Small-scale soft continuum robots capable of active steering and navigation in a remotely controllable manner hold great promise in diverse areas, particularly in medical applications. Existing continuum robots, however, are often limited to millimeter or centimeter scales due to miniaturization challenges inherent in conventional actuation mechanisms, such as pulling mechanical wires, inflating pneumatic or hydraulic chambers, or embedding rigid magnets for manipulation. In addition, the friction experienced by the continuum robots during navigation poses another challenge for their applications. Here, we present a submillimeter-scale, self-lubricating soft continuum robot with omnidirectional steering and navigating capabilities based on magnetic actuation, which are enabled by programming ferromagnetic domains in its soft body while growing hydrogel skin on its surface. The robot’s body, composed of a homogeneous continuum of a soft polymer matrix with uniformly dispersed ferromagnetic microparticles, can be miniaturized below a few hundreds of micrometers in diameter, and the hydrogel skin reduces the friction by more than 10 times. We demonstrate the capability of navigating through complex and constrained environments, such as a tortuous cerebrovascular phantom with multiple aneurysms. We further demonstrate additional functionalities, such as steerable laser delivery through a functional core incorporated in the robot’s body. Given their compact, self-contained actuation and intuitive manipulation, our ferromagnetic soft continuum robots may open avenues to minimally invasive robotic surgery for previously inaccessible lesions, thereby addressing challenges and unmet needs in healthcare.

Further elaborating on the recent progress in materials, fabrication, and theory for ferromagnetic soft robots ( 28 – 31 ), we present submillimeter-scale ferromagnetic soft continuum robots that can navigate through highly constrained environments, such as narrow and tortuous vasculature, based on active, omnidirectional steering upon magnetic actuation ( Fig. 1B ). The robot’s body is composed of soft polymer matrices with evenly dispersed hard magnetic microparticles ( Fig. 1C ) and thus can be easily fabricated into submillimeter scale based on printing or injection molding ( Fig. 1E ). To cope with the substantial friction experienced while navigating highly unstructured environments ( 5 , 32 ), we grew hydrogel skin ( 33 ), a thin (10 to 25 μm) layer of hydrated cross-linked polymers, onto the robot’s surface. This hydrogel skin effectively decreased the surface friction due to its high water content. Enabled by the theoretical framework based on continuum mechanics developed for ferromagnetic soft materials ( 31 ), we also present our model-based material design strategies to optimize the actuation performance of our soft continuum robots. Combining all these features, we demonstrate the capability of navigating through complex and constrained environments, such as a cerebrovascular phantom with multiple aneurysms, which are difficult to navigate with bulky robotic catheters or passive manual instruments. Incorporating a functional core in the robot’s body, we further demonstrate additional functionalities, such as steerable laser delivery, in realistic in vitro environments with relevance to clinical challenges.

An emerging class of magnetically actuated soft robots, which we define as ferromagnetic soft robots, has recently been proposed ( 28 – 30 ) with great promise for biomedical applications, because they can potentially address the abovementioned limitations of conventional soft robots. They are composed of so-called ferromagnetic soft materials, in which magnetized or magnetizable microparticles are uniformly dispersed in soft polymeric matrices. Exploiting magnetic body torques and/or forces generated from the embedded particles under externally applied magnetic fields, ferromagnetic soft robots can be actuated remotely while at the same time controlled accurately based on quantitative models ( 28 – 30 ). Furthermore, the use of ferromagnetic microparticles as distributed actuation sources enables the miniaturization of ferromagnetic soft robots readily to submillimeter scales.

Recently, burgeoning efforts have been made to use soft-bodied robots in the medical domain, with great expectations for enhanced safety due to their inherent compliance ( 1 , 3 , 23 , 24 ). Despite the purported advantage, the field of soft robotics is still faced with a set of key challenges. First, existing soft robots based on pneumatic or hydraulic actuations are mostly heavily tethered, which limits their use in realistic medical applications that typically require tether-free actuation ( 25 ). Second, most soft robots are difficult to accurately control based on quantitative models, largely because their actuation mechanisms often rely on highly nonlinear deformation or instabilities ( 26 ). Third, conventional soft robots are difficult to miniaturize below millimeter scales, because their fabrication schemes are often unfavorable to such small size ( 1 , 27 ).

