Simulating emergent behavior of autonomous swarm systems using agent-based modeling

1. Introduction

Swarming has become a familiar term, especially in the context of unmanned autonomous vehicles or drones in combat operations in current conflicts worldwide. A recent report identified 38 sovereign states with armed drone programs. Eleven of which had used armed drones in combat. ¹ A swarm of drones carrying explosive ordnances is asymmetric warfare, providing a battlefield equalizer for a less-capable force against a more superior adversary. This point is underscored in the current conflict with Ukrainian forces effective in using autonomous drones in combat against Russian forces to gain tactical success. ² Autonomous drone battlefield achievements have definitively piqued interest in the military for their expanded use. The simultaneous deployment of many drones may produce favorable decisive outcomes, if not a psychological advantage over an enemy. However, numbers alone may be an insufficient deterrent, not a long-term solution. Similarly, for commercial drone delivery applications, using a swarm may improve the customer experience with quicker delivery times but introduce safety issues within a complex traffic management schema. ³

Effective swarming requires applying some form of derived intelligence from the collective group. Nature presents an intriguing phenomenon where individual entities cooperate and collaborate to form intelligent swarms. Biological swarms are known to perform complex tasks not expected from any unit in this collective. ⁴ Examples of this behavior include the foraging for food actions of ants or the rhythmic flight patterns exhibited by a flock of birds. ⁵ The observed collective behavior results from the interaction of individuals with each other and their environment. This behavior occurs at the higher level of a network hierarchy and is called emergence, a specific feature characteristic of complex adaptive systems.^6,7 Emergence or emergent behavior is a non-trivial phenomenon resulting from the functional interaction between constituent parts of a larger system including its operating environment. It is usually associated with the collective behavior of an adaptive “system-of-systems” and characterizes many distinct complex systems from biological entities to physical human-engineered systems. The system-level outcome is generally unanticipated beforehand and not deducible from the properties of the individual parts of this system. ⁸ “When a large number of similar entities interact among each other and with their environment at a low scale, unexpected outcomes at higher spatial-temporal scales might spontaneously arise,” ⁹ with examples including the behavior natural swarms of birds, ants, termites, and fish as well as complex, adaptive physical systems like mass transit systems. ¹⁰

Interest in engineering practical application of mechanical swarm systems is increasing with the military continuing research on the operational utility of emergent swarms. But civilian institutions and private industry are also pursuing specific use cases in commercial applications, including logistics and product delivery.^11,12 Early studies by the Air Force Research Laboratory (AFRL) highlighted the attributes of emergent behaviors that are attractive for autonomous systems applications. ¹³ These attributes included (1) decentralized—eliminating the need for centralized operation, giving maneuver freedom to the individual unit; (2) inherently implicit—eliminating the need for direct control, simplifying the command structure, potentially eliminating significant human interaction and management; (3) inherently resilient—emergent behavior is tolerant of imperfections and variants in individual units; and (4) inherently scalable—very robust to increase or decrease of individual autonomous units in the mix. In a military setting, swarms may allow a larger, faster, and better coordinated lethal force. ¹⁴

System planners and developers can use the collective behavior of an autonomous swarm system for specific applications. However, operating these systems becomes problematic when large numbers of autonomous drones are employed. Each drone generally uses dedicated command links, and with many operating simultaneously, this arrangement becomes complex and cumbersome. What if these systems are simple in design, making them economically sustainable, easier to operate, and use emergence to produce favorable results toward a specific objective? Several researchers have suggested that emergent behavior can be a “blessing in some complex problem-solving situations.” ¹⁵

However, the AFRL paper also highlighted potential drawbacks, including non-deterministic execution, probabilistic results, and chaotic collective behavior. These negative characteristics cause significant uncertainty in the outcome and may even be catastrophic, mainly when applied to a military or urban logistics setting. In other words, group behavior is hard to predict and may not be repeatable, and recognizing and predicting emergence remain a technical challenge for researchers and developers.^13,16– 18

These uncertainties continue to hinder the large-scale use of swarm systems in practical applications, with concerns over system control and safety.^19,20 Previous studies examined self-organization as an optimization factor in many swarm operations, ²¹ but this remains a nascent research area worthy of further examination, considering recent technological and computational advancements. There is a need to characterize how the emergent behavior of swarm systems can result in favorable outcomes in achieving an operational objective.

Arguably, there is merit in modeling different swarm behaviors in a single environment to examine and compare emergent behaviors attributable to different types of swarm configurations in a specific scenario or use case. The earlier research from the authors documented in previous studies^22,23 generated a limited number of simulation cases, enabling a relatively quick assessment of the utility of using agent-based modeling (ABM) for the characterization and comparative analysis of swarm types. This early work yielded insights into emergent behavior from 200 cases in the previous study ²² and 6000 cases in the previous study. ²³ The research presented in this paper with over 40,000 simulation cases in three separate operational scenarios enables a more robust comparative analysis of swarm-type performance within given environmental and operational conditions.

This research uses ABM to compare the behaviors of two distinct types of autonomous swarms. The research objective is to visualize and characterize emergent behaviors of each type to determine performance features useful in various operational conditions. Two different swarming types are examined: a modified version of Uri Wilensky’s “Ant Lines,” ²⁴ inspired by Alfred Bruckstein’s earlier work, ²⁵ and a simplified adaptation of Craig Reynolds’“Boids” ²⁶ for a scenario in which unmanned aerial system (UAS) platforms (referenced as drones in this paper) form swarms to overwhelm an adversary’s target location.

