Real-Time Shape Reconstruction for Clamped Homogenous Kirchhoff Rods and Dual-Arm Manipulation of Linear Deformable Slender Objects

Control and path planning of continuum robots, as well as robotic manipulation of slender deformable objects rests on a computationally efficient and sufficiently accurate estimation of the shape of the deformable elements, and elastic rods in particular. Ideally, such an estimation should be able to reconstruct the deformation field from the pose of the terminal ends of the rods only. Various approaches that use numerical shooting methods or collocation methods to solve the governing equations were proposed. Such methods aim at computing the shape exactly on the expense of significant computational complexity. In this paper, a highly efficient iterative algorithm is introduced for shape reconstruction of linearly elastic rods undergoing large deformations modeled with the Kirchhoff theory. The method is derived from a 3rd-order approximation of the exact solution. The rods are assumed to be homogenous and to have a constant cross section. The solution method shows an excellent accuracy while being highly efficient at the same time, which makes is applicable for real-time during operation and control.