Variable-stiffness–morphing wheel inspired by the surface tension of a liquid droplet
受液滴表面张力启发的可变刚度变形轮

To effectively overcome obstacles while maintaining the advantage of a general wheel driving on flat ground, a mechanism for realizing switchable wheel modes was proposed (Fig. 1A). When the wheel drives on a normal flat ground, it maintains a circular shape in a high-modulus state (Fig. 1Ai); thus, it functions as a normal wheel, displaying energy-efficient and stable movement during locomotion without shaking. When the wheel is used on rough terrain with large obstacles, where a typical wheel does not effectively function, the modulus of the proposed wheel decreases. This makes the wheel easily deformable, allowing it to adapt to the shape of the obstacle (Fig. 1Aii). The smart chain structure, which consists of a chain-like structure located at the outermost side of the wheel, is connected to a wire spoke structure (Fig. 1B). As shown by the green line in Fig. 1B, one wire is connected to one side of the chain block, passes through the hub structure colored in red in Fig. 1B, and is fixed to the opposite side of the chain block. The total length of the wire spoke structure is constant and equal to its total length in the initial configuration. Therefore, the length of the wire spoke from the chain block to the hub structure can be changed on the basis of the hub-gap distance between the front- and rear-hub structures (Fig. 1B). As the hub-gap distance increases to l_{h_large}, the length of the wire spoke between the chain block and hub structure decreases, and the chain block is forced inward. Then, the modulus of the wheel increases in a circular shape, and the wheel functions as a normal wheel (Fig. 1Ci). Otherwise, when the hub-gap distance decreases to l_{h_small}, the inward-directional force acting on the chain block also decreases. Then, the wheel enters a soft state, allowing it to deform effectively to adapt to the shape of the obstacle and overcome it (Fig. 1Cii).
为了有效克服障碍，同时保持一般车轮在平坦地面上行驶的优势，提出了一种实现车轮模式可切换的机制（图1A）。当车轮在正常的平坦地面上行驶时，它在高模量状态下保持圆形（图1Ai）;因此，它的功能就像一个普通的轮子，在运动过程中表现出节能和稳定的运动，而不会晃动。当轮子在具有大障碍物的崎岖地形上使用时，典型的轮子无法有效发挥作用，所建议的轮子的模量会降低。这使得轮子很容易变形，使其能够适应障碍物的形状（图1Aii）。智能链结构由位于车轮最外侧的链状结构组成，与线辐结构相连（图1B）。如图1B中的绿线所示，一根导线连接到链块的一侧，穿过图1B中红色的轮毂结构，并固定在链块的另一侧。线辐条结构的总长度是恒定的，并且等于其初始配置中的总长度。因此，根据前轮毂结构和后轮毂结构之间的轮毂间隙距离，可以改变从链块到轮毂结构的线辐条长度（图1B）。随着轮毂间隙距离增加到l_{h_large}，链块与轮毂结构之间的辐条线长度减小，链块被迫向内。然后，轮子的模量以圆形增加，轮子的功能与普通轮子相同（图1Ci）。 否则，当轮毂间隙距离减小到l_{h_small}时，作用在链块上的内向力也减小。然后，车轮进入柔软状态，使其有效地变形以适应障碍物的形状并克服它（图1Cii）。

The variation in stiffness is derived from the sum of the inward-directional force at the chain block, and it can be compared with the surface tension of a liquid droplet (Fig. 1D). The molecules located at the surface of a liquid are pulled inward because of the nonuniform cohesive forces, whereas the net force becomes zero for molecules inside the liquid. The sum of each cohesive force for the molecules at the surface can be described as the attraction force toward the center of the liquid. This attraction force can be matched to the force acting on the chain block because of the wire spokes in the wheel. As the length of a wire spoke from the chain block to the hub structure decreases at a large hub-gap distance, the force acting on the chain block can be separated into the tangential force that brings adjacent blocks closer to each other and the radial force that causes the blocks to move in the direction of the wheel center. This tangential force acting on the block is similar to the surface tension of a liquid, and the net force of each chain block plays a role in the contraction of the wheel structure and affects the shape and stiffness of the wheel. The dependence on contraction force on the variable stiffness of the wheel is similar to the dependence on surface tension on the variable contact angle of a liquid droplet (Fig. 1E). Because of the gravitational force, a liquid droplet is deformed at the surface, and this magnitude of deformation is generally represented as the contact angle. As the contact angle increases and approaches 180°, the shape of the liquid droplet approaches a circular shape with minimized deformation because of gravity. These shape variations of the liquid droplet depending on the surface tension are similar to the shape variations of the wheel depending on the surface tension controlled by the tension of the wire spoke structure.
刚度的变化是由链块处的向向力之和得出的，可以与液滴的表面张力进行比较（图1D）。由于内聚力不均匀，位于液体表面的分子被向内拉，而液体内部分子的净力变为零。表面分子的每种内聚力之和可以描述为朝向液体中心的吸引力。由于轮子中的钢丝辐条，这种吸引力可以与作用在链块上的力相匹配。由于从链块到轮毂结构的线辐长度在轮毂间隙距离大时减小，作用在链块上的力可以分为使相邻链块彼此更近的切向力和使链块沿轮心方向移动的径向力。这种作用在块上的切向力类似于液体的表面张力，每个链块的合力在车轮结构的收缩中起作用，并影响车轮的形状和刚度。收缩力对车轮可变刚度的依赖性类似于液滴可变接触角对表面张力的依赖性（图1E）。由于万有引力的作用，液滴在表面发生变形，这种变形的大小通常表示为接触角。随着接触角的增加并接近 180°，液滴的形状接近圆形，由于重力造成的变形最小。 液滴的这些形状变化取决于表面张力，这与轮子的形状变化（取决于由线辐结构的张力控制的表面张力）相似。

The smart chain structure is located above the soft supporting structure, which supports the initial position of the smart chain structure (Fig. 2A). The soft supporting structure can be replaced with any material or structure that is effective in realizing a large amount of deformation. The modulus of the soft supporting structure influences the range of stiffness variation of the wheel by varying the tension of the wire spoke. Individual smart chain structures are assembled by using a pin at the hole, similar to the method for assembling a general chain.
智能链结构位于软支撑结构上方，支撑着智能链结构的初始位置（图2A）。软支撑结构可以用任何能有效实现大变形的材料或结构代替。软支撑结构的模量通过改变辐条钢丝的张力来影响车轮的刚度变化范围。单个智能链结构是通过在孔上使用销钉组装的，类似于组装一般链条的方法。

