Stealthy movements and concealed swarms of swimming micro-robots

Swimming micro-robots capable of navigating through fluid environments are at the forefront of minimally invasive therapeutics and theranostics.¹ They hold great promise for a wide range of biomedical applications including targeted drug delivery, micro-surgery, remote sensing, and localized diagnostics.^1–4 The past decade has seen a great leap forward in science and engineering of these miniaturized untethered robots.⁵ Particularly, remarkable progress has been made toward exploring various propulsion mechanisms,^6–8 design and fabrication approaches,⁹ imaging technologies for real-time motion tracking,¹⁰ and manipulation techniques for navigation and motion control.^11–14 

However, optimal strategies in swarm control remain largely unexplored for swimming micro-robots.⁷ As a result, little is understood about their potential ability as a group to optimize their fitness and functionality. A few recent studies have only shown a glimpse of such potentials in the realm of micro-scale swimmers. For instance, actively controlled cooperation between artificial micro-swimmers has been reported to significantly improve (both the capacity and precision of) micro-manipulation and cargo transport.¹⁵ It has also been recently shown¹⁶ that a pair of interacting micro-swimmers can boost each other’s swimming speed through the ambient fluid. This observation (termed “hydrodynamic slingshot effect”) implies that by forming a swarm, swimming micro-robots can collaborate and travel faster as a group than single individuals. Now, the more intriguing question is whether by forming a swarm, swimmers are also able to smartly cancel out each other’s disturbing effects in the fluid environment. In other words, is it possible to form a stealth swarm minimally disturbing the ambient fluid? If so, to what extent can such cooperation between the agents be effective in stifling the swarm’s hydrodynamic signature?

Here, we unveil synergistic cooperation of micro-swimmers (in suppressing one another’s disturbing flows) that leads to the formation of stealth swarms. We refer to this mode of swarming as the concealed mode, which can reduce the swarm’s net induced disturbances by more than 99% (or 50%) in three-dimensional (3D) (or two-dimensional) movements. This is equivalent to quenching the swarm’s hydrodynamic signature (and, thus, shrinking its associated detection region) by an order of magnitude in range. Through numerical experiments, we then demonstrate how such a concealed swarm can actively gather around a favorite spot, point toward a target, or track a desired trajectory in space, while minimally disturbing the surrounding environment.

Dynamics of the incompressible flow around swimming objects is governed by the Navier–Stokes equations,

subject to boundary conditions imposed by their body deformations. Here, ρ and η are the density and dynamic viscosity of the surrounding fluid, P denotes the pressure field, u is the velocity field, and F represents the external body force per unit volume. The relative importance of inertial to viscous effects can also be quantified by the Reynolds number, Re = ρUL/η, where U and L denote the characteristic velocity and length, respectively. For micro-scale swimmers (also known as micro-swimmers) swimming in water (ρ ≈ 10³ kg/m³ and η ≈ 10⁻³ Pa s), the corresponding Reynolds number is always very small (i.e., Re ≪ 1). Common examples include: (i) typical bacteria, such as Escherichia coli, with lengths of ∼1 μm–10 μm and a swimming speed of ∼10 μm s⁻¹,¹⁷ for which the Reynolds number is ∼10⁻⁵–10⁻⁴ when swimming in water; (ii) the green algae Chlamydomonas reinhardtii with a characteristic length of L ∼ 10 μm and a swimming speed of U ∼ 100 μm s⁻¹,¹⁸ which result in the Reynolds number Re ∼ 10⁻³. Thereby, it is appropriate to study micro-swimmers in the context of low Reynolds number regimes (Re ≪ 1), where the fluid inertia is negligibly small compared to the fluid viscosity, and the viscous diffusion dominates the fluid transport. The Navier–Stokes equations then simplify to the Stokes equation,

which has no explicit time-dependency. This, along with its linearity, makes the Stokes equation invariant under time-reversal. As a result, sequence of body deformations (or swimming strokes) that are reciprocal (i.e., invariant under time-reversal) do not generate a net motion at the Stokes regime. This means that typical swimming strategies used by larger organisms (e.g., fish, birds, or insects) are not effective at micro- and nano-meter scales. Therefore, motile micro-organisms have evolved alternative propulsion mechanisms to break the time-symmetry while retaining periodicity in time.¹⁹ Their swimming strategies are often based on drag anisotropy on a slender body in the Stokes regime and include cork-screw propulsion of bacteria, flexible-oar mechanism of spermatozoa, and asymmetric beats of bi-flagellate alga.¹⁹ 

