A Flexible Surgical Robot with Hemispherical Magnet Array Steering and Embedded Piezoelectric Beacon for Ultrasonic Position Sensing

This article proposes a flexible surgical robot featuring strong magnetic steering achieved by a hemispherical magnet array actuation, and high‐accuracy ultrasonic position sensing achieved by a beacon total focusing method (b‐TFM). The hemispherical magnet array with magnetic focusing is described and its array parameters are optimized through finite element analysis to increase the magnetic field for actuation. The magnetic field strength at 100 mm for the array with the same mass as the cylindrical magnet is about 1.8 times higher than that of the cylindrical magnet. Using the magnet array actuation, the flexible robot exhibits the capability of agile steering to navigate along a predefined trajectory. In addition, a 1 mm × 1 mm lead zirconate titanate (PZT) patch is embedded into the tip of the flexible robot as a beacon for b‐TFM ultrasonic imaging to detect the position of the robot. Therefore, the entire navigation process can be executed under the supervision of the ultrasonic position sensing system, and the maximum error is 0.8 mm when the steering radius is 100 mm.


Introduction
In the realm of minimally invasive surgery, surgeons can execute surgical procedures through either direct manual guidance or remote control of surgical robots, which subsequently manipulate end instruments. [1]Robotic systems with high precision and flexibility have the capacity to mitigate subjective factors, including emotion and fatigue of the surgeon, thus enhancing the stability of surgical procedures. [2]n contrast to conventional open surgery which necessitates sizable incisions to gain direct access to the target pathological structures, minimally invasive surgical robots employ extended rigid surgical instruments inserted into the target tissues through small incisions. [3]This approach facilitates a safe and expeditious execution of the surgical procedure while minimizing trauma to surrounding tissues, ultimately leading to a reduction in postoperative complications and pain.As an illustration, within the field of neurosurgery, surgical robots find application in tasks such as biopsy procedures and precise targeted drug delivery. [4]onetheless, employing a single surgical trajectory for rigid surgical instruments may potentially lead to unintended damage or perforation of adjacent normal tissues along the instrument's path. [5]esearch into flexible surgical robots aims to tackle the aforementioned challenges by amalgamating the precision control attributes of rigid surgical instruments with the adaptability afforded by flexible materials, enabling interactions with patients along a safer trajectory.Specifically, the application of an external magnetic field to manipulate a flexible structure equipped with permanent magnets at its tip has proven to be an efficacious control method within the domain of flexible surgical robots. [6] magnetically steerable needle featuring a flexible body is designed to access multiple targets within brain tissue, thereby adhering to intricate preoperative trajectories for instrument placement and lesion removal within the brain. [7]Moreover, a magnetic micro-driller is devised for precise nasal navigation in the management of primary acquired nasolacrimal duct obstruction. [8]Compared to utilizing only magnetic field to both steer and propel a magnetic tip, [9] using it solely for magnetic guidewire steering requires a smaller external magnetic field.
An ultrahigh-magnetic-field magnetic actuation framework has been developed to facilitate the operation of a magnetic guidewire within MRI scanners, enhancing the integration of magnetic control technology with clinical equipment.But the system may have limited applicability in surgical scenarios. [10]he provision of an external magnetic field to actuate a flexible robot can also be accomplished through the utilization of either permanent magnets or electromagnets.In comparison to electromagnets, the selection of permanent magnets as a magnetic source proves to be more easily controllable, cost-effective, and compact, requiring compact space. [11]Previous studies have employed multi-degree-of-freedom magnetic actuation based on permanent magnets in various medical applications, including remote magnetically operated neurosurgical interventional platforms, [12] magnetic endoscopy for colonoscopy procedures, [13] and magnetically guided catheter ablation for atrial fibrillation treatment. [14]Notably, permanent magnets offer distinct advantages, including the maintenance of constant magnetization strength and independence from the need for an electric current to generate the magnetic field. [15]onetheless, it's important to note that the magnetic field strength produced by permanent magnets does not exhibit a linear distribution.Instead, it follows an exponential increase towards the direction of the magnetic source. [16]Consequently, control mechanisms reliant on permanent magnets often encounter limitations, including a narrow operational range for the magnetic field and inherent control instability.In cases where precise positioning and control parameters are not meticulously configured, there exists the potential for medical incidents, wherein the micro-robot may inadvertently harm the patient during the treatment process.Achieving an optimal magnetic field distribution necessitates a thoughtful design approach involving modifications to the permanent magnet array, including adjustments in size, shape, number, and arrangement of the permanent magnets.Drawing inspiration from the Halbach array [17] and optimal Halbach permanent magnet designed to maximize the forces applied on magnetic nanoparticles at deep tissue locations, [18] the magnetic array introduced in this article offers the distinct advantage of generating a magnetic focus with smaller magnets.This results in strong magnetic field within the target area and facilitates a more straightforward magnetic control mode.
Furthermore, the operational procedures of flexible robots inherently entail several safety concerns, including challenges related to controllability and the absence of robust sensing capabilities. [19]Additionally, the fact that flexible materials often exhibit a nonlinear response to strain, [20] adds complexity to the simulation of flexible robots.Consequently, it is often imperative to incorporate visual feedback equipment [21] to enhance the applicability of surgical robots.External cameras can be employed for observing and controlling the distal position of magnetic flexible robots, [22] but they are not suitable for surgical scenarios.A ferromagnetic soft continuum robot manufactured using polydimethylsiloxane (PDMS) and neodymium-iron-boron (NdFeB) particles has successfully achieved miniaturization.It requires the use of X-ray imaging to provide visual feedback if used in in-vivo experiments, which may increase radiation risks to the patients. [23]In this article, we propose a novel approach to provide positional feedback during flexible robot navigation by integrating ultrasonic imaging.26] Therefore, these systems suffer from limited repeatability, low signal-to-noise ratios (SNR) and constrained field of view, often impeding clear target detection.For instance, a strategy utilizing ultrasonic image feedback for real-time magnetic navigation of an untethered micro-robot within a large working space is introduced. [27]Nonetheless, the ultrasonic images exhibited imaging artifacts, which limited operational accuracy.To address this challenge, this article endeavors to enhance imaging quality by bifurcating the process of ultrasonic signal transmission and reception.Specifically, we propose the integration of a piezoelectric beacon into the flexible robot for signal transmission, coupled with a separate device designated for signal reception, aimed at enhancing the imaging quality.
Figure 1a presents a magnetically controlled flexible surgical robot control system incorporating ultrasonic position sensing, where the flexible robot, a tethered permanent magnet structure is under the control of the surgeon.This innovative surgical paradigm finds applicability in neurosurgery for precisely tracking complex predefined pathways within brain tissue to reach the intended target, as well as in magnetic navigation intervention therapy for vascular diseases, as depicted in Figure 1b,c respectively.In this article, we undertake a comprehensive exploration of our control system, elucidating the operational mode of the ultrasonic position sensing module and the underlying principles of magnetic steering.Subsequently, employing finite element software, we conduct simulations involving various permanent magnet arrangements to identify a compact structure capable of generating an effective strong magnetic field.Additionally, we perform a comparative analysis of the performance of cylindrical magnets versus the permanent magnet array devised in this article.To validate our innovations, we proceed to fabricate a prototype of the magnetically controlled flexible robot alongside a compact magnetic array.Finally, we subject the robot to testing within a soft tissue phantom to ascertain the feasibility of the proposed magnetic navigation technology and the beacon total focusing method (b-TFM) ultrasonic imaging.

