# Bearing-Constrained Leader-Follower Formation of Single-Integrators with
Disturbance Rejection: Adaptive Variable-Structure Approaches

Thanh Truong Nguyen, Dung Van Vu, Tuynh Van Pham, Minh Hoang Trinh T. T.
Nguyen is with Bosch Global Software Technologies, Hanoi, Vietnam. E-mail:
<EMAIL_ADDRESS>V. Vu is with Unmanned Aerial Vehicle Center,
Viettel High Technology Industries Corporation, Hanoi, Vietnam. E-mail:
<EMAIL_ADDRESS>V. Pham is with Department of Automation
Engineering, School of Electrical and Electronic Engineering, Hanoi University
of Science and Technology (HUST), Hanoi, Vietnam. E-mail:
<EMAIL_ADDRESS>H. Trinh is with AI Department, FPT University,
Quy Nhon City, Binh Dinh, Vietnam. Corresponding author. E-mail:
<EMAIL_ADDRESS>
###### Abstract

This paper studies the problem of stabilizing a leader-follower formation
specified by a set of bearing constraints and being disturbed by unknown
uniformly bounded disturbance. A set of leaders are positioned at the desired
positions, while each follower agent is modeled by a single integrator with
disturbance of which the upper bound is unavailable for the control design.
Adaptive variable-structure formation control laws using only displacements or
bearing vectors are provided to stabilize the agents to a desired formation.
Thanks to the adaptive mechanism, the control laws require neither the
information of the bearing Laplacian nor the directions and the upper bounds
of the disturbance. It is further proved that when the leaders are moving with
the same bounded uniformly continuous velocity, the moving target formation
can still be achieved under the proposed control laws. Simulation results are
also given to support the stability analysis.

###### Index Terms:

adaptive control, variable-structure control, formation control, bearing
rigidity, bearing-only measurements

##  I Introduction

In the last decade, formation control has attracted much interest from the
robotics and control systems communities [1, 2]. Formations of unmanned robots
have been proposed for civilian and military applications such as the
platooning truck lineup, the squad of small and medium-sized drones for
highway monitoring, error checking in solar cell fields, supporting crop
farming, search and rescue operations, unmanned diving equipment for seabed
mapping exploration, or satellite formations for positioning and remote
sensing applications. In addition, research results from formation control are
also applicable to its dual problem - sensor network localization problem.

Let the agents be positioned in the $d$-dimensional space ($d\geq 2$) with an
arbitrary shape called an initial configuration. The formation control problem
focuses on designing control rules for each agent to reach a desired formation
predefined by a set of geometric constraints. Each agent (automatic guided
vehicles, unmanned aerial vehicles, unmanned underwater vehicles,…) is
hypothesized to be fully controlled by its own inner loop controller.
Therefore, in the formation control problem, each agent can be described by a
simple single integrator model. An essential requirement in formation control
is that the control law must be decentralized/distributed [3]. To this end,
each agent is assumed to be an independent system capable of measuring and
communicating (via wireless communication) with other agents on some geometric
variables. These variables can be the position in the global coordinated
frame, the relative position, the distance, the relative angle and the bearing
(direction) vector between nearby agents. Based on the variables measured and
controlled by each agent, formation control are divided into position-,
displacement-, distance-, bearing-, and angle-based controls, etc… The
formation control problem’s level of complexity is inversely proportional to
the amount of information each agent can obtain, measure, and exchange.

Currently, formation control laws based on only bearing vectors (directional
information) are getting more attention. The research is inspired from the
observation that animals can self-localize, navigate, and perform formation-
typed collective behaviors solely by their vision. Researches suggests that
that fairly simple visual-based guidance rules used by animals (fishes or
birds) can unfold these sophisticated phenomena [4]. The bearing vectors can
be obtained from an agent-mounted camera, which provides information about the
relative direction between the agents in the swarm. Compared with the
remaining schemes, the bearing-based control reduces the number of sensors
used by each agent as well as the requirement for a global reference system.
Furthermore, using the camera, which is a passive sensor, makes this solution
suitable for military applications with prohibited signal transmission [5, 6,
7].

The theoretical basis of the bearing-based formation control algorithms in
$d$-dimensional space ($d\geq 2$), was developed in [8, 9, 10, 11, 12]. It is
worth mentioning that some initial studies on the bearing/directional rigidity
theory in the two- or three-dimensional space can be found in [13, 14, 15, 16,
17, 18, 19]. Another development of bearing rigidity theory considering the
difference in the reference system, also known as the rigidity theory in
$SE(d)$, was developed in [20, 21, 22, 23]. Recently, the angle rigidity
theory has also been proposed by different research groups, typically the
works [24, 25, 26, 27]. As robustness is an importance issue of any multi-
agent systems, there have been several works considering consensus and
formation control under disturbances [28, 29, 30]. In the context of bearing-
constrained formation control, the authors in [26, 31] considered the angle-
only formation control and tracking problem, however, the interaction between
agents has to be a Laman triangulated graph. Although disturbances/noises can
be actively included for additional objectives such as escaping from an
undesired unstable formation [6], or formation maneuver [32], the presence of
unknown disturbances usually makes the target formation unachievable or causes
undesired formation’s motions. Several control strategies for bearing-
constrained robust formation acquisition/tracking have recently been proposed
in the literature [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. However, the
works [33, 34, 35, 36, 39, 37, 41, 42, 43] assumed either that the leaders’
velocity and the disturbance’s magnitude are constant, or their upper bounds
are known by the agents. The work [44] proposed an elevation-based bearing-
only formation control with disturbance rejection for single- and double-
integrators. However, the method in [44] works only with minimally rigid
formations. The recent work [45] studied bearing-only formation control with
fault tolerant and time delays. The faults are modeled as a disturbance of
unknown control direction, which can be compensated by a control action with
an appropriate control gain. The authors in [46] proposed a robust adaptive
design method to attenuate the effects of the disturbances to a specific
performance requirement. [40] considered the bearing-only formation tracking
problem with unknown leaders’ velocity. However, [40] considered only
formation with directed acyclic leader-follower topology. A finite-time
bearing-only control law is proposed based on the inverse of the minimum
eigenvalue of the sum of the local projection matrices. Formation tracking via
bearing-only estimation for time-varying leader-follower formations was also
proposed in [47, 48, 49, 50, 51].

This paper focuses on the bearing-based leader-follower formation control
problems with single-integrator modeled agents perturbed unknown and bounded
uniformly continuous disturbances. By bearing-based, we assume that the
geometric constraints which define the target formation are given as a set of
bearing vectors. There are several leaders in the formation, whose positions
already satisfy a subset of bearing constraints. The remaining agents, called
followers, can measure either (i) the relative positions (displacement-based
control) or (ii) the bearing vector (bearing-only control) to their neighbors.
The interaction topology between agents is not restricted into an acyclic
graph, but applicable to any infinitesimal bearing rigid formation having at
least two leaders.

Unlike [52, 53], where a disturbance-free finite-time bearing-only formation
controls were studied or a small adaptive perturbation was purposely included
to globally stabilize the target formation in finite time, the unknown
disturbances (or perturbation) in this work are originated from unmodeled
dynamics or the outside environment. The problem is firstly solved under the
assumption that the agents can measure the relative displacements. The
solution for relative-displacement gives hints for the more challenging task
of stabilizing the desired formation using when agents can only sense the
relative directions (bearing vectors). Intuitively, since no information on
the distances is available, in order to suppress the disturbances with unknown
magnitude, an adaptive gain included to the usual bearing-only control law
should be increased whenever all bearing constraints are still not satisfied.
This intuition is mathematically realized by adaptive variable-structure
control, which can provide fast convergence and robustness with regard to
disturbances [54, 55, 56]. The main novelty of the proposed control laws is
providing a distributed adaptive mechanism which alters the magnitude of the
control law with regard to the errors of the desired and the actual bearing
constraints. By doing so, the control input eventually over-approximates the
magnitude of the disturbance, rejects the disturbance and stabilizes the
target formation without requiring any inter-agent communication nor a-priori
information on the upper bound of the disturbance and the formation’s rigidity
index.111Specifically, the smallest eigenvalue of the grounded bearing
Laplacian is not needed for stabilizing the formation under unknown
disturbances. Modifications of the control laws are proposed to alter the
adaptive gains upon the disturbance’s magnitude or to stabilize the target
formation in case the upper bound of the unknown disturbance is a polynomial
of the formation’s error. Moreover, when the leaders move with the same
bounded uniformly continuous velocity, their motions also act as disturbances
to the followers’ model. Thus, the proposed adaptive control laws can also
make agents to achieve the desired moving formation. As a result, for
formations of single-integrators, the proposed control laws give a unified
solution to two problems: leader-follower formation control with unknown
disturbance rejection and formation tracking with unknown leaders’ velocity.

