Magnetic Miniature Actuators with Six-Degrees-of-Freedom Multimodal Soft-Bodied Locomotion

Magnetic miniature robots (MMRs) are mobile actuators that can exploit their size to noninvasively access highly confined, enclosed spaces. By leveraging on such unique abilities, MMRs have great prospects to transform robotics, biomedicine, and materials science. As having high dexterity is critical for MMRs to enable their targeted applications, existing MMRs have developed numerous soft‐bodied gaits to locomote in various environments. However, there exist two critical limitations that have severely restricted their dexterity: 1) MMRs capable of multimodal soft‐bodied locomotion have only demonstrated five‐degrees‐of‐freedom (five‐DOF) motions because the sixth‐DOF rotation about their net magnetic moment axis is uncontrollable; 2) six‐DOF MMRs have only realized one mode of soft‐bodied, swimming locomotion. Herein, a six‐DOF MMR is proposed that can execute seven modes of soft‐bodied locomotion and perform 3D pick‐and‐place operations. By optimizing its harmonic magnetization profile, the MMR can produce 1.41–63.9‐fold larger sixth‐DOF torque than existing MMRs with similar profiles, without compromising their traditional five‐DOF actuation capabilities. The proposed MMR demonstrates unprecedented dexterity: it can jump through narrow slots to reach higher grounds; use precise orientation control to roll, two‐anchor crawl, and swim across tight openings with strict shape constraints; and perform undulating crawling across three different planes in convoluted channels. An interactive preprint version of the article can be found at: https://www.authorea.com/doi/full/10.22541/au.164087652.25227465.

the elastomer-based ones also possess higher stiffness that can allow them to transmit forces more effectively to the environment and other objects. [28,30,64] By having adequate stiffness, the elastomer-based MMRs can induce sufficient impact from the substrate to produce a jumping locomotion too, [10] and such complex gaits have not been demonstrated by the droplet-based soft MMRs. Due to the advantages of elastomer-based soft MMRs, we will be focusing on this class of actuators, and soft MMRs will only be referred to those that have elastomeric bodies in the subsequent text.
In recent years, the creation of soft MMRs that can generate multiple modes of locomotion represents a significant advancement for small-scale actuators. [10,53,72] For instance, the amphibious soft MMRs of Hu et al. are able to execute seven modes of soft-bodied locomotion, which can enable them to navigate across terrestrial and aquatic environments. [10] Likewise, the soft MMRs of Ren et al. can use three different modes of locomotion to move in fluidic environments. [53] Despite these advancements, soft MMRs with multimodal locomotive gaits are restricted to having five-degrees-of-freedom (five-DOF) motions because the sixth-DOF rotation about their net magnetic moment axis is not possible. [10,53,72] As a result, these MMRs can only rotate about two axes and translate along three axes, [42] and their dexterity is still far from being optimized. While soft MMRs with six-DOF do exist, such actuators have so far only realized one mode of swimming locomotion. [43] This is because existing six-DOF actuators are currently limited to having a 1D magnetization profile for their soft components, [43] so that their changes in geometry and magnetization profile are easy to account for during deformation. Although it is easy to implement six-DOF control on soft MMRs with 1D magnetization profile, such simple profiles have also severely constrained the dexterity of these actuators. The feasibility of executing six-DOF control on sophisticated MMRs, which have 2D or 3D magnetization profiles for their soft components, has not been explored yet. Due to these aforementioned limitations, the dexterity of all the existing soft MMRs is still severely restricted, and these actuators will not be able to overcome difficult barriers that have strict shape constraints. As a result, existing MMRs would not have sufficient agility to effectively negotiate across highly unstructured environments, and this poses a critical limitation for their targeted applications. In view of these challenges, it is therefore highly desirable to create dexterous MMRs that can concurrently possess six-DOF and perform multiple modes of soft-bodied locomotion.
Here, we propose a six-DOF soft MMR that can roll, twoanchor and undulating crawl, jump, swim with two different gaits, and climb a meniscus. Because the proposed MMR has six-DOF, its orientation can be precisely controlled when it executes these soft-bodied locomotive gaits, thereby allowing it to achieve very high dexterity. As the proposed six-DOF MMR can perform seven modes of soft-bodied locomotion, its dexterity will also be significantly higher than existing six-DOF soft MMRs that can only execute one type of swimming gait. [43] Our proposed actuator is made up of a central magnetic beam with two identical buoyant components attached to its free ends ( Figure 1A(i) and see Section S1, Supporting Information, for the fabrication process and material properties of the MMR). The buoyant components are used to make the proposed soft MMR more neutrally buoyant so that it can easily swim against gravity in aquatic environments. To investigate the feasibility of implementing six-DOF control for a soft MMR with 2D magnetization profile, a single-wavelength, harmonic magnetization profile ðMÞ is programmed into our compliant beam component ( Figure 1A(ii)). We have selected a harmonic magnetization profile because this type of profile has proven to be effective for enabling multimodal locomotion. [10,53] To maximize the producible sixth-DOF torque of our soft MMR, the phase shift angle of its magnetization profile, φ, has been optimized to À90°( Figure 1A(ii) and Section S3, Supporting Information). With the selected φ, our proposed MMR can theoretically produce 1.41-63.9-fold larger sixth-DOF torque than existing soft MMRs that have a harmonic magnetization profile, i.e., those with φ of 0°[ 63,66,67,78] and 45° [ 10,53,79,80] (Section S3, Supporting Information). Because the magnitude of our MMR's achievable net magnetic moment,m, is equal to all other soft MMRs with a harmonic magnetization profile (Section S3, Supporting Information), this implies that all of these MMRs will have the same actuation capabilities for their traditional five-DOF motions. [42,43] The enhancement in the sixth-DOF torque for our proposed MMR has therefore been achieved without compromising their traditional five-DOF actuation capabilities. To have a fair comparison for our actuation analysis in Section S3, Supporting Information, we assume that all of these MMRs have the same geometries and material properties, and the key difference between them is that they possess different values of φ in their M. We have also constrained the harmonic magnetization profile of all the MMRs to be single wavelength in our analysis (Section S3, Supporting Information) because this has been a critical requirement for enabling multimodal soft-bodied locomotion. [10,53] Using our optimized harmonic magnetization profile, the proposed MMR can be actuated to produce its desired softbodied locomotion and concurrently achieve optimal six-DOF motions. As our proposed soft MMR has adequate stiffness, it can also execute 3D pick-and-place operations.

