Flocking Dynamics of Multi-Agent Systems Based on an Extended Cucker–Smale Model with Nonlinear Coupling and Binding Forces

Abstract

This paper develops an extended Cucker–Smale model that integrates nonlinear velocity alignment with state-dependent binding forces to achieve stable, collision-free flocking in multi-agent systems. Our framework introduces two dedicated control mechanisms: a velocity-dissipative term $\left(K_1\right)$ for accelerated convergence, and a distance-regulating term $\left(K_2\right)$ for formation cohesion and collision avoidance, which collectively ensure stable flocking. Rigorous Lyapunov analysis establishes provable guarantees for asymptotic velocity alignment and collision safety under verifiable initial energy conditions. Numerical simulations validate the theoretical predictions for a 20-agent swarm; scalability analysis demonstrates effective coordination in systems of up to 100 agents and reveals that velocity synchronization improves substantially—with errors decreasing by nearly two orders of magnitude—as $K_2$ increases from 0.05 to 0.50. A Pareto-optimal parameter region ($K_2\in [0.15,0.30]$) is identified, which achieves sub-centimeter-per-second alignment accuracy while maintaining energy consumption below 35% of the baseline. The proposed framework provides a theoretically rigorous yet practically viable solution for applications demanding guaranteed safety and precise coordination, such as UAV formations, robotic swarms, and autonomous vehicle platoons.

1. Introduction

Flocking behaviour is observed in both natural systems, such as bird flocks [1] and fish schools [2], and engineered systems, such as UAV formations [3] and robotic swarms [4]. This collective behaviour enables coordination through local interactions without requiring global information. Multi-agent systems (MAS) are central to control theory and artificial intelligence, with applications spanning UAV coordination, transportation, and robotic collaboration [5,6,7,8,9,10,11,12,13,14,15]. Flocking represents a fundamental behaviour in MAS. The Cucker–Smale model [16] established the theoretical foundation for asymptotic velocity consensus under ideal communication topologies [17,18].

Key persistent challenges in flocking research include guaranteeing collision avoidance in dense formations and maintaining structural stability in dynamic or disturbed environments. Recent research advances to address these issues encompass the use of binding forces for conformational integrity [19], singular weight functions [20,21], the analysis of dynamic topologies under communication delays [22,23,24], nonlinear velocity coupling [25], cooperative-competition mechanisms [26], distributed delay tolerance conditions [27], hierarchical network structures [28], infinite graph models [29], and event-triggered control strategies [30]. However, a significant gap remains: existing models often lack a unified framework that seamlessly integrates velocity alignment with geometric formation control. Furthermore, they typically offer limited systematic parameter sensitivity analysis, which is crucial for practical implementation and tuning.

To clarify the distinct contributions of this work relative to the existing literature, we provide the following comparative positioning:

• Nonlinear Cucker–Smale Extensions: While prior work [25] introduced nonlinear velocity coupling, it did not incorporate binding forces for formation control. Our model unifies nonlinear coupling with state-aware binding mechanisms, enabling simultaneous velocity alignment and geometric structure regulation.
• Collision Avoidance Strategies: Methods such as artificial potential fields [19] provide empirical collision avoidance but frequently lack rigorous stability proofs. Our binding force formulation, particularly the $K_2$ term, delivers provable collision avoidance (Theorem 2) while preserving formation cohesion.
• Energy-Aware Coordination: Most flocking models focus solely on convergence metrics, neglecting control effort. We introduce an energy-efficiency metric and conduct comprehensive parameter optimization to identify Pareto-optimal trade-offs between synchronization accuracy and energy consumption—an aspect largely unexplored in prior work.
• Theoretical Analysis: In contrast to simulation-based studies, we furnish rigorous Lyapunov-based proofs for both velocity consensus and collision avoidance within a unified nonlinear framework. Our analysis explicitly relates the control parameters $(K_1,K_2)$ to convergence rates and safety guarantees.
• Parameter Sensitivity: Although parameter studies exist, they typically examine isolated effects. Our work provides a systematic multi-metric analysis (encompassing velocity error, spatial dispersion, collision rate, and heading stability) that reveals intricate parameter interactions and their practical implications for controller design.

1.1. Positioning Relative to Existing Literature

To clarify the distinct contribution of this work, we present a critical comparison with key related approaches. The primary contribution of this work is threefold: (1) the mathematical unification of nonlinear alignment and binding mechanisms with provable stability guarantees; (2) energy-aware optimization that bridges theory and practice; (3) comprehensive sensitivity analysis that provides actionable insights for parameter tuning.

1.2. Experimental Validation Strategy

To empirically validate the theoretical advantages outlined above, Section 4.4 provides a comprehensive quantitative comparison with the classical Cucker–Smale formulation. The comparative analysis demonstrates a 73.6% improvement in velocity alignment precision, a 54.2% reduction in spatial dispersion, and guaranteed collision avoidance—concrete metrics that substantiate the claimed advantages over existing approaches. This experimental validation bridges theoretical analysis with practical performance assessment, thereby addressing the need for benchmark comparisons highlighted in prior literature reviews.

1.3. Research Objective

Based on the identified gaps in the existing literature, this study aims to develop a provably stable flocking control framework that simultaneously ensures velocity alignment, formation cohesion, and collision avoidance through the synergistic action of nonlinear coupling and state-aware binding forces. The work establishes both theoretical guarantees (Theorems 1–3) and practical implementation guidelines derived from systematic parameter analysis.

2. Preliminaries

2.1. Nonlinear Velocity Coupling Function Properties

Assumption 1.

The nonlinear velocity coupling function $\Gamma :{\mathbb{R}}^d\to {\mathbb{R}}^d$ satisfies the following properties:

1. 

Oddness and Semi-Positive Definiteness: For all $v\in {\mathbb{R}}^d$,

$$\Gamma (-v)=-\Gamma \left(v\right)\mathrm{and}\langle \Gamma \left(v\right),v\rangle \ge 0.$$

(1)

2. 

Growth Condition: There exist constants $C_1>0$ and $\gamma \in (0.5,1.5)$ such that for all $v\in {\mathbb{R}}^d$,

$$\langle \Gamma \left(v\right),v\rangle \ge C_1{\parallel v\parallel }^{2\gamma }.$$

(2)

Physical Interpretation and Examples

The nonlinear operator $\Gamma (·)$ governs the velocity alignment mechanism. Condition (1) ensures physical consistency: reversing the velocity difference reverses the direction of the coupling force. Condition (2) guarantees sufficient alignment strength to ensure convergence. Two admissible examples include:

• Saturating nonlinearity: ${\Gamma }_{\mathrm{sat}}\left(v\right)={\displaystyle \frac{v}{1+{ϵ\parallel v\parallel }^2}}$ with $ϵ>0$, which models actuator saturation.
• Power-law nonlinearity: ${\Gamma }_{\mathrm{pow}}\left(v\right)={\parallel v\parallel }^{p-1}v$ with $p\in (1,3)$, representing generalized damping.

The condition $\langle \Gamma \left(v\right),v\rangle \ge C_1{\parallel v\parallel }^{2\gamma }$ has a clear physical interpretation: it guarantees that the velocity alignment mechanism always performs positive work on the system, thereby dissipating kinetic energy associated with velocity disagreements. The exponent $\gamma$ controls the growth rate of this dissipative effect. When $\gamma >1$, alignment strengthens super-linearly with velocity differences, potentially accelerating convergence. When $\gamma <1$, alignment is sub-linear, which can provide a smoother response and reduce sensitivity to measurement noise.

These examples demonstrate that the proposed framework encompasses both standard linear couplings ($\Gamma \left(v\right)=v$) and more complex nonlinear behaviors relevant to practical systems with saturation limits, actuator constraints, or non-Newtonian interaction dynamics.

2.2. Asymptotic Flocking Properties

Definition 1.

A multi-agent system $𝒫={\left\{(x_i,v_i)\right\}}_{i=1}^N$ is said to exhibit asymptotic flocking if the following conditions hold:

1. 

Velocity alignment:

$$\underset{t\to \infty }{lim}\parallel v_j\left(t\right)-v_i\left(t\right)\parallel =0\forall 1\le i,j\le N$$

(3)

2. 

Group formation:

$$\underset{t\ge 0}{sup}\parallel x_j\left(t\right)-x_i\left(t\right)\parallel <\infty \forall 1\le i,j\le N$$

(4)

2.3. Barbalat’s Lemma

Lemma 1 (Barbalat’s Lemma).

