Fuzzy adaptive finite-time formation control of unmanned ground vehicles with performance and feasibility constraints

Abstract

This paper investigates the problem of fuzzy adaptive finite-time formation tracking control for unmanned ground vehicles (UGVs) systems with two different constraints. By utilizing the distance tracking error between the actual trajectory and the desired trajectory of each UGV, as well as the constraints of bearing angle, to achieve the performance and feasibility constraints, respectively. Furthermore, the fuzzy logic systems (FLSs) are used to approximate unknown nonlinear functions. Due to the limitations of the field of view and communication distance of UGVs, the controller is designed by using finite time stability theory and universal barrier function. Under the proposed control scheme, the stability of the closed-loop system can be ensured, feasibility and performance constraints are also achieved. Finally, the effectiveness of the proposed control design is verified through simulation results.

Introduction

In recent years, UGVs have received more applications and attention in both civilian and military fields.^1–3 In,^4,5 the author proposed path planning for the UGVs system, enabling UGVs to autonomously navigate. The author considered the situation of the UGVs encountering obstacles in complex environments in.^6,7 The author proposed a trajectory planning and tracking strategy in, ⁶ the purpose is to ensure the safety of UGVs in uncertain environments, in ⁷ designed a fuzzy neural network obstacle avoidance algorithm through multi-sensor information fusion. In, ⁸ the characteristics of unmanned aerial vehicle (UAV)/UGVs are used, heterogeneous systems can collaborate to track and complete many complex tasks.

Formation control is crucial for multiple UGVs systems, with the goal of forming a target formation based on predetermined geometric shapes. The formation control of UGVs has several distinct advantages such as high efficiency and distributed communication, which has led people to pay more and more attention to it in.^9–11 Generally, the target of formation control of multiple UGVs is to prompt all UGVs to follow the required trajectories to keep a desired structure.^12,13 Significantly, when designing controllers for multiple UGV formations, many issues need to be addressed. The main method contained leader-follower method,^14–16 virtual leader/virtual structure method, ¹⁷ behavior base method^18,19 and graph rigidity base method.^20,21 The leader-follower method is the most effective method in formation control. At the same time, many problems need to be considered in formation control.

More importantly, performance is the primary issue in formation control, which includes the precise operation and safety of multiple UGV systems. In,^22–26 security issues are usually defined to some extent as collision avoidance issues, which requires that the distance between agents should not be too small. However, the detection range may vary in practical operation and different environments. Meanwhile, the design of the formation controller is also feasible. In,^27–31 barrier Lyapunov functions are used to satisfy the constraint of heading angle. Meanwhile, it is worth noting that in,^32–34 fuzzy logic systems (FLSs) and neural networks (NNs) are effective tools for solving the problem of multi-agent systems. Therefore, the author proposed a multi-agent formation control decentralized adaptive control scheme in, ³² which includes artificial potential energy functions and robust control method. In, ³³ the author studied the adaptive optimization formation control problem of stochastic multi-agent systems. In control design, fuzzy logic systems were applied to approximate nonlinear functions.

Finite time convergence is a very important characteristic. In, ³⁴ a finite time adaptive neural network control scheme through dynamic surface technology was developed for nonlinear systems. In,^35,36 the authors proposed a finite time control method to ensure the finite time tracking performance of the system and the boundedness of all signals. The author studied the distributed finite time formation tracking control problem of multiple quadcopter UAVs under external disturbances in. ³⁷ In, ³⁸ the finite time formation tracking control of multi-agent systems with obstacle avoidance function was studied. In, ³⁹ the problem of prescribed performance adaptive asymptotic tracking control for a class of nonlinear systems with time-varying parameters was studied. In, ⁴⁰ the issue of adaptive fuzzy prescribed performance tracking control is considered.

In this paper, a novel adaptive fuzzy finite-time formation control design method is developed for a set of multiple UGVs. In the control design, combining finite-time stability theory and backstepping control design, all constraints that can meet the safe and accurate operation of multiple UGVs systems and the feasibility of controllers can be ensured. The main contributions of this paper can be highlighted as follows:

1. This article proposes an adaptive finite time formation tracking control scheme for UGVs systems. It is noted that references^6,8,41 have also studied the formation tracking control problem of UGVs systems, but the above developed control methods cannot ensure that each UGV tracks the leader’s trajectory within a finite time.

2. Different from the existing related works.^42,43 this article not only proposes a performance function for maintaining connectivity and collision avoidance , but also considers different constraints to ensure the safe operation of UGVs and the precise operation of UGV systems.

The distribution of other parts of this article is as follows. In Section “Problem formulations,” the description of the problem, including dynamic models and communication range constraints was provided. In Section “Formation controller design,” the formation controller was designed, while utilizing Lyapunov theory to ensure the stability of the system. In Section “Simulation studies,” the effectiveness of the proposed method was verified through simulation results. Section “Conclusions,” draw conclusions.

Problem formulations

UGVs dynamics

Considering the planar motion, about the L-F UGVs formation structure model, the label $i$ denotes the follower, the label $j$ denotes the leader. The kinematic equations of the unmanned ground vehicles system are expressed as ⁴⁴ :

$$\{\begin{array}{c}{\stackrel{·}{p}}_{\mathrm{xi}}={\wp }_i\cos {\partial }_i\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{p}}_{\mathrm{yi}}={\wp }_i\sin {\partial }_i}\\ \multicolumn{1}{c}{{\stackrel{·}{\partial }}_i={\lambda }_i}\end{array}$$

(1)

where $i=1,2,\dots ,N$ represents the $i$ th UGV. $(p_{\mathrm{xi}},p_{\mathrm{yi}})$ and ${\partial }_i$ are coordinates of UGV and heading angle of UGV, respectively. ${\lambda }_i$ and ${\wp }_i$ are angular velocity and linear velocity of UGV, respectively. The dynamic equations from system (1) can be defined as

$$\begin{array}{c}{\stackrel{·}{\wp }}_i=m_{\wp i}{\tau }_{\wp i}+f_{\wp i}({\wp }_i)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\lambda }}_i=m_{\lambda i}{\tau }_{\lambda i}+f_{\lambda i}({\wp }_i)}\end{array}$$

(2)

where $f_{\wp i}({\wp }_i)$ and $f_{\lambda i}({\wp }_i)$ are unknown functions are associated with variables ${\wp }_i$ and ${\lambda }_i$ . ${\tau }_{\wp i}$ and ${\tau }_{\lambda i}$ are actual controllers of UGV. The parameters $m_{\wp i}$ and $m_{{\lambda }_i}$ are positive constants, $m_{\wp i}=\frac{r_i}{2(m_{1i}+m_{2i})}$ , $m_{{\lambda }_i}=\frac{r_i}{2b_i(m_{1i}-m_{2i})}$ , with UGV wheel radius $r_i$ , let assume system parameters $m_{1i}$ , $m_{2i}$ , and $r_i$ are known constants due to the uncertainty of the model (Figure 1).

Figure 1.

The L-F UGVs formation structure.

