Anisotropic shape-shifting robotic matter enabling versatile collective behaviors

Abstract

Collective robotic systems can exhibit emergent behaviors through local interactions among individual robots, yet most existing platforms rely on sophisticated individuals or complex control strategies to achieve diverse functionalities. Here, we present TriHex, a minimalist robotic unit capable of shape-shifting between triangular and hexagonal shapes using single-motor actuation. Despite its simplicity, collectives of TriHex units display remarkable behavioral diversity through two control dimensions: system configuration and individual unit’s shape-shifting phase. By adjusting these parameters, a static tessellated assembly’s stiffness can be programmed. In discrete assemblies, the system transitions between solid-like and liquid-like states and achieves programmable viscosity. When arranged in cellular configurations with a pre-designed phase difference, the system realizes controllable porosity. Additionally, shape changes enable coordinated collective locomotion, reconfiguration and integrating capabilities of object gripping and transportation. Analytical models and numerical simulations are developed that successfully capture and predict the TriHex collective’s behavior. This work demonstrates that complex collective behaviors can emerge from relatively simple individual units, offering insights for designing scalable and adaptive robotic matter.

TriHex robots employ a single-motor mechanism to dynamically transform between triangular and hexagonal shapes, enabling diverse and adaptive collective behaviors in robotic materials.

Introduction

Simple local interactions can generate emergent collective behaviors in diverse systems, from low-intelligence natural systems such as shaken grains¹, slime molds² and embryonic cells³ to synthetic active matter including colloidal^4–6, self-propelled^7,8 and growing particles⁹. This principle of emergence has been successfully replicated in robotic swarms, where units mediate simple parameters to produce sophisticated behaviors such as phase transitions^10–12, aggregation^13,14, phototaxis^15–18, self-assembly^19,20 and reconfiguration^21–24. Although these systems can exhibit diverse collective behaviors, most were designed for a single behavioral type or a small number of predefined modes, limiting their adaptability and functionality in dynamic, real-world environments.

To address this limitation, recent work introduced the concept of “robo-matter”^11,25–28, a system that integrates material properties into robotic functionalities. From a materials perspective, such systems can modulate their physical states and alter their mechanical properties. From a robotics perspective, they can achieve locomotion, self-reconfiguration, and self-organization. These high-level behaviors are generated by modulating the robo-matter unit’s properties at individual units¹¹, between neighboring units¹², and algorithmic control strategies^29,30.

A simple way to achieve these behaviors is to use shape-shifting particles, which influence their mutual interactions and enable the emergence of versatile behaviors. This principle is demonstrated in biological systems: cellular deformation has been observed across diverse processes, including wound healing³¹, oocyte genesis³², and biofilm morphing³³. Physical models further show that deformation drives phase transitions³⁴, creep dynamics³⁵, and pattern formation⁹. In metamaterials, controlled deformation enables programmable morphing^36,37, stiffness^38–40, and compressibility⁴¹.

Collective robotic systems formed by active shape-shifting units have also been studied. Some systems are constructed as continuous lattice structures^42,43, but their rigid interconnected framework constrains their topological reconfiguration. Other systems are amorphous and loosely coupled, achieving collective motion through isotropic shape-shifting of individual units^{15,20,44–48}, with their directional locomotion induced by external stimuli. On the other hand, some systems incorporate asymmetric or polarized unit designs, which increase the complexity of interactions between the units and enable collective behaviors such as reconfiguration^49–51, complex locomotion^17,52 and self-assembly⁵³. These studies showcase the potential for versatile behaviors in collective systems through shape-shifting particles.

Here, we introduce a minimally actuated robot called TriHex (Fig. 1a, b), capable of anisotropic shape-shifting between a hexagon and a triangle through opposing motions at adjacent vertices (see Supplementary Movie 1). Each unit's shape transformation is described by a normalized phase variable ranging from 0 to 1, where phase 0 corresponds to the hexagonal configuration and phase 0.5 to the triangular configuration (Fig. 1c). The configurations enabled by the magnetic coupling can be represented in two modes: ordered configurations, where units are connected vertex-to-vertex and edge-to-edge with a fixed topology, and disordered configurations, where units interact loosely with varying connection types including vertex-to-edge contacts, allowing passive deformation and complex interaction modes that enrich their collective dynamics.

Fig. 1. Overview of the TriHex system and its capabilities.

a A collective of TriHex units arranged in a group configuration. Different colors indicate marker labels applied to individual units for group identification and tracking purposes; all units are identical in structure and function. b TriHex unit prototypes shown in triangular and hexagonal shapes. c Schematic of a TriHex unit’s anisotropic shape-shifting cycle between hexagonal and triangular shapes. Numbers indicate the phase corresponding to each shape. The color filled on each unit represents its current phase. d Versatile collective behaviors emerging from individual unit's shape-shifting and collective configuration.

Using this platform, we demonstrate that the system can produce a wide range of emergent behaviors (Fig. 1d). In the disordered configurations, shape-shifting enables solid-liquid transitions, where fluid-like behavior can be regulated by different shape-shifting strategies. In ordered configurations, static tessellations provide programmable stiffness, while coordinated shape-shifting enables programmable porosity, reconfiguration, and collective migration. These results highlight two distinctive aspects of the TriHex platform. First, unlike prior single-actuator systems that rely on isotropic shape modulation¹⁵, TriHex encodes anisotropy at the unit level through a mechanically simple underactuated mechanism. Second, by combining assembly configuration and individual shape-shifting as two system-level control parameters, TriHex gives rise to a broader range of collective functional modes than other robotic matter systems^11,27.

Results

Robot prototype

The TriHex unit is a minimally actuated robot capable of shape-shifting between hexagonal and triangular configurations, with key components including the edge panels, Hall effect sensors, sphere magnets, spring boundaries, and a communication module (Fig. 2a). The unit’s boundary is defined by spring-connected vertices and edge panels, enabling passive deformation under external forces. Neodymium sphere magnets embedded in the vertices and edge panels enable loosely coupled inter-robot magnetic interactions, allowing units to attach and detach dynamically during collective behaviors.

Fig. 2. Hardware design of the TriHex unit.

a Photograph of two connected TriHex units with key components labeled. b Top-view photographs of the TriHex unit in the hexagonal (edge length 50.9 mm), intermediate, and triangular (edge length 124.7 mm) states, with type $\mathcal{P}$ and type $\mathcal{Q}$ vertices labeled in the intermediate state. c Schematic of the TriHex unit in the triangular configuration illustrating the underactuated tendon-driven mechanism, with insets showing the dual-layer spool assembly driven by a single motor and the vertex components. d Schematics of the dual-cavity edge panel design illustrating the magnetic behavior in the disconnected and connected states. e Hall sensor response during connection and disconnection cycles, showing distinct deviations between connected and disconnected states.

