2.1 Bio-inspired swarm control and Reynolds’ Rules

The foundational model for decentralized coordination is derived from Reynolds’ “boids” simulation, which demonstrated that complex aggregate motion emerges from three local heuristics: separation, alignment, and cohesion [15]. In robotic swarms, these rules are implemented by sensing neighbors within a finite radius and updating motion commands based on local positions and velocities [30]. Concretely, separation enforces a minimum inter-agent distance to prevent collisions; alignment synchronizes velocity vectors; and cohesion drives agents toward the local centroid to maintain group integrity.

While attractive for their simplicity, directly porting these abstraction-heavy rules to physical platforms such as Segways poses significant challenges. Real-world implementation must account for sensing noise, communication latency, and non-holonomic kinematic constraints, which prevent instantaneous lateral motion. To address these physical realities, researchers have explored adaptive weighting schemes, in which the influence of separation versus cohesion varies dynamically with environmental density [30], and evolutionary optimization methods to automatically tune flocking parameters [31]. These heuristic approaches frequently lack rigorous mathematical proofs of stability, leaving the system vulnerable to oscillatory behavior in cluttered spaces.

2.2 Lagrangian swarm models

To provide a rigorous mathematical basis for coordination, the literature distinguishes between Eulerian (continuum) and Lagrangian (agent-based) modeling approaches [14]. In the Eulerian approach, the swarm is considered a continuum described by its density in one-, two-, or three-dimensional space. In contrast, the Lagrangian approach is an individual-based method that explicitly studies the state (position, instantaneous velocity, and instantaneous acceleration) of each individual and its relationship with other individuals in the swarm. Within this framework, the velocity and acceleration of each agent can be directly influenced by its spatial coordinates. The planar Lagrangian swarm model developed in this research is based on the principle that swarming results from an interplay between long-range attraction and short-range repulsion among individuals. A common design paradigm combines these short-range repulsion potentials with long-range attraction potentials [30]. Repulsive terms, often modeled as inverse barrier functions, increase sharply at small separations to guarantee collision avoidance, while attractive terms guide agents toward the swarm centroid or target. Tuning the relative gains of these potentials governs the swarm’s density and formation shape.

2.3 Navigation and obstacle avoidance strategies

2.4 Research gap and contributions

Table 1 presents a comparative analysis of key studies in the field, highlighting the specific gaps addressed by this research. While prior works have addressed UAV swarms or single Segways, there remains a lack of rigorous, acceleration-based control frameworks for Segway swarms operating in environments with heterogeneous (disk and line) shaped static constraints.

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Table 1. Comparative Analysis of Key Studies and Advancements.

https://doi.org/10.1371/journal.pone.0356369.t001

The comparison highlights the distinct contributions of the present study. By moving from velocity to acceleration control and from simple disk worlds to complex line-based geometries, this work bridges the gap between theoretical swarm stability and practical urban mobility requirements.

3.1 The scenario: Decentralized Urban swarms

Consider a high-density urban environment where a fleet of personal transporters must navigate a shared, constrained workspace. Unlike macroscopic traffic flow models, this system is analyzed through a Lagrangian framework, treating the swarm as a collection of discrete, microscopic agents. The agents in this system are Segway Personal Transporters, characterized by underactuated, nonholonomic kinematic constraints.

The specific problem addressed is the safe and coordinated navigation of a homogeneous swarm of Segway robots operating within a constrained two-dimensional workspace . The environment is populated by heterogeneous static constraints, geometrically modeled as disks (representing pillars, wells, or roundabouts) and line segments (representing walls, curbs, or corridors), as well as independent dynamic obstacles representing moving pedestrians or uncooperative vehicles. The fundamental challenge is to derive a distributed control law that follows the interplay between long-range attractive forces and short-range repulsive forces. This control law must drive the swarm from an arbitrary set of initial states to a desired equilibrium configuration, a process formally described as consensus-based rendezvous, while strictly satisfying kinematic feasibility and safety guarantees in a time-varying environment.

3.2 System modeling assumptions

To ensure the feasibility of the control design while maintaining practical relevance, the following assumptions are established based on standard swarm theoretic norms:

1. Lagrangian Formulation: The swarm is treated as a set of distinct agents indexed by , where the state of each agent is governed by its own ordinary differential equations.
2. Complete Interaction Topology: The communication network is modeled as a complete graph (). Every agent i has access to the state information of every other agent j, allowing for the instantaneous computation of the swarm centroid to enforce cohesion.
3. Semi-Dynamic Environment: The obstacle field comprises both time-invariant geometric constraints (static disks and lines) and time-varying, independent dynamic obstacles.
4. Flat Terrain: The motion is restricted to the horizontal plane, neglecting vertical displacement or pitch/roll dynamics beyond the scope of planar navigation.