Despite these advantages, existing continuum robots are often limited to relatively large scales due to miniaturization challenges inherent in their conventional actuation mechanisms, such as pulling mechanical wires or controlling embedded rigid magnets for manipulation. Tendon-driven continuum robots ( 7 – 10 ) with antagonistic pairs of wires are difficult to scale down to submillimeter diameters due to increasing complexities in the fabrication process as the components become smaller ( 11 – 13 ). The miniaturization challenges have rendered even the most advanced form of commercialized continuum robots, mostly for cardiac and peripheral interventions ( 14 ), unsuited for neurosurgical applications due to the considerably smaller and more tortuous vascular structures ( 6 ). Magnetically steerable continuum robots ( 15 – 19 ) have also remained at large scale because of the finite size of the embedded magnets required to generate deflection under applied magnetic fields. Recenty, a submillimeter-scale device (500 μm in diameter) with tiny magnets embedded in a soft polymer rod has been proposed for potential use in cardiac interventions, demonstrating magnetic steering and navigation in a coronary artery phantom ( 20 , 21 ). However, navigating through cerebrovascular structures with fully soft-bodied continuum robots has not been realized so far. Furthermore, the inherent limitations associated with the use of such rigid magnets, particularly at submillimeter scale, are epitomized by the fact that several products of magnet-tipped microguidewires seeking U.S. Food and Drug Administration (FDA) premarket approval were later recalled because of the concern that tiny magnets at the tip could break off ( 22 ), which may lead to undesired clinical problems.

( A ) Pathologic conditions in hard-to-reach areas across the human body where small-scale soft continuum robots with active steering and navigating capabilities have utility. ( B ) Illustration of the active steerability of a submillimeter-scale soft continuum robot navigating a complex vasculature with an aneurysm. ( C ) Schematic illustration of the magnetically responsive tip of the continuum robot with programmed magnetic polarities resulting from the hard magnetic particles embedded in the robot’s body made of soft polymer matrix. The hydrogel skin provided a hydrated, self-lubricating layer on the robot’s surface, and the silica shell coated around the embedded magnetic particles prevented their corrosion at the hydrated interface. ( D ) Ferromagnetic composite ink based on PDMS + NdFeB (20 volume %) before and after magnetization. When magnetized, the previously freely flowing ink became a thixotropic paste with shear yield stress due to the interaction between embedded magnetic particles. ( E ) Fabrication methods based on (i) printing/extrusion and (ii) injection molding. For printing, the magnetized ink was extruded through a micronozzle. For injection modeling, the ink was injected into a micromold in which a concentric functional core is placed. ( F ) Schematic illustration of hydrogel skin formation onto the outer surface of the fabricated ferromagnetic soft continuum robot.

Small-scale soft continuum robots capable of navigating through complex and constrained environments hold promise for medical applications ( 1 – 3 ) across the human body ( Fig. 1A ). Several continuum robot concepts have been commercialized so far, offering a range of therapeutic and diagnostic procedures that are safer for patients owing to their minimally invasive nature ( 4 – 6 ). Surgeons benefit from remotely controlled continuum robots, which allow them to work away from the radiation source required for real-time imaging during operations ( 5 , 6 ).

RESULTS

Ferromagnetic composite ink Ferromagnetic materials in general develop strong induced magnetization under applied magnetic fields. Unlike soft magnetic materials, such as pure iron, which easily lose the induced magnetization once the external field is removed, hard magnetic materials, such as neodymium-iron-boron (NdFeB), are characterized by their ability to retain high remnant magnetization against the external field once they are magnetically saturated due to their high coercivity (fig. S1A) (31). The main body of our soft continuum robot was made of an elastomer composite that contains magnetizable microparticles (5-μm-sized on average; fig. S2C) of a NdFeB alloy (28–30). The soft polymer matrix of the robot’s body was composed of either silicone [polydimethylsiloxane (PDMS)] or thermoplastic polyurethane (TPU) elastomers, depending on desired mechanical properties. As the initial step of the fabrication process, our ferromagnetic composite ink was prepared by homogeneously mixing nonmagnetized NdFeB particles with uncured PDMS resin or TPU dissolved in solvent at a prescribed volume fraction. To impart desired rheological properties to the mixture for ease of fabrication later, we magnetized the whole mixture upon preparation by applying a strong impulse of magnetic fields to magnetically saturate the dispersed NdFeB particles. This turned the previously freely flowing mixture into a thixotropic paste (Fig. 1D) with shear-yielding (fig. S1B) and shear-thinning (fig. S1C) properties due to the strong interaction between the permanently magnetized NdFeB microparticles. The acquired rheological properties after magnetization were not only crucial for fabrication, as detailed in the following section, but also conducive to preventing phase separation of our composite ink due to sedimentation of the dispersed particles over time (fig. S1, D to F). The suppressed phase separation guaranteed microstructural uniformity (fig. S2B), which allowed us to postulate a homogeneous continuum when modeling the macroscopic behavior of our material to quantitatively predict the response of our soft continuum robot upon magnetic actuation (see the Supplementary Materials for details).