Actuation Principles
To describe the physics of our proposed soft MMR, we first attach a material reference frame to this actuator ( Figure 1A(ii)). The axes and vectors expressed in this reference frame will be denoted by a subscript fMg. When the MMR is undeformed, itsm will be a null vector (Section S2B, Supporting Information). However, when a magnetic field,B, is applied along the positive z fMg -axis, the MMR will deform and assume a "U"-shaped configuration ( Figure 1B(i) and see Section S2A, Supporting Information, for the physics that describes this deformation). In this deformed configuration, the MMR will possess an effectivem that is parallel to the appliedB. Conversely, the MMR will assume an inverted "V"-shaped configuration if the direction ofB is reversed ( Figure 1B(ii)). In the inverted "V"-shaped configuration, them of the soft MMR will be along the negative z fMg -axis, parallel to the appliedB ( Figure 1B(ii)). By adjusting the magnitude ofB, we can also actively control the curvatures of the MMR in these deformed configurations. Specifically, the curvatures will become sharper and gentler when the magnitude ofB is increased and decreased, respectively   ii) The front view of the MMR that illustrates the harmonic magnetization profile of the beam with a phase shift angle (φÞ of À90°. A material reference frame is attached to the MMR and its axes are denoted by the subscript fMg. B) The two types of deformed configurations of the soft MMR: i) the MMR assumes a "U"-shaped configuration when the magnetic field ðBÞ is applied along the positive z fMg -axis. The deformed MMR will produce a corresponding net magnetic moment (mÞ that is parallel toB. ii) The MMR will adopt an inverted "V"-shaped configuration when the direction ofB is reversed. In this configuration, the MMR will possess an effectivem along the negative z fMg -axis, parallel toB. The theoretical deformation of the magnetic beam is represented by the yellow curves, and the rotation about the MMR's m represents the sixth-DOF rotation. C) The spatial reference frames used for describing the MMR's orientation: i) the global, ii) intermediate, and iii) local reference frames, and their axes are denoted by the subscript fGg, fIg, and fLg, respectively. For the intermediate and local reference frames, their respective z-axis is aligned to the MMR'sm. D) The proposed six-DOF control strategy of our MMR. We determine the requiredB andB grad that can make the MMR's desired orientation into a minimum potential energy configuration. Scale bar: 2 mm.
(Section S2A, Supporting Information). In general, our theoretical model of the MMR's deformation characteristics in Section S2A, Supporting Information, agrees with the experimental data ( Figure 1B), and this enables us to control the geometries of the MMR well. Having precise control over the curvatures of the MMR will be a critical requirement for the actuator to enable its desired modes of soft-bodied locomotion.
To analyze the proposed MMR's achievable motions in its deformed configurations, a spatial reference frame is attached to its body ( Figure 1C(iii)). This reference frame represents the local reference frame of the actuator, and the axes and vectors expressed in this frame are denoted by a subscript fLg. Because it will be easier to analyze the proposed MMR if its sixth-DOF axis is parallel to one of the principal axes in the local reference frame, [43,54] here we assign z fLg -axis to be aligned with the MMR'sm in its "U"-shaped configuration ( Figure 1C(iii)). Based on our local reference frame assignment, the MMR's rotation about the z fLg -axis is therefore its sixth-DOF motion. [43,54] This frame assignment also indicates that the MMR'sm will be along the negative z fLg -axis when the actuator deforms to its inverted "V"-shaped configuration. To fully describe the orientation of the proposed MMR, we also introduce an intermediate reference frame and a global reference frame ( Figure 1C where R x and R y represent the standard x-axis and y-axis rotational matrices, respectively (Section S2B, Supporting Information). The variables, α and β, represent the corresponding desired angular displacements about these axes ( Figure 1C(i)-(ii)), while the vector,ṽ, represents an arbitrary vector. It is noteworthy that the z fIg -axis of the MMR defines the desired orientation of itsm, implying that the direction of the sixth-DOF axis can be fully defined based on the values of α and β. In a similar way, the transformation between the intermediate and local reference frames can be described mathematically as where R z represents the standard z-axis rotational matrix (Section S2B, Supporting Information) and θ represents the desired sixth-DOF angular displacement of the MMR ( Figure 1C(ii)-(iii)).
