# Deflection Modeling and Curvature Manipulation of a Variable-Stiffness Flexible Needle  
Guozhao Wei  
School of Mechanical and Electronic Engineering, Shandong Jianzhu University, Jinan 250101, China; wgr_zinf@sdzju.  
The active needles designed for steering can be categorized as follows: precurved styles, programmable bevel needles, tendon-actuated needles, shape memory alloy (SMA)-actuated needles, and magnetically driven flexible needles. The four-probe programmable bevel steerable needle proposed by [PERSON] et al. [9,10] is inspired by the ovipositor of the wood wasp, and these four probes can be freely extended and retracted through an interlocking mechanism. The relative motion of the probes was used to change the steering of the flexible needles, and the offset length was used to adjust the bending curvature. In addition, they optimized the cross-sectional geometries of the catheter and employed thermal drawing technology to produce a new prototype with a diameter of 1.3 mm and a radius of curvature of 109.113 mm. [PERSON] et al. [11,12] designed a tendon-driven omnidirectional steerable stylet that was fabricated by micromachining a superelastic nitinol tube with an outer diameter of 1.17 mm and a wall thickness of 0.1 mm. With a maximum bending angle of 20\({}^{\circ}\), the steerable stylet is actuated by two nitinol tendons arranged at 180\({}^{\circ}\). Similarly, [PERSON] et al. [13] developed a flexible biopsy robot consisting of a tendon-driven catheter and an inner sliding flexible needle as well as their driving systems. The 4-degree-of-freedom catheter has a maximum bend of 140\({}^{\circ}\) and a positioning accuracy of 0.72 mm. The internal structure and constitution of the tendon-driven needles are usually complex due to the use of tendon wires or rods [14]. [PERSON] et al. [15] combined several soft and solid tubes to form an actively bending surgical needle and drove three 120\({}^{\circ}\) distributed SMA wires to realize deflection in all directions. The wires are looped at the needle tip, providing double bending force for the needle and facilitating electrical connections at the bottom end of the needle. [PERSON] et al. [16] developed a percutaneous robot that features a flexible wrist made of an SMA spring. At low temperature, the SMA behaves like a spring backbone that allows the needle to bend under tendon traction (up to 90\({}^{\circ}\)), thus minimizing the risk of cardiac perforation. In contrast, when heated, the SMA stiffens, allowing insertion into the pericardium for drainage. SMAs typically require a longer heating time to reach the phase transition temperature, which increases the puncture time. [PERSON] et al. [17, 18, 19] proposed a curvature-controllable steerable needle consisting of two parts: a cannula and a bevel-tipped stylet. The curvature of the needle's path is controlled by the offset between the bevel tip and cannula. Gelatin puncture experiments revealed a linear relationship between the curvature and the offset. They then conducted trajectory prediction and fuzzy control studies based on a kinematic model. However, analysis of the possible forces and factors that may be involved in the deflection mechanism seems to have been put on hold and is not clear.  
In this paper, a deflection model for a stiffness-tunable flexible needle under external force is established, and the method for manipulating the curvature is clarified. Two additional variable-stiffness needle structures are proposed and analyzed comparatively to evaluate cross-section effects on deflection. Then, simulation and puncture validation experiments are conducted to investigate the effects of puncture speed and flexible needle stiffness on bending curvature.