Harnessing the Oloid Shape in Magnetically Driven Robots to Enable High-Resolution Ultrasound Imaging

This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 1 Harnessing the Oloid Shape in Magnetically Driven Robots to Enable High-Resolution Ultrasound Imaging Nikita J. Greenidge1*, Benjamin Calmé1, Alexandru C. Moldovan2, Bartas Abaravicius3, James W. Martin1, Nils Marahrens1, Jon Woolfrey1, Bruno Scaglioni1, Damith S. Chathuranga1, Srinjoy Mitra3, Sandy Cochran2, Pietro Valdastri1 Magnetic fields enable remote manipulation of objects and are ideal for medical applications, as they pass through human tissue harmlessly. This capability is promising for surgical robots, allowing navigation deeper into the human anatomy and accessing organs beyond the reach of current technologies. However, magnetic manipulation is typically limited to a maximum of two degrees-of-freedom orientation, restricting complex motions, especially those including rolling around the main axis of the magnetic robot. To address this challenge, we introduce a robot design inspired by embodied intelligence and the unique geometry of developable rollers, leveraging the oloid shape. The oloid, with its axial asymmetry and sinusoidal motion, facilitates rolling when precisely controlled by an external magnetic field. We present a versatile closed-loop control model to ensure precise magnetic manipulation of an oloid-shaped robot. This capability was validated in endoluminal applications through the integration of a 28 MHz micro-ultrasound array to perform virtual biopsies – non-invasive real-time histological imaging. Extensive in vitro and in vivo tests using a porcine model showed the robot's ability to execute sweeping motions, identify lesions, and generate detailed 3D scans of gastrointestinal subsurface tissue. This research not only restores a critical movement capability to magnetic medical robots but also enables additional clinical applications deep within the human body. INTRODUCTION The application of magnetic manipulation to medical robots, such as robotic catheters (1, 2), flexible endoscopes (3–5), and capsule endoscopes (NaviCam® (6)), has streamlined device design by eliminating the need for complex internal actuation mechanisms (7). This approach enables miniaturization and enhances adaptability for navigating intricate anatomical pathways within the body. Magnetic manipulation involves the use of a controlling magnetic field source to induce a force, 𝐅∈ ℝ3(newton), and torque, 𝝉∈ℝ3(newton·meter), on a magnetic object, allowing control over its position and orientation. In medical applications, where the robot is typically considerably smaller than its distance from the controlling field source, the robot behaves as a simple north-south magnetic dipole with a symmetric field around its magnetization axis (denoted by XI in Figure 1). As such, magnetic manipulation of objects is typically limited to a maximum of two degrees-of-freedom (DoFs) in orientation and three DoFs in position. Related works (8–11) have explored the use of magnetic force to produce off-axis rigid body torques to control roll around the object’s magnetization axis through various methods detailed in the Results section. However, these techniques, including newer soft magnet methods (12, 13), remain unsuitable in clinical applications. They have only been demonstrated in fluid 1STORM Lab, University of Leeds, Leeds, United Kingdom 2University of Glasgow, Glasgow, United Kingdom 3University of Edinburgh, Edinburgh, United Kingdom *Corresponding Author. Email: [email protected] environments with low force and torque demands, and on micro-scale robots controlled by electromagnetic coil systems. Furthermore, they rely on complex fieldgenerating setups with at least eight magnetic control inputs (1, 14–16). A detailed comparison of these approaches, including power consumption and workspace size, is available in Table S1 and the Supplementary Discussion. In terms of generating the controlling magnetic field, electromagnetic coil systems provide high control precision and adaptability, which are crucial in some applications. However, these systems typically have large physical footprints, limited workspaces, consume substantial power, require cooling, and are expensive to implement (15, 17). Robotically manipulated single External Permanent Magnet (EPM) systems offer several advantages for slower, larger-scale applications such as flexible endoscopy. These systems require no energy to sustain a static magnetic field, making them energy-efficient and suitable for prolonged use. They can generate strong fields over larger workspaces while being more compact, portable, and cost-effective compared to electromagnetic coil systems (3, 18). The MFE system enables painless (19), automated, and remote colonoscopy procedures (20) while retaining the same functionalities as standard flexible endoscopes. Although its feasibility has been validated during clinical trials (19), it has yet to demonstrate diagnostic capabilities that surpass those of standard flexible endoscopes. Greenidge et al. This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 2 Figure 1: Oloid-shaped Magnetic Endoscope (OME) for recovering a lost DoF in magnetic medical robots, enabling virtual biopsies during endoscopy. (A) The magnetic manipulation system relies on the use of cylindrical permanent magnets where the field generated by the External Permanent Magnet (EPM) pulls and orients the Internal Permanent Magnet (IPM) during the procedure. The lost DoF (“roll”) in magnetic manipulation around a magnet’s magnetization axis (XI) is shown by the arrow on the IPM. (B) A joystick is used to control the robotic system where the operator