Introduction

Motivation

Over the past few years, scholars delving into the realm of autonomous systems and automation have been focusing on decentralized control, particularly within the framework involving multi-agent setups. Effectively coordinating decentralized collaboration among multi-agent systems (MAS) represents a pivotal task. Attaining global group behavior solely through local interactions is the goal. In multi-agent systems (MAS), control issues typically involve achieving consensus, forming coalitions, containing conflicts, aggregating information, and coordinating rendezvous [1,2,3,4]. Formation problems can be categorised into formation acquisition [5], formation tracking [6, 7], formation manoeuvring [8], and formation interception [1].

The collaborative behavior known as formation control within multi-agent systems (MAS) entails agents autonomously coordinating to achieve specific spatial positions. This formation control approach is commonly used in collaborative and cooperative tasks, such as search and rescue, object transport, and agricultural irrigation [1, 2, 9]. Virtual rigid body maneuver control of the formation is necessary. This control involves translations, rotations, or a combination of both. When focusing on a singular constraint, these scenarios are frequently termed as tracking problems. Another linked challenge entails agents apprehending and encircling a mobile objective according to a predetermined arrangement.

During the task of acquiring formation, the agents move to a predetermined position in the space where they are located and eventually form a stationary geometric structure. The aim of executing the task focused on tracking formations is to ensure the precise following and alignment of the formation with a designated reference motion trajectory. This is provided that the formation acquisition task has been completed and achieved [5]. The available methods for formation control are categorized into position, displacement, and distance-based methods [3]. Reference [4] uses theoretical ideas based on rigid graphs to provide a theoretical basis for self-driving vehicle formation, filling a gap in the existing theory. Position-based and displacement-based methods are the most commonly used methods. Displacement-based formation control is a highly researched direction. Yet, when contrasting the implementation of distinct strategies for controlling formations by utilizing the attributes of sensed and controlled factors, the distance-centric approach exhibits superior sensing capabilities compared to the position-oriented and displacement-oriented approaches [3].

References [10,11,12] have many inspirations for this article. Among them, [10] was inspired by the swarm intelligence algorithm in the biological world and integrated multiple algorithms to perform without communication between drones. Reference [11] discussed and studied relevant parameters of swarm intelligence, and the results of [12] confirmed that learning adaptability does not specifically refer to a set of task parameters for group learning and optimizing group behavior of a certain type.

Formation control is a fundamental subject that is widely discussed. The algorithms are designed for various types of formation agents. Several general models can be categorized for agent formations. For instance, two common models are the single integrator and double integrator models. Reference [5] presents a method for decentralized formation control that utilizes a single-integral agent model to achieve exponential stability in the dynamic error between agents’ distances. In the reference [13], a fresh approach to cluster control is presented, utilizing principles from rigid graph theory in a two-dimensional space for decentralized formation control within multi-agent systems. This method utilizes a model featuring dual integration capabilities. The results regarding formations can also be classified according to the physical kinematic model of the agents as well as the physical dynamics model. There are two main models in this regard: the incomplete underdrive model, which considers the speed limitations of a moving vehicle. Fully driven Euler–Lagrange models can contain robotic manipulators, spacecraft, and omnidirectional mobile robots. It is important to note that the morphology necessary for formation is determined by the offset vectors between the subjects. These vectors represent displacement or relative position. In practical scenarios, each agent is required to manage its distinct relative position vector, established and predefined within a coherent global frame of reference. For displacement-based control of cluster formation, aligning exclusively within the local frame of reference of the agent is essential. This can be challenging in practice.

Previous works

The article [4] presents distance-based formation control among agents. References [5, 14, 15] provide early findings on distance-based formation control. The aforementioned content, centered around agents using single-integral kinematic models, predominantly concerns research outcomes regarding formation acquisition control. By summarizing the aforementioned research content and findings, the realization of the formation tracking task emerges as feasible. Reference [8] presents a distance-focused control approach designed to tackle maneuvering challenges within multi-agent formations and intercepting targets. The proposed approach employs the dual integrator agent model and the theory of rigid graph theory to introduce cluster and target interception controllers.

The article delves into a proposed rigidity-based controller for single integrators, addressing formation maneuvering and employing an approach rooted in rigid graphs that emphasizes distance-related aspects. Provides cluster speed over time directly to all agents for all controllers. In reference [16], when tackling the control challenge within swarm formations, accounting for the leader’s extra speed, the entire formation adjusts to follow the leader while ensuring the maintained distance among agents. In the case of the single integrator model, the control strategy predominantly employs standard gradient descent to establish the acquisition component, and it integrates an integral term to maintain a zero steady-state error in the speed command. In reference [17], the author delves into the challenge of swarming within the virtual leader’s domain. Recognizing the evolving dynamics of the virtual leader and its dynamically changing topology, a method is introduced to control it. This method involves a new form of switching control law for each agent, relying on the state information of its neighboring agents and external reference signals. In the article [18], the author examines the multi-agent consensus challenge, encompassing an active leader and an evolving interconnection topology. The article presents a localized controller centered around neighborhoods to pursue the changing and unmeasurable leader, alongside methods for state estimation. Once the variable leader’s acceleration is determined, each agent can accurately trail the leader. In [19], the aim was to model the fully driven Euler–Lagrangian equations by ordering only locally interacting followers to track a dynamic leader constrained by a generalized coordinate vector changing over time. Two consensus tracking algorithms are designed. In the first design, a continuously distributed estimator based on speed and an adaptable control system is proposed primarily for aircraft maintaining a consistent speed. In the second design, a sliding film control algorithm is proposed with a constant pilot speed. In [20], a method for solving distributed coordinated tracking using variable structure methods is proposed, paying attention to both consensus tracking and group tracking algorithms and contemplating a dynamic virtual leader, which forms a subset consisting of followers’ neighbors.

A decomposition architecture is proposed in [21]. This architecture is passively executed and mainly divides dynamic groups into two decoupled systems, which are related to multiple rigid body formations and maneuvers. In single integrator and dual integrator states, there are no speed and no acceleration situations under fixed and switched network topologies and finally distributed consensus tracking under no speed measurement and no acceleration measurement situations. The more popular formation control method is mainly the distance-based formation control method. Its essence is deeply connected to the theory of rigid graphs [4]. This fundamental concept is unavoidable as its introduction has assured compliance with distance constraints for the required formation. Finally, the full kinetic model was considered in [22].

Main contributions

Building on previous research, this paper centers on implementing a distance-based approach using a complete kinematic model in the form of a unicycle. The dynamic formation tracking and interception control based on rigid graphs were researched and explored. Through the combination of distance-based rigid graph theory and dynamic formation, the formation interception problem in the presence of obstacles has been successfully solved. Before this, relevant research was extremely rare. For [23, 24] a decentralized control scheme is provided to achieve collision avoidance and tracking based on the known trajectory. Although the formation maneuvering problem and the target tracking problem are mentioned in [8], the target tracking and interception problems in the multi-agent dynamic formation under the non-holonomic dynamics model are not considered. In [25], flocking and target interception control issues under non-holonomic moving agent formations were mentioned, but they did not consider target tracking and interception issues under dynamic formations. In the context of the target interception problem, it is assumed that only the leader possesses knowledge of the moving target, with the target speed remaining uncertain. This can result in diminished robustness of the outcome alongside weakened adaptability and tracking. Drawing inspiration from previous studies [26], this issue can be resolved by utilizing the variable structure class of observers and estimating speed via the continuity of formation graphs. Furthermore, designing the adaptive control of dynamic formations can prove to be a more efficient solution when intercepting targets in formations amidst challenging environments, this makes future research on multi-agent dynamic formation control more important.

The contributions of this paper can be encapsulated in the following summary.

  1. 1.

    Based on [1], the problem of intercepting targets in complex scenarios was studied. For example, situations may arise where the size or geometry of a formation needs to be adjusted over time to avoid obstacles. This necessity requires the use of dynamic formation controllers to effectively solve the target interception problem.

  2. 2.

    Inspired by [26], a control law integrating an asymptotic velocity estimator and adaptive control is introduced to bolster the system’s resilience.

  3. 3.

    In contrast to [25], this paper presents a comprehensive kinematic model, which incorporates dynamic formations.

The subsequent segments of this article are organized as follows: Section “Preliminaries and notations” introduces graph theory concepts and models of unicycle dynamics. Sections “Problem statement” and Dynamic formation tracking intercept design” detail the essential groundwork for addressing the dynamic formation interception challenge. Section “Results from simulation experiments” delineates the outcomes derived from simulating the experiments, while Sect. “Conclusion” presents the derived conclusions.

Fig. 1
figure 1

Modeling dynamics of agents

Full size image

Preliminaries and notations

Notations

In this article, \({\mathbb {R}}\) is used to represent real numbers, and the real number vector size is \(n\times 1\), and \({\mathbb {R}}_n\) is used to describe it, and the size is specified to be \(n\times m\) real numbers Matrix, represented by \({{\mathbb {R}}^{n \times m}}\). The n-dimensional vectors are represented by \( 1_n \), and the \(n \times n\) unit matrices are represented by \({\mathcal {I}}_n\). The Kronecker product is denoted by \(\otimes \), and the diagonal matrix is \(diag(\cdot )\). For any vector \({\mathcal {X}}\in {\mathbb {R}}^{n}\), its Euclidean norm is defined as \(\left\| {\mathcal {X}} \right\| : = {({{\mathcal {X}}^T}{\mathcal {X}})^{\frac{1}{2}}}\).