To enable precise orientation control for the proposed MMR, we controlB and its independent spatial gradients,B grad , in such a way that the desired orientation of the MMR becomes a minimum potential energy configuration (Section S2B-C, Supporting Information). Under the influence of these actuating magnetic signals, the deformed MMR will constantly experience three axes of restoring torques until it self-aligns to the desired orientation ( Figure 1D and Section S2B, Supporting Information). The presence of the restoring torques will also help the proposed MMR to reject external disturbances, enabling this actuator to maintain its desired orientation at all times (Section S2B, Supporting Information). Of the three axes of restoring torques, two of them are induced by the phenomenon in which the MMR'sm will tend to align with the appliedB. [23,30,42] By exploiting this phenomenon, we can therefore control the MMR's two axes of angular displacements, α and β, via specifying the direction ofB (Section S2B, Supporting Information). However, controlling the direction ofB will not allow us to exert the third restoring torque on the MMR, i.e., the restoring torque about its sixth-DOF axis. [42,43,54,81] The sixth-DOF restoring torque can instead be induced on the MMR via controllingB grad , and this torque will enable us to precisely control θ (Section S2B-C, Supporting Information). By controllingB grad , we can also apply desired magnetic forces on the MMR and allow it to translate along three axes so as to achieve another three-DOF (Section S2B, Supporting Information). A notable advantage of our control scheme is that we can fully decouple all the magnetic torques and forces applied to the MMR, effectively allowing the soft actuator to achieve six-DOF motions (Section S2B-C, Supporting Information). Together with controlling the time-varying deformations of the MMR, such six-DOF control will be critical for empowering our proposed actuator to execute multiple modes of soft-bodied locomotion with precise orientation control. Our six-DOF control strategy can be implemented successfully because we have modeled how the critical robotic parameters of our MMR will vary as the actuator changes its geometry andM during deformation (Section S2B-C and S3, Supporting Information). Examples of such robotic parameters will include but not limited to the MMR'sm and producible normalized sixth-DOF torque (Section S2B-C and S3, Supporting Information).

Experimental Results
We first evaluated the sixth-DOF torque, T z,fLg , producible by our proposed soft MMR. Because the producible T z,fLg of the MMR would vary when this actuator underwent different amounts of deformation (Section S3, Supporting Information), here we constrained the MMR to its undeformed configuration for easier characterization purposes. As the buoyant components of the proposed MMR were passive and they could not produce T z,fLg for the MMR, these components were removed in this experiment. The undeformed magnetic beam component of the MMR was fixed at the free end of another fixed-free beam in this experiment ( Figure 2A). By observing how the fixed-free beam would deform as we varied the applied actuating magnetic signals to the proposed MMR, we could deduce the producible T z,fLg of our actuator. This could be achieved by using a camera to record the angular deflection of the fixed-free beam's free end, θ tip , and subsequently applying the Euler-Bernoulli equation to deduce the generated T z,fLg (Figure 2A and Section S4B, Supporting Information). Based on this methodology, we established the relationship between the MMR's generated T z,fLg and the applied magnetic signals ( Figure 2B). Each data point in the graph of Figure 2B had five trials, and the gradient of the best fit line represented the MMR's producible T z,fLg after it was normalized according to the strength of the actuating signals. The normalized sixth-DOF torque of our MMR was experimentally evaluated to be 1.16 Â 10 À7 N m 2 T À1 , and there was a 13.7% deviation from our simulation predictions in Section S3, Supporting Information (predicted value: 1.02 Â 10 À7 N m 2 T À1 ). In general, the experimental data agreed with the prediction from our simulations, and the deviation could be caused by minor fabrication errors and parallax measurement errors from the camera. Another potential source of experimental error was that the MMR may not remain perfectly in its undeformed configuration during the experiments, and this might cause the evaluated normalized sixth-DOF torque to be larger than expected (Section S3, Supporting Information). As our theoretical model could also predict the MMR's deformation characteristics well ( Figure 1B and Section S2A, Supporting Information), the data in Figure 2B supported our hypothesis that the proposed MMR could indeed produce 1.41-63.9-fold larger T z,fLg than existing soft MMRs with single-wavelength, harmonic magnetization profiles [10,53,63,66,67,[78][79][80] (Section S3, Supporting Information). After investigating the producible sixth-DOF torque of our proposed MMR, we proceeded to evaluate the locomotion of this actuator.