It is important to note that the functions $f_{\wp i}({\wp }_i)$ and $f_{\lambda i}({\wp }_i)$ are unidentified in the dynamic equations (2). According to the FLSs ${\hat{f}}_{\wp i}({\wp }_i|{\hat{W}}_{\wp i})={\hat{W}}_{\wp i}^{\mathrm{T}}{\varphi }_{\wp i}({\wp }_i)$ and ${\hat{f}}_{\lambda i}({\lambda }_i|{\hat{W}}_{\lambda i})={\hat{W}}_{\lambda i}^{\mathrm{T}}{\varphi }_{\lambda i}({\wp }_i)$ to approximate the functions $f_{\wp i}({\wp }_i)$ and $f_{\lambda i}({\wp }_i)$ , then we have

$$f_{\wp i}=W_{\wp i}^{*\mathrm{T}}{\varphi }_{\wp i}({\wp }_i)+{\sigma }_{\wp i}({\wp }_i)$$

$$f_{\lambda i}=W_{\lambda i}^{*T}{\varphi }_{\lambda i}({\wp }_i)+{\sigma }_{\lambda i}({\wp }_i)$$

where ${\wp }_i=[{\wp }_i,{\lambda }_i{]}^{\mathrm{T}}$ is an approximate variable, ${\sigma }_{\wp i}({\wp }_i)$ and ${\sigma }_{\lambda i}({\wp }_i)$ are the minimal approximation errors. ${\sigma }_{\lambda i}^{*}>0$ and ${\sigma }_{\wp i}^{*}>0$ are extremely small constant, in which satisfy $|{\sigma }_{\lambda i}(\wp i)|<{\sigma }_{\lambda i}^{*}$ and $|{\sigma }_{\wp i}(\wp i)|<{\sigma }_{\wp i}^{*}$ .

• Lemma 1. ⁴⁵ For any $x\in R$ , the following inequality holds:

$$-m^2\le -\frac{1}{4}{m}^2-{\left|m\right|}^{\frac{3}{2}}+\frac{1}{4}$$

Leader-follower formation under communication range constraint

In the structure of L-F model of UGV, $d_{\mathrm{ij}}$ and ${\phi }_{\mathrm{ij}}$ represent the distance and the bearing angle between the $i$ th UGV and the $j$ th UGV, respectively, which are defined as

$$\begin{array}{c}{d}_{\mathrm{ij}}=\sqrt{{(p_{\mathrm{xi}}-p_{\mathrm{xj}})}^2+{(p_{\mathrm{yi}}-p_{\mathrm{yj}})}^2}\hfill \\ \multicolumn{1}{c}{{\phi }_{\mathrm{ij}}=a\tan 2({\tilde{p}}_{\mathrm{yi}},{\tilde{p}}_{\mathrm{xi}})}\end{array}$$

(3)

$$\begin{array}{c}\left[\begin{array}{c}{\tilde{p}}_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{{\tilde{p}}_{\mathrm{yi}}}\end{array}\right]=\left[\begin{array}{cc}\cos {\partial }_{\mathrm{ij}}\hfill & \sin {\partial }_{\mathrm{ij}}\hfill \\ \multicolumn{1}{c}{-\sin {\partial }_{\mathrm{ij}}}& \cos {\partial }_{\mathrm{ij}}\hfill \end{array}\right]\left[\begin{array}{c}{p}_{\mathrm{xj}}-p_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{p_{\mathrm{yj}}-p_{\mathrm{yi}}}\end{array}\right]\hfill \end{array}$$

(4)

where $i$ represents the $i$ th UGV, $j$ represents the $j$ th UGV. $a\tan 2({\tilde{p}}_{\mathrm{yi}},{\tilde{p}}_{\mathrm{xi}})$ is an arctangent function.

In the information exchange process of the L-F model, any UGV can gain the orientation and position of direct leader. Thus, the boundedness on the field of view and collision avoidance between UGVs are very significant questions in practical formation control. Accordingly, so as to realize range constraints, the following angle and distance constraints are expressed as

$$\begin{array}{c}0<d_{\mathrm{xi}}<d_{\mathrm{ij}}(t)<d_{\mathrm{Xi}}\forall t\ge 0\hfill \end{array}$$

(5)

$$\begin{array}{c}-\frac{\pi }{2}<-{\phi }_{\mathrm{Xi}}<{\phi }_{\mathrm{ij}}(t)<{\phi }_{\mathrm{Xi}}<\frac{\pi }{2}\forall t\ge 0\hfill \end{array}$$

(6)

where $d_{\mathrm{Xi}}$ represents the maximum value of communication and $d_{\mathrm{xi}}$ is the minimum distance for collision avoidance. ${\phi }_{\mathrm{Xi}}$ represents the maximum angle value for maintaining communication. The definition of formation tracking errors are as follows

$$\begin{array}{c}{d}_{\mathrm{ije}}(t)=d_{\mathrm{ij}}(t)-d_{\mathrm{dij}}(t)\hfill \end{array}$$

(7)

$$\begin{array}{c}{\phi }_{\mathrm{ije}}(t)={\phi }_{\mathrm{ij}}(t)-{\phi }_{\mathrm{dij}}(t)\hfill \end{array}$$

(8)

where ${\phi }_{\mathrm{dij}}(t)$ and $d_{\mathrm{dij}}(t)$ represent the desired bearing angle and the desired distance between $i$ th UGV and $j$ th UGV, respectively.

During the operation, performance constraint needs to be satisfied. Therefore, for the $i$ th UGV, the distance tracking error $d_{\mathrm{ije}}(t)$ must satisfy the following inequality:

$$\begin{array}{c}{d}_{\mathrm{xi}}-d_{\mathrm{dij}}(t)<d_{\mathrm{ije}}(t)<d_{\mathrm{Xi}}-d_{\mathrm{dij}}(t)\hfill \end{array}$$

(9)

Additionally, we introduce the bearing angle with a feasibility constraints:

$$\begin{array}{c}-{\phi }_{\mathrm{Xi}}-{\phi }_{\mathrm{dij}}(t)<{\phi }_{\mathrm{ije}}(t)<{\phi }_{\mathrm{Xi}}-{\phi }_{\mathrm{dij}}(t)\hfill \end{array}$$

(10)

The steady-state and transient performance of tracking error is an significant parameter in the process of designing controllers. The error can not only be limited within a certain constraint range, but also fluctuate between the upper and lower limits. In order to facilitate the formation of (9) and (10), the following asymmetric restrictions are given:

$$\begin{array}{c}-R_{\mathrm{ni}}(t)P_{\mathrm{ni}}(t)<n_{\mathrm{ije}}(t)<P_{\mathrm{ni}}(t)\forall t\ge 0\hfill \end{array}$$

(11)

$$\begin{array}{c}{R}_{\mathrm{di}}=\frac{d_{\mathrm{xi}}-d_{\mathrm{dij}}(t)}{d_{\mathrm{dij}}(t)-d_{\mathrm{Xi}}},R_{\phi i}=\frac{{\phi }_{\mathrm{Xi}}+{\phi }_{\mathrm{dij}}(t)}{{\phi }_{\mathrm{Xi}}-{\phi }_{\mathrm{dij}}(t)}\hfill \end{array}$$

(12)

where $R_{\mathrm{ni}}(t)>0$ is a constrained function, $n=d,\phi$ and $P_{\mathrm{ni}}(t)$ is positive performance function through design, $P_{\mathrm{ni}}(t)$ can be represented as

$$\begin{array}{c}{P}_{\mathrm{ni}}(t)=(P_{\mathrm{ni},0}-P_{\mathrm{ni},\infty })e^{-k_{\mathrm{ni}}t}+P_{\mathrm{ni},\infty }\hfill \end{array}$$

(13)

where $n=d,\phi$ , $P_{\mathrm{di},0}=d_{\mathrm{Xi}}-d_{\mathrm{dij}}(t)$ , $P_{\phi i,0}={\phi }_{\mathrm{Xi}}-{\phi }_{\mathrm{dij}}(t)$ , $P_{\mathrm{ni},0}>P_{\mathrm{ni},\infty }>0$ , and $k_{\mathrm{ni}}>0$ are given quantities. To realize the standardization of errors boundary, define the error variables,

$$\begin{array}{c}{\delta }_{\mathrm{ni}}(t)=\frac{n_{\mathrm{ije}}(t)}{P_{\mathrm{ni}}(t)},n=d,\phi \hfill \end{array}$$