Rigidity theory of graphs

Examine systems with n multi-agents, applying algebraic graph theory to describe the communication relationships between them [27]. Represented by an undirected graph \({G}=({\mathcal {V}},{\mathcal {E}},{\mathcal {A}})\), where \({\mathcal {V}} =\{1,2,\ldots , n\}\) represents the set of vertices, and \({\mathcal {E}}\subset {\mathcal {V}}\times {\mathcal {V}}\) represents the set of undirected edges connecting different vertices. Here, vertices symbolize agents, and the undirected edges signify communication paths between these agents. The adjacency matrix \({\mathcal {A}} = [{{\mathfrak {a}}_{{\mathfrak {i}}{\mathfrak {j}}}}] \in {{\mathbb {R}}^{n \times n}}\) is a matrix that contains only non-negative values. It is assigned a weight \({{\mathfrak {a}}_{{\mathfrak {i}}{\mathfrak {j}}}} = 1\) if the node \(({\mathfrak {i}},{\mathfrak {j}})\in {\mathcal {E}}\) exists, and \({\mathfrak {a}}_{\ddot{y}}=0\) if it does not. In addition, the symbol \(({\mathfrak {i}},{\mathfrak {j}})\in {\mathcal {E}}\) indicates that information can be transferred between agents \({\mathfrak {i}}\) and \({\mathfrak {j}}\). For G undirected graph, if the vertex set \(({\mathfrak {i}},{\mathfrak {j}})\in {\mathcal {E}}\) also contains \(({\mathfrak {j}},{\mathfrak {i}} )\). The number of undirected edges can be expressed as \(\ell \in \{1,\ldots ,n(n-1)/2\}\). The neighbor set of vertex \({\mathfrak {i}}\) can be represented by \({{\mathcal {N}}}_{{\mathfrak {i}}}\), where \({{\mathcal {N}}_{\mathfrak {i}}} = \left\{ {{\mathfrak {j}} \in {\mathcal {V}}:({\mathfrak {i}},{\mathfrak {j}}) \in {\mathcal {E}}} \right\} \). With \(p_{{\mathfrak {i}}}\in {\mathbb {R}}^{2}\) denoted as the position of vertex \({\mathfrak {i}}\), the frame \({\mathcal {F}}\) is a pair \(\left( {G},p\right) \), where \(p=(p_{1},p_{2},\ldots ,p_{n})\) is a graph and its configuration. For any order of undirected edges in \({\mathcal {E}}\), the undirected edge function is \(\psi _{G}:{\mathbb {R}}^{2n}\in {\mathbb {R}}^{\ell }\): \({\psi _ {G}}(p) = ( \ldots ,{\left\| {{p_{\mathfrak {i}}} - {p_{\mathfrak {j}}}} \right\| ^2}, \ldots ),({\mathfrak {i}},{\mathfrak {j}} ) \in {\mathcal {E}}\).Determine the stiffness matrix \({\mathcal {R}}:{\mathbb {R}}^{2n}\rightarrow {\mathbb {R}}^{\ell \times 2n}\) as the Jacobi matrix for \({\mathcal {F}}=\left( {G},p\right) \) [27].

$$\begin{aligned} R(p) = \frac{1}{2}\frac{{\partial {\psi _{G}}}}{{\partial p}} \end{aligned}$$
(1)

In two-dimensional space, if the frame \({\mathcal {F}}\) achieves infinite rigidity and fulfills the condition \(rank({\mathcal {R}})=2n-3\), given \({\vee }(t)\in {\mathbb {R}}^{2}\), the expression \({\mathcal {R}}(p)(1_{n}\otimes {\vee })=0\) is then produced.

Agent kinematics and dynamics modelling

Imagine a system comprising n agents. Figure 1 depicts the \({\mathfrak {i}}\) th agent. To provide a clearer depiction of the ensuing content, Establish the initial hypothesis as follows:

Assumption 1

Formed agents move at a constant speed in a working environment.

Assumption 2

The effect of drag is negligible and can be ignored.

Assumption 3

For a system of formation agents, \({\mathfrak {i}}\), \({\mathfrak {i}}\in {\mathcal {V}}\) can receive information from neighbours \({{\mathcal {N}}}_{{\mathfrak {i}}}\) and leaders.

Consider this comprehensive dynamical model [28]:

$$\begin{aligned}&\zeta ={{\mathcal {J}}}(\theta _{{\mathfrak {i}}}){\vee }_{\mathfrak {i}}, \end{aligned}$$
(2)
$$\begin{aligned}&{\bar{M}}_{\mathfrak {i}}{\dot{\zeta }}_{\mathfrak {i}}+{\bar{D}}_{\mathfrak {i}}\zeta _{\mathfrak {i}}={\bar{\tau }}_{\mathfrak {i}}, \end{aligned}$$
(3)
$$\begin{aligned}&{\bar{M}}_{\mathfrak {i}}{\mathcal {J}}(\theta _{\mathfrak {i}}){\dot{\vee }}_{\mathfrak {i}}+{\bar{M}}_{\mathfrak {i}}\dot{{\mathcal {J}}}(\theta _{\mathfrak {i}}){\vee }_{\mathfrak {i}}+{\bar{D}}_{\mathfrak {i}}{{\mathcal {J}}}(\theta _{\mathfrak {i}}){\vee }_{\mathfrak {i}}={\bar{\tau }}_{\mathfrak {i}} \end{aligned}$$
(4)

where \({\bar{M}}_{\mathfrak {i}}=diag(m_{\mathfrak {i}}, \bar{{\mathcal {I}}}_{\mathfrak {i}})\), the mass of the \({\mathfrak {i}}\) th agent is \(m_{\mathfrak {i}}\). \(\bar{{\mathcal {I}}}_{{\mathfrak {i}}}\) represents the rotational inertia of the \({\mathfrak {i}}\) th agent about \(C_{{\mathfrak {i}}}\). \({\bar{D}}_{{\mathfrak {i}}}\in {\mathbb {R}}^{2\times 2}\) is the constant damping matrix. \({{\bar{\tau }}} \in {{\mathbb {R}}^2}\) represents the external actuator that applies force or torque to the \({\mathfrak {i}}\) th agent. The \({\mathfrak {i}}\) th member of a multi-agent formation is denoted by the subscript \({\mathfrak {i}}=1,2,\ldots ,n\). \({\zeta _{\mathfrak {i}}} = {[{\vee },{\omega _{\mathfrak {i}}}]^T} \in {{\mathbb {R}}^2}\) denotes the generalised reference position vector of the \({\mathfrak {i}}\) th agent in the Earth’s fixed reference coordinate system \(\left\{ {\mathcal {E}}\right\} \). At the same time, \({\vee }_{{\mathfrak {i}}}\) represents the linear speed of the \({\mathfrak {i}}\) th agent in the \({\theta }_{{\mathfrak {i}}}\) direction, while \({\omega _{\mathfrak {i}}}\) characterizes the rotational speed of the \({\mathfrak {i}}\) th agent around \(C_{{\mathfrak {i}}}\) along its vertical axis.

The matrices \(M_{{\mathfrak {i}}}(\theta _{{\mathfrak {i}}})\) and \(C_{{\mathfrak {i}}}(\theta _{{\mathfrak {i}}},\dot{\theta _{{\mathfrak {i}}}})\) in use equation (3) are defined as follows:

$$\begin{aligned}{} & {} \left. M_{{\mathfrak {i}}}=\left[ \begin{matrix}{m_{{\mathfrak {i}}}\textrm{cos}^{2}\theta _{{\mathfrak {i}}}+\frac{\bar{{\mathcal {I}}}_{{\mathfrak {i}}}}{L_{{\mathfrak {i}}}^{2}}\textrm{sin}^{2}\theta _{{\mathfrak {i}}}}&{}{\left( m_{{\mathfrak {i}}}-\frac{\bar{{\mathcal {I}}}_{{\mathfrak {i}}}}{L_{{\mathfrak {i}}}^{2}}\right) \textrm{sin}\theta _{{\mathfrak {i}}}\textrm{cos}\theta _{{\mathfrak {i}}}}\\ {\left( m_{{\mathfrak {i}}}-\frac{\bar{{\mathcal {I}}}_{{\mathfrak {i}}}}{L_{{\mathfrak {i}}}^{2}}\right) \textrm{sin}\theta _{{\mathfrak {i}}}\textrm{cos}\theta _{{\mathfrak {i}}}}&{}{m_{{\mathfrak {i}}}\textrm{sin}^{2}\theta _{{\mathfrak {i}}}+\frac{\bar{{\mathcal {I}}}_{{\mathfrak {i}}}}{L_{{\mathfrak {i}}}^{2}}\textrm{cos}^{2}\theta _{{\mathfrak {i}}}}\\ \end{matrix}\right. \right] \end{aligned}$$
(5)
$$\begin{aligned}{} & {} C_{{\mathfrak {i}}}= \left. \left[ \begin{array}{ll}{-\left( m_{{\mathfrak {i}}} - \frac{\bar{{\mathcal {I}}}_{{\mathfrak {i}}}}{L_{{\mathfrak {i}}}^{2}}\right) {\dot{\theta }}_{{\mathfrak {i}}}\sin \theta _{{\mathfrak {i}}}\cos \theta _{{\mathfrak {i}}}}&{}{m_{{\mathfrak {i}}}{\dot{\theta }}_{{\mathfrak {i}}}\textrm{cos}^{2}\theta _{{\mathfrak {i}}}+\frac{\bar{{\mathcal {I}}}_{{\mathfrak {i}}}}{L_{{\mathfrak {i}}}^{2}}{\dot{\theta }}_{{\mathfrak {i}}}\textrm{sin}^{2}\theta _{{\mathfrak {i}}}}\\ {-m_{{\mathfrak {i}}}{\dot{\theta }}_{{\mathfrak {i}}}\textrm{sin}^{2}\theta _{{\mathfrak {i}}}-\frac{\bar{{\mathcal {I}}}_{{\mathfrak {i}}}}{L_{{\mathfrak {i}}}^{2}}{\dot{\theta }}_{{\mathfrak {i}}}\textrm{cos}^{2}\theta _{{\mathfrak {i}}}}&{}{\left( m_{{\mathfrak {i}}}-\frac{\bar{{\mathcal {I}}}_{{\mathfrak {i}}}}{L_{{\mathfrak {i}}}^{2}}\right) {\dot{\theta }}_{{\mathfrak {i}}}\sin \theta _{{\mathfrak {i}}}\cos \theta _{{\mathfrak {i}}}}\end{array}\right. \right] \nonumber \\ \end{aligned}$$
(6)

