We use this model in numerical simulations of stochastic active particles, where we observe cooperative transport when particles have a negative curvity, corroborating the experimental observations (see Fig. 3 and Supplementary Movies 1,2,5, and 6). Surprisingly, in both experiments and simulations the transport propensity increases with increasing payload size (see Fig. 4). We analytically derive a condition for transport, which depends on the geometrical curvature of the payload as well as the intrinsic curvity of self-propelled particles. The condition is consistent in both simulations and experiments, offering a geometrical criterion for cooperative transport.
Stochastic self-propelled robots were built following a modified bristle bot design. A robot (sizing 5-6 cm in diameter) is driven by two vibration motors mounted on a tripod with one stiff leg and a pair of asymmetric soft legs (see Fig. 1 and SI Section I). Vibrations induce noisy forward motion which defines the robot's heading, (see Fig. 1A, C). In the experiments, a large circular passive payload was placed in the middle of a symmetrical arrangement of robots, which were then turned on, setting the swarm into motion. With the traditional design (soft legs placed at the back), robots sporadically push the passive payload which exhibits Brownian-like motion -- with each collision, robots turn away from the payload (Fig. 1C-E, and Supplementary Movie 5). In contrast, when the soft legs are placed at the front, the swarm spontaneously breaks symmetry and propels the payload in a near ballistic trajectory (see Fig. 1A, B, E and Supplementary Movie 1). Here, with each collision, robots tend to turn into the payload and progressively push it until reaching the perimeter of the arena (150 cm diameter). The cooperative transport emerges autonomously, and does not require an external, directional cue, nor manual pre-arrangement. We further find the effect to increase with payload diameter (2a = 7-32 cm), swarm size (N = 1 - 53 robots) in both a custom-made and a modified commercial multi-robot platform (see Fig. 4 and SI Section IB 4).
High-speed video imaging offered insight into the origin of force-alignment (see Fig. 2, Supplementary Movies 3, 4, and SI Section I A 2). While moving, the robot's stiff and soft legs interact differently with the ground -- the stiff leg has higher restitution, and spends a longer duration in the air, whereas the softer legs show only moderate hopping. This difference leads to a differential fore-aft friction which lies at the heart of the mechanical origin of the force-alignment -- robots with soft legs at the back align with an external force (descend downhill), whereas robots with soft legs at the front align against the force (ascend uphill). The difference between the two designs is revealed in the presence of an external force or boundaries. In the absence of such, their dynamics are qualitatively indistinguishable. Force-alignment is generic to self-propelled particles regardless of the locomotion mechanism and should be expected in general both on the macroscopic and the microscopic scale. In the next section, we derive the mechanical origin of force-alignment in granular hoppers from first principles.
The microscopic origin of force-alignment is revealed by considering the instantaneous acceleration, , of a vibrationally propelled robot under an external body force, , acting in the plane of motion. Below we outline the coarse-graining of the rapid hopping dynamics, and derive effective equations of motion of a granular active particle dominated by inertia and dry friction. The equations only require the mean value of the different parameters, with no particular significance to the order of the three phases. We will reproduce previous work that assumed overdamped dynamics, and force alignment based on symmetry, however we will show that their effective parameters (mobility and force-alignment) are controlled by inertial quantities (mass and moment of inertia).
We consider the motion to have three characteristic phases: I -- rest, II -- aerial, and III -- pivot, with a mean overall duration T (see Fig. 2). A robot starts at rest (I) with all contact points on the ground, thereby the external force is perfectly balanced by static friction and there is no motion (). The robot then jumps forward (along ) with an instantaneous horizontal speed of v, having a typical time aloft of τ. While in the aerial phase (II) the robot accelerates by the external force . For simplicity, we treat contact with the substrate as having perfect static friction, and when the robot lands it loses all its momentum. Combining phases I and II results in a coarse-grained velocity proportional to the sum of active velocity and the external force (Eq. (1)), where the nominal speed is and the mobility is . This formalism is similar to Drude's model that leads to linear Ohm's conductivity where charge carriers in a conductor lose their momentum during collisions. Inevitably, contact friction is not equal on all legs, and empirically we find that the robot spends a longer duration, τ, on the softer legs acting as a pivot. During the pivot phase (III), static friction with the contacting feet restrict linear motion (), however, the robot can rotate, as it experiences a torque . The torque is the result of the external force acting on the center of mass which in general is displaced from the axis of rotation, , where I is the moment of inertia around the rotation axis, and the lever arm, δ, is the offset of the center of mass from the axis along the orientation vector (see Fig. 2B, C). The offset, δ, can be positive or negative, respectively resulting in positive or negative force-alignment.