To formalize the problem, let z₁ and z₂ denote the orthogonal spatial coordinates representing the horizontal and vertical axes of the 2D Cartesian workspace , respectively. The following definitions are established:

Definition 3.1 (Swarm Target). The target for the Segway swarm’s centroid is a disk with center and radius . It is described as the set:

(1)

Definition 3.2 (Agent State). The state of the ith Segway in the swarm, for , is described by its Cartesian position , its orientation angle , and the angular velocities of its wheels, and .

Definition 3.3 (Static Disk Obstacle). The kth solid stationary obstacle is a disk with center and radius . It is described as the set:

(2)

Definition 3.4 (Static Line Obstacle). The mth line obstacle is the line segment from the point to the point and is described as the set:

(3)

where .

Definition 3.5 (Swarm Centroid). The centroid of the swarm of Segway robots is denoted by , where

(4)

Definition 3.6 (Dynamic Obstacle). The hth dynamic obstacle, for , is modeled as a moving rigid body residing in a disk with a time-varying center and radius . It is described as the set:

(5)

Definition 3.7 (Dynamic Obstacle Target). The target for the hth dynamic obstacle is a stationary disk with center and radius . It is described as the set:

(6)

3.3 Kinematic model of segway robot

The system under consideration is a swarm composed of autonomous Segway robots. The kinematic model governing the ith agent, as illustrated in Fig 2, is a modification of the formulation established in [22].

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Fig 2. Schematic diagram of the ith Segway robot modified from [22].

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The position of the ith Segway robot is defined by the coordinates , which represent the robot’s center of mass (located at the handlebar). The orientation is given by the heading angle relative to the horizontal axis. Geometrically, the agent is characterized by a wheel track width of b and a wheel radius of r/2. The angular velocities of the left and right wheels for the ith agent are denoted by and , respectively. To ensure operational safety, each agent is enclosed within a circular protective region of radius for collision avoidance.

The motion of the Segway agents is governed by non-holonomic constraints arising from the assumption of pure rolling without slip. Physically, this imposes two specific restrictions on the wheel-ground interaction:

1. No Lateral Slip: The instantaneous velocity of the wheel contact point in the lateral direction must be zero, preventing the robot from sliding sideways.
2. Pure Rolling: The longitudinal velocity of the wheel axis is determined strictly by the angular rotation of the wheels.

These constraints couple the robot’s translational velocity components to its orientation and wheel velocities. Since the system state tracks the center of mass (offset by distance d from the wheel axis), the no-slip condition enforces a kinematic coupling where the robot’s lateral motion is inextricably linked to its angular velocity. Solving these constraint equations yields the specific kinematic model described by the system of ordinary differential equations:

(7)

where:

• , : Angular accelerations of the left and right wheels of the ith Segway, respectively.
• c = r / (2b), where r is the wheel radius and b is the distance between the wheels.
• d is the distance from the handle to the centre of the axle connecting the wheels.

The control inputs for each Segway are the wheel accelerations, and .

To model the moving entities within the workspace, each dynamic obstacle is treated as an independent point-mass system controlled by its acceleration. The motion of the hth dynamic obstacle, where , is described by its Cartesian position and its translational velocity components .

The movement of each dynamic obstacle is governed by its own set of control inputs, specifically the accelerations and in the x and y directions, respectively. The kinematic model is defined by the following system of ordinary differential equations:

(8)

3.3.1 Global kinematic model.

The complete workspace consists of the autonomous swarm and the independent dynamic obstacles. By aggregating the non-holonomic kinematics of the n Segway agents and the translational kinematics of the p dynamic obstacles, the global kinematic model for the entire augmented system is defined by the following system of ordinary differential equations:

(9)

for all swarm agents and all dynamic obstacles .

3.3.2 System state vectors.

To represent the complete environment, comprising the autonomous swarm and the independent dynamic obstacles, an augmented global state vector must be defined.

The state vector for the ith Segway swarm agent remains:

(10)

The state vector for the hth dynamic obstacle is defined as:

(11)

The global state of the entire system, denoted by the augmented vector , concatenates the states of all Segway agents and all dynamic obstacles:

(12)

3.3.3 Initial and equilibrium configurations.

The system initializes at time from a specified condition . The initial state vector for the ith Segway agent is defined as:

(13)

The initial state for the entire swarm, , consists of the stacked individual initial vectors:

(14)

Similarly, the initial state for the hth dynamic obstacle is:

(15)

and the initial state for all p dynamic obstacles is the stacked vector:

(16)

The global initial state for the complete augmented system is the concatenation of the swarm and dynamic obstacle initial states:

(17)

The desired final configuration is the stable equilibrium point, , where all motion in the environment ceases and control objectives are satisfied. For the ith Segway agent, this state is given by:

(18)

where and represent the final position and orientation satisfying the formation requirements. The equilibrium state for the entire swarm is:

(19)

This state incorporates a stable equilibrium where the swarm centroid aligns with the global target, implying:

(20)

For the hth dynamic obstacle, the equilibrium state means it has reached its specific target with zero velocity:

(21)

The collective equilibrium for all dynamic obstacles is:

(22)

Finally, the global equilibrium point for the entire augmented system is the stacked vector:

(23)

At this state , the swarm satisfies its formation and target constraints, all dynamic obstacles rest at their respective targets, and all velocities in the system are strictly zero.