where the matrix \({\mathcal {J}}(\theta _{{\mathfrak {i}}})\in {\mathbb {R}}^{2\times 2}\) represents the coordinate transformation for agent \({\mathfrak {i}}\) from the reference frame \(\left\{ B\right\} \) to \(\left\{ {\mathcal {E}}\right\} \). Its notation is as follows:

$$\begin{aligned} {\mathcal {J}}({\theta _{\mathfrak {i}}}) = \left( {\begin{array}{*{20}{c}} {\cos {\theta _{\mathfrak {i}}}}&{}{\sin {\theta _{\mathfrak {i}}}} \\ { - \frac{{\sin {\theta _{\mathfrak {i}}}}}{{{L_{\mathfrak {i}}}}}}&{}{\frac{{\cos {\theta _{\mathfrak {i}}}}}{{{L_{\mathfrak {i}}}}}} \end{array}} \right) \end{aligned}$$
(7)

where \(L_{{\mathfrak {i}}}\ne 0\), Using the Eq. (2) this can be rewritten as:

$$\begin{aligned} {\vee }_{{\mathfrak {i}}}={{\mathcal {J}}}^{-1}(\theta _{{\mathfrak {i}}})\zeta _{{\mathfrak {i}}} \end{aligned}$$
(8)

Perform a time-derivative on use Eq. (8):

$$\begin{aligned} \begin{aligned} {\dot{\vee }}_{{\mathfrak {i}}}&=\dot{{\mathcal {J}}}^{-1}(\theta _{{\mathfrak {i}}})\zeta +{{\mathcal {J}}}^{-1}(\theta _{{\mathfrak {i}}}){\dot{\zeta }} \\&={{\mathcal {J}}}^{-1}(\theta _{{\mathfrak {i}}})[{\dot{\zeta }}+{{\mathcal {J}}}(\theta _{{\mathfrak {i}}})\dot{{\mathcal {J}}}^{-1}(\theta _{{\mathfrak {i}}})\zeta ] \end{aligned} \end{aligned}$$
(9)

Substituting use Eqs. (2) (3) into (9) yields:

$$\begin{aligned} {\bar{M}}_{{\mathfrak {i}}}{\mathcal {J}}(\theta _{{\mathfrak {i}}}){\dot{\vee }}_{{\mathfrak {i}}}+{\bar{M}}_{{\mathfrak {i}}}\dot{{\mathcal {J}}}(\theta _{{\mathfrak {i}}}){\vee }_{{\mathfrak {i}}}+{\bar{D}}_{{\mathfrak {i}}}{\mathcal {J}}(\theta _{{\mathfrak {i}}}){\vee }_{{\mathfrak {i}}}={\bar{\tau }}_{{\mathfrak {i}}} \end{aligned}$$
(10)

Multiplying both ends of the obtained use Eq. (10) identically by \({\mathcal {J}}^{T}(\theta _{{\mathfrak {i}}})\) yields

$$\begin{aligned} {M_{\mathfrak {i}}}({\theta _{\mathfrak {i}}}){{{\dot{\vee }}} _{\mathfrak {i}}} + {C_{\mathfrak {i}}}({\theta _{\mathfrak {i}}},{{{\dot{\theta }}} _{\mathfrak {i}}}){ \vee _{\mathfrak {i}}} = {\tau _{\mathfrak {i}}} - {\text { }}{D_{\mathfrak {i}}}({\theta _{\mathfrak {i}}}){ \vee _{\mathfrak {i}}} \end{aligned}$$
(11)

where \(M_{{\mathfrak {i}}}={{\mathcal {J}}}^{T}(\theta _{{\mathfrak {i}}}){\bar{M}}_{{\mathfrak {i}}}{{\mathcal {J}}}(\theta _{{\mathfrak {i}}})\), \(C_{{\mathfrak {i}}}={{\mathcal {J}}}^{T}(\theta _{{\mathfrak {i}}}){\bar{M}}_{{\mathfrak {i}}}\dot{{\mathcal {J}}}(\theta _{{\mathfrak {i}}})\), \(D_{{\mathfrak {i}}}={{\mathcal {J}}}^{T}(\theta _{{\mathfrak {i}}}){\bar{D}}_{{\mathfrak {i}}}\dot{{\mathcal {J}}}(\theta _{{\mathfrak {i}}})\), \(\tau _{{\mathfrak {i}}}={{\mathcal {J}}}^{T}(\theta _{{\mathfrak {i}}}){\bar{\tau }}_{{\mathfrak {i}}}\).

Notice that the transformed mathematical model of nonlinear agents uses Eq. (11) to satisfy the following properties, which will aid future work.

Property 1

For \(M_{\mathfrak {i}}(\theta _{\mathfrak {i}})\) inertia matrix, it satisfies both positive definiteness and symmetry, while for \(\mu \in {\mathbb {R}}^{2}\), it follows the following inequality:

$$\begin{aligned} {k_{{m_{{\mathfrak {i}}1}}}}{\left\| \mu \right\| ^2} \leqslant {\mu ^T}{M_{\mathfrak {i}}}({\theta _{\mathfrak {i}}})\mu \leqslant {k_{{m_{{\mathfrak {i}}2}}}}{\left\| \mu \right\| ^2} \end{aligned}$$
(12)

where \({k_{{m_{{\mathfrak {i}}1}}}},{k_{{m_{{\mathfrak {i}}2}}}}\) are constants, and \({k_{{m_{{\mathfrak {i}}2}}}} \geqslant {k_{{m_{{\mathfrak {i}}1}}}} \geqslant 0\).

Property 2

The matrix \(\frac{3}{2}{\dot{M}_{\mathfrak {i}}}({\theta _{\mathfrak {i}}}) - 3{C_{\mathfrak {i}}}({\theta _{\mathfrak {i}}},{{{\dot{\theta }}} _{\mathfrak {i}}})\) is skew-symmetric. This means that for \(\forall \mu \in {\mathbb {R}}^{2}\), \({\mu ^T}(\frac{3}{2}{\dot{M}_{\mathfrak {i}}}({\theta _{\mathfrak {i}}}) - 3{C_{\mathfrak {i}}}({\theta _{\mathfrak {i}}},{{{\dot{\theta }}} _{\mathfrak {i}}}))\mu = 0\) the matrix is antisymmetric.

Property 3

It is observed that by adding the variable \(\xi _{{\mathfrak {i}}}\), use Eq. (11) becomes linear, resulting in.

$$\begin{aligned} {M_{\mathfrak {i}}}({\theta _{\mathfrak {i}}}){{\dot{\mu }}} + {C_{\mathfrak {i}}}({\theta _{\mathfrak {i}}},{{{\dot{\theta }}} _{\mathfrak {i}}})\mu + {D_{\mathfrak {i}}}({\theta _{\mathfrak {i}}}){\vee _{\mathfrak {i}}} = {{\mathcal {Y}}_{\mathfrak {i}}}({\theta _{\mathfrak {i}}},{{{\dot{\theta }}} _{\mathfrak {i}}},{\vee _{\mathfrak {i}}},\mu ,{{\dot{\mu }}} ){\xi _{\mathfrak {i}}} \end{aligned}$$
(13)

For any \(\mu \in {\mathbb {R}}^{2}\), where \(\xi _{{\mathfrak {i}}}\in {\mathbb {R}}^{6}\) represents the constant unknown parameter vector, and \({\mathcal {Y}}_{{\mathfrak {i}}}\in {\mathbb {R}}^{2\times 6}\) represents the known regression matrix.

Problem statement

This article primarily centers on the integration of dynamic formation, tracking, and interception within multi-agent systems amidst obstacles. The objective of studying the interception within formations aims to enable agents to encircle and intercept a mobile target attempting to evade within a predefined arrangement. Meanwhile, the dynamic adjustment of formation size aims to expedite the passage through obstacles by adapting the formation’s dimensions based on varied obstacle distributions. Let’s regard the n th agent as the leader while considering all other agents as followers.

For simplicity, the problem statement can be transformed into the following steps.

  1. 1.

    Guide the agent to adopt the predefined formation \({{\mathcal {F}}^{\bot }}({\mathcal {T}}) = ({{G}^{\bot }},{p^{\bot } }({\mathcal {T}}))\) signifying the completion of the formation acquisition task within the dynamic formation mission;

  2. 2.

    The directive to the leader entails tracking a moving target while navigating obstacles, while the follower adheres to the leader, maintaining a predetermined formation essentially addressing the challenge of dynamic formation tracking;

  3. 3.

    Finally, the leader rejoins the target and the follower maintains a preset formation and closes in on the moving target, completing the dynamic formation interception problem.

