Concentric Tube Robots: Stability Analysis, Optimal Design, and Shape Sensing

Abstract

Minimally invasive surgery can involve navigating inside small cavities or reaching around sensitive tissues. Robotic instruments based on concentric tube technology are well suited to these tasks since they are slender and can be designed to take on shapes of high and varying curvature along their length. One limitation of these robots, however, is that elastic instabilities can arise when manipulating the robots by rotating or translating the bases of the tubes. As the tubes rotate and translate with respect to each other, elastic potential energy associated with tube bending and twisting can accumulate

if a configuration is not locally elastically stable, then a dangerous snapping motion may occur as energy is suddenly released.

To enhance the elastic stability of the concentric tube robots, this paper presents two researches: i) optimal design of tube pair, ii) local stability test to avoid unstable configurations. While prior work has considered tubes of piecewise-constant pre-curvature, the first research in this paper proposes varying tube pre-curvature as a function of arc length as a means to enhance stability. Stability conditions for a planar tube pair are derived and used to define an optimal design problem. This framework enables solving for pre-curvature functions that achieve a desired tip orientation range while maximizing stability and respecting bending strain limits. Analytical and numerical examples of the approach are provided. The second research provide a local stability condition and test to determine if a configuration is a stable equilibrium or not. This condition applies to arbitrary robot designs with any external loads. The local stability test based on this condition is validated by comparison with known stability results, and its utility is demonstrated by application to stable path planning.

Though those two researches address the elastic instability issue of concentric tube robots, they both are based on the theoretical kinematics of the robots. Robot control requires the rapid computation of this kinematics, which involves solving complex mechanics-based models. Furthermore, shape computation based on kinematic input variables can be inaccurate due to parameter errors and model simplification. An alternate approach is to compute the shape in real time from a set of sensors positioned along the length the robot that provide measurements of local curvature, e.g., optical fiber Bragg gratings. In this point of view, the third research in this paper proposes a general framework for selecting the number and placement of such sensors with respect to arc length so as to compute the forward kinematic solution accurately and quickly. The approach is based on defining numerically efficient shape reconstruction models parameterized by sensor number and location. Optimization techniques are used to solve for the sensor locations that minimize shape and tip error between a reconstruction model and a mechanics-based model. As a specific example, several reconstruction models are proposed and compared for concentric tube robots. These results indicate that the choice of reconstruction model as well as sensor placement can have a substantial effect on shape accuracy.