3.3.4 System representation.

The instantaneous accelerations governing both the Segway agents and the dynamic obstacles are determined by a combined feedback control law. This control law, denoted as , aggregates the required inputs for every agent within the workspace.

For the ith Segway agent, the control inputs are the angular accelerations of the right and left wheels, and . For the hth dynamic obstacle, the control inputs are the translational accelerations and . The unified control vector is defined as:

(24)

The dynamics of the complete environment consisting of the Segway swarm kinematics and the dynamic obstacle motions represented by equation (9) are represented by an augmented vector field . By defining the system’s vector field such that is embedded within , the complete autonomous motion of the system can be expressed in the compact state-space form:

(25)

3.4 Control objectives

To solve the described problem, the following specific control objectives are established:

1. To formulate a novel set of nonlinear, time-invariant, continuous acceleration-based controllers that govern the motion of each agent within the autonomous Segway swarm.
2. To achieve reliable, coordinated navigation of the swarm from a given set of initial configurations to a desired final swarm centroid configuration in an environment containing both static geometric constraints and moving entities.
3. To employ the LbCS framework to formally guarantee inter-agent collision avoidance, static obstacle avoidance, and dynamic obstacle evasion, while simultaneously enforcing swarm cohesion.
4. To validate the system’s performance through rigorous mathematical proofs of stability, asymptotic controllability, and convergence for the entire multi-agent system.

4.1 Design of acceleration controllers

The swarm of Seqway’s are cluttered with line and dynamic obstacles. The swarm should successfully navigate the workspace to reach the equilibrium. The primary objective of this section is to design the feedback control law , embedded within the vector field . The controllers are synthesized to guarantee that the swarm system dynamics remain stable and that the state converges to the desired equilibrium configuration . To achieve this, the acceleration controllers for each agent are systematically derived using a composite Lyapunov function, constructed from a set of attractive and repulsive potential functions. The following functions are the foundational components for constructing the total potential.

4.1.1 Target attraction and auxiliary functions.

To guide the swarm’s centroid to its target , the following target attraction function is utilized:

(26)

4.1.2 Swarm cohesion function.

To maintain long-range attraction and the cohesion of the swarm, the function attracts each Segway towards the swarm’s centroid:

(27)

The control parameter determines the eccentricity of the swarm’s formation by weighting the attraction in the y-direction relative to the x-direction. Adjusting allows the swarm to assume different spatial configurations, such as circular, elliptical, or linear formations.

4.1.3 Inter-Agent Collision Avoidance.

To ensure short-range repulsion, the function

(28)

is defined to prevent collision between the ith and jth agents.

4.1.4 Stationary obstacle avoidance functions.

• Disk obstacle avoidance: For the ith swarm agent to avoid the kth static disk obstacle, the function (29)
  will be utilized.
• Line obstacle avoidance: To confine the swarm agents within designated boundaries, such as pavement edges, walls of buildings, or corridor walls, these boundaries are modeled as straight-line segments. Consistent with the Minimum Distance Technique (MDT) [29], the ith agent avoids the closest point on the mth line segment.

Let the mth line segment be defined by the endpoints and . The coordinates of the closest point, , are determined parametrically as:

(30)

Using these coordinates, the following avoidance function is developed to ensure the ith agent maintains a safe distance from the mth line:

(31)

The projection parameter ensures the closest point lies strictly within the segment boundaries. It is defined as:

(32)

where is the calculated projection of the agent’s center onto the line vector and is given by

(33)

4.1.5 Centroid-based line obstacle avoidance.

To ensure the entire swarm avoids line obstacles, the function creates a repulsive potential based on the swarm’s centroid. The function

(34)

is used, where represents the closest point on the mth line obstacle to the swarm’s centroid. These coordinates are determined parametrically as

(35)

The projection parameter ensures the closest point lies strictly within the segment boundaries. It is defined as:

(36)

where is the projection of the centroid onto the line vector and is given by

(37)

4.1.6 Artificial obstacle (Velocity Barrier) functions.

To enforce physical velocity limits, artificial obstacles are used. For a maximum angular wheel velocity of , which is assumed to be uniform for all Segways in the swarm, the barrier functions are:

(38)

These barrier functions translate the physical speed limits of the Segway’s motors into a mathematical construct that the LbCS can interpret as an obstacle. As visualized in Fig 4, this creates a “Free Space” of allowable angular velocities for the right and left wheels, bounded by a prohibited “Obstacle Space” representing speeds beyond the maximum limit, .

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Fig 4. Visualization of the artificial velocity barriers.

The central “Free Space” represents the safe operational range for each wheel’s angular velocity, while the shaded “Obstacle Space” represents prohibited speeds that the controller is designed to avoid.

