In Pinciroli et al. (2008), a position-based algorithm was proposed to achieve 2D triangular lattices in a constellation of satellites in a 3D space. This strategy combines global attraction towards a reference point with local interaction among the agents to control both the global shape and the internal lattice structure of the swarm. In Casteigts et al. (2012), a position-based approach was presented that combines the common radial virtual force [also used in Spears et al. (2004), Hettiarachchi and Spears (2005), Torquato (2009)] with a normal force. In this way, a network of connections is built such that each agent has at least two neighbours; then, a set of geometric rules is used to decide whether any or both of these forces are applied between any pair of agents. Importantly, this approach requires the acquisition of positions from two-hop neighbours. In Zhao et al. (2019), a position-based strategy is presented to achieve triangular and square patterns, as well as lines and circles, both in 2D and 3D; the control strategy features global attraction towards a reference point and re-scaling of distances between neighbours, with the virtual forces changing according to the goal pattern. Therein, a qualitative comparison is also provided with the distance-based strategy from Spears et al. (2004), showing more precise configurations and a shorter convergence time, due to the position-based nature of the solution. Finally, a simple position-based algorithm for triangular patterns, based on virtual forces and requiring communication between the agents, is proposed in Trotta et al. (2018) to have unmanned aerial vehicles perform area coverage.

In Li et al. (2009), a displacement-based approach is presented based on the use of a geometric control law similar to the one proposed in Lee and Chong (2008). The aim is to obtain triangular lattices but small persisting oscillations of the agents are present at steady state, as the robots are assumed to have a constant non-zero speed. In Balch and Hybinette (2000a, b), an approach is discussed inspired by covalent bonds in crystals, where each agent has multiple attachment points for its neighbours. Only starting conditions close to the desired pattern are tested, as the focus is on navigation in environments with obstacles. In Song and O’Kane (2014) the desired lattice is encoded by a graph, where the vertices denote possible roles the agents may play in the lattice and edges denote rigid transformations between the local frames or reference of pairs of neighbours. All agents communicate with each other and are assigned a label (or identification number) through which they are organised hierarchically to form triangular, square, hexagonal or octagon-square patterns. Formation control is similarly addressed in Coppola et al. (2019). The algorithm proposed therein is made of a higher level policy to assign positions in a square lattice to the agents, and a lower level control, based on virtual forces, to have the agents reach these positions. The algorithm can be readily applied to the formation of square geometric patterns, but not to triangular ones. Notably, the reported convergence time is relatively long and increase with the number of agents. Finally, a solution to progressively deploy a swarm on a predetermined set of points is presented in Li et al. (2019). The algorithm can be used to perform both formation control and geometric lattice formation, even though the orientation of the formation cannot be controlled. Moreover, this strategy requires local communication between the agents and the knowledge of a common graph associated to the formation.

A popular distance-based approach for the formation of triangular and square lattices, named physicomimetics, was proposed in Spears and Gordon (1999) and later further investigated in Spears et al. (2004); Hettiarachchi and Spears (2005). In these studies, triangular lattices are achieved with long-range attraction and short-range repulsion virtual forces only, while square lattices are obtained through a selective rescaling of the distances between some of the agents. The main drawback of the physicomimetics strategy (Spears and Gordon, 1999; Spears et al., 2004; Hettiarachchi and Spears, 2005; Sailesh et al., 2014) is that it can produce the formation of multiple aggregations of agents, each respecting the desired pattern, but with different orientations. Another problem, described in Spears et al. (2004), is that, for some values of the parameters, multiple agents can converge towards the same position and collide.

Similar approaches are also used to obtain triangular lattices when using flocking algorithms (Olfati-Saber, 2006; Wang et al., 2022b, a). An extension to achieve the formation of hexagonal lattices was proposed in Sailesh et al. (2014), but with the requirement of an ad hoc correction procedure to prevent agents from remaining stuck in the centre of a hexagon.

In Torquato (2009), an approach exploiting Lennard-Jones-like virtual forces is numerically optimised to locally stabilise a hexagonal lattice. When applied to mobile agents, the interaction law is time-varying and requires synchronous clocks among the agents. A stability proof for the formation of triangular (or 3D lattices) under the effect of virtual forces control algorithm, was recently published in Giusti et al. (2023). In Lee and Chong (2008), a different distance-based control strategy, derived from geometric arguments, was proposed to achieve the formation of triangular lattices. An analytical proof of convergence was given to the desired lattice exploiting Lyapunov methods. Robustness to agents’ failure and the capability of detecting and repairing holes and gaps in the lattice are obtained via an ad hoc procedure and verified numerically. A 3D extension was later presented in Lee et al. (2010).