The stiffness of the wheel can be changed on the basis of two major mechanisms: the spoke length variation at the ground contact position and variation in the distance between each smart chain structure depending on the direction of rotation. In the case of the first mechanism, the required length of the spoke structure at the ground contact position increases as the depth of deformation increases. When the circular structure with radius R is pushed down to a distance d, which is defined as the depth of deformation (Fig. 2Bii), the point P_int at which the wheel touches the ground and begins to bend away from the ground occurs. Then, l_o is defined as the outer length of the overlapped part of the circular structure with the ground. If the circumference of the circular structure is constant and if there is no wrinkling or partial roll-up, then the contact length between the ground and circular structure becomes not l_o′, which is the intersected length of the line where they overlap, but l_o (Fig. 2Biii). The increased contact length between the circular structure and ground, from l_o′, is given by 2Δl_o, where
车轮的刚度可以通过两种主要机制来改变：与地面接触位置的辐条长度变化和每个智能链结构之间距离的变化，具体取决于旋转方向。在第一种机构的情况下，辐条结构在地面接触位置所需的长度随着变形深度的增加而增加。当半径为R的圆形结构被向下推到距离d时，该距离定义为变形深度（图2Bii），车轮接触地面并开始从地面弯曲的点P_int出现。然后，l_o定义为圆形结构的重叠部分与地面的外部长度。如果圆形结构的周长是恒定的，并且没有起皱或部分卷起，则地面和圆形结构之间的接触长度不是l_o′，即它们重叠的线的相交长度，而是l_o（图2Biii）。从 l_o′ 开始，圆形结构与地面之间增加的接触长度由 2Δl_o 给出，其中

Consequently, the position where the wheel begins to bend away from the ground changes from P_int to P_d, which is shown as a red line in Fig. 2Biii. The distance variation between the center of the circular structure and P_d is R′, so the length of the wire spoke of the wheel needs to increase with ΔR as follows
因此，车轮开始从地面弯曲的位置从 P_int 变为 P_d，这在图 2Biii 中显示为一条红线。圆形结构中心与P_d之间的距离变化为R′，因此轮子的线辐长度需要随着ΔR的增加而增加，如下所示

The ΔR value generally increases as the depth of deformation d increases until d < R, which means that the maximum elongated length of the wire spoke influences the depth of deformation d on flat ground. If the circumference of the circular structure consists of a block with a thickness t (Fig. 2C), then the rotation center of each block is determined by the way that the blocks are assembled and how they contact each other. The center of rotation of the block can be located at its middle (Fig. 2C), meaning that each block is assembled using a pin at the middle of the block relative to the block thickness. Otherwise, this center of rotation can be located at the top or bottom side of the block (fig. S1). When the depth of deformation d is reached, the overlapped length of the middle line of the block, which is also the line of the neutral axis of deformation, is defined as l_m (Fig. 2Cii). Similar to Fig. 2B, the contact line between the ground and the wheel increases from l_m′ by 2Δl_m (Fig. 2Ciii); thus, the length of the wire spoke of the wheel needs to increase to R′ by ΔR, as shown in Eq. 2. In this case, the thickness of block t does not affect ΔR. When the rotation center of a block is at the top or bottom side (fig. S1), it is similar to a block rolling against the surface of another block. Therefore, each block cannot be overlapped with each other (fig. S1iii). In this case, the contact length (l_s′) is increased by 2Δl_{s_1}, which is given by
ΔR 值通常随着变形深度 d 的增加而增大，直到 d < R，这意味着在平坦的地面上，线辐的最大伸长长度会影响变形深度 d。如果圆形结构的周长由厚度为t的块体组成（图2C），则每个块体的旋转中心由块体的组装方式以及它们之间的接触方式确定。块的旋转中心可以位于其中间（图 2C），这意味着每个块都是使用块中间相对于块厚度的销钉组装的。否则，该旋转中心可以位于块的顶部或底部（图 S1）。当达到变形深度d时，块体中线的重叠长度，即变形中性轴线的重叠长度定义为l_m（图2Cii）。与图2B类似，地面和车轮之间的接触线从l_m′增加了2Δl_m（图2Ciii）;因此，轮子的线辐条的长度需要增加 Δ R，如式 2 所示。在这种情况下，块 t 的厚度不会影响 ΔR。当一个块的旋转中心位于顶部或底部时（图S1），它类似于一个块在另一个块的表面上滚动。因此，每个块不能彼此重叠（图 S1iii）。在这种情况下，接触长度 （l_s′） 增加 2Δl_{s_1}，由下式给出

The length of the wire spoke needs to be increased by ΔR₁ as follows
线辐条的长度需要增加 ΔR₁，如下所示