Problem of dynamic formation detection

This configuration is solvable given its connection to dynamic agent formations. In this study, an undirected graph is adopted as a dynamic mechanism to depict the intercommunication within a multi-agent system by making assumptions around individual agents possessing different local coordinates. Studying how an agent maintains a predefined geometric shape in a two-dimensional plane is the research goal of this topic. where the required distance over time between agent \({\mathfrak {i}}\) and agent \({\mathfrak {j}}\) is

$$\begin{aligned} {\mathcal {F}}({\mathcal {T}}) \rightarrow { Iso}({{\mathcal {F}}^{ \bot }})\quad { as} \quad {\mathcal {T}}\rightarrow \infty \end{aligned}$$
(14)

The utilization of Eq. (14) is tantamount to

$$\begin{aligned}&{d_{{\mathfrak {i}}{\mathfrak {j}}}}({\mathcal {T}}) = \left\| {p_{\mathfrak {i}}^\bot ({\mathcal {T}}) - p_{\mathfrak {j}}^\bot ({\mathcal {T}})} \right\| \nonumber \\&\quad \geqslant {d_{ safe}},{d_{{\mathfrak {i}}{\mathfrak {j}}}}({\mathcal {T}}) > 0,{\mathfrak {i}},{\mathfrak {j}} \in {{\mathcal {V}}^\bot } \end{aligned}$$
(15)

Among these, \(d_{ safe}\) represents the minimum safe distance between distinct agents, resolving collision avoidance among multiple agents. The anticipated time-varying distances \({d_{{\mathfrak {i}}{\mathfrak {j}}}}\) are assumed to be both bounded and continuous functions, and likewise are their derivatives \({\dot{d}_{{\mathfrak {i}}{\mathfrak {j}}}}\). The agent’s formation framework is \({{\mathcal {F}}^{\bot }}({\mathcal {T}}) = ({{G}^\bot },{p^\bot }({\mathcal {T}})))\), encompassing \(p=(p_{1},p_{2},\ldots ,p_{n})\). Suppose that when \({\mathcal {T}}=0\), the agents aren’t confined by the anticipated distance between them, expressed as \(\left\| p_{{\mathfrak {i}}}({\mathcal {O}})-p_{{\mathfrak {j}}}({\mathcal {O}})\right\| \ne d_{{\mathfrak {i}}{\mathfrak {j}}},{\mathfrak {i}},{\mathfrak {j}}\in {\mathcal {V}}^{\bot }\). In utilizing Eq. (11), the objective of the control is to pursue a configuration with \({\tau _{\mathfrak {i}}}, {\mathfrak {i}} = 1,2, \ldots ,n\), and consequently, \({\mathfrak {j}} \in {{\mathcal {N}}_{\mathfrak {i}}}({{\mathcal {E}} ^\bot }):\)

$$\begin{aligned} \left\| {{p_{\mathfrak {i}}}({\mathcal {T}}) - {p_{\mathfrak {j}}}({\mathcal {T}})} \right\| \rightarrow {d_{{\mathfrak {i}}{\mathfrak {j}}}}({\mathcal {T}}),as{\text { }}{\mathcal {T}} \rightarrow \infty ,\forall {\mathfrak {i}},{\mathfrak {j}} \in {{\mathcal {V}}^\bot } \end{aligned}$$
(16)

Dynamic formation tracking problem of agents

In this problem, the entities must travel at a set speed which is only known by a portion of the entities, according to Sect. “Problem of dynamic formation detection”.

$$\begin{aligned} {\dot{p}_{\mathfrak {i}}}({\mathcal {T}}) - {{ \vee }_0}({\mathcal {T}}) \rightarrow 0\quad { as} \quad {\mathcal {T}} \rightarrow \infty ,{\text { }}{\mathfrak {i}}{{ = 1,2,}} \ldots ,n \end{aligned}$$
(17)

In this context \({{\vee }_0}({\mathcal {T}}) \in {{\mathbb {R}}^2}\) represents any function of time that is continuously differentiable and determines the desired tracking speed. Make the following assumptions: \({{\vee }_0}({\mathcal {T}}), {{\dot{\vee }}_0}({\mathcal {T}}) \in {{\mathcal {H}}_\infty }\), where \({\left\| {{{{\dot{\vee }}}_0}({\mathcal {T}})} \right\| _{{{\mathcal {H}}_\infty }}} \leqslant {\varrho _0 }\), where \({\varrho _0}\) is determined to be a definite positive constant. The group of agents that can directly traverse \({\vee }_{0}\) is represented as \({{\mathcal {V}}_0} \subset {{\mathcal {V}}^\bot }\). Considering that it was previously pointed out in the attribute 3 that the design using Eq. (13) is performed with the unknown variable \({\xi _{\mathfrak {i}}}\), therefore \({\mathfrak {i}}\) can be defined parameter estimates:

$$\begin{aligned} {{\tilde{\xi }} _{\mathfrak {i}}} = {{\hat{\xi }} _{\mathfrak {i}}} - {\xi _{\mathfrak {i}}} \end{aligned}$$
(18)

In this context, \({\hat{\xi _{\mathfrak {i}}}}({\mathcal {T}}) \in {{\mathbb {R}}^6}\) embodies the estimation for the \({\mathfrak {i}}\) th parameter within the vector \({\xi _{\mathfrak {i}}}\) as proposed within the dynamic framework. This parameter vector is subsequently employed in the adaptive control formulation.

Dynamic formation interception of agents

In this section, use Eq. (14) outlines the process for obtaining the desired design formation. Let \({p_{\mathcal {T}}} \in {{\mathbb {R}}^2}\) represent the target’s positional data. It is considered a continuously differentiable quadratic function over time, satisfying \({p_{\mathcal {T}}}({\mathcal {T}}), {\dot{p}_{\mathcal {T}}}({\mathcal {T}}), {\ddot{p}_{\mathcal {T}}}({\mathcal {T}}) \in {{\mathcal {H}}_\infty }\), with \({\left\| {{{\ddot{p}}_{\mathcal {T}}}({\mathcal {T}})} \right\| _{{{\mathcal {H}}_\infty }}} \leqslant {\varrho _{{\mathcal {T}}1}}\) and \({\varrho _{{\mathcal {T}}1}}\) are acknowledged as known positive values. To pursue a moving target, a scheme is employed with a leader-follower configuration, featuring the n th agent as the leader while the remaining agents serve as followers. In the role of a leader, their responsibility lies in tracking the target, while in the role of a follower, they have to sustain and persist in tracking the formation described in Sect. “Problem of dynamic formation detection”. Hence, solely the leader can directly maintain measurement of its position \({p_{\mathcal {T}}} - {p_n}\) relative to the target, while also having access to the measurement of the target’s velocity \({p_{\mathcal {T}}}\). The definition of the formation shape \({{\mathcal {F}}^\bot }\) needs to satisfy the corresponding constraints: \(p_n^\bot \in { conv}\left\{ {p_1^\bot ,p_2^\bot , \ldots , p_{ n - 1}^\bot } \right\} \), and for the symbol \({ conv}\left\{ \cdot \right\} \) in the above definition, it is expressed as a convex hull. The primary objective of this problem is to gradually approach \({p_{\mathcal {T}}}({\mathcal {T}})\) infinitely close to \({ conv}\left\{ {p_1^\bot ,p_2^\bot , \ldots , p_ {n - 1}^\bot } \right\} \) over time [29]. This implies:

$$\begin{aligned} {p_{\mathcal {T}}}({\mathcal {T}}) \in { conv}\left\{ {p_1^\bot ,p_2^\bot , \ldots ,p_{n - 1}^\bot } \right\} \quad { as} \quad {\mathcal {T}} \rightarrow \infty \end{aligned}$$
(19)

Dynamic formation tracking intercept design

In this subsection, the focus is on forming a group of agents, where a designated n th agent acts as a leader, accompanied by followers. The system incorporates a distance-based adaptive dynamic formation tracking intercept controller. The controller design employed the Lyapunov direct method and backstepping method as tools.

Multi-agent dynamic formation tracking and interception control design

Considering a multi-agent system comprising n agents, assuming that Eq. (11) is known for each agent, it becomes feasible to rewrite the equation as follows:

$$\begin{aligned}&{\dot{p}}={\vee }_{{\mathfrak {i}}}, \end{aligned}$$
(20)
$$\begin{aligned}&{M_{\mathfrak {i}}}({\theta _{\mathfrak {i}}}){{{\dot{\vee }}} _{\mathfrak {i}}} + {D_{\mathfrak {i}}}({\theta _{\mathfrak {i}}}){ \vee _{\mathfrak {i}}} = {\tau _{\mathfrak {i}}} - {C_{\mathfrak {i}}}({\theta _{\mathfrak {i}}},{{{\dot{\theta }}} _{\mathfrak {i}}}){ \vee _{\mathfrak {i}}} \end{aligned}$$
(21)

The \({\vee }_{{\mathfrak {i}}}\in {\mathbb {R}}^{2}\) appearing in the above formula is expressed as the agent \({\mathfrak {i}}\) in the \(\{X_{0},Y_{0}\}\) frame of reference relative speed below. The relative position of the agents is described by the following equation:

$$\begin{aligned} {{\tilde{p}}_{{\mathfrak {i}}{\mathfrak {j}}}} = {p_{\mathfrak {i}}} - {p_{\mathfrak {j}}},({\mathfrak {i}},{\mathfrak {j}}) \in {{\mathcal {E}}^\bot } \end{aligned}$$
(22)

Let \({{\vee }_{\mathcal {T}}}: = {\dot{p}_{\mathcal {T}}}\), the interception error between the agent and the target be denoted as:

$$\begin{aligned} {e_{\mathcal {T}}} = {p_{\mathcal {T}}} - {p_n} - c \end{aligned}$$
(23)

Where \(c\in {\mathbb {R}}^{m}\) represents a constant vector, the control mechanism is composed of one target velocity position, denoted as \({\vee }_{{\mathcal {T}}}\). Furthermore, assume that the communication network takes the form of an undirected connected graph and calls it \({{G}_\ell } = ({{\mathcal {V}}_\ell },{{\mathcal {E}} _\ell }) \). As the required rigidity shapes an undirected and connected graph, the ensuing communication graph is regarded as equivalent to a rigid graph. Use \({\hat{\vee }_{\mathcal {T}}}\) to denote the target speed estimate. Shown in the following equation:

$$\begin{aligned} \dot{\hat{\vee }}_{{\mathcal {T}}{\mathfrak {i}}} = - \alpha {\text {sgn}} \left( \sum \limits _{{\mathfrak {j}} \in {{\mathcal {N}}_{\mathfrak {i}}}({{\mathcal {E}}^*})} {({{\hat{\vee }}_{{\mathcal {T}}{\mathfrak {i}}}} - {{\hat{\vee }}_{{\mathcal {T}}{\mathfrak {j}}}})} + {b_{\mathfrak {i}}}({\hat{\vee }_{{\mathcal {T}}{\mathfrak {i}}}} - {{\vee }_{\mathcal {T}}})\right) \end{aligned}$$
(24)