Phase III gives the microscopic basis for force-alignment on which our model rests. Combining the instantaneous dynamics of phases I-III results in coarse-grained equations of motion
where κ acts as an effective charge-like parameter of an active particle, and defined from the microsopic properties of the hopper
Being a key result of our model we name κ curvity, as it has units of curvature, and stems from the particle's activity (for details see SI Section III). Similarly to an electric charge, κ is signed, and its sign is controlled by an internal symmetry. The sign and magnitude of the curvity follow δ, the signed offset of the center of mass.
Despite not having any formal viscus drag, Eq. (1) has the same structure as the equations used to describe drag-dominated micro-swimmers acting in the low Reynolds number regime, where inertial quantities are justifiably neglected, and velocity is proportional to the external force through a mobility constant (μ). Previous work on granular active matter already assumed that dry macroscopic objects can be described using overdamped dynamics. The derivation above shows that while the equations of motion take the same structure, they are controlled by effective parameters that directly depend on inertial quantities, like mass (m) and moment of inertia (I). Equations (1) is also found in the extensively used model of Active Brownian Particles (ABP). Our derivation shows that the rotation of the active particle results from an external force, and does not require self-propulsion. Previous work used similar equations to describe particles that undergo velocity alignment as self-aligning active particles (SAAP). Originally introduced to offer a mechanism for flocking whereby a bird's heading tends to align on its velocity. There, and in subsequent work the alignment strength described using positive quantities such as 'relaxation-time' or 'alignment rate', and more recently, 'alignment-length'. That description successfully captured important collective behavior such as flocking (positive curvity). Since the alignment parameter can be negative, it is more naturally treated as a curvature (signed inverse length). For negative curvity (κ < 0), there is no-self alignment: when subjected to a strong force (μf > v), a particle's heading will settle against its velocity (Eqs. (1), (2)). Moreover, the microscopic derivation shows that the curvity does not depend on self-propulsion (Eq. (3)): an external force can rotate an active particle even at zero nominal speed (v = v = 0, see SI). Therefore we call particles which follow Eqs. (1),(2) Force-Aligning Active Brownian Particles (FAABP), as their alignment stems from the external force (rather than self-propulsion).
An important consequence of the microscopic model presented here is to identify that κ is signed and to offer a powerful design rule. For example, in the point mass limit (I = mδ), the curvity is inversely proportional to the offset κ ∝ 1/δ, and when the offset is negative (center of mass is behind the soft legs, δ < 0), robots turn against an external force. Describing the robots' mass distribution as a disc-shaped core (battery and electronics) embedded in a ring shaped frame (3D printed chassis), shows that the mechanical model predicts the measured curvity to within a factor of 1.8 (see Movies 3-4, and Supplementary Figs. 1 and 2, and Section I A 4 in the SI). The curvity can be also computed for an arbitrary shape (e.g., rod-like) and mass distribution, by evaluating the robot's moment of inertia relative to the pivot axis, as the balance of the increased lever arm (κ ∝ δ), with the increased moment of inertia (κ ∝ 1/I). The derivation above also shows that the effective parameters (v, μ, and κ) are not independent of one another, and how they are linked by the robot's inertial properties. It is interesting to note that previous work showed that some ciliated and flagellated micro-swimmers tend to swim up. Specifically E.coli and Paramecium showed an increasing radius of curvature of their trajectories with their own size, even on the micro-scale. Super diffusion of a passive particle was even observed in bath of E.Coli. When combined, these suggest a potential extension of the collective dynamics described here to the domain of micro-swimmers.
We tested numerically swarms of FAABPs by adding orientational noise to Eq. (2) and short-range repulsion, in a simulation engine using 5th-order Runge-Kutta integration. The orientational noise has zero mean, , with a Gaussian distribution of width 〈ξ 〉 = 2ΔtkT (Δt is the simulation time step, and kT sets the magnitude of the noise, thereby tuning their persistence length, l), with particles modeled as soft discs of radius b (see SI Section II for details).
