Method of actuating a shape changeable member, shape changeable member and actuating system The present invention relates to a method of actuating a shape changeable member of actuatable material. The invention further relates to a shape changeable member and to a system comprising such a shape changeable member and a magnetic field apparatus. Miniature members are envisaged that can facilitate access to and inspection of narrow, remote and unexplored environments. However, restricted mobility of miniature members may hinder the realization of targeted applications in highly unstructured landscapes. To date the miniature members produced are generally tethered members, this means that they are generally connected to a further structure and can be adapted to move relative to the further structure. While there also exists untethered miniature members, these miniature members are generally only capable of carrying out one mode of locomotion and are specifically designed to carry out that one single mode of locomotion. The use of members that can only carry out limited number of modes of locomotion however hinders the movement capabilities and applications that can be carried out by such members.

In view of the foregoing it is an object of the present invention to make available shape changeable members that can more readily access remote and unexplored environments and that are capable of carrying out multiple modes of locomotion.

This object is satisfied in accordance with a method of actuating a shape changeable member as defined in claim 1 .

By way of example, such a method of actuating a shape changeable member, in which the shape changeable member has a height, a width as well as a length, and is untethered; comprises the steps of:

- actuating the shape changeable member by means of at least one stimulus to bring about a change of shape of the member at least over the length of the shape changeable member in at least one spatial direction and thereby inducing a torque into the shape changeable member; - effecting a locomotion of the member by means of the induced change of shape of the member in the at least one spatial direction; and

- moving the shape changeable member in the at least one spatial direction. The use of untethered members increases the movement capabilities of such members. Moreover, the use of at least one stimulus to bring about a variety of changes of shape of the member to induce different moments into the member to effect different types of locomotion increases the variety of modes of moving the member. The different modes of moving the member translate to different types of locomotion which were previously not possible using one and the same member.

Thereby miniature mobile members capable of accessing and navigating effectively in narrow remote environments can be made available. These miniature members may enable unprecedented applications in microfactories, bioengineering, environmental remediation, and healthcare. Moreover, one and the same miniature device can be made available that can inherently carry out the different modes of locomotion.

In this connection it should be noted that the moment can also be referred to as torque and the induced torque is generally small when the induced change of shape of the member causes a small deformation of the member. Moreover, it should be noted that the torque vanishes when the net magnetic moment of the deformed member is aligned with an applied stimulus, such as an applied magnetic field.

Preferably the induced torque is used to steer the shape changeable member in the at least one direction.

It is preferred if a type of locomotion of the shape changeable member is selected from the group of members consisting of swimming, walking, flipping, crawling, rolling, diving, immersion, emersion, jumping, landing and climbing, e.g. meniscus climbing.

Dexterous miniature members that can swim, walk, roll, jump, or crawl have been proposed in recent years. However, these members are only capable of performing one specific locomotion mode, and are inadequate to escape from liquid environments to solid terrains. In contrast to this the shape changeable member can be actuated to carry out several modes of locomotion one after the other. This means that the member can, for example, move through a liquid, leave the liquid and then move on a solid surface, and vice versa.

For example, untethered, amphibious, millimeter-scale soft (rubber like) members capable of multiple modes of locomotion can be made available that move across and/or through liquid and solid unstructured terrains.

In this connection, soft means that the member is inherently flexible, such as a member comprising rubber and is not a hard/rigid member such as a piece of non- flexible metal or wood .

The members can be actuated by locomotion mode-specific external magnetic fields, the magneto-elastic sheet-shaped members with a pre-programmed magnetization profile and hydrophobic surface can swim inside and on the surface of liquids, climb liquid menisci, roll, walk on ground, jump directionally over obstacles, and crawl within narrow spaces. Such members can reversibly transit from the surface to the bulk of a liquid, and from a liquid surface to rigid ground.

Advantageously the change of shape comprises at least one of a deformation, a contraction, a flexing, an undulation and a stretching of the actuatable material of the shape changeable member. In this way the member can change its shape in a variety of ways with each change of shape being capable of being used for certain types of locomotion. By way of example, the shape changeable member can be deformed to adopt a sine like or cosine like shape to effect a walking type of locomotion, in particular wherein a rotating magnetic field B sequence of small magnitude is applied to the shape changeable member to enable an undulating locomotion of the shape changeable member, with an applied magnetic field preferably having a small magnitude |B| being selected in the range of 2 to 5 mT and especially of 3 mT.

By way of a further example, the shape changeable member can be deformed to adopt a part-circular shape, such as a semi-circular shape, to effect a rolling or meniscus climbing type of locomotion, in particular wherein a rotating magnetic field B sequence of large magnitude is applied to the shape changeable member to enable a rolling locomotion of the shape changeable member, with the applied magnetic field preferably effecting a rolling type of locomotion in a clockwise sense, and with an applied magnetic field preferably having a large magnitude |B| selected in the range of 12 to 20 mT and especially of 15 mT, in particular where- in, to effect a meniscus climbing type of locomotion a large constant magnetic field B is applied followed by a rotating magnetic field B sequence of large magnitude with an applied magnetic field preferably having a large magnitude |B| being selected in the range of 10 to 20 mT and especially of 1 1 .7 mT.

By way of a further example, the shape changeable member can be deformed to undulate and generate a travelling wave on the member to effect a swimming type of locomotion or crawling type of locomotion. By way of a further example, the shape of the shape changeable member can be flipped from a first shape into a second shape and back into the first shape to effect a jelly-fish swimming type of locomotion, wherein the first and second shapes preferably resemble that of a part-circle. By way of yet a further example, the shape changeable member can be pulsed with the at least one stimulus in order to effect the jumping type of locomotion.

By way of yet a further example to effect an immersion type of locomotion a magnetic field having a rotating B sequence is applied, with the magnetic field effecting a rotation of the shape changeable member in a counter-clockwise sense to reduce a contact with a liquid surface, with the applied magnetic field preferably having a magnitude selected in the range of 15 to 30 mT, especially of 20mT.

By way of yet a further example to effect a landing type of locomotion a magnetic field having a rotating B sequence of variable magnitude applied, with the initial B- sequence having a high magnitude which is then reduced to a low magnitude, with the high magnetic field preferably being selected in the range of 10 to 15 mT and the low magnitude being selected in the range of 3 to 8 mT. By way of yet a further example a rocking sequence of the magnetic field is applied to perform the walking type of locomotion, in particular the walking type of locomotion is effected by use of four phases in which four different kinds of magnetic field are applied. By way of yet a further example to effect a jelly fish-like swimming motion mode of locomotion of the shape changeable member the applied magnetic field is sequentially flipped from a positive magnetic field to a negative magnetic field causing an inversion of the shape of the shape changeable member. Preferably the stimulus is one of an external stimulus and an internal stimulus.

Examples of an internal stimulus can e.g. be pressurized air that is provided at the member in order to inflate or deflate certain regions of the member to bring about an actuation of its shape.

Examples of an external stimulus are the application of a magnetic field, an electric field, heat, or laser light. E.g. direct illumination of the laser on the member which causes ejection of material and motion of the member.

Examples of members that can be actuated by external stimuli that do not include the application of a magnetic field as the main source of stimulus are thermally swelling materials, light-sensible materials locally heated by laser light. For example, shape-memory polymers can be used that can deform when they are stimu- lated by heat. By pre-programming these polymers, different regions of the polymer can create different curvatures.

It is however preferred if the external stimulus is present in the form of a time varying magnetic field applied in at least one dimension, preferably two dimensions, most preferably in three dimensions.

A time varying magnetic field is simple to realize and can be implemented in a cost effective manner and can be used to move a variety of members through various types of media.

Moreover, a time varying magnetic field can not only be changed in its magnitude but also in its direction and this variation in field strength and direction can be utilized by a shape changeable magnetic member that has a pre-determined magnetization profile to change its shape over its length and height in quick succession in order to be moved by the magnetic field using different forms of locomotion.

Preferably a magnitude of the time varying magnetic field applied is selected in the range of 0 to 500 mT. Such magnetic fields can be used to move a member both in liquid environments as well as in dry environments.

Advantageously the magnitude of the time varying magnetic field applied is varied with a frequency selected in a suitable range of e.g. 0 to 1 kHz, preferably of 0 to 50 Hz. These frequencies have found to be advantageous to bring about the locomotion of the shape changeable member. It is preferred if a direction of the time varying magnetic field applied is varied with a frequency selected in the range of 0 to 1 kHz. These frequencies have been found to be advantageous to bring about the change of shape of the member and to induce the various types of locomotion.

It is preferred if the shape changeable member can be actuated in order to additionally carry out at least one function. In this way, for example, a robot can be made available that is capable of executing pick-and-place tasks and shape de- formation-triggered release of cargos.

For example, the at least one function comprises at least one of a gripping function, a holding function, a dropping function, a clamping function and picking up function.

In accordance with a further aspect the invention also relates to a shape changeable member of actuatable material, the shape changeable member being unteth- ered and having a width, a height and a length, wherein the shape changeable member is configured to be actuated by means of at least one stimulus in order to bring about a change of shape of the member at least over the length of the shape changeable member; wherein the shape changeable member is configured to be actuated in accordance with a method of actuating the shape changeable member as discussed in the foregoing. In this connection it should be noted that the shape changeable member can be furthered using similar features to those discussed in the foregoing in relation to the method in accordance with the invention.