Partial replacement can be applied to the expression above

$$\begin{aligned} \sum \limits _{{\mathfrak {j}} \in {{\mathcal {N}}_{\mathfrak {i}}}({{\mathcal {E}}^\bot })} {({{\hat{\vee }}_{{\mathcal {T}}{\mathfrak {i}}}} - {{\hat{\vee }}_{{\mathcal {T}}{\mathfrak {j}}}})} = \sum \limits _{{\mathfrak {j}} = 1}^n {{{\mathfrak {a}}_{{\mathfrak {i}}{\mathfrak {j}}}}({{\hat{\vee }}_{{\mathcal {T}}{\mathfrak {i}}}} - {{\hat{\vee }}_{{\mathcal {T}}{\mathfrak {j}}}})} \end{aligned}$$

The term \({{\mathfrak {a}}_{{\mathfrak {i}}{\mathfrak {j}}}}\) denotes an element within the matrix representing adjacency \({\mathcal {A}}\), \({b_{\mathfrak {i}}} = \left\{ \begin{gathered} 1,\,{ if}\,{\mathfrak {i}} \in {\vee _0} \\ 0,{ otherwise} \\ \end{gathered} \right. \), and in Sect. “Dynamic formation interception of agents”, Assuming \({\left\| {{{{\dot{\vee }}}_{\mathcal {T}}}({\mathcal {T}})} \right\| _{{{\mathcal {H}}_\infty }}} \leqslant {\varrho _{\mathcal {{\mathcal {T}}}1}}\). Thus, \({{\dot{\vee }}_{\mathcal {T}}}({\mathcal {T}})\) is constrained to \({\varrho _{{\mathcal {T}}1}}\). Let the velocity of every agent be estimated as \({\tilde{\vee }_{{\mathcal {T}}{\mathfrak {i}}}} \triangleq {\hat{\vee }_{{\mathcal {T}}{\mathfrak {i}}}} - {{\vee }_{\mathcal {T}}}\). Applying equation (24), the evolution of the estimation error can be further determined as follows: \({\dot{\tilde{\vee }}_{{\mathcal {T}}{\mathfrak {i}}}}({\mathcal {T}}) = - \alpha {{\text {sgn}}} (\sum \nolimits _{{\mathfrak {j}} \in {{\mathcal {N}}_{\mathfrak {i}}}({{\mathcal {E}}^\bot } )} {{ {\mathfrak {a}}_{{\mathfrak {i}}{\mathfrak {j}}}}({{\hat{\vee }}_{{\mathcal {T}}{\mathfrak {i}}}} - {{ \hat{\vee }}_{{\mathcal {T}}{\mathfrak {j}}}})} + {b_{\mathfrak {i}}}({\hat{\vee }_{{\mathcal {T}}{\mathfrak {i}}}} - {\vee _{\mathcal {T}}})) - {{{\dot{\vee }}}_{\mathcal {T}}}\).

A more succinct form of the aforementioned equation:

$$\begin{aligned} \begin{aligned} {\dot{\tilde{\vee }}}({\mathcal {T}})&= -\alpha {\text {sgn}} ({\mathcal {H}} \otimes {{\mathcal {I}}_m})\tilde{\vee } \\&\quad - \alpha {\text {sgn}} ({\mathcal {B}} \otimes {{\mathcal {I}}_m})\tilde{\vee } - {1_n} \otimes {{{\dot{\vee }}}_{\mathcal {T}}} \\&= -\alpha {\text {sgn}} ({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee } - {1_n} \otimes {{{\dot{\vee }}}_{\mathcal {T}}} \end{aligned} \end{aligned}$$
(25)

The substitution formula \({\hat{\vee }_{{\mathcal {T}}{\mathfrak {i}}}} - {\hat{\vee }_{{\mathcal {T}}{\mathfrak {j}}}} = { \tilde{\vee }_{{\mathcal {T}}{\mathfrak {i}}}} - {\tilde{\vee }_{{\mathcal {T}}{\mathfrak {j}}}}\) is used for this conversion. Where \(\tilde{\vee }({\mathcal {T}}) = {(\tilde{\vee }_1^{T}({\mathcal {T}}), \ldots , \tilde{\vee }_n^{T}({\mathcal {T}}))^{T}} \in {{\mathbb {R}}^{2n}}\), \({\mathcal {H}}\) corresponds to the Laplacian Matrix of \({{G}_\ell } = ({{\mathcal {V}}_\ell }, {{\mathcal {E}}_\ell })\), \({\mathcal {B}} = { diag}({b_1}, \ldots , {b_n})\), and \({\mathcal {M}} \triangleq {\mathcal {H}} + {\mathcal {B}}\). For \({{G}_\ell }\), it is treated as a connected communication graph (undirected), and \({\mathcal {M}}\) is a matrix with positive definiteness and symmetry. Examining Eq. (25) shows a discontinuity on the opposite side. Therefore, non-smooth analysis is used to analyze the stability of Eq. (25). Please consider that the locally bounded and numerically known symbolic function guarantees the existence of Filippov solutions for the differential Eq. (25), regardless of variations in initial conditions. That is, \(\dot{\tilde{\vee }}\in {\mathcal {K}}[f](\tilde{\vee },{\mathcal {T}})\). The set \({\mathcal {K}}[ \cdot ]\) possesses the following attributes: non-empty, compact, convex, and is an upper semi-continuous set-valued mapping. Thus, in differential inclusion, Eq. (25) can be written as \({\dot{\tilde{\vee }}}({\mathcal {T}})\mathop \in \limits ^{a.e.} {\mathcal {K}}\left[ { - \alpha {\text {sgn}} ({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee } - {1_n} \otimes {{{\dot{\vee }}}_{\mathcal {T}}}({\mathcal {T}})} \right] \). Consider such a Lyapunov function \({{\vee }_e}(\tilde{\vee }) = \frac{1}{2}{\tilde{\vee }^T}({\mathcal {M}} \otimes {{\mathcal {I}}_2})\tilde{\vee }\), the set-valued derivative concerning \({{\vee }_e}\) is expressed as follows [30]:

$$\begin{aligned} \begin{aligned}&{{\dot{\vee }}_e} \mathop \in \limits ^{a.e.} \frac{{\partial {{\vee }_e}}}{{\partial \tilde{\vee }}}{\mathcal {K}}[f](\tilde{\vee },{\mathcal {T}}) \\&\quad \subset - \alpha {{\tilde{\vee }}^T}({\mathcal {M}} \otimes {{\mathcal {I}}_m}){\text {sgn}} (({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee }) \\&\qquad - {{\tilde{\vee }}^T}({\mathcal {M}} \otimes {{\mathcal {I}}_m})({1_n} \otimes {{{\dot{\vee }}}_{\mathcal {T}}}) \end{aligned} \end{aligned}$$
(26)

Where \(\mathop \in \limits ^{a.e.} \) means “almost everywhere”. This can be further derived using the Eq. (26)

$$\begin{aligned} {{\dot{\vee }}_e}&{\text { = }} - \alpha {{\tilde{\vee }}^T}({\mathcal {M}} \otimes {{\mathcal {I}}_m}){ NSgn}(({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee })\nonumber \\&\quad - {{\tilde{\vee }}^T}({\mathcal {M}} \otimes {{\mathcal {I}}_m})({1_n} \otimes {{{\dot{\vee }}}_{\mathcal {T}}}) \nonumber \\&{\text { = }} - \alpha {\left\| {({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee }} \right\| _{\text {1}}}\nonumber \\&\quad - {({1_n} \otimes {{{\dot{\vee }}}_{\mathcal {T}}})^T}({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee } \nonumber \\&= - \alpha {\left\| {({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee }} \right\| _{\text {1}}} + {\dot{\vee }}_{\mathcal {T}}^T\sum \limits _{{\mathfrak {i}} = 1}^{mn} {{{[({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee }]}_{\mathfrak {i}}}} \nonumber \\&\leqslant - \alpha {\left\| {({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee }} \right\| _{\text {1}}} + {\left\| {{{{\dot{\vee }}}_{\mathcal {T}}}} \right\| _1}{\left\| {({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee }} \right\| _{\text {1}}} \nonumber \\&\leqslant - (\alpha - \varrho ){\left\| {({\mathcal {M}} \otimes {{\mathcal {I}}_m})\tilde{\vee }} \right\| _{\text {1}}} \end{aligned}$$
(27)

where \({ NSgn}({\mathcal {X}}): = [{ NSgn}({{\mathcal {X}}_1}), \ldots , NSgn({{\mathcal {X}}_n})], \forall {\mathcal {X}} \in {{\mathbb {R}}^{2n}}\) and

$$\begin{aligned} { NSgn}({\mathcal {X}}_{\mathfrak {i}}) = {\left\{ \begin{array}{ll} \ \ \ 1, &{} \text {for} \ {\mathcal {X}}_{\mathfrak {i}} > 0 \\ {[}-1,1], &{} \text {for} \ {\mathcal {X}}_{\mathfrak {i}}=0\\ \ -1, &{} \text {for} \ {\mathcal {X}}_{\mathfrak {i}}<0 \end{array}\right. }\end{aligned}$$
(28)

