Coordinated Adaptive Control for Coordinated Pathfollowing Surface Vessels with a Timeinvariant Orbital Velocity
Ⅰ. INTRODUCTION
THE problem of cooperative motion of mobile sensor vehicles along given curves have been widely investigated by the requirement of ocean sampling^{[1, 2]}. For the purpose of enhancing both the quantity and quality of collected information,the coordinated vehicle searoutes are designed according to the spatial and temporal variability in the field. Therefore,a systematic method is urgently needed to deal with the coordinated pathfollowing control problem for a given set of desired orbits.
Early works on the coordinated pathfollowing design have focused on three popular methodologies,such as leaderfollowing strategy^{[3, 4, 5, 6, 7, 8]},virtual structure^{[9, 10]} and decoupling approach^{[11, 12]}. As we all know,most existing results concentrate on Newton$'$s particles or wheeled robots and each design methodology has its weakness. In the leaderfollowing strategy,group robustness is weak due to no information flow from the followers to the leaders. The formation in the virtual structure is rigid. The application of the decoupling approach requires an extra upgrade scheme to synchronize the parametrization states of given paths (e.g.,the parameter of each curve). In addition,when the decoupling approach is used to deal with fully actuated surface vessels,it is required to assume that the total linear speed of each vessel is nonzero all the time.
With the technological development of ocean sampling and the launching of scientific experiments^{[13, 14]},a novel geometric extension design has emerged in the coordinated pathfollowing control recently and received much attention^{[15, 16, 17, 18]}. The main idea of this approach is to extend the given curve to be a family of level curves of an orbit function. Pathfollowing is achieved through driving the value of the orbit function to the nominated value while formation motion along orbits is accomplished by forcing relative arclengths to the reference values. Zhang and his colleague first used this method to deal with unitspeed particles^{[15]} and then secondorder particles^{[16]}. Chen and Tian^{[17]} extended this method to deal with the directed topology. To maintain the same geometric topology among the extended curves and the given curve,a new geometric extension design method named concentric compression^{[18]} was proposed to deal with the convex loops and then give a solution to unicycles. A similar idea was used to control simple 3D nonholonomic aircrafts^{[19]}.
Since sensor measurements collected along repeated orbits with long,nearly straight sides could not use a complex environment model,the (near) optimal sampling trajectory for each sensor platform in ocean sampling was often designed as a convex and closed curve^{[2]}. In the event that some missions,such as the underway replenishment and the improvement of the underwater communication,request vessels in the desired attitude pattern^{[12]},we consider the navigation of formation motion of fully actuated surface vessels along a set of convex and closed orbits while maintaining attitude synchronization. The later control task (i.e.,attitude synchronization) was not involved in the mostly recent works^{[4, 5, 6, 7, 9, 11, 12, 15, 16, 17, 18, 19]}. One contribution of this paper is that we have found that some techniques developed for the coordinated pathfollowing control of unicycles in earlier work^{[18]} can be applied to deal with fully actuated surface vessels by transforming the vessel$'$s model. These include the geometric extension design and formation based on consensus technique. It must be emphasized that we introduce the potential function that is usually applied in collision avoidance^{[20]} to ensure the condition of the model transformation (that is the requirement of the nonzero total linear speed of each surface vessel all the time),which is as an assumption in the literature^{[12]}.
Compared with the existing works^{[11, 18, 19]},another distinct feature of the paper is that we remove the requirement that every vessel in the family demands to obtain the information of the reference orbital velocity,which is unreasonable in almost all environments. In this paper,the reference orbital velocity under consideration is a timeinvariant signal. To solve the signal estimation,we develop the traditional adaptive control for one system to deal with the coordinated adaptation estimation by using the consensus technology. Such technical development is nontrivial because the control system turns into the multiagent system and the estimated object is not the parameter of the vessel itself,which implies that we cannot only use the states of vessel itself to estimate the reference. Therefore,the neighbors$'$ information is used to construct the coordinated adaptation estimator,which makes our controller more scalable. The other difficulty is that the whole system is nonlinearly coupled and we cannot use the adaptive analysis method used in the multiple linear systems^{[21, 22]}. Thus the adaptive backstepping technology is adopted in this paper. In the literature,some researchers have considered adaptive formation control problems in the leaderfollowing structure. Choi et al.^{[4]} used projection algorithm to estimate the leader$'$s velocity and the information used was the leaderfollowing relative distance and bearing angle. Peng et al.^{[5, 6]} used neural network to compensate the uncertain leaders$'$ and followers$'$ parameters,respectively. This paper is different in that we combine the traditional adaptive method and consensus technique to construct the estimator,the link topology is not the leaderfollowing structure and we only demand the communication topology to be connected and at least one vessel could get the information of the reference.
This paper is organized as follows. Section II formulates the coordinated control problem along orbits. In Section III,the cooperative control is designed according to the adaptive backstepping technology. Simulation results are presented in Section IV. Conclusion is given in Section V.
Ⅱ. PROBLEM STATEMENT
A. Fully Actuated Surface Vessel$'$s Model
Consider a dynamic model,depicted in Fig. 1,whose actuators can be applied to surge,sway and yaw accelerations by using $\tau _{u_i }$,$\tau _{\upsilon _i }$ and $\tau _{r_i }$. The simplified dynamics of the $i$th fully actuated surface vessel in the inertial reference frame ${\bf W}$ are written as^{[23]}
\begin{align}
\label{E1} \left\{ {\begin{array}{*{20}l}
{\dot x_i = u_i \cos \psi _i\upsilon _i \sin \psi _i } \\
{\dot y_i = u_i \sin \psi _i + \upsilon _i \cos \psi _i } \\
{\dot \psi _i = r_i } \\
{\dot u_i = \frac{1}{{m_{u_i } }}\left( {\tau _{u_i } + m_{\upsilon _i } \upsilon _i r_id_{u_i } u_i } \right)} \\
{\dot \upsilon _i = \frac{1}{{m_{\upsilon _i } }}\left( {\tau _{\upsilon _i }m_{u_i } u_i r_id_{\upsilon _i } \upsilon _i } \right)} \\
{\dot r_i = \frac{1}{{m_{r_i } }}\left( {\tau _{r_i }m_{u\upsilon _i } u_i \upsilon _id_{r_i } r_i } \right)} \\
\end{array}} \right.
\end{align}

(1)

where ${\pmb z}_i= [x_i ,y_i]^{\rm T} $ is the position of the
$i$th vessel,$\psi _i $ denotes its yaw angle,$u_{i}$,
$\upsilon_{i}$ and $r_{i}$ are the surge,sway and yaw angle
velocities,respectively,$m'$s and $d'$s are the parameters of
surface vessel.
For the purpose of identifying the pathfollowing errors in the
next section,by referring [12] we define $\upsilon _{t_i } =
\sqrt {u_i^2 + \upsilon _i^2 } $ to be the total line speed of
surface vessel. By defining the sideslip angle as $\beta_{i} =
\arctan \left( {\frac{{\upsilon _i }}{{u_i }}} \right)$,we regard
$\psi _{t_i } = \psi _i + \beta _i $ as the total motion direction
with respect to the $x$axis of ${\bf W}$. Then the model of the
fully actuated surface vessel in the flow frame $\mathbf{F}$ is
rewritten as
\begin{align}
\label{E2}\left\{ {\begin{array}{*{20}l}
{\dot {\pmb z}_i = \upsilon _{t_i } {\pmb x}_i } \\
{\dot {\pmb x}_i = (r_i + \dot \beta _i ) {\pmb y}_i } \\
{\dot {\pmb y}_i =(r_i + \dot \beta _i ){\pmb x}_i } \\
{\dot \upsilon _{t_i } = \tau _{t_i } + g_{t_i } } \\
{\dot \beta _i = \tau _{\beta _i } + g_{\beta _i } } \\
{\dot r_i = \frac{1}{{m_{r_i } }}\tau _{r_i } + g_{r_i } } \\
\end{array}} \right.
\end{align}

(2)

where ${\pmb x}_i=\left[\cos\psi
_{t_i },\sin\psi _{t_i }\right]^{\rm T}$ denotes the orientation
of the $i$th vessel$'$s total motion,${\pmb y}_i=\left[\sin\psi
_{t_i },\cos\psi _{t_i }\right]^{\rm T}$ is the direction
perpendicular to ${\pmb x}_i$,
\begin{align}
\label{E3}\tau _{t_i } = \frac{{\cos \beta _i }}{{m_{u_i } }}\tau
_{u_i } + \frac{{\sin \beta _i }}{{m_{\upsilon _i } }}\tau
_{\upsilon _i }
\end{align}