In addition, the block next to that contacting the surface starts to separate from the ground (marked in green in fig. S1iv), and the required length of the wire spoke is increased from R′ by ΔR₂, expressed as
此外，与表面接触的块旁边的块开始与地面分离（在图 S1iv 中用绿色标记），并且导线辐条所需的长度从 R′ 增加 ΔR₂，表示为where θ is the angle of the deformed block from the ground, and the wire spoke is assumed to be connected to the middle of the block at P_{d_2}. Each block is rotated about the edge of the block marked with a purple dot; then, l_s is increased by 2Δl_{s_2}. Therefore, the required length of the wire spoke at the given depth of deformation is affected by the thickness of block t and the angle of block deformation θ, unlike in the case in Fig. 2C. Because of this complex and sensitive effect on the variation in the length of the wire spoke, we established the rotation axis near the center of the block, similar to that shown in Fig. 2C, not at the edge of the block. As a result, the deformation height of the wheel with a 140-mm radius in Fig. 2D can be controlled by varying the spoke length where the block begins to bend away from the ground, marked as point P_d in Fig. 2C. The maximum length of the wire spoke can be controlled on the basis of the deviation in the hub-gap distance, and, therefore, the length of the wire spoke near position P_d can be controlled on the basis of the hub-gap distance. The longest measured value among the length of wire spoke obtained near point P_d was used as a standard in Fig. 2E; this ambiguity arises from the discontinuity of the chain block and difficulties in distinguishing where the slope changes. The hub-gap distance variation was defined as zero when the distance reached a maximum in the initial state, and the distance variation values increased as the hub structures moved closer together (Fig. 2E). The spoke length deviation was derived from the difference between the length variation of the wire spoke and its initial length when the hub-gap deviation is zero. Because of the length variation of the wire spoke associated with the variation in hub-gap distance, the deformation height of the wheel can be changed (Fig. 2F). After the hub-gap distance reached the target value, the wheel was placed on the ground, and it started to deform until reaching an equilibrium position because of the gravitational force. Thus, the deformation height was defined as the deviation between the initial radius of the wheel in the floating state in air without the tension of the wire spoke and the current height of the wheel center after deformation (Fig. 2F). The deformation height was mainly influenced by the hub-gap distance, not by the dead weight applied to the wheel. This is because the soft supporting layer of the wheel in Fig. 2F was sufficiently soft. Thus, the magnitude of wheel deformation, controlled by the hub-gap distance, reached its limit before the soft supporting structure reached its equilibrium position because of the reaction force generated by the deformation. If the value of the applied dead weight was lighter than the experimental values, then the effect of the soft supporting structure became dominant. Consequently, the deformation height of the wheel reached its maximum when the soft supporting structure had sufficiently deformed and before the limit position controlled by the hub-gap distance. In other words, the effect of the soft supporting structure became dominant as the relative modulus of the soft supporting structure increased compared with that for an applied gravitational force.The deformation height was measured using the wheel model with a soft supporting structure of higher modulus (the moduli of the soft supporting structures in Fig. 2, F and G, are presented in fig. S2), as shown in Fig. 2G, to verify the effect of the relative modulus of the soft supporting structure. When the hub-gap distance variation was near zero, the effect of the external gravitational force became negligible for different deformation heights. This means that the variation in surface tension because of the hub-gap distance primarily affected the characteristics of the wheel when in a high–surface tension state. As the hub-gap distance variation increased, the influence of the applied weight became dominant for the deformation height, whereas the influence of the hub-gap distance variation itself on the deformation height decreased. This is because the possible range of deformation height was controlled by the hub-gap distance, but the deformation of the soft supporting structure reached an equilibrium position as the reaction force increased from deformation, even before the maximum deformation height was attained. These trends can be observed in fig. S3. As the hub-gap distance variation decreased, the deformation heights of each wheel using different soft supporting structures became similar. This is because the influence of surface tension became more dominant as the hub-gap distance variation decreased. Conversely, as the hub-gap distance variation increased, the differences in deformation height between the wheels with different soft supporting structures became more noticeable. To verify the effect of distance variations during the deformation process, the distance variations from the wheel center to each block were measured while the wheel was slowly placed on the ground from the floating state (fig. S4) at a fixed hub-gap distance. As the magnitude of deformation increased, the distance of the blocks located right below the wheel center, such as block numbers 16 and 18, decreased (fig. S4B), whereas the distance of the blocks located at the section where the blocks were bent and moved away from the ground increased. In particular, the distance of block number 22 rapidly increased as it approached the available maximum deformation (fig. S4C). This is because the position of block 22 was the nearest position at which the block was lifted from the ground when the wheel deformation reached a maximum for the given hub-gap distance conditions.
其中 θ 是变形块与地面的角度，假设线辐条在 P_{d_2}处连接到块的中间。每个块都围绕标有紫色点的块的边缘旋转;则，l_s 增加 2Δl_{s_2}。因此，与图2C中的情况不同，在给定的变形深度下，在给定的变形深度下，线辐条所需的长度受块t的厚度和块变形角度θ的影响。由于这种对辐条线长度变化的复杂和敏感的影响，我们在块的中心附近建立了旋转轴，类似于图2C所示的旋转轴，而不是在块的边缘。因此，图2D中半径为140 mm的车轮的变形高度可以通过改变辐条长度来控制，在辐条长度下，块体开始从地面弯曲，在图2C中标记为点P_d。线辐条的最大长度可以根据轮毂间隙距离的偏差来控制，因此，线辐条在位置P_d附近的长度可以根据轮毂间隙距离来控制。图2E以在点P_d附近获得的导线辐条长度中最长的测量值作为标准;这种模糊性是由于链块的不连续性和难以区分坡度变化的位置而产生的。 当距离在初始状态下达到最大值时，轮毂间隙距离变化定义为零，而随着轮毂结构靠得更近，距离变化值增大（图2E）。辐条长度偏差是由线辐条的长度变化与其初始长度之差得出的，当轮毂间隙偏差为零时。由于线辐的长度变化与轮毂间隙距离的变化有关，因此可以改变车轮的变形高度（图2F）。在轮毂间隙距离达到目标值后，将轮子放在地面上，由于重力的作用，轮子开始变形，直到达到平衡位置。因此，变形高度被定义为车轮在空气中浮动状态下的初始半径与变形后车轮中心的当前高度之间的偏差（图2F）。变形高度主要受轮毂间隙距离的影响，而不受施加在车轮上的自重的影响。这是因为图2F中车轮的柔软支撑层足够柔软。因此，由于变形产生的反作用力，由轮毂间隙距离控制的车轮变形幅度在软支撑结构达到其平衡位置之前就达到了极限。如果施加的自重值比实验值轻，则软支撑结构的影响占主导地位。因此，当软支撑结构充分变形且在轮毂间隙距离控制的极限位置之前，车轮的变形高度达到最大值。 换言之，随着软支撑结构的相对模量与施加重力时相比增加，软支撑结构的效应变得占主导地位。使用模量较高的软支撑结构的车轮模型（图2中软支撑结构的模量，F和G，如图S2所示）测量变形高度，以验证软支撑结构的相对模量的影响。当轮毂间隙距离变化接近于零时，对于不同的变形高度，外重力的影响可以忽略不计。这意味着，当处于高表面张力状态时，由于轮毂间隙距离导致的表面张力变化主要影响了车轮的特性。随着轮毂间隙距离变化的增加，外加重量对变形高度的影响变为主导，而轮毂间隙距离变化本身对变形高度的影响减小。这是因为变形高度的可能范围由轮毂间隙距离控制，但随着变形引起的反作用力的增加，软支撑结构的变形达到了平衡位置，甚至在达到最大变形高度之前。这些趋势可以在图 S3 中观察到。随着轮毂间隙距离变化的减小，采用不同软支撑结构的各轮子变形高度变得相似。这是因为随着轮毂间隙距离变化的减小，表面张力的影响变得更加明显。 相反，随着轮毂间隙距离变化的增加，不同软支撑结构的车轮间变形高度差异更加明显。为了验证变形过程中距离变化的影响，在固定的轮毂间隙距离下，当车轮从浮动状态缓慢放置在地面上时，测量了车轮中心到每个块的距离变化（图S4）。随着变形幅度的增加，位于车轮中心正下方的块的距离（例如块编号 16 和 18）减小（图 S4B），而位于块弯曲并远离地面的部分的块的距离增加。特别是，22号块的距离在接近可用的最大变形时迅速增加（图S4C）。这是因为在给定的轮毂间隙距离条件下，当车轮变形达到最大值时，块 22 的位置是块从地面抬起的最近位置。