only has to consider the desired direction based on the camera frame. (C) The operator console is used to visualize the camera feed for navigation and to inspect the 3D reconstructed virtual biopsies. (D) The robotic system, which includes a robotic manipulator, is used to manipulate the EPM and therefore the magnetic field to control the OME. (E) The sweeping motion is used to demonstrate the clinical viability of recovering the rolling motion for diagnostic sweeps. (F) The OME with the subsurface micro-ultrasound visualization sensor. See Movie S1 for a visual representation of the concept overview. At the cutting edge of GastroIntestinal (GI) endoscopic technology is the concept of virtual biopsies where high-fidelity diagnostic sensors are used to perform in situ histopathology. In the context of GI cancer screening, where early and accurate detection is critical (21), the ability to perform virtual biopsies could eliminate the delays, costs, and complications associated with traditional histological analysis, allowing screening, diagnosis, and therapy to occur in a single procedure (22). A modality used alongside standard flexible endoscope is micro-ultrasound (µUS) or highfrequency US, typically delivered via mini-probe endoscopic ultrasound systems like the 20 MHz UM-3R (Olympus America Inc.), that are passed through the flexible endoscope’s working channel. While effective for in situ cancer staging (23), positioning these probes precisely is essential to produce artifact-free imaging (24) which can be difficult in manual manipulation. Additionally, using a µUS probe occupies the working channel, limiting its use for tasks such as margin assessment during therapeutic procedures where access to the working channel is required for other purposes. Another example of GI µUS is transrectal µUS, such as the ExactVuTM probe (25) which can perform 29 MHz µUS of the prostate and has been externally controlled robotically to create 3D µUS images (26). However, these probes are specifically designed for rectal imaging and cannot reach deep within the GI tract, resulting in an unmet need. Previous research on MFEs integrated a single µUS transducer capable of capturing histologically relevant images of the colon wall (3). However, the absence of roll control limited its ability to target specific areas, restore contact if misaligned, or perform radial sweeping motions. Designing 360° curved arrays presents substantial manufacturing challenges, as bending delicate thin transducer elements often leads to high failure rates (27). Some approaches introduce motors to rotate sensors (28); however, these compromise the simplicity and safety of magnetic manipulation, increase power consumption, and fail to address the overall dexterity of magnetic medical robots. These challenges underscore the need for enhanced dexterity in a clinically applicable manner, without additional actuation modes. Greenidge et al. This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 3 Inspired by embodied intelligence and the geometry of developable rollers, this work introduces a clinically applicable approach for generating torque around the magnetization axis in magnetic medical robots. This is achieved by geometrically coupling the existing two DoFs magnetic torque using just five magnetic control inputs for roll control, while still maintaining independent control over the original two DoFs. Developable rollers, observed in applications such as classical and quantum optics (29), sphericon-shaped magnetic milli-robots (30), and fluid mixing are known for their unique meandering rolling motions (Figure 2B). This innovation specifically leverages the oloid shape to achieve axial rotation, utilizing its axial asymmetry and interaction with the environment (see Movie S1). This method, demonstrated on the MFE platform, is agnostic to how the controlling magnetic field is generated, extending roll recovery to any magnetic manipulation system with at least five magnetic control inputs, including electromagnetic coil systems. To validate this approach, we developed and evaluated a differential geometry-based control model for sourceagnostic magnetic manipulation of an Oloid-shaped Magnetic Device (OMD) on various clinically relevant surfaces. In line with its motivation in GI endoscopy, an Oloid-shaped Magnetic Endoscope (OME) was designed and its ability to perform rolling and sweeping motions alongside existing DoFs demonstrated. To enable virtual biopsies, a 32-element, 28 MHz µUS linear array (Figure 2C(iii)) was integrated to capture highresolution subsurface images. Autonomous µUS sweeping and 3D subsurface image reconstruction were achieved using a custom coupling detection algorithm, validated through both in vitro and in vivo testing in a porcine model. Finally, the system’s ability to provide clinicians with in-situ lesion margin and staging information was evaluated in vivo by performing virtual biopsies of an artificially introduced polyp. RESULTS The DoF Limitation in Magnetic Manipulation Magnetic manipulation in medical robotics is simplified through the dipole model where magnets are represented as magnetic dipoles (see Figure 3). This simplification remains accurate as the distance between the controlling source and the robot generally exceeds two times the size of the internal magnet (31). As shown in Figure 3B, a magnetic object (internal dipole) with magnetic moment, 𝐦𝐼∈ℝ3(𝑎𝑚𝑝𝑒𝑟𝑒· 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑒𝑡𝑒𝑟) placed in an external magnetic field 𝐁𝐸∈ ℝ3(𝑡𝑒𝑠𝑙𝑎) experiences both an alignment torque 𝛕𝑚 and gradient-induced force 𝐅𝑚. These are generated by the external dipole to minimize the system’s potential energy. Conventionally, the