The rest of this paper is organized as follows. Section II presents
theoretical background on bearing rigidity theory and formulates the problems.
Sections III and IV propose the formation control/tracking laws using only
displacements and/or only bearing vectors, respectively. Section VI provides
numerical examples. Lastly, section VII concludes the paper.

_Notations._ In this paper, the set of real numbers is denoted by
$\mathbb{R}$. Scalars are denoted by small letters, and vectors (matrices) are
denoted by bold-font small (capital) letters. For a matrix $\mathbf{A}$, we
use $\text{ker}(\mathbf{A})$, $\text{im}(\mathbf{A})$ to denote the kernel and
the image of $\mathbf{A}$, and rank$(\mathbf{A})$ denotes the rank of
$\mathbf{A}$.

##  II Problem statement

###  II-A Bearing rigidity theory

Consider a set of $n$ points in $d$-dimensional space ($n\geq 2,~{}d\geq 2$)
positioned at $\mathbf{p}_{i}\in\mathbb{R}^{d}$, with
$\mathbf{p}_{i}\neq\mathbf{p}_{j},~{}\forall i\neq j,~{}i,j\leq n$. A
framework in the $d$-dimensional space (also known as a formation)
($\mathcal{G},\mathbf{p}$) is given by an undirected graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ (where $\mathcal{V}$ is the vertex set
of $|\mathcal{V}|=n$ vertices and $\mathcal{E}$ is the edge set of
$|\mathcal{E}|=m$ edges) and a configuration
$\mathbf{p}=\text{vec}(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\in\mathbb{R}^{dn}$.
The neighbor set of a vertex $i\in\mathcal{V}$ is defined by
$\mathcal{N}_{i}=\\{j\in\mathcal{V}|~{}(i,j)\in\mathcal{E}\\}$. The graph
$\mathcal{G}$ is connected if for any two vertices $i,j\in\mathcal{V}$, we can
find a sequence of vertices connected by edges in $\mathcal{E}$, which starts
from $i$ and ends at $j$.

Let the edges in $\mathcal{E}$ be indexed as $e_{1},\ldots,e_{m}$. For each
edge $e_{k}=(i,j)\in\mathcal{E},~{}k=1,\ldots,|\mathcal{E}|=m$, the bearing
vector pointing from $\mathbf{p}_{i}$ to $\mathbf{p}_{j}$ is defined by
$\mathbf{g}_{ij}\equiv\mathbf{g}_{k}=\frac{\mathbf{z}_{ij}}{\|\mathbf{z}_{ij}\|}$,
with $\mathbf{z}_{ij}\equiv\mathbf{z}_{k}=\mathbf{p}_{j}-\mathbf{p}_{i}$ is
the displacement vector between $i$ and $j$. It is not hard to check that
$\|\mathbf{g}_{ij}\|=1$, where $\|\cdot\|$ denotes the 2-norm. An edge
$e_{k}=(i,j)$ is oriented if we specify $i$ and $j$ as the start and the end
vertices of $e_{k}$, respectively. According to an arbitrarily indexing and
orienting of edges in $\mathcal{E}$, we can define a corresponding incidence
matrix $\mathbf{H}=[h_{ki}]\in\mathbb{R}^{m\times n}$, where $h_{ki}=-1$ if
$i$ is the start vertex of $e_{k}$, $h_{ki}=+1$ if $i$ is the end vertex of
$e_{k}$, and $h_{ki}=0$, otherwise. Then, we can define the stacked
displacement vector
$\mathbf{z}=[\ldots,\mathbf{z}_{ij}^{\top},\ldots]^{\top}=\text{vec}(\mathbf{z}_{1},\ldots,\mathbf{z}_{m})=\bar{\mathbf{H}}\mathbf{p}$,
where $\bar{\mathbf{H}}=\mathbf{H}\otimes\mathbf{I}_{d}$.

For each bearing vector $\mathbf{g}_{ij}\in\mathbb{R}^{d}$, we define a
corresponding projection matrix
$\mathbf{P}_{\mathbf{g}_{ij}}=\mathbf{I}_{d}-\mathbf{g}_{ij}\mathbf{g}_{ij}^{\top}$.
The projection matrix $\mathbf{P}_{\mathbf{g}_{ij}}$ is symmetric positive
semidefinite, with a unique zero eigenvalue and the remaining eigenvalues are
1. Moreover, the kernel of $\mathbf{P}_{\mathbf{g}_{ij}}$ is spanned by
$\mathbf{g}_{ij}$, i.e.,
ker$(\mathbf{P}_{\mathbf{g}_{ij}})=$im$(\mathbf{g}_{ij})$.

Two formations ($\mathcal{G},\mathbf{p}$) and
($\mathcal{G},\mathbf{p}^{\prime}$) are bearing equivalent if and only if:
$\mathbf{P}_{\mathbf{g}_{ij}}(\mathbf{p}_{j}^{\prime}-\mathbf{p}_{i}^{\prime})=\mathbf{0}_{d},~{}\forall(i,j)\in\mathcal{E}$.
They are bearing congruent if and only if
$\mathbf{P}_{\mathbf{g}_{ij}}(\mathbf{p}_{j}^{\prime}-\mathbf{p}_{i}^{\prime})=\mathbf{0}_{d},~{}\forall
i,j\in\mathcal{V},i\neq j$. A formation ($\mathcal{G},\mathbf{p}$) is called
globally bearing rigid if any formation having the same bearing constraints
with $(\mathcal{G},\mathbf{p})$ is bearing congruent with
$(\mathcal{G},\mathbf{p})$. Let
$\mathbf{g}=\text{vec}(\mathbf{g}_{1},\ldots,\mathbf{g}_{m})\in\mathbb{R}^{dm}$,
the bearing rigidity matrix is defined by

$\displaystyle\mathbf{R}_{b}(\mathbf{p})=\frac{\partial\mathbf{g}}{\partial\mathbf{p}}=\text{blkdiag}\left(\frac{\mathbf{P}_{\mathbf{g}_{k}}}{\|\mathbf{z}_{k}\|}\right)\bar{\mathbf{H}}\in\mathbb{R}^{dm\times
dn}.$

A formation is infinitesimally bearing rigid in $\mathbb{R}^{d}$ if and only
if $\text{rank}(\mathbf{R}_{b})=dn-d-1$, this means
$\text{ker}(\mathbf{R}_{b})=\text{im}([\mathbf{1}_{n}\otimes\mathbf{I}_{d},\mathbf{p}-\mathbf{1}_{n}\otimes\bar{\mathbf{p}}])$,
where
$\bar{\mathbf{p}}=\frac{1}{n}(\mathbf{1}^{\top}_{n}\otimes\mathbf{I}_{d})\mathbf{p}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{p}_{i}$
is the formation’s centroid. An example of infinitesimally bearing rigid
framework is shown in Figure 1.

