The Kinematics of Constant Curvature Continuum Robots through Three Segments

This letter presents an investigation into the mathematical relationships between the positions and orientations at the segment tips of a piecewise constant curvature (PCC) continuum robot with up to three segments. For one-segment, a reachability criterion is proposed, which simplifies the calculation of the neighboring orientation. For two-segments, a reachability criterion is proposed and the redundancy of its inverse kinematics solution is found, establishing a circle of tip locations. For three-segments, the redundancy of the inverse kinematics includes tips that lie on a sphere providing a closed-form solution to the inverse kinematics problem. These relationships are derived from the unique characteristics of the bisecting plane of a single segment. The degenerate cases for the solutions are also addressed. These outcomes stem from a specific PCC parametrization, with implications extending to the general PCC model. Note that this study is grounded solely in simulation.