Path Planning for Flexible Needles Using Second Order Error Propagation

In this paper we propose a computationally efficient method for the steering of flexible needles with a bevel tip in the presence of uncertainties for the case when there are no obstacles in the environment. Based on the stochastic model for the needles, we develop a new framework for path planning of a flexible needle with a bevel tip. This consists of three parts: (a) approximation of probability density functions for the needle tip pose; (b) application of a second order error propagation algorithm on the Euclidean motion group; and (c) application of the path-of-probability (POP) algorithm. The probability density functions are approximated as Gaussians under the assumption that the uncertainty in the needle insertion is fairly small. The means and the covariances for the probability density functions are estimated using the error propagation algorithm that has second order accuracy. The POP algorithm is adapted to the path planning for the flexible needles so as to give the appropriate steering plan. Combining these components and considering 5 degree-of-freedom targets, the new method gives the path of the flexible needle that hits the target point with the desired hitting direction.