The shape-shifting of the unit is achieved through an underactuated tendon-driven mechanism powered by a single motor (Fig. 2b, c). A dual-layer spool with strings wound in opposite directions drives the six vertices in two opposing directions simultaneously: during the hexagonal-to-triangular transition, type $\mathcal{P}$ vertices extend outward to become triangle vertices, while type $\mathcal{Q}$ vertices retract inward to become edge midpoints, and vice versa. This asymmetric actuation drives the unit’s anisotropic shape-shifting, with the two vertex types exhibiting different stiffness due to the tendon-driven mechanism. Consequently, in fully hexagonal or triangular states, extended vertices are compressible by external forces, whereas retracted vertices are constrained by the base structure and resist compression. In intermediate states, both vertex types are partially extended and compressible. The full force-displacement characterization is provided in Supplementary Information Section 1.1. Additionally, a microcontroller regulates motor speed using encoder feedback, ensuring consistent shape-shifting cycles across all units in the collective. The edge panels feature a specialized dual-cavity design with a sphere magnet in the upper cavity and a disc magnet in the lower cavity. The sphere magnet self-aligns vertically when disconnected and reorients horizontally upon edge-to-edge contact (Fig. 2d). This allows reliable detection of edge attachment and detachment based on the resulting change in local magnetic field through the hall sensors distributed around each unit (Fig. 2e). Detailed hardware specifications and control strategies are presented in the Methods section.

Programmable stiffness

The TriHex unit is capable of transitioning between hexagonal and triangular shapes, both of which can form an ordered, gap-free tiling of TriHex units connected edge-to-edge, which will be referred to as a tessellation. This tessellation deforms passively under external forces, as each unit’s vertex is actuated by a tendon-driven mechanism along a guide track, and an elastic boundary connects adjacent vertices. Each unit functions as a cell-like component that collectively defines the properties of the entire system.

To characterize the mechanical behaviors of the tessellated system, compression tests were performed on various configurations (Fig. 3a). Our results indicate that by exploiting the unit’s various shape states and anisotropy, the tessellated structure exhibits programmable mechanical behavior, where different assembly configurations yield distinct linear stress-strain relationships. Comparing the triangular with the hexagonal tessellations (Fig. 3b, c), hexagonal systems exhibit higher stiffness than triangular ones. This difference arises from the geometry-dependent stiffness of the two vertex types: the effective tendon length at each vertex determines its stiffness, with shorter tendons corresponding to higher stiffness. In the hexagonal state, type $\mathcal{P}$ vertices are fully retracted and thus reach their maximum stiffness, whereas in the triangular state, type $\mathcal{Q}$ vertices are fully retracted and reach theirs.

Fig. 3. Mechanical response and adaptive behavior of the TriHex collective.

a Snapshots from compression tests of hexgonal and trianglar tessellations. b Load responses of triangular tessellations with different numbers of layers. c Load responses of hexagonal tessellations with uniform orientation and different numbers of layers. d Load responses of four-layer tessellations with different orientations, where $\mathcal{P}$ and $\mathcal{Q}$ denote the types of outward-facing vertices of the units in the two outermost layers. e Snapshots from creep experiments with and without active shape-shifting (the colored dots on top of each unit serve as labels without additional meaning). f Time-displacement curves from creep tests comparing non-shape-shifting and random shape-shifting conditions. g Chain-constrained morphing process with the system encircled by a plastic chain. h Circularity evolution under three shape-shifting strategies: random shape-shifting with an average period of 60 s/cycle, random shape-shifting with an average period of 30 s/cycle, and compression-based shape-shifting with an average period of 60 s/cycle.

Moreover, the tests on multilayer configurations reveal distinct scaling behaviors between the two shapes. The Young’s moduli of the triangular tessellations maintain almost constant as the number of layers increases (Fig. 3b), while the hexagonal tessellations show variations as layers increase (Fig. 3c). This suggests that the triangular configurations enable efficient stress transmission across layers due to their straight inter-layer connections and stable vertex interactions, while the hexagonal arrangements exhibit higher absolute Young’s moduli due to the increased vertex stiffness.

Furthermore, we investigated the effect of unit orientation on mechanical properties, which arises from the different stiffness of the two vertex types $\mathcal{P}$ and $\mathcal{Q}$. The experiments show that configurations with outer vertices of type $\mathcal{P}$ exhibit larger Young’s modulus at small strains in four-layer tessellations (Fig. 3d), consistent with the results from three-layer systems (Supplementary Fig.5). While type $\mathcal{Q}$ outer vertices produce linear elastic responses, type $\mathcal{P}$ outer vertices exhibit yielding behavior with slope transitions in the stress-strain curves. Mixed vertex configurations display intermediate load responses. These findings demonstrate that the orientations play a significant role in determining mechanical performance regardless of system size, indicating that the material’s mechanical properties can be systematically programmed through the arrangement of the units.

While tessellated configurations provide controllable mechanical properties, as demonstrated above, the magnetic couplings between the TriHex units also enable loose aggregates in which individual units can separate and reconnect dynamically. In such configurations, each unit’s shape imposes topological constraints on the collective assembly, and each unit’s shape-shifting enables the assembly to reconfigure.

To characterize these aggregates, creep experiments were conducted in which the system was subjected to a constant static load of 3 N for 700 s while being constrained in a 0.1 mm polyethylene terephthalate (PET) strip (Fig. 3e) (see Supplementary Movie 2). The results in Fig. 3f reveal distinct behaviors depending on the shape-shifting capabilities. Without shape-shifting, the robots undergo progressive compaction and interlocking, resulting in a small average strain of 0.04, exhibiting solid-like behavior. In contrast, when robots change their shapes actively as they are compressed, internal forces are redistributed, and the vertex extensions allow individual units to intrude into the boundary springs of neighboring robots, enabling separation and rearrangement behaviors that resemble a liquid-like state, leading to an increased average strain of 0.33. In this liquid-like state, the creep rate decreases significantly at later stages, indicating that the system has undergone sufficient rearrangement under the PET strip constraint. These results demonstrate that the system can achieve solid-liquid transitions through active shape-shifting. To further validate this, we conducted switching experiments in which shape-shifting was toggled on and off during the creep test. The results show that the creep strain stops increasing when shape-shifting is disabled and continues to increase when it is re-enabled, confirming that active shape-shifting is essential for the liquid-like behavior (see Supplementary Information Section 2.4).