magnetization axis of a magnetic object aligns with its local coordinate frame such that 𝐗𝑰 is parallel to 𝐦𝐼. However, since magnetic alignment torque is defined by the cross product 𝛕𝑚= 𝐦𝐼×𝐁𝑬 , when 𝐦𝐼 is parallel to 𝐁𝑬, 𝛕𝑚= 𝟎. As a result, magnetic alignment torque can only be generated around axes perpendicular to 𝐦𝐼, meaning it is not possible to generate magnetic alignment torque around an object’s magnetization axis to control the roll angle (ϕ). A visual representation and practical demonstration of this phenomenon are provided in Movie S1 along with a detailed mathematical explanation in the Supplementary Discussion. Related work has explored the use of magnetic force to produce off-axis rigid body torques to control roll around the object’s magnetization axis. This is achieved by coupling force and torque control through the inclusion of multiple discrete magnets (8, 9, 32) or by using a single magnet with a non-uniform magnetization or anisotropic shape (10–13). Figure 2: Experimental setup and design overview of the Oloid-shaped Magnetic Devices. (A) Benchtop experimental setup for the roll control experiments. (B) Illustration of the oloid shape rolling on a horizontal plane, Τ (based on Dirnbock et al. (33)) (C(i)) Oloid-shaped magnetic device, (C(ii)) Oloid-shaped magnetic endoscope and, (C(iii)) Oloid shaped magnetic endoscope with integrated micro-ultrasound array. Greenidge et al. This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 4 Figure 3: Block diagram control schematic for closed-loop control of the Oloid. (A) Roll control (purple): The differential geometry model of the oloid is illustrated according to (33) with (X,Y,Z) representing the local oloid coordinate frame {O} and (I,II,III) the fixed frame {F}. The developed surface corresponding to the projection of the oloid’s generators is shown in grey. The centers of the oloid’s defining circles are marked by c1 and c2 with contact points on the plane denoted by P1 and P2. (B) Magnetic field control: Shows the dipole-dipole model approximation where 𝒑𝐸 and 𝒑𝐼 are point dipoles and 𝒎𝐸 and 𝒎𝐼 represent the magnetic moments. Magnetic forces Fx, Fy and Fz and aligning torques τy and τz act on the internal dipole. Magnetic field lines represent the magnetic field BE generated by the external dipole, which becomes uniform near the internal dipole. No torque τx is shown, as torque cannot be generated around the internal dipole’s magnetization axis XI. (C) Robot control (blue): Single EPM system enabling precise magnetic field manipulation to control the Oloid’s motion. The Oloid Coupling existing DoFs to regain roll control in magnetic manipulation required a geometry with axial asymmetry. For bi-directional rolling, the geometry also needed at least one plane of symmetry, allowing roll actions in two distinct, opposing directions. The oloid, distinguished from others in the developable roller family, is formed by joining two perpendicular, equal intersecting circles with a distance between their centers (c1 and c2) equal to their radii (shown in Figure 3). This unique shape lacks vertices and maintains continuous surface contact during rolling due to its developable, flattenable (developed) surface (Figure 3). As a ruled surface, it is generated by straight lines (generators) connecting its circles at points 𝐏1∈ℝ3 and 𝐏2∈ℝ3, along a directrix. This leads to its parametric equation: 𝐫(𝑢,𝑣)=𝐏1(𝑢)+ 𝑣(𝐏2(𝑢)−𝐏1(𝑢)) (1) where −2π 3≤𝑢≤2π 3, 0≤ 𝑣 ≤1 The script to generate the 3D surface of the oloid using Equation 1 has been made available in our accompanying Data repository. When an oloid rolls on a flat plane, one generator line contacts the plane becoming the instantaneous axis of rotation, with angular velocity, ω ∈ℝ3 (rad/s), parallel to this line and tangent to the plane, expressed as: 𝛚∥𝐏2−𝐏1⇒𝛚×(𝐏2−𝐏1)=𝟎 (2) To control rotation around a robot's magnetization axis using geometric misalignment and two DoFs magnetic torque, the magnetization axis must not be parallel to the object’s angular velocity. This exists in the oloid over the controllable range, in direct contrast with a shape like the cylinder where its angular velocity is always parallel to its central axis. See “Roll Generation in the Oloid” in the Supplementary Discussion for further details. OME versus MFE DoFs To embody the unique rolling abilities of the oloid into the OME design (Figure 2C(ii)), its pure form was adapted to meet clinical requirements and to incorporate essential endoscope features. This required evaluating the oloid’s functional areas and their relation to the range of roll motion. By merging the key elements of the oloid with the cylindrical MFE, a hybrid design was achieved that adhered to clinical size constraints, as detailed in “Oloid Shape Integration” in Materials and Methods. Dexterity was assessed through a direct comparison between the MFE which has a cylindrical shape and the Greenidge et al. This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 5 OME, evaluating their ability to perform independent tilt and yaw motions, as well as coupled rolling motion (see "Magnetic Actuation of the Endoscope" in Materials and Methods for details). These experiments were conducted on lubricated Perspex using the MFE’s robotic system as illustrated in Figure 2A. As shown in Figure 4 and Movie S2, the OME not only enabled controlled rolling motion—albeit coupled with tilt and yaw—but also allowed for more independent tilt and yaw control compared to the MFE’s cylindrical design. The MFE experienced uncontrollable roll during manipulation, minimized in past designs through offset IPM placement for corrective torque and minimalenergy orientation. In GI endoscopy, due to its tubular nature, tilt