Design and Fabrication
The magnetically controlled flexible robot systemin Figure 1a comprises three core components: a flexible robot, a magnetic navigation module, and an ultrasonic position sensing module.
In Figure 2, we present the structure of the flexible robot featuring an integrated miniature piezoelectric ultrasonic beacon.This needle assembly comprises several key components: a Φ0.5 mm nitinol guidewire, a Φ2 mm needle tip with embedded magnet and piezoelectric beacon, and a Φ1.2 mm silicone sleeve serving as a seamless connection between the aforementioned parts, fully encasing the guidewire.The needle tip, with external dimensions measuring Φ2 mm Â 14 mm, is fabricated using metal 3D printing technology.Within its internal cavity (Φ1.2 mm Â 10 mm), resides a Φ1 mm Â 10 mm NdFeB cylindrical permanent magnet, which imparts an axial magnetic torque to the robot.The driving force propelling the robot forward or backward is provided by a wire feeder.The nitinol wire was utilized to enhance the mechanical support and pushability of the silicone sleeve.
As shown in Figure 2b, a 1 mm Â 1 mm Â 0.2 mm lead zirconate titanate (PZT) patch (TJ-45HD, TJ Piezoelectric Specialty Co., Ltd.) is integrated into the needle tip, employed as a piezoelectric beacon to generate ultrasonic waves.Its resonant frequency is determined by impedance analysis of the PZT patch using an impedance analyzer (IM3570, HIOKI, Japan) at room temperature, which is 2 MHz.A 5-cycle 2 MHz sine signal with a V pp of 24 V is applied to the piezoelectric beacon to generate ultrasonic waves.To capture the ultrasonic signals produced by the beacon, we employed a Φ0.8 mm movable ultrasonic probe (I2-6N, EasyNDT CO., Ltd, China) mounted on a UR robot as shown in Figure 1a.
The magnetic steering system primarily comprises two key elements: an external magnetic actuation and a built-in permanent magnet within the flexible robot.Traditional cylindrical permanent magnets function based on the fundamental principle of magnetic attraction between opposite poles and repulsion between like poles.By adjusting the orientation of the external permanent magnet, the robot can execute controlled steering at specific angles through attraction or repulsion, as illustrated in Figure 3a.In this article, we propose a permanent magnet array actuation based on Halbach array design (its design principle can be seen in 4. Magnetic Array Actuation Design Simulation).As shown in Figure 3b, by embedding a certain number of small cubic permanent magnets on the hemispherical surface and adjusting the attitude of each array unit,magnetic focusing can be realized on the array axis.When the array axis does not coincide with the axis of the flexible robot, the magnetic torque induces the motion of the built-in permanent magnet, steering the robot.When coinciding, the magnetic field focused on the axis constrains the flexible robot to be in the current state, so that the flexible robot can follow the motion of the permanent magnet array well.Compared with the traditional single large cylindrical magnet actuation, the magnet array proposed in this article has the advantages of focused strong magnetic field on the axis and flexible adjustment.The permanent magnet array is fixed at the end of the UR robot by a 3D printed fixture, which can provide the flexible robot with two degrees of freedom of motion: translation in the plane perpendicular to the axis.
Figure 4 provides an insight into the control framework governing this flexible robotic system's interaction with the human body.The surgeon guides the robot to steer by means of the magnetic actuation driven by the UR robot through a user interface.During the execution of predetermined procedures within the target tissue, the ultrasonic signal generation, acquisition, and processing (SGAP) unit controls the generation of a sinusoidal voltage signal applied to the beacon, thereby producing an ultrasonic signal.This signal propagates in the surrounding media and it is eventually acquired by the externally mounted ultrasonic probe and to be processed by the b-TFM algorithm to generate high-quality ultrasonic images from which the position of the beacon can be extracted.Based on the position information, the deviation between the actual and planned trajectory is determined and fed back to the surgeon in order to adjust the position and attitude of the flexible robot in real time for closed-loop "human-in-the-loop" control.