Printing/injection molding The main body of our soft continuum robot consisted of a magnetically responsive tip (Fig. 1C) followed by a magnetically inactive segment (Fig. 4A). The soft continuum robot could be fabricated by either printing or injection molding, both of which required extruding the thixotropic paste–like ink through a micronozzle by applying pressure (Fig. 1E). The printing technique differs from conventional extrusion of molten thermoplastic polymers in that it did not require any heating to melt and fluidize the ink. The shear-thinning behavior of the magnetized ink ensured that the composite ink could be easily extruded when pressurized, and the presence of yield stress helped the deposited ink maintain its shape (30) instead of spreading and becoming flat (Fig. 1D). When additional mechanical support or functionalities were required, a functional core could be incorporated into the robot’s body through the injection molding. For this process, a microtube was used as a mold, into which we injected the thixotropic composite ink while locating a concentric functional core inside the mold. Once the printing or injection was complete, the printed or molded ink underwent thermal curing (PDMS-based composite) or solvent evaporation (TPU-based composite) upon heating to solidify into the robot’s body. During the heating process, the presence of yield stress helped the unsolidified ink maintain its shape on the printing substrate or remain stable in the mold instead of flowing and escaping due to the decrease in viscosity at the elevated temperature. Thereafter, the magnetically active tip was uniformly magnetized again, along the axial direction, to have programmed magnetic polarities required to create deflection upon magnetic actuation (Fig. 1C).

Hydrogel skin The hydrogel skin on the outer surface of the robot (Fig. 1C) consisted of cross-linked hydrophilic polymers [polydimethylacrylamide (PDMAA)] that were grafted onto the elastomer chains on the robot’s surface. For the hydrogel coating procedure, we followed the previously reported protocol (33). First, the solidified robot’s body was treated with an organic solution based on ethyl alcohol that contains hydrophobic photoinitiators (benzophenone). Exposure to this organic solution induced swelling-driven absorption of the photoinitiators into the robot’s surface. The treated body was then immersed into a hydrogel monomer (DMAA) solution (Fig. 1F) containing hydrophilic photoinitiators (Irgacure 2959). Upon exposure to ultraviolet (UV) radiation (Fig. 1F), the hydrogel monomers were polymerized by the hydrophilic initiators while covalently grafted onto the surface-bound elastomers by the activated benzophenone, leaving a thin hydrogel-polymer interpenetrated layer on the surface. The thickness of the hydrogel skin was measured to be 10 to 25 μm from fluorescence microscope images taken from coated and uncoated samples with planar geometry (1-mm-thick sheet) (Fig. 2, A to D). The microscopic images identified the presence of the hydrogel skin on the coated samples. Fig. 2 Hydrogel skin as a lubricating layer. Cross-sectional views of (A) the coated specimen of PDMS + NdFeB (20 volume %) with hydrogel skin visualized by absorbed fluorescein and (B) the uncoated specimen without hydrogel skin. The dashed line in (B) indicates the boundary of the cross-section of the uncoated specimen. Top views of (C) the coated specimen with hydrogel skin and (D) the uncoated specimen. The fluorescing specks visible in the uncoated sample were due to residual fluorescein adsorbed onto the surface. (E) Schematic of testing setup for measuring friction coefficients using a rheometer. (F) Schematic of testing setup for measuring force required to pull a cylindrical specimen (diameter of 8 mm) at a constant speed under applied normal force by the pair of grips. (G) Semi-log plot of the pulling force measured over time during the pullout test performed at 200 mm min−1 for both coated and uncoated specimens under two different normal force conditions (2 and 5 N). Friction coefficients measured from both coated and uncoated samples under different (H) shear rates and (I) normal pressure. (J) Friction coefficients measured from prolonged shearing of both coated and uncoated samples up to 60 min at shear rate of 0.5 s−1 under normal pressure of 6 kPa. The error bars in (H) to (J) indicate the SDs of the mean values obtained from five different measurements. The resulting hydrogel skin reduced the surface friction, which was characterized by the friction coefficients measured from a rheometer testing while applying different levels of shear rates and normal pressure (Fig. 2E). The measurements showed a 10-fold decrease in the friction coefficient (Fig. 2, H and I) as a result of the lubricious hydrogel skin in all given conditions. Furthermore, the coated hydrogel skin remained stable and undamaged even after prolonged shearing over an hour, exhibiting sufficient mechanical robustness (Fig. 2J). We also measured forces required to pull cylindrical specimens with and without hydrogel skin at a constant speed (200 mm min−1) under different normal forces (2 and 5 N) applied by a pair of grips (Fig. 2F). The results showed a substantial decrease in the pulling force as a consequence of the self-lubricating hydrogel skin (Fig. 2G). When the applied normal force was 2 N, the hydrogel skin reduced the pulling force by a factor of 15 (from 2.65 to 0.18 N). As the normal force was increased from 2 to 5 N, the force required to pull the same uncoated specimen at the same rate increased by 150%. Compared with this, the required force to pull the coated specimen increased by only 60%, which illustrates how effectively the self-lubricating hydrogel skin reduces the surface friction under the increased load.