To quantify how individual shape-shifting affects the system’s viscosity in the liquid-like state, chain-constrained morphing experiments were conducted where fourteen TriHex units were enclosed by a 3D-printed Polylactic acid (PLA) plastic chain with a length of 130 cm. In these experiments, the active robots were initially arranged into a long oval shape (Fig. 3g), and the system’s circularity was tracked for 10 min. Due to an intrinsic tendency to minimize stress by maximizing the chain’s enclosed area, the robotic system gradually evolved toward a more circular configuration(see Supplementary Movie 3), as a circle provides a symmetric stress distribution and a maximum area-to-perimeter ratio.

The results of the chain-constrained morphing experiments under different strategies (see Methods, Algorithm 1) are shown in Fig. 3h. The fluidity of the system is governed by two key factors: the average shape-shifting period and the control strategy. The average shape-shifting period is defined as the average time required to complete one full cycle from a hexagonal to a triangular shape and back to a hexagonal shape. To isolate its effect, we tested chain-constrained morphing under random shape-shifting with two average periods: 30 s/cycle and 60 s/cycle. The resulting curves show that a shorter period leads to faster convergence towards a circular configuration, confirming a decrease in the system’s viscosity. Furthermore, we compared random shape-shifting with a compression-based strategy, in which each TriHex unit detects local mechanical stress via its Hall sensor and shape-shifts in response to compression. With the same average period, the compression-based strategy achieves faster convergence to a circular configuration. This demonstrates that units with high stress asymmetry undergo more efficient stress relaxation, enhancing the system’s liquid-like behavior. Together, these experiments reveal that the system’s viscosity can be modulated by adjusting the average shape-shifting period or adopting a decentralized sensor-based strategy, enabling programmable flow behavior.

Robotic porous metamaterials

In our system, the shape of individual units imposes topological constraints on their relative positions and creates detachments during shape-shifting. Under certain configurations, the same geometry with different combinations of vertex types can exhibit different morphing behaviors, forming the basis for edge detachments and void formation patterns. These programmable void patterns enable the system to function as a porous metamaterial.

Considering a three-unit hexagonal tessellation with different unit orientations where each unit connects to two neighbors, which is illustrated in Fig. 4a, this tessellation produces various edge detachment and void formation behaviors as each unit transitions to a triangular shape. The system can be arranged with three $\mathcal{P}$, three $\mathcal{Q}$ or mixed vertex types facing the middle, each producing distinct morphing results. The $\mathcal{QQQ}$ configuration exhibits complete symmetry—all six outer edges detach simultaneously, creating a triangular void with the same shape and size as a triangular unit, dramatically expanding the collective’s area. The mixed configuration $\mathcal{QQP}$ produces partial disconnections with asymmetric voids. In contrast, the configurations $\mathcal{QPP}$ and $\mathcal{PPP}$ exhibit nondeterministic morphing results. This pattern reflects the fundamental tendency that when each unit transitions from a hexagon to a triangle, its type $\mathcal{Q}$ vertices detach more frequently while its type $\mathcal{P}$ vertices tend to maintain their connections.

Fig. 4. Shape morphing behaviors and porosity control in the TriHex collective.

a Schematics of the morphing results in a three-unit tessellation with different unit orientations, where each symbol denotes the vertex types ($\mathcal{P}$ or $\mathcal{Q}$) of the three units facing the shared center: $\mathcal{QQQ}$, $\mathcal{QQP}$, $\mathcal{QPP}$, and $\mathcal{PPP}$. b Morphing of a fully aligned six-unit aggregate. Yellow-filled circles indicate the $\mathcal{QQQ}$ configuration and green filled triangles indicate the $\mathcal{QPP}$ configuration. c Morphing of a mixed-orientation six-unit aggregate. Red-filled circles indicate the $\mathcal{QQP}$ configuration and blue-filled triangles indicate the $\mathcal{QPP}$ configuration. d Schematics of six unit cells under different combinations of phase difference Δϕ and phase ϕ following the $\mathcal{P}$-to-$\mathcal{P}$ and $\mathcal{Q}$-to-$\mathcal{Q}$ connection rule. e Snapshots of a thirteen-unit cellular assembly with Δϕ = 0, where voids are filled with red TPU to visualize the porosity variation from the fully closed state to the fully open state. f,g Poisson ratio and porosity as a function of phase ϕ from simulations of 350-unit aggregates following the $\mathcal{P}$-to-$\mathcal{P}$ and $\mathcal{Q}$-to-$\mathcal{Q}$ connection rule. Each curve corresponds to a different value of Δϕ, indicated by curve color. h Large-scale simulation of a 650-unit aggregate (25 × 26 cells) demonstrating reversible switching between a fully closed opaque state and an open state with approximately 30% porosity.

This selective edge detachment principle can be scaled to larger and more complex aggregates (see Supplementary Movie 4). In a fully aligned six-robot hexagonal tessellation shown in Fig. 4b, the $\mathcal{QQQ}$ configurations develop voids while the $\mathcal{PPP}$ configurations remain connected when transitioning from hexagonal to triangular shapes, with the aggregate’s porosity increasing from 0 to 33% in a six-unit system and to approximately 50% in large-scale aggregates and vice versa when transitioning back to hexagonal shapes. In a mixed-orientation system with $\mathcal{QQQ}$ and $\mathcal{QQP}$ configurations (Fig. 4c), each local cluster also follows the disconnection rules as mentioned above.

However, the morphing results are non-deterministic in the mixed-orientation tessellations, limiting utility as only fully aligned configurations can be reliably maintained. To achieve stable and controllable porosity modulation across a continuous range, we introduce a hexagonal cellular assembly composed of six robotic units arranged in a honeycomb pattern (see Supplementary Movie 5). This cellular structure enables scalable assemblies through periodic tiling of the six-unit cells.