This research advances current knowledge of swarm behavior as a novel application of ABM to examine two swarm models simultaneously in a single computational environment. This research contributes to the body of work in swarm intelligence by providing the following: (1) understanding the principles driving swarm dynamics and identifying the factors making them effective in a specific use case by modeling autonomous drone swarms and exploring their emergent behaviors; (2) adding technical content to the emerging theoretical framework of swarm intelligence and complex adaptive systems behavior by examining swarm performance in various operational environments; (3) evaluating patterns in data resulting and determining statistically significant differences between swarm types and operational environments to show the potential utility of emergence as a system-level design feature.

This research paper follows a standard structure. The next section discusses related technical work, especially the swarming models used in this research; this is followed by the research methodology, including a discussion on validation. The simulation results and interpretation are presented next. Finally, the paper discusses limitations and conclusions.

2. Related research

Parunak and Brueckner define swarming as a “useful self-organization of multiple entities through local interaction.” In swarm systems, rudimentary strategies control individual units, while complex behaviors emerge at a higher level through interactions among autonomous units and their environment. Individuals interact locally at the unit level, and the sum of all interactions creates a higher-level condition that defines the swarm’s emergent behavior. ⁷

Swarm engineering is an emerging discipline aiming to define systematic and well-founded procedures for modeling, designing, realizing, verifying, validating, operating, and maintaining swarm autonomous systems. ²⁷ Many researchers desire these autonomous mechanical systems to behave like their biological equivalents ²⁸ and characterize swarm formation or aggregations as structured or behavioral. ⁵ They are structured when the formation acts under rigid frameworks and rules similar to a Leader–Follower (L-F) situation (e.g., when birds fly in a V-formation to achieve the benefit of aerodynamic flight). The behavioral model mimics biological features implemented on each drone with similar characteristics. Simple strategies control individual units, while complex behaviors achieved at a higher level result from interactions among autonomous units and between these entities and their environment. A swarm system is generally controlled by a single control mechanism with a single location for receiving and processing data and information. Alternatively, the swarm system may operate in a decentralized fashion. The latter is most applicable in many real operations where complete autonomy is required. ⁴

ABM is excellent for analyzing the behaviors of swarm systems using computational modeling to describe how an agent behaves. ²⁴ As a modeling framework, it “implements complex internal properties of agents and their non-trivial behavioral rules,” ²⁹ and “ABM is particularly applicable when agent adaptation and emergence are important considerations.” ³⁰ ABM use in modeling swarm natural behavior is familiar, as previous efforts focused on simulating complex outcomes with human social interactions and biological behaviors.^31,32

A common mathematical underpinning for modeling the movement of large groups of individuals is Reynolds’ Boids model and the Social Force Dynamics model. In his 1987 seminal article “Flocks, Herds, and Schools: A Distributed Behavioral Model,” Craig Reynolds introduces the term “Boids,” reflecting a simplistic simulation of the collective behavior of bird flocks and other biological entities exhibiting similar group actions. ²⁶ The model assumes flocking is “simply the result of the interaction between the behavior of individual entities.” Reynolds introduces three key steering behaviors for simulating the collective motion of these groups: (a) Collision Avoidance or Separation (i.e., individuals within the group avoid crowding each other and try to maintain a certain distance from their neighbors); (b) Velocity Matching or Alignment (i.e., individuals try to align their direction and speed with their neighbors, resulting in a cohesive movement); and (c) flock Centering or Cohesion (i.e., individuals attempt to move toward the center of mass of their neighbors, promoting group cohesion).

Reynolds’ model relies on individual interaction (neighbor to neighbor) with separate entities without knowledge of the entire group. They do not care about other entities outside their immediate proximity; the actions of their close neighbors modulate their actions. This simple interaction, in turn, results in complex and realistic-looking flocking behavior of the entire group. Figure 1 represents the steering behaviors from Reynolds’ model within the Boids’ visual sphere (field-of-view). The Boids’ initial condition is set at position P_i with the initial velocity vector (magnitude and direction), V_i, and subjected to the steering vectors associated with cohesion, separation, and alignment. Advancing the Boid at this initial condition by Δt results in P_i + 1 = P_i + ΔtV_i + 1, where V_i + 1 = V_i + C_s S_i + C_k K_i + C_m M_i with C as user-defined coefficients influencing corresponding steering vectors, S (separation), K (cohesion), and M (alignment). Figure 2 depicts the movement of a flock maneuvering around an obstacle while displaying swarm behavior in accordance with their steering control vectors.

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Figure 1.

Boids steering vectors.

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Figure 2.

Boids model representation of flocking agents around obstacles.

An alternative to the Boids model is from the work of Dirk Helbing and Peter Molnar. Helbing and Molnar posited that pedestrian motion, or behavior is driven more by the individual’s internal motivations than the environment in a classic article written in 1995. ³³ They postulated several forces at play impacting the outcome of pedestrian motion in limited space. One force is acceleration to keep up with others and achieve a desired velocity state. Another force assumes that individuals maintain a certain distance from others and borders along the motion path. The last force represents attractive effects. This vector allocation is analogous to Reynolds’ flock equation, which involves an initial condition of position and velocity and is affected by an associated set of steering forces imparted on individual Boids in the flock. ³⁴ Combining all these forces is what Helbing and Molnar call the Social Force, which is the temporal change of the velocity vector of each pedestrian in motion, F_P (t) = dv_P/dt.