Silica shells around the particles Ferromagnetic alloys have a highly corrosive nature due to the high content of iron. To prevent corrosion of the embedded NdFeB particles at the hydrated interface with the water-containing hydrogel skin, we coated the particles with a thin shell of silica (Fig. 1C) based on the condensation reaction of tetraethylorthosilicate (TEOS), which nucleated around the particles to form a cross-linked silica layer (fig. S2A). The resulting silica shell was identified to be 10 nm thick from transmission electron microscope (TEM) imaging (fig. S2D) and further verified by Fourier transform infrared spectroscopy, which indicated the presence of Si─O─Si bonds (fig. S2E). The effectiveness of the silica shell in preventing the corrosion of NdFeB particles could be verified by performing a leaching test for both coated and uncoated particles with a weak acidic solution (0.2 mM HCl, pH 3). The results show highly oxidized uncoated particles but no visible change in silica-coated particles, which illustrates the anticorrosion effect of the silica shell formed around the NdFeB particles (fig. S2F). Because of the marginal thickness of the silica shell compared with the size of microparticles, the silica coating resulted in a slight increase in volume, which was roughly estimated to be around 1% when assuming a uniform silica layer around a spherical particle.

Design for optimal actuation Both magnetic and mechanical properties of the robot’s body made of ferromagnetic soft composites varied with the particle loading concentration. Here, we describe our material design strategy to optimize the actuation performance of the proposed ferromagnetic soft continuum robot. On the basis of our theoretical framework developed for ferromagnetic soft materials (30, 31), we first provide the fundamental equations for quantitative description of the deformation of ferromagnetic soft materials upon magnetic actuation. We denote the magnetic moment density (or magnetization) at any point of a ferromagnetic soft material in the reference (undeformed) configuration by a vector M. Under an applied magnetic field, denoted by a vector B, the ferromagnetic soft material can deform. The deformation at any point of the material is characterized by the deformation gradient tensor F. The application of the magnetic field on the embedded magnetic moment in the material generates the magnetic Cauchy stress σmagnetic = −B ⊗ FM that drives the deformation (30, 31), where the operation ⊗ denotes the dyadic product, which takes two vectors to yield a second-order tensor. Meanwhile, the deformation of the material generates the elastic Cauchy stress σelastic, which is also a function of F defined by hyperelastic constitutive models such as the neo-Hookean model (30, 31). The total Cauchy stress in the material σ = σelastic + σmagnetic is then substituted into the equilibrium equation in eq. S3, from which the deformation (i.e., F) can be evaluated at every material point in equilibrium. Although alternate approaches based on magnetic body forces and torques have been proposed to calculate the deformation of ferromagnetic soft materials (28, 29), the current approach based on the magnetic stress can be readily implemented in commercial finite element software packages such as Abaqus. In addition, the magnetic stress can readily recover the magnetic body force and torque densities used in other approaches (see the Supplementary Materials for details). Because the magnetically responsive tip of our soft continuum robot is axially magnetized (i.e., M along the axial direction), the tip tends to bend along the applied magnetic field B (Fig. 1C) due to the magnetic body torques generated from the embedded magnetized particles. To find the optimal particle concentration that yields the largest bending under given conditions and geometry, without loss of generality, we consider a beam of length L and diameter D under uniform magnetic field B that is being applied perpendicularly to M (Fig. 3A). In addition, to use a tractable analytical solution, we further assume that the magnetically active tip undergoes small bending, where the deflection (denoted δ in Fig. 3A) is below 10% of the tip length L. Then, we can reach the following analytical expression for the deflection of the magnetically active tip (details are available in the Supplementary Materials) δ L = 16 9 ( M B G ) ( L D ) 2 (1)where M and B are the magnitudes of the magnetization and the applied magnetic field, respectively, and G denotes the shear modulus of the material, which is considered as a neo-Hookean solid in the current analysis. Equation 1 relates the material properties (magnetization M and shear modulus G), geometry (beam length L and diameter D), and actuating field strength B to the normalized deflection. From Eq. 1, we notice that, for small bending, the deflection of the beam is linearly proportional to a dimensionless quantity, MB/G, while quadratically dependent on the aspect ratio L/D. The dimensionless quantity MB/G can be interpreted as the actuating field strength normalized by the material properties. Given that both M and G are dependent on the particle volume fraction, Eq. 1 implies that there will likely be an optimal point