If a differentiable function $f\left(t\right)$ satisfies the following conditions:

1. 

$f\left(t\right)$ is lower bounded,

2. 

$\dot{f}\left(t\right)$ is uniformly continuous,

3. 

${lim}_{t\to \infty }{\int }_0^t\dot{f}\left(\tau \right)d\tau$ exists (i.e., the integral converges),

then ${lim}_{t\to \infty }\dot{f}\left(t\right)=0$.

3. Model Construction and Theoretical Analysis

3.1. Construction of Extended Model

To enhance collision avoidance and structural stability in complex environments, the classical Cucker–Smale model is extended with nonlinear velocity coupling and state-aware mechanisms:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{x}_i}{dt}}=v_i\hfill \\ {\displaystyle \frac{d{v}_i}{dt}}={\displaystyle \frac{\lambda }{N}}\sum _{j=1}^N\psi (\parallel x_j-x_i\parallel )\Gamma (v_j-v_i)\hfill \\ +{\displaystyle \frac{\sigma }{N}}\sum _{j=1}^N\left[{\displaystyle \frac{K_1\langle v_j-v_i,x_j-x_i\rangle }{2r_{ij}^2}}(x_j-x_i)+{\displaystyle \frac{K_2(r_{ij}-2R)}{2r_{ij}}}(x_j-x_i)\right]\hfill \end{array}\right.$$

(5)

where $x_i,v_i$ are the position and velocity of the agent respectively, $\psi (·)$ is the communication weight function, $\Gamma (·)$ is the nonlinear velocity coupling function, $\lambda$ and $\sigma$ are the coupling strengths, R is the preset distance between particles, $r_{ij}=\parallel x_j-x_i\parallel$, $K_1$ is the dissipative force gain parameter, $K_2$ is the repulsive force gain parameter.

Physical Interpretation and Derivation of the Force Terms

To provide an intuitive physical foundation for the proposed model, the two newly introduced force terms governed by parameters $K_1$ and $K_2$ are described below.

1. 

The ${\mathit{K}}_{\mathbf{1}}$ Term: Viscous Damping Force

• Physical Role: This term acts as a viscous damper within the system. Its magnitude is proportional to the projection of the relative velocity onto the inter-agent axis, $\langle v_j-v_i,x_j-x_i\rangle$, with $K_1$ serving as the damping coefficient. When agents approach each other ($\langle v_j-v_i,x_j-x_i\rangle <0$), the force is repulsive, decelerating the approach. When they separate ($\langle v_j-v_i,x_j-x_i\rangle >0$), it becomes attractive, inhibiting excessive swarm dispersion. Thus, $K_1$ primarily functions to dissipate internal kinetic energy, suppress oscillations, and expedite convergence to a stable relative configuration.
• Physical Derivation: The $K_1$ term originates from Rayleigh’s dissipation function $\mathcal{R}={\displaystyle \frac{1}{2}}{K}_1{\displaystyle \frac{{\langle v_j-v_i,x_j-x_i\rangle }^2}{2r_{ij}^2}}$, representing the power dissipated due to relative motion along the inter-agent direction. The corresponding dissipative force, ${\mathbf{F}}_{\mathrm{diss}}=-\partial \mathcal{R}/\partial {\mathbf{v}}_i$, yields precisely the $K_1$ term in Equation (5). This derivation confirms $K_1$ as a genuine physical damping coefficient, not merely an empirical parameter.

2. 

The ${\mathit{K}}_{\mathbf{2}}$ Term: Conservative Binding/Spring Force

• Physical Role: This term behaves as an ideal spring that regulates the inter-agent distance $r_{ij}=\parallel x_j-x_i\parallel$ towards a predefined safety distance $2R$. Here, $K_2$ is the spring constant (stiffness). For $r_{ij}>2R$, the force is attractive, maintaining formation cohesion. For $r_{ij}<2R$, it becomes strongly repulsive, forming the core collision avoidance mechanism. Consequently, $K_2$ dictates the “hardness” of the safety distance enforcement. A higher $K_2$ increases sensitivity to spacing deviations, enhancing collision prevention at the potential cost of induced oscillations due to stronger forces.
• Physical Derivation: The $K_2$ term is derived from the conservative potential $U_{\mathrm{bind}}\left(r_{ij}\right)={\displaystyle \frac{K_2}{4}}{(r_{ij}-2R)}^2$, which models a harmonic spring with equilibrium length $2R$. The resulting force, ${\mathbf{F}}_{\mathrm{bind}}=-{\nabla }_{{\mathbf{x}}_i}{U}_{\mathrm{bind}}=-{\displaystyle \frac{K_2}{2}}(r_{ij}-2R){\displaystyle \frac{{\mathbf{x}}_j-{\mathbf{x}}_i}{r_{ij}}}$, matches the $K_2$ term in Equation (5), establishing $K_2$ as a physical spring constant.

Theoretical Interplay Between ${\mathit{K}}_{\mathbf{1}}$ and ${\mathit{K}}_{\mathbf{2}}$

The parameters $K_1$ and $K_2$ govern distinct yet interacting physical processes that collectively determine the system’s behavior. Their relationship is elucidated from an energy perspective:

• Dynamic Role of $K_1$: From the energy derivative $\dot{E}\left(t\right)$ in Theorem 1, the $K_1$ term contributes $-{\displaystyle \frac{\sigma K_1}{4N}}{\sum }_{i,j}{\displaystyle \frac{{\langle {\widehat{v}}_i-{\widehat{v}}_j,x_i-x_j\rangle }^2}{r_{ij}^2}}\le 0$, explicitly demonstrating its dissipative nature. Thus, $K_1$ controls the rate of kinetic energy dissipation and, consequently, the convergence speed.
• Static Role of $K_2$: The $K_2$ term appears in the potential energy $E_{\mathrm{pot}}={\displaystyle \frac{\sigma K_2}{8N}}{\sum }_{i,j}{(r_{ij}-2R)}^2$, which dictates the equilibrium configuration. Theorem 2 provides the group formation bound $\parallel {\widehat{x}}_i-{\widehat{x}}_j\parallel \le 2R+\sqrt{8NE\left(0\right)/\left(\sigma K_2\right)}$, showing that $K_2$ fundamentally limits the maximum inter-agent distance.
• Coupled Dynamics: The interplay between the dissipative ($K_1$) and conservative ($K_2$) forces creates a second-order dynamic system for each agent pair. The effective damping ratio is proportional to $K_1/\sqrt{K_2}$, theoretically explaining the observed performance trade-off: high $K_1$ accelerates convergence but may induce overshoot; high $K_2$ enforces strict spacing at the potential cost of slower convergence. This is a direct consequence of the coupled dissipative-conservative dynamics, not merely an empirical observation.

In summary, $K_1$ and $K_2$ govern the dynamic convergence behavior and the static geometric configuration of the swarm, respectively. Strategic tuning of this parameter pair enables a balanced trade-off among swift synchronization (governed by $K_1$), formation stability and collision safety (governed by $K_2$), and control energy consumption. This understanding forms the basis for the parameter optimization study in subsequent sections. By introducing these two physically interpretable forces, the abstract algebraic model is bridged with a mechanical framework more aligned with practical systems (e.g., multi-robot teams, UAV swarms), thereby providing an intuitive foundation for real-world control applications.

Connection to Real-World Systems: This physically grounded model has direct mappings to practical domains:

• UAV Swarms: $K_1$ represents the gains for aerodynamic damping and velocity tracking, while $K_2$ models the controllers for collision avoidance and formation-keeping.
• Robotic Teams: $K_1$ corresponds to the damping inherent in actuators, and $K_2$ to the impedance control parameters that ensure safe physical interaction.
• Autonomous Vehicles: $K_1$ governs the damping in braking/acceleration response, whereas $K_2$ implements the spacing control logic in adaptive cruise control systems.

The explicit decomposition into dissipative ($K_1$) and conservative ($K_2$) components aligns with fundamental analytical mechanics principles, thereby enhancing the model’s physical interpretability and engineering relevance.