As shown in Fig. 4d, we find that by partitioning the units into two phase groups where adjacent units alternate in phase, and connecting each unit to its neighbors following a consistent rule (either $\mathcal{P}$-to-$\mathcal{P}$ and $\mathcal{Q}$-to-$\mathcal{Q}$, or $\mathcal{P}$-to-$\mathcal{Q}$ throughout), edge connectivity is maintained throughout the shape-shifting cycle. This is geometrically guaranteed by the three-fold rotational symmetry of the TriHex unit (see Supplementary Information Section 3.1), which ensures consistent edge alignment regardless of the phase difference between adjacent units (Δϕ) or the elapsed phase through a cycle (ϕ). This principle extends naturally to larger metamaterials with controllable porosity, as demonstrated in Fig. 4e.

To further characterize the expansion-contraction oscillations of this porous system, we evaluate the temporal evolution of its Poisson ratio (Fig. 4f) and porosity (Fig. 4g) in simulation with 350 units. These simulations show that the Poisson ratio is near -1, indicating that the system exhibits similar contraction and expansion behavior along both axes, characteristic of auxetic behavior. Furthermore, the temporal properties of the metamaterial’s porosity can be programmed by adjusting the phase difference between adjacent units. We further demonstrate the scalability of this design through simulations of a 650-unit aggregate (25 × 26 cells), where coordinated shape-shifting enables reversible switching between a fully closed, opaque configuration with near-zero porosity and an open state with approximately 30% porosity, progressively revealing the “TriHex” lettering beneath the assembly (Fig. 4h). This reversible switching between opaque and open states, driven purely by mechanical actuation of identical units, suggests potential applications in morphing apertures, adaptive shading surfaces, and reconfigurable optical filters.

Collective migration

Individual TriHex units cannot move as their threefold symmetry cancels out the forces generated by asymmetric shape-shifting, resulting in no net displacement. However, when the units are connected along their edges, the mechanical constraints coupled with the shape-shifting break this balance, thus enabling collective migration whose behavior is governed by the aggregate’s initial phases and configuration (see Supplementary Movie 6).

As illustrated in Fig. 5a, a symmetric configuration consists of three units where both upper and lower units attach to the central unit by connecting the vertices of the same types. When the central unit maintains a 25% phase advance relative to the upper and lower units, the system achieves an efficient butterfly-style crawling motion (Fig. 5b). The migration trajectories from multiple experimental trials demonstrate a robust gait performance with an average velocity of 66.0 mm/min translationally. We then examine an asymmetric three-unit configuration where the upper unit connects to the central unit via vertices of different types while the lower unit connects via vertices of the same type (Fig. 5c, d). The results reveal that both the migration direction and the system’s rotation can be modulated by adjusting the lower unit’s phase between a 25% advance and a 25% delay relative to the central unit, while constraining the upper and central units to opposite phases.

Fig. 5. Collective migration and object transportation in TriHex systems.

a System configuration within a single period of a three-unit system with a symmetric configuration and an initial phase of [0.75, 0, 0.75]. b Migration snapshots and trajectory comparison of the gait shown in panel a. In the snapshots, dimmed units indicate initial positions and brightened units indicate final positions, and the orange line represents the migration trajectory. c Migration snapshots and trajectory comparison of a three-unit system with an initial phase of [0.75, 0, 0.5] in asymmetric configuration, presented in the same format as panel b. d Migration snapshots and trajectory comparison of a three-unit system with an initial phase of [0.25, 0, 0.5] in asymmetric configuration, presented in the same format as panel b. e Experimental and model-predicted trajectories corresponding to the configurations shown in panels b–d. Red lines represent the mean experimental trajectories, light red shaded regions represent individual experimental trials, and brown lines represent the model-predicted trajectories. f Schematic and experimental snapshots of the migration of a six-unit collective with a chain-like configuration. The orange line represents the trajectory of the system’s center of mass. In the schematic, gray units indicate static units without active shape-shifting. g Schematic and experimental snapshots of the migration of a twelve-unit collective with a circular configuration. The orange line represents the trajectory of the system’s center of mass. h Schematic of a nine-unit system comprising a gripper subsystem and an arm subsystem. i Object transportation demonstration. Orange lines represent trajectories of the cargo.

To elucidate the underlying physical mechanisms, we developed a dynamic model that agrees well with our experimental observations (see Supplementary Movie 6). The theoretical predictions for translational and rotational motions exhibit consistent directional trends with experimental data (Fig. 5e). However, the model predicts smaller rotational variations than what were observed in the experiments because it neglects the redistribution of frictional forces caused by adhesion-induced torques.

Due to the weak adhesive forces between the TriHex units, the stability of the aggregations decreases with increasing unit number, leading to more frequent edge detachments. To overcome this, configurations with coordinated gaits were designed to achieve efficient migration in extended multi-unit assemblies. The gait design minimizes relative rotation between adjacent units by jointly considering their connection polarity and phase difference, while also constraining the chain length to ensure sufficient traction force throughout the assembly. As illustrated in Fig. 5f, a chain-like configuration comprised of six units demonstrates stable translational motion with an average speed of 39.1 mm/min despite intermittent detachments during its migration cycles. Beyond chain-like arrangements, circular configurations also exhibit robust migration behaviors. Fig. 5g shows a twelve-unit circular assembly achieving stable migration with an average velocity of 32.3 mm/min.

To achieve more advanced capabilities such as object transportation, we created a multifunctional collective by combining a gripper for object manipulation with two arms for migration (see Supplementary Movie 7). In the gripper subsystem, when one unit serves as a base, adjacent units grasp and release objects through coordinated shape-shifting; while in the two-arm subsystem, two symmetric chains function as arms to propel the entire system (Fig. 5h). Fig. 5i shows this nine-unit system executing coordinated grasping and migration tasks. Initially, the three-unit gripper module undergoes synchronized shape-shifting to capture the target object. The six-unit arms subsystem then starts shape-shifting with a predefined gait, generating the necessary driving forces for migration while firmly grasping the object, thereby successfully transporting loads up to 50% of its own weight. This design highlights how integrated functionalities enable a collective of identical units to accomplish complex tasks.

Reconfiguration through edge switching

When two TriHex units are connected in a triangular configuration via vertices of the same type, they share three vertices along a certain side. During their active transitions to hexagonal shapes, the system encounters a choice in edge connectivity: either ${\mathcal{P}}_1{\mathcal{Q}}_1$ (state 1) or ${\mathcal{P}}_2{\mathcal{Q}}_1$ (state 2) remains connected while the other breaks, as illustrated in Fig. 6a. This edge-switching mechanism enables reconfiguration behaviors.