and yaw are the primary DoFs, with roll desired only for specific tasks like tissue scanning or tool or camera manipulation. See Figure S2 in Supplementary Figures for x, y translational experiments and the orientation-time graphs used to generate the radar plots in Figure 4. Generating Rolling Motion with the Oloid in OpenLoop To control the oloid's rolling motion, we used an analytical differential geometry model detailed in "The Oloid Model" in Materials and Methods. This model defined the line of contact during rolling and calculated the corresponding transformation matrix for the oloid's local coordinate frame. We hypothesized that adjusting the applied magnetic field according to this sequence would replicate the desired rolling motion in an OMD in open-loop. We developed an OMD (Figure 2C(i)) with a 3D-printed oloid-shaped shell (20 mm radius, 60 mm length) that accommodated an IPM with integrated localization. This localization data tracked the roll and compared it to the model's predictions. Assuming perfect magnetic coupling, the EPM was programmed to follow the transformation matrix sequence. Initial tests on a high-friction silicone substrate (Figure 5A and Movie S3) emulated the model’s non-slip condition. Tests were repeated on bumpy foam, flat foam, and Perspex (Figures 5B-D and Movie S3) to evaluate performance across different surfaces over a 180° range (see Figure S3 for extended results). The silicone surface closely matched the model's non-slip condition, showing the highest correlation with predicted motion, while Perspex, which was unable to maintain the nonslip condition, showed the least correlation. A scaleddown demonstration was performed in an electromagnetic coil with a mini-OMD (Figure S4) to assess the model’s versatility and scalability. These results indicate that although this open-loop setup can achieve rolling motions in a range of conditions, a closed-loop control system was crucial for precise control, particularly when environmental conditions differed from model assumptions. Closed-Loop Control of the Oloid In real-world medical settings, magnetic coupling cannot be reliably assumed, necessitating device localization. Additionally, manufacturing imperfections and environmental factors lead to deviations from the model, prompting the development of a differential geometry-based closed-loop control system for the oloid (detailed in Materials and Methods). Here the oloid's closed-loop controllability and therefore its potential for innovation in medical devices was evaluated. The OMD was tested on surfaces simulating Figure 4: Comparison of 3-DoF orientation control between the Oloid Magnetic Endoscope (OME) and the Magnetic Flexible Endoscope (MFE). The figure illustrates the coupling between positive and negative roll, tilt, and yaw for the MFE (green) and OME (purple). Radar plots for each DoF display the average absolute deviations in roll, tilt, and yaw (measured in degrees) across three repetitions (Figure S2). The scale bars represent 10 mm. See Movie S2 for related multimedia. Greenidge et al. This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 6 Figure 5: Open-loop control of the Oloid-shaped Magnetic Device (OMD) on various surfaces. The oloid’s rolling behavior on (A) silicone, (B) bumpy foam, (C) flat foam and (D) Perspex, compared to the predicted roll from the oloid model. Each snapshot shows the EPM position and orientation for reference. The scale bars represent 30 mm. See Movie S3 for related multimedia Figure 6: Closed-loop control of the Oloid-shaped Magnetic Device (OMD) with step input on various surfaces. Rolling performance is shown on (A) a flat, non-lubricated surface, (B) a curved, non-lubricated surface, (C) a flat, lubricated surface, (D) a curved, lubricated surface. Snapshots at times t1, t2, and t3 progress from left to right. The scale bars represent 30 mm. See Movie S4 for related multimedia. Greenidge et al. This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 7 the internal GI tract structures, such as mucus-lubricated curved (colon, esophagus) and flat (stomach) surfaces. Initial tests on a non-lubricated flat surface established a performance baseline (Figure 6A). Subsequently, more complex conditions were introduced, including lubricated (Figure 6C & D) and curved surfaces (Figure 6B & D). On the initial flat surface, a 0° to 180° step function input demonstrated a full range of motion. For lubricated and curved surfaces, a 0° to 90° step function input was applied. The results, exhibited in Figure 6 (see Figure S5 for extended results) and Movie S4, highlight the system's adaptability and precise control across all test conditions. The lubricated surfaces enabled the oloid’s ability to decouple roll from translation and generate pseudo-on-axis roll effectively. In vivo Rolling and Sweeping Motions For a practical in vivo demonstration of the system's clinical relevance, we selected a porcine model due to the similarity of porcine and human GI anatomy. The primary goal of the in vivo trials was to validate the OME’s ability to perform controlled rolling and sweeping motions in realistic conditions of friction and tissue interaction. Two distinct experiments were designed to support these capabilities: one to observe the OME’s sweeping motion across the top half of the lumen of the colon and the other to assess its pure rolling motion within a ±50° range. The results, displayed in Figure 7 and Movie S5, include snapshots from a separate standard endoscope camera (see Figure S8) capturing the sweeping and rolling motions of the OME. Notably, the sweeping motion, which combined horizontal translation and roll motion to produce an arch-like effect, achieved a range of ±60°. This combined motion enabled radial scanning by the sensor, with rolling adjusting the probe's orientation and translation moving the endoscope across the surface. Pre-clinical Validation – Virtual Biopsy 3D Reconstruction The primary motivation of this work was to enable virtual biopsies in MFEs, to enhance diagnostic capabilities beyond those of standard flexible endoscopes. Virtual biopsies allow for detailed tissue analysis, such as assessing lesion malignancy and margins, without the need for physical biopsies. The OME is sensor-agnostic, however, for demonstration purposes, we integrated a 32-element 28 MHz µUS array called the OME-U (see the “Ultrasound Integration” section). By combining our autonomous sweeping algorithm (see the “Autonomous Sweeping” section) with precise six DoF localization, the system generated comprehensive µUS imaging datasets. These datasets integrated high-quality 2D ultrasound images with positional data, allowing for the creation of high-fidelity 3D reconstructions of target areas. This process is outlined in the “3D Reconstruction” section. Preliminary validation was conducted on a benchtop setup with a silicone phantom and the OME-U (Figure 2C(iii)). The phantom included copper bands as echogenic subsurface targets. Signals were accurately captured and reconstructed, confirming system precision (see Figures S6, S7 and Supplementary Methods).Further validation was conducted in vivo by performing an autonomous sweep over healthy tissue followed by a simulated flat polyp in the same region of the porcine colon, created by injecting submucosal lifting agent (see Figure 8A and Movie S6). The 3D reconstructed volumes were visualized dynamically using MATLAB (Figure 8C and Movie S6), allowing operators to rotate, translate, and zoom. An isosurface representation feature enabled detailed inspection of tissue features (Figure 8D) through customizable visibility thresholds (see “3D Reconstruction”). Figure 7: In vivo sweeping and rolling of the Oloid Magnetic Endoscope (OME). Selected views from the standard endoscope camera and mirrored simulated front views show the motion of the OME’s sensor area and camera during (A) sweeping and (B) rolling motions. The black circle with blue and green arrows represents the OME’s onboard camera and its frame, while the purple arrow indicates the planned motion of the sensor area. The scale bars represent 20 mm. See Movie S5 for related multimedia. Greenidge et al. This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 8 Figure 8: In vivo subsurface 3D reconstruction of micro-ultrasound images for achieving virtual biopsies with the Oloid Magnetic Endoscope (OME). (A) Endoscopic view showing stages 1-3 of polyp creation. (B) 2D Ultrasound images: (B(i)) without polyp and (B(ii)) with polyp, (C) 3D Reconstruction of the ultrasound scans: (C(i)) without polyp and (C(ii)) with polyp, with the red square indicating the position of the 2D images (B(i)) and (B(ii)) within the 3D scan. (D) 3D Isosurface rendering highlights extracted features of interest: (D(i)) without polyp and (D(ii)) with polyp. See Movie S6 for related multimedia. This was particularly effective in visualizing the flat polyp structure within the 3D volume (Figure 8D(ii)), showcasing the diagnostic potential of the approach. While elements at depth can be observed even in the no polyp case, these features are less intense and inconsistent, whereas the polyp case shows a discontinuity in two layers that meet at both ends. The accuracy of the 3D reconstruction was assessed by comparing the reconstructed volume of the polyp against the injected volume, showing a 9.6% overestimation (injected volume: 1 ml, reconstructed volume: 1.106 ml). DISCUSSION This work introduces an approach to closed-loop roll control in magnetic medical devices, particularly MFEs, using an oloid-shaped design. This shape provides additional dexterity, enabling controlled rolling without additional power consumption or actuation modes, ideal for endoscopes including untethered capsule endoscopes. A precise closed-loop control scheme that integrates MFE localization was developed, addressing scenarios where perfect magnetic coupling cannot be assumed. Roll control was effectively demonstrated in benchtop trials and subsequently in in vivo trials. This study has demonstrated the approach across different scales (OMD, OME, and mini OMD) and control sources (single EPM and electromagnetic coil system), in both tethered and untethered configurations. Although our differential geometry-based model provides a framework for broad clinical applications using any magnetic field source with at least five magnetic control inputs, comprehensive parametric studies are needed to validate utility in other applications. For example, extending this approach to more dynamic surfaces than the colon will require realtime model parameter estimation. The design and in vivo testing of the OME within the colon demonstrated the capacity of the oloid to be integrated into a device that met the specific design requirements for colonoscopy achieving safe rolling motion and an application-specific sweeping motion. Our results also demonstrated that the OME could roll, tilt, and yaw with greater independence and stability Greenidge et al. This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science Robotics on 26/03/2025, DOI: 10.1126/scirobotics.adq4198. 