Such minimalist, programmable and versatile soft members could enable applications in environmental monitoring and minimally invasive medical operations inside the human body.

The proposed members with versatile and robust multimodal locomotion capability may be tailored to diverse applications, including environmental monitoring and minimally invasive medical operations. Further progress is envisaged by implementing a remote member localization method and improving the member's dex- terity, precision, safety, and functionality using a closed-loop feedback control by monitoring the member while this is moved e.g. through the human body.

Preferably the stimulus is an external stimulus in the form of a magnetic field applied in at least one dimension and most preferably in three dimensions, with the magnetic field interacting with a pre-programmed magnetization profile of the shape changeable member, with the member preferably having a hydrophobic surface. Providing a magnetized member that moves in a magnetic field is a beneficial way of implementing the various modes of locomotion required to e.g. facilitate drug delivery in the human body.

A viable solution to develop miniature devices with multimodal and multi-terrain locomotion capability is to employ soft active materials, such as ecoflex, shape memory alloys, liquid metals, silicone rubbers, silicone-based materials, polyure- thanes, soft gels (hydrogels, oil-based gels, aerogels) as well as natural polymers, such as DNA or proteins that are embedded with ferromagnetic, paramagnetic or diamagnetic particles, which endow machines with extra degrees-of-freedom and support programmable shape changes while assuring safe interaction with delicate environments.

Moreover, magnetic actuating fields can harmlessly penetrate most biological materials, making them particularly promising for biomedical applications.

In a further aspect the present invention relates to an actuating system comprising a shape changeable member in accordance with the foregoing and a magnetic field apparatus, wherein the magnetic field apparatus is a source of the at least one stimulus.

The advantages discussed in the foregoing in connection with the method of actuating as well as in connection with the shape changeable member likewise apply to the system. It is preferred if the magnetic field apparatus is configured to generate a uniform 3D time-varying magnetic field at a center of the system. In this way the shape changeable member can be actuated on by a magnetic field in order to effect the different types of locomotion. Such as system can then be used to e.g. steer miniature robots to different positions of application.

Preferably the magnetic field apparatus comprises at least two, preferably three and especially four, pairs of magnetic coils, wherein each of the pairs of coils is arranged on a common axis, with the respective common axes being arranged at least approximately perpendicular to one another at a center of the system. Such a coil arrangement can generate very accurate time-varying magnetic fields that are suitable to moving the shape changeable member. It should be noted in this connection that if four pairs of coils are used then the fourth pair of coils is configured to control a gradient of the applied magnetic field.

Advantageously the shape changeable member is configured to be actuated by means of the pairs of magnetic coils in such a way that the shape changeable member carries out at least one type of locomotion within the applied magnetic field.

It is preferred if each coil of each pair of magnetic coils is arranged at least substantially equidistant from the center. In this way a homogenous magnetic field can be generated that can be varied in a time dependent manner both with regard to the direction of the magnetic field applied and with the magnitude of the applied magnetic field.

In this way the pairs of magnetic coils can be configured to produce a time-varying magnetic field at the center.

In order to produce a time-varying magnetic field at the center, each pair of coils can be actuated to produce a magnetic field at the center whose field strength can be suitably varied, preferably in the range of 0 to 500 mT, in particular of 0 to 50 mT. Indeed each pair of coils can be actuated to produce a magnetic field at the center whose field strength can be varied with a frequency in the range of at least 0 to 1 kHz, in particular of 0 to 50 kHz.

In fact each pair of coils can be actuated to produce a magnetic field having a var- ying direction in one of the following planes spatial planes X-Y, X-Z, Y-Z. By changing the direction in the different planes in which the field is applied means that the shape changeable member can be moved in all three directions of space.

Preferably the change in direction of the magnetic field applied can be varied with a frequency selected in the range of 0 to 1 kHz, in particular of 0 to 50 kHz.

Advantageously each coil of each pair of coils is configured to carry a current in the range from 0 to 20A for a coil comprising 3 layers of wires with 60 wires per layer and a diameter of 1 .05 cm. Preferably, if three pairs of magnetic coils are used, two pairs of the magnetic coils are provided with an at least substantially equal size and each coil of the third pair of magnetic coils has a larger size than a respective coil of the other two pairs of magnetic coils.

In this way a simple configuration is made available to produce a uniform magnetic field. Advantageously a diameter of each of the four coils of the two pairs of magnetic coils of at least substantially equal size is selected in the range of 50 to 65 cm and a diameter of each of the two coils of the third pair of magnetic coils of at least substantially equal size has a size that is selected in the range of 100 to 140 cm. Preferably the shape changeable member is configured to be steered by a magnetic field applied in at least one of the Y-direction and the Z-direction and is further configured to be changed in shape by a magnetic field applied in the X- and Y- direction and is configured to be moved by the magnetic fields applied in the X- and Y- direction.

Advantageously the magnetic field is applied relative to a local X-Y frame of the shape changeable member and through a presence of the magnetic field deforms a shape of the shape changeable member in the X-Y frame, wherein the shape changeable member is steering by a magnetic field applied in at least one of the Y-direction and the Z-direction.

The invention will be explained in detail in the following by means of embodiments and with reference to the drawing in which is shown: Fig. 1 a) a design of a magneto elastic-shape changeable actuatable member and b) the magnetic response of the magneto elastic-shape changeable actuatable member;

Fig. 2 various types of modes of locomotion and associated transition

modes of the magneto elastic-shape changeable actuatable member, with a) illustrating swimming, b) meniscus climbing, c) immersion, d) landing, e) rolling, f) walking, g) crawling and h) jumping; Fig. 3 multimodal locomotion over liquid-solid hybrid terrain by the magneto elastic-shape changeable actuatable member, with a) illustrating roll- ing, diving and swimming, b) immersion, c) emersion, d) meniscus climbing, landing, jumping, walking and e) walking and crawling; a) approach and pick up of a load by the magneto elastic-shape changeable actuatable member and b) transport and placement of the load by the magneto elastic-shape changeable actuatable member;

a magnetization process of a soft magneto-elastic shape changeable actuatable member, with a) illustrating an elastomeric beam loaded with NdFeB microparticles wrapped around a cylindrical glass tube (perimeter: 3.7 mm) and magnetized by a uniform 1 .5 T magnetic field produced by a vibrating sample magnetometer, and b) illustrating the programmed magnetization profile in the unwrapped member, for β _κ = 45°;

a six-coil electromagnetic setup to actuate the magneto-elastic shape changeable actuatable member;

a quasi-static analysis of a soft shape changeable actuatable member;

a rotating B sequence enabling the undulating motion of the magneto-elastic shape changeable actuatable member, with (a) illustrating the required B in dependence on time to undulate the member 10 to the right, with the variables B _x and B _y representing the magnetic field along the x- and y-axis of the coordinate system, shown in (b) with Fig. 8(b) illustrating two principle shapes;

further illustrations as shown in Fig. 8a) and b) for a further type of locomotion;

further illustrations as shown in Fig. 8a) and b) for a further type of locomotion;

further illustrations as shown in Fig. 8a) and b) for a further type of locomotion;

further illustrations as shown in Fig. 8a) and b) for a further type of locomotion;

further illustrations as shown in Fig. 8a) and b) for a further type of locomotion;

further illustrations as shown in Fig. 8a) and b) for a further type of locomotion;

snapshots of straight and directional jumping by the magneto-elastic shape changeable actuatable member, with a) illustrating a first type of straight jumping, b) a second type of straight jumping, c) a type of directional jumping and d) the magnetization profile of the shape changeable magnetic member used in a to c, and its magnetic response at large |B|;

Fig. 16 a vibrational analysis of a shape changeable actuatable member;

Fig. 17 boundary conditions for the Type-1 straight jump of Fig. 15a; and Fig. 18 a design of a modified shape changeable actuatable member for a magnetically triggered cargo release, with a) illustrating an extra strap added to the member 10, b) the extra strap locked into a hole on the body of the member with cargo arranged therebetween and c) the opening of the strap.

Fig. 1 a shows a design of a shape changeable actuatable member 10 present in the form of a strip of material . The shape changeable actuatable member 10 can also be referred to as a magneto-elastic member 10. The member 10 has a length I in the x-direction, a height h in the y-direction and a width w in the z-direction. In the present example the member 10 is formed from a continuum of infinitesimal segments ds of which nine segments s are shown for the purpose of illustration along the x-direction, with the segments s (ds) each comprising the same material but having a different direction of magnetization as indicated by arrows. The final segment Sf in the x-direction indicates a phase shift β _κ of the magnetization profile, this phase shift is important as will be discussed in the following.

The member 10 is magneto-elastic and fabricated from a hydrophobic silicone rubber sheet loaded with embedded 5 μηη NdFeB particles. Fig. 5 and the subse- quent description details steps that can be carried out to form such a member 10. The member 10 is magnetized as a single-period sinusoidal function for its m _x and nriy component along the sheet's length I and equally along the sheet's width w and thickness h. Fig. 1 b shows the member 10 of Fig. 1 in four different states of excitation. The different states of excitation are brought about by way of remote magnetic actuation by means of a multi-coil electromagnet system (see Fig. 6 in this regard). This multi-coil electromagnet system enables untethered operations, i.e. the member 10 is not connected to a host, substrate or the like and can indeed move freely in space.