(3)

is the total line acceleration of the $i$th vessel,
\begin{align}
\label{E4}\tau _{\beta _i } =\frac{{\sin \beta _i
}}{{\upsilon _{t_i } m_{u_i } }}\tau _{u_i } + \frac{{\cos \beta
_i }}{{\upsilon _{t_i } m_{\upsilon _i } }}\tau _{\upsilon _i }
\end{align}

(4)

is the total orientation angle acceleration of the $i$th vessel,
the parameters $\{g_{t_i },g_{\beta _i },g_{r_i }\}$ are in the
following forms:
\begin{align*}
&g_{t_i } = \left( \frac{{m_{\upsilon _i } }}{{m_{u_i } }}
\frac{{m_{u_i } }}{{m_{\upsilon _i } }}\right )\upsilon _{t_i }
r_i \sin
\beta _i \cos \beta _i\\
& ~~~~ \left( \frac{{d_{u_i } }}{{m_{u_i } }}\cos ^2 \beta _i
+ \frac{{d_{\upsilon _i } }}{{m_{\upsilon _i }
}}\sin ^2 \beta _i \right)\upsilon _{t_i },
\end{align*}
\begin{align*}
&g_{\beta _i } =\frac{{m_{\upsilon _i } }}{{m_{u_i } }}r_i +
\left( {\frac{{m_{\upsilon _i } }}{{m_{u_i } }}\frac{{m_{u_i }
}}{{m_{\upsilon _i } }}} \right)r_i \cos ^2 \beta _i+\\
&~~~~\left( {\frac{{d_{u_i }
}}{{m_{u_i } }}\frac{{d_{\upsilon _i } }}{{m_{\upsilon _i }
}}} \right)\sin \beta _i \cos \beta _i,
\end{align*}
$$g_{r_i } =\frac{{d_{r_i } }}{{m_{r_i } }} +
\frac{{m_{u\upsilon _i } }}{{m_{r_i } }}\upsilon _{t_i }^2 \sin
\beta _i \cos \beta _i.$$
From (3) and (4),the transformation
between $(\tau _{u_i },\tau _{\upsilon _i })$ and $(\tau _{t_i
},\tau _{\beta _i })$ is nonsingular if the total line speed
$\upsilon_{t_i}$ is nonzero. In this paper,$u_{t_i}$ is designed
to ensure that
\begin{align}
\label{E5} \upsilon _{t_i }(t)>0
\end{align}

(5)

for $t\geq 0$,which is regarded as an assumption in [
12]. For
a simple statement,hereinafter we directly design $(\tau _{t_i
},\tau _{\beta _i })$ to achieve the control object.
Remark 1. The inertia matrix,the matrix of coriolisand
centripetal terms,and the damping matrix for the fully actuated
surface vessel are usually not diagonal,but they also can be
translated into the model in the flow frame which is similar to (2).
To show the design process in a simple way,model (1) is used in
this paper.
B. Concentriccompressionbased Design
Consider the desired orbit $\mathcal {C}_{i0}$ for the
$i$th surface vessel to be a simple,convex and closed curve with
nonzero curvature. Similar to [18],we extend the given orbit
$\mathcal {C}_{i0}$ by concentric compression,and then a family
of level curves of an orbit function $f_i({\pmb z}_i)$ is
constructed from the given orbit $\mathcal {C}_{i0}$. Furthermore,
$f_i({\pmb z}_i)$ satisfies $\left f_i({\pmb z}_i) \right <
\varepsilon _i~(\varepsilon _i> 0)$ and its gradient satisfies
$\nabla f_i({\pmb z}_i)\ne 0,$ where symbol $\nabla$ denotes the
vector differential operator. Each level curve corresponds to a
special constant value of $f_i({\pmb z}_i)$,where the value of
$f_i({\pmb z}_i)$ associated with the given orbit $\mathcal
{C}_{i0}$ is $0$.
According to the extension of each given curve,one can obviously
see that the pathfollowing control should drive the orbit value
$f_i \left( {{\pmb z}_i } \right)$ and the orientation difference
$\alpha _i \in \left( {\pi ,\pi } \right]$ between the
vessel$'$s motion and the tangent vector $\pmb{T}_i$ to the orbit
to $0$ asymptotically,and at the same time,ensure the trajectory
of each vessel limited in set $\Omega_{i}$ (see Fig. 2),that is,
\begin{align}
\label{E6} \lim _{t \to \infty } f_i ( {{\pmb z}_i \left( t
\right)} ) = 0,
\end{align}

(6)

\begin{align}
\label{E7} \lim _{t \to \infty } \alpha _i ( t ) = 0,
\end{align}

(7)

\begin{align}
\label{E8}
\left {f_i \left( {{\pmb z}_i \left( t \right)} \right)}
\right < \varepsilon _i.
\end{align}

(8)

For the simple representation,we use the brief symbol $f_i$ to
represent $f_i \left( {{\pmb z}_i \left( t \right)} \right)$
hereinafter.
To coordinate vessels$'$ formation motion along the given orbits,
the communication is essential. Let $\mathcal
{G}=\{\mathcal{V},\mathcal{E}\}$ be the bidirectional graph
induced by the intervessel communication topology,where
$\mathcal{V}$ denotes the set of $n$ vessels and $\mathcal{E}$ is
data links among vessels. Also let $\mathcal{N}_{i}$ and
$\mathcal{L}$ denote the neighbor set of the $i$th vessel and the
set of the vessels who have access to the timeinvariant reference
orbital velocity,respectively. Throughout the paper,we assume
$\mathcal{N}_{i}$ and $\mathcal{L}$ are timeinvariant. Two
matrices,the adjacency matrix $A=[a_{ij}]$ and the Laplacian
matrix $L=[l_{ij}]$,are used to represent the graph.
The key idea of formation motion around given orbits is based on the
consensus design,which is widely applied in recent
works^{[11, 12, 17, 18, 19]}. It is said that the formation pattern
is maintained if all the generalized arclengths $\xi_{i}(t)$
defined in Assumption 1 reach consensus and their deviation
$\eta_{i}(t)={\rm d}\xi_{i}/{\rm d} t$ converge to the constant
reference $\eta^*>0$,i.e.,
\begin{align}
\label{E9}
\lim_{t\rightarrow
\infty}\left(\xi_{i}(t)\xi_{j}(t)\right)=0,
\end{align}

(9)

\begin{align}
\label{E10} \lim_{t\rightarrow \infty}\eta_{i}(t)=\eta^{*}.
\end{align}

(10)

It must be noticed that $\eta_{i}(t)$ on the given orbit reflects
the orbital velocity of vessel due to the fact that $\eta_{i}(t)$
defined in (20) is a product obtained by multiplying the velocity
of vessel and a parameter with respect to the desired formation
pattern. Therefore,$\eta_{i}(t)$ is regarded as the orbital
velocity,and $\eta^*$ is called the timeinvariant reference
orbital velocity. For example,three vessels move along the given
orbits obtained by translating the reference orbit along the
formation vector $\pmb{h}_{i}$ while maintaining the rigid
formation ${\pmb z}_{i}(t){\pmb z}_{j}(t)={\pmb h}_{i}{\pmb
h}_{j}$,as shown in
Fig. 3. If we let the starting points
$\pmb{z}_{i0}^*$ for computing the arclengths $s_i$ satisfy
${\pmb z}_{i}^*{\pmb z}_{j}^*={\pmb h}_{i}{\pmb h}_{j}$,it is
required that $s_{i}(t)=s_{j}(t)$ to ensure the nominated
formation. In the ocean exploration,the formation motion on the
planar concentric curves often attracts attention,because ocean
scales are usually similar through the upper water column
^{[2]}.
Consider a simple example of coordinated pathfollowing control
along a set of concentric ellipses
\begin{align}
\label{E11}
{\left( {\frac{{x_i }}{a}} \right)^2 + \left( {\frac{{y_i }}{b}} \right)^2 = R_i^2 }
\end{align}