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In the total potential function, these barriers are implemented as inverse functions (e.g., ). As a wheel’s angular velocity () approaches its maximum (), the value of approaches zero, causing the inverse term to grow towards infinity. This creates a steep “energy wall” that generates a strong repulsive force in the controller’s calculations. This force proactively commands deceleration, ensuring that angular velocity limits are never violated and that all control actions remain within the robot’s physical capabilities.

4.2 Dynamic obstacle avoidance and navigation functions

To model a realistic smart city environment, an asymmetric (one-way) avoidance scheme is implemented. The Segway swarm agents actively detect and avoid the dynamic obstacles. However, the dynamic obstacles that represent entities, such as pedestrians or manually driven vehicles, navigate toward their respective targets independently. They avoid static boundaries and each other, but operate with zero knowledge of the Segway swarm, meaning they do not yield to the autonomous robots.

4.2.1 Swarm to dynamic obstacle avoidance.

For the ith Segway agent to actively avoid the hth dynamic obstacle in its path, the following repulsive avoidance function is incorporated into the swarm’s control architecture:

(39)

for and . As the dynamic obstacle approaches the Segway’s operational radius, this function approaches zero, generating a strong repulsive force that alters the Segway’s trajectory.

4.2.2 Dynamic obstacle target attraction and auxiliary functions.

To ensure the convergence of the hth dynamic obstacle to its designated target, the following radically unbounded function is employed:

(40)

To guarantee that the nonlinear acceleration controllers of the hth dynamic obstacle vanish solely at its target equilibrium position, the corresponding auxiliary function is defined as:

(41)

4.2.4 Dynamic obstacle velocity constraints.

Translational velocity constraints for the dynamic obstacles are incorporated via artificial barrier functions. Let and denote the maximum allowable velocities. The avoidance functions are defined as:

(45)

4.3 Total potential

A Lyapunov function, , combines the component functions with positive gains. The function is designed using inverse barrier functions for repulsion and auxiliary functions for scaling, ensuring the potential vanishes only at the desired equilibrium state. The Lyapunov function for the Segway swarm system and dynamic obstacle with an environment is:

(46)

where

(47)

is an auxiliary function ensuring all the repulsive potentials become zero only when the swarm centroid reaches its target, and the segway agents’ velocity reaches:

4.4 Controller design

The control inputs are derived to ensure the system satisfies the Lyapunov stability criterion . The derivation commences with the time derivative of the Lyapunov function along the trajectories of the kinematic model defined in (9). Using the chain rule:

(48)

Substituting the kinematic equations from (9) and grouping the terms by the respective velocity components yields:

(49)

where and map the spatial partial derivative terms for the ith Segway agent:

(50)

To enforce asymptotic stability for the augmented system, the analytically derived time derivative of the total potential is set equal to a negative semi-definite form. By introducing strictly positive convergence parameters for the Segway agents, and for the dynamic obstacles (), the target condition is established as:

(51)

By equating the velocity-grouped terms from the analytical expansion to this negative semi-definite target, the individual acceleration components can be isolated. For example, equating the right wheel velocity term for the ith Segway yields:

(52)

Solving this algebraically for all control inputs results in the final decentralized nonlinear feedback control laws. The acceleration controllers governing the right and left wheels of the ith Segway agent are:

(53)

(54)

Concurrently, the translational acceleration controllers dictating the autonomous navigation of the hth dynamic obstacle are extracted as:

(55)

(56)

5.1 Simulation framework and units

The simulations were conducted within a dimensionless framework to ensure the generality of the results. All physical quantities are expressed in terms of Simulation Units (SU). For example, a distance is measured in spatial SU, and time is measured in temporal SU. A linear velocity reported by the simulation represents the movement of spatial SU per temporal SU.

This dimensionless approach allows the results to be mapped to a specific real-world physical scenario by defining a characteristic length, (e.g., in meters), and a characteristic time, (e.g., in seconds). A dimensionless linear velocity from the simulation, , can then be converted to a physical velocity, , using the relationship:

(57)

All angular quantities within the simulation, such as the orientation , are expressed in radians. A simulated angular velocity, (e.g., for or ), represents an angular displacement in radians per temporal SU. This can be converted to a physical angular velocity, , by scaling with the inverse of the characteristic time:

(58)

This allows the generalized mathematical model to predict behavior in concrete situations with meaningful physical units (e.g., m/s or rad/s). Unless otherwise specified, all values reported in the following scenarios are in their direct Simulation Units.

5.2 Scenario 1: Consensus-based navigation in heterogeneous clutter with no Dynamic obstacles

This scenario evaluates the distributed control framework within a high-density environment designed to test cooperative maneuvering and formation control. A homogeneous swarm of n = 6 Segway agents is initialized with an arbitrary spatial distribution and tasked with performing a consensus-based rendezvous, resulting in a leader-follower formation.

The simulation workspace is populated with heterogeneous static constraints, comprising distinct geometric primitives: circular obstacles (pillars), convex polygons (triangles, hexagons, pentagons), and non-convex U-shaped barriers. This configuration compels the multi-agent system to deform its spatial arrangement to navigate narrow spaces before re-establishing formation integrity at the target. The specific initial states, obstacle geometric parameters, and consensus gains are detailed in Table 2.