The first state shows a member 10 with β _κ = 45° in its rest state when no magnetic field B is applied, a scale bar is also indicated and has the size of 1 mm. The member 10 shown in the present example has a size of 3.7 χ 1 .5 0.24 mm ³ . Typical sizes of the member can be selected in the range of 0.01 to 10 0.01 to 10 x 0.01 to 2 mm ³ . It should be noted in this regard that also non-uniform shaped members 10 can be produced; however for the ease of simplicity a member 10 of strip shape is described in the following.

The second state shows an applied magnetic field of 3 mT field strength that is aligned in the X-Y plane with a principal direction of a = 225°. In the second state the magnetic member 10 adopts a sinusoidal shape.

The third state shows an applied magnetic field of 3 mT field strength that is aligned in the X-Y plane with a principal direction of a = 135°. In the third state the magnetic member 10 adopts a cosine like shape, i.e. by changing the phase of alignment of the magnetic field by 90° has caused a shift in the general shape of the member by 90°.

If one were to continue changing the principal direction of the magnetic field of the same amplitude one can observe a traveling wave traversing through the member 10. In this way the member 10 can be caused to undulate and thereby a swimming like type of locomotion can be effected (see also Fig. 3a)

In this regard it should be noted that a time varying magnetic field can be applied with a field strength selected in the range of 0 to 500 mT, with a frequency of the magnitude being variable in a range selected as 0 to 1 kHz. Likewise a direction of the applied magnetic field can be changed with a frequency selected in the range of 0 to 1 kHz.

The fourth state depicted shows the application of a significantly larger magnetic field in comparison to the third state for the same principal direction of magnetiza- tion. In this magnetic field the member 10 is distorted into a horse shoe like shape or a semicircle like shape when |B| is large.

The four states show that the shape of the member 10 can be changed by changing the principle direction and/or the strength of the applied magnetic field. Indeed by generating desired magnetic fields net magnetic moments arise in the member 10. The electromagnets allow the soft member 10 to thus not only produce time- varying shapes, but also various magnetic moments that can invoke a variety of modes of locomotion. The magnetic moments align with an externally applied magnetic field B, so that magnetic torques can steer the member 10 in three- dimensions (3D) in all modes of locomotion.

Fig. 2 shows various types of modes of locomotion and associated transition modes of the magneto elastic-shape changeable actuatable member 10. The scale bars shown in Fig. 2 relate to an actual size of 1 mm.

Fig. 2a shows a mode of locomotion of the member 10 that resembles Jellyfishlike swimming, when the member 10 is completely immersed in water. Indeed it has been found that the member 10 can overcome gravity by swimming upwards. The swimming is brought about by a time-asymmetrical reversal of B along the vertical direction. The maximum field strength used is |B| _max = 20 mT. Thus, a periodic magnetic field B with a time-varying magnitude along the vertical direction causes the jellyfish-like swimming motion of the member 10, and propels the member 10 to reach the water surface (Fig. 3c). Upon emersion, the soft member 10 strongly pins to the water-air interface by exposing its hydrophobic body surface to air.

Fig. 2b shows a mode of locomotion of the member 10 that resembles water me- niscus climbing. A counter-clockwise magnetic torque progressively adapts the pose of the member 10 to the meniscus profile and raises its center of mass. The maximum magnetic field used in this example is |B| _max : 1 1 .7 mT.

By applying the different magnetic field a programmed shape change occurs from the shape shown in Fig. 2a that allows the member 10 to exploit the capillary action of a water surface and move upward along the slope of a concave meniscus (Fig. 2b). Meniscus climbing is typically adopted by insects when they cannot swim across frictionless liquid barriers, thus, the member 10 can adopt an insect like mode of locomotion.

It has namely been found that a control of the magnitude of B and of the direction of B makes the member 10 curve upwards and gradually rotate to match the local curvature of the water surface. This curving and rotation permits the member 10 to undergo capillary displacement without extra energy expenditure. Conversely, a fast sequence of downward bending and rotation disengages the member 10 from the water-air interface as shown in Fig. 2c.

Fig. 2c shows a mode of locomotion of the member 10 that resembles immersion, i.e., a transition from the surface of a water pool into the bulk water of a water pool by a combination of curling and rigid-body rotation. The maximum magnetic field used in this example is |B| _max : 20 mT.

Fig. 2d shows a mode of locomotion of the member 10 that resembles landing, i.e., a transition from a water surface onto solid ground. A magnetic field that changes clockwise B peels the member 10 off the water surface and lets it stand on the platform. The maximum magnetic field used in this example is |B| _max : 1 1 .7 mT. On meniscus climbing, the member 10 can thus efficiently rotate to peel away from the water surface to stand on an adjacent solid substrate. The low wettability (re- ceding water contact angle -78°) of the microscopically rough elastomer surface of the member 10 leaves the receding triple contact lines unpinned and makes water easily dewet the surface of the member 10 during its displacement.

Fig. 2e shows a mode of locomotion of the member 10 that resembles rolling by means of a high magnitude, and low frequency rotating B field. The maximum magnetic field used in this example is |B| _max : 15 mT. By rolling, i.e. rigid-body rotation around the z-axis enabled by means of the high magnitude, low frequency B, the member 10 can advance directionally over a rigid substrate and also dive from a solid onto a liquid surface (see Fig. 3a). However, the curled member 10 cannot roll across substrate gaps wider than the diameter of the member 10. In order to bridge gaps the member 10 changes into a walking mode of locomotion.

Fig. 2f shows a mode of locomotion of the member 10 that resembles walking. The direction and magnitude of the magnetic field B control the gait of the member 10 and modulate the stride between its legs, respectively. The maximum magnetic field used in this example is |B| _max : 7.7 mT. The dashed lines mark the net displacement after one walking cycle. Walking is a particularly robust way to move over unstructured terrain morphologies, and affords precise tuning of stride granularity. Directional walking can be achieved by sequentially and periodically adapt- ing the tilting angle and curvature of the member 10 through the control of the orientation and magnitude of the magnetic field B, respectively.

By way of example the walking to the right shown in Fig. 2f will be described in more detail by using the following sequence of the magnetic field B (see also Fig. 13a):

1 . Start walking (Frame I): The magnetic field B is oriented at a=135° with a magnetic field strength |B| = 7.7 mT. Under this condition, the member 10 assumes a vertically symmetric pose with both feet touching the ground (see also Fig. 12b). 2. Rise of the front foot (Frames I and II): B rotated from a = 135° to 124° while |B| = 7.7 mT. This lifts up the front foot (with respect to the moving direction) while keeping the back foot fixed.

3. Stretch and lowering of the front foot (Frames II and III): While a is kept at 124°, |B| is linearly decreased from 7.7 mT to 0 mT. This lowers the front foot and extends the horizontal span of the member 10 by an amount d while the member

10 pivots on the back foot.

4. Shrink and rising of the back foot (Frames III and IV): While a is kept at 146°, |B| increases from 0 mT to 7.7 mT. This makes the member 10 curl to shrink its span and lift up the back foot while keeping the front foot fixed as a pivot.

5. Lowering of the back foot (Frames IV and V) (Lower the back foot): B is rotated from a =146° to 135° while |B| = 7.7 mT. This closes the control cycle by returning the member 10 to its starting pose at a net rightward displacement d with respect to its initial position (compare to frame I).

Jellyfish-like swimming (Fig. 13)

The jellyfish-like swimming shown in Fig. 2a is realized by the following steps: 1 . The member is acted on to implement a fast power stroke.

2. The member is then acted on to implement a slow recovery stroke.

By this periodic flipping of B along the vertical direction to give a fast power (downward) stroke, the robot (member), functioning at a Reynolds number Re -32.5, is shown to swim upward against the gravitational field, and even against the downward magnetic field gradient eventually present at the vicinity of the bot- torn coil of the working space.

Meniscus climbing (Fig. 9)

The meniscus climbing shown in Fig. 2b is realized by the following steps:

1 . The member is acted on to assume a vertically symmetric pose with upward curvature. The member achieves in 400 ms a stable position along the meniscus known to depend on both the initial meniscus profile and the curvature of the device.

2. B is subsequently rotated clockwise toward to progressively displace the position of the robot along the meniscus and raise its center of mass. Once the de- vice reaches contact with the solid substrate, B is turned off and the robot rests on the edge of the adjacent platform (Frame I in Fig. 2d).

Immersion (Fig. 10) The immersion shown in Fig. 2c is realized by the following steps:

1 . The member is acted on to initially curl downward (Frame II in Fig. 2c).

2. The member is then acted on to rotate counterclockwise to reduce the contact with the water-air interface to a minimum (Frame II in Fig. 2c). At the end of this step, one ending of the beam is still attached to the water surface (Frame III in Fig. 2c).

3. A quick 180° flipping of the external B field disengages the ending of the member from the water surface and let the robot sink (Frame IV and V in Fig 2c). Landing (Fig. 1 1 )

The landing shown in Fig. 2d is realized by the following steps:

1 . The member is acted on to keep rotating clockwise while using the edge of the platform as the pivot point.