(a)

(b)

Figure 1: An infinitesimally bearing rigid framework
$(\mathcal{G},\mathbf{p}^{*})$ in $\mathbb{R}^{3}$. (a) the graph raph
$\mathcal{G}$; (b) a desired configuration $\mathbf{p}^{*}$ where
$\mathbf{p}_{i}^{*},i=1,\ldots,20,$ are located at the vertices of a
dodecahedron.

In bearing-based formation, we usually use an augmented bearing rigidity
matrix
$\tilde{\mathbf{R}}_{b}=\text{blkdiag}(\|\mathbf{z}_{k}\|\otimes\mathbf{I}_{d})\mathbf{R}_{b}=\text{blkdiag}(\mathbf{P}_{\mathbf{g}_{k}})\bar{\mathbf{H}}$,
which has the same rank as well as the same kernel as $\mathbf{R}_{b}$ but
does not contain information of the relative distances between the agents
$\|\mathbf{z}_{k}\|$. Further, we define the bearing Laplacian
${\mathbf{L}}_{b}(\mathbf{p})=\tilde{\mathbf{R}}_{b}^{\top}\tilde{\mathbf{R}}_{b}$
which is symmetric, positive semidefinite. For an infinitesimally rigid
formation, $\tilde{\mathbf{L}}_{b}$ has exactly $d+1$ zero eigenvalues and
ker$(\tilde{\mathbf{L}}_{b})=$ker$(\mathbf{R}_{b})$.

###  II-B Problem formulation

Consider a system of $n$ agents in the $d$-dimensional space ($d\geq 2$), of
which the positions are given by
$\mathbf{p}_{1},\ldots,\mathbf{p}_{n}\in\mathbb{R}^{d}$. We assume that there
exist $2\leq l<n$ stationary leader agents in the formation and the remaining
$f=n-l$ agents are followers. Suppose that the axes of the local reference
frames of $n$ agents are aligned.

Defining the vectors $\mathbf{p}^{\rm
L}=[\mathbf{p}_{1}^{\top},\ldots,\mathbf{p}_{l}^{\top}]^{\top}\in\mathbb{R}^{dl}$
and $\mathbf{p}^{\rm
F}=[\mathbf{p}_{l+1}^{\top},\ldots,\mathbf{p}_{n}^{\top}]^{\top}\in\mathbb{R}^{df}$.
Since the leaders are stationary, $\dot{\mathbf{p}}^{\rm L}=\mathbf{0}_{dl}$.

The follower agents are modeled by single integrators in the $d$-dimensional
space with the disturbances:

$\displaystyle\dot{\mathbf{p}}_{i}(t)=\mathbf{u}_{i}(t)+\mathbf{d}_{i}(t),\,i=l+1,\ldots,n,$
(1)

where $\mathbf{p}_{i}$ and $\mathbf{d}_{i}$ denote the position and the
disturbance of agent $i$, respectively. The disturbance vector
$\mathbf{d}(t)=[\mathbf{d}_{1}(t)^{\top},\ldots,\mathbf{d}_{n}(t)^{\top}]^{\top}$
is bounded and uniformly continuous. The upper bound of the disturbance is
denoted as $\sup_{t\geq 0}\|\mathbf{d}(t)\|_{\infty}=\beta>0$, and it is
unknown to the agents.

The desired formation $(\mathcal{G},\mathbf{p}^{*})$, where
$\mathbf{p}^{*}=\text{vec}(\mathbf{p}^{*}_{1},\ldots,\mathbf{p}^{*}_{n})$, is
defined as follows:

######  Definition 1 (Desired formation).

The desired formation satisfies

  1. (i)

Leaders’ positions: $\mathbf{p}_{i}^{*}=\mathbf{p}_{i},~{}\forall
i=1,\ldots,l$, and

  2. (ii)

Bearing constraints:
$\mathbf{g}_{ij}^{*}=\frac{\mathbf{p}_{j}^{*}-\mathbf{p}_{i}^{*}}{\|\mathbf{p}_{j}^{*}-\mathbf{p}_{i}^{*}\|},\;\forall(i,j)\in\mathcal{E}$.

It is assumed that the formation $(\mathcal{G},\mathbf{p}^{*})$ is
infinitesimally bearing rigid in $\mathbb{R}^{d}$. By stacking the set of
desired bearing vectors as
$\mathbf{g}^{*}=[\ldots,(\mathbf{g}_{ij}^{*})^{\top},\ldots]^{\top}$, we have

$\displaystyle\mathbf{L}_{b}(\mathbf{p}^{*})\mathbf{p}^{*}$
$\displaystyle=\mathbf{L}_{b}(\mathbf{g}^{*})\mathbf{p}^{*}$
$\displaystyle=\begin{bmatrix}\mathbf{L}_{ll}(\mathbf{g}^{*})&\mathbf{L}_{lf}(\mathbf{g}^{*})\\\
\mathbf{L}_{fl}(\mathbf{g}^{*})&\mathbf{L}_{ff}(\mathbf{g}^{*})\end{bmatrix}\begin{bmatrix}\mathbf{p}^{L}\\\
\mathbf{p}^{\rm F*}\end{bmatrix}=\mathbf{0}_{dn}$
$\displaystyle\Longleftrightarrow\mathbf{L}_{fl}(\mathbf{g}^{*})\mathbf{p}^{L}$
$\displaystyle+\mathbf{L}_{ff}(\mathbf{g}^{*})\mathbf{p}^{\rm
F*}=\mathbf{0}_{d{(n-l)}}.$ (2)

Under the assumption that $(\mathcal{G},\mathbf{p}^{*})$ is infinitesimally
bearing rigid in $\mathbb{R}^{d}$ and $l\geq 2$, it has been shown in [9] that
$\mathbf{L}_{ff}(\mathbf{p}^{*})$ is invertible. Thus, the desired formation
is uniquely determined from the the leaders’ positions and the bearing vectors
by $\mathbf{p}^{\rm
F*}=-(\mathbf{L}_{ff}(\mathbf{g}^{*}))^{-1}\mathbf{L}_{fl}(\mathbf{g}^{*})\mathbf{p}^{L}$.

To achieve a target formation, the agents need to sense some geometric
variables relating to the formation. Two types of relative sensing variables,
namely, the displacements
$\mathbf{z}_{ij}=\mathbf{p}_{j}-\mathbf{p}_{i},~{}\forall
j\in\mathcal{N}_{i}$, and the bearing vectors
$\mathbf{g}_{ij}=\frac{\mathbf{p}_{j}-\mathbf{p}_{i}}{\|\mathbf{p}_{j}-\mathbf{p}_{i}\|}$,
$\forall j\in\mathcal{N}_{i}$, will be considered in this paper.

######  Problem 1.

Let the follower agents be modeled by (1) and the sensing variables are the
relative displacements. Design control laws for agents such that
$\mathbf{p}(t)\to\mathbf{p}^{*}$ as $t\to\infty$.

######  Problem 2.

Let the agents be modeled by (1) and the sensing variables are the bearing
vectors. Design control laws for agents such that
$\mathbf{p}(t)\to\mathbf{p}^{*}$ as $t\to\infty$.

##  III Displacement-based formation control

In this section, we consider the bearing-based formation control under
disturbance when the agents can measure the displacement vectors with regard
to their neighbors. We begin with an adaptive variable structure control law
which can provide asymptotic convergence of the target formation. Then, we
modify the adaptive variable structure control law to deal with different
assumptions of the disturbances as well as the control objectives.