The second main mechanism that controls the stiffness of the wheel is the variation in the distance between each chain structure. The proposed smart chain structure was designed to have different distances between each block depending on the direction of rotation (Fig. 3). When the block rotates in the negative direction, where the largest deformation occurs in the inward direction (marked by the red circle in Fig. 3A) as the edges of obstacles are encountered, the distance between each block remains constant, but the angle of rotation increases. This movement direction was defined as the direction of negative curvature, and the contact surface between each smart chain structure was a circular shape in this direction. On the other hand, on the basis of the position at which concentrated and large inward directional deformation occurred, the smart chain block rotated in the positive direction near both sides of the wheel (marked by the green circle in Fig. 3A) as the distance between each block increased. This characteristic is because of the specially designed external shape of the smart chain block. On the upper side of the chain block, there is a small bump that alters the distance as the block rotates in the positive direction. As the angle of rotation increased, each block in the contact state continued to follow the protrusion path, and then the distance between each block increased. The pins are located at the hole in each smart chain structure to prevent the blocks from being disassembled (Fig. 3B). The hole shape is elliptical, and the pin can freely move in the lateral direction. When the block rotated in the direction of positive curvature, the distance between each block increased until the pin was stuck in the reduced volume of the overlapping area of pin holes of each block. For rotation in the direction of negative curvature, the distance between each block was maintained because the rotation axis of each block coincided with a circular shape. If a block rotated more than 90°, then the pin prevented further rotation because of the increased distance between each block. Therefore, the pin and the shape of the hole determined the range of rotation of each block.
控制车轮刚度的第二个主要机制是每个链条结构之间距离的变化。所提出的智能链结构被设计为每个块之间的距离取决于旋转方向（图3）。当块体沿负方向旋转时，当遇到障碍物的边缘时，最大的变形发生在向内方向（图3A中的红色圆圈标记），每个块之间的距离保持不变，但旋转角度增加。该运动方向被定义为负曲率方向，每个智能链结构之间的接触面在该方向上呈圆形。另一方面，根据发生集中和大向变形的位置，随着每个块之间距离的增加，智能链块在轮子两侧附近沿正方向旋转（图3A中的绿色圆圈标记）。这个特性是因为智能链块的外部形状是特殊设计的。在链块的上侧，有一个小凸起，当链块沿正方向旋转时，该凸起会改变距离。随着旋转角度的增加，处于接触状态的每个块继续遵循突出路径，然后每个块之间的距离增加。引脚位于每个智能链结构的孔中，以防止块被拆卸（图 3B）。孔形状为椭圆形，销钉可以在横向自由移动。 当块沿正曲率方向旋转时，每个块之间的距离增加，直到销钉卡在每个块的销孔重叠区域的减小体积中。对于负曲率方向的旋转，由于每个块的旋转轴与圆形重合，因此保持了每个块之间的距离。如果一个块旋转超过 90°，则销钉会阻止进一步旋转，因为每个块之间的距离会增加。因此，销钉和孔的形状决定了每个块的旋转范围。

This distance variation of the block mechanism was applied because of the different physical characteristics of a liquid droplet and a wheel. In the case of a liquid droplet, its total volume is conserved even when the shape of the liquid droplet changes, whereas the volume of the soft supporting structure in the wheel can be compressed, and the total volume of the wheel was decreased when an inward directional force was applied to the wheel. Because of these factors, the principle associated with the minimization of the surface area for a given volume in a liquid droplet because of surface tension, which is the reason liquid droplets have a spherical shape, was not applied to the wheel. However, if the distance between each block was changed depending on the rotation conditions, then the total surface area (or total length of the wheel circumference) was changed. Therefore, the potential energy derived from the contraction force of a wire spoke was not minimized when the total surface area increased because of the rotation of the blocks. This was comparable to a liquid that shows minimum potential energy at the minimum surface area. The shape of the smart chain structure was designed to minimize the wheel circumference when the wheel was circular. If any deformation started to occur from the initial circular shape, then the direction of positive curvature was generated as a result, leading to an increase in the total potential energy. For example, when concentrated inward directional deformation occurred (fig. S5), the chain block, situated relatively close to both sides from the position where the concentrated deformation occurred, rotated in the positive direction. Therefore, the total potential energy increased as the length of the circumference increased. The effect of the distance variation of each block was measured as deformation occurred at the fixed hub-gap distance (fig. S6). The wheel was slowly placed on the ground from the floating state, and the total length of the blocks was measured by adding the individual distance between each block. As a result, the total length of the blocks decreased as the center displacement increased until the wheel reached the state of maximum possible deformation. This is because the blocks are partially separated from each other when the wheel floats without an external load; then, the blocks were pushed close to each other as the center displacement increased. These gaps between blocks were diminished after 20 mm of center displacement, and the total length was maintained at a constant thereafter. When the wheel deformation approached its maximum value, the blocks located around the position where the wheels deformed in the direction away from the ground were deformed in the direction of positive curvature, so the distance between the blocks increased.
由于液滴和轮子的物理特性不同，应用了块机构的这种距离变化。在液滴的情况下，即使液滴的形状发生变化，其总体积也是守恒的，而轮子中软支撑结构的体积可以被压缩，当向内定向力施加到轮子上时，轮子的总体积减小。由于这些因素，由于表面张力（即液滴具有球形的原因）而使液滴中给定体积的表面积最小化的原理没有被应用于轮子。但是，如果根据旋转条件改变每个块之间的距离，则总表面积（或车轮周长的总长度）也会改变。因此，当总表面积由于块的旋转而增加时，由线辐条的收缩力得出的势能并没有最小化。这与在最小表面积上显示最小势能的液体相当。智能链结构的形状设计为当轮子为圆形时，将轮子周长降至最低。如果从最初的圆形开始发生任何变形，则会产生正曲率方向，从而导致总势能增加。例如，当发生集中的向内定向变形时（图S5），从集中变形发生的位置相对靠近两侧的链块向正方向旋转。因此，总势能随着圆周长度的增加而增加。 当在固定的轮毂间隙距离处发生变形时，测量了每个块的距离变化的影响（图 S6）。将轮子从浮动状态慢慢放置在地面上，并通过将每个块之间的单独距离相加来测量块的总长度。结果，随着中心位移的增加，块的总长度减小，直到车轮达到最大可能变形的状态。这是因为当轮子在没有外部负载的情况下漂浮时，块会部分地彼此分离;然后，随着中心位移的增加，这些块被推得很近。在中心位移 20 mm 后，块之间的这些间隙减小，此后总长度保持恒定。当车轮变形接近其最大值时，位于车轮在远离地面的方向上变形的位置周围的块体在正曲率方向上变形，因此块之间的距离增加。