These inherently available natural micro-swimmers not only inspire the design of fully synthetic micro-robots⁷ but also may be functionalized and directly used as steerable swimming micro-robots.⁸ Here, we are interested in flocks of such bio-inspired and bio-hybrid swimming micro-robots, where each agent is either a real micro-organism or meticulously synthesized to mimic one. Therefore, to model their induced disturbances, we will treat each individual swimmer as a swimming micro-organism.

In the Stokes regime, self-propelled buoyant micro-swimmers exert no net force and no net torque on the ambient fluid. Flagellated micro-organisms, for instance, use their flagella—flexible external appendages—to generate a net thrust and propel themselves through ambient fluids. This propulsive force—generated mainly owing to the drag anisotropy of slender filaments in the Stokes regime²⁰—is, however, balanced by the drag force acting on the cell body [see, e.g., Fig. 1(a)]. Hence, in the most general form, the far-field of the flow induced by each micro-swimmer can be well described by the flow of a force dipole. To be more precise, the flow of a force dipole composed of the trust force (which is generated by the swimmer’s propulsion mechanism) and the viscous drag acting on its body. Note that the model dipole is contractile for swimmers with front-mounted flagella (i.e., “pullers” such as C. reinhardtii) and extensile for those with rear-mounted flagella (i.e., “pushers” such as E. coli). Schematic representations of the force dipoles generated by archetypal puller and pusher swimming micro-organisms, as well as the direction of the induced flow fields, are shown in Fig. 1(a). This simple model has been validated and widely used in the literature.^21,22 In the case of E. coli bacteria, for example, the validity of this model has been further confirmed by comparing it to the flow field experimentally measured around an individual swimming cell.²³ 

Let us consider a model micro-swimmer swimming toward direction e, through an unbounded fluid domain. The disturbing flow induced by the swimmer can be modeled as the flow of a force dipole located at the instantaneous position of the swimmer (x₀). Thrust and drag forces of equal magnitude are exerted in opposite directions (±f₀e) to the ambient fluid at x₀ ± el/2, where the characteristic length l is on the order of swimmer dimensions. For each point force f exerted at point x_p in an infinite fluid domain, the governing equation will turn into

where $δr$ is the Dirac delta function. Equation (3) can be analytically solved in several ways,²⁴ and the resultant velocity field is known as Stokeslet,

where r_p = x − x_p and G is the corresponding Green’s function. A complete set of singularities in the Stokes regime can then be obtained²⁴ by taking derivatives of the fundamental solution presented in (4). The induced flow field of a model force dipole, ±f₀e, located at the instantaneous position of the swimmer (x₀), can, therefore, be mathematically expressed as

where r = x − x₀, for any generic point x in space. Note that the dipole strength, $D≈f0l$⁠, has a positive (negative) sign for pusher (puller) swimmers, and its value can be inferred from experimental measurements. For instance, the values of f₀ = 0.42 pN and l = 1.9 μm have been experimentally obtained²³ for E. coli, in agreement with resistive force theory¹⁷ and optical trap measurements.²⁵ 

Here, we use velocity scale U_s = f₀/8πηl, length scale L_s = l, and time scale T_s = L_s/U_s to non-dimensionalize the reported quantities. Therefore, a dimensionless disturbing flow induced by a micro-swimmer reads as

where c₀ = +1 (−1) for pushers (pullers) and bar signs denote dimensionless quantities. It is worth noting that the near-field of the flow induced by a micro-swimmer can be described more accurately via including an appropriately chosen combination of higher order terms from the multipole expansion.^26,27 However, here we are interested in the span of swimmers’ induced disturbances and their consequent detection region, for which the far-field of the flow is of primary interest.