2. When the second leg of the member reaches the platform (Frame V, Fig. 2d), the external B field is turned off.

Rolling (Fig. 8)

The rolling shown in Fig. 2e is realized by using the following steps:

1 . A large magnitude B (15 mT) along a field angle of 157° to deflect the robot into a distorted cosine shape resembling a semicircle-like shape (Fig. S5b).

2. The extermal B starts rotating, a rigid-body magnetic torque, Xrigid-body, is induced by the robot, which can be described mathematically as:

^T rigid-body = M _net X B. (S9) The external B starts rotating and the agent , the robot's M _ne t will be aligned with its final direction, and T _rigid _ _body will be reduced to a null vector.

Walking (Fig. 12)

The rightwards walking shown in Fig. 2f is realized by the following steps

1. Start of walk (Frame I): B oriented at a=135° with |B| = 7.7 mT. Under this condition, the robot assumes a vertically symmetric pose with both feet touching the ground (Fig. S8b).

2. Rise the front foot (Frame I and II): B rotated from a = 135° to 124° while |B| = 7.7 mT. This lifts up the front foot (with respect to the moving direction) while keeping the back foot fixed.

3. Stretch and Lower front foot (Frame II and III): While a is kept at 124°, |B| is linearly decreased from 7.7 mT to 0 mT. This lowers the front foot and extends the horizontal span of the robot by an amount d while the robot pivots on the back foot.

Shrink and Rise the back foot (Frame III and IV): While a is kept at 146°, |B| increases from 0 mT to 7.7 mT. This makes the robot curl to shrink its span and lift up the back foot while keeping the front foot fixed as pivot.

Lower the back foot (Frame IV and V) (Lower the back foot): B is rotated from a =146° to 135° while |B| = 7.7 mT. This closes the control cycle by returning the robot to its starting pose at a net rightward displacement d with respect to its initial position (compare to frame I).

Crawling and undulating swimming (Fig. 7)

The crawling mode of locomotion (Fig. 2g) require undulating, time-varying shapes. The agent achieves its undulation by using a rotating B of low magnitude and relatively high frequency. The deformation produced by the time-dependent undulation can be described mathematically as: mAl ³ B

cos I— x I cos I— t j + sin I— x I sin I— 1 1 where the variable T represents the period of the rotating B. The same motion is also applied to undulating swimming. Jumping

The jumping shown in Fig. 2h is realized by using the following steps:

1 . In Type-1 straight jumping, the agent is first pre-bent by the magnetic field to curl upward and stand on its center (frame I of Fig. 2h).

2. The direction of B is then quickly reversed to force the agent to flatten out (frames II and III of Fig. 2h) and hit the substrate by its feet.

Fig. 2g shows a mode of locomotion of the member 10 that resembles crawling inside a cylindrical tube (inner diameter: 1 .62 mm). A clockwise rotating magnetic field B (B = 3 mT at 15 Hz) propels the motion to the right by body undulation. The dashed lines mark net displacement after five crawling cycles. When facing narrow openings in standing obstacles that may prevent walking, the member 10 can effectively resort to small amplitude body undulations to crawl through the opening. The crawling motion is encoded by a low magnitude rotating magnetic field B that produces reversible longitudinal traveling waves along the magneto-elastic member 10. A similar control sequence additionally makes the member 10 swim efficiently on liquid surfaces akin to a Taylor swimming sheet (see Fig. 3a), as it does in the member 10 with m featuring multiple sinusoidal periods. Fig. 2h shows a mode of locomotion of the member 10 that resembles directional jumping. The member 10 is initially tilted at 9° by the applied magnetic field B. The magnetic field B is then flipped clockwise and maintained for 25 ms. The maximum magnetic field used in this example is |B| _max : 18.9 mT. The arrows indicate the orientation of B in the XY-plane and the instantaneous net magnetization vector of the member 10, respectively.

Fronted by obstacles too high to roll or walk over, the soft member 10 can thus also resort to jumping by imparting an impulsive impact on a rigid substrate. Both straight jumping (see Fig. 15) and directional jumping strategies (see Fig. 2h and Fig. 15) can be achieved. To steer the jumping mode of locomotion, the control sequence of the magnetic field B prompts both the rigid-body rotation of the member 10, which specifies the jump direction, and elastic deformations to maximize the momentum of the member 10 before striking the substrate. Alongside the geometrical member design and the control parameters of the magnetic field B, the net magnetic moment of the deformed member additionally allows the modulation of jump height by magnetic gradient pulling. Magnetic gradient pulling alone could theoretically levitate the member 10; however, the required gradient magnitude is too large to be made practically feasible. Fig. 3 presents a hybrid liquid-solid environment that a member 10 can fully explore only through the combination of all of its modes of locomotion. A long-ranged water meniscus adjacent to a smooth, hydrophilic solid platform prompts capillary climbing by the member 10, which subsequently jumps directionally beyond an otherwise insurmountable, fixed obstacle. Navigation across the unstructured envi- ronment of e.g. a surgical human stomach phantom further illustrates the robustness of the multimodal member locomotion towards future potential biomedical applications.

The environment shown in Fig. 3 features laser-milled, poly(methyl methacrylate) platforms (water contact angle < 10°) in a Plexiglas box (52 χ 32 χ 25 mm ³ ) containing de-ionized water. The various modes of locomotion shown in Fig. 3 are sequentially captured in separate videos of which frames are shown, the scale bars used in these Figs, scale to 1 mm. In the following the procedures to implement the multimodal mode of locomotion of the member 10 shown in Fig. 3 will be described. In Fig. 3a, the member 10 first uses its rolling locomotion to get onto the water surface. After dipping into the water, the member 10 drifts slightly away from the platform along the meniscus and then starts swimming after it has stabilized. A clockwise magnetic field B (|B| = 3 mT, 25 Hz) is applied to let the member 10 swim to the right. Subsequently, the member 10 sequentially bends downwards on the water surface to disengage from the water surface. The member 10 is then and rotated counterclockwise to sink into the water, and subsequently swims back to the water surface to engage again at the water/air interface (Fig. 3b-c).

Fig. 3d shows the member 10 using four different modes of locomotion on land. After climbing onto the solid substrate by way of meniscus climbing the member 10 is present on the solid platform. The net magnetization of the member 10 is subsequently used to rotate the member 10 to the required initial pose for directional jumping to move the member 10 beyond the otherwise unsurmountable obstacle. After this, the member 10 starts to walk.

It has been observed that the member 10 incurs slight slipping before and after jumping across the obstacle due to the magnetic gradient force. As in this case, slip may happen because the currently implemented control loop is open: The member 10 traverses a large working space, and for each position in the space the suitable magnetic actuation matrix is chosen by the human controller. Thus, a small gradient force associated with a control error may be present. Specifically, the slipping happens when the member 10 has only one foot in contact with the substrate, that is, when the friction is smallest and even a small gradient force can displace the member 10.

It is worth noting that the walking can make the member 10 advance against an opposing magnetic gradient pulling force. Moreover, as the net magnetization of the member 10 can steer the member 10, it can be seen that the walking member 10 is initially misaligned with the desired advancing direction, and starts to align with the horizontal component of the magnetic field B after a few periods of walking.

In Fig. 3e, the member 10 first walks towards a glass capillary 14 mimicking a tunnel (inner diameter: 1 .62 mm). The tunnel impedes the walking gait of the member 10, such that it gets stuck at the tunnel entrance, because the cross-section of the glass capillary 14 does not allow the member 10 to lift up its front foot to proceed with further walking. To overcome this obstacle the member 1 0 then switches to the crawling mode of locomotion to pass through the tunnel. At this point the input is switched to a clockwise rotating magnetic field B, and the member 1 0 starts to crawl through the tunnel. After passing through the tunnel, the walking locomotion is resumed and the member 1 0 finally leaves the working space.

It has further been discovered that the magneto-elastic member 1 0 can additionally accomplish functional, shape change-enabled tasks. As an example, gripping and transportation of an object is shown in Fig. 4a.

Fig. 4 shows various frames taken from a video depicting the different shapes the member 1 0 adopts while carrying out a pick up and place task. Fig. 4a shows how the member 1 0 approaches a cargo item 14 (nylon, 1 x0.8x 1 .5 mm ³ ) by walking on a flat rigid surface, picks up the cargo by curling (frame V, |B| = 20 mT), transports away the cargo 14 by rolling and maintaining the curled shape, and releases the cargo 14 by uncurling at a new position.

Fig. 4b shows a dynamic and selective cargo 14 release. A paper tissue

(0.5x0.5x0.1 mm ³ , used as an example of a model drug container) is bound to the member 1 0 by an extra appendage. After pre-bending the member 1 0 (frame I), B is quickly reversed (frame 11- IV) to open the appendage and release the cargo (frames V-VIII; cargo 14 indicated by the dashed line). The maximum magnetic field applied in these examples is |B| _max : 1 9 mT. The loading of the cargo 14 does not hinder the locomotive multimodality of the member 1 0. The scale bars shown relate to a scale of 1 mm.