In a situation in which the wheel was pressed against an obstacle, negative curvature was generated where the largest deformation occurred at the edge of the obstacle, and positive curvature was generated at the same time; thus, the total length of the blocks increased (fig. S7). The magnitude of curvature because of adaptation to the obstacle increased as the deformation distance, which was defined from the wheel center to the edge of the obstacle, increased. As a result, the total length variation increased as the depth of deformation increased (fig. S7C). To verify the effects of distance variation on the basis of the rotation angle and direction in detail, the distance between each adjacent individual block was measured on the basis of the relative rotation angle between each block (fig. S8).
在车轮被压在障碍物上的情况下，在障碍物边缘发生最大变形的地方产生负曲率，同时产生正曲率;因此，块的总长度增加了（图 S7）。由于适应障碍物而产生的曲率大小随着变形距离（定义为从车轮中心到障碍物边缘）的增加而增加。因此，随着变形深度的增加，总长度变化也随之增加（图S7C）。为了详细验证距离变化对旋转角度和方向的影响，根据每个块之间的相对旋转角度测量每个相邻块之间的距离（图 S8）。

To evaluate the dependence of stiffness variation on the hub-gap distance, the wheel with a 140-mm radius was installed in the experimental setup (Fig. 4A). Different sizes of indenters could be installed in the experimental setup, and the reaction force associated with each indenter was measured while the indenter was transferred downward. The indenter with a wide shape was used to mimic the scenario in which the wheel was at rest on typical flat ground. The indenter with a narrow shape was used to mimic the scenario at which the wheel was deformed by an obstacle, similar to applying a concentrated force associated with encountering an obstacle. First, the reaction force was measured with the wide-shaped indenter. The reaction force displayed a different trend before and after a 4-mm hub-gap distance variation was reached (Fig. 4B). When the hub-gap distance variation was smaller than 2 mm, the force-displacement slope changed, with ~1.5 mm of displacement (Fig. 4C); then, the first part of the graph can be divided into two regions: the first region of high stiffness and the second region of low stiffness. The first region represents the normal state in which the wheel maintains its circular shape in a stable manner. When the applied pressure became larger than the boundary value of the first region, the chain block structure of the wheel that was in contact with the ground was partially buckled inward, and the second region was reached. When the hub-gap distance variation was larger than 4 mm, the characteristics of the wheel differed compared with those at 0-and 2-mm hub-gap distance variations. The stiffness of the wheel was lower in the first region and higher in the second region after the transition point was reached (Fig. 4D). When a force started to be applied to the wheel, the soft supporting structure started to deform until the desired deformation height was reached at the given hub-gap distance. After the wheel reached its desired deformation height, the tension of the wire spoke structure started to rapidly increase because of the mechanism shown in Fig. 2C, and then the stiffness of the wheel increased, as shown in the second region. Because of the dominant role of the wire spoke structure at the transition point, the hub-gap distance mainly influenced the required displacement and force at the transition position (Fig. 4E). In addition, as the hub-gap distance variation increased, more deformation was required before the stiffness started to increase substantially. The force required to maintain each hub-gap distance was measured as shown in fig. S9.
为了评估刚度变化对轮毂间隙距离的依赖性，在实验装置中安装了半径为 140 mm 的车轮（图 4A）。在实验装置中可以安装不同尺寸的压头，并在压头向下移动时测量与每个压头相关的反作用力。具有宽形状的压头用于模仿车轮在典型平坦地面上静止的情况。具有狭窄形状的压头用于模拟车轮被障碍物变形的情况，类似于施加与遇到障碍物相关的集中力。首先，用宽形状的压头测量反作用力。在达到 4 mm 轮毂间隙距离变化之前和之后，反作用力表现出不同的趋势（图 4B）。当轮毂间隙距离变化小于2 mm时，力-位移斜率发生变化，位移为~1.5 mm（图4C）;然后，图形的第一部分可以分为两个区域：第一个高刚度区域和第二个低刚度区域。第一个区域代表轮子以稳定的方式保持其圆形的正常状态。当施加的压力大于第一区域的边界值时，与地面接触的车轮链块结构部分向内屈曲，达到第二区域。当轮毂间隙距离变化大于4 mm时，车轮特性与0和2 mm轮毂间隙距离变化时的特性不同。 在达到过渡点后，车轮的刚度在第一个区域较低，在第二个区域较高（图4D）。当开始向车轮施加力时，软支撑结构开始变形，直到在给定的轮毂间隙距离处达到所需的变形高度。在车轮达到其所需的变形高度后，由于图2C所示的机构，线辐结构的张力开始迅速增加，然后车轮的刚度增加，如第二个区域所示。由于线辐结构在过渡点处占主导地位，轮毂间隙距离主要影响过渡位置所需的位移和力（图4E）。此外，随着轮毂间隙距离变化的增加，在刚度开始大幅增加之前需要更多的变形。如图 S9 所示，测量了维持每个轮毂间隙距离所需的力。

