# Distributed Fixed-Time Consensus Control for Multiple AUV Systems with Input
Saturations

Mien Van, Yuzhu Sun, Stephen Mcllvanna, Minh-Nhat Nguyen, Federico Zocco,
Zhijie Liu, Hsueh-Cheng Wang  This work was partly supported by the Natural
Environment Research Council, United Kingdom [grant number NE/V008080/1] and
by the Royal Society [grant number IEC/NSFC/211236 and grant number
RGS/R1/221356]. (Corresponding author: Mien Van.) Mien Van, Yuzhu Sun, Minh-
Nhat Nguyen, Stephen Mcllvanna and Federico Zocco are with the School of
Electronics, Electrical Engineering and Computer Science, Queen’s University
Belfast, Belfast, United Kingdom (e-mail: m.van@qub.ac.uk), Zhijie Liu is with
School of Intelligence Science and Technology, University of Science and
Technology Beijing, China (email: liuzhijie2012@gmail.com). Hsueh-Cheng Wang
is with with the Department of Electrical and Computer Engineering, National
Chiao Tung University (NCTU), Taiwan.

###### Abstract

This study proposes a new distributed control method based on an adaptive
fuzzy control for multiple collaborative autonomous underwater vehicles (AUVs)
to track a desired formation shape within a fixed time. First, a formation
control protocol based on a fixed-time backstepping sliding mode control is
designed, in which the consensus cooperative tracking errors for each AUV will
be formulated. Then, to compensate for the saturated control torques, an
adaptive auxiliary variable is introduced. Finally, a fixed-time adaptive
fuzzy logic control (FLC) is derived to approximate the unknown dynamics, in
which the adaptive laws of the FLC is derived such that the adaptive signals
and errors can be convergent within a fixed time. The fixed time convergence
is desired in practice because it provides an exciting property that the
global convergence of the whole system is independent with the initial states
of the AUVs. The computer simulation results for a consensus formation control
of four AUVs show that the proposed formation control can provide high
tracking performance with lower and smoother control efforts.

###### Index Terms:

Multiple collaborative AUVs, Control of AUVs, Fixed-time convergence, Fuzzy
logic system.

##  I Introduction

Robotics have been extensively applied for many challenging applications,
ranging from manufacturing, agriculture, space and ocean applications [1],
[2]. In some applications, the use of single robots or autonomous vehicles has
limited efficiency, due to the limitation of sensing, endurance and payload
carrying. To increase the efficiency of robots and autonomous vehicles for
these applications, a concept of multiple collaborative robotics or swam
robotics have been introduced [3]. For underwater environment, multiple
collaborative autonomous underwater vehicles (AUVs) have shown their great
efficiency for many challenging applications like seabed monitoring, wind
turbine inspection, marine debris monitoring and cleaning, etc [4]. However,
controlling multiple AUVs working collaboratively is not a trivial task
because the effects of nonlinear dynamics, communication delay between AUVs,
and the effects of underwater environmental disturbances, i.e., waves,
currents, etc., become more severe in underwater environment [5].

Many elegant control methods have been investigated for increasing the
tracking accuracy and robustness of multi-agent systems. Optimal controllers
using distributed optimization have been developed [6]. A safe optimal
controller based on control barrier function (CBF) has been proposed in [7].
Another approach based on reinforcement learning has been developed for the
collaborative control of multi-agent systems [8]. Model predictive control
(MPC) has been explored for multi-agent systems generally [9], [10], and for
multiple AUV systems specifically [11]. Although optimal controllers and MPCs
provide a good tracking performance when the full knowledge of the system can
be known in advance, it is difficult to handle the disturbances and/or
uncertainty components. In order to handle disturbance/uncertainty components,
robust controllers have been extensively developed [12]. Robust controllers
are particularly efficient for the control system of AUVs due to their
robustness against the nonlinear effects of underwater working conditions. Due
to its strong robustness against disturbances (i.e., matched disturbances),
sliding mode control (SMC) techniques have been developed [14], [15]. Despite
the advantages of high robustness, SMC generates high chattering, which can
cause significant oscillations within the operation of mechanical systems. To
reduce the chattering for SMC, a distributed bio-inspired SMC has been
proposed for multiple AUVs [16]. However, the conventional SMCs do not provide
finite time/fixed time convergence for the systems. To provide a finite time
convergence, finite time consensus control methods have been developed for
multi-agent systems [17], [18]. To obtain both finite time convergence and
higher robustness, finite time sliding mode controllers have also been
introduced [19], [20]. Finite time controllers have also been employed for
single AUV system [21] and multiple AUV systems [22], [23]. In [24], a
terminal SMC has been developed for the formation tracking control of multiple
AUVs. The main drawback of the finite time controllers is that the convergence
time of the system is dependent on the inital states of the systems. This
issue, unfortunately, prevents the applicability of finite time controllers
for many practical applications because, in practice, some initial states of
some agents are unavailable or unknown. To overcome this drawback, fixed-time
controllers have been studied recently [25], [26]. The use of fixed-time
controllers can provide a fixed-time convergence, which is independent with
the initial states, for the multi-agent systems [27], [28], [29].

One of the issues that reduces the tracking performance of robotic systems and
multi-agent systems is the effects of unknown components such as unknown
system parameters, friction terms, and faults, etc [30]. This becomes even
more severe for AUVs due to the severe effects of external environmental
disturbances, especially for multiple collaborative AUV systems [31]. To
approximate the unknown components, many learning techniques have been
extensively developed. An iterative learning method has been employed for
multi-agent systems [32]. An adaptive NN has been developed for multi-agent
systems [33], [34], and for multiple AUVs [35]. Adaptive fuzzy logic
controllers have also been developed to take the knowledge of human about the
dynamic system into the design to increase the approximation performance of
the FLC [36], [37]. Adaptive fixed-time FLCs have been developed to preserve
the advantages of both fixed-time convergence property and the approximation
capacity of FLC in [38], [39]. However, the adaptive laws of the existing
fixed-time FLCs do not provide fixed-time convergence for the system. In
practice, it is desired that all the adaptive laws of the system can be
convergent within a fixed-time to guarantee the global convergence of the
system within a fixed-time. This is the main motivation of this paper.

Input saturation is another important consideration in the design of practical
controllers for single agent and multi-agent systems since, in practice, the
control efforts of actuators (i.e., motors) are limited [40], [41]. Many
efforts have been spent to find an effective mechanism to mitigate the effects
of input saturation. In general, to reduce the effects of saturated control
torques, an auxiliary design system can be employed [42], [43].

In summary, there are existing research gaps for formation tracking control
for multiple AUVs, which will be addressed in this study: (i) the fixed-time
convergence of the design of controllers for multiple AUV systems, (ii) the
fixed-time convergence of the adaptive laws of the adaptive FLCs, (iii) the
input saturation problem needs to be addressed within the design of
distributed formation control of multiple AUV systems. To address the research
gaps, a new fixed-time distributed formation tracking control for multiple AUV
systems is proposed. The distributed fixed-time consensus formation will be
derived based on a backstepping SMC method. To approximate the unknown
components, an adaptive fixed-time FLC will be developed, in which the
adaptive laws of FLC will be derived such that it can be convergent within a
fixed-time to guarantee a global fixed-time convergence for the system.
Furthermore, an auxiliary adaptive function will be introduced into the fixed-
time controller to compensate for the effects of the overhead control efforts.
The effectiveness of the new control algorithm will be tested on a consensus
formation of four AUVs and compared with the counterpart distributed SMC based
on a computer simulation. To highlight the novelties of this paper, we compare
the proposed method with the existing approaches as follows:

  * •

Unlike the existing distributed consensus formation controllers for multiple
AUV systems [15], this paper develops a fixed-time distributed formation
algorithm for AUVs using a backstepping SMC method to preserve the merits of
Lyapunov stability of the backstepping control, high robustness of SMC and
bounded convergence time of the fixed-time control theory.

