Insect‐Scale Biped Robots Based on Asymmetrical Friction Effect Induced by Magnetic Torque

Multimodal and controllable locomotion in complex terrain is of great importance for practical applications of insect‐scale robots. Robust locomotion plays a particularly critical role. In this study, a locomotion mechanism for magnetic robots based on asymmetrical friction effect induced by magnetic torque is revealed and defined. The defined mechanism overcomes the design constraints imposed by both robot and substrate structures, enabling the realization of multimodal locomotion on complex terrains. Drawing inspiration from human walking and running locomotion, a biped robot based on the mechanism is proposed, which not only exhibits rapid locomotion across substrates with varying friction coefficients but also achieves precise locomotion along patterned trajectories through programmed controlling. Furthermore, apart from its exceptional locomotive capabilities, the biped robot demonstrates remarkable robustness in terms of load‐carrying and weight‐bearing performance. The presented locomotion and mechanism herein introduce a novel concept for designing magnetic robots while offering extensive possibilities for practical applications in insect‐scale robotics.


Introduction
Insect-scale robots have demonstrated significant potential in fields, such as biomedicine and industry, [1] due to their agile locomotion, [2] substantial load capacity, [3] and exceptional robustness. [4]he majority of robots, particularly crawlers and walkers, employ two ways to break the symmetry of friction and allow for translational locomotion.One designs an asymmetric substrate for the locomotion of a robot, [5] enabling crawling or walking.Upon external stimulation from light, temperature, and electric field, the robot alternately bends its front and rear feet to generate unidirectional friction for forward movement.However, the environmental adaptability of these robots is significantly limited because of their strong reliance on the structures of the substrate.The more general approach involves designing the robot structures to achieve friction asymmetry, such as bioinspired designs of asymmetric feet. [6]The design allows for one foot to anchor to the substrate while the other slides forward.The mechanism enables robots to achieve forward movement on a flat surface by alternately swinging their feet.Theoretically, in the case of a robot with weight G evenly distributed on a pair of feet, the sliding foot experiences a frictional resistance of no less than μG/2, while the anchoring foot experiences a frictional thrust exceeding μG/2 (μ indicates the friction coefficient between the foot and the substrate, Figure S1, Supporting Information).Therefore, the robot's ability to move forward relies on the disparity between the two forces, which directly depends on the ability of the anchoring foot to firmly anchor to the substrate.The smoothness of the substrate plays a pivotal role in determining the stability of the robot's locomotion.More importantly, whether the asymmetric friction effect is achieved through substrate structure design or robot structure design, the resulting locomotion of the robot is limited to be unidirectional.
Robots via magnetic field actuation have been widely adopted owing to their untethered locomotion, dexterous control, and rapid response. [7]The robot locomotion can be easily achieved through gradient magnetic force, regardless of its shape and work conditions. [8]Moreover, the robots with the design of programmed magnetization not only realize high mobility in multiple modes but also enable adaptation to complex substrate structures and environments. [9]Achieving efficient and stable locomotion is essential for the current generation of magnetic robots, particularly in order to enable their extensive utilization in real-world applications.Therefore, the exploration and development of new locomotion modes based on magnetic actuation have the potential to address these challenges and facilitate faster applications of robots in biomedicine and industry.
In this work, we reveal and define a mechanism that enables magnetized robot locomotion through magnetic actuation, named asymmetrical friction effect induced by magnetic torque (AFEMT).As an external magnetic field actuates the magnetized robot to rotate parallel to the substrate surface, it simultaneously induces a sideways rolling torque on the robot.The results lead to a decrease in the support force of the sliding foot while increasing the support force of the anchoring foot.The locomotion mechanism is thoroughly examined in terms of static and dynamics models, respectively.The advancement of the mechanism allows magnetized robots to overcome limitations in shape and structure.Drawing inspiration from human walking and running locomotion, we propose a biped robot with a pair of feet.The locomotion capabilities of the biped robot extend beyond basic movement patterns such as forward, reverse, cornering, and turning in place.It is also equipped with the ability to span obstacles, climb steps and slopes, traverse pipes, and walk on biological tissue surfaces.By applying an oscillating magnetic field with a strength of 5 mT and a frequency of 19 Hz, the maximum speed of the biped robot can reach 25.33 BL s −1 .Moreover, it can achieve running speeds exceeding 15 BL s −1 on rough planar substrates through an applied magnetic field with an oscillating frequency of 10 Hz.The programmed control of the applied magnetic field enables the biped robot to rapidly move following a patterned trajectory.In addition, the biped robot exhibits high load capacity, as it can smoothly transport cargo weighing more than four times its weight.The flexible body results in exceptional robustness, allowing the biped robot to walk stably even after being flattened by a heavy load.