In order to produce a programmable magnetic soft composite member 1 0, a negative base mold is produced (not shown), for example of rectangular shape to form a rectangular shaped magnetized member. A decision is then made if the member 1 0 should have a varying magnitude of magnetization with varying direction of magnetization over its length I or whether it should have a constant magnitude of magnetization with varying direction of magnetization. If a constant magnitude of magnetization is desired then the base mold is filled with magnetizable material, i.e. an active component (actuatable component) is filled into the negative mold. An example of the active component can comprise a mixture of NdFeB and Eco- flex.

The member 1 0 comprising the active component is then installed on a jig and then subjected to an external magnetic field to create the different directions of magnetization of the segments s of the member 10. In order to permanently magnetize the member 10, the member 10 is subjected to a strong magnetic field having a strength of around 1.5 T. This is typically achieved by placing and orienting the member and the jig within the magnetic field generated by the electric coils.

If a non-uniform magnitude of magnetization is required then the base mold is filled with a passive component in order to form a rectangular shaped member. For this purpose a mixture of Al and Ecoflex is used as the passive component and simply poured into the mold in liquid form and allowed to cure.

Having calculated the ideal shape required to form a magnetized component of desired variable magnitude and orientation, a negative component mold for the magnetized component (also not shown) is formed in the passive component, for example by cutting out excess material of the passive component with a laser cut- ter to form a shaped band having a uniform or non-uniform width. The shape of the band of is based on a programmed magnetization profile.

In a further step the magnetizable material, i.e. the active component, is filled into the negative mold formed by the passive component. An example of the active component can comprise a mixture of NdFeB and Ecoflex. The active component is poured into the component mold formed within the passive component and is cured in the component mold and thereby adheres to the passive component.

The member 10 comprising the passive component and the active component is then installed on a jig and subjected to an external magnetic field to create the different directions of magnetization of the segments s of the member 10. Again the member 10 is subjected to a strong magnetic field having a strength of around 1 .5 T to permanently magnetize the member 10. In particular the body of the magneto-elastic member 10 is fabricated by loading an Ecoflex-10 polymer matrix (density: 1 .04 g/cm ³ ) with neodymium-iron-boron microparticles (MQP-15-7, Magnequench; average diameter: 5 μιτι, density: 7.61 g/cm ³ ) according to a mass ratio of 1 : 1 . The resulting soft magnetic elastomer has a density of 1 .9 g/cm ³ and a Young's modulus of 6.4x10 ⁴ Pa, as measured by a tensile testing machine (Instron 5943, Instron Inc.). During the fabrication, the pre- polymer is cast onto a flat poly(methyl methacrylate) plate to form a 240 μηη-thick film. After thermal polymer curing, the device is laser cut with the desired geometry. Fig. 5a shows an example jig 16 that is formed by a glass rod. Other forms of jigs can be used to form specific profiles of magnetization. By using a non-uniform width negative component mold into which the active material is introduced a magnetic member 10 of non-uniform magnitude of magnetization is formed in which segments s of the member 10 also have varying directions of magnetization. In the present example a magnetic field of 1 .5T was applied to magnetize the member 10.

It should be noted in this connection that a magnetic field of at least 1 .2 T is re- quired in order to saturate the magnetization of NdFeB.

A detailed description of the embedding process can be found in the publication by G. Z. Lum et al., titled "Shape-programmable Magnetic Soft Matter" and published in Proceedings of the National Academy of Sciences vol. 113, E6007-E6015 (2016), in particular in connection with Fig. 5 and the associated description of said publication. The publication by G. Z. Lum et al. also details the mathematical equations required to compute the shape of the negative component mold in order to form the member 10. European patent application EP16175341 .3 provides further details of this method of manufacture.

The glass rod forming the jig 16 shown in Fig. 5a shows an elastomeric member 10 comprising NdFeB microparticles that is wrapped around the cylindrical glass tube (perimeter: 3.7 mm) and that is magnetized by a uniform 1 .5 T magnetic field produced by a vibrating sample magnetometer. The relative orientation _R defines the phase shift in the magnetization profile. Fig. 5b shows the programmed magnetization profile in the unwrapped member 10 of Fig. 5a, for _R = 45°.

The member 10 dimensions used in this work are shown in Fig. 5b, where I, w, and h are the length, width, and height of the member 10, respectively.

To create the phase shift, β _κ , in the magnetization profile m (Fig. 5b), the rod is rotated by the angle β _κ during the member 10 magnetization. If not otherwise mentioned, a β _κ of 45° is assumed in the following.

It should be noted that the native surface of the composite elastomeric material is hydrophobic and microscopically rough. Static advancing (1 16°±3°) and receding (78°±2°) water contact angles were measured by the sessile droplet method through an automated goniometer routine that continuously increased and decreased the water droplet volume, respectively (Kruss DSA100). Surface roughness (R _a =0.63±0.02 μιτι and R _z = 4.37±0.28 μιτι) was measured by laser interfer- ometry (Keyence VK-X200). The members 10 surface was not subjected to any treatment before the experiments.

Fig. 6 shows an example of a six-coil C electromagnetic setup M to actuate the magneto-elastic member 10. The global coordinate system in X, Y and Z is indicated. The magnetic actuation setup is composed of three orthogonal pairs of custom-made electromagnets. The input currents driving the electromagnets are specified by software control signals through a custom electronic board. In the present setup a calibrated magnetic actuation matrix was used to map the control current input to the actual magnetic field in the working space. The effective working space along the X, Y and Z coordinates is restricted to 30 mm around the center of the reference system to exclude the effect of nonlinear components of the actuation matrix in close proximity of each coil. In particular Fig. 6 shows three pairs of coils C respectively arranged at a common axis, with the three common axes intersecting perpendicular to one another at the center of the apparatus. The spacing of each coil from the center is the same and in the specific example amounts to 37.5 mm. Through passing a current through the coils magnetic fields having a magnitude of 0 to 500 mT can be generated at the center of the arrangement. The magnetic field strength can be varied with a frequency of 1 kHz and the direction of the magnetic field can be varied with a frequency of 1 kHz.

In the global center of the working space, the relation between the current in coil and the magnetic field is:

B = Al; in which B = [B _X , B _Z , B _Z ] ^T and B _x , B _y , B _z are magnetic fields in +X,+Y,+Z directions and are in mT (millitesla); / = [I _X , I _Z , I _Z ] ^T and l _x , l _y , l _z are the current flowing through the coil pair, which contains two coils, in each direction and are in A (ampere). Positive current value is defined to produce positive magnetic field in this direction. For our system, the actuation matrix A is:

Based on the prescribed magnetization profile (Fig. 5b), a description will now be given to describe how the member 10 changes its shape when different external magnetic fields, B, are applied. Furthermore, the steering method is illustrated for the soft member 10. The analysis in this section provides a simple theoretical understanding for the experiments discussed in the following.

Quasi-static model

In order to discuss the quasi-static shapes that the soft member 10 can assume when subjected to B, the magnetization profile m, along the members 10 length, s, is assumed to be:

The variables m and <¾ represent the magnitude and spatial angular frequency of m(s), respectively, with <¾ =— . Using the above m(s) and without any loss of generality, the members 10 local frame is used to describe the actuating magnetic field B (Fig. 5b). As the magnetic field B interacts with the magnetization profile, the soft member 10 experiences a locally varying stress that allows it to create desirable deflections. Fig. 7 shows a quasi-static analysis for the soft member 10. When the member 10 deforms, the bending moment acting on the infinitesimal element at steady state is j _m Ads.

These static deflections can be described via the Euler-Bernoulli equation, where the rotational deflection along the member 10, 6(s), can be expressed explicitly by the actuating magnetic torque, _m , as:

3Θ

T _m A ds = El - ds

where

T _m = [0 0 l]{Rm x B} (S2) The variables = hw, I = h ³ w /12 and E represent the member 10's cross- sectional area, second moment of area and Young's modulus, respectively, while R is the standard z-axis rotational matrix, given as:

Physically, the left side of Eq. S2 represents the magnetic actuation that deforms the member 10, while its right side represents the resultant deflections. Since magnetic forces are not applied onto the beam, the shear forces along the beam will be zero at steady-state and so can be neglected for the quasi-static analyses (see Fig. 7). For the locomotion modes that have small deflections, such as swimming and crawling, Eq. S2 can be simplified by approximating R with a simple 3 x 3 identity matrix, s * x (along the body frame's x-axis), and Θ ¾ ^ where y refer to the vertical deflection. The governing equation then becomes:

Using Eq. S4, the shape of the member 10 can be described via the linear combination of two principal shapes, resembling respectively a cosine and a sine function. The first principal shape can be enforced when the direction of the magnetic field B is parallel to [cos β _κ sin/? _R 0] ^T , i.e., B = B[cos/? _R sin /? _R 0] ^T , and as presented in (1 ), Eq. S4 can be double integrated to become:

mAl ³ B (2π \

y ^{W =} 8^El ^C0S {T ^X ) ^(S5) In a similar manner, when B is parallel to [-sin(/? _R ) cos(/? _R ) 0] ^T , i.e.,

B = B[sin(/? _R ) - cos(/? _R ) 0] ^T , Eq. S4 can be double integrated to become:

While Eqs. S4-S6 can fully describe the achievable shapes for the swimming and crawling locomotion (see Fig. 1 b), they can only be used as an approximation for the other modes of locomotion, with the other modes of locomotion making use of large deflections. Despite the limitations, Eq. S5 can still be used to approximate large beam (member) deflections, since a large magnitude magnetic field B parallel to [cos /? _R sin /? _R 0] ^T creates a distorted cosine shape that resembles a semi- circle (Fig. 1 b). While the cosine function cannot fully describe the semicircle-like shape, Eq. S5 suggests that a larger curvature can be achieved when the amplitude of the cosine function is increased. The capability to create the large curvature semicircle-like shape is essential to achieve most locomotion and transition modes. Steering strategy

In the un-deformed state, the soft member 10 does not have a net magnetic moment. The member 10 can still produce a net magnetic moment upon deformation. As the net magnetic moment M _ne t of the member 10 tends to align with the applied magnetic field B, the direction of the magnetic field B can be controlled to steer the member 10 while it moves. M _ne t can be expressed as:

To achieve the multiple modes of locomotion, the magnetic field B is used to steer M _n et and to allow the member 10 to move along an intended direction or to pro- duce the necessary tilting (e.g., for walking).