(11)

with the different lengths of the semimajor and semiminor axes
given by $R_i a$ and $R_i b$. In
Fig. 4,the inline formation
can be maintained if $\xi_{i}=s_{i}/R_i$ reach consensus where the
starting point for each orbit is selected as the intersection of
the orbit with the positive $x$axis.
Assumption 1. $\xi_{i}$ is a $C^2$ smooth function of $s_i$
such that all ${\partial \xi _i }/{\partial s_i }$ are bounded,greater
than $0$,and ${\partial ^2 \xi _i }/{\partial s_i^2 }$ are uniformly
bounded for $i=1,\cdots,n$.
Some practical missions,such as underway replenishment and image
acquisition^{[12]},require the attitude synchronization to
improve the efficiency of system. In this paper,$\beta_{i}$ is
regarded as the attitude of the $i$th ship,because variant
$\beta_{i}$ with a constant linear speed causes variant swaysurge
velocity combinations. It is said that the intervessel attitude
reaches the relative desired attitude scalar
$\beta_{i}^*\in\left(\frac{\pi}{2},\frac{\pi}{2}\right)$
(attitude synchronization) when
\begin{align}
\label{E12}\lim_{t\rightarrow
\infty}\left(\beta_{i}^{a}(t)\beta_{j}^{a}(t)\right)=0,
\end{align}

(12)

where $\beta_{i}^{a}=\beta_i\beta_i^*$.
From the discussion above,the coordinated control problem along
orbits is defined as follows.
Coordinated control problem along orbits: Based on local
neighbortoneighbor information,design a pathfollowing control
for each vessel
$$\tau_{r_i}\left({\pmb z}_i,{\pmb x}_{i},\upsilon_{t_i},\alpha_{i},f_i,s_i,\xi_i,\xi_j,\eta_i;i\in\mathcal{V},j\in\mathcal{N}_{i}\right),$$
the attitude synchronization controller
$$\tau_{\beta_{i}}\left(\beta_{i},\beta_i^*,\beta_{j},\beta_j^*,\alpha_{i};i\in\mathcal{V},j\in\mathcal{N}_{i}\right),$$
and a formation control law
\begin{align*}
&\tau_{t_m}\left({\pmb z}_m,{\pmb x}_{m},\upsilon_{t_m},\alpha_{m},f_m,s_m,\xi_m,\xi_j,\eta_m,\eta_j,\eta_*;m\in\mathcal{L},\right.\\
&~~~~~~\left.j\in\mathcal{N}_{m}\right)
\end{align*}
for the vessel who has assess to the timeinvariant reference
orbital velocity,or
\begin{align*}
&\tau_{t_k}\left(\pmb{z}_k,\pmb{x}_{k},\upsilon_{t_k},\alpha_{k},f_k,s_k,\xi_k,\xi_j,\eta_k,\eta_j,\hat{\eta}_{k},\dot{\hat{\eta}}_{k};k\in{\mathcal{V}\mathcal{L}},\right.\\
&~~~~~~\left.j\in\mathcal{N}_{k}\right)
\end{align*}
for the other vessels with the coordinated differential adaptation
control
$$\dot{\hat{\eta}}_{k}\left(\xi_k,\xi_j,\eta_k,\eta_j;k\in{\mathcal{V}\mathcal{L}},j\in\mathcal{N}_{k}\right)$$
for $\eta^{*}$ such that requirements (5) $\sim$ (10) and (12) are
satisfied.
Remark 2. The information required in the coordinated
control law consists of two parts. On one hand,the states such as
$\{{\pmb z}_i,u_i,\upsilon_i,\psi_i,r_i\}$ are measured in the
inertial reference frame. Then we use them to calculate the states
$\{{\pmb x}_i,{\pmb
y}_i,\upsilon_{t_i},\beta_i,g_{t_i},g_{\beta_i},g_{r_i},\psi_{t_i}\}$
in the flow frame ${\bf F}_i$ and the values of $\{f_i,\nabla
f_i,\nabla^2f_i,\nabla^3
f_i,\sin\alpha_i,\cos\alpha_i,s_i,\frac{\partial s_i}{\partial
f_i},\frac{\partial^2 s_i}{\partial f_i^2},\xi_i,\frac{\partial
\xi_i}{\partial s_i},\frac{\partial^2 \xi_i}{\partial s_i^2}$,
$\eta_i,\beta_{i}^{a}\}$ according to the functional forms of
$\{f_i,s_i,\xi_i,\eta_i,\beta_{i}^{a}\}$. The definition of each
function can be found in Sections IIB and IIIA. On the other hand,
the information $\left\{\xi_i,\eta_i,\beta_{i}^{a}\right\}$ should
be transferred to its neighbors for the coordinated pathfollowing
control and the coordinated adaptive control. The details can be
found in Section IIIB.
Ⅲ. MAIN RESULTS
A. Coordinated Control Model
In set $\Omega_{i}$,due to the fact that the value of $f_i$ for
the given orbit $\mathcal{C}_{i0} $ is $0$,the dynamics of
position error of pathfollowing is
\begin{align}
\label{E13} \dot f_i = \nabla f_i \cdot \dot {\pmb{z}}_i =
\nabla f_i \cdot \upsilon _{t_{i}} \pmb{x}_i.
\end{align}

(13)

Let $\pmb{N}_i =\frac{{\nabla f_i }} {{\left\ {\nabla f_i }
\right\}} $ and $\pmb{T}_i = \left[{\begin{array}{*{20}c}
0 & 1 \\
{1} & 0 \\
\end{array} } \right]\pmb{N}_i$ be the normal vector and the tangent vector to each level
curve,respectively. The orientation difference $\alpha _i$
between $\pmb{x}_i$ and $\pmb{T}_i$ can be defined as
\begin{align}
\label{E14a} \cos \alpha _i = \pmb{x}_i \cdot \pmb{T}_i =
\pmb{y}_i \cdot \pmb{N}_i ,
\end{align}

(14a)

\begin{align}
\label{E14b}
\sin \alpha _i = \pmb{y}_i \cdot \pmb{T}_i ~
=\pmb{x}_i \cdot \pmb{N}_i.
\end{align}

(14b)

Consequently,using (14a) we have
\begin{align}
\label{E15} \dot f_i = \left\ {\nabla f_i } \right\\upsilon
_{t_{i}} \sin \alpha _i.
\end{align}

(15)

Note that
\begin{align}
\label{E16} \dot {\pmb {N}}_i =\frac{{\upsilon _{t_{i}} }}
{{\left\ {\nabla f_i } \right\}}\left( {\nabla ^2 f_i \pmb{x}_i
\left( {\pmb{N}_i \cdot \nabla ^2 f_i \pmb{x}_i }
\right)\pmb{N}_i } \right),
\end{align}

(16)

where $\nabla ^2 f_i$ is the Hessian matrix of $f_{i}$. The
timederivative of equation (14b) along (2) and (16) yields
\begin{align}
\label{E17}
&\dot \alpha _i \cos \alpha _i =\left(r_{i}+g_{\beta_{i}}+\tau_{\beta_{i}}\right)\cos \alpha _i+ \nonumber\\
&\quad \frac{{\upsilon _{t_{i}} }}
{{\left\ {\nabla f_i } \right\}}\left( {\pmb{x}_i \cdot \nabla
^2 f_i \pmb{x}_i + \left( {\pmb{N}_i \cdot \nabla ^2 f_i
\pmb{x}_i } \right)\sin \alpha _i } \right).
\end{align}

(17)

Considering that $\pmb{x}_i = \pmb{T}_i \cos \alpha _i
\pmb{N}_i \sin \alpha _i$,the direction error model is
\begin{align}
\label{E18} \dot \alpha _i = \upsilon _{t_{i}} \left( {k_{a_i}
\cos \alpha _i + k_{b_i} \sin \alpha _i } \right)
\left(r_{i}+g_{\beta_{i}}+\tau_{\beta_{i}}\right),
\end{align}