Our main contributions can be listed as follows.

1. We introduce a novel distributed displacement-based local control strategy to solve geometric pattern formation problems in swarm robotics that requires no communication among the agents or any need for labelling them. In particular, to achieve triangular and square lattices, we employ two virtual forces controlling the norm and the angle of the agents’ relative position, respectively.

When compared to the control strategies in the existing literature, our approach (i) is able to achieve both triangular and square lattices rather than just triangular ones [e.g., as in Lee and Chong (2008), Casteigts et al. (2012)] (ii) yields more precise and robust square lattices with respect to distance-based algorithms (e.g., Spears et al., 2004; Sailesh et al., 2014), with only a minimal increase in sensor requirements (a compass); and (iii) does not require the more costly sensors and communication devices used for position-based strategies (e.g., Zhao et al., 2019), nor labelling of the agents (Song and O’Kane, 2014; Coppola et al., 2019).

Definition 1

(Swarm). A (planar) swarm S : = { 1,2 , … , N } is a set of N ∈ N > 0 identical agents that can move on the plane. For each agent i ∈ S , x i ( t ) ∈ R 2 denotes its position in the plane at time t ∈ R .

Moreover, r i j ( t ) : = x i ( t ) − x j ( t ) ∈ R 2 is the relative position of agent i with respect to agent j, and θ _ij (t) ∈ [0, 2π] is the angle between r _ij and the horizontal axis (see Figure 1).

Definition 2

(Neighbourhood). Given a swarm and a sensing radius R s ∈ R > 0 , the neighbourhood of agent i at time t is

Definition 3

(Adjacency set). Given a swarm and some finite R min , R max ∈ R > 0 , with R _min ≤ R _max , the adjacency set of agent i at time t is (see Figure 2).

Notice that if R _max ≤ R _s then A i ⊆ N i .

Definition 4

(Links). A link is a pair ( i , j ) ∈ S × S such that j ∈ A i ( t ) (or equivalently i ∈ A j ( t ) ). Moreover, E ( t ) is the set of all links existing at time t.

Clearly, it is possible to associate to the swarm a time-varying graph G ( t ) = ( S , E ( t ) ) (Latora et al., 2017); S and E ( t ) being the set of vertices and edges, respectively ¹ .

Finally, given any two links (i, j) and (h, k), we denote with θ i j h k ( t ) ∈ [ 0,2 π ] the absolute value of the angle between the vectors r _ij and r _hk .

Definition 5

(Lattice). Given some L ∈ {4, 6} and R ∈ R > 0 , a (L, R)-lattice is a set of points in the plane that coincide with the vertices of an associated regular tiling (Engel et al., 2004); R is the distance between adjacent vertices and L is the number of adjacent vertices each point has.

In Definition 5, L = 4, and L = 6 correspond to square and triangular lattices, ² respectively, as portrayed in Figure 2. We say that a swarm self-organises into a (L, R)-lattice if (i) each agent has at most L links, and (ii) ∀ ( i , j ) ∈ E and ∀ ( h , k ) ∈ E it holds that θ i j h k is some multiple of 2π/L. To assess whether a swarm self-organises into some desired (L, R)-lattice, we introduce the following two metrics.

Definition 6

(Regularity metric). Given a swarm and a desired (L, R)-lattice, the regularity metric e _θ (t) ∈ [0, 1] is e θ t : = L π ⋅ θ err t , (3) where, omitting the dependence on time,

The regularity metric e _θ , derived from Spears et al. (2004), quantifies the incoherence in the orientation of the links in the swarm. In particular, e _θ = 0 when all the pairs of links form angles that are multiples of 2π/L (which is desirable to achieve the (L, R)-lattice), while e _θ = 1 when all pairs of links have the maximum possible orientation error, equal to π/L. (e _θ ≈ 0.5 generally corresponds to the agents being arranged randomly.)

Definition 7

(Compactness metric). Given a swarm and a desired (L, R)-lattice, the compactness metric e _L (t) ∈ [0, (N − 1 − L)/L] is e L t : = 1 N ∑ i = 1 N | A i t | − L L . (5)

The compactness metric e _L measures the average difference between the number of neighbours each agent has and the one they are ought to have if they were arranged in a (L, R)-lattice. According to this definition, e _L reaches its maximum value, e _L, max = (N − 1 − L)/L, when all agents are concentrated in a small region, and links exist between all pairs of agents, while e _L = 1 when all the agents are scattered loosely in the plane, and no links exist between them, and, e _L = 0 when all the agents have L links (typically we will require that e _L is below some acceptable threshold, see Section 5.1.1). It is important to remark that, if the number N of agents is finite, e _L can never be equal to zero, because the agents on the boundary of the group will always have less than L links (Figure 2). This effect gets less relevant as N increases. Note that a similar metric was also independently defined in Song and O’Kane. (2014). We remark that the compactness metric inherently penalizes the presence of holes in the configuration and the emergence of detached swarms, as those scenarios are characterized by larger boundaries.