(14)

$$\begin{array}{c}-R_{\mathrm{ni}}(t)<{\delta }_{\mathrm{ni}}(t)<1\forall t\ge 0n=d,\phi \hfill \end{array}$$

(15)

To solve the above constraint problems, then realize the interflow and tracking trajectory of UGVs within an required range, let introduce the following functions as

$$\begin{array}{c}{\eta }_{\mathrm{ijd}}=\frac{{\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}{\delta }_{\mathrm{di}}}{({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}})({\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}})}\hfill \end{array}$$

(16)

$$\begin{array}{c}{\eta }_{\mathrm{ij}\phi }=\frac{{\Omega }_{\mathrm{kijH}}{\Omega }_{\mathrm{kijL}}{\delta }_{\phi i}}{({\Omega }_{\mathrm{kijH}}-{\delta }_{\phi i})({\Omega }_{\mathrm{kijH}}-{\delta }_{\phi i})}\hfill \end{array}$$

(17)

• Remark 1: It is worth noting that functions ${\eta }_{\mathrm{ijd}}$ and ${\eta }_{\mathrm{ij}\phi }$ are universal barrier functions, which are handy for designing virtual controllers to solve tracking problems. This is due to its unique property that ${\eta }_{\mathrm{ijd}}$ and ${\eta }_{\mathrm{ij}\phi }$ will become $+\infty$ or $-\infty$ when $d_{\mathrm{ije}}$ and ${\phi }_{\mathrm{ije}}$ are close to the boundaries of constraint ${\Omega }_{\mathrm{ijH}}$ , ${\Omega }_{\mathrm{ijL}}$ and ${\Omega }_{\mathrm{kijH}}$ , ${\Omega }_{\mathrm{kijL}}$ . Therefore, we only need to ensure the stability of universal barrier functions, then ${\delta }_{\mathrm{di}}$ and ${\delta }_{\phi i}$ can be restricted within the constraint boundary.

Formation controller design

In this section, an adaptive fuzzy finite-time formation control design method will be developed to ensure the specified transient and steady-state performance in asymptotically stable error systems.

• Step 1: To enable all UGV followers of the formation to track the leader’s reference path, designing the actual controllers and the virtual controllers for $i$ th UGV through backstepping method. To realize the desired formation, tracking error is expressed as follows

$$\begin{array}{c}{e}_{\wp i}={\wp }_i-{\alpha }_{\wp i}\hfill \end{array}$$

(18)

$$\begin{array}{c}{e}_{\lambda i}={\lambda }_i-{\alpha }_{\lambda i}\hfill \end{array}$$

(19)

where ${\alpha }_{\wp i}$ and ${\alpha }_{\lambda i}$ are virtual control inputs.

• Step 2 : Based on systems (1)–(2) and L-F formation models (3)–(4), differentiating (7) and (8) has

$$\begin{array}{c}{\stackrel{·}{d}}_{\mathrm{ije}}={\wp }_j\cos {\mathrm{\Psi }}_{\mathrm{ij}}-{\wp }_i\cos {\partial }_{\mathrm{ij}}\hfill \end{array}$$

(20)

$$\begin{array}{c}{\stackrel{·}{\phi }}_{\mathrm{ije}}=\frac{{\wp }_i}{d_{\mathrm{ij}}}\sin {\partial }_{\mathrm{ij}}-\frac{{\wp }_j}{d_{\mathrm{ij}}}\sin {\mathrm{\Psi }}_{\mathrm{ij}}-{\lambda }_i\hfill \end{array}$$

(21)

where ${\mathrm{\Psi }}_{\mathrm{ij}}={\phi }_{\mathrm{ij}}+{\partial }_{\mathrm{ij}}-{\partial }_j$ , ${\wp }_j$ , and ${\partial }_j$ are linear velocity and heading angle for leader. Due to the limited sensors capability of UGVs makes it hard to exactly determine the above variables. Nevertheless, the UGVs onboard sensors can gain bearing angle ${\phi }_{\mathrm{ij}}$ and relative the distance $d_{\mathrm{ij}}$ , and obtain velocities ${\wp }_j$ and ${\lambda }_i$ through velocity sensors.

Consider the following Lyapunov function as $V_1$ for Step 1:

$$\begin{array}{c}{V}_1=\frac{1}{2}{\eta }_{\mathrm{ijd}}^2+\frac{1}{2}{\eta }_{\mathrm{ij}\phi }^2\hfill \end{array}$$

(22)

Therefore, the ${\stackrel{·}{V}}_1$ is

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1={\gamma }_{\mathrm{ijd}}[\frac{1}{P_{\mathrm{di}}}({\wp }_j\cos {\mathrm{\Psi }}_{\mathrm{ij}}-{\wp }_i\cos {\phi }_{\mathrm{ij}})-\frac{{\stackrel{·}{P}}_{\mathrm{di}}{\delta }_{\mathrm{di}}}{P_{\mathrm{di}}^2}]}\\ \multicolumn{1}{c}{+{\gamma }_{\mathrm{ij}\phi }[\frac{1}{P_{\phi i}}(\frac{{\wp }_i}{d_{\mathrm{ij}}}\sin {\phi }_{\mathrm{ij}}-\frac{{\wp }_j}{d_{\mathrm{ij}}}\sin {\mathrm{\Psi }}_{\mathrm{ij}}-{\lambda }_i)}\\ \multicolumn{1}{c}{-\frac{{\stackrel{·}{P}}_{\phi i}{\delta }_{\phi i}}{P_{\phi i}^2}]}\end{array}$$

(23)

where ${\gamma }_{\mathrm{ijd}}=\frac{{\Omega }_{\mathrm{ijH}}^2{\Omega }_{\mathrm{ijL}}^2{\delta }_{\mathrm{di}}({\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}}^2)}{{({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}})}^3{({\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}})}^3}$ and ${\gamma }_{ij\phi }=\frac{{\Omega }_{kijH}^2{\Omega }_{kijL}^2{\delta }_{\phi i}({\Omega }_{kijH}{\Omega }_{kijL}+{\delta }_{\phi i}^2)}{{({\Omega }_{kijH}-{\delta }_{\phi i})}^3{({\Omega }_{kijL}+{\delta }_{\phi i})}^3}$ are known continuous functions.