at which the normalized deflection is maximized. Fig. 3 Optimal design of ferromagnetic soft robot. (A) Schematic of the ferromagnetic soft continuum robot with uniform magnetization M along the axial direction deflecting toward the direction of the uniform magnetic field B applied perpendicularly to the body. The unconstrained length and the outer diameter of the robot are denoted L and D, respectively. δ indicates the deflection of the free end, and θ indicates the deflection angle. (B) Magnitude of magnetization, denoted M, linearly varying with the volume fraction of the embedded magnetic particles. (C) Shear modulus (denoted G) of the ferromagnetic composite at different particle concentrations. (D) Prediction of the variation of M/G, a characteristic quantity that determines the degree of deflection for small bending, with the particle volume fraction under given applied field strength for a given geometry. The unit of this quantity, T−1, or equivalently A·m N−1, was intentionally omitted for simplicity. (E) Actuation angle predicted from finite element simulation and experimental measurements plotted against the applied field strength normalized by material properties (M and G) for a particular composition (20 volume %) with different aspect ratios: L/D = 10, 15, 20. (F) The variation of actuation angle with particle concentration at different actuation field strengths: B = 10, 20, 40, 80 mT, predicted from simulation results when L/D = 10. (G) Prediction of the variation of M2/G, a quantity that characterizes the energy density in a deflected body for small bending case, with the particle volume fraction under given applied field strength for a given geometry. The unit of this quantity, A2/N, was intentionally omitted for brevity. The average energy density predicted by finite element simulation for (H) small and (I) large bending cases, as a function of particle concentration. The magnetization of the ferromagnetic soft composite is linearly proportional to the volume fraction of NdFeB particles (Fig. 3B) and hence can be expressed as M = M p ϕ (2)where M p denotes the magnetization of the magnetic particles and ϕ denotes the particle volume fraction. Unlike the magnetization, the shear modulus increases nonlinearly as the particle concentration increases (Fig. 3C). This nonlinear dependence of shear modulus can be predicted by a simple analytical expression in Eq. 3, a so-called Mooney model (34), under the assumption that the increase in the shear modulus of particle-filled elastomer composites is analogous to the increase in the viscosity of particle suspensions (35) G = G o exp ( 2.5 ϕ 1 − 1.35 ϕ ) (3)where G o denotes the shear modulus of a pure elastomer with no particle. No significant difference was observed in both magnetization (Fig. 3B) and shear modulus (Fig. 3C) between the composite based on uncoated particles (PDMS + NdFeB) and the composite based on silica-coated particles (PDMS + NdFeB@SiO 2 ), which may be attributed to the marginal change in the particle volume due to the marginal thickness of the silica shell as discussed earlier. The small difference in shear modulus between the two types of composites also implies that the affinities of silicone elastomers with metal oxide (of uncoated particles) and silicon oxide (of silica-coated particles) surfaces are not substantially different. By substituting Eqs. 2 and 3 into Eq. 1, we can identify the critical concentration ϕ c at which the deflection is maximized for given conditions (field strength B and the geometric factor L/D). The critical volume fraction is calculated to be 0.207 (or 20.7 volume %), independent of M p and G o (Fig. 3D). Note that this critical concentration is obtained for small bending scenarios as described above. For large bending, the simulation and experimental results in Fig. 3E indicate that the actuation angle θ (defined in Fig. 3A) monotonically increases as a function of the normalized field strength MB/G for different aspect ratios L/D. Therefore, we can anticipate that the critical concentration predicted from the small-deflection analysis will remain effective for large bending cases as well. This is further validated by the simulation results for large bending presented in Fig. 3F, which show the actuation angle varying with the material composition and the applied field strength for a fixed geometry. The results indicate that the actuation angle reaches its maximum, for given applied field strengths, at the critical volume fraction (20.7 volume %) predicted above for the small bending case. As the applied field strength increases, however, the actuation angle begins to saturate while approaching 90°, making the curves around the peak flat (Fig. 3F). When producing mechanical work out of magnetic actuation is of greater importance than the large deflection, we can optimize the actuation performance in terms of the energy density, which corresponds to the amount of work (per unit volume) that one can extract from the continuum robot. For small bending, the equivalent force generated at the free end of the beam can be calculated as F = MBA (see the Supplementary Materials), where A denotes the cross-sectional area of the beam. Combining this with Eq. 1, we can find an analytical expression