Magnetic Actuation Principle
According to Kelvin's formula, the magnetic force on an infinitesimal element (dV ) in a magnet with a magnetizing strength of ⃑ M at the strength of an external magnetic field is where ⃑ H ext denotes the external magnetic field strength.The total force on the magnet is obtained by integrating the differential force on the magnet volume V.
The built-in magnet of the flexible robot is a commercial cylindrical permanent magnet, which is uniformly magnetized and its magnetization strength is known.Therefore, in this system, the factor that affects the magnitude of the magnetic force is the strength of the external magnetic field.We can adjust the size, shape, number, and distribution of permanent magnets to obtain the external magnetic field for steering the flexible robot.

Ultrasonic Positioning Sensing Principle
In this section, an ultrasonic positioning sensing algorithm named b-TFM [28] is introduced.This algorithm is adapted from phased ultrasonic arrays imaging [29] in which the array elements act as both transmitters and receivers.In contrast to the TFM algorithm, the b-TFM algorithm leads to high accuracy, particularly when detecting small, point-like targets.
Figure 5 shows the schematic diagram of the piezoelectric ultrasonic b-TFM algorithm, and Figure 6 shows the block diagram of the b-TFM imaging process.Within the system, the piezoelectric beacon is securely positioned inside the flexible robot, serving as a beacon for ultrasonic transmission.As delineated in Figure 7a, a 5-cycle 2 MHz sine signal with a V pp of 24 V is applied to the beacon, inducing high-frequency vibrations and generating ultrasound waves transmitting into the surrounding space.
Here we use a single-point ultrasonic probe in continuous motion to capture the ultrasonic signals.This is achieved by sequentially moving the single-point probe to each receiver element's position with a constant pitch as shown in Figure 5.Each time the probe aligns with the target position, the beacon transmits a pulse signal, which the probe subsequently receives and transmits to the SGAP unit.Alternatively, this process could be achieved using an array of receivers without the need for the scanning motion.Figure 7b depicts the wave signal received by the receiver during a single ultrasonic detection event, while Figure 7c illustrates the corresponding filtered signal where the filtering can be seen to significantly improve the SNR.
In the post-processing of the aforementioned ultrasonic data, the b-TFM algorithm initiates by segmenting the target imaging  area into a grid.Subsequently, it calculates the distances between the grid pixels and the array elements, employing the x and y coordinates of the center of each pixel.These distances are then divided by the speed of sound within the measurement medium, yielding the propagation delay.The computer orchestrates ultrasonic image by aggregating the time-domain data set stemming from the transmitter-receiver pair.Leveraging a collection of recorded ultrasonic data emitted by the beacon and received by receiver m, the intensity at each pixel within the imaging area is determined by the following expression iðx, yÞ ¼ where x R m , y R m denotes the coordinates of the receiver m, hðtÞ T,R m denotes the Hilbert transform of the processed timedomain signal, and an example of the processed signal is shown in Figure 7d.By amalgamating data from all receivers, the resultant pixel intensity values within the designated target region are aggregated into a matrix, mirroring the grid's dimensions within the imaging area.The peak intensity within this matrix   www.advancedsciencenews.com www.advintellsyst.comcorresponds to the position of the beacon, as demonstrated in the ultrasound image presented in Figure 6.This serves as the basis for calculating the position of the flexible robot.

Magnetic Array Actuation Design
As previously mentioned, this article explores two distinct actuating modes for magnetic navigation: the cylindrical permanent actuation and the permanent magnet array actuation.In the subsequent sections, we delve into intricate explanations of the actuating principles and comprehensive assessments of the operational performance of these two modes.

