Inverse Kinematics of Concentric Tube Steerable Needles

Prior papers have introduced steerable needles composed of precurved concentric tubes. The curvature and extent of these needles can be controlled by the relative rotation and translation of the individual tubes. Under certain assumptions on the geometry and design of these needles, the forward kinematics problem can be solved in closed form by means of algebraic equations. The inverse kinematics problem, however, is not as straightforward owing to the nonlinear map between relative tube displacements and needle tip configuration as well as to the multiplicity of solutions as the number of tubes increases. This paper presents a general approach to solving the inverse kinematics problem using a pseudoinverse solution together with gradients of nullspace potential functions to enforce geometric and mechanical constraints.