Locomotion Mechanism for Robots with Random Shape Structures
Here, we position a cuboid robot with a uniformly distributed mass G 1 on a flat substrate surface that is parallel to the horizontal plane (x-y plane), as shown in Figure 1A.A local coordinate (OXYZ) is established and fixed with the cuboid robot, and its origin coincides with the mass center of the biped robot.The Euler angles, namely the heading angle , rolling angle , and pitching angle , are defined to represent the rotation of the local coordinate relative to the axes of rotation at any given moment (Figure S2, Supporting Information).We define the magnetic moment M of the cuboid robot parallel to the O-Y axis in the local coordinate.Therefore, the magnetic moment M of the cuboid robot can be quantified as where M represents the magnitude of the magnetic moment.An external magnetic field B a is utilized to control the locomotion of the cuboid robot.The actuating field is expressed as in the global coordinate where  denotes the angle between the applied magnetic field and the x-y plane in the global coordinate, and  is the deflection angle between the projection of the applied magnetic field in the x-y plane and the o-x axis (Figure S3, Supporting Information).Therefore, the applied magnetic field can be expressed in the local coordinate where C II I is the rotation matrix from the global coordinate to the local coordinate.With the presence of the external magnetic field, the magnetic moment of the cuboid robot tends to align itself with the actuating field due to the magnetic torque.The magnetic torque  can be expressed as where i, j, and k are unit vectors along the O-X, O-Y, and O-Z axes, respectively.P 1 , P 2 , and P 3 is expressed as and where B A , denoting the magnitude of the magnetic field in the local coordinate, is equivalent to B a .
In the first case, an external magnetic field B a is applied in parallel to the x-y plane.When the applied magnetic field is deflected in the x-y plane, it induces a magnetic torque  to actuate the cuboid robot in locomotion, which can be expressed as Theoretically, if the value of magnetic torque  exceeds that of the frictional resistance  f , it will cause rotational locomotion of the cuboid robot around its perpendicular axis passing through its center of mass (Texts S1-S3, Supporting Information, along with Video S1, Supporting Information).In the first case, the cuboid robot is capable of executing rotational movements solely within its current position.In the second case, it exhibits forward motion capabilities.B) Force analysis for the supporting edge of a cuboid robot.C) Applied magnetic field oscillating in the circular conical surface.D) Gait locomotion conjecture of the cuboid robot actuated by an applied magnetic field oscillating periodically on the circular conical surface.E) Deflection of the cuboid robot under the applied magnetic field B a with an elevation angle  to x-y plane.The inserted graph shows the magnetization of the cuboid robot.F) Gait locomotion of the cuboid robot actuated by an oscillating applied magnetic field.G,H) Gait locomotion of the robots with various geometries, including a "butterfly" and a "penguin."Scale bar: 2 mm.
For the other case, an external magnetic field B a is applied parallel to the y-z plane ( = /2) but at an elevation angle  to the x-y plane.A magnetic torque  would overcome the gravity and actuate the rotational locomotion of the cuboid robot around the supporting edge.After the rotational locomotion, the value of magnetic torque  can be expressed by where  denotes the pitching angle of the cuboid robot (Text S4, Supporting Information).At the moment, the values of magnetic torques  Y and  Z are both zero.The magnetic torque  Y remains zero when the deflection angle  of the applied magnetic field changes, as the magnetic moment of the cuboid robot aligns with the O-Y axis.However, the magnetic torques  Z becomes nonzero.The value can be expressed as which can be decomposed into a component in the vertical direction and a component in the direction of the vertical O-X axis in the horizontal plane.The two components can be expressed as and Therefore, the magnetic field  Z not only drives the cuboid robot to rotate around the perpendicular axis but also leads to a sideways roll of the cuboid robot.The force analysis indicates that the support force value at the proximal end of the supporting edge F s-p surpasses that at the distal end F s-d , further resulting in greater friction at the proximal end F f-p compared to the distal end F f-d (Figure 1B).As a result, the applied magnetic field enables the cuboid robot to rotate around a supporting point at the proximal end, thereby achieving forward movement (Text S5, Supporting Information).By employing the aforementioned method, the gait motion of the cuboid robot can be realized when the applied magnetic field oscillates periodically on the circular conical surface (Figure 1C), as shown in Figure 1D.In this work, the deflection angle of the oscillation magnetic field can be expressed as where  , , and t denote the amplitude, angular frequency, and time of the oscillating angle, respectively.The above mechanism, derived from theoretical analysis, can be validated through experimental verification.Figure 1E shows that a cuboid robot with a magnetic moment parallel to the horizontal plane (x-y plane) is placed on a flat substrate.When the applied magnetic field deviates from horizontal and tilts upward at an angle , the cuboid robot undergoes a rotation by an angle  with respect to the supporting edge.After the applied magnetic field oscillates for half a cycle in the form shown in Figure 1C, the cuboid robot advances by a distance of s 1 (Figure 1F).Additionally, robots with various geometries can move forward in the locomotion pattern, including a "butterfly," a "penguin," or even a random shape structure (Figure 1G,H, and Video S2, Supporting Information).However, when the contact between the robot and the substrate has only one supporting point, the robot can only rotate in place.As shown in Figure S4 and Video S3, Supporting Information, cylinder and ellipsoid robots exhibit periodic oscillations under the actuation of an applied magnetic field, but they do not move forward.
According to the theoretical analysis and experimental verifications, we find that the locomotion pattern described above can be achieved under the following conditions in the space rectangular coordinates (it is assumed that the x-y plane is parallel to the horizontal plane and the o-z axis is perpendicular to the horizontal plane): 1) when a robot with mass is subjected to an external magnetic field, the magnetic moment components of the robot in both the o-z axis and x-y plane are non-zero (i.e.,  ≠ n/2 where n is an integer); 2) the component of the external magnetic field in x-y plane is non-zero (i.e.,  ≠ /2 + m where m is an integer); and 3) the robot must have at least two supporting points with the non-smooth surface.When the applied magnetic field initiates rotation around the o-z axis, the magnetic moment of the robot always lags behind due to friction, resulting in a non-zero magnetic torque that causes sideways roll and alters the distribution of support force from the substrate.We name the phenomenon the AFEMT.
When a cuboid robot moves forward based on AFEMT, the speed of movement is contingent upon the distance between its supporting points.However, due to the typically rough and uneven characteristics of the substrate surface, uncertainty arises regarding the position of supporting points (Figure S5, Supporting Information).Consequently, while the cuboid robot may still achieve forward locomotion, its velocity becomes unpredictable.As shown in Figure S6, Supporting Information, the robots are observed to move forward for two distinct distances, s 1 and s 2 , in the initial and latter halves of the cycle, respectively.