Cylindrical Permanent Magnet Actuating Mode
The magnet actuation within the magnetic navigation system, as depicted in Figure 1a, comprises of four Φ80 mm Â 20 mm N52 cylindrical permanent magnets stacked axially.To provide a comprehensive comparison, we analyze its magnetic flux density mode in contrast to that of a single Φ80 mm Â 20 mm N52 permanent magnet using finite element analysis, as illustrated in Figure 8.In the figure, the white lines represent magnetic susceptibility lines, while the red arrows indicate the direction of the magnetic field.
Upon stacking the four magnets, the radial magnetic field exhibits minimal change, whereas the axial magnetic field experiences a substantial enhancement.This augmentation enables the flexible robot to effectively engage in magnetic navigation over extended distances.Nevertheless, it's important to note that the enhancement of the axial magnetic field does not exhibit exponential growth in tandem with increasing magnet volume.
In practical application, the flexibility of the needle encounters mechanical constraints and tissue reaction forces at the sleeve-tip joint.To ascertain if the cylindrical permanent magnet can deliver adequate magnetic torque for steering the flexible robot, we conducted a thorough experimental characterization of the cylindrical magnet's performance utilizing a Tesla meter.Figure 9a showcases the magnitude of the magnetic field strength at angles of 0°, 30°, and 60°relative to the axis of the permanent magnet.The surface of the Tesla meter sensor is perpendicular to the schematic line indicated in the figure .Remarkably, all three trends depict similarities.At a distance of 20 mm from the permanent magnet, the magnetic field strength can peak at 250 mT.However, as the distance exceeds 140 mm, the magnetic field strength gradually diminishes, eventually approaching zero.In order to achieve precise control of the external actuation magnet in a flexible manner, the axis of the permanent magnet is strategically aligned with the intended working direction.
Subsequently, agar gel (0.4% concentration) was employed to experimentally assess the response of the needle tip when subjected to varying strengths and orientations of magnetic fields within the flexible robotic system.We systematically set the angle (α) between the axis of the cylindrical magnet and the flexible robot at values of 0°, 30°, 60°, and 90°, respectively.Taking into consideration the magnetic field strength distribution at various positions of the permanent magnet (as depicted in Figure 9a), we meticulously adjusted the distance and orientation between the external permanent magnet and the flexible needle with the aid of the UR robot.Then we applied magnetic fields in attraction and repulsion modes, each having magnitudes of 20, 40, 60, and 80 mT to the flexible robot.These magnetic fields were designed to evaluate the behavior of the flexible robot under different magnetic forces.Upon halting the motion of the flexible robot, we measured the deflection angle (θ) of the needle tip axis from its original position, observing the system's response in a stable state.
The outcomes are plotted in Figure 9b, where positive values of the deflection angle (α) signify an attraction mode, while negative values indicate a repulsion mode.As depicted in the figure, there exists a direct relationship between the magnitude of the applied magnetic field and the deflection of the needle tip.Notably, the deflection becomes more pronounced when θ is set at 90°.Furthermore, our findings reveal that when the cylindrical permanent magnet interacts with the flexible robot, the attraction mode yields a significantly more pronounced deflection effect compared to the repulsion mode.Consequently, the attraction mode proves to be more manageable and favorable for practical applications.Examining the data within the figure, it becomes evident that, within the attraction mode, noticeable deflection of the needle tip only occurs when the magnetic field strength reaches 30 mT.This implies that the operational range of permanent magnets is somewhat limited in actual magnetic navigation scenarios.