Swarm models used for studying command and control of autonomous systems range from a simple L-F method to other path-following methods including biologically inspired stigmergy. ³⁵ – ³⁷ For this research, the L-F and a modified Boids model are modeled to compare performance in a single computational framework. The L-F model is chosen for its intrinsic simplicity while the Flocking model provides a well-documented model based on Boids. The two models are used in this simulation as representative of a cross-section of a number of other swarm models examined by other researchers described earlier to provide a basic modeling basis for building a computational agent-based framework for comparative research.

As discussed, the L-F is patterned after Wilensky’s Ant Lines model motivated by Bruckstein’s mathematical derivation of ant behavior. ²⁵ In “Why the Ant Trails Look so Straight and Nice,” Bruckstein developed a mathematical explanation of ant behavior, influenced by Richard Feynman’s observation of ant movement in search of food. ³⁸ In Bruckstein’s model, each ant moves directly toward the one ahead of it with a velocity vector pointing directly toward the ant ahead of it matching its velocity and direction to maintain equal spacing to avoid collisions.

The individual ant actions can be represented, mathematically, with a non-linear differential equation, d/dt(P_n + 1(t) = (1/|P_n + 1(t) − P_n (t)|) (P_n(t) − P_n + 1(t)), where the denominator in this equation is the Euclidian distance between the points P_n(t) − P_n + 1(t)). With an initial path at t = 0 and the time sequence for n = 1,2, 3, …, a series of pursuit paths P_{n is} created for all values of n starting at the origination location at (0, 0) and ending at the destination position at (L, 0). By solving the differential equation, Bruckstein posited that the sequence of pursuit paths, P_n converges into a straight line between (0, 0) and (L, 0) over time with subsequent n number of follower ants. Solving this equation is a very difficult undertaking, ³⁹ but Wilensky’s Ant Lines model demonstrates this process visually, using an ABM platform.

Figure 3 shows Bruckstein’s straight-line theorem for ant behavior graphically showing how ants achieve a straight-line trajectory over time with successive ants in an L-F construct.

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Figure 3.

Bruckstein’s representation of ant path-following behavior.

In this model, the “pioneer” ant, depicted as A_n, starts from position (0, 0) but may follow a curved indirect path toward its destination at (L, 0). This path is shown as P_n(t), where t is the time step. This path is graphically depicted in Figure 3 as the green curve with waypoints in yellow circles (1- 2- 3- 4- 5- 6- 7). The second ant, A_n + 1, influenced by the vector properties of the leading ant, goes directly to the location of A_n at t + 1 avoiding the initial curve at way point # 2. This more optimized path is depicted as P_n (t + 1) as the red dashed curved in the figure with waypoints 1- 3- 5- 7. Subsequent ants (A_n + 2, A_n + 3, ...) further optimizes the path leading to a path that is a straight line from location (0, 0) to location (L, 0) in the end.

The Flocking model naturally fit Reynolds’ Boids model representation, discussed earlier, with formation control of each drone dictated by the spatial arrangements between the drones. ⁴⁰ Agent actions are entirely driven by where individual agents are located in geometrical space with respect to one another in their respective groups.

To summarize the computational experimentation set-up, the first swarming approach considered is L-F, patterned after Wilensky’s Ant Line model and the second is the Flocking swarm model, with characteristics adopted from Reynolds’ Boids artificial life prototype. ²⁶ The L-F type is a self-assembly or morphogenesis model, while the Flocking type is described as the aggregation model. ²⁷ The L-F model exhibits a structured framework, while the Flocking model represents the behavioral setting discussed in the previous study. ⁵ The L-F drones keep a relative distance from the leader. In contrast, distances between drones drive the Flocking drones with the ability to self-organize and reconfigure the group based on the rules specified in the Boids model to maintain separation, coherence, and alignment. ⁴¹

3. Methodology

This research employs a computational experimental framework to address research questions related to characterizing swarm behavior. A physical demonstration is impractical at this research stage, and the availability of open-source simulation platforms allows for an experimental design using ABM. Figure 4 shows the simulation environment with the originating base and target locations. The threats are depicted as red patches across this environment, as discussed. This figure shows three different environmental configurations: no threat, low threat, and high threat denoted by the increasing number of red boxes. The multi-agent modeling and simulation environment used is NetLogo. ⁴¹

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Figure 4.

Simulation set-up showing originating base (yellow) and target (orange) with threats (red).