Design of Biped Robots in the Static Model
Studies on bipeds, such as humans, have demonstrated that the torque for alternating the forward movement of the left and right feet in walking and running locomotion is counterbalanced by the torsion of the upper body, as illustrated in Figure 2A.Simultaneously, lateral torque induces alternating support between the feet.The supporting foot experiences friction from the ground in a forward direction, propelling the body forward.The ancient Chinese employed a method of land measurement known as "measuring land by steps," which serves as evidence for the robustness and predictability of human locomotion. [10]Drawing inspiration from the concept, we have enhanced the cuboid robot based on AFEMT with a pair of feet to achieve robust and predictable movement.Figure 2B shows that the biped robot is equipped with a pair of feet, namely a right foot and a left foot, which are separated by a distance denoted as L. Detailed structural characteristics and magnetic moment of the biped robot are presented in Figure S7, Supporting Information.In the absence of an external magnetic field, the biped robot presents a sitting position due to gravity.When an external magnetic field with an elevation angle  relative to the horizontal is applied, the biped robot presents a standing position.The pitching angle between the magnetic moment of the biped robot and the There is a small protrusion on the back of the biped robot to facilitate the distinction between its front and rear.C) Pitching angle  of the biped robot actuated by an applied magnetic field with an elevation angle .D) Theoretical and experimental results of variations in the pitching angle of the biped robot actuated by the applied magnetic field in various elevation angles.The strength of the applied magnetic field ranges from 2 to 7 mT.E) Pitching angle factor |cossin| as a function of the elevation angle .F) Support force on the left and right foot obtained through theoretical analysis as a function of the deflection angle.The elevation angle  of the applied magnetic field is 10°.G) Corresponding threshold of the deflection angle required for initiating biped robot movement as a function of the frictional coefficients μ.H) Rotation angles  of the biped robot actuated by an applied magnetic field with a deflection angle.I) Theoretical and experimental results regarding variations in lag angle  with magnetic field strength B a and elevation angle .J) Comparison between the biped robot's locomotion process and human walking.Scale bar: 5 mm.horizontal plane, denoted as , can be theoretically predicted (Text S6, Supporting Information).
Therefore, by adjusting the elevation angle of the applied magnetic field, it becomes possible to regulate the biped robot's head-up and head-down postures, as illustrated in Figure 2C.Due to gravitational effects, there may exist deviations between the elevation angle of the applied magnetic field and the pitching angle of the biped robot.Figure 2D presents both theoretical predictions and experimental test results illustrating how variations in the elevation angle affect changes in the pitching angles of the biped robot.The experimental tests indicate that the pitching angle reaches its maximum value, that is, 45°, presenting a sitting position before the elevation angle of the applied magnetic field reaches ±30 .Given that the biped robot and substrate are in surface-surface contact during this sitting position, it may lead to unstable and unpredictable locomotion.Hence, it is recommended to avoid excessively large elevation angles.Additionally, theoretical predictions have indicated that weaker magnetic fields are more prone to inducing the posture of the biped robot to become a sitting position.
Consequently, the applied magnetic field subsequently undergoes rotation around the o-z axis toward the z-x plane, resulting in a magnetic torque  Z .For the biped robot, the magnetic moment  Z can similarly be decomposed into a component in the vertical direction and another component perpendicular to the O-X axis direction in the horizontal plane.The former induces a tendency of rotational motion along the vertical direction and the latter induces a tendency of the sideways roll.Initially, due to frictional resistance, there is no rotation of the biped robot at the beginning of the magnetic field oscillation.As the deflection angle of the magnetic field increases, both the values of the two components increase accordingly.Equations ( 9) and ( 10) of the two components express that they not only exhibit a positive correlation with the deflection angle but also are influenced by the pitching angle factors cossin and coscos, respectively.An increase in elevation angle  results in a noticeable reduction in the factor coscos, while within the range of ±30°, an increase in elevation angle leads to an increment in the factor cossin, as shown in Figure 2E.Therefore, there is a continuous increment in support and friction on the supporting foot while those of the sliding foot decrease until the initiation of robot rotation.
According to the equilibrium equation, the support forces of the sliding and supporting feet may be mathematically expressed as (Text S7, Supporting Information) and where G 2 and V 2 denote the weight and volume of the biped robot, respectively.As shown in Figure 2F, the support force value on the supporting foot F sr always exceeds that on the sliding foot F sl , during the period from when the biped robot tends to rotate.The maximum support force value exerted on the support-ing foot allows it to exceed the weight of the biped robot itself, while the minimum support force value exerted on the sliding foot may be zero.For instance, when an applied magnetic field with a strength of 5 mT is present and the deflection angle of the magnetic field reaches 76.8°, the support force value on the biped robot's supporting foot equals its weight G 2 , whereas no support force is exerted on its sliding foot.Furthermore, it should be noted that changes in elevation angle  of the applied magnetic field significantly influence variations of support force, as shown in Figure S8, Supporting Information.When the deflection angle of the applied magnetic field surpasses a threshold, enabling magnetic torque  Z to overcome frictional resistance, the biped robot initiates rotation (Text S8, Supporting Information).
The deflection angle of the applied magnetic field is defined as the starting angle  at which the biped robot initiates its locomotion on various substrates.The friction coefficients μ between the biped robot and different material substrates exhibit variations, resulting in diverse thresholds.As shown in Figure 2G and Figure S9, Supporting Information, an increase in the friction coefficient μ leads to an increase in the corresponding threshold of the deflection angle required for initiating biped robot movement.
When the biped robot initiates rotation, the supporting foot serves as the axis while the sliding foot slides forward, causing the center of the biped robot to move forward.As shown in Figure 2H, the biped robot undergoes a rotational angle  when the deflection of the applied magnetic field is at an angle /2 − .Due to frictional resistance, there always exists a lag angle  between the angles of rotational angle and deflection angle.Figure 2I presents theoretical predictions and experimental test results regarding variations in lag angle  with the magnetic field strength B a and elevation angle .Theoretical predictions indicate that an elevation angle of 5°results in larger lag angles, whereas an elevation angle of 15°leads to smaller ones.Therefore, the elevation angle of the applied magnetic field is selected to ≈15°to achieve the optimal biped robot locomotion.It is worth noting that the experimental measurements of the lag angle in the biped robot exhibit a significant deviation from the theoretical predictions when the elevation angle of the magnetic field reaches 20°.The experimental analysis suggests that adhesion between the biped robot and the substrate is inevitable due to the high surface energy of its materials.A greater elevation angle of the magnetic field results in an increased contact area, thereby enhancing adhesion and consequently leading to an augmented lag angle of the biped robot (Figure S10, Supporting Information).
At this stage, we have completed the analysis of the locomotion mechanism for the biped robot in the static model.When exposed to an oscillating external magnetic field, the biped robot demonstrates forward locomotion.Figure 2J presents a comparison between the biped robot's locomotion process and human walking, highlighting similarities in gait patterns.The movement of the biped robot can be divided into two phases, namely stance, and swing, resembling human gait characteristics.During the stance phase, the right foot of the biped robot sequentially undergoes initial contact, loading response, mid-stance, terminal stance, and pre-swing stages.In the swing phase, sequential stages include the initial swing, mid-swing, and terminal swing for the right foot.Thus, utilizing the AFEMT mechanism enables stable forward locomotion in a manner similar to natural gait.