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Table 2. System parameters and environmental configuration for Scenario 1.

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The simulation results are presented in Fig 5. Fig 5(a) illustrates the collision-free spatial evolution of the agents in the plane. The agents exhibit emergent leader-follower behaviors to navigate the obstacle field, prioritizing collision avoidance during the transient phase before asymptotically converging to the desired linear formation.

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Fig 5. Swarm navigation performance in Scenario 1.

(a) Complete spatial trajectories showing the transition from arbitrary initial poses (green circles) to the desired linear target formation. The blue lines track individual agents successfully executing collision-free maneuvers around heterogeneous static constraints (including disks, polygons, and a non-convex U-shaped barrier), while the dashed black line traces the swarm centroid’s path. (b) Time evolution of the heading angles . The significant oscillations observed during the transient phase (t < 8000 SU) quantitatively reflect the active steering adjustments mandated by the non-holonomic kinematics for lateral obstacle avoidance. The subsequent asymptotic synchronization to steady-state values confirms robust orientation consensus as the swarm completely clears the clutter and aligns with the global target.

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Fig 5(b) presents the time history of the agent heading angles, . The oscillations observed in the transient phase (t < 8000) are attributed to the active steering maneuvers required by the non-holonomic constraints of the Segway platform; lateral displacement for obstacle avoidance necessitates orientation changes. The evolution of these heading angles illustrates the emergent behavior resulting from the interplay of the continuous Lyapunov potential functions. The initial sharp spikes in correspond to the activation of the short-range repulsive potentials due to static obstacles and the strict enforcement of inter-agent separation. These repulsive potentials force the Segway agents into rapid steering maneuvers to prevent collisions. As time evolves and the swarm clears the cluttered region, long-range attraction dominates. The cohesion potential forces the agents to cluster around the centroid, ensuring centroid convergence. Furthermore, the heading angles asymptotically synchronize to a common bearing, indicating mutual alignment. This orientation consensus signifies that the agents have successfully transitioned from independent avoidance maneuvers to a unified, cooperative trajectory, ultimately achieving global target convergence and reaching their equilibrium point as a well-spaced, cohesive group.

The robustness of the control framework is validated through the quantitative energy analysis presented in Fig 6. Fig 6(a) depicts the time evolution of the composite Lyapunov function, . Significantly, the decay remains strictly monotonic throughout the interaction with the heterogeneous constraints; while the obstacle-avoidance potentials generate local repulsive forces, they do not violate the energy-dissipation condition. The asymptotic convergence of to zero confirms that the system reaches the desired equilibrium state without exhibiting oscillatory behavior. Fig 6(b) confirms the negative semi-definiteness of the Lyapunov derivative, . The trajectory remains strictly non-positive () throughout the simulation horizon. In accordance with LaSalle’s Invariance Principle, this guarantees graphically that the system states converge to the largest invariant set where (the equilibrium manifold), proving that the obstacle avoidance terms do not induce limit cycles.

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Fig 6. Quantitative stability verification for the multi-agent system in Scenario 1.

(a) Evolution of the total Lyapunov potential . The system initializes with high potential energy () due to the distributed starting states, followed by a strict, monotonic decay. The asymptotic convergence to zero as SU confirms the swarm reaches the global equilibrium without entrapment in local minima. (b) Time evolution of the Lyapunov derivative . The trajectory exhibits an immediate, sharp negative peak () during the initial transient phase (t < 200 SU), reflecting the aggressive energy dissipation during early obstacle avoidance and formation assembly. Crucially, the curve remains strictly non-positive () throughout the entire simulation, satisfying asymptotic stability.

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The time evolution of the linearized velocities for the six swarm agents is depicted in Fig 7. The variations correspond to the active acceleration and deceleration maneuvers necessitated by the heterogeneous obstacle field, specifically the negotiation of non-convex U-shaped barriers and polygonal constraints. The sudden transient fluctuations in the velocity profiles illustrate the dynamic interplay between the attractive and repulsive potential functions. As agents approach static obstacles, a short-range repulsive force is generated, helping them navigate without collisions. The attraction potentials actively allow the forward momentum to ensure that the strict non-holonomic velocity bounds of the Segway platforms are not violated. This rapid velocity change is a direct consequence of the Lyapunov-based design, which actively enforces braking during tight avoidance maneuvers to guarantee rigorous collision avoidance before safely resuming steady-state convergence toward the target. As the agents clear the clutter and establish the leader-follower formation (approx. t > 10000 SU), the velocities monotonically decay. The asymptotic convergence of all agent velocities to zero () as confirms graphically that the swarm achieves a stable equilibrium at the target configuration.

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Fig 7. Time evolution of the linearized velocities for the six agents in Scenario 1.