Among them, ${\alpha }_{{\wp }_i}$ and ${\alpha }_{\lambda i}$ are defined as follows, respectively.

$$\begin{array}{c}\multicolumn{1}{c}{{\alpha }_{\wp i}=\frac{1}{\cos {\phi }_{\mathrm{ij}}}[\frac{{\beta }_{1d}{\delta }_{\mathrm{di}}({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}})({\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}})}{{\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}}}\\ \multicolumn{1}{c}{+\frac{{\beta }_{2d}{\left|{\delta }_{\mathrm{di}}\right|}^{\frac{1}{2}}\mathrm{sign}({\delta }_{\mathrm{di}})({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}}{)}^{\frac{3}{2}}{({\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}})}^{\frac{3}{2}}}{{{\Omega }_{\mathrm{ijH}}}^{\frac{3}{2}}{{\Omega }_{\mathrm{ijL}}}^{\frac{3}{2}}}}\\ \multicolumn{1}{c}{+{\hat{\overline{\wp }}}_j\tanh \left(\frac{{\gamma }_{\mathrm{ijd}}{\hat{\overline{\wp }}}_j}{P_{\mathrm{di}}\iota }\right)-\frac{{\stackrel{·}{P}}_{\mathrm{di}}{\delta }_{\mathrm{di}}}{P_{\mathrm{di}}}]}\end{array}$$

(24)

$$\begin{array}{c}\multicolumn{1}{c}{{\alpha }_{\lambda i}=\frac{{\beta }_{1\phi }{\delta }_{\phi i}({\Omega }_{\mathrm{kijH}}-{\delta }_{\phi i})({\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i})}{{\Omega }_{\mathrm{kijH}}{\Omega }_{\mathrm{kijL}}}}\\ \multicolumn{1}{c}{+\frac{{\beta }_{2\phi }{\left|{\delta }_{\phi i}\right|}^{\frac{1}{2}}\mathrm{sign}({\delta }_{\phi i})({\Omega }_{\mathrm{kijH}}-{\delta }_{\phi i}{)}^{\frac{3}{2}}{({\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i})}^{\frac{3}{2}}}{{{\Omega }_{\mathrm{kijH}}}^{\frac{3}{2}}{{\Omega }_{\mathrm{kijL}}}^{\frac{3}{2}}}}\\ \multicolumn{1}{c}{+\frac{{\wp }_j}{d_{\mathrm{ij}}}\sin {\phi }_{\mathrm{ij}}+\frac{{\hat{\overline{\wp }}}_j}{d_{\mathrm{ij}}}\tanh \left(\frac{{\gamma }_{\mathrm{ij}\phi }{\hat{\overline{\wp }}}_j}{P_{\phi i}{d}_{\mathrm{ij}}\iota }\right)-\frac{{\stackrel{·}{P}}_{\phi i}{\phi }_{\mathrm{ije}}}{P_{\phi i}}}\end{array}$$

(25)

where $0<\left|z\right|-z\tanh (z/\iota )\le \mu \iota$ , in which $\mu =0.2785$ and $\iota >0$ . ${\wp }_j$ represents the leaders linear velocity, ${\overline{\wp }}_j$ represents the maximum value of ${\wp }_j$ and ${\hat{\overline{\wp }}}_j$ represents the estimated value of ${\wp }_j$ , the relative correlation of them is ${\tilde{\overline{\wp }}}_j={\overline{\wp }}_j-{\hat{\overline{\wp }}}_j$ . Furthermore, the $\mathrm{sign}$ function is a common function in mathematics used to determine the polarity of a number. It is usually represented as $\mathrm{sign}(z)$ , and when $z$ is greater than zero, the result of $\mathrm{sign}(z)$ is 1; when $z$ equals zero, the result of $\mathrm{sign}(z)$ is 0; when $z$ is less than zero, the result of $\mathrm{sign}(z)$ is −1.

Substituting (24) and (25) into (23), one yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=-\frac{{\beta }_{1d}{\delta }_{\mathrm{di}}^2{\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}({\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}}^2)}{{({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}})}^2{({\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}})}^2}}\\ \multicolumn{1}{c}{-\frac{{\beta }_{2d}{\left|{\delta }_{\mathrm{di}}\right|}^{\frac{3}{2}}{{\Omega }_{\mathrm{ijH}}}^{\frac{1}{2}}{{\Omega }_{\mathrm{ijL}}}^{\frac{1}{2}}({\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}}^2)}{{({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}})}^{\frac{3}{2}}{({\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}})}^{\frac{3}{2}}}}\\ \multicolumn{1}{c}{-\frac{{\beta }_{1\phi }{\delta }_{\phi i}^2{\Omega }_{\mathrm{kijH}}{\Omega }_{\mathrm{kijL}}({\Omega }_{\mathrm{kijH}}{\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i}^2)}{{({\Omega }_{\mathrm{kijH}}-{\delta }_{\phi i})}^2{({\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i})}^2}}\\ \multicolumn{1}{c}{-\frac{{\beta }_{2\phi }{\left|{\delta }_{\phi i}\right|}^{\frac{3}{2}}{{\Omega }_{\mathrm{kijH}}}^{\frac{1}{2}}{{\Omega }_{\mathrm{kijL}}}^{\frac{1}{2}}({\Omega }_{\mathrm{kijH}}{\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i}^2)}{{({\Omega }_{\mathrm{kijH}}-{\delta }_{\phi i})}^{\frac{3}{2}}{({\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i})}^{\frac{3}{2}}}}\\ \multicolumn{1}{c}{+\frac{{\gamma }_{\mathrm{ijd}}{\wp }_j\cos {\mathrm{\Psi }}_{\mathrm{ij}}}{P_{\mathrm{di}}}-\frac{{\gamma }_{\mathrm{ij}\phi }{\wp }_j}{P_{\phi i}{d}_{\mathrm{ij}}}\sin {\mathrm{\Psi }}_{\mathrm{ij}}}\\ \multicolumn{1}{c}{-\frac{{\gamma }_{\mathrm{ijd}}}{P_{\mathrm{di}}}{\hat{\overline{\wp }}}_j\tanh \left(\frac{{\gamma }_{\mathrm{ijd}}{\hat{\overline{\wp }}}_j}{P_{\mathrm{di}}l\iota }\right)-{\gamma }_{\mathrm{ijd}}\frac{e_{\wp i}\cos {\phi }_{\mathrm{ij}}}{P_{\mathrm{di}}}}\\ \multicolumn{1}{c}{-\frac{{\gamma }_{\mathrm{ij}\phi }{\hat{\overline{\wp }}}_j}{P_{\phi i}{d}_{\mathrm{ij}}}\tanh \left(\frac{{\gamma }_{\mathrm{ij}\phi }{\hat{\overline{\wp }}}_j}{P_{\phi i}{d}_{\mathrm{ij}}\iota }\right)-{\gamma }_{\mathrm{ij}\phi }\frac{e_{\lambda i}}{P_{\phi i}}}\end{array}$$

(26)

Based on formulas (16) and (17), the following results can be obtained

$$\begin{array}{c}\multicolumn{1}{c}{-\frac{{\beta }_{1d}{\delta }_{\mathrm{di}}^2{\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}({\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}}^2)}{{({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}})}^2{({\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}})}^2}\le -{\beta }_{1d}{\eta }_{\mathrm{ijd}}^2}\\ \multicolumn{1}{c}{-\frac{{\beta }_{2d}{\left|{\delta }_{\mathrm{di}}\right|}^{\frac{3}{2}}{{\Omega }_{\mathrm{ijH}}}^{\frac{1}{2}}{{\Omega }_{\mathrm{ijL}}}^{\frac{1}{2}}({\Omega }_{\mathrm{ijH}}{\Omega }_{\mathrm{ijL}}+{\delta }_{\mathrm{di}}^2)}{{({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}})}^{\frac{3}{2}}{({\Omega }_{\mathrm{ijH}}-{\delta }_{\mathrm{di}})}^{\frac{3}{2}}}\le -{\beta }_{2d}{\left|{\eta }_{\mathrm{ijd}}\right|}^{\frac{3}{2}}}\\ \multicolumn{1}{c}{-\frac{{\beta }_{1\phi }{\delta }_{\phi i}^2{\Omega }_{\mathrm{kijH}}{\Omega }_{\mathrm{kijL}}({\Omega }_{\mathrm{kijH}}{\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i}^2)}{{({\Omega }_{\mathrm{kijH}}-{\delta }_{\phi i})}^2{({\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i})}^2}\le -{\beta }_{1\phi }{\eta }_{\mathrm{ij}\phi }^2}\\ \multicolumn{1}{c}{-\frac{{\beta }_{2\phi }{\left|{\delta }_{\phi i}\right|}^{\frac{3}{2}}{{\Omega }_{\mathrm{kijH}}}^{\frac{1}{2}}{{\Omega }_{\mathrm{kijL}}}^{\frac{1}{2}}({\Omega }_{\mathrm{kijH}}{\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i}^2)}{{({\Omega }_{\mathrm{kijH}}-{\delta }_{\phi i})}^{\frac{3}{2}}{({\Omega }_{\mathrm{kijL}}+{\delta }_{\phi i})}^{\frac{3}{2}}}\le -{\beta }_{2\phi }{\left|{\eta }_{\mathrm{ij}\phi }\right|}^{\frac{3}{2}}}\end{array}$$