Rolling could be considered as one of the fastest modes of terrestrial locomotion. [6,10] Our proposed MMR could execute this locomotion by first applying aB of 13 mT along its positive z fLgaxis so that it could assume a "U"-shaped configuration ( Figure 3A). It is advantageous to adopt such a configuration because this would be a favorable shape for the MMR to roll. [10] Subsequently, the MMR would be able to start rolling when we continuously rotate the magnetic field in the y fLg z fLg or z fLg x fLg planes (jBj ¼ 13mT). Because our proposed MMR had six-DOF, this actuator could choose to roll along its length (rotating about the x fLg -axis) or along its width (rotating about the y fLg -axis) ( Figure 3A and Video S1, Supporting Information). The rolling direction of the MMR could also be steered via controlling its angular displacement about the z fLg -axis withB grad ( Figure 3B and Video S1, Supporting Information). In comparison, rolling along the length of the MMR would be faster than rolling along its width because the length of our actuator was longer than its width ( Figure 1A and Section S4C, Supporting Information). This is assuming that the rotating frequencies ofB were identical for both rolling scenarios. Despite being slower, rolling along the width of the MMR had its unique merits too. Because the curvatures of the actuator could be controlled to be much gentler for this mode of rolling ( Figure S11, Supporting Information), the MMR could potentially squeeze through smaller openings if it was rolling along its width. By having two modes of rolling, the proposed MMR was therefore very robust. On obstacle-free terrains, it could choose to roll quickly along its length (2.46 mm s À1 with a 0.25 Hz rotatingB). Alternatively, it could choose to roll along its width if it had to negotiate across barriers with narrow openings in the environment (1.85 mm s À1 with a 0.25Hz rotatingB). While existing rolling MMRs were also steerable, they had only demonstrated rolling along their lengths. [10,29,53,82] As a result, our proposed MMR would be much more dexterous than existing rolling MMRs [10,29,53,82] because it would have greater versatility to roll across environments that have difficult barriers. To demonstrate the remaining DOF of our proposed MMR, we commanded it to translate along its three principal axes in the local reference frame ( Figure 3C and Video S1, Supporting Information). Before executing these translations, we first applied aB along the z fLg -axis of the MMR so that it could deform into a "U"-shaped configuration to possess an effectivem. CorrespondingB grad was then applied to exert the desired magnetic forces on the MMR to translate it along the x fLg -, y fLg -, and z fLg -axes ( Figure 3C and Video S1, A) The magnetic beam component of the MMR was fixed at the free end of a larger fixed-free beam. As the sixth-DOF torque of the magnetic beam (T z,fLg Þ was generated, the fixed-free beam's angular deflection at the free end, θ tip , would be recorded by a camera. In this experiment, the magnetic beam was constrained to be in its undeformed configuration. B) The experimental plot of the deduced T z,fLg against the actuating magnetic signals ∂B x,fLg ∂y fLg . Five trials were conducted for each data point and the error bars represented the corresponding standard deviation of the data points. The gradient of the best fit line was computed to be1.16 Â 10 À7 N m 2 T À1 , and it represented the normalized sixth-DOF torque of the proposed MMR (in its undeformed configuration). Scale bar: 2 mm.

MMR Trajectory
Rotation about Figure 3. The rolling, translations, and two-anchor crawling locomotion of the proposed MMR. Except for (B), the remaining subfigures were created by superpositioning the snapshots of the MMR at different timestamps in Video S1 and S2, Supporting Information. A) The rolling locomotion of the MMR: i) rolling along the length of the MMR by continuously rotating it about the x fLg -axis. ii) Rolling along the width of the MMR by continuously rotating it about the y fLg -axis. This mode of rolling could be accomplished with gentler curvatures compared to rolling along the MMR's length. B) The MMR's sixth-DOF rotation (rotating about the z fLg -axis). This rotation enabled the MMR to steer its rolling locomotion. A red line was added to one of the MMR's free ends to better illustrate this rotation. C) Translating the MMR in a paraffin oil reservoir. The MMR was commanded to translate along its x fLg -, y fLg -, and z fLg -axes as shown in (i), (ii), and (iii), respectively. Together with the rolling locomotion, these translations indicated that the proposed MMR had six-DOF, i.e., having three types of rotations and three types of translations. D) By controlling the angular displacements about its x fLg -axis, the MMR could use the two-anchor crawling locomotion to ascend an inclined slope of 20°. E) Using the two-anchor crawling locomotion, the MMR could move across a narrow path with strict shape constraints. This was possible because the MMR could precisely control its angular displacement about the y fLg -axis to adopt a favorable shape for crossing this narrow passage. As the MMR could also precisely control its sixth-DOF angular displacement about the z fLg -axis, it could also execute a sharp turn at the junction of a confined "L"-shaped path. The legend shows the local reference frame of the MMR as well as the types of arrows representing rotations about different axes and the MMR's trajectory. Scale bar: 2 mm.
www.advancedsciencenews.com www.advintellsyst.com Supporting Information). During the translations along the x fLg -and y fLg -axes, an upward magnetic force was also applied to the MMR so that it could reduce the normal force and thus friction induced by the substrate (Figure 3C(i)-(ii)). A notable observation in the experiments shown in Figure 3A-C and Video S1, Supporting Information, was that the rotations and translations of the proposed MMR could be fully decoupled, implying that this actuator was able to attain six-DOF control. It would be advantageous to be able to apply magnetic forces on the proposed MMR because these forces could potentially enhance the efficiency of the actuator's locomotion. [10] For instance, we could apply upward forces on the MMR so that it could reduce its friction with the ground, jump higher, or swim against gravity easier. However, translating the MMR solely by magnetic forces might be detrimental because it would be challenging to