Magnetic Array Actuating Mode
As illustrated in Figure 8, when employing the same magnet material, a larger magnet size yields more substantial magnetic potential and consequently generates higher magnetic field strength.However, the mere superposition of large cylindrical permanent magnets does not proportionally increase the magnetic field strength.This approach results in inefficient use of the magnetic field potential and is, therefore, suboptimal.In this article, we introduce an innovative structure involving multiple small permanent magnets arranged in an array configuration, and optimize the size, shape, number, and arrangement of these permanent magnet elements to concentrate and maximize the magnetic field along the central axis of the array.The result is a more efficient and focused magnetic field, offering superior performance compared to the conventional approach of using larger magnets.
The Halbach magnet arrangement method, a well-established approach originally designed for linear motors, is characterized by its ability to manipulate the magnetization direction of permanent magnets to achieve magnetic field focusing.Our magnet array is founded based on the foundational principles of the Halbach array.Adjustments are made without altering the magnetizing direction itself, rather, it involves changing the orientation of the permanent magnets as illustrated in Figure 10a.This configuration comprises five 10 mm cubes, with the black arrow direction indicating the magnetization orientation of the permanent magnet.The resulting magnetic array effectively concentrates the magnetic field on one side of the array to a significant extent.Comparing this array configuration with an equivalently sized bar magnet (depicted in Figure 10b), it becomes evident that the magnetic field is stronger along the central axis.This enhancement results in a 30 mT magnetic field coverage that can extend up to 57 mm, marking a 1.3-fold improvement in magnetic field strength when compared to the bar magnet of equivalent volume.
The linear arrangement indeed enhances the magnetic field on one side of the array, however, there is potential for further improvement in magnetic field focusing.To establish a stable magnetic focus within the target workspace, it is advisable to rearrange the magnets towards a hemispherical distribution.This adjustment aims to augment the magnetic field strength not only along the central axis but also at the center of the entire array, thereby optimizing magnetic field concentration and effectiveness.
Figure 11a illustrates the schematic diagram of the hemispherical array configuration, with each permanent magnet having a side length denoted as "a".The sphere is preset with a radius of 6a, and the angle (β) between the center of each adjacent permanent magnet and the line connecting the sphere's center is defined.The permanent magnets are arranged in layers across the sphere, and the angle between the magnetization direction of the permanent magnets in the "P" layer and the sphere's center is "(PÀ1)β ".Moving on to Figure 11b, we present a schematic depiction of the magnetic field focusing achieved by a hemispherical Halbach array.This illustration highlights the generation of magnetic focusing precisely at the spherical center and along its central axis.The parameters of this magnetic array are meticulously determined through finite element simulation, ensuring precise and optimized performance.
Considering the fundamental characteristics of permanent magnets, the magnetic field strength within an array is directly proportional to both the number of layers and the quantity of permanent magnets contained within each layer.To further investigate the impact of the angle β on array performance, we conducted simulation with four layers of permanent magnets situated on a hemispherical surface.These layers were configured with magnet counts of 1, 4, 8, and 12, respectively.In Figure 12, we present the flux density patterns of multiple hemispherical arrays at two different angles, namely 14°(Figure 12a) and 18°(Figure 12b).When β is set at 14°, the magnetic field strength at the focal point reaches 30.5 mT, whereas at an angle of 18°, it attains 24 mT.Our findings clearly indicate a negative correlation between the magnetic field strength at the focal point and the angle β.In simpler terms, when all other conditions remain constant, a smaller β results in a more tightly packed arrangement of permanent magnets within the array, thereby enhancing the magnetic focusing effect at the focal point.
When a stronger magnetic field is required, increasing the number of layers on the hemispherical surface of the array becomes a viable approach to generate a more potent magnetic field along the central axis.For instance, in the realm of neurosurgery, this array of permanent magnets can be employed to envelop the brain externally, harnessing the magnetic field for intracranial flexible robotic navigation.Given that the human head has a circumference ranging from approximately 54 cm to 58 cm, with a diameter of 18 cm or less, we can consider a permanent magnet unit with a = 20 mm.The angle between adjacent layers is set at 14°, and we sequentially augment the number of layers, ensuring the inclusion of as many permanent magnets as possible in each layer.This expansion results in the creation of a hemispherical array.However, due to size constraints, the hemispherical array can accommodate a maximum of seven layers of permanent magnets.In this configuration, layers P = 2 to P = 7 contain 4, 12, 18, 24, and 30 permanent magnets, respectively.Figure 12c,d visually depict the magnetic flux density patterns for layers P = 5 and P = 7, respectively.
Table 1 presents an overview of the cylindrical magnet and the hemispherical array's properties with varying numbers of layers in relation to magnetic field performance.It includes an analysis of the array's mass, radius, magnetic field strength at focal point, and the extent of coverage of the 30 mT magnetic field when considering layers P = 4 through P = 7. (We utilize the criterion of reaching a magnetic field strength of 30 mT as an indicator of effective deflection capacity for the flexible needle according to Figure 9.) Based on the simulation data, as the layer number P increases, both the magnetic field strength at focal point and the coverage of the 30 mT magnetic field expand.When P = 4, the magnetic field strength at the focal point for the array with the similar mass as the cylindrical magnet is about 1.8 times higher than that of the cylindrical magnet.Additionally, the 30 mT magnetic field coverage is 1.4 times larger compared to that of the cylindrical magnet.When P = 7, the magnetic field strength at 100 mm along the axis reaches 104 mT, and the coverage of the 30 mT magnetic field extends to 159 mm.Consequently, in clinical settings demanding elevated magnetic field strength, such as when dealing with hard piercing mediums, the magnetic array can be employed to enhance the force exerted by the built-in magnet of the flexible needle.