(27)

Substituting (27) into (26), one yields that

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1\le -{\beta }_{1d}{\eta }_{\mathrm{ijd}}^2-{\beta }_{2d}{\left|{\eta }_{\mathrm{ijd}}\right|}^{\frac{3}{2}}-{\beta }_{1\phi }{\eta }_{\mathrm{ij}\phi }^2-{\beta }_{2\phi }{\left|{\eta }_{\mathrm{ij}\phi }\right|}^{\frac{3}{2}}}\\ \multicolumn{1}{c}{+\frac{{\gamma }_{\mathrm{ijd}}{\wp }_j\cos {\mathrm{\Psi }}_{\mathrm{ij}}}{P_{\mathrm{di}}}-\frac{{\gamma }_{\mathrm{ij}\phi }{\wp }_j}{P_{\phi i}{d}_{\mathrm{ij}}}\sin {\mathrm{\Psi }}_{\mathrm{ij}}}\\ \multicolumn{1}{c}{-\frac{{\gamma }_{\mathrm{ijd}}}{P_{\mathrm{di}}}{\hat{\overline{\wp }}}_j\tanh \left(\frac{{\gamma }_{\mathrm{ijd}}{\hat{\overline{\wp }}}_j}{P_{\mathrm{di}}\iota }\right)-{\gamma }_{\mathrm{ijd}}\frac{e_{\wp i}\cos {\phi }_{\mathrm{ij}}}{P_{\mathrm{di}}}}\\ \multicolumn{1}{c}{-\frac{{\gamma }_{\mathrm{ij}\phi }{\hat{\overline{\wp }}}_j}{P_{\phi i}{d}_{\mathrm{ij}}}\tanh \left(\frac{{\gamma }_{\mathrm{ij}\phi }{\hat{\overline{\wp }}}_j}{P_{\phi i}{d}_{\mathrm{ij}}\iota }\right)-{\gamma }_{\mathrm{ij}\phi }\frac{e_{\lambda i}}{P_{\phi i}}}\end{array}$$

(28)

• Step 3: In accordance to (18) and (19), the time derivatives of $e_{\wp i}$ and $e_{\lambda i}$ are defined as follows

$${\stackrel{·}{e}}_{\wp i}=W_{\wp i}^{*T}{\varphi }_{\upsilon i}({\wp }_i)+{\sigma }_{\wp i}({\wp }_i)+m_{\wp i}{\tau }_{\wp i}-{\stackrel{·}{\alpha }}_{\wp i}$$

(29)

$$\begin{array}{c}{\stackrel{·}{e}}_{\lambda i}=W_{\lambda i}^{*T}{\varphi }_{\lambda i}({\wp }_i)+{\sigma }_{\lambda i}({\wp }_i)+m_{\lambda i}{\tau }_{\lambda i}-{\stackrel{·}{\alpha }}_{\lambda i}\hfill \end{array}$$

(30)

where $m_{\wp i}$ and $m_{\lambda i}$ are known system parameters.

Construct the Lyapunov function $V$ for Step 3 as follows

$$\begin{array}{c}\multicolumn{1}{c}{V=V_1+\frac{1}{2}(e_{\wp i}^2+e_{\lambda i}^2+{\beta }_{1\wp j}^{-1}{\tilde{\overline{\wp }}}_j^2}\\ \multicolumn{1}{c}{+{\tilde{W}}_{\wp i}^T{T}_{\wp i}^{-1}{\tilde{W}}_{\wp i}+{\tilde{W}}_{\lambda i}^T{T}_{\lambda i}^{-1}{\tilde{W}}_{\lambda i})}\end{array}$$

(31)

Hence, $\stackrel{·}{V}$ is

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}={\stackrel{·}{V}}_1+e_{\wp i}[W_{\wp i}^{*T}{\varphi }_{\wp i}({\wp }_i)+{\sigma }_{\wp i}({\wp }_i)+m_{\wp i}{\tau }_{\wp i}-{\stackrel{·}{\alpha }}_{\wp i}]}\\ \multicolumn{1}{c}{+e_{\lambda i}[W_{\lambda i}^{*T}{\varphi }_{\lambda i}({\wp }_i)+{\sigma }_{\lambda i}({\wp }_i)+m_{\lambda i}{\tau }_{\lambda i}-{\stackrel{·}{\alpha }}_{\lambda i}]}\\ \multicolumn{1}{c}{-{\beta }_{1\wp j}^{-1}{\tilde{\overline{\wp }}}_j{\stackrel{·}{\hat{\overline{\wp }}}}_j-{\tilde{W}}_{\wp i}^T{T}_{\wp i}^{-1}{\stackrel{·}{\hat{W}}}_{\wp i}-{\tilde{W}}_{\lambda i}^T{T}_{\lambda i}^{-1}{\stackrel{·}{\hat{W}}}_{\lambda i}}\end{array}$$

(32)

The adaptive laws of ${\hat{\overline{\wp }}}_j$ , ${\hat{W}}_{\wp i}$ , and ${\hat{W}}_{\lambda i}$ are designed as follows:

$$\begin{array}{c}{\stackrel{·}{\hat{\overline{\wp }}}}_j={\beta }_{1{\wp }_j}(\left|\frac{{\gamma }_{\mathrm{ijd}}}{P_{\mathrm{di}}}\right|+\left|\frac{{\gamma }_{\mathrm{ij}\phi }}{P_{\phi i}{d}_{\mathrm{ij}}}\right|-{\beta }_{2{\wp }_j}{\hat{\overline{\wp }}}_j)\hfill \end{array}$$

(33)

$$\begin{array}{c}{\stackrel{·}{\hat{W}}}_{\wp i}={\Gamma }_{\wp i}[{\varphi }_{\wp i}({\wp }_i)e_{\wp i}-{\beta }_{W\wp i}{\hat{W}}_{\wp i}]\hfill \end{array}$$

(34)

$$\begin{array}{c}{\stackrel{·}{\hat{W}}}_{\lambda i}={\Gamma }_{\lambda i}[{\varphi }_{\lambda i}({\wp }_i)e_{\lambda i}-{\beta }_{W\lambda i}{\hat{W}}_{\lambda i}]\hfill \end{array}$$

(35)

where ${\tilde{W}}_{\mathrm{ki}}=W_{\mathrm{ki}}^{*}-{\hat{W}}_{\mathrm{ki}}$ Design ${\tau }_{\wp i}$ and ${\tau }_{\lambda i}$ as follows

$$\begin{array}{c}{\tau }_{\wp i}=\frac{-k_{\wp i}{e}_{\wp i}-k_{1\wp i}{\left|e_{\wp i}\right|}^{\frac{1}{2}}\mathrm{sign}(e_{\wp i})+\frac{{\gamma }_{\mathrm{ijd}}\cos {\mathrm{\Psi }}_{\mathrm{ij}}}{P_{\mathrm{di}}}-{\hat{W}}_{\wp i}^T{\varphi }_{\wp i}({\wp }_i)}{m_{\wp i}}\hfill \end{array}$$

(36)

$$\begin{array}{c}{\tau }_{\lambda i}=\frac{-k_{\lambda i}{e}_{\lambda i}-k_{1\lambda i}{\left|e_{\lambda i}\right|}^{\frac{1}{2}}\mathrm{sign}(e_{\lambda i})+\frac{{\gamma }_{\mathrm{ij}\phi }}{P_{\phi i}}-{\hat{W}}_{\lambda i}^T{\varphi }_{\lambda i}({\wp }_i)}{m_{\lambda i}}\hfill \end{array}$$

(37)

• Theorem 1: For the UGV system (1), if controller (36) and (37), virtual controller (24) and (25), and parameter adaptive law (33) (34) and (35) are used, the overall control scheme ensures the following performance:

• 1) During operation, the constraints (9) and (10) must not be violated.