Motion Attitude Control of Biped Robots in the Dynamics Model
During the locomotion process, the motion attitude of the biped robot is influenced by various factors including the angular velocity and strength of the applied magnetic field, as well as the biped robot's moment of inertia.Therefore, establishing a dynamics model enables a comprehensive analysis of these multiple factors on the biped robot and facilitates better control over its motion attitude.
In the kinematics model (Text S9, Supporting Information), the biped robot's attitude can be represented by Euler angles, including the heading angle , rolling angle , and pitching angle , which were introduced previously.The rate of change in the attitude angle is related to the rotational angular velocity of the biped robot as follows.
where  and  denote the three attitude angles and angular velocity of the biped robot.Therefore, the angular velocity and angular acceleration of the biped robot can be expressed as and where  X and  X ,  Y and  Y , and  Z and  Z denote the angular velocity and the angular acceleration values of the biped robot rotating around the O-X, O-Y, and O-Z axes, respectively, as shown in Figure 3A.
In the dynamics model (Text S10, Supporting Information), the biped robot's motion complies with the theorem of the moment of momentum, which can be expressed by the Euler equation as where H and M(t) denote the momentum moment and the total external moment on the biped robot.
Based on the aforementioned dynamics model, we conduct an analysis of three fundamental motion modes exhibited by the biped robot, namely variations in pitching angle, rotation around the vertical center axis, and rotation around a single foot (Text S11, Supporting Information).
As shown in Figure 3B, the biped robot initially assumes a standing position.Upon the application of an external magnetic field at an elevation angle , the biped robot undergoes a change in its attitude.The corresponding kinetics equation governing this process can be expressed as where J XX is the biped robot's moment of inertia around the line passing through two support points.z 0 is the distance from the center of gravity to the axis of rotation.However, during actual motion, this rotational motion is impeded by air and ground friction, ultimately reaching a steady state.Consequently, Equation (18) becomes where Γ D is the added damping phase which is implemented to achieve a steady state for the biped robot's vibrations.In the present study, we assume that in general [11] Γ where abs() is a sign for taking absolute value.The focus of our analysis is on examining the vibrations of the biped robot's pitching angle, considering values of C 1 and C 2 ranging from 0.1 to 100, respectively (Figure S11, Supporting Information).Specifically, we deliberately set the values of C 1 and C 2 at 10 and 1, respectively, in order to investigate the impact of magnetic field strength and the biped robot's rotational inertia on the vibration of the biped robot's pitching angle (Figure 3C).The results illustrate the rapid oscillation of the biped robot followed by stabilization.Importantly, the pitching angle  after stabilization consistently aligns with the results obtained from the static model.
The second motion mode is to keep the biped robot in a standing position with the applied external magnetic field deflected in the horizontal plane, as shown in Figure 3D.The deflection angle of the oscillation magnetic field is mathematically represented by Equation (11), with an amplitude equal to 60 .The corresponding kinetics equation for this process can be expressed as where J ZZ is the moment of inertia of the biped robot around the O-Z axis direction.μ denotes the friction coefficient between the biped robot and the substrate.
In numerical calculations, it is imperative to consider two distinct phases: the starting phase and the rotation phase.During the former, as the magnetic field deflects, both magnetic moment and friction gradually increase from zero until reaching maximum static friction; at this point, angular velocity and acceleration of the biped robot are nil.Once the magnetic moment surpasses maximum static friction, Equation ( 21) applies and rotation commences.
In the rotational motion model of the biped robot, we investigate the impacts of magnetic field oscillation frequency, magnetic field strength, and the biped robot's moment of inertia on its rotational attitude in a steady-state scenario.When keeping the magnetic field strength and moment of inertia constant, variations in the biped robot's motion attitude occur due to changes in oscillation frequency.As shown in Figure 3E and Figure S12, Supporting Information, the deflection angle of the biped robot transitions from a lagging to an overshooting with an increase in the oscillation frequency of the magnetic field.The frequency domain results reveal more pronounced changes, as shown in Figure 3F.After the oscillation frequency of the applied magnetic field exceeds 10 Hz, the biped robot's deflection angle exhibits a significantly higher amplitude compared to that of the applied magnetic field, gradually increasing with an escalating frequency.However, the biped robot will be unable to achieve the second motion mode once the frequency exceeds a certain threshold.As shown in Figure 3G and Figure S13, Supporting Information, the time domain results obtained from the startup of the biped robot indicate that the overshooting effect occurs in its deflection angle, even at angles exceeding 180°.This phenomenon leads to continuous application of magnetic torque in alignment with the rotational speed while the magnetic field returns, resulting in sustained acceleration and rotation of the biped robot.
The influence of magnetic field strength and the biped robot's moment of inertia on its attitude was investigated similarly.As shown in Figure 3H, Figures S14 and S15, Supporting Information, the lag effect becomes more significant at an oscillation frequency of 1 Hz, accompanied by a decrease in magnetic field strength.Additionally, the startup failure of the biped robot occurs at a lower magnetic field frequency when the magnetic field strength is smaller (Figures S16 and S17, Supporting Information).Conversely, as shown in Figure 3I, Figures S18 and S19, Supporting Information, an increase in the moment of inertia of the biped robot amplifies the overshooting effect.At lower frequencies, startup failure is observed due to the increased difficulty in controlling motion for a biped robot with higher moments of inertia using applied magnetic fields (Figure S20, Supporting Information).
In the motion mode of rotating around one foot, we investigate the effects of angular velocity and strength of the rotating magnetic field, as well as the moment of inertia of the biped robot, on the variation in the support force of the sliding foot during the start-up phase.An initial magnetic field with an elevation angle  is applied to the biped robot, followed by the initiation of a rotating magnetic field with an angular velocity  0 , as shown in Figure 3J.To simplify calculations, we assume a constant pitching angle  for the biped robot throughout this process.The corresponding kinetics equation for this process can be expressed as where J z is the moment of inertia of the biped robot around the vertical axis over the support point in the supporting foot.When the magnetic field starts to deflect, the support force on the sliding foot decreases and reaches a minimum as the deflection angle of the magnetic field increases, as shown in Figure 3K.It is noteworthy that at a sufficiently high angular velocity, the support force on the sliding foot becomes zero, indicating complete detachment of the right foot from the supporting substrate, which is referred to as the rising effect.The detachment of the sliding foot from the support surface occurs for both increasing magnetic field strength and rotational inertia, as illustrated in Figure 3L,M.
In summary, we employed the AFEMT mechanism for designing a biped robot that draws inspiration from human walking and running motions.The static model was utilized to validate the applicability of AFEMT in the locomotion process of the biped robot.The dynamics model was employed to analyze the attitude changes of the biped robot under magnetic field control.Subsequently, we will investigate various motion modes and locomotion on complex terrains.

