Fourth-Order Accurate Strain-Parameterized Shape Representation of Beam Elements for Modeling Continuum Robots and Robotic Manipulation of Slender Objects

Soft rods and tubular elements are the main building blocks of continuum robots. Kinetostatic models along with a geometrically exact description of the kinematics on $SE(3)$ are now an established foundation for simulation and control. A key aspect is the reconstruction of the actual shape of a soft slender element. This involves numerically solving nonlinear differential equations on $SE(3)$⁠, which is problematic, in particular for real-time applications. To circumvent this, shape functions are used to approximate the deformation. A widely used approach is based on the constant curvature assumption. This has limited accuracy, however. In this article, an interpolation is presented that leads to a fourth-order accurate approximation of the deformation of a Cosserat beam. This serves as a strain-parameterized shape function. Either the strain at the two ends of the beam or the strain and its derivative at one end are specified. The presented interpolation is relevant also for shape control when handling flexible slender objects with robotic manipulators.