position the actuator precisely with such control strategies. [10,83] In addition to rolling and translations, our proposed MMR could also execute two-anchor crawling locomotion. A significant advantage of this terrestrial locomotion was that it could allow the MMR to position itself more precisely in unstructured environments compared to rolling. [10] This is because this mode of locomotion allowed the MMR to precisely tune its stride length. [10] To implement the two-anchor crawling locomotion, we used the free ends of the MMR to serve as the two contact points with the substrate. This locomotion was cyclic, and each cycle began by first allowing the MMR to assume a "U"-shaped configuration and allowing both of its free ends to be in contact with the substrate ( Figure S12D(i), Supporting Information, the appliedB was along z fLg -axis with a magnitude of 18 mT ). Subsequently, we would anchor the MMR with its rear end so that it could lift the front end of the body upward via a rotation about the x fLg -axis ( Figure S12D(ii), Supporting Information). This could be achieved by rotating the appliedB counterclockwise (15-30°) in the y fLg z fLg plane. Once the front end of the MMR was lifted, we rotatedB clockwise back to its original direction and reduced its magnitude to 4 mT so that the actuator could extend and produce a net displacement forward ( Figure S12D(iii), Supporting Information). Following this extension, the front end of the MMR would resume contact with the substrate and serve as the new anchor point. By rotatingB clockwise in the y fLg z fLg plane (15-30°) while increasing its magnitude to 18 mT, we could lift and contract the rear end of the MMR ( Figure S12D(iv), Supporting Information). Thereafter, we rotatedB counterclockwise back to its original direction to complete one cycle and produce a net displacement ( Figure S12D(v), Supporting Information). As the proposed MMR possessed six-DOF, we could control this actuator's sixth-DOF angular displacement about its z fLg -axis to steer it along the desired direction. In addition, we could also rotate the MMR about its y fLg -axis and tilt its body at an angle as it executed the two-anchor crawling locomotion. By tilting the MMR, this actuator could better conform to the strict shape constraints of a narrow passage so that it could easily cross this passage with the two-anchor crawling locomotion ( Figure 3E and Video S2, Supporting Information). With six-DOF control, the MMR could also control the angular displacements about its x fLg -axis and perform the two-anchor crawling locomotion to ascend an inclined slope of 20 ( Figure 3D and Video S2, Supporting Information), and successfully execute sharp turns by rotating about its z fLg -axis at the junction of a confined "L"-shaped path ( Figure 3E and Video S2, Supporting Information). Although existing MMRs were steerable when they executed the two-anchor crawling locomotion on obstacle-free flat terrains, [10,79,80] they could not use this locomotion to concurrently accomplish the following tasks: 1) climb an inclined slope; 2) perform effective turns on sharp corners; and 3) tilt at an angle to better conform and thus crossing a narrow passage with strict shape constraints. The proposed MMR therefore demonstrated significantly higher dexterity than existing MMRs that could perform two-anchor crawling locomotion. [10,49,73,79,80,82,84] Terrestrial locomotion like rolling and two-anchor crawling might not be applicable if the MMRs had to negotiate across highly confined and enclosed channels. In such scenarios, executing undulating crawling would be an effective locomotion for MMRs to overcome such obstacles. [10,53] When our proposed MMR entered such channels, we could continuously rotateB (15-20 mT) in the y fLg z fLg plane so that the actuator could undulate and generate a traveling wave along its body ( Figure 4 and Video S3, Supporting Information). Due to the generated traveling wave, the proposed MMR was able to induce a net propulsive force from the surroundings to thrust itself forward. As our MMR had six-DOF, it could precisely control the angular displacements about its y fLg -and z fLg -axes to negotiate through a complex channel that required it to sequentially undulating crawl across two perpendicular planes ( Figure 4A and Video S3, Supporting Information). The MMR could also execute undulating crawling to ascend an inclined slope of 20°by controlling the angular displacement about its x fLg -axis, and rotate about its z fLgaxis to make a sharp turn at the junction of a confined "L"-shaped channel ( Figure 4B and Video S3, Supporting Information). As previous MMRs could only demonstrate undulating crawling through straight or planar channels, [10,53,72] the advancement of our proposed actuator was therefore highly significant because it could use this mode of locomotion to negotiate across a much more diverse range of confined, enclosed channels that had convoluted 3D geometries.
Similar to undulating crawling, jumping would be an effective locomotion for the MMR to overcome difficult obstacles. [6,10] Specifically, MMRs could execute such locomotion to jump across barriers that had heights greater than their body length. [6,10] The jumping locomotion could be executed by first deforming the proposed actuator into a "U"-shaped configuration (the appliedB was along z fLg -axis with a magnitude of 20 mT) and then orientating the MMR's y fLg z fLg plane such that its z fLg -axis had a clockwise 5°angle away from the vertical upright direction. Subsequently, we rotatedB clockwise in the z fLg x fLg plane (175°) via a step change so that a strong magnetic torque could be exerted on the MMR to enable a rigid-body rotation while the actuator concurrently underwent a deformation. During this rotation and shape-changing process, the free ends of the MMR would eventually strike the substrate and gain a momentum to jump against gravity ( Figure 5A and Video S4, Supporting Information). To allow the proposed MMR to jump higher, we also applied an upward magnetic force to this actuator viaB grad (jumping height: 6.8 mm/1.06 body length, jumping range: 9.56-11.2 mm/1.49⋅1.75 body length). In general, we could also vary the jumping direction by tuning the MMR's orientation such that its x fLg -axis is parallel to the desired horizontal projectile motion before initiating the jump. A notable advantage of our proposed actuator over existing soft jumping MMRs [10,85,86] was that it could precisely control its