  * •

Unlike the existing adaptive fixed-time fuzzy controllers [38], [39], which do
not guarantee a fixed time convergence for the adaptive laws of FLC, this
paper develops a new adaptive fixed time fuzzy law to guarantee the fixed time
convergence of the adaptive weights of FLC. This ensures a global fixed time
convergence of the whole collaborative multiple AUVs system.

  * •

Unlike the existing consensus formation controllers for multiple AUVs [4],
[22], [15], which do no consider the input saturation issues, this paper
incorporates an adaptive auxiliary function into the fixed-time distributed
consensus controller to handle the problem of saturated control efforts.

##  II FIXED-TIME STABILITY and CONVERGENCE, FUZZY LOGIC, GRAPH THEORY AND
PROBLEM FORMULATION

###  II-A Fixed-time stability

A typical nonlinear system can be represented as follows[44]:

$\displaystyle\dot{\xi}(t)=f(\xi(t)),\quad\xi(t_{0})=\xi_{0},\quad\xi\in\Re^{n}$
(1)

where $f(\cdot):\Re^{n}\rightarrow\Re^{n}$ is a possibly discontinuous vector
field. The fixed time convergence is determined for system (1) when it is
globally finite-time stable and its convergent time is bounded regardless the
initial states of the system, i.e., $\forall{\xi_{0}}\in\Re^{n}$,
$T(\xi_{0})\leq{T_{\text{max}}}$ is satisfied, where ${T_{\text{max}}}$ is a
positive constant.

######  Lemma 1 ([44])

If a positive definite continuous function $V(\xi):\Re^{n}\rightarrow\Re$ for
system (1) satisfies
$\dot{V}(\xi)\leq{-\chi_{1}V^{\varrho}(\xi)-\chi_{2}V^{\varsigma}(\xi)}$ for
some $\chi_{1}>0$, $\chi_{2}>0$, $\varrho>1$, and $0<\varsigma<1$, then system
(1) is determined as a globally fixed-time stable system. The convergence time
can be calculated independently with the initial states of system (1) as
follows:

$\displaystyle
T(\xi_{0})\leq\frac{1}{\chi_{1}(\varrho-1)}+\frac{1}{\chi_{2}(1-\varsigma)}.$
(2)

######  Lemma 2 ([44])

If a positive definite continuous function $V(\xi):\Re^{n}\rightarrow\Re$ for
system (1) satisfies
$\dot{V}(\xi)\leq{-\chi_{1}V^{p}(\xi)-\chi_{2}V^{q}(\xi)}+\varphi$ for some
$\chi_{1}>0$, $\chi_{2}>0$, $p>1$, $0<q<1$, and $0<\varphi<\infty$, then
system (1) is called a practically fixed-time stable system. Futhermore, the
solution of system (1) has a residual set:

$\displaystyle\lim_{t\to{T}}\xi|\|{\xi}\|\leq{\min}\\{\chi_{1}^{\frac{-1}{p}}\left(\frac{\varphi}{1-\kappa}\right)^{\frac{1}{p}},\chi_{2}^{\frac{-1}{q}}\left(\frac{\varphi}{1-\kappa}\right)^{\frac{1}{q}}\\}$
(3)

where $\kappa$ satisfies $0<\kappa<1$. The settling time can be calculated
independenly with the initial states of the system as follows:

$\displaystyle
T(\xi_{0})\leq\frac{1}{\chi_{1}\kappa(1-p)}+\frac{1}{\chi_{2}\kappa(q-1)}.$
(4)

###  II-B Fuzzy Logic System

Given a vector of input, i.e., $Z=(z_{1},z_{2},...,z_{n})^{T}\in\Re^{n}$ and
an output variable, i.e., ${y}=f({Z})\in\Re$, a fuzzy logic system can be used
to map from the input to the output. The fuzzy rules of fuzzy logic system can
be described as:

$Rule\>\textit{j}:\text{If}\>z_{1}\>\text{is}\>A_{1}^{j}\>\text{and}\>...\>\text{and}\>z_{n}\>\text{is}\>A_{n}^{j}\>\text{then}\>y\>\text{is}\>B^{j}$
(5)

where $A_{1}^{j}$, $A_{2}^{j}$,…, $A_{n}^{j}$ and $B^{j}$ represent fuzzy
sets. The fuzzy output can be obtained as:

$y={}^{\sum\limits_{j=1}^{h}{{{w}_{j}}}\left(\prod\limits_{i=1}^{n}{{{\mu}_{A_{i}^{j}}}({{z}_{i}})}\right)}/{}_{\sum\limits_{j=1}^{h}{\left(\prod\limits_{i=1}^{n}{{{\mu}_{A_{i}^{j}}}({{z}_{i}})}\right)}}={{\text{w}}^{T}}\text{
}\\!\\!\Psi\\!\\!\text{ }({Z})$ (6)

where $h$ specifies the number of fuzzy rules used, and
${{\mu}_{A_{i}^{j}}}({{z}_{i}})$ represents the membership function of
${z}_{i}$. $\text{w}={{\left[{{w}_{1}},{{w}_{2}},..,{{w}_{h}}\right]}^{T}}$
represents the fuzzy weights, and $\text{ }\\!\\!\Psi\\!\\!\text{
}({Z})={{\left[{{\Psi}_{1}}({Z}),{{\Psi}_{2}}({Z}),\dots,{{\Psi}_{h}}({Z})\right]}^{T}}$
is a fuzzy basis vector, where its elements ${{\Psi}_{j}}({Z})$ can be
described as

${{\Psi}_{j}}({Z})={}^{\prod\limits_{i=1}^{n}{{{\mu}_{A_{i}^{j}}}({{z}_{i}})}}/{}_{\sum\limits_{j=1}^{h}{\left(\prod\limits_{i=1}^{n}{{{\mu}_{A_{i}^{j}}}({{z}_{i}})}\right)}}.$
(7)

######  Lemma 3 ([38, 39])

Let ${f(Z)}$ be a continuous function on a compact set $\Omega\in{{\Re}^{n}}$,
there exists a fuzzy logic system, i.e., ${{\text{w}}^{T}}\text{
}\\!\\!\Phi\\!\\!\text{ }({Z})$, such that

$\underset{{Z}\in\Omega}{\mathop{\sup}}\,\left|{f(Z)}-{{\text{w}}^{T}}\text{
}\\!\\!\Psi\\!\\!\text{ }({Z})\right|\leq\bar{\varrho}$ (8)

where $\bar{\varrho}$ is the fuzzy minimum approximation error, $\text{
}\\!\\!\Psi\\!\\!{(Z)}={{{\left[{{\Psi}_{1}}({Z}),{{\Psi}_{2}}({Z}),...,{{\Psi}_{h}}({Z})\right]}^{T}}}/{\sum\limits_{j=1}^{h}{{{\Psi}_{j}}({Z})}}$
is the fuzzy basis function vector.