In the case of using a narrow shape indenter, a concentrated applied force on the wheel was generated, which was similar to the deformation of the wheel when pressing against obstacles (Fig. 4F). By decreasing the width of the indenter, situations in which the wheel overcomes sharp obstacles could be simulated. As the deformation magnitude from displacement of the indenter increased, the transition point at which the slope started to decrease was observed. The position of the transition point was shifted to a larger deformation value as the width of the indenter increased, and the slope before the transition point was maintained. Because of the decreasing modulus of the wheel after the transition point, the wheel was easily deformed, with potentially large deformations, according to the shape of the obstacle. This trend is shown in Fig. 4G. As the hub-gap distance increased, the force required to reach the transition position decreased for the same indenter width (Fig. 4H and fig. S10). When the hub-gap distance variation was 4 mm, the transition point appeared when the width of the indenter was more than 50 mm.
在使用窄形状压头的情况下，在车轮上产生集中的施加力，这类似于车轮在压在障碍物上的变形（图4F）。通过减小压头的宽度，可以模拟车轮克服尖锐障碍物的情况。随着压头位移变形幅度的增加，观察到边坡开始减小的过渡点。随着压头宽度的增加，过渡点的位置向较大的变形值移动，并且过渡点前的坡度保持不变。由于过渡点后车轮的模量减小，因此根据障碍物的形状，车轮很容易变形，并且可能会有很大的变形。这种趋势如图4G所示。随着轮毂间隙距离的增加，在相同的压头宽度下，到达过渡位置所需的力减小（图4H和图S10）。当轮毂间隙距离变化为4 mm时，当压头宽度大于50 mm时，过渡点出现。

To evaluate the ability of the wheel to overcome obstacles, the trajectory of the wheel was measured for an obstacle with a square shape. As the hub-gap distance variation increased, the position of the center of the wheel lowered because of the decreasing modulus of the wheel (Fig. 5A). In particular, when the hub-gap distance variation was less than 4 mm, the position of the center of the wheel was higher compared with cases with larger hub-gap distance variations, because the wheel shape was relatively circular without deformation from gravitational force. However, because of the increased stiffness, the wheel was difficult to deform to the shape of the obstacle. Therefore, the wheel could not overcome the obstacles when the hub-gap distance was less than 4 mm. When the wheel moved down a step, the trajectory for a hub-gap distance variation of 0 mm was similar to that for a wheel with a rigid-circular shape, as the wheel rotated around the edge of the step as the rotation axis (Fig. 5B). When the hub-gap distance variation was larger than 2 mm, deformation occurred as the wheel approached the edge of the step. As the wheel approached the edge of the step, the magnitude of the radial directional force on the wheel from the obstacle increased. Consequently, the distance between the edge of the step and the starting position of the deformation increased as the hub-gap distance variation increased. The gravitational force from the additional weight affected the depth of deformation at the obstacle (fig. S11A), and these variables were proportional to each other. However, until the wheel made contact with an obstacle, its height remained nearly constant regardless of the weight. This finding means that, after the deformation caused by an obstacle, the stiffness of the soft supporting structure predominantly affected the characteristics of the wheel. Therefore, the stiffness of the soft supporting structure needs to be relatively high to maintain a similar shape of deformation under high payload conditions (figs. S11B and S12).
为了评估车轮克服障碍物的能力，针对方形障碍物测量了车轮的轨迹。随着轮毂间隙距离变化的增加，由于车轮模量的减小，车轮中心的位置减小（图5A）。特别是，当轮毂间隙距离变化小于4 mm时，与轮毂间隙距离变化较大的情况相比，车轮中心的位置更高，因为轮子形状相对圆形，不受重力影响变形。然而，由于刚度的增加，车轮很难变形到障碍物的形状。因此，当轮毂间隙距离小于 4 mm 时，车轮无法克服障碍物。当轮子向下移动一个台阶时，轮毂间隙距离变化为0 mm的轨迹与刚性圆形车轮的轨迹相似，因为轮子作为旋转轴围绕台阶边缘旋转（图5B）。当轮毂间隙距离变化大于2 mm时，当车轮接近台阶边缘时会发生变形。当车轮接近台阶边缘时，车轮从障碍物处产生的径向方向力的大小增加。因此，随着轮毂间隙距离变化的增加，台阶边缘与变形起始位置之间的距离也随之增加。来自额外重量的重力影响了障碍物的变形深度（图S11A），这些变量彼此成正比。然而，在轮子与障碍物接触之前，无论重量如何，其高度几乎保持不变。 这一发现意味着，在障碍物引起变形后，软支撑结构的刚度主要影响车轮的特性。因此，软支撑结构的刚度需要相对较高，以便在高有效载荷条件下保持相似的变形形状（图S11B和S12）。

The real-time variation in stiffness required to overcome an obstacle was implemented in the proposed wheel (Fig. 5C and movie S1). As an example of ascending motion, the wheel could maintain a circular high-modulus state before contact with an obstacle; then, the modulus of the wheel decreased to achieve an easily deformable state. After climbing on the step and overcoming obstacles, the wheel returned to the high-modulus state. This process could be repeated in descending motion to minimize abrupt z-directional acceleration. If the wheel maintained the high-modulus mode during its descending motion, then the contact area between the wheel and the step obstacles was limited only to the edge of the step. This limitation reduced the stability of the wheel. The high z-directional acceleration because of the rotation axis at the edge of the obstacle, which was evident from the steeper slope of the trajectory just before the wheel contacted the ground, also negatively affected stable locomotion (Fig. 5C). The height of the wheel could also be controlled by continuously changing the modulus of the wheel in real time (Fig. 5D, fig. S13, and movie S2).
在所提出的车轮中实现了克服障碍物所需的刚度的实时变化（图 5C 和视频 S1）。作为上升运动的一个例子，轮子在与障碍物接触之前可以保持圆形高模量状态;然后，车轮的模量减小，达到易变形状态。爬上台阶并克服障碍后，车轮又回到了高模量状态。该过程可以以下降运动的形式重复，以最小化突然的 z 方向加速度。如果车轮在下降运动过程中保持高模量模式，则车轮与台阶障碍物之间的接触面积仅限制在台阶的边缘。这种限制降低了车轮的稳定性。由于障碍物边缘的旋转轴，由于旋转轴线的高 z 方向加速度，这在车轮接触地面之前的轨迹较陡的斜率中很明显，也对稳定的运动产生了负面影响（图 5C）。也可以通过实时连续改变轮子的模量来控制轮子的高度（图5D、图S13和视频S2）。