9 than cylindrical MFEs, which do not have active roll control. Previous work (3) was confined to 2D scans using singleelement transducers, as linear arrays were impractical due to the absence of roll control. By leveraging roll control, our approach enabled robotically controlled, autonomous sweeping to create 3D µUS images with deeper anatomical reach than transrectal µUS. These virtual biopsies offer potential real-time diagnostic insights without requiring physical samples. The OME’s sensor-agnostic design supports the integration of various diagnostic or therapeutic modalities, such as Optical Coherence Tomography (34) or therapeutic lasers (35), making it adaptable for future applications. In addition, roll control could facilitate precise interventions like submucosal dissection, and targeted ultrasound-triggered drug delivery (36, 37). The oloid-based roll control system and autonomous virtual biopsies contribute to the expanding autonomy of MFEs, which includes autonomous navigation (38), polyp detection, and shared control physical biopsy tasks (20). These advancements could allow endoscopists to focus on critical diagnostic and therapeutic decisions while autonomous systems handle routine navigation and tasks (39). This could also reduce the training times for endoscopists and potentially allow multiple procedures to be supervised simultaneously. The single EPM system of the MFE has been validated in human trials with patients of normal Body Mass Index (BMI) (19), however, patients with higher BMI pose challenges due to increased EPM-IPM distances, reducing magnetic force and torque. Increasing the EPM size to generate stronger magnetic fields, can potentially address these limitations. Additionally, the current single EPM system limits vertical (z-axis) control, causing the endoscope to remain in constant contact with the top half of the lumen of the colon. To extend the sweeping range to include the lower half, patient repositioning is required. This design approach is consistent with standard clinical practice, where patient repositioning is a method for ensuring comprehensive colon examination (40). However, implementing more complex field generation systems (1, 14–16) would enable full 360° scanning in scenarios where patient repositioning is not possible. Rolling demonstrations with the electromagnetic coil system revealed this potential, showing successful oloid movement across lower surfaces. Furthermore, the OME’s enhanced dexterity and diagnostic capabilities have the potential to address gender disparities in colonoscopies, as standard flexible endoscope procedures tend to be more challenging in women, leading to higher rates of incomplete procedures and lower adenoma detection rates (41). In conclusion, the oloid shape facilitates clinically applicable torque generation around the magnetization axis in magnetic medical robots, enhancing the dexterity, diagnostic capabilities, and autonomy of MFEs and magnetic medical robots overall. This approach sets the stage for more autonomous and efficient medical procedures deep within the anatomy. With ongoing clinical validation, such advancements have the potential to transform minimally invasive diagnostics and treatments, making them more accessible and effective for a broader range of patients. MATERIALS AND METHODS The Oloid Model For the successful control of magnetically manipulated oloid-shaped devices, a deep understanding of the oloid’s motion was crucial. Although previous studies have modeled the behavior of the oloid—or “two-circle roller”—on flat surfaces (33, 42, 43), our work is based on an adaptation of the differential geometry framework by Dirnbock et al. (33) where full derivations can be found. In this work, the application of the previous model was extended to curved and lubricated surfaces for any defining oloid radius, 𝑟. We assign a fixed coordinate frame {F} with orthonormal vectors 𝐈,𝐈𝐈,𝐈𝐈𝐈 ∈ℝ3 on the plane for the initial position and orientation of the oloid. As the oloid moves along the plane, its motion can be parametrized by the arc length 𝑢 (𝑟𝑎𝑑𝑖𝑎𝑛) of the contact point, 𝐏1, on the edge of one of its circles over the region: 𝑢 ∈(−2π 3,0)∪(0,2π 3) (3) If {O} denotes the coordinate frame with orthonormal vectors 𝐱,𝐲,𝐳∈ℝ3 at the geometric center of the oloid, then the homogeneous transformation matrix of {O} with respect to {F} is: 𝐓𝐹 O=[𝐑𝐹 O𝐭𝐹 O 𝟎 1]∈𝑆𝐸 (3) (4) where, using 𝑠 =sin (𝑢) and 𝑐 =cos (𝑢) for brevity: 𝐭𝐹 O=𝑟√3 9 [ 𝑐𝑠√1+2𝑐 2(1+𝑐)√2(1+𝑐) +sign(𝑢)arccos(𝑢) 𝑐√2 √1+𝑐 15+13𝑐−𝑐2 2(1+𝑐) +ln ( 2 1+𝑐) 3√3(2+𝑐) 2√2(1+𝑐) ] ∈ℝ3 (5) is the translation vector and: Supplementary Materials for Harnessing the oloid shape in magnetically driven robots to enable highresolution ultrasound imaging Nikita J. Greenidge et al. Corresponding author: Nikita J. Greenidge, [email protected] The PDF file includes: Supplementary Discussion: Addressing the DoF Limitation in Magnetic Manipulation Roll Generation in the Oloid Supplementary Figures: 3 DoF Orientation Comparison between OME and MFE Generating Rolling Motion with the Oloid in Open-Loop Closed-loop Control of the Oloid-shaped Magnetic Device (OMD) Supplementary Methods: Benchtop Ultrasound Autonomous Sweeping Setup Contact Detection Algorithm In Vivo Experimental Setup 3D Subsurface Reconstruction Algorithm Benchtop 3D Reconstruction Validation ROS Interface Localization Calibration Other Supplementary Materials for this manuscript include the following: Movies S1 to S6 (https://www.youtube.com/playlist?list=PLWtIpCj5v7Nez_fUVWN239PC-4kG1nhsh) Addressing the DoF Limitation in Magnetic Manipulation In an attempt to minimize the potential energy between an external magnetic field 𝐁𝑬∈ℝ3(T) generated by the controlling source (external dipole) and a magnetic object (internal dipole) with magnetic moment, 𝐦I∈ℝ3(A∙m2) , an alignment torque 𝛕m is induced on the magnetic object and is given by: 𝛕m=𝐦I×𝐁𝑬=𝑆(𝐦I)𝐁𝑬 (1) where 𝑆(⋅):ℝ3↦ℝ3×3 is the skew-symmetric matrix operator. The gradient-induced force exerted on the magnetic object is