The simulation environment contains a 101 × 101 grid, expressing each location in x and y coordinates. In this environment, the drones originate at x_o, y_o, and an area designated as a target location centered at x_T, y_T. The swarms are deployed from the originating location and move toward the target location. Both models are set, with each corresponding drone having specific rules for movement and action. Dispersed between the operating and target bases are randomly placed obstacles (threats) the swarm units must avoid and maneuver around to reach the target successfully. These threats may be assumed to be physical objects like terrain and buildings or represent an active defense system to thwart adversary operations around a critical or sensitive area. There is no a priori information provided to the swarm models and the threat environment is randomly established at the beginning of each simulation. The simulation determines the presence of an obstacle in the path of a drone and, if a threat exists, takes appropriate evasive actions according to its established rules. For the simulated grid environment, threats appear in some 10,201 possible locations (grid squares). The number of threats and their locations on the grid are randomly generated, so each simulated case is unique. The lower the threat level, the fewer deployed threats in the simulated environment. The threat profile is categorized as low, medium, and high for each simulation run, allowing for analysis of each swarm model’s emergent behavior based on the severity of threats in the operational environment. How the swarming approaches are modeled is discussed next.

This research adopted two swarming approaches in a simulation environment first used in the initial research described in the previous studies.^22,23 Figures 5 and 6 depict the rules set and deployment sequence for the L-F and Flocking swarm models, respectively.

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Figure 5.

Representation of the leader–follower model with associated agent rules.

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Figure 6.

Representation of the flocking model with associated agent rules.

As shown in Figure 5, L-F₁ is the Leader drone for the L-F model, with L-F₂ to L-F_N as the trailing followers. The subscript N is the number of drones used per simulation. In this paper, N equals 100 drones. All L-F drones start at a heading of 90° (roughly toward the target), parallel to the X-axis, at T = 0. At the first increment, T + 1, L-F₁ move out randomly inside a 90° cone. No a priori information about the exact target location exists, so the random direction is as good as any to start the deployment process.

A designated delay time, Δt, is also an adjustable variable representing the required time separation between L-F drones. So, L-F₂ deploys at T + 1 + Δt following L-F₁. The L-F drones begin to form a line that, over time, will follow a “pursuit” path that is a straight line from location x₀, y₀ to x_T, y_T. However, the straight-line pursuit path is disturbed by a threat, and the drone turns in a random direction inside a 90° cone and moves forward at a prescribed distance when the path is without a threat.

The Follower drones move toward the direction of the drone ahead of them, as directed in the rules set. The L-F line continues until it crosses the x-axis plane of the x_T coordinate of the target location. At this point, the L-F drones head toward the target location at x_T and y_T to complete the task. It is assumed that by crossing x_T, the target is within the detection range of the individual drones. This is done deliberately to simplify the final logic in locating the target to complete the task.

Similarly, at T = 0, 100 Flocking model drones are based at the originating location at x₀, y₀. As depicted in Figure 6, at T + 1, all drones deploy simultaneously, each moving in random directions but within a 90° cone. In this simplified Boids model, only the alignment vector is considered.

At every step in the simulation, each drone attempts to flock by aligning with each other based on the distance and direction of its closest neighbor. As established by its rules, each Flocking drone constantly searches for a nearby neighbor to align with. This function assigns a minimum separation distance (MSD) between drones. If one drone is within the MSD of another, the rules set directs the drone to turn toward the neighbor and follow the path of that drone. A maximum turn angle value deliberately limits maneuverability, so drones that exceed this limit cannot join their nearest neighbor. The model also assigns a vision variable restricting the field of view observable by each drone. For this simulated environment, the vision levels are the number of patches (squares) in the NetLogo grid observable by the drone. For this research, 1, 5, and 10 patches represented low-, medium-, and high-vision levels. The drone interacts with others within this visual field. Like the L-F drones, the Flocking drones continue to assess the threat environment and maneuvers when threats are along their path. As a result of this dynamic between drones, several separate flocks may form. When the location of each Flocking drone crosses the x-axis plane at x_T, it heads toward the target’s location at x_T, y_T to complete the task. This final action is similar to the L-F model simplifying the final logic of locating the target and completing the task.

A successful mission (i.e., targeting) is achieved when a pre-determined number of drones of one swarm type reach the target location at x_T and y_T. (Note: the number of drones per type and the number of drones required to achieve success are arbitrary variables defined by the user. One hundred drones are assumed for each kind, requiring a user-defined number of drones from either type inside the target location for success). The simulation is suspended, and Time (number of simulation steps in ticks) is recorded for the successful swarm. In this paper, the swarm type able to deposit the required number of drones inside the target location is called “winner.” In this paper, the number of required drones entering the target location is called “Drones-to-Kill” for the specific military use case of targeting described in this paper. However, the delivery of required number of drones to a target area has other application including forest fire suppression, search and rescue, and product delivery.

The results of the simulations enabled the comparison of the performance of the two models against an environment with varying threats. Three simulation scenarios are presented. The first ran 20,000 simulations with all control variables randomized, including the number of drones required to complete the task; the second scenario ran 10,000 simulations with the number of required drones to set at 20 drones, based on the resulting sensitivity analysis from the first scenario; and the third completed another 10,000 simulations with the number of required drones set at 90.

3.1. Model validation and documentation

Model validation is expanded in the next research phase to consider other methods for system representation, given the model’s algorithmic complexity, ensuring an accurate depiction of the real-world system under examination with repeatable results. This process included verification of the agent-based model by conducting model calibration and documentation, ensuring accuracy and repeatability. ⁴² The “speed” parameter for the individual drones (or agents) was calibrated by model type (L-F or Flocking), ensuring equal performance at the outset in a benign operational environment. Fifty thousand simulations compared the “time-to-target” of two agents of each type, and the speed parameter was adjusted to eliminate modeling bias.