• 2) The formation tracking error $d_{\mathrm{ije}}$ is limited within the range of performance functions $P_{\mathrm{di}}(t)$ and $-R_{\mathrm{di}}(t)P_{\mathrm{di}}(t)$ .

• 3) The formation tracking error ${\phi }_{\mathrm{ije}}$ is limited within the range of performance functions $P_{\phi i}(t)$ and $-R_{\phi i}(t)P_{\phi i}(t)$ , respectively.

• 4) The control signals (36) and (37) are both uniformly bounded.

Therefore, the derivative of $V$ is as follows

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -{\beta }_{1d}{\eta }_{\mathrm{ijd}}^2-{\beta }_{2d}{\left|{\eta }_{\mathrm{ijd}}\right|}^{\frac{3}{2}}-{\beta }_{1\phi }{\eta }_{\mathrm{ij}\phi }^2-{\beta }_{2\phi }{\left|{\eta }_{\mathrm{ij}\phi }\right|}^{\frac{3}{2}}}\\ \multicolumn{1}{c}{+\frac{{\gamma }_{\mathrm{ijd}}{\wp }_j\cos {\mathrm{\Psi }}_{\mathrm{ij}}}{P_{\mathrm{di}}}-\frac{{\gamma }_{\mathrm{ij}\phi }{\wp }_j}{P_{\phi i}{d}_{\mathrm{ij}}}\sin {\mathrm{\Psi }}_{\mathrm{ij}}-{\overline{\wp }}_j\left|\frac{{\gamma }_{\mathrm{ij}\phi }}{P_{\phi i}{d}_{\mathrm{ij}}}\right|}\\ \multicolumn{1}{c}{+{\hat{\overline{\wp }}}_j\left|\frac{{\gamma }_{\mathrm{ijd}}}{P_{\mathrm{di}}}\right|-\frac{{\gamma }_{\mathrm{ijd}}}{P_{\mathrm{di}}}{\hat{\overline{\wp }}}_j\tanh \left(\frac{{\gamma }_{\mathrm{ijd}}{\hat{\overline{\wp }}}_j}{P_{\mathrm{di}}\iota }\right)+{\hat{\overline{\wp }}}_j\left|\frac{{\gamma }_{\mathrm{ij}\phi }}{P_{\phi i}{d}_{\mathrm{ij}}}\right|}\\ \multicolumn{1}{c}{-\frac{{\gamma }_{\mathrm{ij}\phi }{{\hat{\overline{\wp }}}_j}_j}{P_{\phi i}{d}_{\mathrm{ij}}}\tanh \left(\frac{{\gamma }_{\mathrm{ij}\phi }{\hat{\overline{\wp }}}_j}{P_{\phi i}{d}_{\mathrm{ij}}\iota }\right)-k_{\wp i}{e}_{\wp i}^2-k_{\lambda i}{e}_{\lambda i}^2}\\ \multicolumn{1}{c}{-k_{1\wp i}{\left|e_{\wp i}\right|}^{\frac{3}{2}}-k_{1\lambda i}{\left|e_{\lambda i}\right|}^{\frac{3}{2}}-e_{\wp i}{\stackrel{·}{\alpha }}_{\wp i}-e_{\lambda i}{\stackrel{·}{\alpha }}_{\lambda i}}\\ \multicolumn{1}{c}{+e_{\wp i}{\sigma }_{\wp i}+e_{\lambda i}{\sigma }_{\lambda i}+{\beta }_{W\wp i}{\tilde{W}}_{\wp i}^T{\hat{W}}_{\wp i}}\\ \multicolumn{1}{c}{+{\beta }_{W\lambda i}{\tilde{W}}_{\lambda i}^T{\hat{W}}_{\lambda i}+{\beta }_{2\wp j}{\tilde{\overline{\wp }}}_j{\hat{\overline{\wp }}}_j}\end{array}$$

(38)

The following inequalities hold

$$\begin{array}{c}\frac{{\gamma }_{\mathrm{ijd}}{\wp }_j\cos {\mathrm{\Psi }}_{\mathrm{ij}}}{P_{\mathrm{di}}}-{\overline{\wp }}_j\left|\frac{{\gamma }_{\mathrm{ijd}}}{P_{\mathrm{di}}}\right|\le 0\hfill \end{array}$$

(39)

$$\begin{array}{c}-\frac{{\gamma }_{\mathrm{ij}\phi }{\wp }_j}{P_{\phi i}{d}_{\mathrm{ij}}}\sin {\mathrm{\Psi }}_{\mathrm{ij}}-{\overline{\wp }}_j\left|\frac{{\gamma }_{\mathrm{ij}\phi }}{P_{\phi i}{d}_{\mathrm{ij}}}\right|\le 0\hfill \end{array}$$

(40)

Applying the Lemma1 and Young’s inequality, the following inequalities can be obtained

$$\begin{array}{c}\multicolumn{1}{c}{{\beta }_{2\wp j}{\tilde{\overline{\wp }}}_j{\hat{\overline{\wp }}}_j\le -\frac{{\beta }_{2\wp j}}{8}{\tilde{\overline{\wp }}}_j^2-\frac{{\beta }_{2\wp j}}{2}{\left|{\tilde{\overline{\wp }}}_j\right|}^{\frac{3}{2}}}\\ \multicolumn{1}{c}{+\frac{{\beta }_{2\wp j}}{8}+\frac{{\beta }_{2\wp j}}{2}{\overline{\wp }}_j^2}\end{array}$$

(41)

$$\begin{array}{c}\multicolumn{1}{c}{{\beta }_{W\wp i}{\tilde{W}}_{\wp i}^T{\hat{W}}_{\wp i}\le -\frac{{\beta }_{W\wp i}}{8}{\tilde{W}}_{\wp i}^2-\frac{{\beta }_{W\wp i}}{2}{\left|{\tilde{W}}_{\wp i}\right|}^{\frac{3}{2}}}\\ \multicolumn{1}{c}{+\frac{{\beta }_{W\wp i}}{8}+\frac{{\beta }_{W\wp i}}{2}\Vert W_{\wp i}^{*}{\Vert }^2}\end{array}$$

(42)

$$\begin{array}{l}{\beta }_{W\lambda i}{\tilde{W}}_{\lambda i}^T{\widehat{W}}_{\lambda i}\le -\frac{{\beta }_{W\lambda i}}{8}{\tilde{W}}_{\lambda i}^2-\frac{{\beta }_{W\lambda i}}{2}{\left|{\tilde{W}}_{\lambda i}\right|}^{\frac{3}{2}}\\ +\frac{{\beta }_{W\lambda i}}{8}+\frac{{\beta }_{W\lambda i}}{2}{\Vert W_{\lambda i}^{\ast }\Vert }^2\end{array}$$