Simulated dynamics of individual FAABPs with a constant force reproduce experimental trajectories of robots moving on an inclined plane (see Fig. 2D-F). FAABPs with a positive curvity (κ > 0) turn in the direction of the force similarly to robots having their center of mass in front of their soft legs (δ > 0), whereas FAABPs with a negative curvity (κ < 0) turn anti-parallel and move against the external force, like the robots with their center of mass behind the soft legs (δ < 0). The singular, zero curvity FAABP (κ = 0), is simply an ABP -- its heading is unaffected as it drifts in the direction of the external force (Fig. 2F).
We tested the effect of a passive particle of radius a on a randomly distributed swarm of FAABPs ranging in sizes between N ∈ [1, 1000], with negative curvities, κ < 0. We observe that after a short transient where the swarm homogeneously accumulates around the passive particle, symmetry is spontaneously broken and particles form an active wake on one side that propels the passive payload (see Fig. 3 and Supplementary Movie 2). In line with the experimental findings, the transport emerges autonomously, despite the initial isotropic random arrangement of the active particles. The passive particle shows elongated trajectories, larger than its size, and larger than the simulation box (for periodic boundary conditions). The direction of transport is different from one run to the other, and the active wake is in a dynamic steady state, constantly exchanging the participating FAABPs. A similar effect is also observed in the non-periodic simulation, excluding the effect of the boundary. Transport is also observed when FAABPs are non-interacting (can pass through one another but not through the passive particle) excluding the role of Motility Induced Phase Transition. This means that cooperation emerges not by direct robot-to-robot interaction, but instead by a proxy -- the passive payload. Transport is completely absent for FAABPs with positive curvity (κ > 0) or for smaller payloads, where the passive particle only shows a diffusive trajectory (see Figs. 3C, D, 4, and Supplementary Movie 6).
Counter-intuitively, cooperative transport is enhanced with increasing payload radius, a. By contrast to thermal fluctuations, where a particle's diffusion decreases with its size, here we find that both the payload's speed (v) and persistence length (l) increase with its size, with an overall increased long term effective diffusion ( ∝ vl). Performing 116 experiments and over 1000 simulations (varying payload size, curvity, robot count, and orientational noise, see SI), we found that in both experiments and simulations, larger payloads are better transported, provided that the curvity of the active particles is sufficiently negative (see Fig. 4). In the experiments, the payload's weight is proportional to the payload's radius (m ∝ a, see SI Section I B 2), for a combined super-linear increase in mass transport. Trajectories of a larger payload in a swarm of robots with negative curvity show more persistent motion compared to that when the curvity is positive or the payload is smaller (Fig. 4A). Experimentally, there is a considerable increase in the average power-law of the mean square displacement of the payload when κa < -1 (see Fig. 4B). Exploring the κ-a phase space in simulation reveals two phases, with an order of magnitude increase in the mean speed of the payload (Fig. 4C). The phase boundary for both experiments and simulations lies at κa = - 1.
We next show that the condition for cooperative transport is geometrical and stems from the interplay of the active particles' curvity, κ, and the curvature of the circular passive particle, 1/a. Despite the deprecate dynamics underlying the numerical and experimental systems the condition for cooperative transport is identical: simulated particles are in the over-damped limit, with drag proportional to the particle's diameter, smooth self-propulsion, Gaussian orientational noise, and strictly two-dimensional motion, whereas in experiments objects are inertial, making an effective active granular gas, with solid friction with the ground and with one another, an intermittent vibrational self-propulsion, a non-Gaussian orientational noise, and quasi-two-dimensional hopping.