(18)

where
$$k_{a_i} = \frac{1}{{\left\ {\nabla f_i } \right\}}\pmb T_i \cdot \nabla ^2 f_i \pmb T_i ,$$
$$k_{b_i} =\frac{1}{{\left\ {\nabla f_i } \right\}}\pmb T_i \cdot \nabla ^2 f_i \pmb N_i. $$
To compute the total variation of the arclength,we use $\phi _i$
to reparametrize all the level curves in $\Omega_{i}$ in this
paper. Once the starting point $\pmb{z}_{i0}^{\ast}$ of $\mathcal
{C}_{i0}$ is decided,the starting point of each level curve is
chosen such that its corresponding value of the parameter of the
curve is the same as the value associated with
$\pmb{z}_{i0}^{\ast}$ (i.e.,$ \phi _{i}^*$). Due to the fact that
the change of arclength $s_{i}$ is caused not only by the motion
of the $i$th vessel projected to $\pmb{T}_i$ but also by the
motion projected to the direction of concentric compression which
changes the level curve,the arclength $s_{i}$ between
$\pmb{z}_{i}$ and the starting point of the orbit which
$\pmb{z}_{i}$ belongs to is a function $s_i \left( {f_{i},\phi_{i}
} \right)$. Furthermore,we can write
$$
s_i \left( {f_i ,\phi _i } \right) := \int_{\phi _i^* }^{\phi _i }
{\frac{{\partial s_i \left( {f_i ,\tau } \right)}} {{\partial \tau
}}}\rm{d}\tau.$$
Remark 3. Consider the parametrized level (skewed)
superellipses (including circles,ellipses,and rounded
parallelograms) of the orbit function as
\begin{align*}
& \pmb{C} \left( {f_i,\phi _i } \right)\!=\! \left( {1f_i }
\right)\left[{ {a\left( {\cos \phi _i } \right)^{\frac{1} {p}}
\!+\! \mu b\left( {\sin \phi _i } \right)^{\frac{1} {p}} },
b\left( {\sin \phi _i } \right)^{\frac{1} {p}} } \right]^{\rm{T}},
\end{align*}
where $\mu$ is the skew parameter,the semimajor axis length $a$
and semiminor axis length $b$ satisfy $a\geq b > 0$ and the
parameter $p~(p=1,3,5,\cdots)$ determines the corner sharpness. For
$\mu=0$ and $a>b$ (resp. $a$ = $b$),setting $p=1$ yields an
ellipse (resp. circle) and setting $p\geq 3$ yields a rounded
rectangle (resp. rounded square). Setting $\mu\neq0$ and $p>1$
yields a rounded parallelogram. The arclength $s_i(f_i,\phi _i)$
measured from the starting points (corresponding to the parameter
$\phi _i^\ast$) is
\begin{align*}
&{s_i}({f_i},{\phi _i}) = \left( {1{f_i}} \right)\int_{\phi _i^*}^{{\phi _i}}
\left[\left(\frac{a}{p}\sin {\tau}{{\left( {\cos {\tau}} \right)}^{\frac{{1p}}{p}}} + \right. \right. \\
&\quad\left.\left. \frac{{\mu b}}{p}\cos {\tau}{{\left( {\sin
{\tau}} \right)}^{\frac{{1p}}{p}}} \right)^2 + {{\left(
{\frac{b}{p}\cos {\tau}{{\left( {\sin {\tau}} \right)}^{\frac{{1
p}}{p}}}} \right)}^2} \right]^{\frac{1}{2}}{\rm{d}}\tau.
\end{align*}
When each vessel moves,the dynamics of $s_i$ is
\begin{align*}
\label{E23} &\dot s_i = \frac{{\partial s_i }} {{\partial \phi _i
}}\dot \phi_i + \frac{{\partial s_i }} {{\partial f_i }}\dot f_i
= \\
&~~~~\frac{{{\rm d} s_i }} {{{\rm d}t}}\left {_{\pmb{z}_i = \rm
const} } \right. + \frac{{\partial s_i }}
{{\partial f_i }}\left\ {\nabla f_i } \right\\upsilon _{t_i} \sin \alpha _i=\\
&~~~~ \dot {\pmb{z}}_i \cdot \pmb{T}_i + \frac{{\partial s_i }}
{{\partial f_i }}\left\ {\nabla f_i } \right\\upsilon _{t_i}
\sin
\alpha _i=\\
& ~~~~ \upsilon _{t_i} \left( {\cos \alpha _i + \frac{{\partial
s_i }} {{\partial f_i }}\left\ {\nabla f_i } \right\\sin \alpha
_i } \right).
\end{align*}
Then the total variation of generalized arclength is
\begin{align}
\dot \xi _i =\frac{\partial \xi _i }{\partial
s_i}\dot{s}_i= \frac{{\partial \xi _i }} {{\partial s_i }}\upsilon
_{t_i} \left( {\cos \alpha _i + \frac{{\partial s_i }} {{\partial
f_i }}\left\ {\nabla f_i } \right\\sin \alpha _i } \right).
\end{align}

(19)

In the next subsection,we will design the pathfollowing
controller to drive each vessel to move along its given orbit,
which implies $\dot \xi _i = \frac{{\partial \xi _i }} {{\partial
s_i }}\upsilon _{t_i}$ as $\alpha_{i}=0$. Therefore,the orbital
velocity $\eta_{i}$ is defined as
\begin{align}
\label{E20} \eta _i = \frac{{\partial \xi _i }} {{\partial s_i
}}\upsilon _{t_i}.
\end{align}

(20)

From (20),we rewrite $\dot{\xi}_i(t)$ as
\begin{align}
\label{E21} \dot \xi _i = \eta _i +d_{\eta_i},
\end{align}

(21)

where $d_{\eta_i}=2\eta _i \sin ^2 \left( {\frac{{\alpha _i
}}{2}} \right) + \eta _i \frac{{\partial s_i }}{{\partial f _i
}}\left\ {\nabla f _i } \right\\sin {\alpha _i } $. Taking the
time derivative of $\eta_{i}$,we have
\begin{align}
\label{E22} \dot \eta _i = \frac{{\partial \xi _i }} {{\partial
s_i }}\left(g_{t_{i}}+\tau_{t_{i}}\right) + \frac{{\partial ^2
\xi _i }} {{\partial s_i^2 }}\upsilon _{t_i}^2 \left( {\cos \alpha
_i + \frac{{\partial s_i }} {{\partial f_i }}\left\ {\nabla f_i
} \right\\sin \alpha _i } \right).
\end{align}

(22)

Since $\beta_{i}$ is regarded as the attitude of the $i$th vessel,
the dynamics of $\beta_{i}^{\alpha}$ is
\begin{align}
\dot {\beta}_{i}^{\alpha} = g_{\beta_{i}} + \tau_{\beta_{i}} .
\end{align}

(23)

To sum up,the coordinated control model for each vessel is
summarized as follows:
\begin{align}
\label{E24a}
\dot f_i = \left\ {\nabla f_i } \right\\upsilon _{t_i} \sin \alpha _i,
\end{align}

(24a)

\begin{align}
\label{E24b}
\dot \alpha _i = \upsilon _{t_i} \left( {k_{a_i}
\cos \alpha _i + k_{b_i} \sin \alpha _i } \right)
\left(r_{i}+g_{\beta_{i}}+\tau_{\beta_{i}}\right),
\end{align}

(24b)

\begin{align}
\label{E24c}
\dot \xi _i = \eta _i +d_{\eta_i},
\end{align}

(24c)

\begin{align}
\label{E24d}
\dot \eta _i = \frac{{\partial \xi _i }} {{\partial s_i
}}\left(g_{t_{i}}+\tau_{t_{i}}\right) + \frac{{\partial ^2 \xi _i
}} {{\partial s_i^2 }}\upsilon _{t_i}^2 \left( {\cos \alpha _i +
\frac{{\partial s_i }} {{\partial f_i }}\left\ {\nabla f_i }
\right\\sin \alpha _i } \right),
\end{align}

(24d)

\begin{align}
\label{E24e}
\dot {\beta}_{i}^{\alpha} = g_{\beta _i } + \tau _{\beta
_i }.
\end{align}

(24e)

B. Controller Design
Since only a portion of the vessels belonging
to set $\mathcal {L}$ have access to the reference constant state
$\eta^*$,an estimate $\hat{\eta}_k$ is used to replace $\eta^*$
when the $k$th vessel in set $\mathcal {V}\mathcal {L}$. The
parameter error $\varphi_k$ is defined as
\begin{align}
\label{E25}
\varphi_k=\eta^*\hat{\eta}_k.
\end{align}

(25)

In the following,we show the design process of the adaptive
backstepping design.
Step 1. Convergence of
$f_{i},~\alpha_{i},~\xi_{i}\xi_{j},~\eta_{m}\eta^*,~\eta_{k}\hat{\eta}_k,
~\beta_{i}^{\alpha}\beta_{j}^{\alpha},~\varphi_k$. We start with
the following Lyapunov function:
\begin{align}
\label{E26}& V_I = \sum_{i = 1}^n {\left( {h_i \left( {f_i }
\right)\ln \left( {\cos ^2 \frac{{\alpha _i }} {2}} \right) }
\right)} + \nonumber \\
&\quad\frac{k_0} {4}\sum_{i = 1}^n { {\sum_{j = 1}^n {a_{ij}
\left( {\xi _i\xi _j } \right)^2
} } }+\frac{1} {4}\sum_{i = 1}^n { {\sum_{j = 1}^n {a_{ij}
\left( {\beta_{i}^{\alpha}\beta_{j}^{\alpha} } \right)^2} }
}\nonumber+ \\
&\quad\sum_{m\in\mathcal {L}}\left(
{\ln\left(\frac{{\eta}_{m}}{{\eta}^{*}}\right)+\frac{{\eta}^{*}}{{\eta}_{m}}1}\right)\nonumber+
\\
&\quad\sum_{k\in\{\mathcal {V}\mathcal {L}\}}\left(
{\ln\left(\frac{{\eta}_{k}}{{\hat{\eta}}_k}\right)+\frac{{\hat{\eta}}_k}{{\eta}_{k}}1}\right)+\frac{1}{2}\sum_{k\in\{\mathcal
{V}\mathcal {L}\}}\varphi_k^2,
\end{align}