The ability to overcome obstacles was mainly affected by the hub-gap distance variation, such as the height of obstacles (Fig. 6). The stability of climbing was defined as follows
克服障碍物的能力主要受轮毂间隙距离变化的影响，如障碍物的高度（图6）。攀爬的稳定性定义如下where l_block is the length of the unit smart chain block and d_slip is the slip distance at which the wheel slides off the ground before starting to climb the step-shaped obstacle. As the stability value approached 1, the slip distance decreased, and then the wheel climbed the obstacle in a stable manner. As the hub-gap distance variation increased, the decreased wheel modulus allowed the wheel to stably climb higher step-shaped obstacles (Fig. 6A). However, when the hub-gap distance variation exceeded a certain value and the step height was higher than 100 mm, the stability decreased because of the relatively low height of the wheel before the wheel contacted the obstacle. When the weight applied to the wheel increased, the hub-gap distance variation needed to be increased for stable climbing (Fig. 6B). When the hub-gab distance variation was small, the deformation generated in the wheel from the obstacle was not completely restored after overcoming the obstacle (fig. S14). This characteristic could be minimized by using a high-modulus soft supporting structure (fig. S15). The average traction force of the wheel was measured as 23.4 N with a 5.3-kg applied weight and 32.7 N with a 7.3-kg applied weight (fig. S16). Under high-speed driving conditions, the vibration characteristics of the wheel, using two different soft supporting structures, were evaluated on the basis of changes in the hub-gap distance variation (figs. S17 to S20 and movie S3). Regarding durability, the tensile strength of the single wire spoke module was also measured, and its average value was 1.193 kN (fig. S21).
其中 L_块是单元智能链块的长度，D_滑块是轮子在开始爬升阶梯形障碍物之前滑离地面的滑移距离。当稳定性值接近1时，滑移距离减小，然后车轮以稳定的方式爬升障碍物。随着轮毂间隙距离变化的增加，车轮模量的降低使车轮能够稳定地爬升更高的阶梯形障碍物（图6A）。然而，当轮毂间隙距离变化超过一定值且台阶高度高于100 mm时，由于车轮接触障碍物前的高度相对较低，稳定性下降。当施加在车轮上的重量增加时，需要增加轮毂间隙距离变化以实现稳定爬坡（图 6B）。当轮毂-轮毂间距变化较小时，车轮在克服障碍物后产生的变形并未完全恢复（图S14）。通过使用高模量软支撑结构，可以最小化这一特性（图S15）。车轮的平均牵引力测量为 23.4 N（施加重量为 5.3 kg）和 32.7 N（施加重量为 7.3 kg）（图 S16）。在高速行驶条件下，根据轮毂间隙距离变化的变化，评估了使用两种不同软支撑结构的车轮的振动特性（图S17至S20和视频S3）。在耐久性方面，还测量了单线辐条模块的抗拉强度，其平均值为 1.193 kN（图 S21）。

In the simulation results, the trajectories of the wheel when ascending and descending step-shaped obstacles displayed characteristics similar to those in the experimental results (Fig. 6, D and E, and movie S4). The detailed shape of deformation (Fig. 6F) and the position variation of each chain block (fig. S23 and Supplementary Methods) were also verified. In addition, the state transition between the circular high-modulus and deformable low-modulus states was verified in the simulation (fig. S24 and Supplementary Methods).
在仿真结果中，车轮在上升和下降阶梯状障碍物时的轨迹表现出与实验结果相似的特征（图6，D和E，以及视频S4）。还验证了变形的详细形状（图6F）和每个链块的位置变化（图S23和补充方法）。此外，在仿真中验证了圆形高模量态和可变形低模量态之间的状态跃迁（图S24和补充方法）。

A variable-stiffness wheel was implemented in a four-wheeled vehicle system for performance evaluation (Fig. 7, A and B; movie S5; and fig. S25). The surface tension of all four wheels could be adjusted individually, and a black sponge was used as the soft supporting structure because of the increased weight of the system. The four-wheeled vehicle system could operate with wheels in a circular high-modulus state on flat ground, and the vehicle could overcome rocks with irregular shapes by transitioning the wheels to the deformable low-modulus state (Fig. 7A). In a similar sequence, the four-wheeled vehicle could overcome a 180-mm-high obstacle, which was 1.2 times higher than the wheel radius (Fig. 7B).
在四轮车辆系统中实现了可变刚度轮，用于性能评估（图7，A和B;视频S5;和图S25）。所有四个车轮的表面张力都可以单独调节，由于系统的重量增加，使用黑色海绵作为软支撑结构。四轮车辆系统可以在车轮处于圆形高模量状态的情况下在平坦地面上运行，车辆可以通过将车轮过渡到可变形的低模量状态来克服形状不规则的岩石（图7A）。在类似的序列中，四轮车辆可以克服 180 毫米高的障碍物，该障碍物是车轮半径的 1.2 倍（图 7B）。

The wheel was also implemented in a two-wheeled wheelchair system (fig. S26). In this case, the size of the wheel was doubled, considering the size of the wheelchair user. The weight of the vehicle system was ~120 kg, which was four times heavier than the weight of the four-wheeled system. Because the sponge structure had limitations in terms of increasing stiffness, a urethane honeycomb structure was applied as the soft supporting structure. The traction force (fig. S27) and the vibration characteristics of the large-sized wheel were evaluated at high speeds of up to 30 km/hour (fig. S28 and movie S6), and its durability was also tested for more than an hour at a speed of 15 km/hour (movie S7). The dynamics of the two-wheeled wheelchair system was studied (Supplementary Methods), and a control algorithm was developed (Supplementary Methods). The resulting control input torque applied to the wheelchair can be described as
轮子也实现在两轮轮椅系统中（图 S26）。在这种情况下，考虑到轮椅使用者的尺寸，轮子的尺寸增加了一倍。车辆系统的重量为~120 kg，是四轮系统的重量的四倍。由于海绵结构在增加刚度方面存在局限性，因此采用聚氨酯蜂窝结构作为软支撑结构。在高达30 km/h的高速下评估了大型车轮的牵引力（图S27）和振动特性（图S28和视频S6），并在15 km/小时的速度下测试了其耐久性超过一个小时（视频S7）。研究了两轮轮椅系统的动力学特性（《补充方法》），并开发了控制算法（《补充方法》）。施加在轮椅上的控制输入扭矩可以描述为

