# Photogrammetric Measurement and Analysis of the Shape Profile of Pneumatic Artificial Muscles
## 1 Introduction
ease of computation. Only [PERSON] et al. [9] has noted that volume measurements appear to show that the cylindrical approximation may be sufficient for modeling the internal volume of the PAM. Our experience with PAMs has found that the overall shape profile of a PAM is variable, and often exhibits undulating curvature along the full length of the PAM. Having a quantitative method of measuring the shape profile of PAMs would be highly beneficial for improving the accuracy of models that rely on PAM geometry. Having a more accurate measurement of the shape profile would also aid in characterizing the PAM's internal volume which is often used in defining an actuator's system requirements such as the working fluid flow rate and storage volume requirements. First-cut estimations of internal volume are often made using the cylindrical approximation of the PAM, while end-effects corrections are sometimes included to account for the reduced volume near each end-fitting with unknown improvements to accuracy. Some researchers have turned to quantifying the internal volume through experimental approaches [9]. While experimental methods should provide accurate calculations of internal volume, they often require specialized test equipment. To date, experimental methods have required either submerging the PAM in water, or pressurizing the PAM using water from a closed graduated cylinder [9], and then tracking the change in water level as the PAM contracts in length. The specialized equipment required for characterizing the volume using these methods not only subject the PAM to fluid that its material composition may not be compatible with, but also requires adjustment, replacement, or even redesign of the test setup components depending on the size and length of the PAM to be tested. Conversely, an experimental method that can test PAMs of all sizes while not requiring specialized equipment is desirable. It is evident that previous PAM research efforts have relied on multiple different assumed shape profiles without consensus on which ones best fit the observed profiles, and without any conclusive measurement of their accuracy. This paper seeks to provide an experimental method that enables quantitative analysis of the shape profile of PAMs for improved accuracy in modeling efforts. In this research, photogrammetry, the science of making dimensional measurements of physical objects from photographic images [11], is adapted for use in the measurement of the PAM's shape profile. An overview of the photogrammetric methods applied in this research are presented first. A test setup for image acquisition is introduced followed by a method of image analysis to render the shape profile measurements from the acquired images. The photogrammetric methods are demonstrated through example by testing a 22.2 mm (7/8 in) diameter, 236 mm (9-9/32 in) long PAM. The results from testing the PAM are presented, followed by analysis of the PAM using the photogrammetric data. The obtained shape profile data is compared to the profiles assumed by the cylindrical approximation. The validity and accuracy of the cylindrical approximation is tested. If more detailed and accurate modeling forms are required, the development of a fit equation is presented for use that accurately replicates the curvature of the PAMs shape profile. Potential improvements to current modeling efforts are also presented, including a demonstration of how easily inaccurate initial geometric conditions can be applied to models without having photogrammetric data available to reference. The profile data is then used to present a simple and accurate method of characterizing the internal volume of PAMs. Finally, the ability of the photogrammetric test method to be universally adopted is demonstrated through shape profile characterizations of PAMs ranging in length from 236 mm to 508 mm, and diameters of 22.2 mm and 57.2 mm.