For the sake of brevity, in what follows we will omit dependence on time when that is clear from the context.

Consider a planar swarm S whose agents’ dynamics is described by the first order model x ̇ i t = u i t , ∀ i ∈ S , (6) where x _i (t) was given in Definition 1 and u i ( t ) ∈ R 2 is some input signal determining the velocity of agent i ³ .

We want to design a distributed feedback control law u i = g ( { r i j } j ∈ N i , L , R ) to let the swarm self-organise into a desired triangular or square lattice, starting from any set of initial positions in some disk of radius r, while guaranteeing the control strategy to be:

1. robust to failures of agents and to noise;

We will assess the effectiveness of the proposed strategy by using the performance metrics e _θ and e _L introduced above (see Definition 6 and Definition 7).

To solve this problem we propose a distributed displacement-based control law of the form u i t = u r , i t + u n , i t , (7) where u _r,i and u _n,i are the radial and normal control inputs, respectively. The two inputs have different purposes and each comprises several virtual forces. The radial input u _r,i is the sum of attracting/repelling actions between the agents, with the purpose of aggregating them into a compact swarm, while avoiding collisions. The normal input u _n,i is also the sum of multiple actions, used to adjust the angles of the relative positions of the agents.

Note that the control strategy in (7) is displacement-based because it only requires each agent i (i) to be able to measure the relative positions of the agents close to it (in the sets N i and A i ), and (ii) to possess knowledge of a common reference direction. Next, we describe in detail each of the two control actions in (7).

The radial control input u _r,i in (7) is defined as the sum of several virtual forces, one for each agent in N i (neighbours of i), each force being attractive (if the neighbour is far) or repulsive (if the neighbour is close). Specifically, we set u r , i = G r , i ∑ j ∈ N i f r ‖ r i j ‖ r i j ‖ r i j ‖ , (8) where G r , i ∈ R ≥ 0 is the radial control gain. Note that u _r,i is termed as radial input because in (8) the attraction/repulsion forces are parallel to the vectors r _ij (see Figure 1). The magnitude and sign of each of these forces depend on the distance, ‖r _ij ‖, between the agents, according to the radial interaction function f r : R ≥ 0 → R . Here, we select f _r as the Physics-inspired Lennard-Jones function (Brambilla et al., 2013), given by f r ‖ r i j ‖ = min a ‖ r i j ‖ 2 c − b ‖ r i j ‖ c , 1 , (9) where a , b ∈ R > 0 and c ∈ N are design parameters. In (9), f _r is saturated to 1 to avoid divergence for ‖r _ij ‖ → 0. f _r is portrayed in Figure 3A.

For any link (i, j), we define the angular error θ i j err ∈ − π L , π L as the difference between θ _ij and the closest multiple of 2π/L (see Figure 1), that is, θ i j err : = θ i j − 2 π L arg min q ∈ Z θ i j − q 2 π L , (10)

Then, the normal control input u _n,i in (7) is chosen-as u n , i = G n , i ∑ j ∈ A i f n θ i j err r i j ⊥ r i j , (11) where G n,i ∈ R ≥ 0 is the normal control gain. Note that each of the normal virtual forces is applied in the direction of r i j ⊥ , that is the vector normal to r _ij , obtained by applying a π/2 counterclockwise rotation (see Figure 1). The magnitude and sign of these forces are determined by the normal interaction function f n : − π L , π L → − 1 , 1 , given by f n θ i j err = − L π θ i j err . (12) f _n is portrayed in Figure 3B.

We consider a swarm consisting of N = 100 agents (unless specified differently). To represent the fact that the agents are deployed from a unique source (as typically done in the literature, see e.g., Spears et al. (2004), their initial positions are drawn randomly with uniform distribution from a disk of radius r = 2 centred at the origin ^{4, 5} .

Initially, for the sake of simplicity and to avoid the possibility of some agents becoming disconnected from the group, we assume that R _s in (1) is large enough so that ∀ i ∈ S , ∀ t ∈ R ≥ 0 , N i t = S \ i ; (13) i.e., any agent can sense the relative position of all others. Later, in Section 5.3, we will drop this assumption and show the validity of our control strategy also for smaller values of R _s. All simulation trials are conducted in Matlab ⁶ , integrating the agents’ dynamics using the forward Euler method with a fixed time step Δt > 0. Moreover, the speed of the agents is limited to V _max > 0. The values of the parameters used in the simulations are reported in Table 2.