In the following a description of the magnetic field B required to achieve each of the different modes of locomotion expressed by the shown member 10 will be given. Moreover, additional experimental details necessary to combine all the modes of locomotion together will also be provided. Unless stated to the contrary the member 10 described in the following has a /3R of 45°.

Low-magnitude (< 5 mT) rotating B: Undulating swimming and Crawling The undulating swimming and crawling modes of locomotion require similar undulating, time-varying shapes. Like the Taylor swimming sheet, the soft member 10 achieves its undulation by using a rotating magnetic field B of low magnitude and relatively high frequency. Fig. 8 shows the rotating magnetic field B sequence enabling the undulating motion of the magneto-elastic member 10. Fig. 8a shows the required B plotted against time to undulate the member 10 to the right. The variables B _x and B _y represent the magnetic field along the x- and y-axis of the coordinate system, shown in Fig. 8b. Fig. 8b shows the two principle shapes as described by Eq. S5-S6 (re- produced from frame I in Fig. 1 b). The scale bars shown scale to 1 mm.

Based on Eq. S5-S6, when the field angle of the magnetic field B, represented by a, is 135° or 225° (Fig. 8b), a small magnetic field |B| (3 mT) enables the member 10 to produce a shape that resembles a cosine or a sine function, respectively. As a is subsequently rotated, the member 10 produces a traveling wave along its body. The deformation produced by the time-dependent undulation can be described mathematically as: mAl ³ B \ (2π \ (2π \ (2π \ (2π \]

y ^~ 8π ΕΪ ^C0S [ ^~x ) ^C0S { ^' Y ^t ) + sin l— l sin l— t ) (S8) where the variable T represents the period of the rotating magnetic field B.

High magnitude (> 7 mT), low frequency (< 15 Hz) B: Rolling, Immersion, Meniscus Climbing and Landing

Fig. 9 shows the rotating sequence of the magnetic field B enabling the rolling mode of locomotion. Fig. 9a shows the required B plotted against time (normalized to one period) to roll clockwise. The variables B _x and B _y represent the magnetic field along the x- and y-axis of the coordinate system, shown in Fig. 9b. Fig. 9b shows the semi-circle shape required for the rolling locomotion (reproduced from frame I in Fig. 2e). The scale bars indicated an actual size of 1 mm.

To perform the rolling locomotion shown e.g. in Fig. 2e, a large magnitude magnetic field B (15 mT) is first applied along a field angle of 157° to deflect the mem- ber 10 into a distorted cosine shape resembling a semicircle-like shape (see Fig. 9b). The deflected member 10 produces a M _ne t described by Eq. S7, allowing the member 10 to be steered via a gradual rotation of the direction of the magnetic field B. During this period of time, a rigid-body magnetic torque, Xrigid-body, is induced by the member 10, which can be described mathematically as:

^T rigid-body = M _net X B. (S9) Once the rotation of the magnetic field B is stopped, the M _ne t Of the member 10 will be aligned with its final direction, and T _rigid _ _body will be reduced to a null vector.

Fig. 10 shows the sequence of the magnetic field B to perform the meniscus climbing mode of locomotion. Fig. 10a shows the required magnetic field B plotted against time to climb the meniscus on the right. The variables B _x and B _y represent the magnetic field along the x- and y-axis of the coordinate system, shown in Fig. 10b. When the magnetic field B has a field angle of 315°, the member 10 produces a shape that resembles a semi-circle. The member 10 pins on the water surface with its feet under deformation. The scale bar shown depicts an actual size of 1 mm.

This rigid-body torque is necessary for meniscus climbing. The required magnetic field B sequences to perform meniscus climbing are shown in Fig. 10a. The initial magnetic field B (|B|=1 1 .7 mT, a = 315°) is imposed to let the member 10 assume a vertically symmetric pose with upward curvature (Fig. 10b). In 400 ms the member 10 achieves a stable position along the meniscus known to depend on both the initial meniscus profile and the curvature of the device. The magnetic field B is subsequently rotated clockwise toward a = 157° to progressively displace the position of the member 10 along the meniscus and raise its center of mass. Once the device reaches contact with the solid substrate, the magnetic field B is turned off and the member 10 rests on the edge of the adjacent platform (see also Frame I in Fig. 2d).

Fig. 1 1 shows the rotating B sequence required to perform immersion. Fig. 1 1 a shows the required B plotted against time (normalized to one period) to rotate counter-clockwise and immerse into the water. The variables B _x and B _y represent the magnetic field along the x- and y-axis of the coordinate system, shown in Fig. 1 1 b. Fig. 1 1 b shows the initial curving of the member 10 for immersion (reproduced from frame I in Fig. 2c). The scale bar shown depicts an actual size of 1 mm.

The sequence of the magnetic field B for immersion is shown in Fig. 1 1 . In this mode of locomotion the member 10 rotates counterclockwise to reduce the contact with the water-air interface to a minimum, right before a quick flip in the direction of B intervenes to disengage it and let it sink (0.9 time unit in Fig. 1 1 a).

Fig. 12 shows the rotating B sequence to perform landing. Fig. 12a shows the required magnetic field B plotted against time to escape the capillary attraction of the water. The variables B _x and B _y represent the magnetic field along the x- and y-axis of the coordinate system, shown in 12b. Fig. 12b shows the initial curving of the member 10 for landing (reproduced from the frame I of Fig. 2d).

During the landing mode of locomotion water can easily dewet the surface of the member 10 because its high receding contact angle leaves the receding contact lines unpinned. The magnitude of the applied magnetic field B decreases at the 0.2 time unit in Fig. 12a from 1 1 .7 mT to 5 mT. Before this time point, most of the member 10 has peeled away from the water surface and a large magnetic field B is no longer necessary.

Fig. 13 shows the rocking sequence of the magnetic field B required to perform the walking mode of locomotion. Fig. 13a shows the required magnetic field B plotted against time to walk rightward. The actuation is divided into four phases, corresponding to Frame I -V in Fig. 2f. Note that the actuation is periodic, so that Frame I corresponds to Frame V. Fig. 13b shows how when the magnetic field B has a field angle of 315° the member 10 produces a shape that resembles a semi- circle (reproduced from the frame I of Fig. 2f). The scale bar shown depicts an actual size of 1 mm.

Fig. 14 shows the flipping magnetic field B sequence enabling the jelly fish-like swimming motion mode of locomotion of the magneto-elastic member 10. Fig. 14a shows the required B plotted against time to make the member 10 swim like a jellyfish upwards inside water. The variable B _y represents the magnetic field along y- axis of the coordinate system, shown in Fig. 14b. B _x is zero throughout the period. Fig. 14b shows the two shapes assumed by the member 10 during swimming. The scale bar shown depicts an actual size of 1 mm.

The jellyfish-like swimming shown in Fig. 14 in bulk liquid is implemented by periodic flipping of the magnetic field B along the vertical direction to give a fast power (downward) stroke and slow recovery stroke. In this way the member 10, function- ing at a Reynolds number Re -2.3, is shown to swim upward against the gravitational field, and even against the downward magnetic field gradient eventually present at the vicinity of the bottom coil of the working space.

Flipping the magnetic field B can also make the member 10 jump. To this end Fig. 15 shows snapshots of frames taken to illustrate straight and directional jumping by way of the magneto-elastic member 10. The first six video frames and time- lapse (last column) are shown for each jumping sequence. The same member 10 with _R = -90° was used in all cases. The recording speed was adapted to the dynamics of each jump type. Fig. 15a shows Type-1 straight jumping (550 fps). There is almost no net magnetization in the second frame as the member 10 is almost flat. In this example the maximum magnetic field B _max is12 mT.

Fig. 15b shows Type-2 straight jumping (355 fps). In this example the maximum magnetic field B _max is 18.9 mT. Fig. 15c shows directional jumping (720 fps), which is also shown in Fig. 2e. The direction of the magnetic field B (18.9 mT) applied is switched from 189° to 0° and maintained for 25 ms. In the frames where the magnetic field B is applied, the instantaneous net magnetization vector of the member 10 is indicated by the arrow. Fig. 15d shows the magnetization profile used for the member 10 in (a-c), and its magnetic response under large |B|. The scale bar shown depicts an actual size of 1 mm.

