Real-time position control of concentric tube robots

A novel approach to constructing robots is based on concentrically combining pre-curved elastic tubes. By rotating and extending the tubes with respect to each other, their curvatures interact elastically to position and orient the robot's tip, as well as to control the robot's shape along its length. Since these robots form slender curves, they are well suited for minimally invasive medical procedures. A substantial challenge to their practical use is the real-time solution of their kinematics that are described by differential equations with split boundary equations. This paper proposes a numerically efficient approach to real-time position control. It is shown that the forward kinematics are smooth functions that can be precomputed and accurately approximated using Fourier series. The inverse kinematics can be solved in real time using root finding applied to the functional approximation. Experimental demonstration of real-time position control using this approach is also described.