A challenge for ABM is thoroughly describing the models and the simulation process. There is recognition of the need for standardization ⁴³ in how the models are explained and presented. One standard that has gained wide acceptance for model documentation is the Overview, Design Concepts, and Details (ODD) protocol,^44,45 and this methodology is used in this research. A summary of the protocol section describing agent and environment control variables is shown in Table 1.

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Table 1.

Control Variables for Simulated Drones and Operational Environment From the ODD Protocol.

Feature	Control variable	Leader-follower	Flocking
Environment	Number of Drones	✓ 100	✓ 100
	Drones to Kill	✓ 1st Scenario: Variable 10–90 (in increments of 10)✓ 2nd Scenario: Fixed at 20✓ 3rd Scenario: Fixed at 90	✓ 1st Scenario: Variable 10–90 (in increments of 10)✓ 2nd Scenario: Fixed at 20✓ 3rd Scenario: Fixed at 90
	Threat Level	✓ Varies (Range: 1–10)	✓ Varies (Range: 1–10)
Agent	Time Delay	✓ Range: 1 or 2	N/A
	Vision	N/A	✓ Range: 1–10
	Max Turn	N/A	✓ Range: 1–20
	Minimum Separation	N/A	✓ Range: 1–10

As previously discussed, the simulated drones use these control variables to define the environmental conditions they experience and highlight the technical limitations of their onboard sensor capabilities. These variables are simplified for these simulations but have significance in the design complexity of the corresponding physical entities. The L-F model uses Time Delay, Δt, to synchronize the deployment of the follower drones with its leader. For example, while shorter delay times are desirable, there could be limitations in the ability to launch successive drone units, as the deployment system may require more time to reconstitute before the next drone is released. The Vision variable used by the Flocking model may dictate the necessary complexity of the onboard sensor of the physical drone for collision avoidance and threat detection. Similarly, the Max Turn and the Minimum Separation variables impact the design and engineering of the drone’s propulsion system. Understanding the performance of these swarm models and the impact of these input variables allows for a more optimal match of technical capability and operational environment to ensure mission task completion and success.

4. Simulation results

This section provides examples of simulation outcomes to familiarize the reader with the visualized products from the ABM tool. The following figures illustrate emergent behaviors from each swarm type under specific scenarios sampled from thousands of simulated cases. Each figure compares swarm-type behavior as a function of the threat environment, specifically one under benign threat conditions and another with a higher threat profile. Section 4.1 provides exploratory data analysis results. Sampled results from each scenario present a sensitivity analysis of performance as a function of changes to the control variables. Section 4.2 is a brief interpretive discussion of the observed emergent outcomes, providing an initial assessment of emergent behavior based on performance and observed visualized characteristics. Figure 7 displays the performance of the swarm models in a benign environment from two different simulations. L-F drones are depicted as green airplanes, while the Flocking drones are blue airplanes. The L-F drones form a relatively straight flight path with drones stacked one after the other in a smooth line toward the target location. Bruckstein’s pursuit path is observed in this simulation as the line continually becomes smoother and straighter without threats. ²⁵ Path optimization is achieved as successive drones are deployed from the originating base, enhancing performance efficiency.

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Figure 7.

Leader–follower and flocking swarm model operating in a benign threat environment.

The Flocking drones, deployed simultaneously from the originating base at random headings within a 90° cone, align with each other, following the rules set for MSD as described above. The absence of threat boxes in the environment facilitates faster alignment, resulting in a concentration of Flocking drones near the target location. As discussed, the simulation stops once a pre-determined number of drones of one type (of the 100 drones) have entered the target location and the winning model’s completion time is recorded. This figure also highlights how the tight grouping of the Flocking drones contributed to their rapid convergence around the target location. This formation is attributable to the absence of threats, allowing uninterrupted execution of the Flocking model’s rule-based movements in space.

Figure 8 is an expanded view of another representative simulation depicting the movement patterns of both Flocking and L-F drones under benign and high-threat environments. A line trace is included for several representative drones of both types to help visualize individual drone spatial movement from the originating base toward the target location.

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Figure 8.

(a) L-F model path optimization. (b) Path traced for Flocking and L-F models in higher threat.

Figure 8(a) isolates the L-F swarm, showing the flight patterns of the leader drone and the followers. The overall path is optimized as successive drones learn the best path through dimensional space. Notable in this figure is the erratic behavior of the leader drone, L-F₁, as it departs the originating base. The adjustments made by L-F₂ and the following drones successively result in an increasingly straighter line and more direct path to the leader’s x, y position at this point in time.

The Flocking drones are also shown concurrently, early in their deployment flow, and beginning to align following their rule set. At this point in the simulation, the Flocking drones appear to be ahead of the L-F lead drone in this particular simulation situation. In addition, since the Flocking drones are dispersed simultaneously, their spatial distribution appears chaotic early in the deployment but becomes more concentrated as the rule set for each Flocking drone is followed, and they form a collective group alignment headed toward the target location.