###  II-C Graph theory

Consier a directed graph $G=\\{\Lambda,\Xi\\}$ is used to describe the
formation shape among a group of AUV vehicles, where
$\Lambda=\\{\nu_{1},\nu_{2},...,\nu_{N}\\}$ denotes $N$ AUV followers,
$\Xi\subseteq{\Lambda\times{\Lambda}}$ is the set of edges.
$A=[\alpha_{ij}]\in\Re^{N\times{N}}$ denotes the weight of the edges, where
$\alpha_{ij}=\alpha_{ji}>0$ if there is an edge between AUVs $i$ and $j$,
i.e., $(\nu_{j},\nu_{i})\in{\Xi}$, and $\alpha_{ij}=\alpha_{ji}=0$ otherwise.
Let $B=\text{diag}\\{b_{1},...,b_{N}\\}$, where $b_{i}>0$ indicates that the
follower AUV $i$ can receive the direct command signals from the AUV leader;
under other conditions $b_{i}=0$. The Laplacian matrix
$L=[l_{i,j}]\in\Re^{N\times{N}}$ with $l_{i,j}=-\alpha_{i,j}$ for $i\neq j$,
and $l_{i,i}=\sum_{j=1}^{N}{\alpha_{i,j}}$. It is assumed that the graph $G$
is undirected and connected, and the desired trajectory information from the
virtual leader will be transfered to at least one AUV, and thus not all the
elements of B indentify to zero. Therefore, $L+B>0$.

###  II-D Dynamics of AUVs and Control Objective

In this paper, a control method that can form the operations of $N$ AUVs with
the dynamics described in (LABEL:D1) in a consensus manner will be derived.

$\displaystyle\dot{\eta}_{i}=J_{i}(\eta_{2,i})\upsilon_{i},$ (9)
$\displaystyle
M_{i}\dot{\upsilon}_{i}+C_{i}(\upsilon_{i})\upsilon_{i}+D_{i}(\upsilon_{i})\upsilon_{i}=u_{i}\left(\tau_{i}(t)\right)+d_{i}(t,\eta_{i},\upsilon_{i})$

where $\eta_{i}=[\eta_{1,i},\eta_{2,i}]^{T}\in\Re^{6\times{1}}$,
$\eta_{1,i}=[x_{i},y_{i},z_{i}]^{T}\in\Re^{3\times{1}}$,
$\eta_{2,i}=[\phi_{i},\theta_{i},\psi_{i}]^{T}\in\Re^{3\times{1}}$ denote the
position and orientation of $i$-th AUV, respectively.
$\upsilon_{i}=[\upsilon_{1,i},\upsilon_{2,i}]^{T}\in\Re^{6\times 1}$,
$\upsilon_{1,i}=[\upsilon_{x,i},\upsilon_{y,i},\upsilon_{z,i}]^{T}\in\Re^{3\times
1}$,
$\upsilon_{2,i}=[\omega_{x,i},\omega_{y,i},\omega_{z,i}]^{T}\in\Re^{3\times
1}$ represents the translational and rotational velocities of $i$-th AUV,
respectively. $u_{i}\left(\tau_{i}(t)\right)\in\Re^{6\times 1}$, which will be
described in (33), represents the control effort subject to saturation
nonlinearity for the $i$-th AUV. The description of the inertia matrix
$M_{i}\in\Re^{6\times 6}$, the Coriolis and centripetal matrix
$C_{i}(\upsilon_{i})\in\Re^{6\times 6}$, the hydrodynamic matrix
$D_{i}(\upsilon_{i})\in\Re^{6\times 6}$ and the Jacobian matrix
$J_{i}(\eta_{2,i})$ can be found in [16].
$d_{i}(t,\eta_{i},\upsilon_{i})\in\Re^{6\times 1}$ denotes the lumped model
uncertainty and disturbance component in the system.

Control Objective: The objective of a distributed consensus controller is to
design an appropriate controller for each AUV with the dynamics (LABEL:D1) so
that the group of AUVs can: (i) form a desired formation shape, and (ii)
follow a predefined trajectory, which is known as a virtual leader, within a
fixed time. The desired formation shape of a group of AUVs can be determined
by a specific relative postures, i.e., position and orientation, between AUVs.

##  III Fixed-time Backstepping Sliding Mode Control Design for Consensus
Formation Tracking Control

Let $\eta_{i}^{d}$, $\dot{\eta}_{i}^{d}$ and $\ddot{\eta}_{i}^{d}$ be the
desired position, velocity and acceleration of the virtual leader. Define the
position and orientation tracking errors between the objective trajectories
and the reference trajectory for $i$-th AUV
$(i\in\Gamma,\Gamma=\\{1,...,N\\})$ as follows:

$\displaystyle\varepsilon_{1,i}=\sum_{j\in{\Gamma}}\alpha_{ij}(\eta_{i}-\eta_{j}-\delta_{ij})+b_{i}(\eta_{i}-\eta_{i}^{d}-\delta_{id})$
(10)
$\displaystyle\dot{\varepsilon}_{1,i}=\sum_{j\in{\Gamma}}\alpha_{ij}(\dot{\eta}_{i}-\dot{\eta}_{j})+b_{i}(\dot{\eta}_{i}-\dot{\eta}_{i}^{d}).$

Here, $\alpha_{ij}\geq{0}$ and $b_{i}\geq{0}$ are defined as in section II.C.
$\delta_{ij}$ indicates the relative position and orientation between $i$-th
AUV and $j$-th AUV ($j\in{\Gamma}$). $\delta_{id}$ denotes the relative
posture between the $i$-th AUV and the reference trajectory (i.e., the virtual
leader). All the AUVs are expected to have the same velocity and acceleration
as the desired reference trajectory.

Differentiating the velocity of tracking error $\dot{\varepsilon}_{1,i}$ with
respect to time, we have:

$\displaystyle\ddot{\varepsilon}_{1,i}=\sum_{j\in{\Gamma}}\alpha_{ij}(\ddot{\eta}_{i}-\ddot{\eta}_{j})+b_{i}(\ddot{\eta}_{i}-\ddot{\eta}_{i}^{d})$
(11)

where $\ddot{\eta}_{i}$ and $\ddot{\eta}_{j}$ represent the acceleration of
$i$-th AUV and its neighbors $j\in{\Gamma}$, respectively. Based on
(LABEL:D1), the dynamic model of the $i$-th AUV can be expressed as:

$\displaystyle\ddot{\eta}_{i}=$
$\displaystyle\Phi_{i}\left(\upsilon_{i},\eta_{i}\right)\upsilon_{i}+J_{i}\left(\eta_{2,i}\right)\Pi_{i}u\left(\tau_{i}(t)\right)$
(12) $\displaystyle+J_{i}(\eta_{2,i})\Pi_{i}d_{i}(t,\eta_{i},\upsilon_{i})$

where,
$\Pi_{i}=M_{i}^{-1}$ and
$\Phi_{i}(\upsilon_{i},\eta_{i})=\dot{J}_{i}(\eta_{2,i})-J_{i}(\eta_{2,i})\Pi_{i}C_{i}(\upsilon_{i})-J_{i}(\eta_{2,i})\Pi_{i}D_{i}(\upsilon_{i})$.
For facilitating the design of controllers later, the following matrices are
defined:
$\bar{\Phi}(\upsilon,\eta)=\text{diag}\\{\Phi_{1}(\upsilon_{1},\eta_{1}),...,\Phi_{N}(\upsilon_{N},\eta_{N})\\}$,
$\bar{\Pi}=\text{diag}\\{\Pi_{1},...,\Pi_{N}\\}$,
$\bar{J}(\eta_{2})=\text{diag}\\{J_{1}(\eta_{2,1}),...,J_{N}(\eta_{2,N})\\}$,
$u\left(\tau(t)\right)=[u_{1}\left(\tau_{1}(t)\right),u_{2}\left(\tau_{2}(t)\right),...,u_{N}\left(\tau_{N}(t)\right)]^{T}$,
$d=[d_{1}(t,\eta_{1},\upsilon_{1}),d_{2}(t,\eta_{2},\upsilon_{2}),...,d_{N}(t,\eta_{N},\upsilon_{N})]^{T}$.