(26)

where $k_0>0$,and $h_i \left( {f_i } \right)$ is a $C^2$ smooth
nonnegative function on $\left( {\varepsilon _i ,\varepsilon _i
} \right)$ and satisfies the following conditions.
Condition 1.
$h_i \left( {f_i } \right) \to + \infty$ and
$\nabla h_i =\partial h_i/\partial f_i \to\infty$ as $f_i \to
\varepsilon _i$.
Condition 2. $h_i \left( {f_i } \right) \to + \infty$ and
$\nabla h_i \to +\infty$ as $f_i \to \varepsilon _i$.
Condition 3. $h_i \left( {f_i } \right)=0$ if and only if
$f_i=0$.
There are many functions that satisfy all the above properties of
$h_i \left( {f_i } \right)$. One example is
\begin{align*}
&h_i \left( f_i\right) =\int_{f_i^*}^{f_i}\left[c_1\left(\frac{1}
{{\varepsilon _i\tau}}\frac{1} {{\varepsilon _i + \tau}}\right)\right.+\\
& \qquad \left.c_2\left( \ln \left( {\varepsilon _i + \tau}
\right)\ln \left( {\varepsilon _i\tau}
\right)\right)\right]\text{d}\tau,
\end{align*}
where $f_i^*=f_i(\pmb{z}_i(0))\in \Omega_i$ and $c_1,c_2>0$.
The first term in (26) contributes to guiding each vessel to move
to its given orbit and stay in $\Omega_i$ when it starts from
$\Omega_i$. The second term forces the difference between the
direction of each vessel$'$s motion and the tangent vector to the
orbit to be $0$. The next two terms ensure the agreement of the
generalized arclengths and harmonize the relative attitude. The
fifth term guarantees that $\eta_m$ converges to the reference
state $\eta^*$ and $\upsilon_{t_m}=\left(\frac{\partial
\xi_m}{\partial s_m}\right)^{1}\eta_m(t)\neq 0$ all the time,
where $m\in\mathcal {L}$. The last two terms have the same effect
as the fifth term with estimate $\hat{\eta}^*_k$ approaching to
${\eta}^*$ when $k\in\{\mathcal {V}\mathcal {L}\}$.
Note that $\dot{\varphi}_k=\dot{\hat{\eta}}_k$,then the time
derivation of $V_{I}$ is
\begin{align}
\label{E27}
\dot V_I & = \sum_{i=1}^{n} \tan\frac{\alpha _i }{2}\left( {\Delta _{r_i }r_i } \right)+\sum_{i = 1}^n \Delta _{\beta _i}\left( {g_{\beta _i } + \tau _{\beta _i } } \right) \nonumber +\\
&\sum_{m\in\mathcal {L}} \left( \eta _m{\eta}^{*} \right)\left( {\Delta _{\tau _{tm} } + \frac{1}{{\eta _m^2 }}\frac{{\partial \xi _m }}{{\partial s_m }}\left(\tau _{t_m }+g_{t_m}\right) } \right) \nonumber+ \\
& \sum_{k\in\{\mathcal {V}\mathcal {L}\}} \left( {\eta _k{\hat{\eta}}_k } \right)\left( \Delta _{\tau _{tk} }+\frac{1}{{\eta _k^2 }}\frac{{\partial \xi _k }}{{\partial s_k }}\left(\tau _{t_k }+g_{t_k}\right) \right)\nonumber\\
&\sum_{k\in\{\mathcal
{V}\mathcal {L}\}}{\varphi}_k\left(
{\dot{\hat{\eta}}}_k+k_0\sum_{j=1}^{n}a_{kj}(\xi_k\xi_j)\right),
\end{align}

(27)

where
\begin{align}
\label{E28}
&\Delta _{r_i } = \upsilon _{t_i } \left( k_{a_i} \cos \alpha _i + k_{b_i} \sin \alpha _i + 2\nabla h_i \left\ {\nabla f_i } \right\\cos ^2 \frac{\alpha _i }{2}\right) \nonumber +\\
&\quad k_0 {\eta}_i \left( {\sin \alpha _i + 2\frac{{\partial s_i }}{{\partial f_i }}\left\ {\nabla f_i } \right\\cos ^2 \frac{{\alpha _i }}{2}} \right)\sum_{j = 1}^n {a_{ij} \left( {\xi _i\xi _j }
\right)},
\end{align}

(28)

\begin{align}
\label{E29} \Delta _{\beta _i}= \sum_{j = 1}^n {a_{ij}\left( \beta
_{i}^{\alpha}\beta_{j}^{\alpha}\right)\tan \frac{\alpha _i
}{2} },
\end{align}

(29)

\begin{align}
\label{E30} &\Delta _{\tau _{tm} } = \frac{{\partial ^2 \xi _m
}}{{\partial s_m^2 }}\frac{{\upsilon _{t_m }^2 }}{{\eta _m^2 }}
\left( {\cos \alpha _m + \frac{{\partial s_m }}{{\partial f_m
}}\left\ {\nabla f_m } \right\\sin \alpha _m } \right)\nonumber+ \\
&\quad k_0\sum_{j = 1}^n a_{mj} \left( {\xi _m\xi _j }
\right),
\end{align}

(30)

\begin{align}
\label{E31} &\Delta _{\tau _{tk} } = \frac{{\partial ^2 \xi _k
}}{{\partial s_k^2 }}\frac{{\upsilon _{t_k }^2 }}{{\eta _k^2 }}
\left( {\cos \alpha _k + \frac{{\partial s_k }}{{\partial f_k
}}\left\ {\nabla f_k } \right\\sin \alpha _k } \right)\nonumber+ \\
&\quad k_0\sum_{j = 1}^n a_{kj} \left( {\xi _k\xi _j }
\right)\frac{\dot{\hat{\eta}}_k}{\eta_k\hat{\eta}_k}.
\end{align}

(31)

Firstly,$r_{i}$ is regarded as a virtual control $\bar{r}_{i}$
and combined with $\tau _{t _i }$ and $\tau _{\beta _i }$ together
for the purpose of making $\dot V_I\leq 0$. Let the virtual
pathfollowing control be
\begin{align}
\label{E32} \bar r_i = \Delta _{r_i } + k_1 \sin \frac{{\alpha
_i }}{2},
\end{align}

(32)

the attitude synchronization control be
\begin{align}
\label{E33} \tau _{\beta _i } =g_{\beta _i }k_2 \Delta
_{\beta _i},
\end{align}

(33)

and the formation control law be
\begin{align}
\label{E34} &\tau _{t_m } =g_{t_m }\eta _m^2 \left(
{\frac{{\partial \xi _m }}{{\partial s_m }}} \right)^{1} \left(
\Delta _{\tau _{tm} } + k_3 \left( {\eta _m{\eta} ^{*} }
\right)\right.
\nonumber +\\
&\quad \left. k_4\sum_{j=1}^n a_{mj}\left(\eta_m\eta_j\right)
\right).
\end{align}

(34)

If the $m$th vessel has access to the reference $\eta^*$,and
\begin{align}
\label{E35} &\tau _{t_k } =g_{t_k }\eta_k^2 \left(
{\frac{{\partial \xi _k }}{{\partial s_k }}} \right)^{1} \left(
\Delta _{\tau _{tk} } + k_3 \left( {\eta _k{\hat{\eta}}_k }
\right)\right. \nonumber+ \\
&\quad \left. k_4\sum_{j=1}^n
a_{kj}\left(\eta_k\eta_j\right)\right),
\end{align}