All types of jumping are achieved by quickly flipping the direction of the magnetic field B via a step or a fast ramp function. In order to effect this a single pair of coils is preferably used to create a (vertical) magnetic field B along the Y-axis of the global reference system and the factors affecting the jumping performance.

When a R = 45° is used, a magnetic field B along the (1 ,1 ) direction in the X-Y plane can be used, though this is prone to more errors since two pairs of coils have to be used.

In Type-1 straight jumping, the member 10 is first pre-bent by the magnetic field to curl upward and stand on its center (frame I of Fig. 15a). The direction of the mag- netic field B is then quickly reversed to force the member 10 to flatten out (frames II and III of Fig. 15a) and hit the substrate with its feet. An example of the modeling framework used to optimize this type of jump is shown in the following. In contrast to this, Type-2 straight jumping does not require pre-bending, as the magnetic field B is only applied to curl the member 10 to jump (Fig. 15b). This difference be- tween Type-2 and Type-1 jumping is characterized by using: 1 ) different ramping time of external magnetic field B; 2) different magnitude of the magnetic field B used; 3) a different magnetic field B spatial gradient; and 4) for two types of m.

Directional jumping (Fig. 15c) is also featured in the multimodal locomotion exper- iment shown in Fig. 3b. The results shown in Fig. 3e are re-plotted in Fig. 15c with more details. For this type of jump, the member 10 is initially deformed into a shape that resembles a cosine function. Subsequently, the direction of the applied magnetic field B is slightly rotated (-9°) about the longitudinal axis of the member 10 to specify the rotational direction during the jump.

Following this a step change in the direction of the magnetic field B is imparted, this simultaneously incepts both rigid-body rotation (clockwise here, to align the net magnetization to the external magnetic field B direction), and shape change (frames l-lll in Fig. 15c). During the rotation, the shape of the member 10 changes to an inverted cosine, increasing the momentum of the member 10 before it strikes the substrate (frames lll-V in Fig. 15c).

It should be noted in this connection that the jumping mode of locomotion of the member 10 is sensitive to the initial pose and shape of the member 10. Exact con- secutive reproduction of the same jump may be hindered by the inherent viscoe- lasticity of the body of the member 10 and by the associated shape hysteresis.

In the following a discussion will also be presented on how the dimensions of the member 10 affect the performance of each mode of locomotion, in particular how the dimensions of the member 10 affect its swimming and crawling speed. In order to facilitate this discussion, simple theoretical models are employed that can approximate the physics of each mode of locomotion. The derivation of these models is a first step towards understanding the full physics of a miniature soft member 10 with multimodal modes of locomotion.

The speed of the swimming and crawling modes of locomotion depends on the highest rotational frequency of the magnetic field B that the member 10 can respond to, the mechanical bandwidth of the member 10, i.e. its non-zero fundamen- tal natural frequency can be computed. This computation can be achieved by using a simple vibrational analysis where small deflections for the member 10 are assumed. Using this assumption, Newton's second law is initially used to analyze an arbitrary infinitesimal element of the member 10, dx, at a time t. Fig. 16 shows a vibrational analysis of the member 10 by showing the forces and torques acting on an infinitesimal element of the member 10.

Based on Newton's force law, the equation of motion of such an element along the vertical axis is given as:

dv

-Cwy +— = pAy (S10) where C, v and p represent the damping coefficient, shear force and density along the member 10's body, respectively. In Eq. S10, -Cwy and pAy represent the damping and inertia forces acting on the element, respectively.

In a similar manner, τ = Ja can be used to obtain the rotational equation of motion for this element about its bending axis:

dv d ² M _h dx _m

OX ΟΧ ^Δ ox

The variable M _b represents the bending moment acting on the element, and based on the Euler-Bernoulli equation, it is also equivalent to EI ^. Therefore, by substituting Eq. S1 1 into Eq. S10, one can obtain the governing equation for the beam:

. _Cw y _ _El ^ ^d y . ^ _{A = pA} y (S12)

The well-established mode-shape analysis is used to evaluate this partial differential equation. A free-vibration analysis is initially used to obtain the mode shapes of the member 10, i.e. the damping effects and magnetic actuation are temporarily removed from Eq. S12. Subsequently, the variable separation method for y is used such that

y{x, t) = F(x)G(t) (S13) where F(x) is solely a function of x and G(t) is only a function of time. By substituting Eq. S13 into Eq. S12 (without the damping effects and magnetic actuation) and rearranging the terms, one obtains:

Based on classical vibration analysis, both sides of Eq. S14 are independent of x and t, and they can be equated to become a function of the R ^th natural frequency, ω _{η Κ} ² , of the member 10 (4):

an

After solving the homogeneous equation in Eq. S16, one can obtain both ω _{η Κ} ² and the R ^th mode shape of the member 10 (F _R ):

F _R = A _R e β + B _R e P + C _R cos xj (517)

where

. EI

β ⁴ =—

PA

and the variables A _R , B _R , CR, and D _R are constants dependent on the boundary conditions of the member 10. Because the member 10 has free ends, the follow- ing free-free boundary conditions are substituted into Eq. S17:

Physically, the boundary conditions in Eq. S18 dictate that there are no shear forces and bending moments acting on the free ends of the member 10. Based on these boundary conditions, one can obtain the following characteristic equation for the member 10:

Once the characteristic equation has been solved numerically, the two lowest natural frequencies of the member 10 are given as: = 4.73 (S20) The first mode, ω _{η 1} , represents the rigid-body motion of the member 10, while the second mode shows the non-zero fundamental natural frequency of the member 10. As the first mode does not describe the shape change of the member 10 with respect to the actuating magnetic field's frequency, the presented frequency response analysis begins with the second lowest natural frequency, i.e.s ω _{η 2} . Based on Eq. S20, this natural frequency can be expressed as:

Using Eq. S21 and the Taylor swimming sheet model, one can approximate the achievable swimming speed, v of the member 10. Although the selected fluidic boundary conditions result in a slight deviation from the Taylor model, i.e. the pre- sented member 10 swims on the water surface and not far away from walls, it is assumed that the following equation can still capture the fundamental dynamics of the swimming locomotion:

where f represents the frequency of the traveling wave (coincident with the rotation frequency of B). As v _swtm can be maximized by applying the highest f and since the highest f is limited by / = -^, one can rewrite Eq. S20 as:

2π

Based on Eq. S23, the swimming speed of the member 10 can be increased when its I and h are increased and decreased, respectively.

Similar to the swimming mode of locomotion, the member 10 can crawl faster when it can produce a faster undulating frequency. However, in order to enhance the crawling speed one would have to increase the second lowest natural frequency of the member 10, and Eq. S21 implies, in contrast to Eq. S23, that /i should be increased while I should be decreased. The rolling mode of locomotion also uses a rotating magnetic field B to control the soft member 10. However, in this case the magnetic field B rotation frequency f has to be much slower (f < 15 Hz in the present example) than the one utilized in the swimming and crawling mode of locomotion to prevent the member 10 from creating a traveling wave along its body. Therefore, it will not be necessary to use vibrational analyses to evaluate the performance of this mode of locomotion. It is assumed that the rolling mode of locomotion is similar to that of a wheel, such that it is more efficient when the member 10 becomes rounder. A rounder shape will require the soft member 10 to produce deformations that have large curvatures. While the cosine function in Eq. S5 cannot fully describe the semicircular shape of the member 10 for the rolling mode of locomotion, this equation suggests that a larger curvature can be achieved when the amplitude of this cosine function, i.e., _8π3Εΐ , can be increased. As m, B, E and other constants are independent of the dimensions of the member 10, one can conclude that a larger curvature can be easily accomplished when — is maximized. Since— = — = 12—, larger cur-

12 ^W ' ¹

vatures can be achieved by increasing / and decreasing h.

Similar to the rolling mode of locomotion, the effectiveness of meniscus climbing is highly dependent on the maximum achievable curvature of the member 10. As a result, one can use Eq. S5 and the above discussion on rolling to approximate how the dimensions of the member 10 can affect the climbing behavior of the member 10. This equation suggests that a larger curvature can be achieved by increasing I and decreasing h.

Jumping

Fig. 17 shows the boundary condition for the Type-1 straight jump (see also Fig. 15a). The analysis can be simplified by considering a fixed-free half beam, since the center point of the member 10 neither rotates nor translates and the motion is symmetrical about this point. As it is very challenging to derive a mathematical model that can accurately describe how the dimensions of the member 10 can affect its jumping performance, only a simple vibrational analysis will be provided as a first step towards understanding this mode of locomotion. The presented analysis is performed to approximate the simplest jumping mode - Type-1 straight jump in Fig. 15a. To proceed with this vibrational analysis, one assumes that the shape of the member 10 begins with a cosine function in stage I (Fig. 17a). Furthermore, after the member 10 is subjected to a step change in B direction (i.e. flipping B for 180°), the shape of the member 10 will tend to change into an inverted cosine function. During this transition, the member 10 experiences an impact with the substrate, allowing itself to jump (stage II).