###  III-A Proposed control law

Consider the Problem 1, the control law is proposed as follows

$\displaystyle\mathbf{u}_{i}$
$\displaystyle=-\sum_{j\in\mathcal{N}_{i}}\gamma_{ij}\mathbf{P}_{\mathbf{g}^{*}_{ij}}\text{sign}(\mathbf{q}_{ij}),\,i=l+1,\ldots,n,$
(3a) $\displaystyle\mathbf{q}_{ij}$
$\displaystyle={\mathbf{P}_{\mathbf{g}^{*}_{ij}}(\mathbf{p}_{i}-\mathbf{p}_{j})},$
(3b) $\displaystyle\dot{\gamma}_{ij}$
$\displaystyle=\kappa\left|\left|\mathbf{q}_{ij}\right|\right|_{1},~{}\forall(i,j)\in\mathcal{E},$
(3c)

where, corresponding to each edge $e_{k}=(i,j)$, the matrix
$\mathbf{P}_{\mathbf{g}^{*}_{ij}}=\mathbf{I}_{d}-\mathbf{g}^{*}_{ij}(\mathbf{g}^{*}_{ij})^{\top}$
can be computed from the desired bearing vector
$\mathbf{g}^{*}_{ij}\equiv\mathbf{g}_{k}^{*}$, the scalar $\gamma_{ij}$ are
adaptive gains, which satisfy $\gamma_{ij}(0)>0$, and $\kappa>0$ is a positive
constant. As the leaders are stationary, $\mathbf{u}_{i}=\mathbf{0}_{d}$ for
$i=1,\ldots,l$. In the following analysis, we will denote $\mathbf{u}^{\rm
L}=\text{vec}(\mathbf{u}_{1},\ldots,\mathbf{u}_{l})=\mathbf{0}_{dl}$,
$\mathbf{u}^{\rm F}=\text{vec}(\mathbf{u}_{l+1},\ldots,\mathbf{u}_{n})$,
$\mathbf{u}=\text{vec}(\mathbf{u}_{1},\ldots,\mathbf{u}_{n})=\text{vec}(\mathbf{u}^{\rm
L},\mathbf{u}^{\rm F})$,
$\boldsymbol{\gamma}=[\ldots,\gamma_{ij},\ldots]^{\top}=[\gamma_{1},\ldots,\gamma_{m}]^{\top}$,
$\boldsymbol{\Gamma}=\text{diag}(\boldsymbol{\gamma})$, and
$\bar{\boldsymbol{\Gamma}}=\boldsymbol{\Gamma}\otimes\mathbf{I}_{d}$.

The system under the proposed control law (3) can be expressed in the
following form:

$\displaystyle\dot{\mathbf{p}}$
$\displaystyle=-\bar{\mathbf{Z}}\left((\tilde{\mathbf{R}}_{b}(\mathbf{p}^{*}))^{\top}\bar{\boldsymbol{\Gamma}}\text{sign}\left(\text{blkdiag}(\mathbf{P}_{\mathbf{g}^{*}_{k}})\bar{\mathbf{H}}\mathbf{p}\right)-\mathbf{d}\right),$
(4a) $\displaystyle\dot{\boldsymbol{\gamma}}$
$\displaystyle=\kappa\left[\|\mathbf{q}_{1}\|_{1},\ldots,\|\mathbf{q}_{m}\|_{1}\right]^{\top},$
(4b)

where $\mathbf{Z}=\begin{bmatrix}\mathbf{0}_{l\times l}&\mathbf{0}_{l\times
f}\\\ \mathbf{0}_{f\times l}&\mathbf{I}_{f}\end{bmatrix}$ and
$\bar{\mathbf{Z}}=\mathbf{Z}\otimes\mathbf{I}_{d}$. For brevity, we will use
the short-hands
$\tilde{\mathbf{R}}_{b}(\mathbf{p}^{*})=\tilde{\mathbf{R}}_{b}(\mathbf{g}^{*})=\tilde{\mathbf{R}}_{b}^{*}$,
$\mathbf{L}_{b}(\mathbf{g}^{*})=\mathbf{L}_{b}^{*}$, and
$\mathbf{L}_{ff}(\mathbf{g}^{*})=\mathbf{L}_{ff}^{*}$ in the subsequent
analysis.

###  III-B Stability analysis

In this subsection, we will analyse the system (4). Since the right-hand-side
of Eq. (4)(a) is discontinuous, the solution of (4)(a) is understood in
Fillipov sense [57]. We will prove that $\mathbf{p}(t)$ converges to
$\mathbf{p}^{*}$ as $t\to+\infty$ under the proposed control law (3).

######  Lemma 1.

Consider the Problem 1. If
$\gamma_{ij}(0)>\gamma_{0}:=\beta\sqrt{\frac{dn}{\lambda_{\min}(\mathbf{L}_{ff}^{*})}}$,
$\forall(i,j)\in\mathcal{E}$. Under the control law (3),
$\mathbf{p}(t)\to\mathbf{p}^{*}$ in finite time.

###### Proof:

Let
$\boldsymbol{\delta}=\mathbf{p}-\mathbf{p}^{*}=\text{vec}(\mathbf{0}_{dl},\mathbf{p}^{\rm
F}-\mathbf{p}^{\rm F*})$, and consider the Lyapunov function
$V=\frac{1}{2}\|\boldsymbol{\delta}\|^{2}$, which is positive definite,
radially unbounded, and bounded by two class $\mathcal{K}_{\infty}$ functions
$\frac{h_{1}}{2}\|\boldsymbol{\delta}\|^{2}$ and
$\frac{h_{2}}{2}\|\boldsymbol{\delta}\|^{2}$, for any $0<h_{1}<1<h_{2}$. Then,
$\dot{V}\in^{a.e}\dot{\tilde{V}}=\bigcup_{\boldsymbol{\xi}\in\partial
V}\boldsymbol{\xi}^{\top}\text{K}[\dot{\boldsymbol{\delta}}]$, where $\partial
V=\\{\boldsymbol{\delta}\\}$. It follows that

$\displaystyle\dot{{V}}$
$\displaystyle=\boldsymbol{\delta}^{\top}\bar{\mathbf{Z}}\left(-(\tilde{\mathbf{R}}_{b}^{*})^{\top}\bar{\boldsymbol{\Gamma}}\text{K}[\text{sign}]\big{(}\text{blkdiag}(\mathbf{P}_{\mathbf{g}^{*}_{k}})\mathbf{z}\big{)}+\mathbf{d}\right)$
$\displaystyle=-\boldsymbol{\delta}^{\top}(\tilde{\mathbf{R}}_{b}^{*})^{\top}\bar{\boldsymbol{\Gamma}}\text{K}[\text{sign}]\big{(}\text{blkdiag}(\mathbf{P}_{\mathbf{g}^{*}_{k}})\mathbf{z}\big{)}+\boldsymbol{\delta}^{\top}\mathbf{d}$
$\displaystyle=-\mathbf{z}^{\top}\text{blkdiag}\left(\mathbf{P}_{\mathbf{g}^{*}_{k}}\right)\bar{\boldsymbol{\Gamma}}\text{K}[\text{sign}]\big{(}\text{blkdiag}(\mathbf{P}_{\mathbf{g}^{*}_{k}})\mathbf{z}\big{)}+\boldsymbol{\delta}^{\top}\mathbf{d}$
$\displaystyle=-\sum_{k=1}^{m}\gamma_{k}(t)\|\mathbf{P}_{\mathbf{g}_{k}^{*}}\mathbf{z}_{k}\|_{1}+\boldsymbol{\delta}^{\top}\mathbf{d}$
$\displaystyle\leq-\underbrace{\min_{k}\gamma_{k}(0)}_{:=\zeta}\sum_{k=1}^{m}\|\mathbf{P}_{\mathbf{g}_{k}^{*}}\mathbf{z}_{k}\|_{1}+\|\boldsymbol{\delta}\|_{1}\|\mathbf{d}\|_{\infty},$
(5)

Note that in the third equality, we have used the fact that
$(\mathbf{p}^{*})^{\top}\tilde{\mathbf{R}}_{b}^{{*}\top}=\mathbf{0}_{dn}^{\top}$,
and the inequality (5) follows from the fact that
$\gamma_{k}(t)\geq\gamma_{k}(0)\geq\min_{k}\gamma_{k}(0)>0$. Based on the norm
inequality for a vector $\mathbf{x}\in\mathbb{R}^{dn}$,
$\|\mathbf{x}\|\leq\|\mathbf{x}\|_{1}\leq\sqrt{dn}\|\mathbf{x}\|,$ we can
further write