To assess the optimality of various swarm arrangements (in stifling disturbing effects), one needs to first quantify the induced fluid disturbances. A measure of distortion (caused by a flock of swimmers to the ambient fluid) can be obtained²⁸ by directly computing the Mean Disturbing Flow-magnitude (MDF) over a surrounding ring (⁠$C$ ) of radius R, i.e.,

where $R¯=R/Ls$ and $u¯net$ is the overall dimensionless flow field induced by the flock. Due to linearity of the Stokes equation (2), the net disturbing flow (⁠$u¯net$⁠) is computed through the superposition of the flow fields (⁠$u¯SD$⁠) induced by individual swimmers forming the flock (i.e., $u¯net=∑i=1Nu¯SDi$⁠). Alternatively, one can quantify swarm’s induced fluid disturbances by computing the Area of the Detection Region (ADR), within which disturbances exceed a predefined threshold. More precisely, the detection region refers to the subset of the space, within which magnitude of the net induced disturbing flow (⁠$u¯net=unet/Us$⁠) exceeds a predefined threshold (⁠$ū$_th = u_th/U_s), i.e.,

which is also consistent with previous numerical studies on swimming micro-organisms.²⁹ The threshold value (u_th) can be tuned based on the characteristics of the specific problem of interest. For the system representing a prey swarm, as an example, it can be inferred from experimental observations on sensitivity of predators’ receptors in sensing flow signatures.

Here, we combine the described theoretical analysis with direct computations and non-linear optimization to perform a systematic parametric study on flocks of N ≥ 2 swimmers with a bottom-up approach. Specifically, we develop a general procedure to determine optimal swarming configurations and systematically investigate their significance in reducing the swarm’s induced fluid disturbances. We then present computational evidence demonstrating how such a concealed swarm can actively gather around a favorite spot, point toward a target, or track a desired trajectory, while minimally disturbing the ambient fluid. As a benchmark, here we consider planar arrangements/movements of pusher swimmers (say, E. coli bacteria) in an infinite fluid domain. Nevertheless, the reported concealed arrangements will be the same for pullers, and our study can be inherently extended to three-dimensional (3D) scenarios (see  Appendix A for details).

Let us consider simple groups of only two and three swimmers. The relative orientation of the swimmers primarily controls the amount of distortion (measured in terms of MDF/ADR) they induce to the surrounding environment. Our results reveal that by swimming in optimal orientations, swimmers can reduce their induced disturbances by more than 50% [Figs. 1(b) and 1(c)] compared to when they simply swim in schooling orientations (i.e., toward the same direction). In fact, there exist a range of optimal swarm configurations, arranging into which will result in minimally disturbing the surrounding fluid. For instance, when two of the agents (in a group of three) swim in directions normal to each other, the swarm arrangement remains optimal regardless of the third one’s swimming direction [see the green dashed line in Fig. 1(c)]. This is due to the axisymmetric nature of the disturbing flow induced by two perpendicular dipoles [Fig. 1(b–II)].

It bears attention that the computed values of induced disturbances (in terms of ADR/MDF) vary depending on the associated hyper-parameters, specifically, the threshold value (⁠$ū$_th) considered in computing the ADR or the radius (R/L_s) of the surrounding ring over which the MDF is computed. However, such dependencies can be avoided by normalizing the computed values against MDF/ADR induced by a reference swarm arrangement [see, e.g., Fig. 1(b)]. The consequent normalized values also represent a cross match for MDF vs ADR [Fig. 1(b)]. This further confirms equivalence of the described measures in assessing the optimality of various swarm arrangements.

For groups including a larger number of swimmers (i.e., those with N > 3 agents), plotting the MDF (or ADR) over the entire parameter-space is not practical. However, we know that any flock of $N∈4,5,…$ swimmers can be divided into sub-groups of only two or three agents. For each of such sub-groups, the optimal region of configurations is then readily available [Figs. 1(b) and 1(c)], assuring about 50% reduction in induced disturbances. This along with the linearity of Stokes equation (2), which describes dynamics of the flow around micro-swimmers, guarantees the existence of optimal swarm arrangements with the same 50% concealing efficiency. Therefore, the optimal region of configurations exists for any flock of N swimmers, and one can extract an optimal arrangement by implementing non-linear optimization over the parameter-space.

