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.

中文翻译：

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柔性针转向和反馈控制的概率路径算法

在本文中，我们开发了一个新的框架，用于带有斜角尖端的柔性针的路径规划。基于针转向的随机模型，针尖姿势的概率密度函数近似为高斯。使用具有二阶精度的误差传播算法来估计均值和协方差。然后我们将概率路径 (POP) 算法应用于具有斜角尖端的柔性针的路径规划。我们展示了我们的规划算法如何用于柔性针的反馈控制。我们还为端口放置问题导出了一个封闭形式的解决方案，以便在没有障碍的情况下为柔性针找到良好的插入位置。此外，