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Bevel-tip flexible needles have greater mobility than straight rigid needles, and can be used to reach targets behind sensitive or impenetrable areas. Accurately planning and executing the optimal motions for such steerable needles is difficult, however, and requires solving inverse kinematics for a nonholonomic system.This paper presents an approach to 3D motion planning for bevel-tip needles in an environment with obstacles. Instead of discretizing the configuration space as in earlier work, we discretize the control space, such that the trajectory of the needle can be expressed analytically without the need for approximate numerical simulation. This results in a fast optimization routine that finds a locally optimal path in a 3D environment with obstacles, requiring just a few seconds of computation time on a standard PC.We introduce two different discretization strategies that lead to differently structured paths and show that both produce valid trajectories from start to goal. To our knowledge, the presented method is the first to address motion planning for bevel-tip needles in a 3D environment with obstacles.

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Abstract:-Steerable needles can be used in medical applications to reach targets behind sensitive or impenetrable areas. The kinematics of a steerable needle are nonholonomic and, in 2D, equivalent to a Dubins car with constant radius of curvature. In 3D, the needle can be interpreted as an airplane with constant speed and pitch rate, zero yaw, and controllable roll angle.We present a constant-time motion planning algorithm for steerable needles based on explicit geometric inverse kinematics similar to the classic Paden-Kahan subproblems. Reachability and path competitivity are analyzed using analytic comparisons with shortest path solutions for the Dubins car (for 2D) and numerical simulations (for 3D). We also present an algorithm for local path adaptation using null-space results from redundant manipulator theory. Finally, we discuss several ways to use and extend the inverse kinematics solution to generate needle paths that avoid obstacles.