Therefore,

$\displaystyle\ddot{\eta}=\bar{\Phi}(\upsilon,\eta)\upsilon+\bar{J}(\eta_{2})\bar{\Pi}u\left(\tau(t)\right)+\bar{J}(\eta_{2})\bar{\Pi}d.$
(13)

###### Assumption 1

The disturbance term $J_{i}(\eta_{2,i})\Pi_{i}d_{i}(t,\eta_{i},\upsilon_{i})$
is bounded by the positive constant $\tilde{\lambda}_{i}$:

$\displaystyle\|J_{i}(\eta_{2,i})\Pi_{i}d_{i}(t,\eta_{i},\upsilon_{i})\|\leq\tilde{\lambda}_{i},{}{}{}i\in\Gamma.$
(14)

The parameter $\tilde{\lambda}_{i}$ typically depends on the internal model
uncertainties and external environmental disturbances (i.e., marine
environment) of the vehicles.

Letting the following variables:

$\displaystyle\bar{\varepsilon}_{1}=[\varepsilon_{1,1},\varepsilon_{1,2},...,\varepsilon_{1,N}]^{T},$
(15)
$\displaystyle\bar{\varepsilon}_{2}=[\dot{\varepsilon}_{1,1},\dot{\varepsilon}_{1,2},...,\dot{\varepsilon}_{1,N}]^{T},$

and

$\displaystyle\ddot{\eta}=[\ddot{\eta}_{1},\ddot{\eta}_{2},...,\ddot{\eta}_{N}]^{T}.$
(16)

Adding the results in (11), (15) and (16) to form the overall error dynamics
as:

$\displaystyle\dot{\bar{\varepsilon}}_{1}=\bar{\varepsilon}_{2},$ (17)
$\displaystyle\dot{\bar{\varepsilon}}_{2}=\left(L+B\right)\left(\ddot{\eta}-\textbf{1}_{N}\otimes\ddot{\eta}^{d}\right),$

where $\otimes$ denotes the Kronecker product between two matrices.
$\textbf{1}_{N}$ stands for an $N\times{1}$ vector with unitary elements.

Then, based on (LABEL:8), a fixed time backstepping SMC can be designed as
follows:

Step 1: The first sliding surface is selected as:

$\displaystyle s_{1}(t)=\bar{\varepsilon}_{1}(t).$ (18)

Differentiating (18) yields

$\displaystyle\dot{s}_{1}(t)=\alpha_{s}(t),$ (19)

where $\alpha_{s}(t)=\dot{\bar{\varepsilon}}_{1}(t)$ is identified as the
virtual control of the system (19).

To stabilise the sliding surface $s_{1}(t)$, the following virtual control
input is designed:

$\displaystyle\alpha_{s}=-\left(k_{1}s_{1}+k_{2}s_{1}^{\gamma}+k_{3}s_{1}^{\iota}\right),$
(20)

where $k_{1}>0$, $k_{2}>$ and $k_{3}>0$ , and $0<\gamma<1$ and $\iota>1$.

Consider a candidate Lyapunov function below:

$\displaystyle V_{1}$ $\displaystyle=\frac{1}{2}s_{1}^{T}{s}_{1}.$ (21)

Adding the result in (20) into the derivative of (21), we obtain:

$\displaystyle\dot{V}_{1}$ $\displaystyle=s_{1}^{T}\dot{s}_{1}$ (22)
$\displaystyle=-s_{1}^{T}\left(k_{1}s_{1}+k_{2}s_{1}^{\gamma}+k_{3}s_{1}^{\iota}\right)$
$\displaystyle=-k_{1}s_{1}^{T}s_{1}-k_{2}\left(s_{1}^{T}s_{1}\right)^{\frac{\gamma+1}{2}}-k_{3}\left(s_{1}^{T}s_{1}\right)^{\frac{\iota+1}{2}}$
$\displaystyle\leq{-k_{2}\left(s_{1}^{T}s_{1}\right)^{\frac{\gamma+1}{2}}-k_{3}\left(s_{1}^{T}s_{1}\right)^{\frac{\iota+1}{2}}}$
$\displaystyle\leq-2^{\frac{\gamma+1}{2}}k_{2}\left(\frac{1}{2}s_{1}^{T}s_{1}\right)^{\frac{\gamma+1}{2}}-2^{\frac{\iota+1}{2}}k_{3}\left(\frac{1}{2}s_{1}^{T}s_{1}\right)^{\frac{\iota+1}{2}}.$

Step 2: Define the second sliding surface:

$\displaystyle s_{2}=\bar{\varepsilon}_{2}-(L+B)\alpha_{s}.$ (23)

The derivative of $s_{2}$ is:

$\displaystyle\dot{s}_{2}$
$\displaystyle=\dot{\bar{\varepsilon}}_{2}-\left(L+B\right)\dot{\alpha}_{s}$
(24)
$\displaystyle=\left(L+B\right)\left(\ddot{\eta}-\textbf{1}_{N}\otimes\ddot{\eta}^{d}-\dot{\alpha}_{s}\right).$

A candidate Lyapunov function is selected as

$\displaystyle V_{2}=\frac{1}{2}s_{2}^{T}{s}_{2}.$ (25)

Adding the result in (24) into the derivative of (25), we obtain:

$\displaystyle\dot{V}_{2}$ $\displaystyle=s_{2}^{T}\dot{s}_{2}$ (26)
$\displaystyle=s_{2}^{T}\left(\left(L+B\right)\left(\ddot{\eta}-\textbf{1}_{N}\otimes\ddot{\eta}^{d}-\dot{\alpha}_{s}\right)\right)$
$\displaystyle=s_{2}^{T}\left((L+B)\left(\bar{\Phi}(\upsilon,\eta)\upsilon+\bar{J}(\eta_{2})\bar{\Pi}u(\tau(t))\right.\right.$
$\displaystyle\left.\left.+\bar{J}(\eta_{2})\bar{\Pi}d-\textbf{1}_{N}\otimes\ddot{\eta}^{d}-\dot{\alpha}_{s}\right)\right).$

Based on (26), the backstepping sliding mode controller can be taken as

$\displaystyle u(\tau(t))$
$\displaystyle=[\bar{J}(\eta_{2})\bar{\Pi}]^{-1}\left(-\bar{\Phi}(\upsilon,\eta)\upsilon+\textbf{1}_{N}\otimes\ddot{\eta}^{d}+\dot{\alpha}_{s}+\tau^{{}^{\prime}}\right),$
(27)

where,

$\displaystyle\tau^{{}^{\prime}}$
$\displaystyle=-\beta_{s}\text{sign}(s_{2})-k_{8}s_{2}-k_{9}s_{2}^{\gamma}-k_{10}s_{2}^{\iota},$
(28)

where $k_{8},k_{9},k_{10}$ are positive constants. $\beta_{s}$ is chosen to be
$\beta_{s}>\tilde{\lambda}$, where
$\tilde{\lambda}=[\tilde{\lambda}_{1},\ldots,\tilde{\lambda}_{N}]^{T}$, which
were defined as in Assumption 1.

