In this paper we develop a new framework for path planning of flexible needles with bevel tips. Based on a stochastic model of needle steering, the probability density function for the needle-tip pose is approximated as a Gaussian. The means and covariances are estimated using an error propagation algorithm which has second-order accuracy. Then we adapt the path-of-probability (POP) algorithm to path planning of flexible needles with bevel tips. We demonstrate how our planning algorithm can be used for feedback control of flexible needles. We also derive a closed-form solution for the port placement problem for finding good insertion locations for flexible needles in the case when there are no obstacles. Furthermore, we propose a new method using reference splines with the POP algorithm to solve the path planning problem for flexible needles in more general cases that include obstacles.
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