Shape Representation and Modeling of Tendon-Driven Continuum Robots Using Euler Arc Splines

Due to the compliance of tendon-driven continuum robots, carrying a load or experiencing a tip force result in variations in backbone curvature. While the spatial robot configuration theoretically needs an infinite number of parameters for exact description, it can be well approximated using Euler Arc Splines which use only six of them. In this letter, we first show the accuracy of this representation by fitting the Euler Arc splines directly to experimentally measured robot shapes. Additionally, we propose a 3D static model that can account for gravity, friction and tip forces. We demonstrate the utility of using efficient parameterization by analyzing the computation time of the proposed model and then, using it to propose a hybrid model that combines physics-based model with observed data. The average tip error for the Euler arc spline representation is $0.43$% and the proposed static model is $3.25$% w.r.t. robot length. The average computation time is $0.56 \,\mathrm{ms}$ for nonplanar deformations for a robot with ten disks. The hybrid model reduces the maximum error predicted by the static model from $8.6$% to $5.1$% w.r.t. robot length, while using 30 observations for training.