Piecewise constant strain kinematic model of externally loaded concentric tube robots*

This paper presents a piecewise constant strain kinematic model for concentric tube robots (CTR) in externally loaded conditions. It discretizes the pre-curved tubes comprising the robot into a finite number of pieces and involves external effects as a set of wrench vectors exerted along the robot backbone. Constant strain lets us describe the pieces with helices in which shear deformation and elongation are neglected. The resulting piecewise helix is the simplest curve that can catch the torsion of tubes that play a crucial role in kinematic behavior. This approximation transforms the conventional boundary value problem (BVP) of CTRs models into a set of nonlinear equations that drastically decreases the model resolution time. The present method uses a Lyapunov function and torsional Jacobian to ensure the distal torsion constraint consistently and, as a result, the solution?s convergence. The paper?s primary purpose is to present a fast, numerically stable, and relatively accurate kinematic model not reliant on measurement data. Experimental results on a two-tube prototype and provided for different tip loading conditions reveal maintaining a balance between adequate accuracy and reasonable running time, about 7 ms for five pieces per section, for real-time applications in the presence of external load.