$\displaystyle\dot{V}$
$\displaystyle\leq-\zeta\sum_{k=1}^{m}\|\mathbf{P}_{\mathbf{g}_{k}^{*}}\mathbf{z}_{k}\|_{1}+\|\boldsymbol{\delta}\|_{1}\|\mathbf{d}\|_{\infty}$
$\displaystyle\leq-\zeta\|\tilde{\mathbf{R}}_{b}^{*}\mathbf{p}\|_{1}+\|\boldsymbol{\delta}\|_{1}\|\mathbf{d}\|_{\infty}$
$\displaystyle\leq-\zeta\|\tilde{\mathbf{R}}_{b}^{*}\boldsymbol{\delta}\|+\|\boldsymbol{\delta}\|_{1}\|\mathbf{d}\|_{\infty}$
$\displaystyle\leq-\zeta\left(\boldsymbol{\delta}^{\top}\boldsymbol{\mathbf{L}}_{b}^{*}\boldsymbol{\delta}\right)^{1/2}+\sqrt{dn}\|\boldsymbol{\delta}\|\|\mathbf{d}\|_{\infty}$
$\displaystyle=-\zeta\left((\boldsymbol{\delta}^{\rm
F})^{\top}\boldsymbol{\mathbf{L}}_{ff}^{*}\boldsymbol{\delta}^{\rm
F}\right)^{1/2}+\sqrt{dn}\beta\|\boldsymbol{\delta}\|.$ (6)

Substituting the inequality $(\boldsymbol{\delta}^{\rm
F})^{\top}\boldsymbol{\mathbf{L}}_{ff}^{*}\boldsymbol{\delta}^{\rm
F}\geq\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})\|\boldsymbol{\delta}^{\rm
F}\|^{2}=\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})\|\boldsymbol{\delta}\|^{2}$
into equation (6), we get

$\displaystyle\dot{V}\leq-\underbrace{(\zeta\sqrt{\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})}-\sqrt{dn}\beta)}_{:=\frac{1}{\sqrt{2}}\varepsilon}\|\boldsymbol{\delta}\|\leq-\varepsilon\sqrt{V}.$
(7)

We prove finite-time convergence of the desired formation by contradiction. If
there does not exist a finite time $T>0$ such that $V(T)=0$, and
$V(t)=0~{}\forall t\geq T$, then it follows from (7) that

$\displaystyle\frac{1}{2}\int_{V(0)}^{V(t)}\frac{dV}{\sqrt{V}}\leq-\frac{{\varepsilon}}{2}\int_{0}^{t}d\tau,$
(8)

or i.e.,

$\displaystyle 0\leq\sqrt{V(t)}\leq\sqrt{V(0)}-\frac{{\varepsilon}}{2}(t-0).$
(9)

When $t$ is large enough, the right hand side of the inequality (9) becomes
negative, which causes a contradiction. This contradiction implies that
$\exists T>0:V(t)=0$ and for $t\geq T$. Thus, we conclude that
$\mathbf{p}(t)\to\mathbf{p}^{*}$ in finite time. ∎

Lemma 1 suggests that if initially, the gains $\gamma_{ij}$ have been chosen
sufficiently large, the desired formation is achieved in finite time. However,
some quantities such as the smallest eigenvalue of the grounded bearing
Laplacian $\lambda_{\min}(\mathbf{L}_{ff}^{*})$ and the number of agents $n$
are usually unavailable. The proposed adaptive mechanism (11d) makes the
agents achieve the desired formation without requiring any a-priori
information on the number of agents $n$, the desired formation’s structure
$\lambda_{\min}(\mathbf{L}_{ff}^{*})$ and the upper bound $\beta$ of the
disturbance.

######  Theorem 1.

Consider the Problem 1. Under the control law (3), the following statements
hold:

  1. (i)

$\mathbf{p}(t)\to\mathbf{p}^{*}$, as $t\to+\infty$,

  2. (ii)

There exists a constant vector
$\boldsymbol{\gamma}^{*}=[\ldots,\gamma_{ij}^{*},\ldots]^{\top}=[\gamma_{1}^{*},\ldots,\gamma_{m}^{*}]$,
such that $\boldsymbol{\gamma}(t)\to\boldsymbol{\gamma}^{*}$, as
$t\to+\infty$,

  3. (iii)

Additionally, if
$\gamma_{k}^{*}>\gamma_{0}:=\beta\sqrt{\frac{dn}{\lambda_{\min}(\mathbf{L}_{ff}^{*})}},~{}\forall
k=1,\ldots,m$, and there exists a finite time $T$ such that
$|\gamma_{k}-\gamma_{k}^{*}|<\min_{k}|\gamma_{k}-\gamma_{0}|,~{}\forall
i=1,\ldots,m,$ then $\mathbf{p}(t)\to\mathbf{p}^{*}$ in finite time.

###### Proof:

(i) Consider the Lyapunov function
$V=\frac{1}{2}\|\boldsymbol{\delta}\|^{2}+\frac{1}{2\kappa}\|\boldsymbol{\gamma}-{\bar{\gamma}}\mathbf{1}_{m}\|^{2}$,
for some ${\bar{\gamma}}>{\gamma_{0}}$. $V$ is positive definite with regard
to
$\mathbf{x}=[\boldsymbol{\delta}^{\top},(\boldsymbol{\gamma}-{\bar{\gamma}}\mathbf{1}_{m})^{\top}]^{\top}$,
radially unbounded, and bounded by two class $\mathcal{K}_{\infty}$ functions
$h_{1}(\|\mathbf{x}\|)=\min\\{0.5,0.5\kappa^{-1}\\}\|\mathbf{x}\|^{2}$ and
$h_{2}(\|\mathbf{x}\|)=\max\\{0.5,0.5\kappa^{-1}\\}\|\mathbf{x}\|^{2}$.
Similar to the proof of Lemma 1, we have

$\displaystyle\dot{V}$
$\displaystyle=-\sum_{k=1}^{m}\gamma_{k}(t)\|\mathbf{P}_{\mathbf{g}_{k}^{*}}\mathbf{z}_{k}\|_{1}+\boldsymbol{\delta}^{\top}\mathbf{d}+\sum_{k=1}^{m}(\gamma_{k}-{\bar{\gamma}})\|\mathbf{P}_{\mathbf{g}_{k}^{*}}\mathbf{z}_{k}\|_{1}$
$\displaystyle\leq-\sum_{k=1}^{m}{\bar{\gamma}}\|\mathbf{P}_{\mathbf{g}_{k}^{*}}\mathbf{z}_{k}\|_{1}+\boldsymbol{\delta}^{\top}\mathbf{d}$
$\displaystyle\leq-({\bar{\gamma}}\sqrt{\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})}-\sqrt{dn}\beta)\|\boldsymbol{\delta}\|\leq
0,$ (10)

which implies that $\boldsymbol{\delta}$, $\boldsymbol{\gamma}$ are bounded,
and $\exists\lim_{t\to+\infty}V(t)\geq 0$. Since $\dot{V}$ is uniformly
continuous, it follows from Barbalat’s lemma that $\dot{V}\to{0}$, or
$\boldsymbol{\delta}\to\mathbf{0}_{dn}$, as $t\to\infty$.

(ii) Since $-\gamma_{k},~{}k=1,\ldots,m,$ is bounded and non-increasing, it
has a finite limit. Thus, there exists $\boldsymbol{\gamma}^{*}$ such that
$\boldsymbol{\gamma}(t)\to\boldsymbol{\gamma}^{*}$, as $t\to+\infty$.