Multimodal Locomotion of Biped Robots
The multimodal locomotion of the biped robot endows it with significant potential to operate in complex conditions.Figure 4A shows sequential snapshots of the biped robot's forward locomotion at quarter-cycle intervals (Video S4, Supporting Information).According to its locomotion mechanism, when the applied magnetic field oscillates clockwise with an elevation angle, the right foot serves as the supporting foot while the left foot functions as the sliding foot.Conversely, when the applied magnetic field oscillates counterclockwise, it is the left foot that acts as the supporting foot and the right foot becomes the sliding foot.In this manner, by alternating between using its left and right feet as either a sliding or supporting component respectively, the biped robot achieves forward locomotion.When the applied magnetic field periodically oscillates at a depression angle, both feet of the biped robot undergo an exact reversal of their supporting and sliding functions.It results in reverse locomotion for the biped robot, as shown in Figure 4B.Additionally, we observe that the distance covered during one-half of the cycle remains nearly identical regardless of whether the biped robot is moving forward or backward.
The biped robot possesses not only the capability to stabilize its forward and reverse locomotion in complex conditions but also the flexibility to make turns.During the biped robot's movement, altering the timing or angle of clockwise and counterclockwise oscillations of the applied magnetic field leads to corresponding variations in sliding distances for its left and right feet.Consequently, it enables easy realization of turning locomotion for the biped robot.As presented in Figure 4C and Video S5, Supporting Information, when the biped robot reaches a turning position after walking straight for a certain distance, there is a decrease in the oscillation angle of the counterclockwise oscillating magnetic field from 120°to 30°.With this alteration, the direction of movement for the biped robot rotates by 90°toward its right side, thereby accomplishing turning locomotion.In a special scenario where the applied magnetic field consistently oscillates in either clockwise or counterclockwise directions, the biped robot can achieve a locomotion pattern that rotates in place with one foot as an axis of rotation (Figure 4D and Video S6, Supporting Information).
In addition to the fundamental locomotion patterns of moving forward and backward, rotating, and turning on a flat substrate, the biped robot is capable of spanning obstacles, ascending and descending stairs, as well as climbing slopes.When faced with a raised obstacle ahead, the most convenient approach is to circumvent it from its side or utilize the legs' height advantage to step over it.If one of the biped robot's feet becomes obstructed by the obstacle, it can effortlessly span over it as presented in Figure 4E and Video S7, Supporting Information.The mechanism for spanning the obstacle has been theoretically analyzed in the previous section and illustrated in Figure S21, Supporting Information.As the oscillation angle of the applied magnetic field progressively increases, so does the magnetic torque responsible for laterally flipping the biped robot.Consequently, there is a continuous augmentation in the support force on one foot while a simultaneous reduction occurs on the other foot until reaching an oscillation angle sufficient to induce sideways flipping of the obstructed foot.Simultaneously, there is a persistent increase in the magnetic torque that induces a tendency of twisting motion in the biped robot.Once lifted to match with obstacle height, the obstructed foot is actuated by magnetic torque to rapidly overtop of obstacle's upper surface, enabling successful spanning.
However, there are three specific scenarios where the biped robot fails to span the obstacle, as shown in Figure S22, Supporting Information.In the first scenario, the oscillation angle of the applied magnetic field is insufficient to generate enough magnetic torque for lifting the sliding foot of the biped robot, that is, it falls below the critical value depicted in Figure 4E.In the second scenario, due to an extremely low friction coefficient between the biped robot and substrate, the supporting foot starts sliding before reaching the critical oscillation angle required for sideways flipping.In the third scenario, although sufficient magnetic torque is generated to flip the biped robot sideways, its center of gravity exceeds the vertical axis of the supporting foot due to a high obstacle, ultimately resulting in complete tipping over.
The locomotion mechanism utilized by the biped robot for ascending steps is identical to that used for spanning obstacles.Figure 4F and Video S8, Supporting Information present the biped robot's continuous ascent of multiple steps followed by its descent.However, the mechanism employed by the biped robot for climbing slopes differs.In scenarios where the slope angles are lower than the elevation angle of the applied magnetic field, the biped robot can continue its forward movement.When faced with a slope angle greater than the elevation angle of the applied magnetic field, the biped robot will rotate in place at the junction between the slope and the level ground, rendering it incapable of climbing, as shown in Figure 4G and Video S9, Supporting Information.For this scenario, increasing the elevation angle of the applied magnetic field can be employed to realize the continued upward locomotion of the biped robot.Nevertheless, the approach induces variations in friction on the biped robot's supporting foot due to oscillation angles, resulting in unstable motion on slopes and potential downward sliding.The issue can be resolved by altering the axis direction of oscillating conical surfaces where the applied magnetic field is located.The force analysis of biped robot behavior on slopes and deflection angle for the axis line of conical surfaces is illustrated in Figure S23, Supporting Information.When the inclination angle of the slope is too steep, the biped robot becomes highly susceptible to rollover as its center of gravity exceeds the vertical axis passing through the supporting foot at maximum oscillation angle (Video S10, Supporting Information).
Additionally, with the advantage of the mechanism, the biped robot demonstrates robust locomotion on complex terrain.As shown in Figure 4H,I, as well as Video S11, Supporting Information, the biped robot is actuated by an external magnetic field to achieve leftward, downward, and rightward movement underwater.Moreover, it can proficiently navigate through a U-shaped tube with a concave lower surface, as shown in Figure 4J,K, and Video S12, Supporting Information.When confronted with unstable surfaces like gravel terrain, the biped robot adeptly follows a predetermined route for its locomotion strategy, as demonstrated in Figure 4L and Video S13, Supporting Information.Specifically, it moves forward initially followed by backward motion before proceeding forward again.Employing this same mode of locomotion enables traversal on the surface of fresh pork small intestine to accomplish pre-defined routes effortlessly, as shown in Figure 4M and Video S14, Supporting Information.

Locomotion Speed of Biped Robots
Without taking into account lagging and overshooting effects, it is evident that the velocity of the biped robot exhibits a proportional relationship with the distance between its two feet, as well as the oscillation frequency and angle of the applied magnetic field (Text S12, Supporting Information).Here, we define the oscillation frequency lower than 1 Hz as low-frequency and higher than 1 Hz as high-frequency.Figure 5A,B illustrates variations in relative moving speed under low-frequency and high-frequency magnetic fields.The distance between the biped robot's two feet is 6 mm and the body length of the biped robot is 8 mm.
For low-frequency magnetic fields with an oscillation angle of 30°, there is a disparity between experimental and theoretical values, with the experimental value appearing smaller.However, as the oscillation angle of the applied magnetic field increases, this deviation diminishes.On one hand, a smaller oscillation angle results in a larger proportion of hysteresis angle relative to the total oscillation angle (Figure S24, Supporting Information), thereby amplifying this effect.On the other hand, increasing the amplitude of the magnetic field leads to higher angular velocity of magnetic field deflection at the same frequency.According to our analysis based on the dynamics model, this increased angular velocity reduces friction applied to the sliding foot and consequently results in a larger deflection angle for the biped robot.
In the case of high-frequency magnetic fields, as the frequency increases, there is a significant deviation between experimental and theoretical values, particularly when considering an oscillation angle of 30°for the applied magnetic field.At an oscillation frequency of 19 Hz for the applied magnetic field, the biped robot  S1, Supporting Information.F) Walking speed of the biped robot with varying ratios of magnetic particles.G) Walking speed as a function of the magnetic field frequency for the biped robots with different distances between the feet.The distances between the biped robot's two feet are 4, 6, and 8 mm, respectively.H) Walking speed as a function of the magnetic field frequency for the biped robots with different body lengths.The body lengths are 4, 6, and 8 mm, respectively.I) Forward locomotion of the biped robot running on various substrates at an oscillation frequency of 10 Hz.Substrates #1 through #10 are oil surface, water surface, copper film, acrylic, glass, resin, office paper, sandpaper (P2000), and sandpaper (P1000), respectively.Scale bar: 5 mm.achieves a maximum speed of 202.63 mm −1 s with a relative velocity of 25.33 BL s −1 (where body length equals 8 mm), while the theoretical value stands at 14.25 BL s −1 .As indicated in the dynamics model analysis, increasing the frequency of oscillation in the magnetic field not only leads to an overshooting effect but also causes a rising effect on the sliding foot of the biped robot.
The combination of these two effects results in experimental test speeds exceeding those predicted by theory.
The analysis of Video S15, Supporting Information captured by the high-speed camera reveals that, as the frequency increases, the sliding foot of the biped robot transitions from ground contact to forward jumping locomotion.Additionally, Video S16, Supporting Information demonstrates that with an increase in magnetic field oscillation frequency, both the rising height and deflection angle of the sliding foot in mid-air also increase.The position states of the sliding foot transition from brief lift-and-drop motion to prolonged lift with deflection angles exceeding those caused by magnetic field oscillation alone.Combining these results with the dynamics model analysis explains why experimental walking speeds surpass theoretical values when magnetic field oscillation frequency is increased.
It is worth noting that at a frequency of 12 Hz, the biped robot's gait exhibits remarkable similarity to that of a human, as shown in Figure 4C and Video S17, Supporting Information.Consistent with the preceding introduction, the biped robot's walking posture can be categorized into two phases during one oscillation cycle of the applied magnetic field: the stance phase and the swing phase.However, if the magnetic field oscillation frequency becomes excessively high, locomotion instability arises in the biped robot.The sliding foot fails to reach its maximum deflection angle before the applied magnetic field starts oscillating in the opposite direction.Phase diagrams (Figure S25, Supporting Information) illustrate different oscillation angles at which the biped robot can move based on varying frequencies of the applied magnetic field.
Further systematic velocity tests of the biped robot were conducted at a frequency of 10 Hz for the applied magnetic field.When the strength of the applied magnetic field is increased from 2 to 7 mT, a marginal enhancement in the biped robot's speed is observed, as shown in Figure 5D.However, with an increase in the moment of inertia of the biped robot (Figure S26 and Table S1, Supporting Information), there is a notable decrease in its speed and an escalating instability in its motion attitude, as shown in Figure 5E.Conversely, varying the magnetic particle content within the prepared material for the biped robot does not result in any significant alteration to its speed, as shown in Figure 5F.It is analyzed that the decrease in the magnetic particle content leads to a decrease in the magnitude of the magnetic moment, although it also leads to a decrease in the density, that is, a decrease in the moment of inertia of the biped robot.
Figure 5G shows the variation of speed for biped robots with different distances between the feet.An increasing trend in locomotion speed can be observed as the distances between the feet increase.When scaling down the biped robot, there is hardly any significant change in its relative speed despite changes in its locomotion velocity, as shown in Figure 5H and Figure S27, Supporting Information.It is worth noting that variations in friction coefficient μ (range from 0.23 to 1.26) between the biped robot and substrates do not significantly affect its locomotion speed (Figure 5I).Specifically, when the oscillation frequency of the applied magnetic field is set at 10 Hz, the locomotion speed of the biped robot remains around 15 BL s −1 (Video S18, Supporting Information).The biped robot can move quickly even on surfaces covered with water (Video S19, Supporting Information).When actuated by a high-frequency oscillating magnetic field, the biped robot experiences limited forward movement on the oil surface.However, reducing the frequency of oscillation allows the biped robot to regain normal locomotion capabilities (Video S20, Supporting Information).Based on analysis from the dynamics model, it is determined that under high-frequency magnetic fields, the friction force acting upon the supporting foot is insufficient to satisfy the inertia required for forward locomotion of the biped robot.This phenomenon can be likened to an individual rapidly sliding their feet on ice without making any significant progress.