Inserting the control input in (27) and (28) into (26), we have:

$\displaystyle\dot{V}_{2}$ $\displaystyle=s_{2}^{T}\dot{s}_{2}$ (29)
$\displaystyle=s_{2}^{T}\left(L+B\right)\left({-\beta_{s}\text{sign}(s_{2})-k_{8}s_{2}-k_{9}s_{2}^{\gamma+1}}\right.$
$\displaystyle\left.{-k_{10}s_{2}^{\iota+1}+\bar{J}(\eta_{2})\bar{\Pi}d}\right)$
$\displaystyle\leq-2^{\frac{\gamma+1}{2}}\left(L+B\right)k_{9}\left(\frac{1}{2}s_{2}^{T}s_{2}\right)^{\frac{\gamma+1}{2}}$
$\displaystyle-2^{\frac{\iota+1}{2}}\left(L+B\right)k_{10}\left(\frac{1}{2}s_{2}^{T}s_{2}\right)^{\frac{\iota+1}{2}}.$

Step 3: We define a compounded candidate Lyapunov function:

$\displaystyle{V}=V_{1}+V_{2}.$ (30)

Differentiating (30) yields:

$\displaystyle\dot{V}$ $\displaystyle=\dot{V}_{1}+\dot{V}_{2}$ (31)
$\displaystyle\leq-2^{\frac{\gamma+1}{2}}k_{2}\left(\frac{1}{2}s_{1}^{T}s_{1}\right)^{\frac{\gamma+1}{2}}-2^{\frac{\iota+1}{2}}k_{3}\left(\frac{1}{2}s_{1}^{T}s_{1}\right)^{\frac{\iota+1}{2}}$
$\displaystyle-2^{\frac{\gamma+1}{2}}(L+B)k_{9}\left(\frac{1}{2}s_{2}^{T}s_{2}\right)^{\frac{\gamma+1}{2}}$
$\displaystyle-2^{\frac{\iota+1}{2}}(L+B)k_{10}\left(\frac{1}{2}s_{2}^{T}s_{2}\right)^{\frac{\iota+1}{2}}$
$\displaystyle\leq{-2^{\frac{\gamma+1}{2}}}\zeta_{1}\left(V_{1}^{\frac{\gamma+1}{2}}+V_{2}^{\frac{\gamma+1}{2}}\right)$
$\displaystyle-{2}^{\frac{\iota+1}{2}}\zeta_{2}\left(V_{1}^{\frac{\iota+1}{2}}+V_{2}^{\frac{\iota+1}{2}}\right)$
$\displaystyle\leq{-2^{\frac{\gamma+1}{2}}}\zeta_{1}\left(V^{\frac{\gamma+1}{2}}\right)-{2}^{\frac{\iota+1}{2}}\zeta_{2}\left(V^{\frac{\iota+1}{2}}\right).$

where $\zeta_{1}=\text{min}\left(k_{2},\left(L+B\right)k_{9}\right)$ and
$\zeta_{2}=\text{min}\left(k_{3},\left(L+B\right)k_{10}\right)$.

Therefore, thanks to Lemma 1, the global fixed-time convergence can be
established for the system (31), and the reaching time can be obtained as:

$\displaystyle
T\leq{\frac{2}{{\zeta_{1}2^{\frac{\gamma+1}{2}}}(1-\gamma)}}+\frac{2}{{2}^{\frac{\iota+1}{2}}\zeta_{2}(\iota-1)}.$
(32)

##  IV Distributed Backstepping Fuzzy Sliding Mode Controller with Input
Saturation

The backstepping SMC presented in section III has two main shortcomings: (i)
the bigger sliding gain is chosen based on Assumption 1, for which if the
disturbance is big, the controller provides a big chattering in the system,
(ii) the saturated control torque effects have not been considered. In this
section, we introduce an auxiliary variable and a fuzzy approximation to
overcome these shortcomings.

The designed control input $\tau_{i}(t)\in\Re^{n}$ is affected by the
saturation nonlinearity and can be expressed as [41]:

$\displaystyle
u_{i}(\tau_{i}(t))=\begin{cases}\text{sign}(\tau_{i}(t))\tau_{\text{max}_{i}},&|\tau_{i}(t)|\geq{\tau_{\text{max}_{i}}},\\\
\tau_{i}(t),&|\tau_{i}(t)|<{\tau_{\text{max}_{i}}},\end{cases}$ (33)

where ${\tau_{\text{max}_{i}}}$ represents the maximum control torque allowed
for joint $i$.

Furthermore, considering the input saturation, the saturated control torque
can be approximated by

$\displaystyle u_{i}(\tau_{i})=g_{i}(\tau_{i})+\varsigma_{i}(\tau_{i}),$ (34)

where $g_{i}(\tau_{i})$ is a smooth function. $\varsigma_{i}(\tau_{i})$ is the
bounded approximation error. $g_{i}(\tau_{i})$ can be chosen as [41]:

$\displaystyle g_{i}(\tau_{i})$
$\displaystyle=\tau_{\text{max}_{i}}\times{\text{tanh}}\left(\frac{\tau_{i}}{\tau_{\text{max}_{i}}}\right)$
(35)
$\displaystyle=\tau_{\text{max}_{i}}\frac{e^{\tau_{i}/\tau_{\text{max}_{i}}}-e^{-\tau_{i}/\tau_{\text{max}_{i}}}}{e^{\tau_{i}/\tau_{\text{max}_{i}}}+e^{-\tau_{i}/\tau_{\text{max}_{i}}}}.$

The approximation error $\varsigma_{i}(\tau_{i})$ is bounded by

$\displaystyle|\varsigma_{i}(\tau_{i})|=|u_{i}(\tau_{i})-g_{i}(\tau_{i})|\leq\tau_{{\text{max}_{i}}}(1-\text{tanh}(1))=\bar{\Delta}_{i}.$
(36)

Let:
$g(\tau(t))=[g_{1}(\tau_{1}(t)),g_{2}(\tau_{2}(t)),...,g_{N}(\tau_{N}(t))]^{T}$,
$\varsigma(\tau(t))=[\varsigma_{1}(\tau_{1}(t)),\varsigma_{2}(\tau_{2}(t)),...,\varsigma_{N}(\tau_{N}(t))]^{T}$.
To compensate for the saturated controller, an adaptive auxiliary variable
$\mu$ is introduced as:

$\displaystyle\dot{\mu}=-\mu+\bar{J}(\eta_{2})\bar{\Pi}\left(g(\tau)-\tau\right).$
(37)

For this controller, Step 1 is as in section III. Step 2 will be re-designed
as follows:

Step 2: The error variable $s_{2}$ can be redefined as

$\displaystyle s_{2}=\bar{\varepsilon}_{2}-(L+B)\alpha_{s}-(L+B)\mu.$ (38)

The derivative of $s_{2}$ can be computed as

$\displaystyle\dot{s}_{2}$
$\displaystyle=\dot{\bar{\varepsilon}}_{2}-(L+B)\dot{\alpha}_{s}-(L+B)\dot{\mu}$
(39)
$\displaystyle=\left(L+B\right)\left(\ddot{\eta}-\textbf{1}_{N}\otimes\ddot{\eta}^{d}-\dot{\alpha}_{s}-\dot{\mu}\right)$
$\displaystyle=\left(L+B\right)\left(\bar{\Phi}(\upsilon,\eta)\upsilon+\bar{J}(\eta_{2})\bar{\Pi}u(\tau)\right.$
$\displaystyle\left.+\bar{J}(\eta_{2})\bar{\Pi}d-\textbf{1}_{N}\otimes\ddot{\eta}^{d}-\dot{\alpha}_{s}-\dot{\mu}\right)$
$\displaystyle=(L+B)\left(\bar{\Phi}(\upsilon,\eta)\upsilon+\bar{J}(\eta_{2})\bar{\Pi}u(\tau)\right.$
$\displaystyle\left.+\bar{J}(\eta_{2})\bar{\Pi}d-\textbf{1}_{N}\otimes\ddot{\eta}^{d}-\dot{\alpha}_{s}\right.$
$\displaystyle\left.+\mu-\bar{J}(\eta_{2})\bar{\Pi}\left(g(\tau)-\tau\right)\right)$
$\displaystyle=(L+B)\left(\bar{\Phi}(\upsilon,\eta)\upsilon+\bar{J}(\eta_{2})\bar{\Pi}\left(\varsigma(\tau\right)+d)\right.$
$\displaystyle\left.-\textbf{1}_{N}\otimes\ddot{\eta}^{d}+\mu+\bar{J}(\eta_{2})\bar{\Pi}\tau-\dot{\alpha}_{s}\right).$