Figure 8(b) shows the path traced for both L-F and Flocking drones in the presence of threats. The trace reveals how both swarm types are navigating around threats, displaying emergent behavior as they adapt to operational environment constraints. Tracking the position and motion of individual units in this simulation enables the measurement of unit performance efficiency for each swarm type. The data serves as the basis for quantifying the degree or level of emergence for each swarm type as proposed by other researchers (e.g., Shannon entropy, Chaos Composite, Bayesian classification).^46,47

Figure 9 shows the behaviors of each swarm model in the presence of increasing threats from separate simulations. The L-F drones are maneuvering successfully around the threats. The L-F drones are forming a line behind the leader, avoiding threat boxes while maintaining proximity with each other and separated by the assigned delay time, Δt. The follower drones form an effective attack line once the L-F leader breaks through the threat boxes, heading toward the target. Each follower only needed to follow the drone ahead to reach the target, even though each drone was still required to maneuver around any threats they encountered individually.

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Figure 9.

Swarm behavior for L-F and flocking in differing threat environments.

In contrast, the wide spatial dispersion of the Flocking drones results from individual drones avoiding threats along their path, making it difficult for group alignment to occur at this time. The drones essentially end up scattered in space as they individually search for clear paths, changing headings in the prescribed random direction while attempting to maintain relative positioning with one another as directed by their rules for alignment.

In several simulated cases, despite this observed dispersion, the Flocking drones reached the target with the necessary number of drones before the L-F model. The Flocking drones dispersed randomly and concurrently, limiting their movement by their given flocking characteristics, but still have the advantage of aligning in groups with several drones attached in formation, arriving at the target location in large numbers. In these cases, the L-F drones’ structured line formation proved a performance liability, resulting in increased time to get the established number of drones on target for success. However, the performance of the L-F model improves as the number of threats increases, while the Flocking drone model is penalized. With more threats, their movements are sufficiently disrupted, resulting in delays to their group alignment, making it challenging to achieve mission success compared to the L-F drones.

4.1. Simulation scenario results

The following sections detail the interpretation of the three scenarios’ simulation results. As discussed, the first scenario consisted of over 20,000 simulations where the variables shown in Table 1 were randomized. The time delay (Δt) randomly switched between 1 and 2 for each simulation for the L-F model. The variables involving Vision, minimum separation, and maximum turn, for the Flocking model, were also randomized within the given range shown on the table. The algorithm for each simulation randomly generates values for each control variable, making each simulation unique. The objective is to assess the sensitivity of each independent variable to the performance of the swarm models, measured as Time, the dependent variable.

Figure 10 shows a histogram for each control variable, with a normally distributed sampling profile producing equal opportunities for each randomly generated value. Two focused simulations were added based on the results of this first scenario. As a result of the exploratory data analysis conducted for Scenario 1, two additional scenarios were added. The Drones-to-Kill variable appeared to have the most impact on performance and, as such, was selected for additional exploitation in Scenarios 2 and 3. For these additional scenarios, 10,000 simulation runs were completed for each, with this variable set at 20 drones and 90 drones, respectively. As defined, this variable sets the number of drones of one swarm type required to enter the target zone for a successful mission and task completion. The first swarm type to achieve this is declared the winning swarm for that specific simulation, and as such, the swarm type achieving this feat is referred to as the “winner” in this section.

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Figure 10.

Histogram of sampled values for each simulation variable for both swarm models.

Figure 11 shows a density plot of time and threat level between the L-F and the Flocking drones for Scenario 1, while Figure 12 shows performance impact as a function of the number of drones required for the mission for this scenario. The density profile is relatively uniform for all threat levels, as shown in Figure 11 for the Flocking model, with high-density regions (yellow) centered at approximately 600 ticks (Time).

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Figure 11.

Density plot of Scenario 1 swarm type performance.

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Figure 12.

Observed performance relative to number of drones required for successful task.

In contrast, the L-F model generally exhibited lower time across all threat levels, with one concentrated region at the higher threat level. The Flocking model is more consistent across all threat levels with narrower density contours and maybe a more efficient choice for a mission with varying threat levels. The L-F model may be the preferred model for mission scenarios requiring faster task completion, with a mean Time advantage of approximately 50 ticks over the Flocking model. The impact of the Relative Threat is limited with no significant difference in Time for the Flocking model, while the L-F model Time advantage is at the highest threat level.

Figure 12 shows the performance impact of Drones-to-Kill, the user-defined metric for mission success. For the Flocking model, the density profile is relatively linear with Time, indicating a more predictable outcome for this model. In contrast, the L-F model shows higher density concentration at lower values of required drones but greater Time dispersion as the number of required drones increases.

Another representation of the data is shown in Figure 13, which shows a distinct time advantage of the L-F model across all levels of relative threat. The shaded area shows dispersion around the mean and relatively tight clustering around this mean. The Flocking model Time increases linearly with the Relative Threat while the L-F model remains flat, especially in the mid- to high-threat regions.

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Figure 13.

Smoothed line plot of L-F and flocking model performance across threat levels.

As discussed, Scenarios 2 and 3 compare performance for both swarm types as a function of the number of drones required for a successful mission, with the required drones at the target set at 20 and 90, respectively, to further examine performance impact.