Multifunctionality of the Biped Robot
Agile locomotion is a crucial functional characteristic for the design and development of robots.Figure 6A illustrates a comparative analysis of relative locomotion speed in relation to body length for untethered walking robots and common arthropods.The corresponding data are depicted in Table S2, Supporting Information.The speed of the biped robot is represented by the red pentagram and can achieve a maximum value of 25.33 BL s −1 (under the working conditions involving an applied magnetic field with a strength of 5 mT, an oscillation angle, and an oscillation frequency of 30°and 19 Hz, respectively).The walking speed surpasses that of most common arthropods and outperforms all previously reported untethered walking robots.
Besides walking at high speeds, the biped robot is capable of carrying a load approximately four times its weight in stable forward and reverse locomotion, achieved through the application of a magnetic field strength of 9 mT and an oscillation frequency of 0.1 Hz (Video S21, Supporting Information).The biped robot itself weighs 0.137 g, while the cargo weighs 0.586 g, as presented in Figure 6B.In addition to its impressive locomotion capabilities, the biped robot demonstrates remarkable robustness and flexibility due to the utilization of soft materials in its construction.As shown in Figure 6C, even when subjected to finger presses that flatten its shape, the biped robot continues to function normally once the load is released (Video S22, Supporting Information).Furthermore, a detailed compression experiment reveals that the biped robot can be compressed from an initial height of 4 mm down to 2 mm with loads ranging from 0 to 1.1 N without any obvious adverse effects on its functionality after releasing the load (Video S23, Supporting Information).
Figure 6D and Video S24, Supporting Information demonstrate the multimodal locomotion of the biped robot actuated by low-frequency oscillations of an applied magnetic field on a complex and diverse surface.The demonstration titled "Lofty mountains and flowing water, infinity of confidants" shows the capabilities of the biped robot as it walks down steps, wades through water, climbs slopes and turns, and walks up steps to reach a high point.At the peak position, the biped robot skillfully performs somersaults to pick up objects imprinted with two Chinese characters pronounced as "zhi" and "yin," symbolizing "confidant."Subsequently, it descends back to its initial location.In this process, solely an applied magnetic field is requisite to actuate the biped robot through oscillations and deflections (Figure 6E), enabling a sequence of locomotion to be accomplished.Additionally, the biped robot is of insect size and can be conveniently stored within a capsule shell (Figure 6F).Subsequently, it is introduced into the human digestive system through ingestion.Despite the irregularities present on the internal surface of the stomach, actuated by an external magnetic field, the biped robot adeptly navigates toward the targeted lesion and subsequently reaches the antrum (Figure 6G, Video S25, Supporting Information).Notably, the biped robot exhibits not only multimodal locomotion by applied magnetic fields with low-frequency oscillations but also programmable controlled locomotion actuated under high-frequency oscillations.For instance, the control of a biped robot to achieve circular locomotion is a common scenario.As shown in Figure 6H, the applied magnetic field initially undergoes a counterclockwise deflection of 70°followed by a clockwise deflection of 106°within one oscillation cycle.As a result, the forward direction of the biped robot experiences a clockwise rotation of 36°.As previously analyzed, when undergoing high-frequency magnetic fields, the lag angle resulting from the biped robot's deflection can be considered negligible.Therefore, after ten oscillation cycles of the applied magnetic field, the biped robot will successfully complete circular motion in the clockwise direction.The diameter of this circular trajectory can be adjusted by modifying the deflection angle.Figure 6I shows the patterned design of the locomotion trajectory named "infinite" at an applied magnetic field with a strength of 5 mT and frequency of 5 Hz respectively.The biped robot performs rapid movements at a speed of 5 BL s −1 .
More complex trajectories can be realized in the same controlling method.As shown in Figure 6J, a "pentacle" trajectory can be accomplished by combining turning and straight locomotions.Specifically, the turning locomotion involves a deflection angle difference of 24°(Figure 6K).Following six consecutive oscillation cycles of the applied magnetic field, the biped robot successfully completes a 144°rotation.Importantly, despite being actuated by a high-frequency oscillating magnetic field, the distance between the start position of the trajectory and its end position after three laps remains significantly smaller than the body length of the biped robot (Video S26, Supporting Information), thus demonstrating stable and controllable locomotion.

