Real-Time Analytic Shape Reconstruction of Deformable Linear Objects modeled as Kirchhoff Rods in Static Equilibrium

Motion planning and control for robotic manipulation of deformable linear objects (DLO) necessitates computationally efficient and sufficiently accurate estimation of the DLO shape. Ideally, such an estimation should be able to reconstruct the deformation field from the pose of the terminal ends of the DLO only, which is particularly relevant for dual-arm manipulation. Numerical shooting and collocation methods accurately solve the associated boundary value problem, but are not applicable in time critical conditions. In this paper, a highly efficient algorithm is introduced for shape reconstruction of DLO undergoing large deformations. The method yields an explicit analytic representation of the displacement field. DLO are modeled as Kirchhoff rod since shear and compression can be neglected for the majority of relevant objects. The method is derived from a 3rd-order approximation of the exact solution. Assuming homogenous material and constant cross section, the shape estimation problem is reduced to a purely kinematic problem, which does not need material parameters. The solution method shows an excellent accuracy while being highly efficient at the same time. The result is equally relevant for continuum robots.