three axes of angular displacements in midair after the jumping locomotion was executed. For example, we had shown that the proposed MMR could control its angular displacements about the z fLgand x fLg -axes while it executed the jumping locomotion in Figure 5A(i) and Video S4, Supporting Information. Similarly, this MMR could also precisely control its angular displacements about the z fLg -and y fLg -axes while it performed the jump in Figure 5A(ii) and Video S4, Supporting Information. By having the ability to precisely control its orientation during a jump, our proposed MMR could therefore have the potential to adjust itself into a favorable shape necessary for jumping across barriers, which have openings with strict shape constraints. Such an ability would be highly desirable, but it had not been achieved by existing jumping MMRs yet. [6,10,85,86] As an amphibious robot, our proposed MMR could also mimic the swimming locomotion of a biological jellyfish to maneuver in water bulk. This soft locomotive gait could be realized by oscil-latingB along the z fLg -axis of the MMR so that the actuator could alternate between its "U"and inverted "V"-shaped configurations (Video S5, Supporting Information). By oscillatingB at a frequency of 6-8 Hz (maximum jBj: 24mT), the proposed MMR could induce a net propulsive force from its surroundings to swim against gravity (average velocity: 7.65-10.4 mm s À1 ). It was possible to swim with this reciprocal gait because the MMR was able to attain an adequately high Reynolds number of 54.9-74.6 to gain sufficient inertia effects (Section S4D, Supporting Information). The swimming direction of the MMR could also be steered by controlling its angular displacements about the x fLg -and y fLg -axes to dictate the propulsion direction ( Figure 5B-C and Video S5, Supporting Information). As it swam, the proposed MMR could also rotate about its sixth-DOF axis (the z fLg -axis) via controllingB grad ( Figure 5D and Video S5, Supporting Information). Because we could precisely control the MMR's three axes of angular displacements, we could potentially command this actuator to adjust its orientation and adopt a favorable shape to easily swim through barriers with openings that had strict shape constraints. Having such dexterity would be a significant advancement over existing five-DOF MMRs that could perform jellyfish-like swimming locomotion. [10,30,64] This is because although existing five-DOF, jellyfish-like MMRs could also steer their swimming direction, they would lack the required sixth-DOF dexterity to swim across complicated barriers. [10,30,64] While the six-DOF soft MMR in our previous work could also fully control its orientation to swim across difficult barriers, it was restricted to having one soft-bodied locomotion. [43] Thus, our previous actuator [43] would not be able to perform the other types of gaits demonstrated by our proposed MMR.
Because our proposed MMR could concurrently possess six-DOF and execute multimodal soft-bodied locomotion, it was able to demonstrate significantly higher dexterity than existing MMRs. [10,43,53,64,72] To demonstrate such capabilities, we commanded our MMR to negotiate across three synthetic, unstructured environments that had challenging obstacles. For the first unstructured environment, the MMR began by jumping through a narrow slot to reach a higher ground (height: 6.5 mm) ( Figure 6A and Video S6, Supporting Information). This was possible because the MMR was able to adjust its orientation during the jump such that it could adopt a favorable shape to fit through the slot. An additional video was taken in slow motion to better illustrate this jumping process ( Figure 6B and Video S6, Supporting Information). Once the actuator arrived at the higher ground, it could use its two-anchor crawling locomotion to precisely maneuver itself to reach the narrow opening of a wall on its right ( Figure 6A and Video S6, Supporting Information). In order to bypass this wall, the MMR had to roll along its width  A) The MMR could use its undulating crawling locomotion to negotiate through a complex channel by using this gait to crawl across two perpendicular planes. During this process, the MMR had to control its angular displacement about the y fLg -axis to transit between the planes. B) The MMR could execute undulating crawling to ascend an inclined slope of 20°by controlling its angular displacements about the x fLg -axis. By rotating about its z fLg -axis, the MMR could perform a sharp turn at the junction of a confined "L"-shaped channel. The legend shows the local reference frame of the MMR as well as the types of arrows representing rotations about different axes and the MMR's trajectory. Scale bar: 2 mm.
www.advancedsciencenews.com www.advintellsyst.com (rotating about the y fLg -axis) so that it could adopt a gentle curvature necessary for squeezing through the wall's narrow opening ( Figure 6A and Video S6, Supporting Information, average speed: 0.37mm s À1 ); otherwise, it would not be able to roll across this rigid barrier. After passing through this wall, the MMR controlled its sixth-DOF angular displacement about the z fLg -axis to steer the rolling direction through the opening of the final wall. Because this opening was much larger than the previous one, the

(viii) (vi) (vii)
Stacked up objects Figure 6. The six-DOF MMR utilized its multimodal soft-bodied locomotion to navigate through various synthetic, unstructured environments with complex barriers and performed a 3D pick-and-place operation. Except for (E), the subfigures were created by superpositioning the snapshots of the MMR at different timestamps in Video S6-S8, Supporting Information. A) The MMR navigated through an unstructured terrestrial environment via the jumping and rolling locomotion. The MMR first arrived at a higher ground by jumping through a narrow slot via precisely controlling its orientation. Subsequently, the MMR rolled along its width with a gentle curvature to squeeze through the first wall's narrow opening and then rolled along its length to bypass the second larger wall with a relatively fast rolling speed. B) The MMR jumped through a narrow slot, with frames captured in slow motion. C) The MMR navigated through an unstructured terrestrial environment via the two-anchor and undulating crawling locomotion. The MMR was first commanded to two-anchor crawl across a path with strict shape constraints and subsequently climbed up a stair-like structure and moved cross a gap.