Fig. 6. Controlled reconfiguration in TriHex collectives via edge switching.

a Edge switching choices during shape-shifting from triangular to hexagonal configuration. State 1: edge ${\mathcal{P}}_1{\mathcal{Q}}_1$ remains connected. State 2: edge ${\mathcal{P}}_2{\mathcal{Q}}_1$ remains connected. b Schematic of a three-unit system consisting of a control unit, a base unit, and a target unit. c Switching probabilities for each configuration and schematics of six control modes ($\mathcal{M}1$--$\mathcal{M}6$) with varying control unit positions and orientations. Without control, switching yields 75% state 1 and 25% state 2. Yellow arrows indicate the rotation direction of the connecting edge between the base unit and the control unit. Purple arrows indicate the rotation direction of the control unit generated by its shape-shifting. d Calculated velocities for state 1 and state 2 in configuration $\mathcal{M}1$. Gray arrows indicate velocity direction and magnitude. Red lines indicate edges that remain connected. e Kinetic energy of state 1 and state 2 for each control mode. f Sequential snapshots of the reconfiguration process in a four-unit system. g Sequential snapshots of the reconfiguration process in a hub-and-spoke configuration.

Experiments reveal that edge switching occurs with approximately 25% probability under no-control conditions. The system prefers to maintain existing connections because the magnets on the original edge have achieved aligned positions, yielding a stronger magnetic force than the alternative edge, which needs time to align.

To understand and control edge switching, we investigated how an additional control robot affects the probability of reconfiguration. As shown in Fig. 6b, a control robot connects to a base robot, which in turn connects to a target robot capable of switching between state 1 and state 2. Fig. 6c presents six different control modes $\mathcal{M}1$ to $\mathcal{M}6$, with their corresponding probabilities of edge switching.

The control robot influences edge switching through three primary mechanisms. First, it rotates passively, influenced by the base robot’s shape-shifting. For instance, in configuration $\mathcal{M}2$, the base robot’s shape-shifting rotates the control robot counterclockwise, generating a clockwise torque on the base robot, which reduces the probability of switching to state 2. In contrast, in $\mathcal{M}4$, the clockwise torque increases the possibility of reaching state 2. Second, the control robot’s own shape-shifting generates additional torque to the system. Configuration $\mathcal{M}6$ demonstrates this effect clearly: the control robot’s shape-shifting produces a clockwise torque that, combined with the torque from the geometric constraints, significantly enhances the probability of reconfiguring to state 2. Additionally, configuration $\mathcal{M}5$ reveals a third factor: unbalanced magnetic forces. When one type $\mathcal{P}$ vertex is connected to more units than the other, its stronger magnetic force creates a preference for its edge to remain connected.

This control mechanism can be explained through an energy perspective. The kinetic energy of each state is calculated assuming the system evolves into that state. Based on the velocity constraints and the shape-shifting speed, the resultant velocity of each unit after the edge detachment can be determined. Fig. 6d shows an example of the velocity calculations of configuration $\mathcal{M}1$. The results in Fig. 6e indicate that larger energy differences between states correlate with higher edge-switching probabilities, confirming that the system transitions preferentially to states with lower kinetic energy. This correlation establishes that the kinetic energy difference serves as an effective predictor for the system’s reconfiguration behavior.

This edge switching mechanism enables complex reconfiguration behaviors in larger assemblies (see Supplementary Movie 8). Fig. 6f demonstrates a four-unit system moving a single unit along the collective’s edge through coordinated edge switching. This concept can be applied to more complex architectures, such as the hub-and-spoke configuration shown in Fig. 6g, where a central unit connects to three linear chains. Through active shape-shifting, the three outer units reconfigure simultaneously, bending the chains. These examples showcase predictable self-reconfiguration and controllable morphing of the collective through edge switching.

Discussion

Existing robotic collectives achieve rich collective behaviors through a range of design strategies. In mobile systems, such as Kilobots¹⁹, Granulobots¹², M-Blocks²⁴, Magbot collectives²⁷, and BOBbots¹⁰, individual units actively drive collective behaviors through self-propulsion or externally guided motion. These systems have demonstrated impressive capabilities, but mobility often introduces additional mechanical complexity and control overhead. Truss Links²⁰ occupy an intermediate regime, in which underactuated bar-shaped modules with prismatic joints achieve topology reconfiguration and self-repair through growing and healing by consuming modules of the same type without introducing new component types. Collective locomotion in this system is further achieved on a declined surface through coordinated shape-shifting of individual units.

A complementary approach relies on fully immobile units whose collective behavior largely emerges from local mechanical interactions. In such systems, the deformation mode and interaction geometry of each unit become key factors in determining collective functionality. Units with isotropic deformation, such as Particle Robots¹⁵, achieve simplicity and scalability but are limited by the symmetry of their deformation, which may restrict the richness of emergent behaviors. Anisotropic actuation breaks this symmetry and enables a broader range of collective phenomena: Smarticles¹⁷ demonstrate that stochastic mechanical interactions through individual anisotropic shape-shifting can achieve collective locomotion with simpler control, while material-like robots¹¹ show that controlling relative velocities between neighboring units can produce programmable stiffness, phase transitions and self-reconfiguration.

Against this background, our work demonstrates that immobile units with a simple single-actuator design can nevertheless achieve rich collective behaviors through anisotropic shape-shifting and local mechanical interactions, as summarized in Table 1. TriHex switches between two distinct polygonal geometries through a single-actuator underactuated mechanism, thereby changing how each unit mechanically interacts with its neighbors. This design unlocks five distinct collective functional modes within a unified platform: programmable stiffness, phase transitions, auxetic-like behavior, collective migration with object transportation and controlled reconfiguration. Notably, the auxetic-like behavior, which is rarely reported in robotic matter systems, extends the collective's behavioral space toward that of mechanical metamaterials. These results suggest that diverse emergent behaviors can arise from morphological simplicity by combining appropriate geometric design and passive mechanical coupling.

Table 1.

Comparison of TriHex with existing robotic matter systems

	Individual Property			Collective Behavior
System	Actuators	Shape-shifting	Mobility	PT	ST	CL	SR	SA
Kilobot 19	2	No	✓			✓	✓	✓
Magbot 27	1	No	✓	✓		✓	✓	✓
Granulobot 12	1	No	✓	✓	✓	✓	✓	✓
BOBbot 10	1	No	✓	✓		✓	✓	✓
M-Block 24	1	No	✓				✓	✓
Truss Link 20	2	Anisotropic				✓	✓	✓
Particle Robot 15	1	Isotropic				✓
Smarticle 17	2	Anisotropic				✓
Material-like 11	1	No		✓	✓		✓
TriHex (Ours)	1	Anisotropic		✓	✓	✓	✓

Collective behaviors: PT Phase Transition, ST Stiffness Tuning, CL Collective Locomotion, SR Self-Reconfiguration, SA Self-Assembly. Checkmarks (✓) denote capability for each function.