Experimental Testing of the Magnetic Array's Actuation Performance
The magnetic arrays depicted in Figure 12c,d exhibit a dense arrangement of magnets, resulting in substantial interaction forces between the permanent magnets.This characteristic poses challenges in terms of practical implementation within a laboratory setting.Therefore, for validation purposes, we have opted for the structure presented in Figure 12a to assess its real-world performance against the finite element simulation results.The magnet fixture is constructed based on an array structure design and manufactured using 3D printing, as illustrated in Figure 13a.The permanent magnets are carefully arranged, starting from the P = 1 layer, and precise magnetization direction is ensured.Throughout the assembly process, the magnets are affixed using hot melt adhesive to secure them in their designated positions, preventing any unintended shifts resulting from attraction or repulsion between the permanent magnets.The demagnetization temperature of NdFeB permanent magnets exceeds 90 °C, whereas the hot melt adhesive operates within the temperature range of 70-85 °C and can rapidly cool and solidify without impacting the magnets' performance.Following assembly, the fixture cover and fixture body are securely connected using screws to prevent dislodgment of the hot melt adhesive or deformation of the array due to gravitational forces.The 30 mT magnetic field coverage of the magnetic array extends to approximately 52 mm along the axis.In contrast, when using the same small magnets with a = 10 mm, either individually or grouped in cubic and planar arrays within the same 52 mm span, the resulting magnetic field strengths are considerably lower as illustrated in Figure 13b,c, measuring 6.5 and 3.5 mT, respectively.So the proposed magnetic array increases the magnetic field strength by about 4.6 times.
The actual magnetic field distribution of the hemispherical magnetic array was experimentally assessed using a Tesla meter, and the measurements are presented in Figure 14.Along the center axis of the array, the magnetic field strength gradually diminishes as the distance from the bottom magnet increases, with the rate of decay exhibiting an initial rapid decline followed by a slower one.To gauge the magnetic force, a N52 NdFeB cubic permanent magnet with a = 10 mm was selected and positioned at various locations along the array axis for measurement.A force Table 1.Comparison of the properties of hemispherical arrays with different layers and the cylindrical magnet.The side length "a" of the magnetic array unit is 20 mm.The angle "β" between adjacent layers is set at 14°. sensor was employed for this purpose.During the measurements, the magnet's bottom surface was situated at distances ranging from 50 to 100 mm from the central axis.The results, displayed in Figure 14b, reveal a positive correlation between the force exerted by the permanent magnet and the magnetic field strength.Subsequently, we conducted measurements of the magnetic field strength within the plane where the focal point is situated.This plane runs parallel to the bottom magnet's surface and is positioned 55 mm away from it.Given the symmetrical distribution of the permanent magnet array, we focused our measurements on Line 0°and Line 45°, as depicted in Figure 15a.At intervals of 10 mm from the focal point, we recorded magnetic field strength values.Experimental data indicates that the magnetic field generated by the array within the focal plane is symmetrically distributed overall.Notably, the magnetic field is strongest at the focal point and gradually decreases as one moves farther away from it.As illustrated in the figures, the magnetic field strengths on Line 0°and Line 45°exhibit slight variations, which can be attributed to the distribution angle of the permanent magnets within the array.Nevertheless, the disparity between the two is negligible, suggesting that the magnetic field strength remains roughly consistent along the circumference at the same radius.
Based on the acquired data, the magnetic field is intensified at the focal point and along the central axis, aligning with the magnetic field pattern depicted in Figure 11b.The magnetic induction within this array follows a path from the periphery toward   the central axis, effectively achieving magnetic focusing.Therefore, when employing the permanent magnet array for magnetic navigation, if the flexible robot deviates from the axis, the array's magnetic field imparts the necessary magnetic torque for steering the built-in magnet.Conversely, when the flexible robot approaches the axis, the focused magnetic field guides it toward the central axis, showcasing a commendable tracking effect as the array moves, as illustrated in Figure 15b.Consequently, this enhances the ease of actuation control in the magnetic navigation process while achieving an effective magnetic focusing effect.

Magnetic Steering Experiment
To illustrate the primary functionalities of the flexible robot and the hemispherical magnetic array devised in this study, we constructed a restricted magnetic navigation environment, as depicted in Figure 16a.The ring paths were used to approximate the surgical trajectory, illustrating the capability of the robotic system to navigate through tortuous pathways with flexibility.Multiple circular rings with a diameter of 10 mm are intricately connected to a square plastic sheet with a side length of 13 cm through threaded fastenings.The vertical distance from the center of each ring to the surface of the sheet is configured to be 20 mm.To enhance clarity, specific coordinates for each path point are shown in Figure 16c.For experimental demonstration, we employed the previously manufactured hemispherical array as a magnetic actuation, supplying a magnetic field to steer the flexible robot through three sets of distinctively shaped rings within an air medium.The rigid needle-shaped tip has a certain impact on the flexibility of the flexible robot to steer along a tortuous path.Generally, the shorter the tip, the better the flexibility.In this study, to accommodate the magnet and piezoelectric beacon, the tip length is set at 14 mm, which can satisfy path navigation where the curvature changes are not very significant.
The user manipulated the array's orientation using a UR5 robot (Universal Robots Co., Ltd, Denmark), which allowed steering the flexible robot toward the array's axis, thus bending it toward the desired direction.The magnet array was securely affixed to the UR5 robot's flange using the dedicated fixture.Additionally, a rigid nitinol wire enclosed by a silicone sleeve was managed manually to facilitate forward propulsion for the flexible robot.As demonstrated in Figure 16b-i, the magnetic field efficiently steered the flexible robot, enabling it to navigate through circular paths resembling "Z", "S" and "Y" shapes.Moreover, the results show that the applied magnetic field strength can make the flexible robot execute effective turns while driving along a tortuous path.