Define system center point:

$$x_c={\displaystyle \frac{1}{N}}\sum _{i=1}^N{x}_i,v_c={\displaystyle \frac{1}{N}}\sum _{i=1}^N{v}_i$$

(6)

Then the relative movement volume:

$${\widehat{x}}_i=x_i-x_c,{\widehat{v}}_i=v_i-v_c$$

(7)

The relative motion dynamics become:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{\hat{x}}_i}{dt}}={\widehat{v}}_i\hfill \\ {\displaystyle \frac{d{\widehat{v}}_i}{dt}}={\displaystyle \frac{\lambda }{N}}\sum _{j=1}^N\psi (\parallel {\widehat{x}}_j-{\widehat{x}}_i\parallel )\Gamma ({\widehat{v}}_j-{\widehat{v}}_i)\hfill \\ +{\displaystyle \frac{\sigma }{N}}\sum _{j=1}^N\left[{\displaystyle \frac{K_1\langle {\widehat{v}}_i-{\widehat{v}}_j,{\widehat{x}}_i-{\widehat{x}}_j\rangle }{2r_{ij}^2}}({\widehat{x}}_j-{\widehat{x}}_i)+{\displaystyle \frac{K_2(r_{ij}-2R)}{2r_{ij}}}({\widehat{x}}_j-{\widehat{x}}_i)\right]\hfill \end{array}\right.$$

(8)

The flocking behaviour depends solely on the relative motions between agents, rather than on the absolute motion of the center of mass. Therefore, the system is analyzed within the center-of-mass reference frame. Consequently, the system dynamics are reduced to examining a system of relative movements.

3.2. Energy Function Definition and System Properties

3.2.1. Total Energy Definition

The total energy of the system is defined as

$$E\left(t\right)=E_k\left(t\right)+E_p\left(t\right)={\displaystyle \frac{1}{2}}\sum _{i=1}^N{\parallel {\widehat{v}}_i\left(t\right)\parallel }^2+{\displaystyle \frac{\sigma K_2}{8N}}\sum _{i,j=1}^N{(r_{ij}\left(t\right)-2R)}^2.$$

(9)

Lemma 2 (Local Existence and A Priori Estimates).

Given initial conditions $\parallel {\widehat{v}}_i\left(0\right)\parallel <\infty$, $\parallel {\widehat{x}}_i\left(0\right)\parallel <\infty$ with $r_{ij}\left(0\right)=\parallel {\widehat{x}}_i\left(0\right)-{\widehat{x}}_j\left(0\right)\parallel >0$ for all $i\ne j$, and parameters $\lambda >0$, $\sigma >0$, $K_1>0$, $K_2>0$, system (8) possesses a unique solution on a maximal time interval $[0,T_{max})$ with $T_{max}>0$. Moreover, there exist constants $M_v,M_x,T_0>0$, depending only on the initial data and parameters, such that for all $t\in [0,T_0]$:

$$\parallel {\widehat{v}}_i\left(t\right)\parallel \le M_v,\parallel {\widehat{x}}_i\left(t\right)\parallel \le M_x,\forall i=1,\dots ,N,$$

where $T_0=min(T_{max},1)$ is a guaranteed time of existence with a priori bounds.

Proof. 

Step 1. Local existence and uniqueness. Define the state vector

$$\mathbf{z}={({\widehat{x}}_1,\dots ,{\widehat{x}}_N,{\widehat{v}}_1,\dots ,{\widehat{v}}_N)}^{\top }\in {\mathbb{R}}^{2dN}.$$

The dynamics (8) can be written as $\dot{\mathbf{z}}=f\left(\mathbf{z}\right)$.

The function f is smooth on the open domain

$$\Omega =\{\mathbf{z}\in {\mathbb{R}}^{2dN}:r_{ij}=\parallel {\widehat{x}}_i-{\widehat{x}}_j\parallel >0\mathrm{for}\mathrm{all}i\ne j\},$$

because all terms in (8) are continuously differentiable when the denominators $r_{ij}$ are nonzero. By assumption, the initial configuration satisfies $\mathbf{z}\left(0\right)\in \Omega$.

By the Picard–Lindelöf theorem, there exists a unique maximal solution $\mathbf{z}\left(t\right)$ defined on $[0,T_{max})$ with $T_{max}>0$.

Step 2. A priori estimates via standard ODE theory. We now establish a priori bounds that do not rely on the energy dissipation property. Let

$$V_0=\underset{i}{max}\parallel {\widehat{v}}_i\left(0\right)\parallel ,X_0=\underset{i}{max}\parallel {\widehat{x}}_i\left(0\right)\parallel ,D_0=\underset{i,j}{max}\parallel {\widehat{x}}_i\left(0\right)-{\widehat{x}}_j\left(0\right)\parallel .$$

Define $T_0=min(T_{max},1)$. Consider the solution on $[0,T_0]$. We will use a continuation argument to obtain bounds.

Since $r_{ij}\left(0\right)>0$ and the solution is continuous, there exists $T_1\in (0,T_{max}]$ such that

$$r_{ij}\left(t\right)\ge {\displaystyle \frac{D_0}{2}}>0\mathrm{for}\mathrm{all}t\in [0,T_1]\mathrm{and}\mathrm{all}i\ne j.$$

This follows from the continuity of the solution and the fact that $r_{ij}\left(0\right)=D_0>0$.

Now, on the interval $[0,T_1]$, we can bound the right-hand side of (8). Let

$$M=max\{\parallel {\widehat{v}}_i\left(t\right)\parallel :i=1,\dots ,N,t\in [0,T_1]\}$$

be the maximum velocity magnitude on this interval (which exists by continuity).

From the structure of Equation (8), we bound each term:

• The communication weight $\psi \left(r_{ij}\right)$ is bounded since $\psi$ is continuous and $r_{ij}\ge D_0/2$:

  $$|\psi \left(r_{ij}\right)|\le {\psi }_{max}=\underset{r\ge D_0/2}{max}\psi \left(r\right).$$
• The nonlinear coupling term satisfies, by Assumption 1 and the continuity of $\Gamma$,

  $$\parallel \Gamma ({\widehat{v}}_j-{\widehat{v}}_i)\parallel \le C_2{\left(2M\right)}^{\gamma }\mathrm{for}\mathrm{some}{C}_2>0.$$
• The $K_1$ and $K_2$ terms are bounded because:

  $$\begin{array}{cc}\hfill ∥{\displaystyle \frac{K_1\langle {\widehat{v}}_j-{\widehat{v}}_i,{\widehat{x}}_j-{\widehat{x}}_i\rangle }{2r_{ij}^2}}({\widehat{x}}_j-{\widehat{x}}_i)∥& \le {\displaystyle \frac{K_1}{2}}·{\displaystyle \frac{\left(2M\right)(2X_0+2M{T}_1)}{{(D_0/2)}^2}}·(2X_0+2M{T}_1),\hfill \\ \hfill ∥{\displaystyle \frac{K_2(r_{ij}-2R)}{2r_{ij}}}({\widehat{x}}_j-{\widehat{x}}_i)∥& \le {\displaystyle \frac{K_2}{2}}·{\displaystyle \frac{2X_0+2M{T}_1+2R}{(D_0/2)}}·(2X_0+2M{T}_1).\hfill \end{array}$$

Combining these bounds, there exists a constant $L>0$, depending on $V_0$, $X_0$, $D_0$, and the parameters, such that

$$∥{\displaystyle \frac{d{\widehat{v}}_i}{dt}}∥\le L\mathrm{for}\mathrm{all}i=1,\dots ,N\mathrm{and}t\in [0,T_1].$$

Step 3. Deriving explicit bounds. From the velocity derivative bound, we have by integration:

$$\parallel {\widehat{v}}_i\left(t\right)\parallel \le \parallel {\widehat{v}}_i\left(0\right)\parallel +{\int }_0^{t}Ld\tau \le V_0+Lt\le V_0+L\mathrm{for}t\in [0,T_1].$$

For positions, integrating the velocity bound gives:

$$\parallel {\widehat{x}}_i\left(t\right)\parallel \le \parallel {\widehat{x}}_i\left(0\right)\parallel +{\int }_0^t\parallel {\widehat{v}}_i\left(\tau \right)\parallel d\tau \le X_0+(V_0+L)t\le X_0+V_0+L.$$

Set $M_v=V_0+L$ and $M_x=X_0+V_0+L+2R$ (the additional $2R$ accounts for the maximum possible relative displacement).

Step 4. Determining the guaranteed existence time. The bounds derived in Step 3 are valid on $[0,T_1]$. Set $T_0=min(T_1,1)$. Then for all $t\in [0,T_0]$, we have:

$$\parallel {\widehat{v}}_i\left(t\right)\parallel \le M_v,\parallel {\widehat{x}}_i\left(t\right)\parallel \le M_x,\forall i=1,\dots ,N.$$

The constant $T_0>0$ is guaranteed because $T_1>0$ (by continuity of the solution and the condition $r_{ij}\left(0\right)>0$) and we take the minimum with 1 to ensure a concrete time.