Yet this simplicity comes with trade-offs. The loosely coupled edges that improve reconfiguration flexibility compromise mechanical stability, leading to unstable motions on high-friction surfaces and occasional edge detachments. Similarly, the tendon mechanism composed of springs and strings preserves design simplicity but limits force transmission among the units. Together, these limitations make the system difficult to scale up.

Currently, the TriHex collective requires manual reconfiguration due to its limited reconfiguration capability, making it harder to switch between behaviors. Moreover, unexpected reconfigurations can occur during migration due to magnetic connection failures. However, this instability holds potential for adaptability, as spontaneous reconfigurations induced by mechanical stress may enable the collective to navigate complex terrain or recover from partial disassembly.

Looking forward, several research directions could enhance the system’s capabilities while preserving design simplicity. Hardware improvements such as stronger magnetic couplings, enlarged contact areas, or variable-stiffness boundary springs could extend operational scale and robustness. In particular, replacing the magnetic coupling with permanent connections could further enhance structural stability for large-scale deployment, such as building metamaterials. Additionally, the current porosity range is constrained by the design, which requires the TriHex unit to maintain a constant area across both hexagonal and triangular configurations; relaxing this constraint through alternative actuation mechanisms or geometric designs could significantly expand the achievable porosity range. Incorporating sensors such as photodetectors^11,15 would enable the system to respond to environmental stimuli, and distance or proximity sensors could further allow units to sense the relative positions of neighboring units, supporting richer collective interactions. Besides, advanced computational models that integrate coupled active-passive mechanics may uncover additional emergent behaviors and guide the optimization of the phase coordination strategies.

Methods

Fabrication and control of TriHex units

Each TriHex unit is driven by a single motor through an underactuated tendon-driven mechanism, employing nylon strings (diameter: 0.5 mm) and springs (wire diameter: 0.4 mm; length: 30 mm for type $\mathcal{P}$ vertices, 45 mm for type $\mathcal{Q}$ vertices). To maintain a constant area across both shapes while accommodating different displacement requirements, the spool layers feature distinct diameters with a ratio of $\sqrt{2}$. The resulting geometries yield edge lengths of 124.7 mm (triangular) and 50.9 mm (hexagonal), both maintaining an area of 6734.2 mm². Each assembled TriHex unit has a mass of 133 ± 5 g.

Twelve neodymium sphere magnets (diameter: 4 mm) embedded in the vertices and edge panels enable loosely coupled inter-robot magnetic interactions. The sphere magnets are free to rotate within their cavities, allowing them to self-align upon mutual approach with an average connecting force of 1.3 N, and enabling the Hall sensor to detect changes in the local magnetic field based on magnet orientation. The edge panels feature a specialized dual-cavity design: an upper cavity houses a sphere magnet while a lower cavity contains a disc magnet (diameter: 2 mm; thickness: 1 mm), as shown in Fig. 2d. When robots are physically separated, the disc magnet constrains the sphere magnet to align vertically, keeping the magnetic field parallel to the Hall sensor (Texas Instruments DRV5055A1QLPG) and ensuring a stable baseline reading. Upon edge-to-edge contact, inter-sphere magnetic attraction dominates, reorienting the poles horizontally to facilitate edge attachment, at which point the magnetic field becomes perpendicular to the Hall sensor and generates a significant deviation from the baseline (Fig. 2e). Since the magnetic pole facing the Hall sensor is not predetermined upon connection, this deviation may be either higher or lower than the baseline. A 3 mm acrylic base plate is attached to each edge to create an isotropic contact plane. Plastic hemispheres (diameter: 6 mm) are mounted at the bottom of each vertex. All structural components are fabricated via 3D printing: the edge panels and the top plate are printed in polylactic acid (PLA), while all other components are printed in a photosensitive resin. With this design, each unit experiences an average sliding friction force of 0.19 N when translating on the Teflon surface. The detailed interaction forces between units, including the magnetic attachment force between vertices, the restoring force of the boundary springs, and the force-displacement relationship of the vertex springs, are characterized and provided in the Supplementary Information 1.1.

Each TriHex unit integrates a microcontroller (Seeeduino XIAO SAMD21) for onboard control and sensor processing. LoRa modules (ALIENTEK ATK-LORA-01) enable command broadcasting and status checking across the collective. A slot-based addressing scheme assigns unique positional indices to each robot, enabling point-to-point communication within the decentralized system (details presented in the Supplementary Information 1.3). The electronics system comprises a 3.7 V lithium-ion battery, a motor driver board, and a DC gear motor with encoder (CHIHAI MOTOR GM12N20). The motor speed is controlled by pulse-width modulation (PWM) signals from the microcontroller. A proportional-derivative (PD) control based on the feedback from the encoder is implemented to achieve consistent shape-shifting cycles across all robots. Detailed control strategies are presented in the Supplementary Information Section 1.2.

Dynamic modeling of shape-shifting-induced migration

To simplify the shape-shifting process for analytical calculations, we model the shape transformation as a two-bar linkage (Fig. 7a). We define O as the geometric center of the robot, $O^{\prime }$ as the equivalent rotation center, and N as the midpoint of the edge. The length of segment $O{O}^{\prime }$ is denoted as d and the length of segment $N{O}^{\prime }$ as l. The angles of $O{O}^{\prime }$ and $N{O}^{\prime }$ with respect to the fixed horizontal axis of the global coordinate system are denoted as α and β respectively.  The angle between NO' and the corresponding edge of the robot remains constant, whereas the relative angle between the two segments varies with the degree of shape transformation of the robot. Detailed information is presented in the Supplementary Information Section 5.1.

Fig. 7. Modeling settings and results.

a Geometric parameters used in the approximation of shape-shifting. b Schematics of shape-shifting in the two-bar linkage model. c Modeling results of migration trajactory under phase coordinates [0.75, 0, 0.5] and [0.25, 0, 0.5]. Red lines represent the trajectories of the center of mass. d Experiment and modeling results of angle displacement under phase coordinates [0.75, 0, 0.75], [0.75, 0, 0.5] and [0.25, 0, 0.5].