Note that objective functions quantifying swarm’s induced disturbances (i.e., MDF/ADR) are non-linear and often subject to constraints (e.g., minimum separation distance between the agents). Thereby, in search of the global minima by starting from multiple points, here we perform sequential quadratic programming using local gradient-based solvers. It is worth mentioning that merely applying gradient-based solvers will only find local optima, depending on the starting point. To avoid this, here the starting points are generated using a scatter-search mechanism,³⁰ which is a high-level heuristic population-based algorithm, designed to intelligently search on the problem domain. Its deterministic approach in combining high-quality and diverse members of the population—rather than extensive emphasis on randomization—makes it faster than other similar evolutionary mechanisms, such as genetic algorithm.³¹ 

As a benchmark, the magnitude of disturbing flow induced by a random suspension of 12 swimmers and the one induced by the same group arranged into an isotropic organized school are compared in Fig. 2 to those induced by concealed swarms of 12 swimmers. It is to be noted that the amount of disturbances induced by a flock of swimmers depends also on the minimum separation distance between the agents. Our numerical results show that the role of this factor is more significant for a concealed swarm than for an organized school (see, e.g., Fig. 3). However, it can still be considered as a minor factor compared to the relative orientation of the swimmers.

There exist many situations (for both motile micro-organisms and swimming micro-robots) that swimmers form an active swarm (i.e., a disordered cohesive gathering) around a desired spot. For instance, this can be a swarm of bacteria around a nutrient source³² or a flock of biological micro-robots performing a localized micro-surgery.^1,33 Remaining concealed in these scenarios can keep a bacterial swarm stealth from nearby predators or help keeping a deployed flock of micro-robots non-disturbing to the host medium.

Note that forming a swarm (as opposed to a random suspension), by itself, keeps the net induced distortions bounded (see, e.g., Fig. 2). However, to have minimal disturbing effects, arrangement of the swimmers (forming a swarm) must lie within the optimal region of configurations at every instant of time. To this end, we choose the proposed measure of net induced disturbances (i.e., MDF) as the objective function (⁠$Z$⁠) to be minimized by the swarm arrangement. Dynamics of an active concealed swarm can then be described as follows: (i) swimmers forming the swarm get into an optimal arrangement (with minimal fluid disturbances); (ii) each swimmer then swims steadily forward (i.e., “runs”) for a fixed period of time (say, τ_r); (iii) the swimmers will then reorient quickly (i.e., “tumble”) into a new optimal arrangement and then (iv) start running once again toward the new directions. This sequence of events occur, in turn, repeatedly. Parameters including swimming speed and frequency of tumbling events (⁠$τr−1$⁠) can be tuned according to the system of interest. As a benchmark, here we present a sample time evolution of an active concealed swarm in Fig. 4. Through the above described dynamics, the swarm remains cohesive, keeps itself confined within a finite region of space (around a desired spot), and is able to stifle the induced disturbances by ∼50% through system evolution (Fig. 4). This is equivalent to shrinking the swarm’s detection region by half.

It is also worth mentioning that the motion of each individual swimmer in the presented system can be seen as a controlled version of the so-called run-and-tumble mechanism. This is inspired by the observed behavior of swimming micro-organisms, such as E. coli bacteria, known as the paradigm of run-and-tumble locomotion.³⁴ Recent observations³⁵ reveal that even C. reinhardtii cells swim in a version of run-and-tumble. From a practical point of view, realization of the smart form of the run-and-tumble mechanism also seems feasible in the context of internally/externally controlled artificial micro-swimmers. The recently proposed Quadroar swimmer,^36,37 for instance, propels (i.e., runs) on straight lines and can perform full three-dimensional (3D) reorientation (i.e., tumbling) maneuvers.³⁸ 