We start by modeling a circular payload as a repulsive, two-dimensional, radially symmetric potential fixed at the origin, , exerting a repulsive force on the active particles: (see Fig. 5A). Active particles interact with this circular obstacle following Eqs. (1) and (2). Γ(r) sets the radial profile of the force and is chosen such that the magnitude of the repulsive force at the payload's perimeter exactly balances the nominal speed of the active particle, Γ(r = a) = 1, effectively setting the payload's size. Inspired by previous work on repulsive particles, keeping the potential profile implicit (exponential decay, soft-core, Yukawa, etc.), makes the result below more general. Circular self-propelled particles in 2D have three degrees of freedom (see Fig. 5B): a radial and azimuthal position , and a heading relative to the x-axis, . The system has rotational symmetry and the dynamics depend only on the orientation of the heading relative to the center of the potential ψ ≡ θ - φ. Plugging the radial force term into the FAABPs' equations of motion (Eqs. (1), (2)) gives a dynamical system described by two non-linear coupled first-order differential equations:
(see SI Section IV A for a detailed derivation). At ψ = 0 the active particle points away from the payload, and at ψ = π, it fronts the payload. When the prefactor in Eq. (5) switches sign (), an active particle is effectively attracted to the repulsive potential (see Fig. 5C). Given the definition of , this can be satisfied when
The condition in Eq. (6) presents a geometrical criterion for cooperative transport: once the curvity is sufficiently negative, instead of being scattered away (ψ → 0), an active particle colliding with the obstacle re-orients sufficiently fast into the receding repulsive hill to continually push against the obstacle (ψ → π). The inequality in Eq. (6) is agnostic to whether the curvity or the curvature is negative, and could be applied more generally. Even if the force alignment is non-negative (positive or zero), a self-propelled particle could display effective attraction to a concave boundary, provided that its curvature is sufficiently negative (1/a < 0). This has been previously observed in self-propelled particles interacting with a concave obstacle, in a single confined active particle interacting with the inner concave walls of a harmonic trap, and more recently, in active particles interacting with one another.
Phase portraits of the dynamical systems above (κa = 1) and below (κa = -2) the transition, show a local topological change at the fixed point where the active particle is pushing against the payload, r/a = 1, ψ = π (see Fig. 5E, F). When Eq. (6) is satisfied, the dynamical system undergoes a bifurcation, and the saddle point turns into a linearly stable sink that attracts active particles (see SI Section IV A 2). In both experiments and simulations, this effective attraction is manifested in an enhanced kissing number N of robots touching the payload (see Figs. 1, 3, 4 and Supplementary Movies 1, 2 and 5, 6), with an increased linear filling fraction, λ ≡ Nb/a (see Fig. 5D). In a system that meets the condition in Eq. (6), the effective attraction and the resulting cooperative transport are robust over a range of orientational noises (see Fig. 6).
We find that the persistence of the payload (l) increases with the number of active particles, and can even surpass the persistence of the active particles themselves (l, Fig. 7A, B). This effect becomes clear when measuring the amplification of the persistence length (l/l): with increasing interaction (κa more negative), the amplification increases faster with increasing swarm size (Fig. 7D). Moreover, at a given interaction strength, the amplified persistence increases super-linearly with the number of robots (N), a hallmark of a cooperative swarm. Overall, both the speed (Figs. 4, 6), and the persistence length (Fig. 7) of the payload increase with its size (a). The effective equations of motions derived above can explain this pronounced effect.
Once the payload starts moving, the dynamics are no longer isotropic. A velocity fluctuation driving the passive particle to the right, (w.l.o.g.), spontaneously breaks symmetry and introduces an explicit dependence on the azimuthal coordinate in Eqs. (4) and (5), as well as an additional dynamical equation for the azimuth itself, leading to a dynamical system with three variables:
(see SI Section IV B 1 for derivation). In the isotropic case (Eqs. (4) and (5)), there was a fixed point for any combination of the heading (θ) and azimuth (φ) that satisfy: ψ ≡ θ - φ = π. When the payload is already moving, this is no longer the case. There are only two fixed points for the azimuthal degree of freedom, either pushing against the payload's motion (φ = 0, unstable) or along its motion (φ = π, stable). This means that when a group is transiting a payload, it preferentially recruits further individuals to push in the same direction, facilitating cooperative transport. Since the direction of the payload guides recruitment, a single 'leader' could, in theory, synchronize the group's behavior, by adjusting the payload's movement. Whereas in previous work, the manual pre-arrangement of a small number of robots dictated the direction of transport. Here, the recruitment effect emerges directly from symmetry-breaking, supporting cooperation at scale.