac{D(x)}{2}\Bigr{)}^{2}dx,&\text{continuous profile}\end{cases}  \tag{12}\]  
\(V_{prof}\) is the volume encased within the outline of the shape profile of the PAM, \(V_{bld}\) is the volume consumed by the incompressible bladder, and \(V_{brd}\) is the volume of the braid. Therefore, Equation (11) states that the internal volume of the PAM is equal to the volume within the shape profile minus the volumes of the bladder and braid that are also contained within the shape profile volume. The bladder and braid volumes are fixed values that can easily be found using their cylindrical dimensions in the PAM's resting length configuration (i.e., \(L_{0}\)). For the PAM used in this research, the braid has an assumed resting diameter of 21.9 mm (0.863 in) equal to that of the end-fitting at each end, and a thickness of 0.28 mm (0.011 in) resulting in a value for \(V_{brd}\) of 4.47 cm\({}^{3}\) (0.273 in\({}^{3}\)). The incompressible bladder has a resting outer diameter of 22.2 mm (7/8 in) and thickness of 1.6 mm (1/16 in), resulting in a bladder volume, \(V_{bld}\), of 24.26 cm\({}^{3}\) (1.481 in\({}^{3}\)). The volume \(V_{prof}\) can be found through integration of the defined shape profile (Equation (12)). The method of integration is dependent on how the shape profile is defined. In this paper, the diameter of the shape profile has been defined as a discrete function \(D_{i}\), with the width of the PAM defined for each pixel \(i\) along the length of the PAM, and as a continuous function, \(D(x)\), defined by the Fourier series fit equations used to describe the shape profile of the PAM. To find \(V_{prof}\) for the discrete definition of the shape profile, integration is performed as a summation of the volume of each pixel-width slice of the PAM. This discrete integration is shown in Equation (12), where \(w_{p}\) is the width of each pixel, and \(L_{p}\) is the current length (in pixels) of the PAM. The continuous function definition of the shape profile provides a much simpler method of calculating the volume of the PAM. As shown in Equation (12), disc integration is performed to calculate the volume \(V_{prof}\) using the continuous function form of the shape profile, \(D(x)\). The cylindrical approximated diameter of the PAM, \(D_{cyl,fit}\), having been shown to accurately represent the average diameter of the PAM, can also be used to calculate the volume \(V_{prof}\) of the PAM.