Step 5. Consistency with the overall proof structure. This lemma establishes the local well-posedness of the system using only standard ODE theory. The existence of $T_0$ and the bounds $M_v$, $M_x$ are derived independently of any energy considerations. The global analysis will be completed in subsequent theorems. □

3.2.2. Energy Dissipation Theorem

Theorem 1 (Energy Dissipation Property).

Consider the multi-agent system governed by equations (8) with parameters $\lambda >0$, $\sigma >0$, $K_1>0$, and a communication weight function satisfying $\psi \left(r\right)>0$ for all $r\in (0,\infty )$. Under Assumption 1, on the maximal interval of existence $[0,T_{max})$ where the solution exists, the total energy $E\left(t\right)$ defined in (9) satisfies:

$${\displaystyle \frac{d}{dt}}E\left(t\right)\le 0\mathrm{for}\mathrm{all}t\in [0,T^{*}),$$

where $T^{*}=min\left(T_{max},inf\{t\ge 0:r_{ij}\left(t\right)=0\mathit{for}\mathit{some}i\ne j\}\right)$ is the first potential collision time, and the inequality holds strictly whenever velocities are not aligned.

Proof of Theorem 1.

Step 1. Setup and definition of the collision-free interval. By Lemma 2, system (8) admits a unique solution on a maximal time interval $[0,T_{max})$ with $T_{max}>0$.

Define the first potential collision time as

$$T^{*}=min\left(T_{max},inf\{t\ge 0:r_{ij}\left(t\right)=0\mathrm{for}\mathrm{some}i\ne j\}\right).$$

If no collision occurs, then $T^{*}=T_{max}$. By definition, on the interval $[0,T^{*})$, we have $r_{ij}\left(t\right)>0$ for all $i\ne j$, ensuring all denominators in the dynamics are well-defined.

Step 2. Energy derivative computation on the collision-free interval. On $[0,T^{*})$, we compute the time derivative of the total energy $E\left(t\right)$. From the definition

$$E\left(t\right)={\displaystyle \frac{1}{2}}\sum _{i=1}^N{\parallel {\widehat{v}}_i\left(t\right)\parallel }^2+{\displaystyle \frac{\sigma K_2}{8N}}\sum _{i,j=1}^N{(r_{ij}\left(t\right)-2R)}^2,$$

we differentiate each component.

For the kinetic energy part:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{2}}\sum _{i=1}^N{\parallel {\widehat{v}}_i\parallel }^2\right)& =\sum _{i=1}^N{\widehat{v}}_i·{\displaystyle \frac{d{\widehat{v}}_i}{dt}}\hfill \\ \hfill & ={\displaystyle \frac{\lambda }{N}}\sum _{i,j=1}^N\psi \left(r_{ij}\right){\widehat{v}}_i·\Gamma ({\widehat{v}}_j-{\widehat{v}}_i)\hfill \\ \hfill & +{\displaystyle \frac{\sigma }{2N}}\sum _{i,j=1}^N\left[{\displaystyle \frac{K_1}{r_{ij}^2}}\langle {\widehat{v}}_j-{\widehat{v}}_i,{\widehat{x}}_j-{\widehat{x}}_i\rangle \langle {\widehat{v}}_i,{\widehat{x}}_i-{\widehat{x}}_j\rangle \right.\hfill \\ \hfill & \left.+{\displaystyle \frac{K_2(r_{ij}-2R)}{r_{ij}}}\langle {\widehat{v}}_i,{\widehat{x}}_i-{\widehat{x}}_j\rangle \right].\hfill \end{array}$$

For the potential energy part:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\sigma K_2}{8N}}\sum _{i,j=1}^N{(r_{ij}-2R)}^2\right)={\displaystyle \frac{\sigma K_2}{4N}}\sum _{i,j=1}^N(r_{ij}-2R){\displaystyle \frac{\langle {\widehat{x}}_j-{\widehat{x}}_i,{\widehat{v}}_j-{\widehat{v}}_i\rangle }{r_{ij}}}.$$

Step 3. Symmetry manipulation and simplification. Applying index symmetry $i\leftrightarrow j$ to the velocity coupling term:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\lambda }{N}}\sum _{i,j}\psi \left(r_{ij}\right){\widehat{v}}_i·\Gamma ({\widehat{v}}_j-{\widehat{v}}_i)& ={\displaystyle \frac{\lambda }{2N}}\sum _{i,j}\psi \left(r_{ij}\right)[{\widehat{v}}_i·\Gamma ({\widehat{v}}_j-{\widehat{v}}_i)+{\widehat{v}}_j·\Gamma ({\widehat{v}}_i-{\widehat{v}}_j)]\hfill \\ \hfill & =-{\displaystyle \frac{\lambda }{2N}}\sum _{i,j}\psi \left(r_{ij}\right)({\widehat{v}}_j-{\widehat{v}}_i)·\Gamma ({\widehat{v}}_j-{\widehat{v}}_i).\hfill \end{array}$$

By Assumption 1, we have $({\widehat{v}}_j-{\widehat{v}}_i)·\Gamma ({\widehat{v}}_j-{\widehat{v}}_i)\ge C_1{\parallel {\widehat{v}}_j-{\widehat{v}}_i\parallel }^{2\gamma }$, hence

$$-{\displaystyle \frac{\lambda }{2N}}\sum _{i,j}\psi \left(r_{ij}\right)({\widehat{v}}_j-{\widehat{v}}_i)·\Gamma ({\widehat{v}}_j-{\widehat{v}}_i)\le -{\displaystyle \frac{\lambda C_1}{2N}}\sum _{i,j}\psi \left(r_{ij}\right){\parallel {\widehat{v}}_j-{\widehat{v}}_i\parallel }^{2\gamma }.$$

Combining all terms and simplifying, we obtain the fundamental inequality:

$${\displaystyle \frac{dE}{dt}}\le -{\displaystyle \frac{\lambda C_1}{2N}}\sum _{i,j}\psi \left(r_{ij}\right){\parallel {\widehat{v}}_j-{\widehat{v}}_i\parallel }^{2\gamma }-{\displaystyle \frac{\sigma K_1}{4N}}\sum _{i,j}{\displaystyle \frac{{\langle {\widehat{v}}_j-{\widehat{v}}_i,{\widehat{x}}_j-{\widehat{x}}_i\rangle }^2}{r_{ij}^2}}.$$

(10)

Step 4. Sign analysis and conclusion. For all $t\in [0,T^{*})$, since $r_{ij}\left(t\right)>0$, the right-hand side of (10) is well-defined. Both terms on the right-hand side are non-positive:

• The first term is non-positive because $\lambda ,C_1>0$, $\psi \left(r_{ij}\right)>0$, and $\parallel {\widehat{v}}_j-{\widehat{v}}_i{\parallel }^{2\gamma }\ge 0$.
• The second term is non-positive as a sum of squares.

Therefore, for all $t\in [0,T^{*})$,

$${\displaystyle \frac{dE}{dt}}\le 0,$$

with strict inequality whenever there exists at least one pair $(i,j)$ with ${\widehat{v}}_i\left(t\right)\ne {\widehat{v}}_j\left(t\right)$.

Step 5. Energy bound. Integrating the inequality from 0 to t for any $t<T^{*}$ yields

$$E\left(t\right)=E\left(0\right)+{\int }_0^t{\displaystyle \frac{dE}{d\tau }}d\tau \le E\left(0\right).$$

Thus, on the collision-free interval $[0,T^{*})$, the energy is non-increasing and bounded above by its initial value.

Step 6. Scope of the theorem. This theorem establishes the energy dissipation property on the interval $[0,T^{*})$ where collisions are absent. Whether $T^{*}=T_{max}$ (i.e., collisions never occur) is a separate question that will be addressed by Theorem 2. The current theorem provides the foundation for that analysis. □

3.3. Collision Avoidance Analysis

Theorem 2 (Collision Avoidance Condition).

Consider the multi-agent system governed by Equation (8) with maximal existence time $T_{max}>0$ as guaranteed by Lemma 2. If the initial energy $E\left(0\right)$ satisfies

$$E\left(0\right)<{\displaystyle \frac{\sigma K_2{R}^2}{2N}},$$

(11)

then for all $t\in [0,T_{max})$ and all $i\ne j$, the inter-agent distance satisfies $r_{ij}\left(t\right)>0$. Moreover, $T_{max}=\infty$, i.e., the solution exists globally in time.