Conclusion
In the process of utilizing an applied magnetic field to actuate a moving magnetic object, an interesting phenomenon has been observed.When the magnetic moment of a magnetized object and the applied magnetic field are in specific directions, the deflection of the applied magnetic field causes forward locomotion of the center point of the object.5a,9k] Theoretical analysis reveals that in this particular scenario, the deflection of the applied magnetic field generates not only a magnetic torque that drives the magnetic object to rotate around the perpendicular axis but also another magnetic torque that leads to a sideways flip.This results in a change in the distribution of the support force on the magnetic object by the supporting surface, that is, an increase in the support force on one side and a decrease in the support force on another side.If the rotational torque surpasses frictional resistance sufficiently to initiate rotation, then greater support force will lead to higher frictional force on one side while lesser support force will lead to lower frictional force on another side.As a consequence, the magnetic object will rotate around a point where greater frictional force is present, leading to forward locomotion at its center point.Based on this observation, we conjectured that when an oscillating applied magnetic field is employed to actuate the magnetic object, it will move forward in the form of a gait.Subsequent experiments verified the conjecture.We define the mechanism that can realize the above locomotion patterns, naming it the AFEMT.
For robots equipped with locomotion capabilities, robust locomotion performance is a crucial characteristic.We have developed a biped robot based on AFEMT and conducted comprehensive analyses of its force and locomotion posture through theoretical calculations and experimental tests.By utilizing an oscillating applied magnetic field, the biped robot achieves a robust gait by alternately moving forward with its left and right feet.Based on the advantages of AFEMT, the biped robot effortlessly overcomes obstacles, ascends steps, and climbs slopes.Moreover, the biped robot demonstrates robust locomotion on complex terrain, including a circular pipe and the uneven inner wall surface of a fresh small intestine.
The relative locomotion speed of the biped robot actuated by a high-frequency magnetic field reaches 25.33 BL s −1 , surpassing the walking speed of most arthropods and all previously reported untethered walking robots.Additionally, we observed that as the frequency of the applied magnetic field increased, the slipping foot of the biped robot was lifted and deflected forward.At an oscillating frequency of 12 Hz, the biped robot exhibits a gait posture in agreement with the human walking.Based on the advantages of AFEMT, the biped robot maintains consistent motion speeds on substrates with varying friction coefficients.Furthermore, driven by programmed applied magnetic fields, it can achieve rapid locomotion along pre-designed trajectories.After completing three laps around such a trajectory, the position error of the biped robot is less than its body length.
Despite the innovative guidance provided by this study for the design of insect-scale magnetic robots, we acknowledge that there are still certain limitations that need to be addressed.The primary structural characteristic of the biped robot is represented by its two legs, which necessitates further optimization.For instance, incorporating a Gomboc structure inspired by a turtle shell at the back of the biped robot would facilitate easier recovery of posture in case of tipping over. [12]Additionally, while the actuation approach of the magnetic field has demonstrated efficacy, it does impose restrictions on the biped robot's mobility due to limited space availability.Therefore, alternative actuation approaches should be explored in future research endeavors.Furthermore, with regards to functionality enhancement, consideration should be given to equipping these robots with specialized tools or instruments such as miniature cameras for non-invasive examination of the digestive tract or even drug delivery systems for targeted treatment within the stomach and intestines. [13]Although programmed magnetic field actuation has already been achieved, combining it with machine learning techniques can enable autonomous navigation on specific terrains.We believe that this AFEMT-based design opens up new possibilities for magnetically actuated robots and holds great potential for widespread application of insect-scale robotics.
Fabrication of the Resin Mold: The mold with robot-shaped concaves was fabricated by a 3D printer (NanoArch S130, BMF Precision, China) and poly(ethylene glycol) diacrylate resin.In order to prevent the residual resin from affecting the elastomer curing process, the resin mold underwent UV light irradiation for 30 min.Hydrophobic treatment was applied to the surfaces of the resin mold.The resin mold was treated with plasma treatment for 20 min followed by silane steam treatment at 85 °C for 8 h using POTS.
Fabrication and Actuation of the Robots: The biped robots can be fabricated using the demolding method, which involves the utilization of magnetic particles (NdFeB) and silicon polymer (Ecoflex 00-30).The fabrication process consists of three main steps, as illustrated in Figure S28, Supporting Information.First, the NdFeB particles and silicon-based materials (Ecoflex 00-30: part A and part B) were mixed in a 1:1 ratio by mass.Subsequently, vacuum degassing was conducted for 5 min in a vacuum chamber to eliminate air bubbles.The mixture was then poured onto the surface of the robot mold, and the entire structures were placed in a vacuum chamber for 5 min to ensure thorough filling of the mold cavity.After degassing, the mixture was cured at 60 °C for 20 min in an oven.Following curing, the magnetization of the robots was achieved using a magnetizer.The NdFeB particle distribution and the hysteresis loops are shown in Figure S29, Supporting Information.Actuation of the robots was accomplished through three-axis Helmholtz coils that generated an oscillating magnetic field, controlled by custom-programmed software with LABVIEW.The substrates, unless otherwise specified, were composed of 3D-printed resin upon which the robots walked and ran.
Various Substrates: Oil surface: Apply a layer of silicone oil to the resin substrate.Water surface: Cover the resin substrate with a layer of deionized water.Copper film: Copper foil tape adhered to an acrylic plane.The thickness of the copper film was 50 μm.Acrylic: The acrylic plate was milky white and had a polished surface.PDMS: The PDMS solution consists of a 10:1 mass ratio of PDMS monomers and curing agents, which was spin-coated on a silicon wafer and cured in the oven at 100 C for 30 min.The thickness of the PDMS film was 500 μm.Glass: Microscope slides polished on one side.Paper: Office A4 paper.P2000: The grit sizes of the sandpapers were 2000.P1000: The grit sizes of the sandpapers were 1000.
Speed Measurement: The speed of the robots was measured through frame analysis of the videos.A camera was utilized to capture the movement of the robots on a substrate imprinted with a scale.Once the robot had moved a certain distance, both the number of frames and distance were computed in order to ascertain its velocity.Each experiment was conducted more than four times, and subsequently, average values were derived.
Friction Coefficient Measurement: The friction coefficient was tested using the Mechanical Tester (Mach-1 v500cst, MA008), which enables displacement control in both axes.Initially, the relationship between compression depth and load was established through vertical displacement loading.Subsequently, the specified displacement was achieved using horizontal displacement loading followed by vertical displacement loading.The sliding friction force was measured until sliding commenced.The force-displacement curves for measuring the friction coefficient between the robot and various substrates are shown in Figure S30, Supporting Information.
Characterization Techniques: The magnetization profile was measured by a magneto-optical sensor (MagViewS, Matesy, Jena, Germany).The magnetic hysteresis of the magnetic elastomer was measured by a PPMS model 6000 quantum Design VSM.The SEM measurement was performed on a JSM-7800F scanning electron microscope.Numerical calculations were performed with the software MATLAB.