for the energy density u as follows u = 16 9 ( M 2 B 2 G ) ( L D ) 2 (4) By substituting Eqs. 2 and 3 into Eq. 4, we find that the energy density reaches its maximum when the particle volume fraction is 29.3 volume % under given conditions in terms of applied field strength B and geometry L/D (Fig. 3G). This analytical prediction is validated by our model-based simulation for small bending (Fig. 3H), which shows how the energy density varies with the particle volume fraction when B = 5 mT and L/D = 10. As the bending becomes larger, however, the peak at which the energy density is maximized shifts to the right, toward the higher volume fractions (Fig. 3I). The peak eventually disappears when the actuation angle saturates, after which the energy density keeps increasing with the particle volume fraction. Qualitatively, this can be understood by considering the exponentially increasing stiffness (Fig. 3C), which dominantly contributes to the energy density when the deformation level remains almost unchanged (Fig. 3F). When the material properties M and G are fixed because of a given particle volume fraction, the actuation performance of our ferromagnetic soft continuum robot under given applied field strength can still be optimized by adjusting the aspect ratio, according to Eqs. 1 and 4 along with the simulation results in Fig. 3. This implies that fine features with high aspect ratios, such as cilia-like soft continuum robots, would require significantly lower field strength to induce the bending actuation. Given that the printing-based method can easily produce very fine features, down to 80 μm in diameter as demonstrated in the previous study (30), such extremely thin, cilia-like soft continuum robots may also be conceived for applications that require manipulating highly delicate structures. Despite these fabrication and miniaturization capabilities, however, we focused more on the design, fabrication, and demonstrations of our ferromagnetic soft continuum robots within the context of functional challenges and requirements for conventional continuum robots to constrain the scope of this study. Hence, our demonstrations are mostly focused on the functional capabilities achieved from the proposed concept of ferromagnetic soft continuum robots—such as active steerability, navigability, and maneuverability—while suggesting additional possible functions enabled by the incorporated functional cores.

Active steering and navigation Hereafter, we demonstrate the main capability of our ferromagnetic soft continuum robots designed for navigating complex and constrained environments, such as vasculature, based on active steering upon the magnetic actuation and additional functionalities enabled by the functional core incorporated in the robot’s body. Figure 4A illustrates the proposed concept of a ferromagnetic soft continuum robot passing through a set of rings using its magnetically responsive tip, which follows the direction in which the actuating field is applied. For experimental demonstration, we used a cylindrical permanent magnet (diameter and height of 50 mm) to apply the actuating magnetic fields at a distance. The basic principle for magnetic actuation and steering was to align the central axis (denoted z axis in fig. S5A) of the magnet along the desired direction to induce bending of the robot’s tip toward the desired direction (fig. S5B). Although the bending actuation in general is driven by magnetic body torques as discussed earlier, the spatial gradients of applied magnetic fields can also give rise to magnetic body forces, which further encourage the robot’s tip to align itself along the magnet’s central axis (fig. S6), as discussed in the Supplementary Materials. Fig. 4 Demonstration of active steering and navigating capabilities. (A) Schematic of the demonstrated ferromagnetic soft continuum robot with a magnetically responsive tip (with uniform magnetization M) and a tapered nitinol core required for active steering and navigation under magnetic actuation. (B) Experimental demonstration of the designed ferromagnetic soft continuum robot (based on PDMS + NdFeB composite) selectively navigating through a set of rings based on magnetic actuation and steering. The magnetic fields for actuation (20 to 80 mT) were generated by a cylindrical permanent magnet (diameter and height of 50 mm) at a distance (40 to 80 mm). The proximal end was pushed to advance the magnetically steered distal tip of the robot during the navigation. The outer diameter of the demonstrated prototype was 600 μm. Detailed dimensions of the demonstration setup are available in fig. S4A. Figure 4B and movie S1 show the experimental demonstration of the fabricated prototype, which selectively navigates through a set of loosely placed rings (see fig. S4A for details) based on steering achieved by manually manipulating a single magnet. The demonstrated prototype was fabricated through injection molding (Fig. 1E) of the PDMS + NdFeB composite ink and was designed to be 600 μm in diameter. To provide mechanical support and pushability required for the demonstrated task, we incorporated a nickel-titanium alloy (nitinol) core in the robot’s body (Fig. 4A). Because the nitinol core was from the tip of a commercial