## 1 Introduction  
Steerable needles represent a new technology in intervention surgery; they can follow a curvilinear trajectory to safely avoid critical organs, while reaching a lesion inside a patient's body [1]. Steerable needles have been widely used for drug delivery, biopsy, and brachytherapy [2, 3, 4]. Thus far, needle-steering systems have encountered several challenges, such as the complex anatomical structure, disturbances resulting from tissue deformation and the movement of important organs, and trauma caused by the steering methodology. According to the statistics reported by [PERSON] et al., needle-steering techniques can be classified into six categories: needle tip actuation by applying an external moment at the needle base [5], a bevel-tip needle [6] that is deflected by tissue reaction forces, a steerable needle composed of concentric tubes [7], a needle with a tendon-driven tip [8], and a magnetic needle controlled using an electromagnetic system [9]. However, the above steering techniques have issues, such as insufficient driving force, tissue damage, and large dimensions. To overcome the aforementioned shortcomings, we proposed a curvature-controllable steerable (CCS) bevel-tip needle in a previous work [10]. The curvature of the CCS needle was mainly controlled by means of style translation, as illustrated in Figure 1, and the trajectory-following capability of the CCS needle was preliminarily tested in [11]. Yet, the steering system has to partially use the duty cycle algorithm [12] (a control strategy which periodically spins the needle shaft to adjust the trajectory curvature for a conventional level-tip needle), such motion causes additional damage to surrounding tissues. In addition, in the trajectory-following test, the movement of important organs and the target point were not considered. Motion planning involves finding feasible trajectories from the insertion point to the target point, while avoiding any collisions with obstacles [13]. The motion-planning task has been investigated extensively by many researchers and it can be a good solution to the existing issues affecting CCS needle-steering systems. [PERSON] et al. [14] represented the trajectory of a needle by using a curvature polynomial and identified its coefficients by using a gradient-based optimization method. In [15], an evaluation was conducted in a gelatin phantom using a programmable needle-steering system, the result showed this algorithm is only suitable for offline path planning. [PERSON] et al. [16] integrated the adaptive fractal tree algorithm with optimized geometric Hermite curves, and it was found that the resulting algorithm could plan trajectories under both curvature and heading constraints. However, the above algorithms did not consider the dynamic nature of the operating environment, and they can be used only for preoperative path planning. For online path replanning, [PERSON] et al. [17] proposed an algorithm based on A* that could use information from a previous path to perform iterative path optimization. [PERSON] et al. [18] developed a lifelong planning A* algorithm for replanning the path of a concentric tube robot; this algorithm improved working efficiency by avoiding the recalculation of the entire graph. The abovementioned graph-based methods are relatively simple, but they require finer environment discretization in three-dimensional (3D) space, which increases the computation time. [PERSON] et al. [19] formulated the planning problem as a Markov decision process based on an efficient discretization of the state space, with optimal steering actions computed using dynamic programming. Though motion uncertainty was considered, the study focused only on planar needle steering. Recently, neural networks have been used as a planning method for robotic systems [20]. In the context of steerable needle systems, [PERSON] et al. [21] implemented the GA3C algorithm for building steerable needle path planning models.