The connection of the robots can be seen as bars that connect, with the connection orientation determined by each robot's topological constraint. For a three-robot system in a chain-like configuration, the collective can be modeled as an eight-bar linkage. We define $\mathcal{R}$ as the set of robots in the system, $\mathcal{C}$ as the set of kinematic constraints between connected robots, and $\mathcal{P}\left(j\right)$ as the ordered set of bars contributing to the position of robot j (j = 1, 2, 3).

The position vector of robot j is given by:

$${\mathbf{p}}_j={\sum }_{k\in \mathcal{P}\left(j\right)}\left(l_k\widehat{\mathbf{r}}\left({\alpha }_k\right)+d_k\widehat{\mathbf{r}}\left({\beta }_k\right)\right),$$ 1

where $\widehat{\mathbf{r}}\left(\theta \right)$ denotes the unit vector at angle θ.

The corresponding velocity vector is:

$${\stackrel{°}{\mathbf{p}}}_j={\sum }_{k\in \mathcal{P}\left(j\right)}\left(l_k{\stackrel{°}{\alpha }}_k{\widehat{\mathbf{r}}}^{\perp }\left({\alpha }_k\right)+d_k{\stackrel{°}{\beta }}_k{\widehat{\mathbf{r}}}^{\perp }\left({\beta }_k\right)\right),$$ 2

where ${\widehat{\mathbf{r}}}^{\perp }\left(\theta \right)$ denotes the unit vector perpendicular to $\widehat{\mathbf{r}}\left(\theta \right)$, obtained by a 90^∘ counterclockwise rotation.

The multi-robot system maintains three types of constraints:

(1) Intra-Robot Constraints:

$${\stackrel{°}{\beta }}_j-{\stackrel{°}{\alpha }}_j={\omega }_j\left(t\right),j\in \mathcal{R}$$ 3

where ω_j(t) represents the prescribed relative angular motion that is a function of the shape-shifting.

(2) Inter-Robot Constraints:

$${\stackrel{°}{\alpha }}_j-{\stackrel{°}{\alpha }}_k=0,\left(j,k\right)\in \mathcal{C}$$ 4

where $\mathcal{C}$ denotes the set of connected robot pairs.

(3) Topological Constraints:

$${\gamma }_j\cdot \left(\pi +{\beta }_j-{\alpha }_j\right)={\gamma }_k\cdot \left(\pi +{\beta }_k-{\alpha }_k\right),\left(j,k\right)\in \mathcal{C}$$ 5

where γ ∈ { − 1, 1} defines the connection orientation of each robot. As shown in Fig. 7b, edge P_iQ_i has γ = 1, and P_i+1Q_i has γ = − 1.

For this system, the Lagrangian equation of motion is:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \stackrel{°}{\mathbf{q}}}\right)-\frac{\partial L}{\partial \mathbf{q}}={\mathbf{Q}}_{\mathrm{fric}}+{\mathbf{J}}^{\top }\mathbf{\lambda }$$ 6

Here, L is the Lagrangian, q = [x, y, θ]^⊤ are the generalized coordinates (position and orientation), Q_fric represents the generalized friction forces, J is the constraint Jacobian derived from $\mathbf{J}\stackrel{°}{\mathbf{q}}=\mathbf{0}$, and λ are Lagrange multipliers that enforce the constraints.

The robot has a rounded bottom surface. For analysis, we assume the mass is uniformly distributed across the contact region. When the TriHex unit slides on the surface, the generalized friction forces are approximated using a discrete summation method. The contact surface is divided into a finite number of discrete contact points, and the friction contribution of each point is computed individually.

The set of contact points across all robots is denoted by r_ij, where $i\in \mathcal{R}$ indexes the robots and j = 1, …, N indexes the contact points on each robot, with corresponding friction forces f_ij acting at each point. The total generalized friction force ${\mathbf{Q}}_{\mathrm{fric}}\in {\mathbb{R}}^n$ is then calculated as:

$${\mathbf{Q}}_{\mathrm{fric}}={\sum }_{i\in \mathcal{R}}{\sum }_{j=1}^N{\mathbf{J}}_{ij}^{\top }{\mathbf{f}}_{ij},$$ 7

where ${\mathbf{J}}_{ij}=\partial {\mathbf{r}}_{ij}/\partial \mathbf{q}\in {\mathbb{R}}^{2\times n}$ is the Jacobian matrix relating the position of the i-th robot’s j-th contact point to the generalized coordinates $\mathbf{q}\in {\mathbb{R}}^n$, and ${\mathbf{f}}_{ij}\in {\mathbb{R}}^2$ is the friction force at that contact point.

For each contact point, the position is given by:

$${\mathbf{r}}_{ij}={\mathbf{p}}_i+{\mathbf{d}}_{ij}\left(\mathbf{q}\right),$$ 8

where ${\mathbf{p}}_i\in {\mathbb{R}}^2$ is the position of the i-th robot’s center, and ${\mathbf{d}}_{ij}\left(\mathbf{q}\right)\in {\mathbb{R}}^2$ is the offset vector from the robot’s center to its j-th contact point.

The Jacobian matrix ${\mathbf{J}}_{ij}\in {\mathbb{R}}^{2\times n}$ is obtained by:

$${\mathbf{J}}_{ij}=\frac{\partial {\mathbf{r}}_{ij}}{\partial \mathbf{q}}=\frac{\partial {\mathbf{p}}_i}{\partial \mathbf{q}}+\frac{\partial {\mathbf{d}}_{ij}}{\partial \mathbf{q}}.$$ 9

The offset vector d_ij(q) points from robot i’s center to a contact point on its perimeter and is time-varying due to the robot’s orientation θ_i(q).

The partial derivative of d_ij with respect to q follows the chain rule:

$$\frac{\partial {\mathbf{d}}_{ij}}{\partial \mathbf{q}}=\frac{\partial {\theta }_i}{\partial \mathbf{q}}\cdot \frac{\mathrm{d}{\mathbf{d}}_{ij}}{\mathrm{d}{\theta }_i}=\frac{\partial {\theta }_i}{\partial \mathbf{q}}\cdot \left({\mathbf{\omega }}_i\times {\mathbf{d}}_{ij}\right),$$ 10

where θ_i is the orientation angle of robot i, $\frac{\partial {\theta }_i}{\partial \mathbf{q}}$ expresses how robot i’s rotation depends on the generalized coordinates, ω_i is the angular velocity of robot i, and ω_i × d_ij gives the instantaneous velocity of the contact point due to rotation.