Here, \({\left\| { \cdot } \right\| _{\text {1}}}\) denotes the 1-norm of the vector. When the condition \(\alpha > \varrho \) is met, \(\dot{{V}}_{e}\) becomes negative, and for \(\tilde{\vee } = 0\), it tends toward consistency and asymptotic stability. Hence, for every corresponding edge in the graph \({{G}_\ell }\), the error in distance between different agents should be:

$$\begin{aligned} {e_{{\mathfrak {i}}{\mathfrak {j}}}} = \left\| {{{{\tilde{p}}}_{{\mathfrak {i}}{\mathfrak {j}}}}} \right\| - {d_{{\mathfrak {i}}{\mathfrak {j}}}}({\mathcal {T}}),({\mathfrak {i}},{\mathfrak {j}}) \in {{\mathcal {E}}^\bot } \end{aligned}$$
(29)

According to the Eq. (29), \({e_{{\mathfrak {i}}{\mathfrak {j}}}} \in [ - {d_{{\mathfrak {i}}{\mathfrak {j}}}},\infty )\), you can inferred:

$$\begin{aligned} \begin{aligned} {{\dot{e}}_{{\mathfrak {i}}{\mathfrak {j}}}}&= {({{{\tilde{p}}}^T}_{{\mathfrak {i}}{\mathfrak {j}}}{{{\tilde{p}}}_{{\mathfrak {i}}{\mathfrak {j}}}})^{ - \frac{1}{2}}}{{{\tilde{p}}}^T}_{{\mathfrak {i}}{\mathfrak {j}}}({{\vee }_{\mathfrak {i}}} - {{\vee }_{\mathfrak {j}}}) - {{\dot{d}}_{{\mathfrak {i}}{\mathfrak {j}}}} \\&= \frac{{{{{\tilde{p}}}^T}_{{\mathfrak {i}}{\mathfrak {j}}}({{\vee }_{\mathfrak {i}}} - {{\vee }_{\mathfrak {j}}})}}{{{e_{{\mathfrak {i}}{\mathfrak {j}}}} + {d_{{\mathfrak {i}}{\mathfrak {j}}}}}} - {{\dot{d}}_{{\mathfrak {i}}{\mathfrak {j}}}} \end{aligned} \end{aligned}$$
(30)

Among them, use Eq. (29) and use Eq. (20) are used. For the convenience of subsequent discussion, the following equations are introduced:

$$\begin{aligned} \begin{aligned} {z_{{\mathfrak {i}}{\mathfrak {j}}}}&= {\left\| {{{{\tilde{p}}}_{{\mathfrak {i}}{\mathfrak {j}}}}} \right\| ^2} - d_{{\mathfrak {i}}{\mathfrak {j}}}^2 \\&= {e_{{\mathfrak {i}}{\mathfrak {j}}}}\left( {{{\left\| {{{{\tilde{p}}}_{{\mathfrak {i}}{\mathfrak {j}}}}} \right\| }^2} + d_{{\mathfrak {i}}{\mathfrak {j}}}^2} \right) \\&= {e_{{\mathfrak {i}}{\mathfrak {j}}}}\left( {{e_{{\mathfrak {i}}{\mathfrak {j}}}} + 2{d_{{\mathfrak {i}}{\mathfrak {j}}}}} \right) ,({\mathfrak {i}},{\mathfrak {j}}) \in {{\mathcal {E}}^\bot } \end{aligned} \end{aligned}$$
(31)

Using use Eq. (29). If and only if \({e_{{\mathfrak {i}}{\mathfrak {j}}}} = 0\), \({z_{{\mathfrak {i}}{\mathfrak {j}}}} = 0\) can be determined, taking into account that \(\left\| {{{{\tilde{p}}}_{{\mathfrak {i}}{\mathfrak {j}}}}} \right\| \geqslant 0\). Choose the following Lyapunov candidate functions [31]:

$$\begin{aligned} {{\vee }_{{\mathfrak {i}}{\mathfrak {j}}}}&= \frac{1}{4}z_{{\mathfrak {i}}{\mathfrak {j}}}^2, \end{aligned}$$
(32)
$$\begin{aligned} {\vee }(e)&= \sum \limits _{{\mathfrak {j}} \in {{\mathcal {N}}_{\mathfrak {i}}}({{\mathcal {E}}^\bot })} {{{\vee }_{{\mathfrak {i}}{\mathfrak {j}}}}({e_{{\mathfrak {i}}{\mathfrak {j}}}})} \end{aligned}$$
(33)

where \(e = ( \ldots ,{e_{{\mathfrak {i}}{\mathfrak {j}}}}, \ldots ) \in {{\mathbb {R}}^\ell },({\mathfrak {i}},{\mathfrak {j}}) \in {{\mathcal {E}}^\bot }\), Calculating Eq. (33) along the time derivative of Eq. (31) yields:

$$\begin{aligned} \begin{aligned} {\dot{\vee }}&= \sum \limits _{({\mathfrak {i}},{\mathfrak {j}}) \in {{\mathcal {E}}^\bot }} {{e_{{\mathfrak {i}}{\mathfrak {j}}}}({e_{{\mathfrak {i}}{\mathfrak {j}}}} + 2{d_{{\mathfrak {i}}{\mathfrak {j}}}})[{{{\tilde{p}}}^T}_{{\mathfrak {i}}{\mathfrak {j}}}({{\vee }_{\mathfrak {i}}} - {{\vee }_{\mathfrak {j}}}) - {d_{{\mathfrak {i}}{\mathfrak {j}}}}{{\dot{d}}_{{\mathfrak {i}}{\mathfrak {j}}}}]} \\&= {z^T}({\mathcal {R}}(p){\vee } - {d_c}) \\ \end{aligned} \end{aligned}$$
(34)

where \({\vee } = ({{\vee }_1},{{\vee }_2}, \ldots {{\vee }_n}) \in {{\mathbb {R}}^{2n}},z = ( \ldots ,{z_{{\mathfrak {i}}{\mathfrak {j}}}}, \ldots ) \in {{\mathbb {R}}^\ell },({\mathfrak {i}},{\mathfrak {j}}) \in {{\mathcal {E}}^\bot }\) and \({d_c} = ( \ldots ,{d_{{\mathfrak {i}}{\mathfrak {j}}}}{\dot{d}_{{\mathfrak {i}}{\mathfrak {j}}}}, \ldots ) \in {{\mathbb {R}}^\ell },({\mathfrak {i}},{\mathfrak {j}}) \in {{\mathcal {E}}^\bot }\).

By defining the variable \(s = {\vee } - {{\vee }_f}\), where \({{\vee }_f} \in {{\mathbb {R}}^{2n}}\) represents the input speed, use the Eq. (34) can be expressed as:

$$\begin{aligned} {{\vee }_m}(e,s) = {\vee }(e) + \frac{1}{2}{s^T}{M}(p)s \end{aligned}$$
(35)

where \({M}(p) = { diag}({{M}_1}({p_1}), \ldots ,{{M}_n}({p_n}))\). The derivative of the Lyapunov function Eq. (35) may be acquired as follows:

$$\begin{aligned} \begin{aligned} {{\dot{\vee }}_m}&= {z^T}({\mathcal {R}}(p){\vee } - {d_c}) + \frac{1}{2}{s^T}\dot{M}(p)s + {s^T}M(p)\dot{s} \\&= {z^T}[{\mathcal {R}}(p)(s + {{\vee }_f}) - {d_c}] + \frac{1}{2}{s^T}\dot{M}(p)s \\&\quad + {s^T}[\tau - C(\theta ,{{\dot{\theta }}} )\dot{p} - D(\theta )\dot{p} - M(\theta ){{{\dot{\vee }}}_f}] \\&= {z^T}{\mathcal {R}}(p)({{\vee }_f} - {d_c}) \\&\quad + {s^T}[\tau - C(\theta ,{{\dot{\theta }}} ){{\vee }_f} - D(\theta )\dot{p} \\&\quad - M(\theta ){{{\dot{\vee }}}_f} + {{\mathcal {R}}^T}(p)z] \\ \end{aligned} \end{aligned}$$
(36)

where \(C(\theta ,{{\dot{\theta }}} ) = { diag}({C_1}(\theta ,{{{\dot{\theta }}} _1}), \ldots ,{C_n}(\theta ,{{{\dot{\theta }}} _n})), D(\theta ) = { diag}({D_1}({\theta _1},), \ldots ,{D_n}({\theta _n}))\) and \(\tau = { diag}({\tau _1}, \ldots , {\tau _n}) \in {{\mathbb {R}}^{2n}}\). According to Property 3, Eq. (13) may be derived by substituting it into Eq. (36) in [32]:

$$\begin{aligned} {{\dot{\vee }}_m}&= {z^T}(R(p){{\vee }_f} - {d_c})\nonumber \\&\quad + {s^T}[\tau - {\mathcal {Y}}(\theta ,{{\dot{\theta }}} ,{\vee },{{\vee }_f},{{\dot{\vee }}_f})\xi + {{\mathcal {R}}^T}(p)z] \end{aligned}$$
(37)

where \({\mathcal {Y}}(\theta ,{{\dot{\theta }}},{\vee },{{\vee }_f},{{\dot{\vee }}_f}) = { diag}({{\mathcal {Y}}_1},{{\mathcal {Y}}_2} \ldots ,{{\mathcal {Y}}_n})\) represents the known regression matrix, which can be regarded as a candidate matrix for the Lyapunov function.

$$\begin{aligned} {{\vee }_a}(e,s,{\tilde{\xi }} ) = {{\vee }_m}(e,s) + \frac{1}{2}{{\tilde{\xi }} ^T}{\varrho ^{ - 1}}{\tilde{\xi }} \end{aligned}$$
(38)