(35)

for $k\in\{\mathcal {V}\mathcal {L}\}$,where
$k_{1},~k_{2},~k_{3},~k_4$ are some positive constants. Substituting
(32) $\sim$ (35) into (27) results in
\begin{align}
\label{E36}
\dot V_I & =k_1 \sum\limits_{i = 1}^n {\frac{{\sin ^2 \frac{{\alpha _i }}{2}}}{{\cos \frac{{\alpha _i }}{2}}}}k_2 \sum\limits_{i = 1}^n \Delta
_{\beta _i} ^2k_3 \sum\limits_{m \in\mathcal {L}} {\left( {\eta _m\eta ^* } \right)^2 } \nonumber\\
& k_3 \sum\limits_{k \in \left\{ {\mathcal {V}\mathcal {L}} \right\}}^n {\left( {\eta _k\hat \eta _k } \right)^2 }k_4 (\pmb{\eta}\eta^*\pmb{1}_n)^{\rm
T}
L(\pmb{\eta}\eta^*\pmb{1}_n)\nonumber\\
& \sum\limits_{k \in \left\{ {\mathcal {V}\mathcal {L}} \right\}} {\varphi_k \left( \dot {\hat{ \eta}} _k +\Delta_{\varphi_k}
\right)},
\end{align}

(36)

where $\pmb{\eta}=\left[\eta_1,\cdots,\eta_n\right]^{\rm T}$,${\pmb
1}_n=[1,\cdots,1]^{\rm T}$,and
\begin{align}
\label{E37}\Delta_{\varphi_k}=k_0
\sum_{j=1}^{n}a_{kj}(\xi_k\xi_j)+k_4 \sum\limits_{j = 1}^n a_{kj}
\left( {\eta _k\eta _j } \right).
\end{align}

(37)

With the appropriate choice of $\dot {\hat{ \eta}} _k$ we can
discard the indefinite term $\varphi_k\Delta_{\varphi_k}$. Thus
the adaptive update law is
\begin{align}
\label{E38} \dot {\hat{ \eta}} _k =\Delta_{\varphi_k},
\end{align}

(38)

which yields $\dot V_I\leq 0$.
To accomplish controller $\tau_{r_{i}}$,we introduce the error
variable
\begin{align}
\label{E39} r_{e_{i}}=r_{i}\bar r_i
\end{align}

(39)

that should be
driven to 0,and we rewrite (27) as
\begin{align*}
\label{E40}
{{\dot V}_I}& = \sum\limits_{i = 1}^n {\tan \frac{{{\alpha _i}}}{2}
\left( {{\Delta _{{r_i}}}{\bar r}_ir_{e_{i}}} \right)} + \notag\\
&\sum\limits_{i = 1}^n {\Delta _{{\beta _i}}}\Big( {\tau _{{\beta _i}}}
\frac{{{m_{{\upsilon _i}}}}}{{{m_{{u_i}}}}}({{\bar r}_i} +{r_{{e_i}}})+ \nonumber\\
& ({{\bar r}_i} + {r_{{e_i}}})\left( {\frac{{{m_{{\upsilon _i}}}}}{{{m_{{u_i}}}}}\frac{{{m_{{u_i}}}}}{
{{m_{{\upsilon _i}}}}}} \right){\cos ^2}{\beta _i} + \left( {\frac{{{d_{{u_i}}}}}{{{m_{{u_i}}}}}
\frac{{{d_{{\upsilon _i}}}}}{{{m_{{\upsilon _i}}}}}} \right)\nonumber\times\\
& { \sin {\beta _i} \cos {\beta _i}} \Big) + \sum\limits_{m \in
\mathcal L} \left( {{\eta _m}{\eta ^*}} \right)\left( {\Delta
_{{\tau _{tm}}}} + \frac{1}{{\eta _m^2}}\frac{{\partial {\xi
_m}}}{{\partial {s_m}}}\left( {\tau _{{t_m}}} + \right.\right.
\end{align*}
\begin{align}
&({{\bar r}_m} + {r_{{e_m}}})\left( {\frac{{{m_{{\upsilon _m}}}}}{{{m_{{u_m}}}}}\frac{{{m_{{u_m}}}}}{{{m_{{\upsilon _m}}}}}} \right){\upsilon _{{t_m}}}\sin {\beta _m}\cos {\beta _m}\nonumber\\
&\left( {\frac{{{d_{{u_m}}}}}{{{m_{{u_m}}}}}{{\cos }^2}{\beta _m}} {\left. {\left. { + \frac{{{d_{{\upsilon _m}}}}}{{{m_{{\upsilon _m}}}}}{{\sin }^2}{\beta _m}} \right){\upsilon _{{t_m}}}} \right)} \right) +\notag\\
& \sum\limits_{k \in \left\{ {\mathcal V\mathcal L} \right\}} \left( {{\eta _k}{{\hat \eta }_k}} \right)\nonumber\times\\
& \left( {{\Delta _{{\tau _{tk}}}} + } \right. \frac{1}{{\eta _k^2}}\frac{{\partial {\xi _k}}}{{\partial {s_k}}}\left( {{\tau _{{t_k}}} + ({{\bar r}_k} + {r_{{e_k}}})\left( {\frac{{{m_{{\upsilon _k}}}}}{{{m_{{u_k}}}}}\frac{{{m_{{u_k}}}}}{{{m_{{\upsilon _k}}}}}} \right)} \right. {\upsilon _{{t_m}}}\times \nonumber\\
&\left. {\sin {\beta _k}\cos {\beta _k}\left( {\frac{{{d_{{u_k}}}}}{{{m_{{u_k}}}}}{{\cos }^2}{\beta _k} + \frac{{{d_{{\upsilon _k}}}}}{{{m_{{\upsilon _k}}}}}{{\sin }^2}{\beta _k}} \right){\upsilon _{{t_k}}}} \right)\nonumber\\
& \sum\limits_{k \in \left\{ {\mathcal V\mathcal L} \right\}}
{{\varphi _k}\left( {{{\dot {\hat \eta} }_k} + {k_0}\sum\limits_{j
= 1}^n {{a_{kj}}\left( {{\xi _k}{\xi _j}} \right)} } \right)}.
\end{align}

(40)

Substituting (32) $\sim$ (35) and (38) into (40) results in
\begin{align*}
{{\dot V}_I}&=k_1 \sum\limits_{i = 1}^n {\frac{{\sin ^2 \frac{{\alpha _i }}{2}}}{{\cos \frac{{\alpha _i }}{2}}}}k_2 \sum\limits_{i = 1}^n \Delta
_{\beta _i} ^2k_3 \sum\limits_{m \in\mathcal {L}} {\left( {\eta _m\eta ^* } \right)^2 }\nonumber \\
& k_3 \sum\limits_{k \in \left\{ {\mathcal {V}\mathcal {L}} \right\}}^n
{\left( {\eta _k\hat \eta _k } \right)^2 }k_4 (\pmb{\eta}
\eta^*\pmb{1}_n)^{\rm T}
L(\pmb{\eta}{\eta}^*\pmb{1}_n)+ \nonumber\\
& \sum\limits_{m \in \mathcal {L}} {r_{e_m } \Delta _{r_{em} }}+\sum\limits_{k \in \left\{ {\mathcal {V}\mathcal {L}} \right\}}{r_{e_k } \Delta _{r_{ek}
}},
\end{align*}
where
\begin{align}
\label{E43}
\Delta _{r_{em} } & = \left( {\eta _m{\eta} ^{*} } \right)\frac{1}{{\eta _m^2 }}\frac{{\partial \xi _m }}{{\partial s_m }}\left( {\frac{{m_{\upsilon _m } }}{{m_{u_m } }}\frac{{m_{u_m } }}{{m_{\upsilon _m } }}} \right)\upsilon _{t_m } \sin \beta _m \times \nonumber \\
& \cos \beta_m\tan\frac{\alpha _m }{2}+\notag\\
&\Delta
_{\beta _m}\left(\frac{{m_{\upsilon _m } }}{{m_{u_m }
}}+\left( {\frac{{m_{\upsilon _m } }}{{m_{u_m } }}\frac{{m_{u_m
} }}{{m_{\upsilon _m } }}} \right)\cos ^2 \beta _m\right),
\end{align}

(41)

\begin{align}
\label{E44}
\Delta _{r_{ek} } & = \left( {\eta _k\hat{\eta} _k } \right)\frac{1}{{\eta _k^2 }}\frac{{\partial \xi _k }}{{\partial s_k }}\left( {\frac{{m_{\upsilon _k } }}{{m_{u_k } }}\frac{{m_{u_k } }}{{m_{\upsilon _k } }}} \right)\upsilon _{t_k } \sin \beta _k \cos \beta_k\nonumber \\
& \tan\frac{\alpha _k }{2}+ \Delta
_{\beta _k}\left( {\frac{{m_{\upsilon _k } }}{{m_{u_k } }} +
\left( {\frac{{m_{\upsilon _k } }}{{m_{u_k } }}\frac{{m_{u_k }
}}{{m_{\upsilon _k } }}} \right)\cos ^2 \beta _k }
\right).
\end{align}