Proof of Theorem 2.

Step 1. Setup and contradiction argument. Assume, for the sake of contradiction, that there exists a first collision time $T_c\in (0,T_{max})$ such that ${lim}_{t\to T_c^{-}}{r}_{ij}\left(t\right)=0$ for some pair $(i,j)$, with $r_{ij}\left(t\right)>0$ for all $t\in [0,T_c)$ and all $i\ne j$.

Step 2. Energy dissipation on $[0,T_c)$. On the interval $[0,T_c)$, the conditions of Theorem 1 are satisfied (since $r_{ij}\left(t\right)>0$ for all $t\in [0,T_c)$). Hence,

$${\displaystyle \frac{dE}{dt}}\le 0\mathrm{and}E\left(t\right)\le E\left(0\right)\mathrm{for}\mathrm{all}t\in [0,T_c).$$

Step 3. Potential energy bound. From the definition of the potential energy component, we have

$$E_p\left(t\right)={\displaystyle \frac{\sigma K_2}{8N}}\sum _{i,j=1}^N{(r_{ij}\left(t\right)-2R)}^2\le E\left(t\right)\le E\left(0\right).$$

In particular, for the pair $(i,j)$ presumed to collide as $t\to T_c^{-}$,

$${\displaystyle \frac{\sigma K_2}{8N}}{(r_{ij}\left(t\right)-2R)}^2\le E\left(0\right)\mathrm{for}\mathrm{all}t\in [0,T_c).$$

Step 4. Distance bound derivation. From the above inequality, we obtain

$${(r_{ij}\left(t\right)-2R)}^2\le {\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}⟹|r_{ij}\left(t\right)-2R|\le \sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}.$$

This implies the two-sided bound:

$$2R-\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}\le r_{ij}\left(t\right)\le 2R+\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}.$$

Step 5. Condition preventing collision. Condition (11) is equivalent to

$$\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}<2R.$$

Under this condition, the lower bound is strictly positive:

$$2R-\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}>0.$$

Therefore, for all $t\in [0,T_c)$,

$$r_{ij}\left(t\right)\ge 2R-\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}>0.$$

Step 6. Contradiction and conclusion. Taking the left-hand limit as $t\to T_c^{-}$ and using the continuity of $r_{ij}\left(t\right)$, we have

$$r_{ij}\left(T_c\right)=\underset{t\to T_c^{-}}{lim}{r}_{ij}\left(t\right)\ge 2R-\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}>0.$$

This contradicts the initial assumption that ${lim}_{t\to T_c^{-}}{r}_{ij}\left(t\right)=0$. Hence, no such first collision time $T_c$ can exist; therefore, $r_{ij}\left(t\right)>0$ for all $t\in [0,T_{max})$ and all distinct pairs $i\ne j$.

Step 7. Global existence. Since collisions are rigorously excluded, the solution remains within the domain $\Omega =\{\mathbf{z}\in {\mathbb{R}}^{2dN}:r_{ij}>0\mathrm{for}\mathrm{all}i\ne j\}$ where the right-hand side of (8) is smooth. By standard ODE continuation theory (see Lemma 2), the solution can be extended indefinitely, which proves global existence, i.e., $T_{max}=\infty$. □

3.4. Asymptotic Clustering Verification

Theorem 3 (Asymptotic Flocking).

Assume the initial energy $E\left(0\right)$ is finite and satisfies condition (11), the communication weight function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfies $\psi \left(r\right)\ge {\psi }_0>0$ for all $r\ge 0$, and the parameters satisfy $\lambda >0$, $\sigma >0$, $K_1>0$. Then the extended Cucker–Smale system achieves asymptotic flocking in the following sense:

1. 

Velocity alignment: The relative velocities converge to zero asymptotically:

$$\underset{t\to \infty }{lim}\parallel {\widehat{v}}_i\left(t\right)-{\widehat{v}}_j\left(t\right)\parallel =0,1\le i,j\le N.$$

2. 

Group formation: The inter-agent distances remain uniformly bounded:

$$\underset{0\le t<\infty }{sup}\parallel {\widehat{x}}_i\left(t\right)-{\widehat{x}}_j\left(t\right)\parallel \le 2R+\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}.$$

Proof of Theorem 3.

We prove the two parts separately.

Part 1: Proof of velocity alignment.

Step 1.1: Energy properties. By Theorems 1 and 2, under condition (11), the system has a global solution and the total energy $E\left(t\right)$ satisfies:

• $E\left(t\right)$ is non-increasing for all $t\ge 0$,
• $E\left(t\right)$ is bounded below by 0,
• Therefore, ${lim}_{t\to \infty }E\left(t\right)=E_{\infty }$ exists and is finite.

Step 1.2: Convergence of the energy integral. From the existence of the limit $E_{\infty }$, we have

$${\int }_0^{\infty }\dot{E}\left(\tau \right)d\tau =\underset{t\to \infty }{lim}{\int }_0^t\dot{E}\left(\tau \right)d\tau =E_{\infty }-E\left(0\right)<\infty .$$

Thus, ${\int }_0^{\infty }\dot{E}\left(\tau \right)d\tau$ converges absolutely.

Step 1.3: Uniform continuity of $\dot{E}\left(t\right)$. From Theorem 1, the energy derivative satisfies

$$\dot{E}\left(t\right)=-{\displaystyle \frac{\lambda C_1}{2N}}\sum _{i,j}\psi \left(r_{ij}\right){\parallel {\widehat{v}}_j-{\widehat{v}}_i\parallel }^{2\gamma }-{\displaystyle \frac{\sigma K_1}{4N}}\sum _{i,j}{\displaystyle \frac{{\langle {\widehat{v}}_j-{\widehat{v}}_i,{\widehat{x}}_j-{\widehat{x}}_i\rangle }^2}{r_{ij}^2}}.$$

We verify that $\dot{E}\left(t\right)$ is uniformly continuous by showing $\ddot{E}\left(t\right)$ is bounded. Differentiating $\dot{E}\left(t\right)$ yields expressions involving:

• ${\widehat{v}}_i\left(t\right)$ and their derivatives ${\dot{\widehat{v}}}_i\left(t\right)$ (accelerations)
• ${\widehat{x}}_i\left(t\right)$ and ${\widehat{v}}_i\left(t\right)$
• Terms with denominators $r_{ij}\left(t\right)$

From Theorem 2, we have the collision avoidance guarantee, which implies a uniform lower bound on inter-agent distances. Specifically, from the proof of Theorem 2, we have

$$r_{ij}\left(t\right)\ge 2R-\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}=:\delta >0\mathrm{for}\mathrm{all}t\ge 0,$$

where the positivity $\delta >0$ follows from condition (11).

Furthermore, from Lemma 2 and the energy bound $E\left(t\right)\le E\left(0\right)$, the positions and velocities remain bounded. Consequently, the accelerations ${\dot{\widehat{v}}}_i\left(t\right)$ from Equation (8) are also bounded, as all terms in (8) involve bounded functions divided by $r_{ij}\left(t\right)\ge \delta >0$.

Since $\dot{E}\left(t\right)$ is a rational combination of bounded functions with denominators bounded away from zero, its derivative $\ddot{E}\left(t\right)$ is bounded. By the Mean Value Theorem, $\dot{E}\left(t\right)$ is Lipschitz continuous, hence uniformly continuous.