Here, \(\varrho \in {{\mathbb {R}}^{6n \times 6n}}\) denotes a positive definite constant, specifically, the diagonal. The Eq. (38) is derived by differentiating concerning time:

$$\begin{aligned} \begin{aligned} {{\dot{\vee }}_a}&={{\dot{\vee }}_m} + {{\tilde{\xi } }^T}{\varrho ^{ - 1}}\dot{\hat{\xi }} \\&={z^T}{\mathcal {R}}(p)({{\vee }_f} - {d_c}) \\&\quad + {s^T}[\tau - {\mathcal {Y}}(\theta ,{{\dot{\theta }}},{\vee },{{\vee }_f},{{{\dot{\vee }}}_f})({\hat{\xi }} - {\tilde{\xi }} )\\&\quad + {{\mathcal {R}}^T}(p)z] + {{{\tilde{\xi }} }^T}{\varrho ^{ - 1}}\dot{\hat{\xi }} \\&={z^T}{\mathcal {R}}(p)({{\vee }_f} - {d_c})\\&\quad + {s^T}[\tau - {\mathcal {Y}}(\theta ,{{\dot{\theta }}},{\vee },{{\vee }_f},{{{\dot{\vee }}}_f}){\hat{\xi }} {\text { + }}{{\mathcal {R}}^T}(p)z] \\&\quad + {{{\tilde{\xi }} }^T}{\varrho ^{ - 1}}(\dot{\hat{\xi }} + \varrho {{\mathcal {Y}}^T}s)\\ \end{aligned} \end{aligned}$$
(39)

The devised control input, \(\tau \), must adhere to the prescribed conditions for intercepting the dynamic formation. This involves using a semi-negative definite function, \({{\dot{\vee }}_a}\) [33] and constructing the adaptive formation interception control algorithm based on Eq. (38).

$$\begin{aligned} \tau&= - {k_\vartheta }s + {\mathcal {Y}}(\theta ,{{\dot{\theta }}} ,{\vee },{{\vee }_f},{{\dot{\vee }}_f}){\hat{\xi }} - {{\mathcal {R}}^T}(p)z , \end{aligned}$$
(40a)
$$\begin{aligned} {{\vee }_f}&= {{\mathfrak {u}}_a} + 1 \otimes h, \end{aligned}$$
(40b)
$$\begin{aligned} h&= {k_1}{e_T} + {\hat{\vee }_T}, \end{aligned}$$
(40c)
$$\begin{aligned} {{\mathfrak {u}}_a}&= {{\mathcal {R}}^ + }(p)( - {k_{v}}z + {d_c}), \end{aligned}$$
(40d)
$$\begin{aligned} \dot{\hat{\xi }}&= - \varrho {{\mathcal {Y}}^T}(\theta , {{\dot{\theta }}}, {\vee }, {{\vee }_f}, {{\dot{\vee }}_f})s. \end{aligned}$$
(40e)

Here, \({k_\vartheta }\), \({k_{v}}\), and \({k_1}\) are constants with positive values. \({{\mathcal {R}}^+}(p)\) denotes the Moore–Penrose pseudoinverse, defined as \({{\mathcal {R}}^T}(p){[{\mathcal {R}}(p) {{\mathcal {R}}^T}(p)]^{-1}}\). \({\hat{\xi }} = ({{\hat{\xi }}_1}, {{\hat{\xi }}_2}, {\ldots }, {{\hat{\xi }}_n})\), and the control law \(\tau \) is specified in Eq. (13).

Proof

$$\begin{aligned} \begin{aligned} {{{\dot{\vee }}}_a}&= - {k_{v}}{z^T}{\mathcal {R}}{{\mathcal {R}}^T}z - {k_\vartheta }{s^T}s \\&= - 4{k_{v}}{\vee }(e) - {k_\vartheta }{s^T}s \end{aligned} \end{aligned}$$
(41)

Since \({\vee }(e)\) is a Lyapunov function, it is obvious that \(- 4{k_{v}}{\vee }(e)\) is negative definite, it is only necessary to rewrite \( - {k_\vartheta }{s^T}s\). Therefore, the equation above can be expressed as:

$$\begin{aligned} {{\dot{\vee }}_a} \leqslant - 4{k_{v}}{\vee }(e) - {k_\vartheta }{\left\| s \right\| ^2} \end{aligned}$$
(42)

It can be clearly understood from the above expressions that \({k_\vartheta },{k_{v}}\) is a positive constant, and from Eq. (42) it is clear that \({{\dot{\vee }}_a}\) is negatively definite, According to the Lyapunov function, it can be seen that \(- 4{k_{v}}{\vee }(e)\) is negative definite, and \({\left\| s \right\| ^2}\) is the square of the Euclidean norm, the sign of the term is positive, so it is negative definite for the entire \({{\dot{\vee }}_a}\). For all cases where \({\mathcal {T}} \geqslant 0\), \({{\dot{\vee }}_a} \leqslant 0\), so for \({\mathcal {T}} \geqslant 0\) there are only two cases in which \({{\vee }_a}\) can be considered, one in which it remains the same; and the other in which it does not increase. Note that \({{\vee }_a}\) has finite limits, since \({\mathcal {T}}\) tends to infinity.

From Eqs. (38) and (42) \({{\vee }_a} \geqslant 0\), \({{\dot{\vee }}_a}\) is less than or equal to 0. It follows that if \({\mathcal {T}} \geqslant 0\), there exists \(z({\mathcal {T}}), s({\mathcal {T}}), {\tilde{\xi }} ({\mathcal {T}}) \in {{\mathcal {H}}_\infty }\), where \({{\mathcal {H}}_\infty }\) denotes the infinite paradigm. From Eqs. (29) and (31), it can be observed that when \({\mathcal {T}}\) is greater than or equal to 0, both \({\tilde{p}}({\mathcal {T}})\) and \(e({\mathcal {T}})\) belong to the set of bounded functions, denoted by \({{\mathcal {H}}_\infty }\). This result implies that Eq. (23) holds and the tracking error \({e_{\mathcal {T}}}({\mathcal {T}})\) approaches zero as \({\mathcal {T}}\) approaches infinity. According to Eqs. (40b) and (40c), \({\hat{\vee }_{\mathcal {T}}}({\mathcal {T}}) \in {{\mathcal {H}}_\infty }\) for \({\mathcal {T}} \geqslant 0\). It’s worth noting that, as assumed between Eqs. (22) and (23), \({{\vee }_{\mathcal {T}}}: = {\dot{p}_{\mathcal {T}}} \in {{\mathcal {H}}_\infty }\). According to the given definition of \(z({\mathcal {T}})\), it is determined that \({\hat{\vee }_{\mathcal {T}}}\) is located in the set \({{\mathcal {H}}_\infty }\). Consequently, this implies Eq. (24), specifically \({\dot{\hat{\vee }}_{\mathcal {T}}} \in {{\mathcal {H}}_\infty }\) and \(\dot{z}({\mathcal {T}}) \in {{\mathcal {H}}_\infty }\). Utilizing Eq. (40a) and acknowledging that \(\theta \) manifests solely through trigonometric functions, one can subsequently establish the introduction of \(\tau \in {{\mathcal {H}}_\infty }\). Furthermore, through the expression in the equation. \({{\mathcal {J}}^{-1}}({\mathcal {T}}) \in {{\mathcal {H}}_\infty }\) and Eq. (11), it becomes feasible to introduce \({{{\bar{\tau }}} _{\mathfrak {i}}} \in {{\mathcal {H}}_\infty }\). Furthermore, \(\ddot{p}({\mathcal {T}}) \in {{\mathcal {H}}_\infty }\) is derived from Eqs. (11) and (12), while \({\dot{\zeta _{\mathfrak {i}}}}({\mathcal {T}}) \in {{\mathcal {H}}_\infty }\) is evident from Eq. (3).

Now check the Eq. (40a) to satisfy the following equation requirements:

$$\begin{aligned} M({p_{\mathfrak {i}}})\dot{s}= & {} - C({p_{\mathfrak {i}}},{\dot{p}_{\mathfrak {i}}})s - D({p_{\mathfrak {i}}},{\dot{p}_{\mathfrak {i}}})s - {k_\vartheta }s \nonumber \\{} & {} + {\mathcal {Y}}(\theta ,{{\dot{\theta }}},{\dot{p}_{\mathfrak {i}}},{{\vee }_f},{{\dot{\vee }}_f}){\hat{\xi }} - {{\mathcal {R}}^T}(p)z \end{aligned}$$
(43)

The Eq. (43) is considered according to Property 3, given that \({\mathcal {Y}}(\theta ,{{\dot{\theta }}}, {\dot{p}_{\mathfrak {i}}},{{\vee }_f},{{\dot{\vee }}_f})\) is introduced as bounded within Property 1. The bounded nature of s can be determined from Eq. (40a), showing its continuity throughout \({\mathcal {T}}\). Above, \(z({\mathcal {T}}),\dot{z}({\mathcal {T}})\in {{\mathcal {H}}_\infty }\) is explained, and according to the Eqs. (42), (33), (32), for \({\mathcal {T}}\geqslant 0\),\(z({\mathcal {T}})\in {{\mathcal {H}}_2}\). Additionally, realize that \({{\vee }_a}\) has consistent continuity over time. So far, the following inferences can be made: \({\mathcal {T}} \rightarrow \infty \), \({{\dot{\vee }}_a} \rightarrow 0\), \(z({\mathcal {T}}) \rightarrow 0\), and \({ e_{\mathcal {T}}} \rightarrow 0\). Given the validity of Eq. (14) and the definition of \({{\mathcal {F}}^{\text {}}}\), where \(p_n^\bot \in conv\left\{ {p_1^\bot ,p_2^\bot , \ldots , p_{n - 1}^\bot } \right\} \), it follows that \({p_n}({\mathcal {T}}) \in conv\left\{ {{p_1}({\mathcal {T}}), {p_2}({\mathcal {T}}), \ldots , {p_{n - 1}}({\mathcal {T}})} \right\} \,{ as}\,{\mathcal {T}} \rightarrow \infty \) as \({\mathcal {T}} \rightarrow \infty \). Hence, with the convergence of \({e_{\mathcal {T}}}\) towards 0 as \({\mathcal {T}}\) approaches infinity, Eq. (19) remains valid at this juncture. To summarize, the proposed algorithm can globally ensure asymptotic stability and provable completeness of dynamic formation interception by controlling the inputs. Finally, since \({\mathcal {T}}\) tends to infinity, it can be deduced that \({{\vee }_{\mathcal {T}}}({\mathcal {T}}) - {\hat{\vee }_{\mathcal {T}}}({\mathcal {T}}) \rightarrow 0\). \(\square \)