denoted by 𝐅m : 𝐅m=∇(𝐁𝑬∙𝐦I) (2) In the magnetic object’s local coordinate frame with orthonormal vectors 𝐗𝑰,𝐘I, and 𝐙I∈ℝ3, Euler angles, ϕ,θ and ψ denote the rotation around these vectors (Figure 3). Conventionally, 𝐗𝑰∥𝐦I (as shown in Figure 1 and 3), but when 𝐦I∥𝐁𝑬 then according to Equation 1, 𝛕m= 𝟎. Consequently, it is not possible to generate magnetic alignment torque around the magnetization axis to control the roll angle (ϕ). Paper Magnetic Workspace Environment Required magnetically controllable inputs Applied field Maximum Power Roll Wang et. al (13) 100 x 100 x 100 mm3 Water 8 N/A 1.4 kW Yes Xu et al (10) N/A Non-Newtonian fluid (paraffin oil, glycerol) 8 20 mT 6 kW Yes Diller et al (8) 20 × 20 × 20 mm3 Silicone oil 8 8.3 mT N/A Yes Giltinan et al (11) 1000 mm3 Silicone oil 8 22 mT 6 kW Yes Taddese et al (44) 300 x 300 x 150 mm3 * In vivo - colon 5 25 mT 1-1.5 kW No This work 300 x 300 x 150 mm3 * In vivo - colon 5 25 mT 1-1.5 kW Yes Table S1: Roll Approaches Comparison. All values were estimated based on the details provided in the respective papers. N/A – information not provided. *This workspace is based on the static range of the EPM which can be moved within a larger workspace (1000 x 800 x 800 mm³) due to the mobility provided by the robot arm. As discussed in the manuscript, other methods have primarily been demonstrated in fluid environments where force and torque requirements are low. These systems typically rely on electromagnetic coil systems, requiring eight magnetically controllable inputs, leading to high power consumption and limited workspaces compared to single EPM systems. Table S1 highlights how the introduction of roll in the single EPM system represents a significant improvement over previous approaches. In electromagnetic coil systems, the workspace and applied field are proportional to power consumption. Roll Generation in the Oloid By concatenating orthonormal basis vectors of ℝ3 we can formulate a rotation matrix, 𝐑O F which can be expressed as: 𝐑O F=[𝐱ˆ𝑂𝐲ˆ𝑂𝐳 ˆO]. If: The oloid is magnetized about the 𝐱-axis 𝐱ˆ𝑂∥𝐦, and It rotates about the line 𝐩2−𝐩1, then Its orientation can be controlled so long as 𝐱ˆO⊮𝐩2−𝐩1. In other words, rotation can be induced so long as 𝐱ˆO is not perpendicular to the surface normal 𝐳 ˆF : 𝐱ˆO⊥𝐳 ˆF⟹𝐱ˆO T𝐳 ˆF≠𝟎 Where 𝐳 ˆF is equivalent to the z-axis of {F}. From inspection of 𝐑O F it must hold that: −sin (𝑢) √2(1+cos (𝑢))≠0∀𝑢 ∈(−2𝜋 3,0)∪(0,2𝜋 3) This is true only if: u=0, or 𝑢 =𝜋 Figure S1: Function plot over the range u. In a cylinder, this is true throughout the entire surface. 3 DoF Orientation Comparison between OME and MFE Figure S2: 3 DoF Orientation Comparison between OME and MFE. The figure illustrates the coupling between roll, tilt and yaw, positive and negative, for both the MFE and OME. It also depicts the variation in orientation during x and y translation for both devices. See Data S2 for the corresponding multimedia. The magnitudes of the points in the radar plot shown in Figure 4 of the manuscript were determined by calculating the average change in each DoF across the three repetitions presented for each movement in Figure S2. As evidenced by the raw data in Figure S2, the starting point for roll is consistently near zero in all the OME’s movements. In contrast, the MFE exhibits arbitrary baseline values for roll initiation. Additionally, there is increased variability during movements not intended to induce roll, such as tilt and translation. Lastly, when subjected to the same input intended to trigger roll in the OME, the MFE's roll response was ineffective, especially for positive roll. Generating Rolling Motion with the Oloid in Open-Loop Using MFE System Extended representation of Figure 5 in the manuscript. Figure S3: Oloid Rolling on Various Surfaces. Rolling performance of the oloid on (A) silicone, (B) bumpy foam, (C) flat foam, and (D) Perspex. Five repetitions shown for each. See Data S5 for the corresponding multimedia. Scale bars, 10 mm. Using Electromagnetic Coil System The oloid control model is generic and not exclusively designed for use with a robotic arm and an external permanent magnet. To support this claim, we demonstrate that, by applying a similar model to the one described in the paper, the oloid shape can perform a rolling motion on a flat surface, using electromagnetic coils. For this demonstration, a MiniMag system (MagnebotiX, Switzerland) shown in Figure S4 (A) was employed. In order to fit this setup and to demonstrate scalability, the Oloid Magnetic Device (OMD) was scaled down to a defining oloid radius of 5 mm called the Mini-OMD. Snapshots of the rolling motion are illustrated in Figure S4 (B). This result demonstrates scalability, a tetherless configuration and that roll can be achieved with the oloid regardless of contact with the upper or lower surfaces. Figure S4: Oloid Rolling in MiniMag. (A) Experimental Setup and (B) Snapshots of a 5 mm Mini OMD rolling on a flat surface. See Data S1 for the corresponding multimedia. Closed-Loop Control of the Oloid-shaped Magnetic Device (OMD) Extended representation of Figure 6 in the manuscript. Figure S5: Closed-loop control of the Oloid-shaped Magnetic Device (OMD), step input on a: (A) flat non-lubricated surface (B) curved non-lubricated surface (C) flat lubricated surface (D) curved lubricated surface. Five repetitions shown for each. Small oscillations are observed especially on the lubricated surfaces from a combination of localization errors and inherent dynamics of the KUKA robot’s control system. In scenarios involving faster environments, such as the lubricated surfaces, these oscillations become more pronounced. To reduce oscillations, KUKA’s Fast Robot Interface (FRI) could be implemented