At t = 0 SU, the agents initialize and rapidly accelerate, with peak velocities reaching approximately 0.16 SU/SU time. The pronounced oscillations observed during the transient phase (t < 500 SU) quantitatively reflect the active acceleration and braking maneuvers required to negotiate the heterogeneous obstacle field, including non-convex U-shaped barriers. As the swarm clears the clutter and establishes the leader-follower formation, the velocities smoothly dampen. The final asymptotic convergence of all agent velocities to zero () as SU confirms the global asymptotic stability of the swarm at the target rendezvous configuration.

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5.3 Scenario 2: Robust navigation in semi-enclosed heterogeneous environments with no dynamic obstacles

Following the validation of the distributed control framework in open heterogeneous environments, Scenario 2 examines the system’s robustness and navigational efficacy within highly constrained, non-convex topologies. Specifically, this scenario evaluates the acceleration controller’s capability to resolve local minima problems inherent to semi-enclosed spaces. A swarm of n = 6 non-holonomic Segway agents is initialized within a warehouse-like enclosure, bounded by static linear constraints that form a geometric trap. To further increase the navigational complexity, the external environment is populated with a dense field of 13 static disk obstacles and 10 linear barriers. This configuration establishes a complex potential energy landscape, effectively segregating the agents from the target centroid at and necessitating advanced obstacle-avoidance and formation-deformation maneuvers. To minimize redundancy, only the parameters that differ from the baseline configuration in Scenario 1, specifically the initial positions, target coordinates, and velocity limits, are detailed in Table 3.

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Table 3. Parameters for Scenario 2, listing only the values changed from Scenario 1.

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The primary objective is to verify that the acceleration controllers derived from the LbCS can successfully guide the agents out of the enclosure and converge towards the target. The framework must ensure a configuration free of local-minima traps while strictly maintaining safety with respect to inter-agent and obstacle collisions. The total potential function for this constrained scenario exhibits strictly monotonic decay, similar to the graphical stability characteristics presented in Scenario 1.

The agent trajectories and positional states illustrated in Fig 8(a) demonstrate the swarm’s emergent capability to adapt its spatial topology to environmental constrictions. Specifically, a formation split-and-merge maneuver is observed as the swarm encounters the dense disk obstacle field at t = 5300 SU. The interplay between the inter-agent repulsion terms and the boundary potentials forces a spontaneous deformation of the swarm, allowing it to avoid the disk obstacle.

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Fig 8. Simulation results for Scenario 2, illustrating swarm navigation through a constricted passage.

(a) Agent trajectories and positional states. The position superimposed at time steps t = 0, 900, 5300, and 50000 SU to illustrate how the six-agent swarm avoids static obstacles and the enclosure boundaries. (b) Time evolution of the heading angles for all six agents. The initial fluctuations result from steering actions required to avoid the enclosure, while the subsequent convergence indicates alignment with the target.

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The time evolution of the agent heading angles, , presented in Fig 8(b), provides further insight into the control effort. The initial phase (t < 5000 SU) is characterized by high-frequency steering adjustments. These oscillations correspond to the “negotiation phase,” in which agents actively balance repulsive forces. As the swarm clears the obstacle field and enters free space (t > 15000 SU), the heading angles asymptotically synchronize to a common angle, indicating that the swarm has achieved orientation consensus.

Finally, the stability of the system is corroborated by the time evolution of the linearized velocities, , as depicted in Fig 9. The initial phase (t < 6000 SU) is characterized by rapid, high-amplitude modulations. These transients correspond to the aggressive acceleration and braking maneuvers required to resolve the conflicting potential fields within the trap and the subsequent obstacle field. As the swarm agents navigate through the narrow enclosure exit, both inter-agent separation and repulsive potentials are effective due to static obstacles. The sharp dips in the velocity profile occur because of these steep, only momentarily present, repulsive potentials. The vanishing velocity profile () confirms graphically that the system achieves a stable equilibrium at the desired configuration without exhibiting limit cycle behavior or residual oscillations.

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Fig 9. Time evolution of the linearized velocities for the six agents in Scenario 2.

At t = 0 SU, the agents initialize from rest and exhibit sharp acceleration spikes (reaching up to approximately 0.05 SU) as they react to the repulsive potentials of the semi-enclosed trap. The phase of high-amplitude modulations (t < 6000 SU) directly corresponds to the aggressive braking and acceleration maneuvers necessary to negotiate the dense field of 13 disk obstacles outside the trap. Once the swarm clears these constraints (t > 7000 SU), the velocities exhibit a smooth, monotonic decay. The final asymptotic convergence to zero () as SU confirms graphically that the system achieves global stability at the designated target configuration.

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5.4 Scenario 3: Distributed assembly and corridor navigation with no dynamic obstacles with no dynamic obstacles

Scenario 3 examines the framework’s versatility through a distributed rendezvous problem, comparable to a decentralized logistics operation in which agents are dispatched from spatially distinct nodes (Shops 1–5) to a single customer location. The objective is to evaluate the effectiveness of the acceleration-based controllers in a dense environment. Fig 10 illustrates the Segway positions at six distinct time steps (t = 0, 120, 1200, 3600, 6000, 15000 SU), with the simulation parameters detailed in Table 4.