Fig. 2
figure 2

Initial desired framework and achieving (desired framework)

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Fig. 3
figure 3

Trajectory of six agents

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Results from simulation experiments

This chapter will provide the validation outcomes regarding dynamic formation tracking and the interception challenges within dynamic formations. To better reproduce the experiment, the configuration part of the simulation experiment is first introduced. The equipment used is a 12th Gen Intel(R) Core(TM) i7-12700 H 2.30 GHz CPU and 16 GB RAM PC, the system environment used is Windows 11, and the experimental environment is Matlab-2022b compilation and simulation platform. The agent labeled as 6 within the multi-agent system primarily assumes the responsibility of tracking the moving target. The remaining members, identified as 1 through 5, are tasked with following Agent No. 6 and adhering to the predefined formation while moving. Configure an orthogonal pentagon as shown in Fig. 2, which qualifies as a minimum or infinite stiffness diagram, where \(rank({\mathcal {R}}(p)) = 2n - 3\in {{\mathbb {R}}^2}\) satisfies. The simulation in Fig. 3 displays the trajectory path of a formation. In the simulation, please contemplate a series of arrangements encompassing 6 agents. To better explain the scene simulation is essential, so a group of circular gray obstacles were randomly generated in the map to set obstacles to increase the complexity of the environment. The leader tracks a target that is in motion and the followers comply with the leader whilst sustaining a predetermined formation. Assuming the vertices of the required inflexible structure, let \(p_1 = (0,1), p_2=(-{\mathfrak {w}}_1,{\mathfrak {y}}_1),p_3 = (-{\mathfrak {w}}_2,-{\mathfrak {y}}_2),p_4 = ({\mathfrak {w}}_2,-{\mathfrak {y}}_2),p_5 = ({\mathfrak {w}}_1,{\mathfrak {y}}_1),p_6=(0,0)\) where \({\mathfrak {w}}_{1} = 0.5877853\), \({\mathfrak {w}}_{2} = 0.5777853\), \({\mathfrak {y}}_{1} = 0.80901700\) and \({\mathfrak {y}}_{2} = 0.2836622\). According to the edge sorting in the graph Fig. 2, the edge set sequence \({\mathcal {E}}=\{(1, 2), (2, 3), (3, 4), (4, 5)\) can be obtained), \((1, 5), (1, 6), (2, 6), (3, 6), (4, 6)\}\). The anticipated distances between each agent are as follows: \(d_{12}=d_{23}=d_{34}=d_{45}=d_{15}=\sqrt{2(1-{\mathfrak {y}}_{1})}, d_{26} = d_{36} = d_{46}=\sqrt{{\mathfrak {y}}_1^2 + {\mathfrak {w}}_1^2} \ \, d_{16} = 1\). The Laplacian matrix linked with the communication graph obtains:

$$\begin{aligned} {L} = \left( {\begin{array}{*{20}{c}} 0&{}\quad {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {0}}&{}\quad {\text {1}}&{}\quad {\text {1}} \\ {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {0}}&{}\quad {\text {1}} \\ {\text {0}}&{}\quad {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {1}} \\ {\text {0}}&{}\quad {\text {0}}&{}\quad {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {1}}&{}\quad {\text {1}} \\ {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {0}}&{}\quad {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {0}} \\ {\text {1}}&{}\quad {\text {1}}&{}\quad {\text {1}}&{}\quad {\text {1}}&{}\quad {\text {0}}&{}\quad {\text {0}} \end{array}} \right) \end{aligned}$$
(44)
Fig. 4
figure 4

Dynamic formation interception: capture the agent positions of \({\mathcal {F}}({\mathcal {T}})\) at different times along the agent’s trajectory

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When the velocity of the moving target is uncertain, assume it to be \({\vee }_{{\mathcal {T}}}=[1,\cos {\mathcal {T}}]^{T}\). Starting point is located at \(p_{{\mathcal {T}}}({\mathcal {O}})=[1,-1]\). The control coefficient is set to \(k_{v} = 10\) in Eq. (40d), \(k_1 = 2\) in Eq. (40c), and \(k_\vartheta = 10\) in Eq. (40a) for the initial estimation of the target’s velocity, \(\hat{{\vee }}_{{\mathcal {T}}}=0\). Technical term abbreviations are explained upon first use. The initial conditions for the position and velocity estimators of the agents are as follows: \(p({\mathcal {O}}) \ = \ [0.3143,\ \ 0.7435,\ \ -0.5218,\ \ 0.1590, -0.8912, \ -1.0580,\ \ 0.7038,\ -0.8357, 0.8027,\ 0.6398,\ 0.0853, 0.0497]^{T}\), the gain for the velocity estimator, represented by \(\text {a}\) in Eq. (24), is set to 0.033 to fulfill the condition \(\alpha > {\varrho _{{\mathcal {T}}1}}\), where \({\varrho _{{\mathcal {T}}1}} = 0.033 = {\left\| {{\dot{\vee }}({\mathcal {T}})} \right\| _{{\mathcal {H}}_\infty }}\). Figure 4 depicts the pursuit of a mobile target by a leader whilst navigating an obstacle, with the follower maintaining the predetermined formation. Subsequently, the leader returns to the target, and the follower continues to uphold the original formation and approaches the mobile target, culminating in the resolution of the dynamic formation interception problem.

Fig. 5
figure 5

The errors (distance) \(:e_{{\mathfrak {i}}{\mathfrak {j}}}({\mathcal {T}}),{\mathfrak {i}},{\mathfrak {j}}\in {{\mathcal {V}}}^{\bot }\) between any two agents

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Fig. 6
figure 6

The errors (distance) \(:e_{{\mathfrak {i}}{\mathfrak {j}}}({\mathcal {T}}),{\mathfrak {i}},{\mathfrak {j}}\in {{\mathcal {V}}}^{\bot }\) between any two agents

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Fig. 7
figure 7

The errors (distance) \(:e_{{\mathfrak {i}}{\mathfrak {j}}}({\mathcal {T}}),{\mathfrak {i}},{\mathfrak {j}}\in {{\mathcal {V}}}^{\bot }\) between any two agents

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Fig. 8
figure 8

Dynamic target interception error (distance) \(:e_{{\mathfrak {i}}{\mathfrak {j}}}({\mathcal {T}}),{\mathfrak {i}},{\mathfrak {j}}\in {{\mathcal {V}}} ^{\bot }\)

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Fig. 9
figure 9

Target interception: control inputs \({\tau _{u{\mathfrak {i}}x}},{\mathfrak {i}}=1,\ldots ,6\)

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The distance errors between any two adjacent and distinct multi-agents among the 6 multi-agents, denoted as \(e_{{\mathfrak {i}}{\mathfrak {j}}}({\mathcal {T}}),{\mathfrak {i}},{\mathfrak {j}}\in {{\mathcal {V}}}^{\bot }\), either remain unchanged or become zero. This is demonstrated in Figs. 5, 6, and 7. Figure 8 illustrates the distance error among the subjects.

Fig. 10
figure 10

Target interception: control inputs \({\tau _{u{\mathfrak {i}}y}},{\mathfrak {i}}=1,\ldots ,6\)

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Figures 9 and 10 present the control inputs’ convergence of each agent to \({\vee }_{{\mathcal {T}}}\) even though the target velocity is unknown.

Conclusion

This study presents a methodology for controlling tracking and intercepting in dynamic formations. It demonstrates the application of a distance and stiffness graph theoretical framework to implement dynamic agents. An Eulerian Lagrangian-like model is used to control multiple intelligences, and an adaptive controller is introduced for execution. Furthermore, the effectiveness of the proposed methodology for tracking and intercepting moving targets in formation is demonstrated. The methodology also allows for automatic adjustment of control parameters based on the dynamic properties of the system, resulting in improved control performance in the presence of complexity, variability, and unknowns. The methodology also allows for automatic adjustment of control parameters based on the dynamic properties of the system, resulting in improved control performance in the presence of complexity, variability, and unknowns. The methodology also allows for automatic adjustment of control parameters based on the dynamic properties of the system, resulting in improved control performance in the presence of complexity, variability, and unknowns. Dynamic formations are better equipped to handle complex environments and obstacles, making them more effective at tracking and interception compared to traditional formation control methods. This has been validated through experimental simulations in a multi-agent system with non-complete motion. This methodology can be applied in the industry to track and intercept objects on the production line using robot formation, thereby improving productivity and reducing labor costs. However, this paper is limited to exploring dynamic formation motion in the horizontal plane.

In the future, greater emphasis should be placed on conducting an in-depth study of the dynamic formation problem of target interception and tracking using formation reconstruction methods to achieve dynamic formation interception in three-dimensional space. It is important to explore the application of this approach to real networked multi-intelligent body systems that involve delayed communication and dynamic interference variations. The design of adaptive algorithms with multi-intelligent body output feedback is an interesting area of investigation. These concepts can be implemented in real robotic systems. The next research plan should be to validate them in simulation under the ROS robotics platform and through validation experiments in real robots or multi-intelligent bodies. Ultimately, these algorithms can be applied to real-life applications.