Obstacles Avoidance Experiments
To enable navigation of flexible robots within soft tissues, we endeavored to control the translation and rotation of the external magnet array, thereby adjusting the position and orientation of the flexible needle.As the flexible robot advances, the user employs the teach pendant to make real-time adjustments to the magnetic array at the UR robot's end, allowing for immediate corrections.To simulate soft tissues for experimental purposes, we utilized a transparent hemispherical silicone mold with a 60 mm diameter, illustrated in Figure 17a.The mold was filled with agar gel (0.4% concentration), which solidified into a colloidal state, mimicking the properties of soft tissues.The hemispherical soft tissue model was designed to effectively showcase the magnetic steering characteristics of the optimized array.Due to the small size of the array, the hemispherical model reduces the acting distance between the array and the flexible robot.
Since the agar gel is transparent, the motion of the needle tip can be observed visually, as shown in Figure 17b,c.It was found to steadily achieve path bending utilizing the hemispherical array to steer the flexible robot.This indicates that the magnetically controlled flexible robot can fulfill the trajectory bending as expected and realize the obstacle avoidance function.Expressed in a clinical environment, the flexible robot can avoid tender non-target tissues during surgery and can reach multiple targets on a preset path to accomplish relevant tasks.However, since the experiment was conducted in a relatively small model, the curvature of the bending path is small, and the stiffness of the nitinol wire is large, which makes the robot path deform to some extent and deviate from the original position in the process of moving forward.However, this problem is not obvious in large curvature bending.And it can be improved by optimizing the mechanical model of the flexible robot.

Magnetic Navigation and Ultrasonic Positioning Sensing Experiments
Three arcs of curvature with radius of 80, 90, and 100 mm were plotted on the reference coordinate paper as the preset paths, which were placed under the agar gel (0.4% concentration).In Figure 18, we present the experimental setups used to showcase magnetic navigation and ultrasonic positioning sensing capabilities.The flexible robot was introduced into the agar gel from a common starting point shared by all three arcs.Throughout the operation, the trajectory of the needle tip's movement was determined through visual observation.The magnetic force was utilized to steer the needle-shaped tip.The user manually controlled the orientation of the external permanent magnet array through the teach pendant, thus changing the position of the flexible robot through magnetic field.When performing the twodimensional plane navigation of the flexible robot, the axis line of the magnetic array should be kept horizontal with respect to the experimental platform.
A wire feeder was employed to provide the driving force for the forward and backward movement of the flexible robot.The wire feeder was driven by a stepper motor (CTM28-0601), with a ball screw length of 200 mm and a pitch of 1 mm.A fixture clamped onto the wire feeder secured a silicone sleeve containing the nitinol wire, providing the driving force for the flexible robot.Higher propulsion speeds might result in bending of the nitinol wire and silicone sleeve, affecting navigation accuracy.And the system's propelling speed was set at 1 mm s À1 .
After reaching the first target, the steering robot was retracted to the starting point, and this process was repeated for all three different curvatures.between the actual path of the needle and the preset path is shown in 19a-c respectively.During navigation with a small curvature, the flexible robot exhibited a more noticeable error, with a maximum error of approximately 2 mm.Ultrasonic positioning sensing was carried out during magnetic navigation.The generation and data acquisition of ultrasound signals were performed using the portable multifunctional unit HS5-540XM (TiePie Engineering CO., Ltd, The Netherlands).As shown in Figure 18, the ultrasonic probe was secured in place using a fixture attached to the flange of the UR5 robot.Throughout the experiment, the UR5 robot systematically moved the probe to predefined positions along the gel surface, simulating the function of a receiver array.Control of the UR5 industrial robot and its actuation system was managed by a computer.During the ultrasonic detection phase, the PZT patch embedded within the needle tip was stimulated to transmit ultrasonic into the surrounding space.The ultrasonic probe captured and digitized ultrasonic signals that were subsequently recorded by the SGAP unit.Each cycle at a single array position and moving the probe to the next position took about 1 s, and the complete position sensing process required approximately 70 s in total.In the future, the adoption of a compact array probe will replace the single-point ultrasonic probe, thus achieving real-time imaging.
The signals were collected at four separate positions.During ultrasonic position sensing, the wire feeder ceased operation and was restarted once the flexible robot's position was detected.Figure 20a displays the result of ultrasonic position sensing at a certain point during the robot's navigation, while Figure 20b-d illustrate the combined outcomes of the four positions superimposed on each other as the robot navigates with varying curvature.The depicted area is illustrated in decibels where 0 dB is the signal maximum, and the centroid of the maximum intensity zone (highlighted in red) serves as the center position of the beacon.This enables the calculation of the needle tip's positional status, providing crucial feedback to the user for real-time motion trajectory adjustments.To evaluate the detection error of ultrasonic positioning sensing, we compared the beacon coordinates in the ultrasonic image with the actual trajectory.The maximum position detection error was approximately 0.8 mm when the steering radius was 100 mm.These results underscore the effectiveness of the ultrasonic positioning sensing method outlined in this article and its potential for closed-loop control in the absence of visual guidance during in-vivo experiments.