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Table 4. System parameters and environmental configuration for Scenario 3.

https://doi.org/10.1371/journal.pone.0356369.t004

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Fig 10. A sequence of snapshots illustrating the multi-agent rendezvous and corridor navigation task in Scenario 3.

At t = 0, SU, the five agents are distributed across distinct nodes (Shops 1–5, indicated by orange circles). Driven by cohesion potential, the agents rapidly aggregate into a cohesive swarm (t = 120 SU) before entering the narrow corridor. A critical key trajectory feature occurs around t = 3600 SU, where the swarm executes a smooth split-and-merge maneuver to bypass the central blue disk obstacle. The sequence concludes with the successful asymptotic convergence of the entire formation at the designated target location (dashed red circle) by t = 15000 SU. The dashed black line traces the collision-free trajectory of the swarm’s centroid.

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The system dynamics exhibit a distinct two-phase kinematic evolution driven by the superposition of the potential functions. In the initial phase ( SU), the formation cohesion term dominates the control effort. The aggregation observed in Fig 10 illustrates the complete interplay of the multiple attractive and repulsive potential functions comprising the Lyapunov formulation. The system is driven by a dual-attraction mechanism: the cohesion potential () pulls individual agents inward to ensure rapid centroid convergence, while the global attraction potential () continuously pulls this swarm centroid toward the final destination. Simultaneously, various repulsive potentials remain continuously active. Short-range inter-agent separation potentials strictly prevent internal collisions as agents densely cluster, while the swarm safely navigates around static obstacles due to the interplay of repulsive potential functions.

Subsequently, the swarm navigates the corridor as a unified entity. Upon encountering the disk obstacle, the system exhibits a split-and-merge maneuver. The interaction between the obstacle-repulsive gradient and the inter-agent potentials forces a separation of the agent formation to avoid the disk obstacle. Following obstacle clearance, the formation effectively regains cohesion and asymptotically converges to the equilibrium, validating the system’s capability to execute complex coordination tasks in structured environments.

The stability of the convergence is corroborated by the linearized velocity profiles shown in Fig 11(a). The velocities exhibit a smooth, monotonic decay driven by the dissipative damping terms (). However, the transient spikes in both velocity and heading angle observed between t = 2000 and t = 6000 SU directly correlate to the split-and-merge maneuver. During this phase, the gradient of the disk obstacle’s repulsive potential forces a lateral divergence, which mathematically necessitates a temporary increase in linear velocity and a sharp deviation in heading to satisfy the under-actuated kinematic constraints before the formation can reunite. The asymptotic convergence to zero ( as ) confirms graphically that the system possesses global asymptotic stability at the target configuration.

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Fig 11. Control effort and state evolution for the five-agent swarm in Scenario 3.

(a) The linearized velocities of the agents. The initial transients (t < 6000 SU) correspond to the acceleration required for formation assembly and the split-and-merge maneuver around the disk obstacle. The subsequent asymptotic decay to zero () confirms the system’s global stability at the target configuration. (b) Time evolution of the heading angles . The significant oscillations observed during the interval SU reflect the active steering adjustments necessary to negotiate the corridor constraints and the disk obstacle. The final convergence to a common steady-state value indicates that orientation consensus is achieved as the swarm stabilizes in the safety zone.

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The system’s dynamic response is further quantified by the temporal evolution of the agent states. Fig 11(b) presents the heading angles . As the swarm exits the corridor constraints, the heading angles asymptotically synchronize to a common steady-state value, confirming that orientation consensus is robust.

5.5 Scenario 4: Swarm navigation in a dynamic environment

To validate the framework’s safety in active urban environments, Scenario 4 evaluates the swarm’s performance when encountering three independent moving entities (). In this asymmetric avoidance task, a swarm of n = 15 Segway agents initializes at the top of the workspace and navigates toward a global target at . The agents must avoid a static environment comprising 7 disks and 11 line barriers, while simultaneously avoiding inter-agent and dynamic obstacle collision.

These dynamic obstacles autonomously navigate toward their respective targets while actively avoiding the static environmental constraints. A one-way avoidance scheme has been used, in which swarm agents avoid only the dynamic obstacles. Evaluating this asymmetric formulation is highly relevant for validating real-world urban scenarios. First, it accurately models encounters with uncooperative or unaware entities commonly found in smart cities, such as distracted pedestrians, manually driven legacy vehicles, or animals. Second, it serves as a rigorous worst-case stress test for the proposed LbCS framework; by forcing the swarm to shoulder the entire physical burden of collision avoidance, it demonstrates the controller’s absolute safety guarantees without relying on the cooperative maneuvering of external agents. Finally, it reflects practical operational hierarchies where the swarm must actively yield the right-of-way to higher-priority or heavily constrained dynamic entities. The swarm possesses complete real-time state information of all entities within the workspace. The Segway agents rely entirely on the dynamic repulsive potential to adjust their trajectories and avoid collisions with these non-reactive moving obstacles. Table 5 summarizes the configuration, environmental constraints, and specific control gains utilized for the multi-agent system in Scenario 4.