By combining the Lagrangian equation with the constraint equations, we solve the resulting ordinary differential equations. The predictions of three-robot collective migration from the dynamic model are shown in Fig. 5c and Fig. 7c, d.

Mechanical characterization experiment

The mechanical characterizations were performed using a specially designed testing system (see the Supplementary Information Section 2.1 for details). In the compression tests, randomly selected TriHex units were placed in a tessellated configuration, whose stress-strain curve was obtained under a loading speed of 5 mm/s. Its cross-sectional area was calculated based on the original width of the system plus the height of the connecting edges. In the creep tests, a maximum force of 3 N was applied and maintained for 700 s. Since the assembly was composed of active shape-shifting units, the system’s resistance force varied periodically. Therefore, the testing machine compressed the robot system unidirectionally at a maximum speed of 3.75 mm/s.

Chain-constrained morphing experiment for fluidity characterization

To characterize how the shape-shifting of individual units affects system viscosity in the liquid-like state, chain-constrained morphing experiments were conducted where robots were enclosed by a chain loop. Under uneven stress distribution imposed by the chain constraint, systems with lower viscosity distribute stress faster through reconfigurations.

In these experiments, fourteen TriHex units were initially arranged into a long oval configuration. The system’s circularity, calculated using the formula 4π × area/perimeter², was tracked continuously for ten minutes.

The system’s fluidity was controlled through an algorithm (Algorithm 1) that modulates shape-shifting through three distinct strategies: random shape-shifting with an average period of 60 s/cycle, random shape-shifting with an average period of 30 s/cycle, and compression-based shape-shifting with an average period of 60 s/cycle. Each control cycle lasts T = 3 s, during which each unit independently decides whether to execute the next shape-shifting step based on the selected strategy.

The robots operate in one of two modes: random or compression-based. In random mode, each robot generates a random waiting time within the three-second cycle before shape-shifting. In the compression-based mode, each robot first selects a sensing period and monitors its six Hall sensors during this interval. The standard deviation of the sensor readings indicates the asymmetry in local compression. When the standard deviation exceeds the threshold σ_th, shape-shifting is triggered immediately, responding to a high stress asymmetry; otherwise, it is executed at the end of the sensing period.

Algorithm 1

Controllable fluidity algorithm

Require Control cycle duration T = 3 s, minimum sensing period T_min = 1 s, standard deviation threshold σ_th = 10, shape-shifting probability p_def, operation mode mode ∈ {random, Compression-based}

while True do

  r ← Uniform(0, 1)

  if r > p_def  then

   Wait for duration T

  else if mode = random then

   $t_{\mathrm{wait}}\leftarrow \mathrm{Uniform}\left(T_{\min },T\right)$

   Wait for duration t_wait

   ExecuteShapeShift()

   Wait for duration T − t_wait

  else if mode = Compression-based then

   $t_{\mathrm{sense}}\leftarrow \mathrm{Uniform}\left(T_{\min },T\right)$

   shifted ← False

   Wait for duration $T_{\min }$

   while t < t_sense do

    H_i ← ReadHallSensor(i) for i = 1, . . . , 6

    D_i ← ∣H_i − 512∣ for i = 1, . . . , 6

    $\sigma \leftarrow \sqrt{\frac{1}{6}{\sum }_{i=1}^6{\left(D_i-\overline{D}\right)}^2}$

    if not shifted and σ > σ_th then

     ExecuteShapeShift()

     shifted ← True

    end if

   end while

   if not shifted then

    ExecuteShapeShift()

   end if

   Wait for duration T − t_sense

  end If

end while

Probability measurement of edge switching

We investigated the selective switching behavior between two connection states during shape transitions using a base two-robot system and an extended three-robot system with an additional control robot. Six different control modes ($\mathcal{M}1$–$\mathcal{M}6$) were tested, with additional configurations detailed in the Supplementary Information Section 5.2.

For each test, the robots were positioned according to the specified configuration. Both the target and base robots were initialized in the regular hexagonal state, and the control robot was set to the phase under test. The whole system began shifting its shapes simultaneously. Each configuration was tested 20 times on a 3-mm-thick Teflon board with a shape-shifting period of 5.2 s. The final connection state was recorded after five shape-shifting cycles.

To ensure binary switching behavior and prevent complex multi-robot connections, some sphere magnets at the vertices were selectively removed from the system. Robots were randomly selected to represent average system behavior. The observed switching probability deviated from theoretical predictions due to several factors: (1) magnetic alignment requires time, creating a bias toward the existing configuration; (2) the strength of the magnets varies among vertices.

Collective migration experiment

This experiment demonstrated the collective migration capability of the robot system on a 3 mm Teflon board. The assembly consisted of multiple robots initially connected in a chain-like configuration with a programmed gait that maintains their connectivity through coordinated shape-shifting.

The robot phases were pre-synchronized through communication protocols, with shape-shifting initiated by broadcast commands. The shape-shifting period was set to 8.2 s, slightly longer than the reconfiguration period to minimize the effect of vibration noises. Each test ran for 20 cycles (164 s in total).

Locomotion trajectories were obtained by visually tracking markers on the top of the robots using the overhead camera footage. The coordinate system was defined with the connection line between the robots with red and green markers as reference, with the red-to-green direction designated as the positive y direction.

Variations between experiments primarily resulted from slight differences in mass distribution among robots. Additionally, the relatively high center of gravity of each robot affected the stability of collective migration and the consistency of trajectories.

Statistical analysis

Data are presented as mean ± standard deviation (SD) unless otherwise specified. The shaded regions in the figures represent the SD.

Author contributions

Conceptualization: Y.Z. and S.L. Prototype design and manufacturing: Y.Z. and S.L. Model and simulation: Y.Z. and C.H. Data analysis and visualization: Y.Z. and C.H. Supervision: S.L. Writing—original draft: Y.Z. and C.H. Writing—review and editing: S.L., Y.Z., and C.H.

Peer review

Peer review information

Nature Communications thanks Jie Zhang and Xingxing Ke for their contribution to the peer review of this work. A peer review file is available.

Funding

This research is sponsored by Tsinghua-Toyota Joint Research Fund (grant number 20253930024).

Data availability

All data needed to evaluate the conclusions in this paper are present either in the main text or in the Supplementary Information. The raw data generated in this study are available in the Zenodo repository⁵⁴.

Code availability

The source code used for simulations and figure generation is available in the Zenodo repository⁵⁴.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-76768-x.