Through altruistic collaborations, micro-swimmers can also remain stealthy while traveling toward a target point or tracking a desired trajectory in space. There is only one caveat here. The objective function (⁠$Z$⁠), to be minimized by the traveling swarm during each consecutive run, must now represent a measure not only for the overall disturbances induced by the swimmers, but also their distances from the target point (or from the desired trajectory). Therefore, we define

where $RMS¯d$ stands for the normalized root mean square of swimmers’ distances from the target point, $MDF¯$ quantifies the overall induced disturbances, and 0 ≤ ε ≤ 1 is the detuning parameter that determines the importance of concealing vs travel time. Note that the ratio ε/(1 − ε) properly covers the entire span of [0, ∞) when ε ∈ [0, 1). In practice, a variety of imaging techniques¹⁰ can be used to feedback the state information (including swimmers’ distance toward the target point) into the agents’ decision-making unit. However, some bio-hybrid systems may rely on local (point-wise) sensing capabilities of swimming micro-organisms (in measuring quantities such as light or chemicals) to obtain state information. To optimally control such systems, a more realistic objective function can be devised ( Appendix B) by replacing $RMS¯d$ in Eq. (9) with $RMS¯θ$⁠. The latter stands for the normalized root mean square of the swimmers’ deviations from their locally desired directions, in the absence of concealing interest. Nevertheless, our numerical experiments reveal that the bottom-line of analysis conducted using such an alternative objective function still remains the same (Fig. 8,  Appendix B).

Sample flocks of micro-swimmers, controlled to travel from a starting point (⁠$A$ ) toward a target point (⁠$B$⁠) in stealth vs fast modes (tuned by ε), are shown in Fig. 5. Note that ε = 0 corresponds to the fastest traveling swarm, for which the swimmers travel in schooling arrangements (i.e., toward the same direction), but it provides no concealing benefits (i.e., MDF = 100% for ε = 0). On the other extreme, i.e., for ε = 1, the swarm will have the highest concealing efficiency (MDF = 49.7%), yet never reaches the target point [Fig. 5(a) and Movie S2]. The trade-off between the travel time and the overall efficiency of concealing is demonstrated with more details in Fig. 5(c). To illustrate, we have monitored the induced fluid disturbances for traveling swarms controlled with various values of ε, during their migration from $A$ to $B$⁠. As ε → 1 (→0), the swarm will travel slower (faster), yet induces less (more) disturbances to the ambient fluid.

Recall that for a traveling swarm controlled by ε = 0, the objective function (9) encodes only a measure of the swimmers’ distances to the target point. This results in reaching the target point in a minimum amount of time [Fig. 5(a)]. However, once an ε > 0 is introduced to the swarm control strategy, the objective function (to be minimized by the traveling swarm) will also include a measure for the swimmers’ overall induced disturbances (in terms of MDF). Thereby, the fluid disturbance (measured in terms of normalized MDF) induced by a controlled traveling swarm rapidly decays with ε and eventually converges to the highest possible concealing efficiency (i.e., MDF = 49.7%) at ε ≈ 0.5 [Fig. 5(c)]. It is remarkable that swarming in such an optimally concealed mode [Fig. 5(b) and Movie S3], that is, the fastest among those with the highest possible concealing efficiency, costs only 23% increase in the trip duration compared to the fastest possible swarm. The associated concealing efficiency for such an optimal traveling swarm is equivalent to ∼50% shrink of the swarm’s detection region throughout its migration from $A$ to $B$⁠.

In the end, it is also worth noting that although the focus of our analysis in this article has been on planar (2D) swarm arrangements/movements, the present study can be readily extended to three-dimensional (3D) scenarios. As a benchmark, we discuss 3D concealed arrangements in  Appendix A (Fig. 6), for which the reduction in the swarm’s induced disturbances exceeds 99%. We then demonstrate how such swarm configurations can help a group of swimmers to remain stealthy (with ≥90% reduction in the MDF) throughout their trip from $A$ to $B$ in a 3D space (Fig. 7). Additionally, a sample concealed swarm of micro-swimmers tracking the desired trajectory through a non-uniform environment is also discussed in  Appendix C (Fig. 9 and Movie S4).