By using a FLC to approximate the lumped uncertainty and disturbance
$\bar{J}(\eta_{2})\bar{\Pi}(\varsigma(\tau)+d)$, the derivative of the error
$s_{2}$ in (39) can be represented as

$\displaystyle\dot{s}_{2}=$
$\displaystyle(L+B)\left(\bar{\Phi}(\upsilon,\eta)\upsilon+W^{*T}\Psi(Z)+\epsilon(t)\right.$
(40)
$\displaystyle\left.-\textbf{1}_{N}\otimes\ddot{\eta}^{d}+\mu+\bar{J}(\eta_{2})\bar{\Pi}\tau-\dot{\alpha}_{s}\right),$

where
$\epsilon(t)=\bar{J}(\eta_{2})\bar{\Pi}\left(\varsigma(\tau)+d(t,\eta,\upsilon)\right)-W^{*T}\Phi(Z)$.
From (14) and (36) and Lemma 3, we can obtain
$||\epsilon(t)||\leq\bar{\epsilon}$, where $\bar{\epsilon}>0$.

A candidate Lyapunov function is defined as

$\displaystyle
V_{3}(s_{2},\tilde{\theta})=\frac{1}{2}s_{2}^{T}s_{2}+\frac{1}{2}\tilde{\theta}^{T}\tilde{\theta},$
(41)

where $\tilde{\theta}=\theta-\hat{\theta}$ is the weight approximation error
with $\theta=\max_{h\in{H}}\|{W^{*}}\|$, and $\hat{\theta}$ is the
approximation of $\theta$.

The derivative of $V_{3}$ in (41) can be computed as:

$\displaystyle\dot{V}_{3}$
$\displaystyle=s_{2}^{T}\dot{s}_{2}-\tilde{\theta}^{T}\dot{\hat{\theta}}$ (42)
$\displaystyle=s_{2}^{T}(L+B)\left(\bar{\Phi}(\upsilon,\eta)\upsilon+W^{*T}\Psi(Z)+\epsilon(t)\right.$
$\displaystyle\left.-\textbf{1}_{N}\otimes\ddot{\eta}^{d}+\mu+\bar{J}(\eta_{2})\bar{\Pi}\tau-\dot{\alpha}_{s}\right)-\tilde{\theta}^{T}\dot{\hat{\theta}}.$

Let
$F_{sum}=\bar{\Phi}(\upsilon,\eta)\upsilon-\textbf{1}_{N}\otimes\ddot{\eta}^{d}+\mu-\dot{\alpha}_{s}$.
Applying inequality principle, we have

$\displaystyle
s_{2}^{T}W^{*T}\Psi(Z)\leq{\frac{1}{2}+\frac{1}{2}s_{2}^{2}\theta\Psi(Z)^{T}\Psi(Z)}.$
(43)

Then, $\dot{V}_{3}$ becomes:

$\displaystyle\dot{V}_{3}$
$\displaystyle\leq{s_{2}}^{T}(L+B)\left({\bar{J}(\eta_{2})\bar{\Pi}\tau+F_{sum}+\epsilon(t)}\right.$
(44)
$\displaystyle\left.{+\frac{1}{2}s_{2}^{2}\hat{\theta}\Psi(Z)^{T}\Psi(Z)}\right)+\frac{1}{2}(L+B)-\tilde{\theta}^{T}\dot{\hat{\theta}}$
$\displaystyle\leq{s_{2}^{T}}(L+B)\left({\bar{J}(\eta_{2})\bar{\Pi}\tau+F_{sum}+\epsilon(t)}\right.$
$\displaystyle\left.{+\frac{1}{2}s_{2}^{2}\hat{\theta}\Psi(Z)^{T}\Psi(Z)}\right)+\frac{1}{2}(L+B)$
$\displaystyle+\tilde{\theta}^{T}\left(\frac{1}{2}(L+B)s_{2}^{2}\Psi(Z)^{T}\Psi(Z)-\dot{\hat{\theta}}\right).$

Based on (44), a distributed control input is designed as:

$\displaystyle\tau$
$\displaystyle=\left(\bar{J}\left(\eta_{2}\right)\bar{\Pi}\right)^{-1}\left(-F_{sum}-\frac{1}{2}s_{2}^{2}\hat{\theta}\Psi(Z)^{T}\Psi(Z)\right.$
(45)
$\displaystyle\left.-\beta_{s}{\text{sign}}(s_{2})-k_{8}s_{2}-k_{9}s_{2}^{\gamma}-k_{10}s_{2}^{\iota}\right),$

where $\beta_{s}$ is selected such that $\beta_{s}>\bar{\epsilon}$.

The adaptive law of the FLC can be selected as

$\displaystyle\dot{\hat{\theta}}=\frac{1}{2}\left(L+B\right)s_{2}^{2}\Psi(Z)^{T}\Psi(Z)-w_{1}\hat{\theta}^{\gamma}-w_{2}\hat{\theta}^{\iota}.$
(46)

Therefore,

$\displaystyle\dot{V}_{3}$
$\displaystyle\leq{s_{2}^{T}(L+B)\left(-k_{8}s_{2}-k_{9}s_{2}^{\gamma}-k_{10}s_{2}^{\iota}\right)}$
(47)
$\displaystyle+\frac{1}{2}(L+B)+\tilde{\theta}^{T}\left(w_{1}\hat{\theta}^{\gamma}+w_{2}\hat{\theta}^{\iota}\right).$

Using inequality:

$\displaystyle\tilde{\theta}\hat{\theta}^{\gamma}\leq{l_{1}\theta^{1+\gamma}-l_{2}\tilde{\theta}^{1+\gamma}},$
(48)

$\displaystyle\tilde{\theta}\hat{\theta}^{\iota}\leq{l_{1}\theta^{1+\iota}-l_{2}\tilde{\theta}^{1+\iota}}.$
(49)

Taking the results in (47), (48) and (49) together, we have:

$\displaystyle\dot{V}_{3}$
$\displaystyle\leq\left(L+B\right)\left(-k_{8}s_{2}-k_{9}s_{2}^{\gamma+1}-k_{10}s_{2}^{\iota+1}\right)$
(50)
$\displaystyle+\frac{1}{2}(L+B)+w_{1}\left({l_{1}\theta^{1+\gamma}-l_{2}\tilde{\theta}^{1+\gamma}}\right)$
$\displaystyle+w_{2}\left({l_{1}\theta^{1+\iota}-l_{2}\tilde{\theta}^{1+\iota}}\right)$
$\displaystyle\leq(L+B)\left({-k_{8}s_{2}-k_{9}s_{2}^{\gamma+1}-k_{10}s_{2}^{\iota+1}}\right)$
$\displaystyle-
l_{2}\left(w_{1}\tilde{\theta}^{1+\gamma}+w_{2}\tilde{\theta}^{1+\iota}\right)+\sigma,$

where:

$\displaystyle\sigma=\frac{1}{2}\left(L+B\right)+l_{1}\left(w_{1}{\theta}^{1+\gamma}+w_{2}{\theta}^{1+\iota}\right).$
(51)

Therefore,

$\displaystyle\dot{V}_{3}$
$\displaystyle\leq{-k_{9}\left(L+B\right)s_{2}^{\gamma+1}}-l_{2}w_{1}\tilde{\theta}^{1+\gamma}$
(52) $\displaystyle-
k_{10}\left(L+B\right)s_{2}^{\iota+1}-l_{2}w_{2}\tilde{\theta}^{1+\iota}+\sigma$
$\displaystyle\leq{-2^{\frac{\gamma+1}{2}}}\nu_{1}{V_{3}}^{\frac{1+\gamma}{2}}{-2^{\frac{\iota+1}{2}}}\nu_{2}{V_{3}}^{\frac{1+\iota}{2}}+\sigma.$

Here,

$\displaystyle\nu_{1}=\text{min}\left(k_{9}\left(L+B\right),l_{2}w_{1}\right)$
(53)
$\displaystyle\nu_{2}=\text{min}\left(k_{10}\left(L+B\right),l_{2}w_{2}\right)$
(54)

Thus, according to Lemma 2, the value $s_{2}$ and $\tilde{\theta}$ will
converge to zero. The convergence time can be calculated as
$T\leq\frac{2}{\nu_{1}{(1-\gamma)}}+\frac{2}{\nu_{2}{(\iota-1)}}$.