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Table 5. System parameters and environmental configuration for Scenario 4.

https://doi.org/10.1371/journal.pone.0356369.t005

Fig 12(a) illustrates the initial setup of the workspace, detailing the initial positions and respective targets for both the Segway swarm and the three dynamic obstacles. Fig 12(b) presents the complete spatial evolution of the system. The blue curves represent the trajectories of the swarm agents, confirming that the swarm successfully avoids the static constraints and converges at the global target while maintaining a strict safety distance from the dynamic obstacles and DO₃. The most critical interaction occurs in Region A, where the swarm’s planned trajectory intersects directly with the first dynamic obstacle (DO₁). As observed in the spatial evolution in Fig 12(b), the proximity of this moving entity triggers the dynamic repulsive potential, forcing the swarm into a rapid local deformation to ensure collision avoidance before resuming its path to the target.

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Fig 12. Global workspace configuration and spatial evolution for Scenario 4.

(a) The initial spatial distribution of the n = 15 Segway agents alongside the three independent dynamic obstacles () and their respective designated targets. (b) The complete trajectory trace of the multi-agent system. The blue curves represent the collision-free paths of the Segway agents, illustrating their successful navigation through heterogeneous static constraints (disk and line barriers) and their active evasion of the dynamic obstacles (red trajectories) prior to converging to global target equilibrium .

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The avoidance maneuver within Region A is detailed via snapshots in Fig 13. As DO₁ approaches the swarm’s centroid between t = 1010 and t = 1300, the repulsive potential forces the leading agents to widen their formation. By t = 1550, the swarm actively splits to create a safe passage for DO₁. This split-and-merge reaction demonstrates that local repulsive functions momentarily override the global cohesion gain to prioritize collision avoidance. Once DO₁ clears the interaction zone at t = 1950, the cohesion potential forces the agents back into a rigid cluster. The swarm recovers its formation completely by t = 2300 and resumes its synchronized approach towards the target.

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Fig 13. Snapshot sequence detailing the real-time dynamic avoidance maneuver in Region A.

(a)-(c) At t = 1010 to 1300 SU, the 15-agent swarm detects the impending collision with the independent dynamic obstacle (DO₁, red circle) approaching along the dashed trajectory. (d)-(f) Between t = 1400 and t = 1750 SU, the dynamic repulsive potential dominates the control effort, forcing the agents to execute a rapid split-and-merge maneuver. The formation actively deforms, creating a safe passage corridor to prioritize collision avoidance over global cohesion. (g)-(i) Once DO₁ clears the interaction zone (t > 1950 SU), the cohesion potential reasserts control. By t = 2300 SU, the agents have successfully reconstituted their rigid formation and resumed their synchronized trajectory toward the global target.

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The kinematic responses of the agents are quantified in Fig 14. The linearized velocity profile Fig 14(a) displays transient fluctuations corresponding directly to the active braking and acceleration required during the dynamic avoidance maneuvers. Specifically, the sharp change in velocity between and SU is mathematically forced by the time-varying dynamic repulsion term . As the dynamic obstacle DO₁ approaches, the controllers aggressively brake and laterally disperse the swarm until the dynamic threat has passed. Simultaneously, strict inter-agent separation and repulsive potentials from static obstacles are actively enforced to prevent collisions. Once the dynamic threat passes, the dual-attraction mechanism reasserts dominance. The cohesion potential drives rapid centroid convergence to reform the formation, while the target convergence potential continues to pull the swarm toward the destination. Ultimately, the agents achieve mutual alignment and resume a synchronized, well-spaced trajectory toward the equilibrium point. Over time, these velocities strictly decay to zero, confirming asymptotic convergence at the equilibrium point. Similarly, the orientation angles Fig 14(b) stabilize after the collision avoidance phase, indicating that the entire multi-agent system achieves its desired final heading.

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Fig 14. Kinematic profiles for the 15-agent Segway swarm in Scenario 4.

(a) Time evolution of the linearized velocities . The pronounced transient fluctuations observed between and SU directly correspond to the aggressive braking and acceleration maneuvers required to evade the independent dynamic obstacles. The subsequent asymptotic decay to zero ( as SU) confirms the system’s ultimate stabilization at the global equilibrium point. (b) Time evolution of the orientation angles . The sharp steering oscillations during the transient phase (t < 3000 SU) reflect the active heading adjustments necessary for the local split-and-merge avoidance maneuvers. Once the dynamic threats are cleared, the orientation angles asymptotically converge to their steady-state values, indicating orientation consensus and formation integrity at the target.

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6.1 Limitations and future work

While the simulation results demonstrate the theoretical robustness of the proposed framework, several limitations inherent to the study design must be acknowledged. These constraints define the trajectory for future research.