(iii) If there exists a finite time $T$ such that
$|\gamma_{k}-\gamma_{k}^{*}|<\min_{k}|\gamma_{k}-\gamma_{0}|,~{}\forall
i=1,\ldots,m,$ then for all $t\geq T$, the inequality (7) holds. Therefore,
the proof of this statement follows directly from the proof of Lemma 1. ∎

######  Remark 1.

Since the control law (3) uses only signum functions, chattering will be
unavoidable. To reduce the magnitude of chattering, we may add a proportion
term
$-k_{p}\sum_{j\in\mathcal{N}_{i}}\gamma_{ij}\mathbf{P}_{\mathbf{g}^{*}_{ij}}\mathbf{q}_{ij}$
into (3a) as follows:

$\displaystyle\mathbf{u}_{i}=-k_{p}\sum_{j\in\mathcal{N}_{i}}\gamma_{ij}\mathbf{P}_{\mathbf{g}^{*}_{ij}}\mathbf{q}_{ij}-\sum_{j\in\mathcal{N}_{i}}\gamma_{ij}\mathbf{P}_{\mathbf{g}^{*}_{ij}}{\rm
sign}(\mathbf{q}_{ij}),\,$

for $i=l+1,\ldots,n$. If there is no disturbance, the proportion term is
sufficient for achieving target formation. When disturbances exist, the
proportion term provides some control effort to the formation acquisition and
disturbance rejection objectives, at a slower rate in comparison with the
signum term.

######  Remark 2.

An issue with the control law (3a)–(3c) is that the control gains
$\gamma_{ij}$ is non-decreasing at any time $t\geq 0$. Thus, if the
disturbance has a high magnitude for a time interval, and then decreases in
time, much control effort will be wasted. To address this issue, we may relax
the objective from perfectly achieving a target formation into achieving a
good approximation of the target formation. More specifically, we may control
the formation under disturbances to reach a small neighborhood of the desired
formation in finite-time while the control magnitude estimates the unknown
upper bound of the disturbance [54]. A corresponding modified formation
control law is then modified as follows:

$\displaystyle\mathbf{u}_{i}$
$\displaystyle=-k_{p}\sum_{j\in\mathcal{N}_{i}}\mathbf{P}_{\mathbf{g}^{*}_{ij}}\mathbf{z}_{ij}-\sum_{j\in\mathcal{N}_{i}}\gamma_{ij}\mathbf{P}_{\mathbf{g}^{*}_{ij}}{\rm
sign}(\mathbf{q}_{ij}),$ (11a) $\displaystyle\mathbf{q}_{ij}$
$\displaystyle={\mathbf{P}_{\mathbf{g}^{*}_{ij}}(\mathbf{p}_{i}-\mathbf{p}_{j})}\equiv\mathbf{q}_{k},$
(11b) $\displaystyle\dot{\gamma}_{ij}$
$\displaystyle=\kappa(\|\mathbf{q}_{ij}\|_{1}-\alpha\gamma_{ij})\equiv\dot{\gamma}_{k},$
(11c) $\displaystyle\gamma_{ij}(0)$ $\displaystyle>0,\forall
e_{k}=(i,j)\in\mathcal{E},$ (11d)

where $i=1,\ldots,n$, and $\alpha,\kappa>0$ are positive constants. For each
$(i,j)\in\mathcal{E}$,

$\displaystyle{\gamma}_{ij}(t)=\underbrace{e^{-\kappa\alpha
t}\gamma_{ij}(0)}_{\geq
0}+\underbrace{\kappa\int_{0}^{t}e^{-\kappa\alpha(t-\tau)}\|\mathbf{q}_{ij}\|_{1}d\tau}_{\geq
0}\geq 0,\,\forall t\geq 0.$

Similar to the proof of Theorem 1, consider the Lyapunov function
$V=\frac{1}{2}\|\boldsymbol{\delta}\|^{2}+\frac{1}{2\kappa}\|\boldsymbol{\gamma}-{\bar{\gamma}}\mathbf{1}_{m}\|^{2}$,
where ${\bar{\gamma}>\gamma_{0}}$. We have,

$\displaystyle\dot{V}$
$\displaystyle=-\sum_{k=1}^{m}\left(k_{p}\|\mathbf{P}_{\mathbf{g}_{k}^{*}}\mathbf{z}_{k}\|^{2}+\gamma_{k}\|\mathbf{q}_{k}\|_{1}\right)+\boldsymbol{\delta}^{\top}\mathbf{d}$
$\displaystyle\qquad\qquad\qquad+\sum_{k=1}^{m}(\gamma_{k}-{\bar{\gamma}})(\|\mathbf{q}_{k}\|_{1}-\alpha\gamma_{k})$
$\displaystyle\leq-\sum_{k=1}^{m}\left(k_{p}\|\mathbf{q}_{k}\|^{2}+\alpha(\gamma_{k}^{2}-{\bar{\gamma}}\gamma_{k})\right)$
$\displaystyle\qquad\qquad-{\bar{\gamma}}\sum_{k=1}^{m}\|\mathbf{q}_{k}\|_{1}+\boldsymbol{\delta}^{\top}\mathbf{d}$
$\displaystyle\leq-\sum_{k=1}^{m}\left(k_{p}\|\mathbf{q}_{k}\|^{2}+\frac{1}{2}\alpha(2\gamma_{k}^{2}-2{\bar{\gamma}}\gamma_{k}+{\bar{\gamma}}^{2})-\frac{1}{2}\alpha{\bar{\gamma}}^{2}\right)$
$\displaystyle\qquad\qquad-\left({\bar{\gamma}}\sqrt{\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})}-\sqrt{dn}\beta\right)\|\boldsymbol{\delta}\|$
$\displaystyle\leq-\sum_{k=1}^{m}\left(k_{p}\|\mathbf{q}_{k}\|^{2}+\frac{1}{2}\alpha(\gamma_{k}-{\bar{\gamma}})^{2}\right)+\frac{m}{2}\alpha{\bar{\gamma}}^{2}$
$\displaystyle\leq-
k_{p}\boldsymbol{\delta}^{\top}\mathbf{L}_{b}^{*}\boldsymbol{\delta}-\frac{\alpha}{2}\|\boldsymbol{\gamma}-{\bar{\gamma}}\mathbf{1}_{m}\|^{2}+\frac{m}{2}\alpha{\bar{\gamma}}^{2}$
$\displaystyle\leq-
k_{p}\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})\|\boldsymbol{\delta}\|^{2}-\frac{\alpha}{2}\|\boldsymbol{\gamma}-{\bar{\gamma}}\mathbf{1}_{m}\|^{2}+\frac{m}{2}\alpha{\bar{\gamma}}^{2}.$
(12)

Let
$\varrho=\min\\{2k_{p}\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*}),\kappa\alpha\\}$,
we have,

$\displaystyle\dot{V}$ $\displaystyle\leq-\varrho
V+\frac{m}{2}\alpha{\bar{\gamma}}^{2}$
$\displaystyle=-\varrho(1-\theta)V-\varrho\theta
V+\frac{m}{2}\alpha{\bar{\gamma}}^{2},$ (13)

for some $\theta\in(0,1)$. Thus, when
$V(t)\geq\Delta:=\frac{m}{2\varrho\theta}\alpha{\bar{\gamma}}^{2}$, we have
$\dot{V}\leq-\varrho(1-\theta)V$, or $V\leq\max\\{V(0),\Delta\\}$. Thus,
$\mathbf{x}=[\boldsymbol{\delta}^{\top},(\boldsymbol{\gamma}-{\bar{\gamma}}\mathbf{1}_{m})^{\top}]^{\top}$
is globally ultimately bounded. Defining the ball
$\mathcal{B}_{\Delta}=\\{\mathbf{x}=[\boldsymbol{\delta}^{\top},(\boldsymbol{\gamma}-{\bar{\gamma}}\mathbf{1}_{m})^{\top}]^{\top}\in\mathbb{R}^{dn+m}|~{}\|\mathbf{x}\|\leq
h_{2}^{-1}(h_{1}(\Delta))\\}$, then $\mathbf{x}$ enters the ball
$\mathcal{B}_{\Delta}$ after a finite time. It follows that
$\|\boldsymbol{\delta}\|=\|\mathbf{p}(t)-\mathbf{p}^{*}\|\leq
h_{2}^{-1}(h_{1}(\Delta))$ after a finite time.