Figure 1 .
Figure1.Locomotion mechanism based on asymmetrical friction effect induced by magnetic torque.A) Schematics of a cuboid robot with two locomotion patterns.In the first case, the cuboid robot is capable of executing rotational movements solely within its current position.In the second case, it exhibits forward motion capabilities.B) Force analysis for the supporting edge of a cuboid robot.C) Applied magnetic field oscillating in the circular conical surface.D) Gait locomotion conjecture of the cuboid robot actuated by an applied magnetic field oscillating periodically on the circular conical surface.E) Deflection of the cuboid robot under the applied magnetic field B a with an elevation angle  to x-y plane.The inserted graph shows the magnetization of the cuboid robot.F) Gait locomotion of the cuboid robot actuated by an oscillating applied magnetic field.G,H) Gait locomotion of the robots with various geometries, including a "butterfly" and a "penguin."Scale bar: 2 mm.

Figure 2 .
Figure 2. Design of the biped robot in the static model.A) Schematics of human walking and running locomotion.B) Design of the biped robot based on AFEMT.The biped robot is composed of two feet, namely the right foot and the left foot.There is a small protrusion on the back of the biped robot to facilitate the distinction between its front and rear.C) Pitching angle  of the biped robot actuated by an applied magnetic field with an elevation angle .D) Theoretical and experimental results of variations in the pitching angle of the biped robot actuated by the applied magnetic field in various elevation angles.The strength of the applied magnetic field ranges from 2 to 7 mT.E) Pitching angle factor |cossin| as a function of the elevation angle .F) Support force on the left and right foot obtained through theoretical analysis as a function of the deflection angle.The elevation angle  of the applied magnetic field is 10°.G) Corresponding threshold of the deflection angle required for initiating biped robot movement as a function of the frictional coefficients μ.H) Rotation angles  of the biped robot actuated by an applied magnetic field with a deflection angle.I) Theoretical and experimental results regarding variations in lag angle  with magnetic field strength B a and elevation angle .J) Comparison between the biped robot's locomotion process and human walking.Scale bar: 5 mm.

Figure 3 .
Figure 3. Motion attitude control of biped robots in dynamics model.A) Force analysis of the biped robot in the dynamic model.B) Illustration depicting the variation in the pitching angle attitude of the biped robot.C) Numerical calculations of pitching angle for varying the biped robot's moment of inertia and magnetic field strengths.The results of calculations in statics are represented by graphical symbols.D) Illustration depicting the rotational motion model around the vertical center axis.E) The rotation angle and angular velocity of the biped robot.F) Frequency domain results of the deflection angle amplitude for the biped robot actuated by a magnetic field with varying frequencies.G) Time domain results of the deflection angles before the angle between the deflection angles of the biped robot and the magnetic field exceeds 180°.H) Frequency domain results of the deflection angle amplitude for the biped robot actuated by a magnetic field with varying strengths.I) Frequency domain results of the deflection angle amplitude for the biped robot with varying moments of inertia.J) Illustration depicting the motion mode of rotating around one foot.K) The rotation angle of the biped robot and the support force on the sliding foot for the biped robot actuated by a magnetic field with varying angular velocities.L) The rotation angle of the biped robot and the support force on the sliding foot for the biped robot actuated by a magnetic field with varying strengths.M) The rotation angle of the biped robot and the support force on the sliding foot for the biped robot with varying moments of inertia.

Figure 4 .
Figure 4. Multimodal locomotion of the biped robot.A) Sequential snapshot of the biped robot for forward locomotion at quarter-cycle intervals.The distance covered in the first half of the cycle is denoted as s 1 , while the distance covered in the second half is denoted as s 2 .B) Sequential snapshot of the biped robot for reverse locomotion at quarter-cycle intervals.The distances of reverse locomotion in the first and second half of the cycle are represented by s 1 and s 2 , respectively.C) Sequential snapshot of the biped robot for turning locomotion.The biped robot rotates by 90°toward its right side when it reaches a turning position.D) Sequential snapshot of the biped robot for rotation pattern on its one foot.E) Sequential snapshot of the biped robot for spanning the obstacle.The height of the obstacle is 1.0 mm.F) Sequential snapshot of the biped robot for climbing the steps.The height of the obstacle is 0.4 mm.G) Sequential snapshot of the biped robot for climbing the slope.The slope is 30°.H,I) Sequential snapshot of the biped robot for walking in a water tank from top and side view.J,K) Sequential snapshot of the biped robot for walking through a U tube from top and side view.L) Sequential snapshot of the biped robot for walking across a sand land from top view.M) Sequential snapshot of the biped robot for walking on a fresh small intestine.Scale bar: 2 mm.

Figure 5 .
Figure 5. Locomotion speed of the biped robot.A) Walking speed of the biped robot under low-frequency magnetic fields.The distance between the biped robot's two feet is 6 mm.B) Walking speed of the biped robot under high-frequency magnetic fields.The distance between the biped robot's two feet is 6 mm.C) Comparison between the human walking gait and the gait of the biped robot at an oscillation frequency of 12 Hz of the applied magnetic field.D) Walking speed of the biped robot under varying magnetic field strength.E) Walking speed of the biped robot with varying moment of inertia.The parameters of the moment of inertia of biped robots #1 to #5 are shown in TableS1, Supporting Information.F) Walking speed of the biped robot with varying ratios of magnetic particles.G) Walking speed as a function of the magnetic field frequency for the biped robots with different distances between the feet.The distances between the biped robot's two feet are 4, 6, and 8 mm, respectively.H) Walking speed as a function of the magnetic field frequency for the biped robots with different body lengths.The body lengths are 4, 6, and 8 mm, respectively.I) Forward locomotion of the biped robot running on various substrates at an oscillation frequency of 10 Hz.Substrates #1 through #10 are oil surface, water surface, copper film, acrylic, glass, resin, office paper, sandpaper (P2000), and sandpaper (P1000), respectively.Scale bar: 5 mm.

Figure 6 .
Figure 6.Multifunctionality of the biped robot.A) Comparison of speeds in relation to body length for untethered walking robots and common arthropods.B) Load-carrying performance of the biped robot.The magnetic field strength and the oscillation frequency are 9 mT and 0.1 Hz, respectively.The biped robot itself weighs 0.137 g, while the cargo weighs 0.586 g.C) Weight-bearing performance of the biped robot under a finger.D) Multimodal locomotion of the biped robot on a designed platform.The demonstration is titled "Lofty mountains and flowing water, infinity of confidants."The body length is 8 mm.E) Locomotion patterns and corresponding magnetic field forms.F) Diagram of the biped robot entering the digestive system by oral administration.The biped robot can be conveniently stored within a capsule shell.G) Multimodal locomotion of the biped robot on an internal surface of a stomach model.The biped robot enters through the cardia and subsequently reaches the lesion and the antrum in turn.H) The rotation of the biped robot in circular motion and the deflection of the applied magnetic field.I) Patterned locomotion trajectory of the biped robot by programmable controlling.The trajectory is a symbol of the "infinite."The body length is 4 mm.J) Patterned locomotion trajectory.The trajectory is a symbol of the "pentacle."The body length is 4 mm.K) The rotation of the biped robot in turning motion and the deflection of the applied magnetic field.Scale bar: 5 mm.