In conclusion, here we revealed that micro-swimmers can form a stealth swarm through controlled cooperation in suppressing one another’s disturbing flows. Specifically, our results unveil the existence of concealed arrangements, which can stifle the swarm’s hydrodynamic signature (and, thus, shrink its detection region) by more than 50% (or 99%) for planar (or 3D) movements. We then demonstrated how such a concealed swarm can actively gather around a desired spot, point toward a target, or track a prescribed trajectory in space. Our study provides a road map to optimally control/lead a swarm of interacting micro-robots in stealth vs fast modes. This, in turn, paves the path for non-invasive intrusion of swimming micro-robots with a broad range of biomedical applications.¹ 

The presented findings also provide insights into dynamics of prey–predator systems. Importance of the fluid mechanical signals (i.e., flow signatures) produced by swimming objects, in dynamics of prey–predator systems, is well appreciated for a broad range of aquatic organisms. Free-living copepods, for example, possess highly sensitive fluid-mechanoreceptors³⁹ capable of detecting disturbing flows as small as 20 μm s⁻¹. These sensors enable the organism to accurately measure fluid disturbances induced by nearby predators (preys), estimate their distance/size, and, thereby, properly trigger escape (catch) behavior.^40,41 Another example is the Gram-negative Bdellovibrio bacteriovorus,⁴² which is a prototypical predator among motile micro-organisms and hunts other bacteria, such as E. coli. Recent experiments⁴³ show that it is, in fact, hydrodynamics rather than chemical clues that lead this predator into regions with a high density of prey. Therefore, quenching the flow signature (and, thus, shrinking the associated detection region) by swarming in concealed modes can potentially have a significant impact on trophic transfer rates among a broad range of aquatic organisms. In particular, stifling the induced disturbances may help an active swarm of prey swimmers gathered around a favorite spot (say, a nutrient source) to lower their detectability (and, thus, predation risk) through shrinking their detection region. Quenching flow signatures induced by a traveling swarm, on the other hand, may help a swarm of predators to remain concealed while attacking a target prey flock.

See the supplementary material for Movies S1–S4. A complete description of these supplementary movies is provided in  Appendix E.

The data that support the findings of this study are available within the article and its supplementary material.

This work was supported by the National Science Foundation, Grant No. CMMI-1562871. The authors would like to thank Dr. Mir Abbas Jalali for valuable discussions.

The focus of this article has been on planar (2D) swarm arrangements/movements. However, the present study can be readily extended to three-dimensional (3D) scenarios. In particular, the same procedure can be used to also find 3D optimal (i.e., concealed) swarm arrangements for any flock of N swimmers. The exception is that to quantify fluid disturbances induced by a 3D swarm arrangement, one needs to either: (i) compute the mean disturbing flow-magnitude (MDF) over the surface of a surrounding sphere or (ii) compute the volume of swarm’s detection region (VDR).

As a benchmark, a 3D concealed arrangement is demonstrated in Fig. 6 for the same flock of 12 swimmers presented in Fig. 2 of the article. We highlight that forming such a 3D concealed swarm suppresses the induced disturbance by more than 99%; cf. this value to the ∼50% reduction achieved for 2D concealed arrangements of the same flock (Fig. 2). This is equivalent to more than 99% shrink of the instantaneous detection region for the swarm. Note that such a dramatic suppression of the induced disturbances, achieved by forming a 3D concealed swarm (versus a 2D one), reflects a sudden drop in the leading order of the swarm’s induced disturbing flows. In fact, for any sub-group of three orthogonally oriented agents, within a 3D swarm, the leading order of induced disturbances switches from being a dipole (decaying as 1/r²) to a quadrapole (vanishing as 1/r³).

Through altruistic collaborations, micro-swimmers can also form a 3D concealed swarm while traveling toward a target point in a 3D space (see Fig. 7). The objective function (to be minimized by the swarm through cooperation of the agents) remains untouched, and one can follow the same procedure (as outlined in the paper) to find the optimally concealed traveling swarm. The caveat here is that the 3D MDF is computed over the surface of a surrounding sphere. As a benchmark, here we show (in Fig. 7) a sample 3D concealed swarm of micro-swimmers traveling from a starting point (⁠$A$ ) toward a target point (at $B$⁠). Our numerical experiments reveal that by setting ε = 0.5, the traveling swarm can shrink its detection region by more than 90% and remain concealed throughout the trip from $A$ to $B$ (see Fig. 7).