Step 3: Define a candidate Lyapunov function:

$\displaystyle{V}=V_{1}+V_{3}.$ (55)

The derivative of the above Lyapunov function is

$\displaystyle\dot{V}$
$\displaystyle\leq{-2^{\frac{\gamma+1}{2}}}\lambda_{min}\\{k_{2},k_{5}\\}V_{1}^{\frac{\gamma+1}{2}}{-2^{\frac{\iota+1}{2}}}\lambda_{min}\\{k_{3},k_{6}\\}V_{1}^{\frac{\iota+1}{2}}$
(56)
$\displaystyle{-2^{\frac{\gamma+1}{2}}}\nu_{1}V_{3}^{\frac{\gamma+1}{2}}{-2^{\frac{\iota+1}{2}}}\nu_{2}V_{3}^{\frac{\iota+1}{2}}+\sigma$
$\displaystyle\leq{-2^{\frac{\gamma+1}{2}}}\chi_{1}\left(V_{1}^{\frac{\gamma+1}{2}}+V_{3}^{\frac{\gamma+1}{2}}\right)$
$\displaystyle-{2}^{\frac{\iota+1}{2}}\chi_{2}\left(V_{1}^{\frac{\iota+1}{2}}+V_{3}^{\frac{\iota+1}{2}}\right)+\sigma$
$\displaystyle\leq{-2^{\frac{\gamma+1}{2}}}\chi_{1}\left(V^{\frac{\gamma+1}{2}}\right)-{2}^{\frac{\iota+1}{2}}\chi_{2}\left(V^{\frac{\iota+1}{2}}\right)+\sigma,$

where $\chi_{1}=\text{min}\\{\nu_{1},\lambda_{min}\\{k_{2},k_{5}\\}\\}$ and
$\chi_{2}=\text{min}\\{\lambda_{min}\\{k_{3},k_{6}\\},\nu_{2}\\}$.

Therefore, according to Lemma 2, the global fixed-time convergence of the
system is guaranteed. The settling time can be calculated as:

$\displaystyle
T\leq{\frac{2}{{\chi_{1}2^{\frac{\gamma+1}{2}}}\kappa(1-\gamma)}}+\frac{2}{{\chi_{2}}2^{\frac{\iota+1}{2}}\kappa(\iota-1)}.$
(57)

###### Remark 1

The employment of $sign$ function in (45) generates a chattering in the
system. In order to reduce the chattering, the controller (45) can be revised
as

$\displaystyle\tau$
$\displaystyle=\left(\bar{J}(\eta_{2})\bar{\Pi}\right)^{-1}\left(-F_{sum}-\frac{1}{2}s_{2}^{2}\hat{\theta}\Psi(Z)^{T}\Psi(Z)\right.$
(58)
$\displaystyle\left.-\beta_{s}(\frac{s_{2}}{||s_{2}||+\epsilon_{1}})-k_{8}s_{2}-k_{9}s_{2}^{\gamma}-k_{10}s_{2}^{\iota}\right),$

where $\epsilon_{1}$ is a small positive number.

##  V Results and Discussions

In this section, we validate the performance of the proposed algorithm. The
dynamic model of each vehicle is described as in (LABEL:D1), where the
parameters are selected as in Table I [16].

TABLE I: Parameters used in the simulation of $i$th AUV ($i\in\\{1,2,3,4\\}$) Parameters | $Value$ | Parameters | $Value$
---|---|---|---
$m_{i}$ | $20$ | $I_{x,i}$ | $20$
$I_{y,i}$ | $30$ | $I_{z,i}$ | $35$
$\iota_{\upsilon{x},i}$ | $-8$ | $\iota_{\upsilon{y},i}$ | $-10$
$\iota_{\upsilon{z},i}$ | $-9$ | $\iota_{\dot{\upsilon}{x},i}$ | $-7$
$\iota_{\dot{\upsilon}{y},i}$ | $-8$ | $\iota_{\dot{\upsilon}{z},i}$ | $-6$
$\iota_{{\omega}{x},i}$ | $-0.2$ | $\iota_{{\omega}{y},i}$ | $-0.25$
$\iota_{{\omega}{z},i}$ | $-0.15$ | $\iota_{\dot{\omega}{x},i}$ | $-20$
$\iota_{\dot{\omega}{y},i}$ | $-30$ | $\iota_{\dot{\omega}{z},i}$ | $-35$

Fig. 1 illustrates the connection between AUVs and the virtual leader, and
$\alpha_{12}=a_{21}=\alpha_{23}=\alpha_{32}=\alpha_{34}=\alpha_{43}=1$. As
illustrated in Fig. 1, in this considered communication topology, the desired
trajectory will be communicated and given to the AUV-1, i.e., $b_{1}=1$.
Therefore, the $L$ and $B$ matrices can be calculated as:

$\displaystyle L=\begin{bmatrix}1&-1&0&0\\\ -1&2&-1&0\\\ 0&-1&2&-1\\\
0&0&-1&1\end{bmatrix},B=\begin{bmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix}.$ (59)

The moving trajectory of the virtual leader is selected as
$\eta^{d}(t)=[30-30e^{-t},5t,2t,0,0,0]^{T}$. The desired posture between AUVs
are given by $\delta_{12}=[0,10,0]^{T}$, $\delta_{21}=[0,-10,0]^{T}$,
$\delta_{23}=[-10,0,0]^{T}$, $\delta_{32}=[10,0,0]^{T}$,
$\delta_{34}=[0,-10,0]^{T}$, and $\delta_{43}=[0,10,0]^{T}$. All the vehicles
have the same orientation. The relative distance between the virtual leader
and AUV 1 is $\delta_{1d}=[20,0,0]^{T}$. It is assumed that the four AUVs will
start from the initial positions: $\eta_{1}(0)=[2,3,3,0.3,0,0.2]^{T}$,
$\eta_{2}(0)=[2.5,3.5,3,0.2,0,0.25]^{T}$, $\eta_{3}(0)=[2,3,3,0.3,0,0.2]^{T}$,
$\eta_{4}(0)=[3,3,2,0.3,0,0.2]^{T}$, and
$\upsilon_{i}=0_{6\times{1}},i\in\\{1,2,3,4\\}$ is set for the initial
velocities of AUVs.