Conclusion
In this study, a magnetically controlled flexible robotic system equipped with ultrasonic position sensing utilizing a piezoelectric beacon is introduced.Initially, an ultrasonic detection scheme named b-TFM is present, which can enhance imaging quality by segregating ultrasonic signal transmission and reception processes.Subsequently, the arrangements of permanent magnets capable of efficient magnetic focusing are simulated based on the Halbach array principle.We find that the magnetic field strength at 100 mm for the hemispherical magnetic array with the same mass as the cylindrical magnet is about 1.8 times higher than that of the cylindrical magnet.Then a prototype of a hemispherical magnetic array is fabricated and its performance is rigorously tested.Finally, a prototype of the flexible robot is manufactured and an experimental platform is established to  assess the robot's magnetic navigation capability.The experimental results affirm that the magnetic control and ultrasonic position sensing methods proposed in this study significantly assist the flexible robot in accomplishing steering and obstacle avoidance tasks.Specifically, during navigation with small curvature under visual guidance, errors are easily generated, with a maximum error of approximately 2 mm.The maximum ultrasonic detection position error is about 0.8 mm, and the detection results can be relayed to the control system to mitigate magnetic navigation errors.
Future research endeavors encompass the fabrication of expansive magnetic arrays for navigation experiments within biological tissues and the exploration of expeditious ultrasonic imaging strategies for flexible robots to attain real-time closedloop "human-in-the-loop" control of the magnetic navigation process.

Figure 1 .
Figure 1.a) Schematic diagram of a magnetically controlled flexible robot system incorporating ultrasonic position sensing.Flexible robot for b) Neurosurgical path tracking and c) Intravascular navigation.

Figure 4 .
Figure 4. System control schematic diagram consisting of five main parts: surgeon, user interface, magnetic unit, patient, and processing unit.

Figure 3 .
Figure 3. Schematic diagram of the principle of magnetic steering system.a) Cylindrical magnet operating mode.b) Permanent magnet array operating mode.

Figure 5 .
Figure 5. Schematic diagram of the piezoelectric ultrasound beacon total focusing method.A set of signals is recorded between the beacon and the array element (receivers) to create a complete data matrix.

Figure 7 .
Figure 7. Signals in b-TFM ultrasound imaging.a) Excitation signal of the piezoelectric beacon.b) Time-domain raw signal received by the array element.c) Filtered signal.d) Absolute amplitude value of the filtered signal after Hilbert transform.

Figure 8 .
Figure 8. Flux density model of the cylindrical permanent magnets.a) Single block.b) Four blocks stacked.

Figure 9 .
Figure 9. Experimental characterization of the cylindrical permanent magnet's performance.a) Magnetic field strength distribution at different positions of the permanent magnet.b) Deflection angle of the flexible robot corresponding to different magnetic field conditions in the agar gel.

Figure 10 .
Figure 10.Flux density mode.a) Linear array.b) Same volume bar magnet.

Figure 11 .
Figure 11.a) Hemispherical array design schematic.b) Schematic depiction of the magnetic field focusing achieved by a hemispherical Halbach array.

Figure 14 .
Figure 14.a) Axial magnetic field strength test.b) Induced force on the permanent magnet in the axis of the array.

Figure 15 .
Figure 15.a) Magnetic field strength in the plane where the focal point is located.b) Schematic diagram of the flexible robot following the motion of the array.

Figure 16 .
Figure 16.a) Magnetic array actuation steering the flexible robot in restricted environments.The separation distance, denoted as "h", between each column of circular rings is set to 35 mm.The flexible robot navigated through circular paths resembling b,c) "Z", d-f ) "S", and g-i) "Y" shapes.
Figure 19 illustrates the experimental figures for magnetically controlled steering.On each figure, there are three curves, and the flexible robot navigates along paths with curvatures of 80, 90, and 100 mm, respectively.The deviation

Figure 17 .
Figure 17.Experiment of flexible robot navigation in soft tissues.a) Experiment setup.b) The flexible robot steers in soft tissue guided by the magnetic field.c) The magnetically controlled flexible robot can realize the obstacle avoidance function.

Figure 18 .
Figure 18.Experimental platform for magnetic navigation and ultrasonic positioning sensing.

Figure 19 .
Figure 19.Experimental figures of a flexible robot executing curvature bending in soft tissue based on visual cues.The three preset paths have radii of a) 80 mm.b) 90 mm.c) 100 mm.

Figure 20 .
Figure 20.Ultrasonic positioning sensing results.a) An ultrasound positioning sensing result at a certain point.Superimposed ultrasound positioning sensing results of four positions when the robot navigated along paths with curvatures of b) 80 mm, c) 90 mm, and d) 100 mm.