(43)

$$\begin{array}{c}\multicolumn{1}{c}{-e_{\mathrm{ki}}{\stackrel{·}{\alpha }}_{\mathrm{ki}}\le \frac{r_{\mathrm{ki}}{e}_{\mathrm{ki}}^2}{2}+\frac{{\overline{\stackrel{·}{\alpha }}}_{\mathrm{ki}}^2}{2r_{\mathrm{ki}}}k=\wp ,\lambda }\end{array}$$

(44)

$$\begin{array}{c}\multicolumn{1}{c}{e_{\mathrm{ki}}{\sigma }_{\mathrm{ki}}\le \frac{l_{\mathrm{ki}}{e}_{\mathrm{ki}}^2}{2}+\frac{{\sigma }_{\mathrm{ki}}^{*2}}{2l_{\mathrm{ki}}}k=\wp ,\lambda }\end{array}$$

(45)

where $r_{\mathrm{ki}}$ , $l_{\mathrm{ki}}$ , ${\overline{\stackrel{·}{\alpha }}}_{\mathrm{ki}}$ and ${\sigma }_{\mathrm{ki}}^{*}$ are positive constant values.

Substituting inequalities (39)–(45) into (38), we can obtain

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -{\beta }_{1d}{\eta }_{\mathrm{ijd}}^2-{\beta }_{2d}{\left|{\eta }_{\mathrm{ijd}}\right|}^{\frac{3}{2}}-{\beta }_{1\phi }{\eta }_{\mathrm{ij}\phi }^2-{\beta }_{2\phi }{\left|{\eta }_{\mathrm{ij}\phi }\right|}^{\frac{3}{2}}}\\ \multicolumn{1}{c}{-(k_{\wp i}-\frac{r_{\wp i}+l_{\wp i}}{2})e_{\wp i}^2-(k_{\lambda i}-\frac{r_{\lambda i}+l_{\lambda i}}{2})e_{\lambda i}^2}\\ \multicolumn{1}{c}{-k_{1\wp i}{\left|e_{\wp i}\right|}^{\frac{3}{2}}-k_{1\lambda i}{\left|e_{\lambda i}\right|}^{\frac{3}{2}}-\frac{{\beta }_{2\wp j}}{8}{\tilde{\overline{\wp }}}_j^2}\\ \multicolumn{1}{c}{-\frac{{\beta }_{2\wp j}}{2}{\left|{\tilde{\overline{\wp }}}_j\right|}^{\frac{3}{2}}-\frac{{\beta }_{W\wp i}}{8}{\tilde{W}}_{\wp i}^2-\frac{{\beta }_{W\wp i}}{2}{\left|{\tilde{W}}_{\wp i}\right|}^{\frac{3}{2}}}\\ \multicolumn{1}{c}{-\frac{{\beta }_{W\lambda i}}{8}{\tilde{W}}_{\lambda i}^2-\frac{{\beta }_{W\lambda i}}{2}{\left|{\tilde{W}}_{\lambda i}\right|}^{\frac{3}{2}}+2\mu \iota +\frac{{\overline{\stackrel{·}{\alpha }}}_{\wp i}^2}{2r_{\wp i}}+\frac{{\overline{\stackrel{·}{\alpha }}}_{\lambda i}^2}{2r_{\lambda i}}}\\ \multicolumn{1}{c}{+\frac{{\beta }_{2\wp j}+{\beta }_{W\wp i}+{\beta }_{W\lambda i}}{8}+\frac{{\sigma }_{\wp i}^{*2}}{2l_{\upsilon i}}+\frac{{\sigma }_{\lambda i}^{*2}}{2l_{\lambda i}}}\\ \multicolumn{1}{c}{+\frac{{\beta }_{2\mathrm{vj}}{\overline{\wp }}_j^2+{\beta }_{W\wp i}{\Vert W_{\wp i}^{*}\Vert }^2+{\beta }_{W\lambda i}{\Vert W_{\lambda i}^{*}\Vert }^2}{2}}\end{array}$$

(46)

where $k_{\mathrm{ki}}$ is chosen as $k_{\mathrm{ki}}>\frac{r_{\mathrm{ki}}+l_{\mathrm{ki}}}{2}>0k=\wp ,\lambda$ .

From the definition of $V$ in formula (31), the derivative form of $V$ can be obtained as follows:

$$\begin{array}{c}\stackrel{·}{V}\le -C_{i}V-D_i{V}^{\frac{3}{4}}+E_i\hfill \end{array}$$

(47)

where

$$\begin{array}{c}\multicolumn{1}{c}{C_i=min\{2{\beta }_{1d},2{\beta }_{1\phi },2k_{\wp i}-r_{\wp i}-l_{\wp i},2k_{\lambda i}-r_{\lambda i}-l_{\lambda i},}\\ \multicolumn{1}{c}{\frac{{\beta }_{1\wp j}{\beta }_{2\wp j}}{4},\frac{{\beta }_{W\wp i}}{4{\lambda }_{max}\left(T_{\wp i}^{-1}\right)},\frac{{\beta }_{W\lambda i}}{4{\lambda }_{max}\left(T_{\lambda i}^{-1}\right)}\}}\end{array}$$

and

$$\begin{array}{c}\multicolumn{1}{c}{D_i=min\{2^{\frac{3}{4}}{\beta }_{2d}{,2}^{\frac{3}{4}}{\beta }_{2\phi }{,2}^{\frac{3}{4}}{k}_{1\wp i}{,2}^{\frac{3}{4}}{k}_{\lambda i},}\\ \multicolumn{1}{c}{2^{-\frac{1}{4}}{\beta }_{1\wp j}^{\frac{3}{4}}{\beta }_{2\wp j},\frac{2^{-\frac{1}{4}}{\beta }_{W\wp i}}{{\lambda }_{max}\left(T_{\wp i}^{-\frac{3}{4}}\right)},\frac{2^{-\frac{1}{4}}{\beta }_{W\lambda i}}{{\lambda }_{max}\left(T_{\lambda i}^{-\frac{3}{4}}\right)}\}}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{E_i=\frac{{\beta }_{2\wp j}+{\beta }_{W\wp i}+{\beta }_{W\lambda i}}{8}+\frac{{\overline{\stackrel{·}{\alpha }}}_{\wp i}^2}{2r_{\wp i}}+\frac{{\overline{\stackrel{·}{\alpha }}}_{\lambda i}^2}{2r_{\lambda i}}+\frac{{\sigma }_{\wp i}^{*2}}{2l_{\upsilon i}}+\frac{{\sigma }_{\lambda i}^{*2}}{2l_{\lambda i}}}\\ \multicolumn{1}{c}{+\frac{{\beta }_{2\mathrm{vj}}{\overline{\wp }}_j^2+{\beta }_{W\wp i}{\Vert W_{\wp i}^{*}\Vert }^2+{\beta }_{W\lambda i}{\Vert W_{\lambda i}^{*}\Vert }^2}{2}+2\mu \iota }\end{array}$$

Considering the boundedness of $V_1$ , it is clear that $d_{\mathrm{ije}}$ and ${\phi }_{\mathrm{ije}}$ are bounded, $d_{\mathrm{ij}}$ and ${\phi }_{\mathrm{ij}}$ are bounded from (5) and (6). Then, we will indicate that all signals are bounded in the controled system. The boundedness of $d_{\mathrm{ije}}$ guarantees that ${\alpha }_{\wp i}$ is bounded. Therefore, ${\wp }_i$ is bounded since the boundedness of $d_{\mathrm{ije}}$ and ${\alpha }_{\wp i}$ . ${\alpha }_{\lambda i}$ is bounded due to the boundedness of ${\phi }_{\mathrm{ije}}$ , ${\wp }_i$ and $d_{\mathrm{ij}}$ . Thus ${\lambda }_i$ is also bounded. We can conclude that ${\hat{W}}_{\wp i}$ , ${\tau }_{\wp i}$ , ${\tau }_{\lambda i}$ , and ${\hat{W}}_{\lambda i}$ are bounded. Therefore, the UGVs system is stable and all signals are bounded.