Step 1.4: Application of Barbalat’s Lemma. We apply Barbalat’s Lemma (Lemma 1) to $f\left(t\right)=E\left(t\right)$. The conditions are satisfied:

• $E\left(t\right)$ is bounded below (by 0)
• $\dot{E}\left(t\right)$ is uniformly continuous (established above)
• ${\int }_0^{\infty }\dot{E}\left(\tau \right)d\tau$ converges (established in Step 1.2)

Therefore, by Barbalat’s Lemma,

$$\underset{t\to \infty }{lim}\dot{E}\left(t\right)=0.$$

Step 1.5: Convergence to velocity alignment. Substituting the expression for $\dot{E}\left(t\right)$, we have

$$\underset{t\to \infty }{lim}\left[{\displaystyle \frac{\lambda C_1}{2N}}\sum _{i,j}\psi \left(r_{ij}\right){\parallel {\widehat{v}}_j-{\widehat{v}}_i\parallel }^{2\gamma }+{\displaystyle \frac{\sigma K_1}{4N}}\sum _{i,j}{\displaystyle \frac{{\langle {\widehat{v}}_j-{\widehat{v}}_i,{\widehat{x}}_j-{\widehat{x}}_i\rangle }^2}{r_{ij}^2}}\right]=0.$$

Since both terms are non-negative and $\lambda C_1>0$, $\sigma K_1>0$, it follows that

$$\underset{t\to \infty }{lim}\sum _{i,j}\psi \left(r_{ij}\right){\parallel {\widehat{v}}_j-{\widehat{v}}_i\parallel }^{2\gamma }=0.$$

By the assumption on the communication weight function, $\psi \left(r\right)\ge {\psi }_0>0$ for all $r\ge 0$. Therefore,

$${\psi }_0\sum _{i,j}\parallel {\widehat{v}}_j-{\widehat{v}}_i{\parallel }^{2\gamma }\le \sum _{i,j}\psi \left(r_{ij}\right){\parallel {\widehat{v}}_j-{\widehat{v}}_i\parallel }^{2\gamma }\to 0.$$

Since $\gamma >0.5>0$, we conclude that

$$\underset{t\to \infty }{lim}\parallel {\widehat{v}}_j\left(t\right)-{\widehat{v}}_i\left(t\right)\parallel =0\mathrm{for}\mathrm{all}i,j.$$

This establishes velocity alignment.

Part 2: Proof of group formation.

The group formation bound follows directly from Theorem 2, which provides the two-sided bound

$$\parallel {\widehat{x}}_i\left(t\right)-{\widehat{x}}_j\left(t\right)\parallel \le 2R+\sqrt{{\displaystyle \frac{8NE\left(0\right)}{\sigma K_2}}}\mathrm{for}\mathrm{all}t\ge 0.$$

Taking the supremum over all $t\ge 0$ yields the stated result. □

4. Numerical Simulation and Parametric Analysis

• Scope of Numerical Validation

The simulation results presented in this section were obtained under idealized conditions—perfect communication, noise-free measurements, and homogeneous agents—to cleanly validate the theoretical predictions. While these conditions are sufficient to demonstrate the fundamental capabilities of the proposed framework, practical implementation would require consideration of additional real-world factors, as discussed in Section 5.

4.1. Simulation Settings

The multi-agent system comprising 20 agents in 3D space was simulated in MATLAB R2021a. Initial velocities were randomly generated within the range $[-2,2]\mathrm{m}/\mathrm{s}$, and initial positions were uniformly distributed within a cube of side length $5\mathrm{m}$ centered at the origin, i.e., ${[-2.5,2.5]}^3{\mathrm{m}}^3$. To prevent initial agent overlap, positions were resampled until the pairwise Euclidean distance between any two agents exceeded the safety threshold $2R$. Numerical integration was carried out using a fourth-order Runge–Kutta method with a fixed time step of $\Delta t=0.1\mathrm{s}$ over a total simulation duration of $T=100\mathrm{s}$. The communication weight function was chosen as $\psi \left(r_{ij}\right)=1/{(1+r_{ij}^2)}^{0.4}$, and the nonlinear velocity coupling function as $\Gamma \left(v\right)=C_1{\parallel v\parallel }^{1.4}v$. Key parameters used in the simulations are listed in

Table 1. Multi-agent flocking simulation parameters.

Parameter	Value
Velocity coupling strength ($\lambda$)	1.2
Binding force strength ($\sigma$)	2.5
Dissipative force gain ($K_1$)	1.5
Repulsive force gain ($K_2$)	0.8
Expected spacing (R)	0.5

.

4.2. Analysis of Swarm Behaviour Evolution

As illustrated in Figure 1, the system undergoes a transition from a state of disordered motion to one of ordered flocking. Initially, the agents’ positions and velocity vectors are randomly distributed. After 100 s of simulation, however, the agents converge into spatially coordinated formations with aligned velocity directions.

Figure 2 shows that the maximum velocity error, ${max}_{i,j}\parallel {\widehat{v}}_i\left(t\right)-{\widehat{v}}_j\left(t\right)\parallel$, decreases exponentially over time, reaching a value on the order of ${10}^{-3}$ m/s after approximately $t=10$ s. This behaviour aligns with the velocity alignment condition, which requires ${lim}_{t\to \infty }{max}_{i,j}\parallel {\widehat{v}}_i\left(t\right)-{\widehat{v}}_j\left(t\right)\parallel =0$.

Similarly, Figure 3 indicates that the maximum inter-agent position difference stabilizes at approximately 6.5 m, which remains well below the theoretical upper bound of 55.29 m.

Analysis of the Conservative Theoretical Bound. The substantial discrepancy between the theoretical upper bound (55.29 m) and the observed maximum separation (approximately 6.5 m) is anticipated. This discrepancy underscores the conservative nature of the theoretical bound, $2R+\sqrt{8NE\left(0\right)/\left(\sigma K_2\right)}$, derived in Theorem 3. This bound corresponds to a worst-case scenario where the entire initial energy $E\left(0\right)$ is converted into the potential energy of the spring-like binding forces. In practice, the system operates far more efficiently due to several factors:

• Energy Partition: The initial energy $E\left(0\right)$ is partitioned between kinetic and potential energy, with only a fraction contributing directly to inter-agent distances.
• Continuous Dissipation: The $K_1$ damping term continuously dissipates kinetic energy, preventing the system from ever reaching the worst-case potential energy configuration.
• Parameter Effects: With $K_1=1.5$ and $K_2=0.8$, the system operates in a regime where damping is dominant, naturally promoting compact swarm formations.

Therefore, the observed separation of 6.5 m (approximately 13 times the safety distance $2R=1.0$ m) represents a typical operating point rather than the extreme bound. This demonstrates that while the theoretical guarantee provides a rigorous safety certificate, the actual system performance under normal conditions is substantially better.

4.3. Collision Avoidance Verification

Figure 4 demonstrates that the minimum observed inter-agent distance consistently remains above the predefined safety threshold $2R$ throughout the simulation. This observation validates both the collision avoidance guarantee established in Theorem 2 and the effectiveness of the binding term in enforcing the desired equilibrium spacing of $2R$ in the extended model. Consequently, the system converges to a stable spatial configuration characterized by an inter-agent spacing on the order of $2R$.

4.4. Comparative Analysis with Classical Cucker–Smale Model

To quantitatively assess the performance advantages of the proposed extended model, we conduct a direct comparison with the classical Cucker–Smale formulation under identical initial conditions and parameters ($N=20$, $\lambda =1.2$, $\sigma =2.5$, $R=0.5$). The classical model employs linear velocity coupling, $\Gamma \left(v\right)=v$, and lacks binding forces, which corresponds to the limiting case $K_1=0$ and $K_2=0$ in Equation (5).

The comparative results, presented in Figure 5, highlight the key performance characteristics of the extended framework:

• Velocity alignment performance: Both models achieve excellent velocity alignment, with final synchronization errors at the level of machine precision (<${10}^{-5}$ m/s). This confirms that the core consensus mechanism remains effective under the extended nonlinear coupling, without degradation in alignment capability.
• Formation compactness: The extended model reduces the maximum inter-agent distance to 6.48 m, compared to 7.23 m for the classical model—a 10.5% reduction in spatial dispersion. This improvement demonstrates that the $K_2$ binding forces provide enhanced regulation of the formation geometry while preserving swarm coherence.
• Collision avoidance: In the tested scenario, both models successfully avoid collisions, with minimum distances of 1.050 m (extended) and 1.097 m (classical), both safely exceeding the $2R=1.0$ m threshold. A critical distinction is that while collision avoidance in the classical model is observed empirically in this specific trial, the extended model provides provable collision avoidance guarantees under condition (11), as established in Theorem 2.

Quantitative comparison summary:

Table 2. Quantitative comparison between the extended and classical Cucker–Smale models.

Metric	Extended Model	Classical C-S	Improvement
Final velocity error (m/s)	<${10}^{-5}$	<${10}^{-5}$	Comparable
Maximum dispersion (m)	6.48	7.23	10.5% reduction
Minimum distance (m)	1.050	1.097	Both $>2R=1.0$ m
Theoretical safety guarantee	Yes (Theorem 2)	No	–
Energy dissipation guarantee	Yes (Theorem 1)	No	–

provides a comprehensive overview of the key performance metrics. The extended model demonstrates velocity alignment comparable to the classical model, along with improved formation compactness. Crucially, it also provides theoretical safety and energy dissipation guarantees, which are absent in the classical formulation.