It is worth noting that by relaxing the control objective, we also further
reduce the chattering behaviors of the formation in both magnitude and
switching frequency. Most control efforts are provided to maintain the
formation error inside a closed ball, of which the radius is jointly
determined by the desired formation (number of bearing constraints $m$ and the
minimum eigenvalue $\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*}$) and the
control parameters (proportional control gain $k_{p}$, adaptation rate
$\kappa$, and the decay rate $\alpha$). Other methods for avoiding chattering
may be softening the sign function by the tanh($\cdot$) function [44], or
considering a deadzone once error is small enough. Nevertheless, all above
mentioned method needs to sacrifice control performance for eradication of
chattering.

In the next remark, we further consider a larger class of the disturbance
acting on the formation. Let the upper bound of the disturbance be a
polynomials of the formation’s error. The main idea is to design adaptive law
for each coefficient term [56].

######  Remark 3.

Suppose that the upper bound of the unknown disturbance acting on the
formation satisfies

$\displaystyle\|\mathbf{d}(t)\|_{\infty}$
$\displaystyle\leq\beta_{0}+\beta_{1}\|\boldsymbol{\delta}\|_{1}+\ldots+\beta_{N}\|\boldsymbol{\delta}\|_{1}^{N}=\sum_{r=0}^{N}\beta_{r}\|\boldsymbol{\delta}\|_{1}^{r},$

$\forall t\geq 0,\mathbb{N}_{+}\ni N$, where $\beta_{1},\ldots,\beta_{N}$ are
unknown positive constants.

The following adaptive formation control law is proposed:

$\displaystyle\mathbf{u}_{i}$
$\displaystyle=-\sum_{j\in\mathcal{N}_{i}}\gamma_{ij}(t)\mathbf{P}_{\mathbf{g}^{*}_{ij}}{\rm
sign}(\mathbf{q}_{ij}),\,i=1,\ldots,n,$ (14a) $\displaystyle\mathbf{q}_{ij}$
$\displaystyle={\mathbf{P}_{\mathbf{g}^{*}_{ij}}(\mathbf{p}_{i}-\mathbf{p}_{j})}\equiv\mathbf{q}_{k},$
(14b) $\displaystyle{\gamma}_{ij}(t)$
$\displaystyle=\hat{\beta}_{0}^{ij}(t)+\hat{\beta}_{1}^{ij}(t)\|\mathbf{q}_{ij}\|_{1}+\ldots+\hat{\beta}_{N}^{ij}(t)\|\mathbf{q}_{ij}\|_{1}^{N}\equiv\gamma_{k}(t)$
(14c) $\displaystyle\hat{\beta}_{r}^{ij}(t)$
$\displaystyle\equiv\hat{\beta}_{1}^{k}(t),\,\forall
e_{k}=(i,j)\in\mathcal{E},~{}k=1,\ldots,m.$ (14d)
$\displaystyle\dot{\hat{\beta}}_{0}^{ij}(t)$
$\displaystyle=\left|\left|\mathbf{q}_{ij}\right|\right|_{1},\,{\hat{\beta}}_{0}^{ij}(0)>0,$
(14e) $\displaystyle\dot{\hat{\beta}}_{1}^{ij}(t)$
$\displaystyle=\left|\left|\mathbf{q}_{ij}\right|\right|_{1}^{2},\,{\hat{\beta}}_{1}^{ij}(0)>0,$
(14f) $\displaystyle\vdots$ $\displaystyle\dot{\hat{\beta}}_{N}^{ij}(t)$
$\displaystyle=\left|\left|\mathbf{q}_{ij}\right|\right|_{1}^{N+1},\,{\hat{\beta}}_{N}^{ij}(0)>0.$
(14g)

For stability analysis, let
$\boldsymbol{\hat{\beta}}_{r}=[\ldots,\hat{\beta}_{r}^{ij},\ldots]^{\top}=[\hat{\beta}_{r}^{1},\ldots,\hat{\beta}_{r}^{m}]^{\top}\in\mathbb{R}^{m}$,
$\forall r=0,1,\ldots,N$, and consider the Lyapunov candidate function

$V=\frac{1}{2}\|\boldsymbol{\delta}\|^{2}+\frac{1}{2}\sum_{r=0}^{N}\|\boldsymbol{\hat{\beta}}_{r}-\bar{\beta}_{r}\mathbf{1}_{m}\|^{2},$

where
$\bar{\beta_{r}}>\beta_{r}\left(\sqrt{\frac{\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})}{dn}}\right)^{r+1},\forall
r=0,\ldots,N$. Then,

$\displaystyle\dot{V}$
$\displaystyle=-\sum_{k=1}^{m}\gamma_{k}\|\mathbf{q}_{k}\|_{1}+\boldsymbol{\delta}^{\top}\mathbf{d}+\sum_{k=1}^{m}\sum_{r=0}^{N}(\hat{\beta}_{r}^{k}-\bar{\beta}_{r})\|\mathbf{q}_{k}\|_{1}^{r+1}$
$\displaystyle\leq-\sum_{k=1}^{m}\sum_{r=0}^{N}\hat{\beta}_{r}^{k}\|\mathbf{q}_{k}\|_{1}^{r+1}+\|\boldsymbol{\delta}\|_{1}\|\mathbf{d}\|_{\infty}$
(15)
$\displaystyle\qquad\quad+\sum_{k=1}^{m}\sum_{r=0}^{N}(\hat{\beta}_{r}^{k}-\bar{\beta}_{r})\|\mathbf{q}_{k}\|_{1}^{r+1}$

It follows that

$\displaystyle\dot{V}$
$\displaystyle\leq-\sum_{r=0}^{N}\bar{\beta}_{r}\sum_{k=1}^{m}\|\mathbf{q}_{k}\|_{1}^{r+1}+\sum_{r=0}^{N}\beta_{r}\|\boldsymbol{\delta}\|_{1}^{r+1}$
$\displaystyle\leq-\sum_{r=0}^{N}\bar{\beta}_{r}\left(\sum_{k=1}^{m}\|\mathbf{q}_{k}\|_{1}\right)^{r+1}+\sum_{r=0}^{N}\beta_{r}\|\boldsymbol{\delta}\|_{1}^{r+1}$
$\displaystyle\leq-\sum_{r=0}^{N}\bar{\beta}_{r}\left(\sqrt{\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})}\|\boldsymbol{\delta}\|\right)^{r+1}+\sum_{r=0}^{N}\beta_{r}\left({dn}\right)^{\frac{r+1}{2}}\|\boldsymbol{\delta}\|^{r+1}$
$\displaystyle\leq-\sum_{r=0}^{N}\underbrace{\left(\bar{\beta}_{r}\left(\sqrt{\lambda_{\min}(\boldsymbol{\mathbf{L}}_{ff}^{*})}\right)^{r+1}-\beta_{r}\left({dn}\right)^{\frac{r+1}{2}}\right)}_{>0}\|\boldsymbol{\delta}\|^{r+1}$
$\displaystyle\leq 0.$ (16)

It follows that $\boldsymbol{\delta}$ and
$\hat{\boldsymbol{\beta}}_{r},~{}\forall r=0,1,\ldots,N,$ are uniformly
bounded. Similar to the proof of Theorem 1, we can show that
$\|\boldsymbol{\delta}\|\to\mathbf{0}_{dn}$, or
$\mathbf{p}(t)\to\mathbf{p}^{*}$, as $t\to\infty$, and
$\lim_{t\to\infty}\hat{\boldsymbol{\beta}}_{r},~{}\forall r=0,1,\ldots,N,$
exists. Further, if
$\hat{\boldsymbol{\beta}}_{r}>\bar{\beta}_{r}\mathbf{1}_{m},~{}\forall
r=0,1,\ldots,N$, where “$>$” is understood to be element-wise, then
$\mathbf{p}(t)\to\mathbf{p}^{*}$ in finite time.

