# A Methodology for the Mechanical Design of Pneumatic Joints Using Artificial Neural Networks
## 1 Introduction  
In recent decades, in the field of robotics, human-machine interaction has increased with both the advent of collaborative robotics [1] and soft robotics [2]. Robots are now allowed to operate in shared workspaces with operators [3, 4], eliminating the need for dedicated spaces with safety barriers. Indeed, human-machine interaction has been extended into confined spaces [5] with, for example, path planning [6, 7] and emotional intelligence [8] algorithms. This extends the adaptability of robots even in unstructured environments, as it endows robots with unexpected capabilities compared to the past through high degrees of freedom, such as stretching [9], bending [10, 11], twisting [12], locomotion [13, 14, 15], jumping [16], swimming [17], climbing [18], and growing [19]. This is possible using silicone rubbers [20], which allow for soft and easily deformable actuators. The integration of these technologies can maximize the performance of a device, thanks to the integration of the advantages of each technology. For instance, utilizing the benefits mentioned for cobuts equipped with soft end-effectors enables the manipulation of delicate objects with extreme dexterity and security [21, 22, 23]. Several types of actuations have been proposed over the years [24], but the pneumatic one is the most adopted [25]. Indeed, it enables actuators to be made lightweight, safe, inexpensive, easy to manage, and with a high power-to-weight ratio. Actuators of this typeare known as soft pneumatic actuators (SPAs), and can be used without [26] or with an external [27] or internal [13] reinforcement to limit and guide their deformation. On the other hand, there are several disadvantages involving the complex identification of analytical models [28; 29] for designing SPAs, since they are affected by non-linearities due to air compressibility, silicone rubbers' viscoelastic behavior, hysteresis [30; 31], friction [32] between silicone and reinforcement when used, and large deformations. Over the years, analytical models have been formulated to describe the behavior of SPAs [33]. Models are often complex and require the identification of parameters. In addition, simplifying assumptions must usually be made or simple geometries must be used to describe and predict the actuator behavior. An approach for achieving a predictive model for SPAs is response surface methodology (RSM) [34; 35]. Firstly, it allows the planning of testing campaigns to minimize the resources used and the time required by identifying a predictive polynomial model of the analyzed response (e.g., bending angles of the SPA) based on the input parameters (geometric or functional parameters of the SPA) [36; 37]. Secondly, it aims to simplify complex analytical models that have already been identified: several simulations of an identified model are run, and with regression methods, the responses fit simpler models. Alternatively, static finite element analysis (FEAs), i.e., simulations with constant and uniform application of pressure to surfaces [13] or transient [38] with fluid-structure interaction, can be employed to predict SPA behavior. However, simulations are computationally expensive and time-consuming for numerical resolution. Integration between FEAs and RSM is helpful to minimize the number of simulations needed to characterize SPAs [39]. A data-driven approach involves machine learning algorithms that correlate input data (independent variables) with output data (dependent variable) without adopting simplifying assumptions for modeling the actuator. Artificial neural networks (ANNs) have been implemented to describe the workspace of SPAs [40; 41; 42]. In addition, ANNs can also control an SPA by continuously learning [43]. Approaches between FEAs and ANNs adopt data for training that are derived directly from the numerical results of simulations [44]. Alternatively, the accuracy of ANN predictions can be improved with a hybrid approach between model-free and an analytical model, called a physics-informed recurrent neural network [45].