in place of the current smart servo control. The FRI offers higher update rates and more precise control, which can significantly minimize oscillatory behavior by enabling faster and more accurate responses to control inputs. Benchtop Autonomous Ultrasound Sweeping Setup The capsule was evaluated using the platform described in Figure S6, which includes the OMEU, the KUKA LBR robotic arm, and the micro-ultrasound (µUS) system. A silicone phantom was used for benchtop trials and the initial characterization of the robotic system. Two display monitors were employed: one for the robot control user interface and the other for ultrasound visualization. The capsule's dexterity was recorded using a Basler Ace camera (acA2040-120uc, Basler AG, Ahrensburg, Germany), as depicted in the red square in Figure S6. Figure S6: Detailed illustration of the experimental setup. Overview of the various components utilized during the benchtop experimentation with (red square) validation point of view for contact detection and roll/sweep performance and (purple square) snapshot depicting feedback on contact quality (1-2: good, 3: none) and visualization of the copper strip (2). The 8 mm thick silicone phantom (Ecoflex 00-30, Smooth-on, PA, USA), utilized to validate different algorithms, was enclosed within a 7 cm inner diameter acrylic tube. Copper strips were integrated into the phantom to create components with high echogenicity, facilitating clear identification with the ultrasound probe. The white area in the second image (enclosed by the purple frame) becomes apparent when the array is oriented towards the copper strip, confirming both the proper functioning of the OME-U and the probe's ability to capture images simultaneously. The green and red bars at the top of the snapshot, enclosed by the purple frames, highlight the quality of contact between the OME and the silicone phantom. ROS Interface Robot Operating System (ROS) was utilized to manage all aspects of our robotic endoscope’s functionality. The KUKA LBR Med R820 robot is managed in joint space through the `iiwa_stack` software package (version 1.3.0) (https://github.com/IFLCAMP/iiwa_stack/wiki), which seamlessly integrates with ROS Melodic (Ubuntu 18.04). MATLAB was utilized for ultrasound data processing, and the processed data were transmitted via the Olympus Decklink Mini Monitor 4K frame-grabber which interfaces with ROS for realtime processing. Figure S11: ROS Interface Localization Calibration The localization system in the endoscope is calibrated using a calibration cube precisely tailored to the endoscope's overall shape. The cube, machined from a solid block of Delrin, ensures sub-degree accuracy by providing dedicated features that secure the endoscope in a reference orientation relative to gravity. This process computes the orientation of the IMU with respect to the endoscope reference frame and the relative orientation of each sensor with respect to the IMU. The calibration only involves orientations, and so, the position of the endoscope during calibration is not critical. The main variable affecting accuracy is the orientation of the surface upon which the cube rests during data acquisition, which should be flat with respect to gravity. The calibration process is robust, with errors up to 5% of the sensors' range having negligible effects on localization. To address potential drift over repeated uses, the system incorporates a secondary 'on-the-fly' gyro bias removal calibration. This feature ensures consistent calibration over more than 10 procedures (typically no more than 30 minutes), with any constant bias easily detected by placing the endoscope in a known pose and verifying static accuracy. These measures ensure the robustness and reliability of the localization system, even under conditions that may introduce minor disturbances. Other Supplementary Material Movies:  Movie S1: Concept Overview of the Oloid-Shaped Magnetic Medical Device.  Movie S2: Oloid Magnetic Endoscope (OME) versus Magnetic Flexible Endoscope (MFE) Degrees of Freedom (DoFs)  Movie S3: Open-Loop Control of the Oloid-Shaped Magnetic Device (OMD).  Movie S4: Closed-Loop Control of the Oloid-Shaped Magnetic Device (OMD).  Movie S5: In Vivo Rolling and Sweeping Motion of the Oloid Magnetic Endoscope (OME).  Movie S6: In Vivo Autonomous Sweeping of the Oloid Magnetic Endoscope with Integrated Micro-Ultrasound (OME-U) and 3D Virtual Biopsy Reconstruction. https://www.youtube.com/playlist?list=PLWtIpCj5v7Nez_fUVWN239PC-4kG1nhsh Data:  Data S1: MiniMag-miniOMD.mp4: Video showing mini-OMD oloid rolling in the MiniMag electromagnetic coil system.  Data S2: DOF_comparison.zip: This folder contains video footage with live Euler angle graphs comparing the degrees of freedom (DOF) of the system under different conditions.  Data S3: 3D_Reconstruction.zip: This folder contains data related to the 3D reconstruction experiments.  Data S4: In_vivo_-_roll_sweep.zip: This folder contains in vivo roll sweep experiments, including .bag files, endoscopic videos and synced demonstration videos.  Data S5: Open_loop_roll.zip: This folder contains videos from the open-loop roll experiments on different surfaces.  Data S6: Oloid_development.m: MATLAB script to apply a transformation to a single point using Theorem 4 to handle the full transformation for the oloid geometry.  Data S7: Parametric_Oloid_Surface.m: MATLAB Script to plot the oloid shape based on parametric equations.  Data S8: Theorem4_sym.m: MATLAB Script with function for Theorem 4 from the paper by Dirnbock and Stachel (1997), "The Development of the Oloid" to generate the transformation matrices for the oloid's motion. Dryad Database: https://doi.org/10.5061/dryad.t1g1jwtbx