In the presented study, we demonstrated that through altruistic collaborations, micro-swimmers can form a concealed swarm while traveling toward a target point or track a desired trajectory in space. These traveling swarms can represent: (i) flocks of swimming micro-robots traveling (in vivo) toward a target point while controlled to be fast/concealed (as tuned by ε); (ii) a swarm of predators attacking a target prey flock (at point $B$⁠) in stealth vs fast modes (tuned by ε); or even (iii) flocks of motile micro-organisms swarming under the influence of an external gradient (the intensity of which being modeled as 1 − ε) from $A$ to $B$⁠, e.g., in chemotaxis of sperm cells toward an egg. However, we note that sensing capabilities of swimming micro-organisms are often limited to the point-wise measurement of various quantities, such as light or chemicals.⁴⁴ This makes them unable to identify their distance toward a desired target point, as required for the model presented in Eq. (9) of the article. Therefore, for a bio-hybrid system which relies on sensing capabilities of swimming micro-organisms, a more realistic objective function (⁠$Z$⁠) can be devised by replacing $RMS¯d$ in Eq. (9) with $RMS¯θ$⁠, which stands for the normalized root mean square of the swimmers’ deviations from their locally desired directions in the absence of concealing interests. The alternative objective function thereby reads as

Note that when concealing is not of interest, the desired direction at the location of each swimmer is the steepest ascent in the external field that leads it toward the target. Thus, $RMS¯θ$ stands for the normalized root mean square of the swimmers’ deviations from the direction representing the maximal gradient (i.e., toward the target) [see Fig. 8(b)]. Mathematically, we define

This requires the swimmers to only identify deviation of their swimming direction from that of the maximal gradient in the external field of interest that can be obtained locally.

The trade-off between the travel time and the overall efficiency of concealing in this case is demonstrated with more details in Fig. 8. The induced fluid disturbances (measured in terms of MDF) are monitored during the trip from $A$ to $B$⁠, while the traveling swarm is controlled with various values of ε. Similar to what was observed for the test cases presented in Fig. 5, as ε → 0 (→1), the swarm travels faster (slower) in space, i.e., the travel time decreases (increases), but it will induce more (less) disturbances to the ambient fluid. Our results reveal that the bottom-line of our analysis also remains the same. In particular, the results reveal that swarming in an optimally concealed mode (via such an alternative local sensory information), with more than 50% reduction in disturbances, may cost only 23% increase in the trip duration compared to the fastest possible trip (Fig. 8). This is equivalent to 50% shrink in the detection region of the swarm throughout its migration from $A$ to $B$⁠.

There exist many situations, for both biological micro-robots and swimming micro-organisms, in which they have to travel through a non-uniform environment. Examples include fluids at the interface of different organs inside the human body with distinct viscosities, or those in vicinity of a mucus zone.⁴⁵ Depending on their propulsion mechanism, motile micro-organisms experience different energy expenditures, and, thus, distinct swimming speeds, while traveling in regions with different rheological properties.^46–51 

Here, we show that through altruistic collaborations, micro-swimmers can also remain stealthy while traveling toward a target point or tracking a desired trajectory in such non-uniform environments. In particular, we are interested to find an optimally concealed traveling swarm that is the fastest among those with highest possible concealing efficiency; see, e.g., the one presented in Fig. 5(b) passing through a uniform environment. Note that a straight line connecting two points in non-uniform environments no longer represents the fastest pathway between them. Therefore, one needs to first find the optimal (i.e., fastest) pathway from the starting point to the target point. Then, a similar procedure (as outlined in Sec. III C) can be used to track the specified trajectory. However, there is a caveat here: $RMS¯d$ in the objective function (⁠$Z$⁠) now stands for the normalized root mean square of swimmers’ distances from the optimal pathway.

As a benchmark, let us consider a simple example of two side-by-side regions (Fig. 9), each with a distinct swimming cost for swimmers. This, for instance, can represent the interface between two distinct liquids. A concealed swarm of three micro-swimmers tracking a prescribed optimal pathway (from $A$ to $B$⁠) through such an inhomogeneous environment is represented in Fig. 9 (see also Movie S4). At the cost of only 30% increase in the travel time, compared to the fastest possible swarm, the detection region of the swarm is significantly stifled so that reduction in the ADR exceeds 50% during the trip. This is also equivalent to minimally disturbing the ambient fluid with 50.4% reduction in the MDF.