Simulation studies

In this section, the initial preparation is to select four UGVs. All parameters are from reference, ⁴⁴ the specific content is shown in Table 1. Among them, directed topology structure graph of the multiple UGVs is shown in Figure 2.

Table 1.

All reference parameters for UGV.

Parameter	Value	Unit	Parameter	Value	Unit
$b_i$	0.75	$m$	${\mathrm{I}}_{\mathrm{ci}}$	15.625	$\mathrm{kg}·m^2$
$a_i$	0.3	$m$	${\mathrm{I}}_{\lambda i}$	0.005	$\mathrm{kg}·m^2$
$r_i$	0.15	$m$	${\mathrm{I}}_{\mathrm{mi}}$	0.0025	$\mathrm{kg}·m^2$
$m_{\mathrm{ci}}$	30	$\mathrm{kg}$	$d_{i1}$	5	$\mathrm{kg}/s$
$m_{\mathrm{wi}}$	1	$\mathrm{kg}$	$d_{i2}$	5	$\mathrm{kg}/s$

Figure 2.

The multiple UGVs formation permutation graph.

Firstly, the leader’s reference trajectory is designed to $x_L=60\sin (0.02t)$ , $y_L=-60\cos (0.01t)+60$ . Other UGVs initial position information are provided

$$p_1=[-4,2,-\pi /4{]}^{\mathrm{T}},p_2=[-2,-3,\pi /3{]}^{\mathrm{T}},$$

$$p_3=[-8.2,1,\pi /9{]}^{\mathrm{T}},p_4=[-6,-4,\pi /6{]}^{\mathrm{T}}.$$

In addition, the desired distance $d_{\mathrm{dij}}(t)=5+0.01\sin (0.01t)$ m, the desired bearing angles $\begin{array}{c}{\phi }_{d10}(t)={\phi }_{d30}(t)=-\pi /4+0.01\sin (0.01t),\hfill \\ {\phi }_{d20}(t)={\phi }_{d40}(t)=\pi /4+0.01\sin (0.01t).\hfill \end{array}$

They can guarautee the steady-state and the transient performance of formation tracking errors $d_{\mathrm{ije}}$ and ${\phi }_{\mathrm{ije}}$ with the constraints (5) and (6). Where $f_{\wp i}=\frac{b_{i}c{\lambda }_i^2}{m_{1i}+m_{2i}}-\frac{{\wp }_i(d_{i1}+d_{i2})}{2(m_{1i}+m_{2i})}-\frac{b_i{\lambda }_i(d_{i1}-d_{i2})}{2(m_{1i}+m_{2i})}$ , $f_{\lambda i}=-\frac{{\wp }_{i}c{\lambda }_i}{b_i(m_{1i}-m_{2i})}-\frac{{\wp }_i(d_{i1}-d_{i2})}{2b_i(m_{1i}-m_{2i})}-\frac{{\lambda }_i(d_{i1}+d_{i2})}{2(m_{1i}-m_{2i})}$ and $c=\frac{r_i^2{m}_{\mathrm{ci}}{a}_i}{2b_i}$ , $m_{1i}=\frac{r_i^2({\mathrm{mb}}_i^2+I)}{4b_i^2}+I_{\lambda i}$ , $m_{2i}=\frac{r_i^2({\mathrm{mb}}_i^2-I)}{4b_i^2}$ , $I=m_{\mathrm{ci}}{a}_i^2+2m_{\omega i}{b}_i^2+I_{\mathrm{ci}}+2I_{\mathrm{mi}}$ and $m=m_{\mathrm{ci}}+2m_{\omega i}$ . The parameters are ${\beta }_{1d}={\beta }_{2d}={\beta }_{1\phi }={\beta }_{2\phi }=0.01$ , $\iota =1$ . The parameters and matrix are ${\beta }_{1\wp j}=1$ , ${\beta }_{2\wp j}=0.01$ , $k_{\wp i}=k_{1\wp i}=k_{\lambda i}=k_{1\lambda i}=0.1$ , ${\beta }_{\mathrm{Wki}}=1,{\Gamma }_{\mathrm{ki}}=1k=\wp ,\lambda$ , the minimum value of collision avoidance distance $d_{\mathrm{xi}}=3$ , the maximum value of collision avoidance distance $d_{\mathrm{Xi}}=6$ , the maximum angle for maintaining communication ${\phi }_{\mathrm{Xi}}=\pi /2$ . The constraint functions are defined as:

$$\begin{array}{c}{P}_{\mathrm{di}}(t)=(3+0.5)\exp (-0.5t)+1.2\hfill \\ P_{\phi 1}(t)=P_{\phi 3}(t)=(3\pi /5-0.3)\exp (-0.3t)+0.7\hfill \\ P_{\phi 2}(t)=P_{\phi 4}(t)=(2\pi /5-0.1)\exp (-0.3t)+0.2\hfill \end{array}$$

The simulation results are shown in Figures 3–10.

Figure 3.

The curves of relative distance errors $d_{\mathrm{ije}}$ .

Figure 4.

The curves of relative bearing errors ${\phi }_{\mathrm{ije}}$ .

Figure 5.

The curves of relative bearing errors ${\phi }_{\mathrm{ije}}$ .

Figure 6.

The curves of controllers ${\tau }_{\wp i}$

Figure 7.

The curves of controllers ${\tau }_{\lambda i}$

Figure 8.

The running trajectory of all UGVs in 3D.

Figure 9.

The curves graph of adaptive laws ${\hat{W}}_{\wp i}$ .

Figure 10.

The curves of adaptive laws ${\hat{W}}_{\lambda i}$ .

The signal-to-noise ratio of the white noise is 5. In Figure 3, the formation tracking error $d_{\mathrm{ije}}$ is limited within the range of performance functions $P_{\mathrm{di}}(t)$ and $-R_{\mathrm{di}}(t)P_{\mathrm{di}}(t)$ . Based on the formation topology, in Figures 4 and 5, the formation tracking error ${\phi }_{\mathrm{ije}}$ is limited within the range of performance functions $P_{\phi i}(t)$ and $-R_{\phi i}(t)P_{\phi i}(t)$ , respectively. The curves of the actual control signals for UGVs 1–4 are shown in Figures 6 and 7. Figure 8 shows the trajectory of the leader-follower formation of five UGVs at t = 2000 s. The curve plots of adaptive laws ${\hat{W}}_{\mathrm{ki}}k=\wp ,\lambda$ are demonstrated in Figures 9 and 10, respectively.

Conclusions

In this paper, we have considered the formation control issue of a group of multiple UGVs systems with different constraints, including performance and feasibility constraints. Thus, we have proposed a control framework to address the fuzzy adaptive finite-time formation control problem of a group of multiple UGVs systems. Meanwhile, in the adaptive backstepping technology, the adaptive formation controller combines the universal barrier Lyapunov functions to ensure that relative distance error and relative angle error converge to zero. Under all constraints, the exponential convergence speed of distance tracking error can be guaranteed.