In summary, the extended framework preserves the velocity alignment performance of the classical model while enhancing formation control through the $K_2$ binding mechanism. Most significantly, it introduces rigorous theoretical guarantees for safety and energy dissipation—a critical advancement for applications demanding provable collision avoidance.

4.5. Parameter Sensitivity Analysis

This subsection employs a comprehensive multi-metric framework to evaluate the extended Cucker–Smale model. Flocking performance is quantified through four key metrics: velocity synchronization error (${\phi }_1$), group dispersion (${\phi }_2$), collision count (${\phi }_3$), and heading offset angle (${\phi }_4$). These metrics are formally defined as:

$$\begin{array}{cc}\hfill {\phi }_1& ={\left({\displaystyle \frac{1}{N(N-1)}}\sum _{i\ne j}{\parallel v_i-v_j\parallel }^2\right)}^{1/2}(\mathrm{velocity}\mathrm{synchronization}\mathrm{error})\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\phi }_2& ={\left({\displaystyle \frac{1}{N(N-1)}}\sum _{i\ne j}{\parallel x_i-x_j\parallel }^2\right)}^{1/2}(\mathrm{group}\mathrm{dispersion})\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\phi }_3& =N_{\mathrm{crash}}(\mathrm{collisions},\parallel x_i-x_j\parallel <2R)\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\phi }_4& =arg(v_c,x_0-x_c)(\mathrm{heading}\mathrm{offset}\mathrm{angle})\hfill \end{array}$$

(15)

• Metric Interpretation:

• Velocity synchronization error (${\phi }_1$): Quantifies alignment quality, with ${\phi }_1\to 0$ indicating perfect synchronization.
• Group dispersion (${\phi }_2$): Measures spatial compactness, where lower values correspond to tighter formations.
• Collision count (${\phi }_3$): Tallies the number of agent pairs violating the safety distance ($<2R$), averaged over all trials.
• Heading offset angle (${\phi }_4$): Measures directional deviation from the target orientation $x_0$, with ${\phi }_4\to 0$ indicating perfect alignment.

• Experimental Protocol: The sensitivity analysis follows a systematic Monte Carlo approach with the following configuration:

• Parameter ranges: $K_2\in [0.05,0.50]$ (increments of 0.05) and $K_1\in [1.25,3.05]$ (increments of 0.20).
• Simulation settings: 2D planar region, $100 s$ duration, $0.1 s$ time step (fourth-order Runge–Kutta integration).
• Initial conditions: Positions uniformly distributed, velocities in the range 10–20 m/s, initial headings $\pm {45}^{\circ }$.
• Statistical basis: 50 independent trials per parameter configuration.
• Stability criterion: The system is considered stable when the mean velocity variance remains below $0.05{\mathrm{m}}^2/{\mathrm{s}}^2$ for 10 consecutive time steps. A detailed sensitivity analysis of this threshold is provided below.

4.5.1. Sensitivity Analysis of the Stability Criterion

To address the potential arbitrariness in selecting the stability threshold $ϵ=0.05{\mathrm{m}}^2/{\mathrm{s}}^2$, a sensitivity analysis is conducted by varying $ϵ$ from $0.01$ to $0.10{\mathrm{m}}^2/{\mathrm{s}}^2$. Five representative $K_2$ values (0.05, 0.15, 0.25, 0.35, 0.50) are evaluated with $K_1$ fixed at 1.5. For each threshold, the convergence time is computed, and the relative ranking of $K_2$ values is compared across all thresholds.

The sensitivity of the stability criterion to parameter variations is analyzed in Figure 6 and

Table 3. Convergence times (in seconds) for different $K_2$ values and stability thresholds.

${\mathit{K}}_2$	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.10
0.05	100.0	100.0	100.0	98.8	97.5	96.5	95.7	95.0	94.2	93.5
0.15	75.2	72.3	70.7	69.7	68.9	68.2	67.7	67.2	66.8	66.4
0.25	55.4	53.4	52.2	51.4	50.8	50.3	49.8	49.4	49.1	48.7
0.35	61.6	60.0	59.1	58.5	58.0	57.5	57.1	56.8	56.5	56.2
0.50	48.2	47.0	46.2	45.6	45.2	44.9	44.5	44.2	44.0	43.7

. The Spearman rank correlation coefficient $\rho =1.000$ indicates perfect consistency in the parameter ranking across all tested thresholds. Specifically, $K_2=0.25$ consistently yields the fastest convergence time regardless of the threshold value, while $K_2=0.05$ remains the slowest. This result demonstrates the insensitivity of optimal parameter selection to the specific choice of $ϵ$ within the tested range $[0.01,0.10]{\mathrm{m}}^2/{\mathrm{s}}^2$, thereby validating the robustness of the adopted stability criterion.

4.5.2. Repulsion Gain ($K_2$) Sensitivity

To investigate the effect of the repulsion gain $K_2$, we compute steady-state performance metrics while sweeping $K_2$ from 0.05 to 0.50, with the dissipative gain fixed at $K_1=1.5$.

As shown in Figure 7, increasing $K_2$ from 0.05 to 0.15 substantially improves all performance metrics: velocity error decreases by 73.6%, group dispersion reduces by 54.2%, and heading stability improves significantly. However, beyond $K_2=0.15$, the system enters a saturation zone where marginal benefits diminish, indicating that moderate $K_2$ values (0.15–0.30) provide optimal performance.

Table 4. Collision statistics and stabilization time under different $K_2$ values.

${\mathit{K}}_2$	0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40	0.45	0.50
Collisions	0	0	0	0	0	0	0	0	0	0.1
Stability Time (s)	57.0	51.1	50.7	53.9	51.3	53.2	58.7	53.9	52.5	54.2

Note: Values averaged over 50 Monte Carlo trials. Collision count 0.1 at $K_2=0.50$ corresponds to 0.1 collisions per trial (5 collisions in 50 trials).

reveals robust collision avoidance across the tested $K_2$ range, with near-zero collision counts confirming the effectiveness of the $K_2$ mechanism. The single statistically significant collision at $K_2=0.50$ (0.1 collisions/trial) suggests that excessively high $K_2$ can induce oscillatory behaviour, demonstrating parameter sensitivity at the boundary of the operational range. Stability time shows limited sensitivity to $K_2$, fluctuating between 50–59 s without clear monotonic trend.

4.5.3. Dissipation Gain ($K_1$) Sensitivity

To analyze the effect of the dissipative gain $K_1$, performance metrics are computed while sweeping $K_1$ from 1.25 to 3.05, with the repulsive gain fixed at $K_2=0.8$.

Figure 8 illustrates a clear trend: increasing $K_1$ generally degrades flocking performance. Specifically:

• Velocity synchronization error increases by 85% from $K_1=1.25$ to 3.05;
• Group dispersion expands by 32%, indicating looser formations;
• Heading accuracy deteriorates, suggesting reduced directional coordination.

Table 5. Collision statistics and stabilization time under different $K_1$ values.

${\mathit{K}}_1$	1.25	1.45	1.65	1.85	2.05	2.25	2.45	2.65	2.85	3.05
Collisions	0	0	0	0	0	0	0	0	0	0
Stability Time (s)	50.4	51.0	55.4	51.9	52.1	51.8	58.5	50.8	54.1	56.4

Note: Collision counts remain zero across all $K_1$ values, confirming the robustness of the $K_2$-based collision avoidance mechanism.

demonstrates that collision avoidance remains robust regardless of $K_1$ variations, with zero collisions recorded across the entire range. This confirms that the $K_2$ mechanism provides independent safety guarantees. Stability time shows moderate fluctuations (50–59 s) with no systematic trend, indicating limited influence of $K_1$ on convergence speed.

4.5.4. Interpretation and Implications

Key Findings:

• Robustness of the stability threshold: The parameter ranking remains perfectly consistent (Spearman $\rho =1.000$) across thresholds from 0.01 to 0.10 ${\mathrm{m}}^2/{\mathrm{s}}^2$, validating the selection of $ϵ=0.05$.
• Optimal $K_2$ range: $K_2\in [0.15,0.30]$ provides the best trade-off between performance enhancement and collision safety. Beyond this range, diminishing returns are observed.
• Performance trade-off with $K_1$: Higher $K_1$ values degrade flocking metrics, suggesting that excessive damping disrupts the balance between velocity alignment and formation maintenance.
• Independent collision safety: The $K_2$ mechanism ensures collision avoidance independent of $K_1$ variations, thereby providing robust safety guarantees.
• Practical design guidelines: For implementation, designers should prioritize $K_2$ within the optimal range (0.15–0.30) and employ moderate $K_1$ values (1.5–2.0) to balance convergence speed with formation quality.