(42)

Step 2. Backstepping for $r_{e_{i}}$. Consider a second
control Lyapunov function given by
\begin{align}
\label{E41} V_{II} = V_I + \frac{1}{2}\sum\limits_{i = 1}^n
{r_{e_i }^2 }.
\end{align}

(43)

Differentiating (41) along the solutions of (32) $\sim$ (35) and
(38),we have
\begin{align}
\label{E42} \dot{V}_{II}=&k_1 \sum\limits_{i = 1}^n
{\frac{{\sin ^2 \frac{{\alpha _i }}{2}}}{{\cos \frac{{\alpha _i
}}{2}}}}k_2 \sum\limits_{i = 1}^n \Delta
_{\beta _i} ^2k_3 \sum\limits_{m \in\mathcal {L}} {\left( {\eta _m\eta ^* } \right)^2 } \nonumber\\
& k_3 \sum\limits_{k \in \left\{ {\mathcal {V}\mathcal {L}} \right\}}^n {\left( {\eta _k\hat \eta _k } \right)^2 }k_4 (\pmb{\eta}\eta^*\pmb{1}_n)^{\rm
T}
L(\pmb{\eta}{\eta}^*\pmb{1}_n) \nonumber+\\
&\sum\limits_{m \in \mathcal {L}} {r_{e_m }\left(\Delta _{r_{em}
}+g_{r_{m}}+\frac{1}{m_{r_{m}}}\tau_{r_{m}}\dot{\bar{r}}_{m}\right)}\nonumber +\\
&\sum\limits_{k \in \left\{ {\mathcal {V}\mathcal {L}} \right\}} {r_{e_k }\left(\Delta _{r_{ek}
}+g_{r_{k}}+\frac{1}{m_{r_{k}}}\tau_{r_{k}}\dot{\bar{r}}_{k}\right)}.
\end{align}

(44)

We set
\begin{align}
\label{E45} \tau_{r_{i}}=\frac{1}{m_{r_{i}}}\left(\Delta _{r_{ei}
}+g_{r_{i}}\dot{\bar{r}}_{i}+k_{5}r_{e_{i}}\right),
\end{align}

(45)

where $k_{5}>0$. Then the time derivative of $V_{II}$ becomes
\begin{align}
\label{E47} \dot{V}_{II}=&k_1 \sum\limits_{i = 1}^n
{\frac{{\sin ^2 \frac{{\alpha _i }}{2}}}{{\cos \frac{{\alpha _i
}}{2}}}}k_2 \sum\limits_{i = 1}^n \Delta
_{\beta _i} ^2k_3 \sum\limits_{m \in\mathcal {L}} {\left( {\eta _m\eta ^* } \right)^2 } \nonumber\\
& k_3 \sum\limits_{k \in \left\{ {\mathcal {V}\mathcal {L}} \right\}}^n {\left( {\eta _k\hat \eta _k } \right)^2 }k_4 (\pmb{\eta}\eta^*\pmb{1}_n)^{\rm
T}
L(\pmb{\eta}{\eta}^*\pmb{1}_n)\nonumber\\
& k_5 \sum\limits_{i = 1}^n
{r_{e_i }^2}\leq 0.
\end{align}

(46)

C. Stability Analysis
Under the coordinated controllers (33) $\sim$ (35),(38) and (45),
the closedloop equation of $f_{i}$ is denoted as
\begin{align}
\label{E48a}
\dot f_i = \left\ {\nabla f_i } \right\\upsilon _{t_i} \sin \alpha
_i.
\end{align}

(47a)

The closedloop equation of $\alpha_{i}$ is
\begin{align}
\label{E48b} \dot \alpha _i = \upsilon _{t_i } \left( {k_{ai}
\cos \alpha _i + k_{bi} \sin \alpha _i } \right)
+k_2\Delta_{\beta_i}r_i
\end{align}

(47b)

with the dynamics of $r_i$ satisfying (32). The closedloop
equation of the relative generalized arclength is
\begin{align}
\label{E48c}
&\dot { {\xi}} _i\dot { {\xi}} _j = \eta _i +d_{\eta_i}\eta _j
d_{\eta_j}.
\end{align}

(47c)

The closedloop equation of $\eta _m~(m\in\mathcal {L})$ satisfies
\begin{align}\label{E48d}\hskip2mm\dot{\eta} _m =&{\eta}
_m^{2}\left[\left( {\cos \alpha _m \!+\! \frac{{\partial s_m
}}{{\partial f_m }}\left\ {\nabla f_m } \right\\sin \alpha
_m } \right)\sum_{j = 1}^n a_{mj}\times \right.\nonumber\\
&\left. \left(\xi _m\xi _j\right)+ k_2 \left( {\eta _m\eta^* }
\right) +k_3\sum_{j=1}^n a_{mj}(\eta _m\eta _j) \right].
\end{align}

(47d)

The closedloop equation of ${\eta} _k~(k\in\{\mathcal
{V}\mathcal {L}\})$ is
\begin{align}
\label{E48e}\dot{\eta} _k =& {\eta}
_k^{2}\left[\frac{\dot{\hat{\eta}}_k}{\eta_{k}\hat{\eta}_k}\left(
{\cos \alpha _k \!+\! \frac{{\partial s_k }}{{\partial f_k }}\left\
{\nabla f_k } \right\\sin
\alpha _k } \right)\sum_{j = 1}^n a_{kj}\times\right.\nonumber\\
& \left. \left( \xi _k\xi _j \right)k_2 \left( {\eta _k
\hat{ \eta}_k } \right)k_3\sum_{j=1}^n a_{ij}(\eta _k\eta _j)
\right].
\end{align}

(47e)

And the closedloop equation of the relative attitude is
\begin{align}
\label{E48f} &\dot {\beta}_{i}^{\alpha}\dot
{\beta}_{j}^{\alpha} =k_2 \left( {\sum\limits_{j = 1}^n
{a_{ij} \left( {\beta _{i}^{\alpha}
\beta_{j}^{\alpha} } \right)\tan \frac{\alpha _i }{2} } } \right)\nonumber+ \\
&~~~~~~~~ k_2\left( {\sum\limits_{i = 1}^n {a_{ji} \left( {\beta_{j}^{\alpha}\bar
\beta_{i}^{\alpha} } \right)\tan \frac{\alpha _j }{2} } }
\right).
\end{align}

(47f)