For the sake of simplicity, in this section we assume that G _r,i = G _r and G _n,i = G _n, for all i ∈ S ; later, in Section 6, we will present an adaptive control strategy allowing each agent to independently vary online its own control gains. To select the values of G _r and G _n giving the best performance in terms of regularity and compactness, we conducted an extensive simulation campaign and evaluated, for each pair (G _r, G _n) ∈ {0, 1, …, 30}×{0, 1, …, 30}, the following cost function, averaged over 30 trials, starting with random initial conditions: C e θ ss , e L ss = e θ ss e θ * 2 + e L ss e L * 2 . (20) The results are reported in Figure 4 for the triangular (L = 6) and the square (L = 4) lattices; in the former case, the pair ( G r * , G n * ) L = 4 minimising C is (22, 1), whereas in the latter case it is ( G r * , G n * ) L = 6 = ( 15,8 ) . Both pairs achieve C ≤ 1, implying e θ ss ≤ e θ * and e L ss ≤ e L * .

In Figure 5, we report four snapshots at different time instants of two representative simulations, together with the metrics e _θ (t) and e _L (t), for the cases of a triangular and a square lattice, respectively. The control gains were set to the optimal values ( G r * , G n * ) L = 6 and ( G r * , G n * ) L = 4 . In both cases, the metrics quickly converge below their prescribed thresholds, as T < 2.75 s . Moreover, note that e _L (t) decreases faster than e _θ (t), meaning that the swarm tends to first reach the desired level of compactness and then agents’ positions are rearranged to achieve the desired pattern. Finally, we note that it is immediate to verify that it is possible to control the orientation of the resulting lattice simply by applying a uniform offset to the agents’ compasses.

In this section, we investigate numerically the properties that we required in Section 4.1, that is robustness to faults and noise, flexibility, and scalability.

As done in related literature (Zhao et al., 2019) (yet for a position-based solution) we compared our control law (7) to the so-called “gravitational virtual forces strategy” (see the Appendix) (Spears et al., 2004), that represent an established solution to geometric pattern formation problems. In Spears et al. (2004), a second order damped dynamics is considered for the agents. Hence, for the sake of comparison, we reduced the model therein to the first order model in (6), by assuming that the viscous friction force is significantly larger than the inertial one.

To select the gravitational gain G and the saturation value F _max in the control law from Spears et al. (2004), we applied the same tuning procedure described in Section 5.2. In particular, we considered (G, F _max) ∈ {0, 0.5, …, 10}×{0, 1, …, 40}, and performed 30 trials for each pair of parameters, obtaining as optimal pair for the square lattice ( G * , F max * ) = ( 35,2 ) (see Figure 10A). All other parameters where kept to the default values in Table 2.

Then, we performed the same scalability test in Section 5.3.4 and report the results in Figure 10B. Remarkably, by comparing these results with ours, we see that our proposed control strategy performs better, obtaining much smaller values of e θ ss , regardless of the size N of the swarm. In particular, the control law from Spears et al. (2004) only rarely achieves e θ ss ≤ e θ * , implying a low success rate.

Next, we test robustness to faults, flexibility, and scalability for the adaptive law (22), similarly to what we did in Section 5.3.

We ran a series of agent removal tests, as described in Section 5.3.1. For the sake of brevity, we report the results of one of such tests with L = 4 in Figures 12A–E. At t = 30 s, 30% of the agents are removed; yet, after a short time the swarm reaggregates to recover the desired lattice.

We then repeated the test in Section 5.3.3, with the difference that this time we set G _r = 18.5 (that is the average between the optimal gain for square and triangular patterns), and set G _n,i according to law (22), resetting all G _n,i to 0 when L is changed. The results are shown in Figure 12F. When compared to the non-adaptive case (Figure 8), here e θ ss and e L ss are smaller (better pattern formation), but T _θ and T _L are larger (slower), especially when forming square patterns. Interestingly, when L = 4, G ̄ n settles to about 5, while when L = 6 it settles to about 0.3, a much smaller value.

Finally, we repeated the test in Section 5.3.4, setting again the sensing radius R _s to 3 R and assessing performance while varying the size N of the swarm; results are shown in Figure 12G. First, we notice that the larger the swarm is, the larger the steady state value of G ̄ n is. Comparing the results with those obtained for static gains shown in Figure 9B, here we observe a slight improvement of performance, with a slightly smaller e θ ss , while we verified that the convergence time is comparable to the one observed for the static policy.