guidewire, the magnetically responsive tip was naturally connected to the commercial guidewire (see Materials and Methods for details). We also fabricated another prototype based on printed TPU + NdFeB composite without a core. The printed segment was also connected to the commercial guidewire. As demonstrated in fig. S3 and movie S2, the TPU-based prototype performed the same functional tasks shown with the PDMS-based prototype in Fig. 4B and movie S1. Because the printed segment does not contain any core for additional support, this prototype was designed to be thicker in diameter (810 μm) to ensure sufficient bending rigidity required for the demonstrated tasks. The navigating performance of the two prototypes is comparable when considering the average time taken to complete the demonstrated task: 50 ± 1.58 s with the first prototype and 54 ± 1.87 s with the second prototype. Given the higher degrees of freedom in terms of design and fabrication, we chose to use the PDMS-based composite for further exploration of possible designs and functionalities of the proposed ferromagnetic soft continuum robots with functional cores. To enable making sharp turns and hence navigating through a tortuous path, we introduced a variation in the bending stiffness of the magnetically responsive part of our soft continuum robot. Now, the continuum robot (diameter of 600 μm) has a short (3-mm-long), softer segment at the distal end of the magnetically active portion. This softer segment was composed of the PDMS + NdFeB composite only and thus substantially softer than the remainder, which contained a stiff nitinol core (diameter of 80 μm). The effective Young’s modulus of the stiffer segment (14 MPa) was calculated to be 10 times that of the softer segment (1.4 MPa) from Eq. 5 in Materials and Methods. Both segments have uniform magnetization (M = 128 kA m−1) along the axial direction. The softer and hence more responsive tip enables multiple modes and degrees of bending depending on the direction and strength of the applied actuating field, as well as the unconstrained length of the magnetically active segment, as predicted from our model-based simulation in Fig. 5 (A to C). When the unconstrained length of the stiff segment equaled that of the soft segment, only the very end tip of the continuum robot reacted effectively to the applied magnetic fields, creating a J-shaped tip (Fig. 5A). This is because the short unconstrained segment has a large bending stiffness due to the small aspect ratio, as predicted in Fig. 3E. As the unconstrained length increased, the bending stiffness of the stiffer segment decreased, which increased the radius of curvature of overall bending upon magnetic actuation (Fig. 5, B and C). Figure 5D and movie S3 show the experimental demonstration of our fabricated prototype navigating through a tortuous path formed by a series of tightly spaced rings (see fig. S4B for details) based on the ability to make sharp turns, which was enabled by the design described above. Fig. 5 Multiple modes and degrees of bending for navigating through a tortuous path. (A to C) Simulation results identifying multiple different modes and degrees of deflection, depending on the unconstrained length of the magnetically responsive tip (consisting of stiff and soft segments) as well as the applied field strength and direction, which help in creating sharp turns when navigating through tortuous paths. (D) Experimental demonstration of navigating through a highly nonlinear path formed by a set of tightly spaced multiple rings. The magnetic fields for actuation (20 to 80 mT) were generated by a cylindrical permanent magnet (diameter and height of 50 mm) at a distance (from 40 to 80 mm). The proximal end was pushed to advance the magnetically steered distal end of the robot during the navigation. The outer diameter of the demonstrated prototype was 600 μm. Detailed dimension of the demonstration setup is available in fig. S4B. To illustrate the potential impacts of the proposed ferromagnetic soft continuum robots in medical applications, we extended the demonstrated steering and navigating capabilities of our soft continuum robots to a more realistic, clinically relevant environment. To this end, we used a real-sized, silicone vascular phantom that replicates a particular cerebrovascular anatomy, called the circle of Willis, as well as the surrounding arteries with multiple aneurysms (localized dilation) at different locations. As can be seen in fig. S7, the vascular structures are highly complex and tortuous, involving several acute-angled corners. The inner diameter of the silicone vessels along the targeted path (from carotid artery to middle cerebral artery in fig. S7B) to be navigated by our continuum robot ranged from 2.5 to 7.5 mm, and the aneurysms to reach along the path were 9 mm (first), 7.5 mm (second), and 5 mm (third) in diameter (fig. S7A). The overall distance navigated by the robot along the targeted path was around 250 mm (fig. S7, B and C). The required task for our ferromagnetic soft continuum robot was to reach all the aneurysms along the targeted path while demonstrating the ability to locate the robot’s distal tip inside each aneurysm based on magnetic actuation and steering capabilities. In