It is also worth noting that by solving the normalized Eikonal equation, $|∇T¯|=C¯(x¯,ȳ)$⁠, using a fast marching level-set method,^52,53 one can find $T¯(x¯,ȳ)$⁠, which is the minimum cost (i.e., the least required time) of reaching to any arbitrary point $(x¯,ȳ)$ in space. Here, the bar signs denote dimensionless quantities, and $C¯(x¯,ȳ)$ is the swimming cost at $(x¯,ȳ)$ normalized by $Us−1$⁠. Values of $T$ are also normalized by the time scale T_s = L_s/U_s. Tracing back from point $B$ to $A$⁠, while always moving normal to the isolines of $T¯$ (see Fig. 9), will then provide the optimal pathway from $A$ to $B$⁠.⁵³ 

It is desired for individual swimmers to form a group and collaborate to cancel out each other’s disturbing effects on the surrounding fluid. Our results further reveal that a traveling concealed swarm can attract nearby individual swimmers (those swimming in its vicinity) and subsequently expand and re-form into a new larger swarm. To provide further insight, the minimal example is demonstrated in Fig. 10 via successive snapshots [(a)–(e)]. It is shown how a single traveling swimmer joins a nearby concealed swarm of two swimmers, and together, they form a new concealed swarm of three swimmers. Note that the only imposed constraint on the motion of swimmers is the upward swimming (cf. gravitaxis). Relaxing this constraint will simply result in a quasi-random walk of the swarm with no preferred direction.

Movie S1: The time evolution of an active concealed swarm of ten swimmers. The agents are initially positioned and oriented randomly (at t/T_s = 0). Thereafter, each agent represents a version of run-and-tumble dynamics (with τ_r/T_s = 5) so that to keep the swarm’s arrangement within the optimal region of configurations at every instant of time. The immediate value of the MDF (computed over a surrounding ring of radius R/L_s = 100) is also noted for each of the presented snapshots. Here, the instantaneous positions of the swimmers are denoted by colored dots, and gray-scale lines represent their trajectories over time.

Movie S2: A traveling flock of three micro-swimmers starting from point $A$ with the intention to reach a target point at $B$⁠. This traveling swarm is controlled by ε = 1 and, as a result, has the highest possible concealing efficiency (i.e., MDF = 49.7%). However, the swarm control has no trace of constraints on the preferred direction, and thus, it never reaches the target. Here, trajectories of the swimmers are shown by blue, green, and red solid lines. The dashed circle marks the instantaneous position of the swarm.

Movie S3: A traveling flock of three micro-swimmers starting from point $A$ with the intention to reach a target point at $B$⁠. This traveling swarm is controlled by ε = 0.5 and reaches the target in stealth mode (i.e., with the most possible concealing efficiency of MDF = 49.7%) at the cost of only 23% increase in its travel time. Here, trajectories of the swimmers are shown by blue, green, and red solid lines. The dashed circle marks the instantaneous position of the swarm.

Movie S4: A concealed swarm of three micro-swimmers tracking the optimal trajectory in a non-uniform environment from a starting point (⁠$A$ ) to the target point at $B$⁠. The detection region of the swimmers is significantly stifled such that reduction in the ADR exceeds 50% during the trip. This is equivalent to minimally disturbing the ambient fluid with 50.4% reduction in the MDF. The total time taken for the swimmers to reach the target point is 210 T_s. The normalized swimming cost is $C¯(x,y)=1$ for x/l < 50 and $C¯(x,y)=2$ for x/l ≥ 50, where l is the characteristic length of the swimmers (i.e., L_s). The optimal (i.e., fastest) pathway from $A$ to $B$ is computed through the fast marching level-set method and shown by a black dashed line. Isolines correspond to $T¯(x,y)$⁠, which is the minimum time required to reach any point (x, y), starting from point $A$⁠. Trajectories of the swimmers (tracking the optimal pathway) are also shown by blue, red, and green solid lines.