The formulation can be reduced to analyze only half of the beam because the jumping motion is symmetrical about the center point of the member 10, which neither rotates nor translates, and its boundary condition can be simplified into a fixed-free type (Fig. 17).

In order to facilitate the formulation in the following, one uses V to represent 1/2 and x' = x - V. Using these variables, one can mathematically represent the boundary conditions for half of the member 10 as:

Although the boundary conditions for the jumping mode of locomotion is different from the swimming mode of locomotion, the vibrational analysis from Eq. S10-17 is still valid. By substituting the boundary conditions in Eq. S24 into Eq. S17, one obtains the following characteristic equation: cos cosh V -1 (S25)

Once the R natural frequency in Eq. S25 has been solved numerically, this value is substituted back into Eq. S17 to obtain the corresponding mode shape:

F _R (x)

(526)

By substituting the obtained mode shapes into Eq. S12, rearranging the terms and normalizing the inertia terms, the following equation is obtained: Because the magnetic actuation on the right side of Eq. S27 can be computed = -^ cos (^x'), Eq. S27 becomes:

p ds

To simplify the discussion, only the fundamental natural frequency and mode shape are used to describe the jumping mode of locomotion. While the exclusion of other mode shapes might introduce errors into this prediction, it is believed that the discussion is still valid, since the fundamental mode shape is the most critical component in Eq. S28. Therefore, the orthogonal property for vibration is used and Eq. S28 is multiplied with F ₁ and subsequently both sides are integrated over ' from x' = 0 to x' = V to obtain the following equation:

The characteristic equation in Eq. S25 numerically evaluates that J^- i' = 1.88, and by using this value one numerically determines J ^" ^' F ² dx = 19.9/', o'l _h ^ ^{~Fldx =} ₂ τ _ι τ) _' ^ANC ' ^COS (^ ^ Fidx = -3.261 . By substituting these values back into Eq. S27 and normalizing the inertia term, one obtains:

At stage I, the member 10's initial shape can be expressed as ^ ₀ [l ^{_ cos} (^ ')] where A ₀ is a positive real number that represents the amplitude of the cosine function. Similar to Eq. S27, one can use the orthogonal property of classical vibration analysis to extract the initial condition for G^t = 0), i.e., one can multiply the initial shape of the member 10 with F ₁ and subsequently integrate the new function over x ' from 0 to I ' . Based on numerical evaluation, one obtains G (t = 0) = G _{1 Q} l, where Gi.o is a positive real number. Because the member 10 does not have an initial speed, one can conclude that the second initial condition can be expressed as G^t = 0) = 0. Therefore, by using these two initial boundary conditions and assuming the system is underdamped (i.e. low magnitude of C) one can solve Eq. S30 to obtain: Gi = - G ₀ V +

+ <p),

Where

12.4E/ C ² C 2ρ/ιω _ώ

^ = ρ-Α Γγ - ρ^ ^{Άη φ = ~} 2ph^ _d ' ¹ * ^{η φ = (S31)} The kinetic energy for the member 10 before it impacts the substrate at time ft, £ _kEj£ , can then be approximated numerically as:

where V represents the volume of the member 10. After striking the substrate, Newton's law of restitution dictates that the kinetic energy of the member 10, E _{kE f} , will become:

£kE, = Y ² E E,i (S33) where γ represents the coefficient of restitution and it is a positive real number that is between 0 and 1 . One assumes that E _{kE f} can be converted fully to gravitational potential energy such that:

pVgz = E _kE (S34) where z is the highest point of the jump. Therefore, by substituting Eq. S32 and S33 into S34, and rearranging the equation, one can approximate z as:

z = l9.9y ² G ² (t = tf) (S35) To maximize z, the magnitude of G {t = tf) and y ² have to be maximized. However, as G ² (t = tf) is highly dependent on other parameters like C,p, E and , it is difficult to deduce the effects of h and I for such a complex function. Likewise, it is also difficult to deduce the mathematical relationship between y ² and the dimensions of the member 10, i.e., its w, h and /. As a result, it is believed to be necessary to perform a numerical optimization to determine the optimal values o h, w and I such that the member 10 can maximize z. Walking and Jellyfish-like Swimming

The net stride of the member 10 in one walking cycle is dependent on the difference between the curvature of the member 10 in Steps 3 and 4. Thus, when a larger curvature can be achieved in Step 4, the member 10 can produce a bigger stride. By using similar analyses as presented in the foregoing, one can deduce that a larger stride can be achieved per cycle when / is increased and h is decreased.

On the other hand, the walking speed also depends on the bandwidth of the mem- ber 10, i.e. the capability to respond to high frequency periodic magnetic fields B. In this instance the bandwidth of the member 10 is approximated with its fundamental natural frequency, so that based on Eq. S21 the bandwidth can be increased when / is decreased and h is increased. Because the criteria to produce a larger stride are contrary to the criteria for higher bandwidth, it is possible to per- form numerical optimization methods to determine the optimal dimensions of the member 10.

In a similar manner, greater swimming propulsion can be generated in one swimming cycle (jellyfish-like swimming) when the speed difference between the power and recovery strokes is increased. The fastest achievable speed is dictated by the bandwidth of the member 10. Therefore, it is beneficial to decrease I and crease h such that the fundamental natural frequency of the member 10 can be maximized. However, decreasing / and increasing h will increase the overall stiffness of the member 10, making it harder to create a large stroke length that is necessary for generating large propulsions. Because the criteria to produce a larger stroke is contrary to the criteria for higher bandwidth, it is assumed that a numerical optimization is necessary to determine the optimal dimensions of the member 10.

Table 2 summarizes the design considerations discussed above for the member 10 under the control of a magnetic coil setup with negligible settling time.

Given the magnetic actuation, it is possible to use the spatial gradients of the magnetic field B to exert magnetic pulling on the member 10 to enhance its locomotion. For instance, after the member 10 has jumped, the deformed member 10 has a net magnetic moment that allows the use of gradient pulling to modulate jump height and distance. Theoretically, magnetic gradient pulling may also levitate a magnetic member 10 and make it traverse across different terrains and obstacles. However, this approach is not practical. First, in absence of closed loop feedback control, the trajectories generated by gradient pulling are generally jerkier than those achievable by the proposed locomotion. Second, the required spatial gradients of the magnet- ic field B for the member 10 to levitate across all obstacles and terrains, estimated to be 3.8 T/m, are difficult to obtain. Moreover, when envisioning the use of the member 10 in healthcare, high spatial gradients of the magnetic field B are considered adverse for e.g. medical application, because they can create undesirable forces on implanted objects within a patient's body, e.g. an implantable cardioverter defibrillator (ICD). Although advancements in the design of ICDs are underway, the shock coils of many ICDs are ferromagnetic and they can experience significant adverse translational motions when they are subjected to such high spatial gradients. In contrast, although the ICD devices will also be affected by the re- quired magnetic actuation, most ICD devices can fully function when the magnitude of the magnetic field B is lower than 500 mT, and this value is well above the magnetic field strength utilized for the soft member 10 (-20 mT) presented herein.

A design of a modified member 10 for selective, magnetically triggered drug re- lease is shown in Fig. 18. To this end Fig. 18a shows that an extra strap 18 is added to the body of the original member 10. For clarity, only the main component of magnetization on the extra strap 18 is marked out in Fig. 18a, as it is the dominant and functional component in releasing a drug that is provided as the cargo 14. The magnetization of the member 10 main body is the same as that shown in Fig. 5b.

During locomotion, the cargo 14 is mechanically bound on the member 10 by inserting the strap head 20 into a hole 22 on the body 24 (Fig. 18b). When a large magnetic field B is applied along the y-axis, B _y (-19 mT), then the magnetic torque bends and unlocks the strap to release the cargo (Fig. 18c). Note that other components of magnetic field (B _x , B _z ) may also be present. Due to the restriction of the hole 22, a smaller B _y (<18 mT in our case) is not sufficient to open the strap 18, which makes normal locomotion possible. The design may be optimized to also carry liquid or powder based loads as cargo 14. in the foregoing approximate quasi-static and dynamic models have been shown to describe how the width w, length /, and thickness h of the member 10 can be designed to optimize each mode of locomotion (see also Table 2). Hence observations were made that w only affects jumping and jellyfish-like swimming, whereas / and h are subject to conflicting requirements for different locomotion modes. Improved meniscus climbing and rolling is predicted when / is increased and h is re- i ³

duced, as this increases the member's bending compliance (oc -^). Conversely, fast and efficient crawling requires limited member compliance to preserve its ability to track high frequency control signals. The walking gait imposes by itself con- flicting requirements on / and h. A more compliant member body is preferred to produce a larger net stride and larger body curvatures, which implies the increase and decrease of / and h, respectively. However, this would also effectively decrease the bandwidth of the member, reducing the maximum frequency of steps in the walking gait. The present study only elicits a feasible design that enables all reported modes of locomotion.

Table 2 shows design considerations of the member 10 for multimodal locomotion. The arrows respectively indicate whether an increase or decrease in the length, height or width have beneficial effects on which type of locomotion in comparison to the member 10 discussed in the foregoing, so as to optimize it for a certain mode of locomotion.