The disturbance term is assumed to be:

$\displaystyle d_{i}(t,\eta_{i},\upsilon_{i})=$
$\displaystyle[2.5\sin(t)-0.5\upsilon_{xi}^{2}-0.7\sin(\upsilon_{xi}\upsilon_{yi}),$
(60) $\displaystyle 2.5\cos(t)+0.1\upsilon_{xi}^{2}+0.5\sin(\upsilon_{yi}),$
$\displaystyle 2.5\sin(t)+0.7\upsilon_{xi}^{2}+0.8\sin(\upsilon_{zi}),$
$\displaystyle 0.5\sin(t)+0.2\upsilon_{\phi{i}}^{3},$ $\displaystyle
0.5\cos(t)-0.2\upsilon_{\theta{i}}^{2},$ $\displaystyle
0.5\sin(t)-0.4\upsilon_{\Phi{i}}^{3},]^{T},$
$\displaystyle(i\in\\{1,2,3,4\\}).$

Note that the above parameters are selected to be quite similar to the
parameters used in [16] to facilite the comparison later. However, in this
experiment, the disturbance term (60) is modeled to be more severe to include
both environmental disturbances and model uncertainties.

The selected parameters, which were chosen based on a trial-and-error
procedure, for the proposed controller in this simulation are $k_{1}=k_{8}=5$,
$k_{2}=k_{3}=k_{9}=k_{10}=0.4$, $w_{1}=w_{2}=1$, $\beta_{s}=20$. The
parameters $\gamma=5/7$ and $\iota=7/5$. The control efforts are saturared by
$\tau_{\text{max}}=300Nm$. The proposed controller uses the below membership
functions, which were tuned based on a trial-and-error procedure:

${{\mu}_{A_{i}^{1}}}=\exp\left({-{{\left({{Z}_{i}}+7\right)}^{2}}}/{4}\;\right),{{\mu}_{A_{i}^{2}}}=\exp\left({-{{\left({{Z}_{i}}+5\right)}^{2}}}/{4}\;\right),{{\mu}_{A_{i}^{3}}}=\exp\left({-{{\left({{Z}_{i}}+3\right)}^{2}}}/{4}\;\right),{{\mu}_{A_{i}^{4}}}=\exp\left({-{{\left({{Z}_{i}}+1\right)}^{2}}}/{4}\;\right),{{\mu}_{A_{i}^{5}}}=\exp\left({-{{\left({{Z}_{i}}+0\right)}^{2}}}/{4}\;\right),{{\mu}_{A_{i}^{6}}}=\exp\left({-{{\left({{Z}_{i}}-1\right)}^{2}}}/{4}\;\right),{{\mu}_{A_{i}^{7}}}=\exp\left({-{{\left({{Z}_{i}}-3\right)}^{2}}}/{4}\;\right),{{\mu}_{A_{i}^{8}}}=\exp\left({-{{\left({{Z}_{i}}-5\right)}^{2}}}/{4}\;\right),{{\mu}_{A_{i}^{9}}}=\exp\left({-{{\left({{Z}_{i}}-7\right)}^{2}}}/{4}\;\right).$

The input of the FLC is $Z_{i}=[\eta_{i},\upsilon_{i}]^{T}$. To reduce
chattering, the controller (58) is used and $\epsilon_{1}=0.01$.

In order to highlight the superior performance of the proposed controller, it
is analysed in a comparison with the distributed SMC [16]. The SMC can be
designed as in Appendix A. The sliding gain of the SMC is selected as
$\beta_{0}=200$. Note that the SMC [16] has not considered the effects of the
input saturation in the design. The tracking performances of the proposed
controller are shown in Figs. 2, 3, 4, 5, while the performances of the SMC
are shown in Figs. 6, 7, 8, 9. In particular, Fig. 2 shows the formation shape
of the four AUVs under the proposed controller. Compared with the formation
shape of the AUVs under the SMC controller shown in Fig. 6, the proposed
controller provides faster and smoother convergence, as shown in Fig. 2. Figs.
3 and 4 show the tracking errors of $\varepsilon_{1,i},(i=1,2,3,4)$ and
$\varepsilon_{2,i},(i=1,2,3,4)$ under the proposed controller, which are
convergent to zero. The errors $\varepsilon_{1,i}$ and $\varepsilon_{2,i}$
under the effects of the SMC are shown in Figs. 7 and 8, respectively.
Comparing Figs. 7 and 8 and Figs. 3 and 4, respectively, show that the SMC
provides faster convergence for the tracking errors. However, this is because
the sliding gain of the SMC was chosen to be a bigger value. Consequently, it
leads to a much higher, possibly physically unrealisable, control efforts, as
shown in Fig. 9. Fig. 5 shows the control efforts of the proposed controller,
which are smoother and bounded by the $\tau_{\text{max}}$.

Figure 1: The communication topology graph for 3 AUVs formation control Figure
2: The formation shape of four AUVs under the proposed controller Figure 3:
Position tracking error $\varepsilon_{1}={e}_{1}$ of the AUVs under the
proposed controller Figure 4: Velocity tracking error
$\varepsilon_{2}={e}_{2}$ of the AUVs under the proposed controller Figure 5:
Control efforts $\tau_{i}(i=1,2,3,4)$ of the AUVs under the proposed
controller Figure 6: The formation shape of four AUVs under the SMC controller
Figure 7: Position tracking errors $\varepsilon_{1}=e_{1}$ of the AUVs under
the SMC controller Figure 8: Velocity tracking errors $\varepsilon_{2}=e_{2}$
of the AUVs under the SMC controller Figure 9: The control efforts
$\tau_{i},(i=1,2,3,4)$ of the AUVs under the SMC controller

##  VI Conclusion

This paper has introduced a new distributed fixed-time consensus formation
control for multiple AUV systems based on an adaptive backstepping fuzzy
sliding mode control. In this scheme, a distributed consensus tracking error
was derived to form a distributed global error dynamics. Then, a fixed-time
backstepping SMC was designed to obtain a fixed-time convergence for the whole
system. However, the backstepping fixed-time SMC had two main shortcomings:
(i) it produces a greater magnitude of chattering due to the selection of
bigger sliding gain, (ii) it has not considered the input saturation effects
in the design. To overcome these shortcomings, an auxiliary variable and an
adaptive fixed-time fuzzy logic approximation have been employed. The computer
simulation on a formation consensus control for four AUVs demonstrated that
the fixed-time controller provided a quick response and stability for the
group of AUVs and can handle the input saturation problem well.

In future works, we will address the multiple constraints problem (i.e.,
working spaces constraints, states constraints and input constraints
simultaneously) and obstacle avoidance problems for multiple collaborative
AUVs. An physical experiment system based on the hardware systems of BlueROV2
robots are being developed, and the experimental results will be shown in
future works.

## Appendix A: Design Distributed sliding mode control

The SMC can be derived as follows [16]:
First, the sliding surface is selected as

$\displaystyle s=k_{1}(L+B)\bar{\varepsilon}_{1}+\bar{\varepsilon}_{2}$ (61)

where $k_{1}$ is a positive constant.

The sliding mode control can be designed as

$\displaystyle\tau$
$\displaystyle=[\bar{J}(\eta_{2})\bar{\Pi}]^{-1}\left(-\bar{\Phi}(\upsilon,\eta)\upsilon+\tau^{{}^{\prime}}\right)$
(62)

where $\tau^{\prime}$ is designed as

$\displaystyle\tau$
$\displaystyle=[\bar{J}(\eta_{2})\bar{\Pi}]^{-1}\left(-k_{1}\bar{\varepsilon}_{2}+\textbf{1}_{N}\otimes\ddot{\eta}^{d}-\beta_{0}\text{sign}(s)\right)$
(63)

To reduce the chattering, the controller (63) is revised as:

$\displaystyle\tau$
$\displaystyle=[\bar{J}(\eta_{2})\bar{\Pi}]^{-1}\left(-k_{1}\bar{\varepsilon}_{2}+\textbf{1}_{N}\otimes\ddot{\eta}^{d}-\beta_{0}\frac{s}{||s||+\epsilon_{1}}\right).$
(64)

The convergence and stability of the SMC can be referred to [16].

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