# Photogrammetric Measurement and Analysis of the Shape Profile of Pneumatic Artificial Muscles
## 1 Introduction  
Pneumatic Artificial Muscles (PAMs) are a form of soft actuator that is applied to an expanding number of applications for their unique characteristics such as their low weight and simple construction, inherent compliance, and high specific force and specific work capabilities. In a design environment where optimization techniques are increasingly used for mechanism development and packaging, it is becoming more important than ever to have accurate model representations of device components, including actuators, to accomplish effective design outcomes. Actuator design requirements typically include force and stroke values, while common actuator design constraints include actuator size, energy efficiency, and storage requirements. Models have been developed which address each of these design constraints and requirements for PAMs, but many of those models lack the accuracy required for their use in a world of increasingly tight design spaces, and improvements to those models has remained stagnant for years. While continual improvements have been made in modeling the force response of PAMs, there has been little recent progress in the development of methods for characterizing the size and overall shape profile of PAMs during operation, parameters commonly used for calculating efficiency, fluid storage, and flow rate requirements. The PAM's shape profile (Figure 1) has long been approximated through visual observations, and has oftenbeen cited as a suspected source of modeling error [1; 2]. This paper seeks to remedy this by providing a new method of accurately measuring the PAM's shape profile. With the simple construction of an elastomeric bladder, a surrounding helically braided load-bearing sleeve, and rigid end-fittings at each end, a pressurized contractile PAM can provide a tensile force along its longitudinal axis. Upon internal pressurization of the PAM, the radial pressure force of the internal fluid is transferred through the bladder to the braided sleeve. The braided sleeve transforms that radial force into an axial force expressed at each end-fitting. The actuation force produced by a PAM is a function of its active length and internal pressure states. The actuation force is at a maximum at its resting length, when the braid angle, defined with respect to the radial axis of the PAM, is at a maximum. As the PAM contracts, the braid angle reduces and the bladder strain increases, resulting in a reduction of actuation force until the PAM reaches a pressure-dependent maximum _free contraction_ in its zero-actuation-force state. The shape profile of a PAM changes with contraction of its length (Figure 2a). At its resting length, the PAM has an apparent constant diameter and braid angle along its length. As a PAM contracts in length, its diameter expands and the braid angle reduces as a function of the position along the length of the PAM. The braid geometry is constrained only at each end-fitting where the diameter is equal to the initial diameter. The shape profile is also highly dependent on the PAM, with PAMs of identical material construction having visible differences in their shape profile (Figure 2b). PAMs have had shape profiles that range from a more cylindrical mid-section shape to ones that have constant variations in curvature along their entire length. The bulbous shape of a PAM is often approximated as a cylindrical profile with a constant diameter along its entire length (Figure 3). Assuming the braid is inextensible, the cylindrically approximated shape enables a simplification of analysis through the so-called _triangle relationship_ that correlates the PAM's diameter, braid angle, and active length. This _cylindrical approximation_ serves as the basis for many existing PAM models, including models for volume and actuation efficiency, as well as for force-response models that have been developed using both force-balance [3] and principle of virtual work approaches [4]. Figure 1: PAM with its braid traced along its edge (green), defining the shape profile of the PAM.