To negotiate across a confined and enclosed channel with convoluted geometries, the MMR activated its undulating crawling locomotion to ascend an upslope of 10°, perform a sharp turn at the junction of an "L"-shaped channel, and transit to crawl across a vertical plane. To better illustrate the crawling locomotion within the transparent complex channel, we included rendered images of the undulating crawling MMR into this subfigure. D) The MMR navigated in a hybrid aquatic-terrestrial environment. The MMR first performed its jellyfish-like swimming locomotion with precise orientation control to bypass a barrier and then rolled across an air-water interface to transit from water to land. E) The MMR performed a 3D pick-and-place operation. Snapshots were taken from Video S9, Supporting Information. i) This operation would require the MMR to grab, transport and stack two objects at the desired placement location (highlighted by the dotted red ellipse). ii) The MMR rolled toward an object and grabbed it.
iii-iv) The MMR held on to the object and transported it to the desired placement location. v-viii) The MMR repeated the same process for the second object so that it could stack both objects on top of each other. Scale bar: 2 mm.
www.advancedsciencenews.com www.advintellsyst.com MMR could pass through by rolling along its length (rotating about the x fLg -axis) and achieved a relatively fast rolling speed to reach the final destination (Video S6, Supporting Information, average speed: 0.46 mm s À1 ). It is noteworthy that existing five-DOF MMRs, which could execute multimodal softbodied locomotion, would not be able to negotiate across this unstructured environment. [10,53,72] This is because such existing soft MMRs would neither be able to fully control their orientation to jump through narrow slots nor roll along their width (shorter side). Next, we commanded the proposed MMR to overcome all the obstacles in the second unstructured terrain ( Figure 6C and Video S7, Supporting Information). The MMR began by first rotating about its y fLg -axis so that it could tilt and adopt a favorable orientation while executing the two-anchor crawling locomotion. This tilting was necessary as it allowed the actuator to two-anchor crawl across a path that had strict shape constraints ( Figure 6C and Video S7, Supporting Information). Subsequently, we steered the MMR while it continued to use the two-anchor crawling locomotion to climb up the stair-like structure and move across the gap to reach the left side of the terrain ( Figure 6C and Video S7, Supporting Information). The actuator was then commanded to negotiate across a confined and enclosed channel, which had convoluted geometries. To move across such a complex channel, the MMR activated its undulating crawling locomotion to ascend an upslope of 10°until it reached a horizontal plane parallel to the ground ( Figure 6C and Video S7, Supporting Information). Subsequently, it continued to use this locomotion in the horizontal plane to perform a sharp turn at the junction of an "L"-shaped channel and transited to crawl across a vertical plane to reach its final destination ( Figure 6C and Video S7, Supporting Information). We would like to highlight that existing soft MMRs would not be able to navigate across this unstructured terrain because they would not be able to tilt while performing the two-anchor crawling locomotion. [10,49,73,79,80,82,84] As a result, they would not be able to cross the narrow path that had strict shape constraints. Furthermore, existing soft MMRs would not have the dexterity to undulating crawl across convoluted, enclosed channels with nonplanar geometries. [10,53,72] The third unstructured environment, which the proposed MMR navigated in, was a hybrid aquatic-terrestrial environment ( Figure 6D and Video S8, Supporting Information). The MMR began at the bed of the water reservoir, and it started to swim against gravity by executing the jellyfish-like swimming locomotion ( Figure 6D and Video S8, Supporting Information, average speed: 9.37 mm s À1 , Reynolds number: 67.2). Because the proposed MMR was able to precisely control its three axes of angular displacements, it was able to steer toward the opening of a barrier and subsequently adopted a favorable orientation which could allow it to negotiate across the opening ( Figure 6D and Video S8, Supporting Information); otherwise, the MMR would not be able to cross this rigid obstacle. Upon reaching the water surface, the proposed MMR adopted a rolling locomotion to peel its surface off the air-water interface and eventually transited from water to land ( Figure 6D and Video S8, Supporting Information). To allow the MMR to peel its surface off the air-water interface more easily, we had coated the buoyant components of the MMR with a thin layer of hydrophobic polymer (Ecoflex 00-10). Nonetheless, because the surface tension of the air-water interface was strong, the MMR had to perform several rounds of rolling before it could eventually break off this surface tension (Video S8, Supporting Information). Another limitation of the MMR was that its free ends were found to be stuck to a water droplet after it landed. Hence, the MMR was unable to recover to its flat configuration and had to remain in the "U"-shaped configuration. In theory, this issue could be mitigated if the surfaces of the MMR could be coated with a thin layer of Teflon to become superhydrophobic because this would prevent the water droplets from sticking to the actuator after it emerged from water. [87,88] We aim to investigate this approach more thoroughly in the future. Despite these limitations, we would like to highlight that existing five-DOF soft MMRs would not have the dexterity to overcome the obstacle in the reservoir because they could not fully control their orientation and adopt a favorable shape to swim through such a tight opening with strict shape constraints. [10,30,64] Although the six-DOF, jellyfish-like MMR in our previous work could theoretically overcome this barrier too, it could not perform a similar rolling locomotion to transit from water to land. [43] The experiments in Figure 6 and Video S6-S8, Supporting Information, therefore demonstrated that our proposed MMR had unprecedented dexterity because it was able to negotiate across difficult barriers that are impassable by existing small-scale devices. [10,43,53,64,72] Because our proposed MMR had adequate stiffness, it could also perform 3D pick-and-place operations. For instance, our MMR could roll toward an object and control its "U"-shaped curvature to grip that object ( Figure 6E(i)-(ii) and Video S9, Supporting Information). Subsequently, the MMR could hold on to the object while transporting it to the desired placement location via the rolling locomotion ( Figure 