addition, direct contact of the robot with the inner wall of the aneurysms should be avoided, given that aneurysms have a high risk of rupture, which can lead to hemorrhagic stroke. With the same prototype presented earlier (Fig. 5 and movie S3), we experimentally demonstrate the capability of our ferromagnetic soft continuum robot to successfully carry out the required tasks in the vascular phantom, which was filled with a blood analog that simulates the friction between commercial guidewires and real blood vessels. As can be seen in Fig. 6 and movie S4, the proposed ferromagnetic soft continuum robot was able to smoothly navigate through the targeted path while completing all the required tasks without any noticeable difficulties or unintended motion. The importance and the effectiveness of the self-lubricating hydrogel skin became evident when comparing the navigating performance of ferromagnetic soft continuum robots with and without the hydrogel skin. As shown in movie S5, the uncoated prototype suffered from the substantial friction acting on the robot while going through the first acute-angled corner, exhibiting unwanted jerky movement. Despite the omnidirectional steering capability that enabled the robot to orient its distal tip toward the desired direction, the significant friction did not allow the robot’s body to proceed. Upon further pushing, followed by another jerk, the robot struck the inner wall of the first aneurysm. After hitting the second aneurysm in an unpredictable manner, the demonstrated prototype without hydrogel skin eventually failed to further proceed through the third acute and narrow corner. Fig. 6 Demonstration of navigating through a 3D cerebrovascular phantom network. The soft continuum robot first passed through the sharp corner with acute angulation (between t = 0 s and t = 5 s). The robot made another sharp turn after reaching the first aneurysm (t = 11 s) based on the magnetic steering capability to reach the second aneurysm (t = 15 s). Then, it made another sharp turn at the acute-angled corner beneath the second aneurysm (t = 18 s) to reach the third aneurysm (t = 25 s) and navigated further downstream (t = 36 s). Magnetic fields for actuation (20 to 80 mT) were generated by a cylindrical (diameter and height of 50 mm) permanent magnet at a distance (40 to 80 mm). The proximal end was pushed to advance the magnetically steered distal end of the robot during the navigation. The outer diameter of the demonstrated prototype was 600 μm. Note that the continuum robot may look thicker than the actual size due to the magnifying effect of the round, thick-walled silicone vessel. Detailed anatomy and dimensions of the phantom model are provided in fig. S7. As discussed earlier, commercialized continuum robots with navigational capabilities are mostly limited to cardiac and pulmonary interventions, due mainly to the relatively large size (a few millimeters in diameter). Unlike those bulky devices, commercial guidewires can reach smaller and narrower areas, when carefully manipulated by skilled interventionalists, and hence are widely used in cerebrovascular and endovascular neurosurgery (6). For these manually controlled, passive guidewires, the distal tip of the device is typically preshaped with a fixed curvature or shapeable into curved or bent shapes, instead of being straight, for steering purposes (36, 37). This pre-bent distal tip can be oriented by manually twisting the proximal end of the device. After orienting the tip toward a desired direction through the twisting manipulation, the proximal end is pushed to advance the whole device forward. Upon this pushing manipulation, the floppy tip conforms to the environment and passively follows the continuous path as the guidewire moves forward. To illustrate the importance of active steering capability of the proposed ferromagnetic soft continuum robots, we also compared the navigating performance of our prototype with that of a commercial guidewire with a comparable diameter in movie S6. We chose a preshaped hydrophilic guidewire with a fixed curvature that is large enough to cross the large gap within the first aneurysm. Note that, however, this demonstration with a certain preshaped device does not represent the guidewire manipulation performed by skilled interventionalists and therefore should be taken merely as an illustrative example. From the demonstration, we first notice that even though the pre-bent shape with large curvature enabled the device to pass the large (first) aneurysm almost effortlessly based on the twist-based steering, the very end tip of the device could act as an anchor that causes large friction and hence some jerky movements while navigating in a narrow vessel. In addition, the fixed curvature of the preshaped tip was not useful for the twist-based steering in highly constraining environments, such as the region near the third (smallest) aneurysm. Although the use of shapeable guidewires may alleviate this issue, additional time required for repetitive reshaping maneuvers and adjustments will likely be unavoidable (36). In this regard, we expect that the proposed soft continuum robot may overcome the current limitations of manual guidewires inherent in their passive steering mechanisms.