Figure 14 compares the results from both scenarios. This figure shows two sets of curves: each set results from 10,000 simulations showing the number of drones of each type completing the mission successfully. The x-axis represents the operating condition for the number of drones at the target at three levels of Relative Threat (Low, Medium, and High). The y-axis is the number of wins by swarm model type. The set on the left is data from Scenario 2 showing the number of drones per swarm type with successful missions (completed the targeting task first) at a value of 20 (20 drones out of 100 required to complete the task). In this scenario, the L-F swarm model is dominant over the Flocking swarm, with the L-F swarm completing approximately 66% of the 10,000 simulation runs before the Flocking swarm model. The increasing Relative Threat levels show a minor deviation of results, but at the high-threat level, the L-F swarm showed a slight increase in mission completion over the Flocking model (+2%). In this environment requiring fewer drones at the target, the L-F swarm model performed consistently regardless of threat level, arguably due to its more structured coordination and control protocol and deliberative actions when fewer drones are required for task completion.

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Figure 14.

Comparing simulation results for required number of drones in Scenarios 1 and 2.

A reversal of results is noted in Scenario 3 when the number of required drones at the target value is changed to 90. The set on the right side of Figure 14 shows how the Flocking swarm model fared against the L-F model, with the Flocking model completing the task in almost 60% of the 10,000 simulations compared to the L-F model. However, as the Threat Level increased from Low to High, there was a noticeable increase in the percentage of L-F drones successfully completing the task compared to the Flocking model. The L-F swarm model win percentage increased by approximately 7% at the highest threat level.

A comparison of Time between the two models also showed decreasing Time with the L-F swarm model as the threat level increased. This is shown in Figure 15, a plot of the mean Time of the winning swarm type in any given scenario. The shift in results gives insight into which model behavior may be more efficient in particular applications. The L-F model may be an appropriate strategy for simpler environments requiring fewer assets, while for more complex scenarios requiring adaptability, the Flocking model is the right approach. As noted earlier, however, there is a penalty in Time with the Flocking model compared to the L-F swarm type.

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Figure 15.

Time versus relative threat comparison for both models in Scenario 3.

The Flocking model also used Vision, maximum Turn, and minimum Separation as control variables. The impact of Vision is discussed below, as this variable had a greater impact than the other two on performance.

Figures 16 and 17 show Vision’s density plot for the required number of drones at 20 and 90, respectively. The larger field of view results in drone alignment that reduces time-to-target, an expected outcome. At a required number of 20, the Flocking model converges quicker than when this variable was set at 90, with the Time of the peak densities centered around 550 at the low end compared to a Time of around 675 at the high end of required drones. The impact of Vision is more pronounced at this higher value of the number of required drones at the target and the lowest Vision level. The density profile is spread over Time, suggesting less coordinated behavior. The density bands are more concentrated at the medium- and high-vision levels, indicating better performance with larger fields of view for the drones. The most prominent performance change is between low and medium-level Vision, while the difference between medium and high vision may not be as significant. For example, if cost is an issue, the design engineers may choose the medium-level sensor since the performance difference may be inconsequential for the mission.

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Figure 16.

Impact of vision on flocking model performance at 20 required drones.

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Figure 17.

Impact of vision on flocking model performance at 90 required drones.

4.2. Interpretation of simulated results

Both models exhibit emergent behavior useful for a specific targeting scenario based on observed performance. As mentioned earlier, research of unmanned aerial vehicle (UAV) systems at AFRL identified the following emergent features as advantageous for swarm operations: (1) decentralized, (2) inherently implicit, (3) inherently resilient, and (4) inherently scalable. ⁷ Table 2 provides a summary of the comparison of both models in this research when measured against the four emergent behaviors listed in the AFRL study. ¹³

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Table 2.

Comparative Summary of Swarm Model Performance Against Four Identified Emergent Features.

Emergent feature	Leader–follower model	Flocking model
Decentralized—Eliminating the need for centralized operation, giving maneuver freedom to the individual unit	Mostly centralized. Followers are subjugated to the Leader drone, limiting full autonomy of individual units in the swarm	Decentralized. Each drone operates based on local interactions.Behavior is not driven by centralized control mechanism
Inherently implicit—Eliminating the need for direct control, simplifying the command structure, potentially eliminating significant human interaction and management	Hierarchical. Swarm system control is limited to the Leader drone and Followers are guided by Leader actions.When successful in completing the task, this model displayed faster time to target performance	Implicit coordination between drones based on three features of cohesion, alignment, and separation.Drone units react to geometric relationships inside the swarm formation.
Inherently resilient—Emergent behavior is tolerant of imperfections and variants in individual units	This model requires a survivable leader droneAt higher threat levels, the L-F drone improved performance with lower number of required drones	Distributed behavior of drones is tolerant to threat environment.More efficient across all threat levels with a tighter variance band, indicating consistency and predictability
Inherently scalable—Very robust to increase or decrease of individual UAS units in the mix	Scalable to the number of drones required for success. More tolerant at higher threat levels	Scalable to the number of drones required for success

The L-F model is more structured in its deployment and dependent on the leader drone’s ability to navigate toward the target. The Flocking model features decentralized behavior, as each drone can operate independently. Individual drones react to situations based on specific rules for threat avoidance and group alignment, resulting in swarm formations.

The Flocking model is more resilient to the loss of individual drones than the L-F model. The Flocking types behave independently and autonomously from each other, even if they require the cooperation of other drones to form an attack swarm to overwhelm a target. L-F drones are also robust but require a resilient and survivable leader drone to succeed.