6E(iii) and Video S9, Supporting Information). Upon arrival, the MMR could reduce its gripping force and release the object ( Figure 6E(iv) and Video S9, Supporting Information). By repeating this process for the second object ( Figure 6E(v)-(vii)), the MMR could successfully stack these objects on top of each other ( Figure 6E(viii) and Video S9, Supporting Information). To evaluate the payload of our soft MMR, we had commanded our MMR to grab various objects and use the rolling locomotion to transport them to a desired placement location ( Figure S13, Supporting Information). Our experiments indicated that the proposed MMR was able to carry objects that had 0.5-to 3-fold of its weight (29.4 μN). We also observed that the MMR would not be able to grab an object firmly when the object's weight was increased to 3.5-fold of its weight. In addition, we had computed the theoretical positioning resolution of our MMR during the pick-and-place operation. Assuming that the MMR was operating on an obstacle-free flat terrain, the angular resolution of the MMR when it was rotating about the x fLg -and y fLg -axes would be 0.57 (Section S4F, Supporting Information). The sixth-DOF angular resolution of the MMR would be 0.36 (Section S4F, Supporting Information). These angular resolutions were computed based on the resolution of our actuating magnetic signals (Section S4A, Supporting Information). During the pick-andplace operation, the shape of the rolling MMR could be approximated as a circle, and the radius of this best fit circle was about 0.95 mm ( Figure S14, Supporting Information). Under no-slip conditions, the theoretical translational resolution of the MMR when it rolled along its length could be computed as 9.5 μm via the product of its radius and y fLg -axis angular resolution (Section S4F, Supporting Information). The positioning resolution, however, might become coarser if the no-slip condition was violated. In the future, we aimed to explore the feasibility of performing 3D pick-and-place operations in conjunction with the MMR's other modes of locomotion. As it was challenging to draw the magnetic actuating signals applied to our MMR in most of the figures and videos, we had instead presented these experimental signals separately in Figure S15-S31, Supporting Information.

Discussion
In this section, we will provide additional discussions on the locomotion of our proposed MMR. Relevant comparisons between the proposed MMR and other similar existing devices will also be made here. To make our discussions more complete, the limitations of our MMR will also be explicitly discussed. Furthermore, we will also suggest feasible future works that may potentially address or mitigate these issues.
Similar to the five-DOF soft MMRs presented by Hu et al., [10] our proposed actuator can also perform undulating swimming and meniscus-climbing on an air-water interface (Section S4E, Supporting Information). Hence, the proposed MMR is able to execute a total of seven types of soft-bodied locomotion. However, possessing six-DOF will not enhance the dexterity of the MMR when it is executing the undulating swimming and meniscus-climbing gaits. This is because these two modes of locomotion require the MMRs to be constrained on an air-water interface. [10] Therefore, the motions of the MMR are restricted to 2D for these gaits, and this has, in turn, eliminated the benefits of having the additional sixth-DOF rotation. In the future, we aim to endow the proposed MMR with more modes of soft-bodied locomotion, which can benefit from having six-DOF. We will also investigate feasible gaits that can eventually allow our proposed MMR to reliably transit from an air-liquid interface to the liquid bulk.
While the proposed MMR had demonstrated unprecedented dexterity, it was controlled manually by a human operator in this first study. In the subsequent studies, we aim to increase the speed of the MMR by implementing closed-loop position control with visual feedback. For lab-on-chip applications, such visual feedback for the MMRs can be obtained via standard cameras. [22] In the future, when MMRs are deployed for their targeted biomedical applications, ultrasound imaging technology would be a promising candidate for providing such visual feedback for these actuators. [10] This is because recent studies have shown that the actuating magnetic fields for MMRs have minimal interference with ultrasound imaging systems. [10] To make our proposed MMRs more favorable for potential medical applications, we aim to further scale down the size of these actuators in the future.
For this first study, our main objective is to investigate the enhancement in dexterity when a soft MMR is able to concurrently possess six-DOF motions and multimodal locomotion. Hence, we did not perform detailed characterization on the proposed MMR to evaluate the maximum speed achievable by each locomotion. We also did not evaluate the maximum height or range that the MMR could jump. Such in-depth experimental investigations can potentially be pursued in subsequent studies. In the future, we also aim to investigate if six-DOF control can be applied to soft MMRs with 3D magnetization profiles. This can potentially allow us to discover new MMRs that can realize higher sixth-DOF torque and yet capable of performing multimodal soft-bodied locomotion. The feasibility of optimizing the MMR's geometries via numerical methods [89][90][91][92] to further enhance its dexterity can similarly be explored in such further studies too.
While our proposed MMR can jump across complex barriers, it is difficult to accurately control the initial jumping velocity. This is because it is very challenging to model the dynamic interaction between the MMR and the substrate when the former strikes the latter to gain the necessary momentum to jump. [6,10] In the future, we aim to investigate whether deep learning techniques can be employed to overcome this limitation. [93] By obtaining experimental data to train the weights of the neural networks, it may be possible to develop numerical models that can accurately predict the MMR's initial jumping velocity. Using such numerical models, we can potentially allow the jumping locomotion to be more deterministic.
In summary, this work proposes a six-DOF MMR that can execute seven modes of soft-bodied locomotion and perform 3D pick-and-place operations. Based on the proposed control strategy, our MMR has demonstrated unprecedented dexterity by negotiating across barriers impassable by existing small-scale devices. We envision that this work can potentially inspire future MMRs to be significantly more dexterous, and this would be a critical step toward making these actuators practical for their targeted biomedical applications.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.