The stiffness variation in the smart chain structure with a 40-kg weight was evaluated (movie S8), and the wheelchair system effectively overcame the obstacle by changing the state of the wheel (Fig. 7C and movie S9). Using the wheelchair system, driving speed and performance were assessed in an atypical outdoor environment (movies S10 and S11), and the state transition of the wheel in real time was demonstrated; notably, the wheelchair system successfully overcame the obstacle (Fig. 7D and Movie 1).
评估了重量为40 kg的智能链条结构的刚度变化（视频S8），通过改变轮子的状态，轮椅系统有效地克服了障碍物（图7C和视频S9）。使用轮椅系统，在非典型的户外环境（电影 S10 和 S11）中评估了驾驶速度和性能，并实时演示了车轮的状态过渡;值得注意的是，轮椅系统成功地克服了障碍物（图7D和视频1）。

The smart chain block was fabricated with a three-dimensional (3D) printer (F370 Stratasys Inc.) with ABS (ABS-M30). Each chain block was connected by an aluminum rod with a radius of 2 mm. A sponge structure and a honeycomb structure were used as the soft supporting structure. The modulus of each sponge was measured (fig. S2). The honeycomb structure was fabricated with a 3D-printed mold (ABS-M30) and with liquid urethane rubber (Vytaflex 60, Smooth-On Inc.). The detailed fabrication process is described in figs. S34 and S35.
智能链块是用带有ABS（ABS-M30）的三维（3D）打印机（F370 Stratasys Inc.）制造的。每个链块都通过半径为 2 mm 的铝棒连接。采用海绵结构和蜂窝结构作为软支撑结构。测量了每个海绵的模量（图S2）。蜂窝结构是用3D打印模具（ABS-M30）和液态聚氨酯橡胶（Vytaflex 60，Smooth-On Inc.）制造的。详细的制造过程在图中进行了描述。S34 和 S35。

The material used for the wire spoke structure was Kevlar fiber with a thickness of 2 mm. For the small-sized wheel that was applied to the four-wheeled vehicle, the diameter of the wheel was 300 mm, and the width was 40 mm. For the large wheel that was applied to the wheelchair system, the wheel diameter was 560 mm, and the width was 90 mm.
用于线辐条结构的材料是厚度为 2 mm 的 Kevlar 纤维。对于应用于四轮车辆的小型车轮，车轮的直径为300 mm，宽度为40 mm。对于应用于轮椅系统的大轮子，轮子直径为 560 毫米，宽度为 90 毫米。

To evaluate the basic characteristics of the wheel, the load cell at the indenter was used to measure the reaction force of the wheel, and the other load cell at the hub structure was used to measure the required force to maintain the desired hub-gap distance (Fig. 4A). The two load cells (CSBA-50 L, CAS Ins.) at the hub structure and one load cell (CSBA-50 L, CAS Ins.) at the indenter were used to measure force. A laser displacement sensor (ODSL 96B M, Leuze) was used to measure the positions of the indenter and hub structure. The data were collected with the NI DAQ module (cDAQ-9172, NI-9215, National Instruments) and LabVIEW software. For the experiment in the flat-ground–condition case, an indenter with a square surface at the bottom with a length of 200 mm and a width of 40 mm was used.
为了评估车轮的基本特性，压头处的称重传感器用于测量车轮的反作用力，轮毂结构处的另一个称重传感器用于测量保持所需轮毂间隙距离所需的力（图4A）。轮毂结构处的两个称重传感器（CSBA-50 L，CAS Ins.）和压头处的一个称重传感器（CSBA-50 L，CAS Ins.）用于测量力。激光位移传感器（ODSL 96B M，Leuze）用于测量压头和轮毂结构的位置。使用NI DAQ模块（cDAQ-9172、NI-9215、National Instruments）和LabVIEW软件收集数据。在平坦地面条件下的实验中，使用了底部具有方形表面、长度为 200 mm、宽度为 40 mm 的压头。

The experimental system for evaluating the wheel was developed as shown in Fig. 5. A rail on which the wheel could move freely in the x-axis (forward) and z-axis (vertical) directions was installed in the system, and two motors that rotate the wheel and change the hub-gap distance were installed as the only parts of the wheel requiring power. The position of each smart chain block and the trajectory of the wheel were analyzed with image tracking software (ProAnalyst, Xcitex).
评估车轮的实验系统如图5所示开发。系统中安装了车轮可以在x轴（向前）和z轴（垂直）方向上自由移动的导轨，并且安装了两个旋转车轮并改变轮毂间隙距离的电机，作为车轮唯一需要动力的部分。使用图像跟踪软件（ProAnalyst，Xcitex）分析了每个智能链块的位置和车轮的轨迹。

For the four-wheeled vehicle system, each wheel was actuated by using two motors: a motor (Maxon 488607 BLDC Motor) to rotate the wheel and a motor (Maxon 647692 BLDC Motor) to vary the hub-gap distance. The detailed configuration is shown in fig. S25. In the case of the two-wheeled wheelchair system, each wheel was actuated by using a motor (Komotek KAND-15DF3N2 AC Servo Motor) to rotate the wheel and a motor (Komotek KAFZ-06D) to change the hub-gap distance. A balancing algorithm was implemented with a controller. The detailed configuration and balancing algorithm are described in Supplementary Methods and figs. S29 to S32.
对于四轮车辆系统，每个车轮都使用两个电机驱动：一个电机（Maxon 488607 BLDC电机）用于旋转车轮，一个电机（Maxon 647692 BLDC电机）用于改变轮毂间隙距离。详细配置如图S25所示。在两轮轮椅系统的情况下，每个轮子都通过使用电机（Komotek KAND-15DF3N2 AC Servo Motor）来驱动轮子旋转，并使用电机（Komotek KAFZ-06D）来改变轮毂间隙距离。利用控制器实现了平衡算法。详细的配置和平衡算法在补充方法和图中进行了描述。S29 至 S32。

We used mean values (± SD) for Fig. 2E and figs. S16 and S27 (n = 5 for each experimental condition) and for Figs. 2 (F and G) and 4 (E and H) and figs. S3, S14, and S15 (n = 3 for each experimental condition). Linear fitting equations and plots in Fig. 4 (C and D) were calculated and drawn using Origin 2023b.
我们使用平均值 （± SD） 进行图 2E 和无花果。S16 和 S27（每个实验条件 n = 5）以及图 2（F 和 G）和 4（E 和 H）以及图 S3、S14 和 S15（每个实验条件 n = 3）。图 4（C 和 D）中的线性拟合方程和绘图是使用 Origin 2023b 计算和绘制的。