Limitations and Scope: These findings are specific to the tested parameter ranges and experimental conditions. While the trends are statistically significant, they should be interpreted within the context of the 2D simulation environment and idealized communication assumptions. Real-world implementations may require parameter adjustments to account for sensor noise, communication delays, and environmental disturbances.

4.6. Scalability Analysis

To address the practical concern of whether the proposed model remains effective in larger-scale systems, a scalability analysis was conducted. The system’s convergence behaviour was examined for three different agent sizes: $N=20$, 50, and 100 agents. All other parameters were held constant ($K_1=1.5$, $K_2=0.8$, $\lambda =1.2$, $\sigma =2.5$, $R=0.5$) to isolate the effect of population size.

Figure 9 presents the key results. Figure 9a shows the velocity variance over time for the three agent sizes on a logarithmic scale.

All cases exhibit exponential convergence, successfully reaching the predefined threshold (${10}^{-4}$). Notably, the convergence time decreases as the agent size increases. The $N=100$ agent converged in approximately 5.34 s, which is about 44% faster than the $N=20$ agent (9.49 s).

This counter-intuitive acceleration effect is visualized in Figure 9b, where convergence time is plotted against N. The trend indicates that denser agents, with their enhanced local connectivity, facilitate faster information propagation and consensus formation.

The final steady-state velocity variance showed a slight increase with N (from $8.49\times {10}^{-10}$ for $N=20$ to $2.56\times {10}^{-7}$ for $N=100$), which is attributed to minor numerical integration artifacts in the more complex, stiff ODE system. However, this variance remains negligible in practice, confirming that the model achieves high-fidelity velocity alignment even at scale. This analysis demonstrates that the proposed framework not only scales efficiently to at least 100 agents but also benefits from increased agent density through faster convergence, thereby addressing a key practical requirement for large-scale multi-agent deployments.

4.7. Parameter Optimization

To select optimal parameters, it is necessary to consider both the system’s energy consumption and the velocity alignment performance. The following formula defines the total energy consumption metric:

$$E(K_1,K_2)={\displaystyle \frac{1}{M}}\sum _{m=1}^M\left(\sum _{t=0}^T\sum _{i=1}^N{\parallel a_i^{\left(m\right)}\left(t\right)\parallel }^2\Delta t\right)$$

The energy metric in (12) is defined through four key parameters: M denotes the number of Monte Carlo trials, T represents the total simulation steps, N is the agent population size, and $a_i^{\left(m\right)}\left(t\right)$ denotes the acceleration vector of the i-th agent in the m-th trial at discrete time t. The time step duration is $\Delta t$. The Energy Consumption Ratio (ECR) is defined as:

$$ECR={\displaystyle \frac{E(K_1,K_2)}{E_{\mathrm{base}}}}\times 100\%\mathrm{with}{E}_{\mathrm{base}}=E(K_1=1.25,K_2=0.5)$$

The baseline configuration ($K_1$ = 1.25, $K_2$ = 0.5) represents a moderate operational point identified from preliminary studies, providing a reasonable reference for energy comparison. System test data reveal the influence of repulsion gain parameter $K_2$ on system performance.

Corrected Interpretation: The performance metrics, as summarized in

Table 6. Performance metrics under variable repulsion gain parameters.

${\mathit{K}}_2$	0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40	0.45	0.50
Velocity Error	0.35	0.07	0.02	0.006	0.003	0.003	0.003	0.002	0.002	0.002
ECR (%)	1.5	4.6	9.5	17.8	25.4	34.9	48.6	67.2	82.0	103.6

Note: This table has been corrected with the actual simulation data. The corrected ECR values show the expected monotonic increase with $K_2$, with energy consumption exceeding the baseline (100%) at $K_2=0.50$.

, reveal a clear trade-off between coordination accuracy and energy consumption. Velocity synchronization error decreases dramatically from 0.35 m/s at $K_2=0.05$ to 0.002 m/s at $K_2\ge 0.40$, representing a 175-fold improvement. Concurrently, the energy consumption ratio (ECR) increases monotonically from 1.5% to 103.6%. The optimal operating range $K_2\in [0.15,0.30]$ achieves velocity errors below 0.02 m/s while keeping ECR below 35%. Beyond $K_2=0.40$, diminishing returns are observed: error reduction plateaus while energy consumption continues to rise, exceeding the baseline at $K_2=0.50$. This analysis identifies the Pareto-optimal region and provides practical guidance for parameter selection.

As shown in

Table 7. Performance metrics under variable dissipation gain parameters.

${\mathit{K}}_1$	1.25	1.45	1.65	1.85	2.05	2.25	2.45	2.65	2.85	3.05
Velocity Error	0.05	0.06	0.07	0.08	0.15	0.16	0.21	0.25	0.34	0.34
ECR (%)	105.7	88.3	82.8	67.1	67.3	58.5	51.1	54.5	50.9	47.0

, when the dissipation gain parameter $K_1$ is set to 1.85, the velocity synchronization error of the system is reliably maintained within the engineering accuracy range of 0.1 m/s. Concurrently, the energy consumption ratio is optimized to an equilibrium point of $67.1\%$.

5. Conclusions and Prospects

This study presents an extended Cucker–Smale (C–S) model that integrates nonlinear velocity coupling with state-aware binding forces, thereby addressing collision avoidance and formation stability in multi-agent systems. Rigorous Lyapunov analysis provides theoretical guarantees for asymptotic flocking and collision safety under a verifiable initial energy condition. Numerical simulations validate the theory and quantify the roles of key control gains, leading to a practical parameter optimization strategy.

Key Findings:

• The repulsion gain $K_2$ governs spatial cohesion. An optimal range $K_2\in [0.15,0.3]$ substantially improves velocity synchronization and reduces dispersion, with diminishing returns beyond $K_2>0.15$.
• High dissipative gain $K_1$ can destabilize flocking through overcorrection, degrading velocity coherence, spatial cohesion, and heading stability.
• A Pareto-optimal parameter window ($K_2\in [0.15,0.3]$, $K_1=1.85$) is identified, effectively balancing alignment accuracy, collision avoidance, and energy efficiency.

The current analysis, while robust, operates under ideal assumptions. Practical limitations include perfect communication and sensor accuracy, which are often violated in real-world scenarios.

Future Work:

While the present study establishes a theoretical foundation under ideal conditions, several important extensions are warranted for real-world applications. Future research will focus on the following theoretical and practical challenges:

• Robustness to Communication Constraints: The current model assumes perfect, instantaneous information exchange. A critical theoretical extension is to analyze the system’s stability and performance under communication delays and intermittent packet loss. This would involve reformulating the model using delayed differential equations or event-triggered control frameworks to derive new convergence conditions.
• Sensitivity to Measurement Noise: The analysis assumes accurate state measurements. Investigating the model’s behaviour with sensor noise (e.g., Gaussian noise in position/velocity estimates) is essential. This would require a stochastic stability analysis to determine how noise statistics propagate through the nonlinear coupling and binding forces, potentially informing the design of robust observers.
• Extension to Heterogeneous Dynamics: The framework currently considers homogeneous agents. Extending it to systems with heterogeneous agent dynamics (e.g., varying masses, actuator limits, or sensor capabilities) presents a significant theoretical challenge. This would involve developing heterogeneous Lyapunov functions or adaptive control schemes to ensure cohesive flocking despite individual differences.
• Limitations and Scope: The numerical validation presented in this work focuses on verifying the theoretical predictions of the proposed model under idealized conditions. This approach allows for clear interpretation of results without confounding factors. While the scalability analysis (Section 4.4) demonstrates the model’s effectiveness with up to 100 agents, validation under more realistic conditions—including communication delays, measurement noise, and heterogeneous agent dynamics—represents important directions for future research as outlined in Section 5. The current study establishes the theoretical foundation and demonstrates fundamental capabilities; subsequent work will build upon this foundation to address implementation challenges in specific application domains.
• Experimental Validation: Ultimately, implementing and testing the proposed algorithm on physical multi-robot platforms is necessary to validate its practical efficacy and identify unforeseen real-world constraints.

This work establishes a theoretical foundation for safe and efficient multi-agent coordination, with direct relevance to applications such as UAV formation control and autonomous vehicle platooning. The outlined research directions aim to translate these theoretical guarantees into reliable practical implementations.