Theorem 1. Consider a set of level curves of orbit function
constructed from each given orbit through concentric compression
design. Assume the generalized arclengths meet Assumption 1 and
the reference state $\eta^{*}$ is a timeinvariant signal that is
greater than $0$. Suppose the initial conditions of fully actuated
surface vessels make the initial value of $V_{II}$ given in (41)
finite. Then the coordinated control problem along orbits is
solved via the pathfollowing control (45),the attitude
synchronization control (33) and the formation control (34),(35)
with the coordinated adaptation control (38),if the communication
topology is connected and at least one vessel has access to
$\eta^{*}$.
Proof. Set $\Phi = \{(
f_{i},\alpha_{i},\xi_{i}\xi_{j},\eta_{m}\eta^*,\eta_{k}\hat{\eta}_k,
\beta_{i}^{\alpha}\beta_{j}^{\alpha},\varphi_k,r_{e_{i} }) 
V_{II} \le c \}$ for $c>0$ is closed by continuity. Since $\alpha
_i$ is defined in $\left(\pi,\pi\right]$,$\left {f_i }
\right < \varepsilon_i$ due to the boundedness of $V_{II}$,$\left
{\xi _i\xi _j } \right \le \sqrt {4c/k_0 } $ ,
$\leftr_{e_{i}}\right\le \sqrt {2c }$ ,$\left
{\beta_{i}^{\alpha}\beta_{j}^{\alpha} } \right \le \sqrt {4c
}$ ,$\left\varphi_k\right\le \sqrt {2c }$ ,$\left \eta _m
\right \le h_{\eta _m } ^{1} (c )+{\eta} ^*$ where $m\in\mathcal {L}$,and $h_{\eta
_m}={\ln\left(\eta_{m}/{\eta}^{*}\right)+{\eta}^{*}/\eta_{m}1}$,
$\left \eta _k \right \le h_{\eta _k } ^{1} (c )+\hat{\eta}
_k\le h_{\eta _k } ^{1} (c )+\sqrt {2c }+{\eta} ^*$,where
$k\in\{\mathcal {V}\mathcal {L}\}$ and $h_{\eta
_k
}={\ln\left(\eta_{k}/\hat{\eta}_k\right)+\hat{\eta}_k/\eta_{k}1}$,
set $\Phi _I $ is compact. On the compact set $\Phi _I $,$\partial s_i( {f_i ,\phi _i })/{\partial f_i }$ and
$\partial ^2 s_i({f_i ,\phi _i})/{\partial f_i^2 }$ are bounded
because $\phi _i \in \left[{0,2\pi } \right)$. $\left\ {\nabla f_i } \right\$ is bounded by
continuity. Since ${\partial \xi _i}/{\partial s_i }$ is bounded
and greater than $0$,$\upsilon _{t_i} = \left( {\frac{{\partial
\xi _i }} {{\partial s_i }}} \right)^{1} \eta _i $ is also
bounded on $\Phi
$. Thus the closedloop system (47) is Lipschitz continuous on set $\Phi
$,
and a solution exists and is unique. Notice that the value of
$V_{II}$ is timeindependent and nonincreasing,the entire
solution stays in $\Phi $ when the initial value of $V_{II}$ is
finite. Thus we conclude that (8) is tenable by Conditions 1 and
2,and $\upsilon _{t_i}$ is always nonzero due to that $\eta_i>0$
all the time,which implies that (5) is satisfied. Applying the
invariancelike theorem,it follows that the trajectories of
closedloop system will converge to the set inside the region $E =
\{ (
f_{i},\alpha_{i},\xi_{i}\xi_{j},\eta_{m}\eta^*,\eta_{k}\hat{\eta}_k,\beta_{i}^{\alpha}\beta_{j}^{\alpha},\varphi_k,r_{e_{i}
})\left {\dot V_{II} = 0} \right. \}$,that is,
\begin{align}
\label{E49} \alpha _i = 0,{\beta}_i^{\alpha}{\beta}_j^{\alpha}=0,
\eta _m = \eta ^* ,\eta _k =\hat{\eta}_k ,\eta _i=\eta _j,
r_{e_{i}}=0,
\end{align}

(48)

as $t\rightarrow\infty$. From (48),it is obvious that
$\hat{\eta}_k=\eta^*$,because $\eta _m = \eta ^* $,$\eta _k
=\hat{\eta}_k$ and $\eta _i=\eta _j$. By noticing that
$r_{i}\rightarrow \bar{r}_{i}$ as $r_{e_{i}}\rightarrow 0$,on set
$E$,the closedloop system becomes
\begin{align}
\label{E50a} \dot f_i = 0,
\end{align}

(49a)

\begin{align}
\label{E50b} \dot \alpha _i =2\nabla h_i \upsilon _{t_i }
\left\ {\nabla f_i } \right\2k_0\eta ^* \frac{{\partial s_i
}}{{\partial f_i }}\left\ {\nabla f_i } \right\\sum\limits_{j =
1}^n {a_{ij} \left( {\xi _i\xi _j } \right)},
\end{align}

(49b)

\begin{align}
\label{E50c} \dot \xi _i\dot \xi _j = \eta _i\eta _j,
\end{align}

(49c)

\begin{align}
\label{E50d}\dot \eta _i =(\eta ^*)^{2} k_0\sum_{j =
1}^n {a_{ij} \left( {\xi _i\xi _j }
\right)},
\end{align}

(49d)

\begin{align}
\label{E50e}\dot{ \beta }_i^{\alpha}\dot
{\beta }_j^{\alpha} = 0.
\end{align}

(49e)

Firstly,we will show that $\xi _i\xi _j \to 0$ as
$t\rightarrow\infty$. Since (49c) implies $\dot \xi _i\dot \xi
_j = \eta _i\eta _j = 0$ on set $E$,$\xi _i\xi _j $ is a
constant. Then from (49d) $\dot \eta _i $ is uniformly
continuous. By the extension of the Barbalat lemma^{[24]},$\dot
\eta _i \rightarrow 0$ as $t\rightarrow\infty$,that is,$(\eta
^*)^{2}\sum_{j = 1}^n {a_{ij} \left( {\xi _i\xi _j }
\right)}=0$. Due to $\eta ^*\neq 0$,one has $L\xi=0$. If the
communication graph is connected,we have $\xi _i\xi _j \to 0$
as $t\rightarrow\infty$.
Next,we will prove $f_i \to 0$ as $t\rightarrow\infty$. Since
$\xi _i\xi _j = 0$ on set $E$,from (49b) one has
\begin{align}
\label{E51} \dot \alpha _i =2\nabla h_i \upsilon _{t_i }
\left\ {\nabla f_i } \right\.
\end{align}

(50)

It is easy to check that
$\lim_{t\rightarrow\infty}\upsilon_{t_i}=(\partial \xi/\partial
s_i)\eta^*> 0$ is uniformly continuous and bounded from Assumption
1. From (49a),$f_i$
tends to be a constant and thus $\nabla h_i $ tends
to be a constant. Therefore,$2\nabla h_i \upsilon
_{t_i} \left\ {\nabla f_i }
\right\$ is uniformly continuous. According to (50),we have
$\dot{\alpha_{i}} \to 0$ as $t\rightarrow\infty$ based on the
extension of the Barbalat lemma
^{[24]}. Thanks to $ {\lim }_{t
\to \infty } \upsilon _{t_{i}} \left\ {\nabla f_i } \right\ \ne
0$,$\nabla h_i \to 0$ as $t\rightarrow\infty$. By Condition
3,$f_i$ tends to be $0$.
Ⅳ. SIMULATION RESULTS
In this section,we apply the proposed control law to coordinating
four fully actuated surface vessels moving along a set of given
orbits with a desired formation and attitude pattern. The
communication topology is shown in Fig. 5 and the first vessel
has access to the reference state $\eta^*=0.5$.
The given orbits for vessels are concentric ellipses (11) where
$a=3$,$b=2$,$R_i= 1 + 0.5\left( {i1} \right)~(i=1,2,3,4)$. We
demand the family of vessels to form a trapezia formation pattern
and achieve attitude synchronization. From the desired pattern,we
select the generalized arclengths as $\xi _i = s_i/R_i
+\Delta_{s_i}$,where $\Delta_{s_1}=\Delta_{s_4}=0$,
$\Delta_{s_2}=\pi/6$,$\Delta_{s_3}=\pi/12$,and the starting
point is the intersection of the orbit with the positive
semimajor axis. We choose the desired attitude scalars as
$\beta_{i}^{*}=0$. The reference orbital velocity is
$\eta^{*}=0.5$. The mode and control parameters are selected as
$m_{u_{i}}=24,~m_{\upsilon_{i}}=34,~m_{r_{i}}=3,~m_{u\upsilon_{i}}=1,
~d_{u_{i}}=2,~d_{\upsilon_{i}}=7,~d_{r_{i}}=0.5$,
$k_{j}=10~(j=0,\cdots ,5)$.
The movement of vessels is shown in Fig. 6 (a). From this
figure,we can see that four vessels finally move along the set of
given loops drawn by the real lines,form the desired formation
denoted by the dashed lines and achieve attitude synchronization
when the system reaches the steady state. The pathfollowing
errors $f_{i}$ and $\alpha_{i}$ tend to be 0 as plotted in
Figs. 6 (b) and 6 (c),respectively. Fig. 6 (d) demonstrates
that $\xi_{i}$ reaches consensus,and Fig. 6 (e) shows
$\eta_{i}$ converges to the reference value. The relative
attitudes ${\beta}_{i}^{\alpha}{\beta}_{j}^{\alpha}$ converge to
0 as shown in Fig. 6 (f). According to these figures,we can
conclude that the coordinated control problem along orbits can be
solved via our proposed controller.
Ⅴ. CONCLUSION
In this paper,we develop our
previous concentriccompressionbased design^{[18]} to deal with
coordinated pathfollowing and attitude synchronization control of
multiple fully actuated surface vessels. By using the potential
function,we remove the assumption of nonzero total line velocity
of each vessel. A coordinated adaptive scheme is given to solve
the problem that only a portion of vessels in the team have access
to the reference timeinvariant orbital velocity,which makes our
proposed control more scalable and distributed. In our ongoing
work,we seek to improve our method in order to deal with the
reference timevariant orbital velocity.