Both are shown to scale regarding the number of drones required for task completion, even in a diverse environment with differing threats. Impact on drone performance based on the number of drones required and the threat level is evident, but a more involved data reduction may reveal an optimum combination of drones producing the most favorable time performance against a particular threat environment.

The number of required drones may be a critical variable for practical applications when speed and mission effectiveness are priorities. In other words, if the user requires task completion in the shortest time but has a limited supply of drones in inventory or operationally available, using L-F as the swarm model may be the most cost-efficient, especially at high-threat levels.

Based on these simulations, one possible system-level interpretation of the data is as follows: (1) the Flocking swarm architecture performs well in dynamic and distributed environments with uncertain threats and minimal time pressure in completing tasks and (2) the L-F swarm model is best suited for structured time-critical environments with high levels of threats that require fewer drones. In situations where the L-F model outperforms the Flocking swarm model, it does so much faster. In addition, there may be situations where a hybrid approach is advantageous. A hybrid system can maximize the probability of mission success by combining the unique strengths and capabilities of each swarm type within the architecture.

5. Research limitations

This article provides a comparative framework for examining different types of swarm models by visually characterizing emergent behavior simultaneously against a specified objective of targeting. Simulations were conducted to assess characteristic emergent behavior based on simple rules. As such, limitations are acknowledged in this research’s construct.

One limitation of this research is the inherent simplicity of agent models, which rely solely on basic movement commands such as “forward” and “turn.” These models do not incorporate individual drone characteristics, such as energy, power, or lifespan, that could contribute to additional emergent behaviors. Another limitation is the lack of active communication to enhance interaction between drones, which is constraining their ability to better adapt to an elevated threat environment.

Another constraint is the absence of “ground truth” data from a physical swarm system to validate model performance and behavior. Obtaining real-world data is technically impractical and cost-prohibitive at this research phase, and collecting it requires material and personnel resources not available in this research. Scalability presents additional challenges, particularly transitioning from a 2D worldview—often used in simulations—to a complex 3D model, more representative of the real world.

Still, considering the research questions posed for this research, the current implementation is sufficient to provide a basis for understanding emergent behavior. The next phase of this continuing research addresses a number of these limitations.

6. Conclusion

The modeling and simulation framework discussed in this paper facilitates comparative research of several drone types to identify the best candidate swarm architecture for a given application. While this early result is promising, they are limited to the simulations presented. Authoritative interpretations about swarm emergent behavior cannot be made until additional research is conducted and validated with real physical data from a live demonstration in real three-dimensional space.

Ongoing research in swarm intelligence highlights the potential of leveraging emergent behavior features of autonomous systems for various applications. Researchers discussed the advantages of accentuating positive features while minimizing destructive behavior in swarms. ¹⁵ However, past efforts to measure emergence focused on singular metrics, with no consensus on the best method.^46,47

This paper introduces a novel approach comparing the performance of two different types of swarms within a unified modeling and simulation environment. Although the specific use case is a military-oriented objective, the findings on emergence are relevant to other applications, including environmental monitoring, ⁴⁸ precision agriculture, ⁴⁹ and supply chain management. ⁵⁰

ABM is an ideal framework for visualizing and characterizing swarm behavior. The computational environment presented in this paper examines the emergent behaviors of the L-F and Flocking models.

The simulations showed that the flocking model is more efficient across all threat levels with a tighter variance band, indicating consistency and predictability when compared to the L-F model. However, the Flocking model lagged in its time performance in all three scenarios.

The authors acknowledge the limitations of the current research, as discussed in the previous section. To resolve, the following improvements are considered for continued research. (1) Completing a more robust validation of the computational framework. Other validation techniques are more effective for agent-based simulations relevant to this research.^43,51 (2) Adding active communications, sensing capability, and increased maneuverability producing higher level swarm intelligence. These increased capabilities may include as additional agent features in the modeling. (3) Introducing more dynamic (mobile) threats capable of destroying drones, enabling swarm resiliency as an emergent feature. The performance of the L-F model is especially noteworthy if the leader drone is eliminated in the process. (4) A third model based on “stigmergy.” This biologically inspired coordination mechanism optimizes the swarm’s path toward the target location using artificial pheromones, mimicking attractive and repulsive forces.^5,31,52,53 These planned enhancements generate enormous data sets. Multivariate analysis techniques like the Latin Hypercube sampling are planned to efficiently reduce data while ensuring a holistic assessment of swarm model dependencies. ⁵⁴

A tool for predicting outcomes is the ultimate goal, leveraging a classifier and machine learning algorithm to support real-time operation. Such a tool would enable planners and decision-makers to visualize and characterize swarm emergence, providing effective decision-making tools for operational planning and resource allocation. Simulated data play a critical role in this process, forming the basis of a dynamic data-driven application system. The system is integral to an operational user’s critical command and control system. ⁵⁵ Researchers and users can perform more robust statistical analyses to assess differences among swarm types by collecting extensive data for various combinations of control variables for all swarm models. The choice of model depends on mission objectives, resource constraints, and time limitations imposed by the user. Results from these simulations offer developmental engineers’ insights for fine-tuning physical drone system parameters, such as payload sensors and propulsion systems, for cost-effective solutions.