##  IV Bearing-only based formation control

In this section, we further assume that the agents can measure only the
relative bearing vectors with regard to their neighbors. We propose a
corresponding adaptive variable-structure bearing-only formation control law
and showed that the desired formation can be asymptotically achieved.
Moreover, due to the adaptive gains, the effects of unknown time-varying
disturbances acting on formation can be completely rejected even when the
followers agents are not given any information of the disturbances’ upper
bound.

###  IV-A Proposed control law

Consider the system of single-integrator agents with disturbance (1). The
bearing-only control law for each follower agent $i\in\\{l+1,\ldots,n\\}$ is
proposed as follows

$\displaystyle\mathbf{u}_{i}$
$\displaystyle=-\gamma_{i}\text{sign}\left(\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\right),$
(17a) $\displaystyle\dot{\gamma}_{i}$
$\displaystyle=\kappa_{i}\left|\left|\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\right|\right|_{1}.$
(17b)

We can express the $n$-agent system under the control law (17a)–(17b) in
vector form as follows:

$\displaystyle\dot{\mathbf{p}}$
$\displaystyle=\bar{\mathbf{Z}}\left(-\bar{\boldsymbol{\Gamma}}\text{sign}\left(\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})\right)+\mathbf{d}\right),$
(18a) $\displaystyle\dot{\boldsymbol{\gamma}}$
$\displaystyle=\kappa[\|\mathbf{P}_{\mathbf{g}_{1}}\mathbf{g}_{1}^{*}\|_{1},\ldots,\|\mathbf{P}_{\mathbf{g}_{m}}\mathbf{g}_{m}^{*}\|_{1}]^{\top},$
(18b)

where
$\boldsymbol{\gamma}=[\gamma_{1},\ldots,\gamma_{n}]^{\top}\in\mathbb{R}^{n}$,
$\boldsymbol{\Gamma}=\text{diag}(\boldsymbol{\gamma})$ and
$\bar{\boldsymbol{\Gamma}}=\boldsymbol{\Gamma}\otimes\mathbf{I}_{d}$.

###  IV-B Stability analysis

This subsection studies the stability of the $n$-agent system (18a)–(18b).
Particularly, we show that the desired formation $\mathbf{p}^{*}$ defined as
in Definition 1 will be eventually achieved as $t\to\infty$. Since the right-
hand-side of Eq. (18a) is discontinuous, we understand the solution of (18a)
in Fillipov sense.

We will firstly prove the following lemma.

######  Lemma 2.

[35, Lemma 2] Suppose no agents coincide in $\mathbf{p}$ or $\mathbf{p}^{*}$.
The following inequality holds

$\displaystyle\mathbf{p}^{\top}\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})$
$\displaystyle\geq 0,$ (19)
$\displaystyle(\mathbf{p}^{*})^{\top}\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})$
$\displaystyle\leq 0,$ (20)
$\displaystyle(\mathbf{p}-\mathbf{p}^{*})^{\top}\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})$
$\displaystyle\geq 0,$ (21)

where the equality holds if and only if $\mathbf{g}=\mathbf{g}^{*}$.

######  Lemma 3.

[35, Lemma 3] Suppose no agents coincide in $\mathbf{p}$ or $\mathbf{p}^{*}$,
then

$\displaystyle\mathbf{p}^{\top}\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})$
$\displaystyle\geq\frac{1}{2\max_{k=1,\ldots,m}\|\mathbf{z}_{k}\|}\mathbf{p}^{\top}\mathbf{L}_{b}^{*}\mathbf{p}.$
(22)

Furthermore, if $\mathbf{g}_{k}^{\top}\mathbf{g}^{*}_{k}\geq 0,~{}\forall
k=1,\ldots,m,$ then

$\displaystyle\mathbf{p}^{\top}\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})$
$\displaystyle\leq\frac{1}{\min_{k=1,\ldots,m}\|\mathbf{z}_{k}\|}\mathbf{p}^{\top}\mathbf{L}_{b}(\mathbf{p}^{*})\mathbf{p}.$
(23)

Next, we prove that the adaptive bearing-only control law (17) guarantees
boundedness of the formation’s error
$\boldsymbol{\delta}=\mathbf{p}-\mathbf{p}^{*}$ in the following lemma.

######  Lemma 4.

Consider the Problem 2. Under the control law (17), $\boldsymbol{\delta}$ is
uniformly bounded.

###### Proof:

Consider the Lyapunov function

$V=\mathbf{p}^{\top}\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})+\sum_{i=l+1}^{n}\frac{(\gamma_{i}-\gamma_{0})^{2}}{2\kappa_{i}},$
(24)

where $\gamma_{0}>\|\mathbf{d}\|_{\infty}$. Then, $V=0$ if and only if
$\mathbf{p}^{\top}\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})=0\Longleftrightarrow\mathbf{g}=\mathbf{g}^{*}$
and $\gamma_{i}=\gamma_{0},\forall i=l+1,\ldots,n$. Since
$(\mathcal{G},\mathbf{p}^{*})$ is infinitesimally rigid and $l\geq 2$, the
equality $\mathbf{g}=\mathbf{g}^{*}$ implies that $\mathbf{p}=\mathbf{p}^{*}$.
Thus, $V$ is positive definite with regard to
$[\boldsymbol{\delta}^{\top},(\boldsymbol{\gamma}-\gamma_{0}\mathbf{1}_{n})^{\top}]^{\top}$.
We have

$\displaystyle\dot{V}$ $\displaystyle\in^{\text{a.e}}\dot{\tilde{V}}$
$\displaystyle=-(\mathbf{g}-\mathbf{g}^{*})^{\top}\bar{\mathbf{H}}\bar{\mathbf{Z}}\left(\bar{\boldsymbol{\Gamma}}\text{sign}\left(\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})\right)-\mathbf{d}\right)+\sum_{i=1}^{n}(\gamma_{i}-\gamma_{0})\left|\left|\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\right|\right|_{1}$
$\displaystyle=-\sum_{i=l+1}^{n}\gamma_{i}\left|\left|\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\right|\right|_{1}-\sum_{i=l+1}^{n}\Big{(}\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\Big{)}^{\top}\mathbf{d}_{i}+\sum_{i=1}^{n}(\gamma_{i}-\gamma_{0})\left|\left|\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\right|\right|_{1}$
$\displaystyle\leq-\sum_{i=l+1}^{n}(\gamma_{0}-\|\mathbf{d}_{i}\|_{\infty})\left|\left|\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\right|\right|_{1}$
$\displaystyle\leq-(\gamma_{0}-\|\mathbf{d}\|_{\infty})\left|\left|\sum_{i=l+1}^{n}\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\right|\right|_{1}$
$\displaystyle\leq-(\gamma_{0}-\|\mathbf{d}\|_{\infty})\left|\left|\bar{\mathbf{Z}}\bar{\mathbf{H}}^{\top}(\mathbf{g}-\mathbf{g}^{*})\right|\right|_{1}\leq
0.$ (25)

It follows that $V(t)\leq V(0),~{}\forall t\geq 0$,
$\mathbf{z}^{\top}(\mathbf{g}-\mathbf{g}^{*})$ and
$(\boldsymbol{\gamma}-\gamma_{0}\mathbf{1}_{n})$ are always bounded.

Further, from the inequality

$\displaystyle\max_{(i,j)\in\mathcal{E}}\|\mathbf{z}_{ij}\|\leq\|\mathbf{z}\|=\|\bar{\mathbf{H}}\mathbf{p}\|$
$\displaystyle=\|\bar{\mathbf{H}}(\mathbf{p}-\mathbf{p}^{*}+\mathbf{p}^{*})\|$
$\displaystyle\leq\|\bar{\mathbf{H}}\|(\|\boldsymbol{\delta}\|+\|\mathbf{p}^{*}\|)$

and equation (22), we have

$\displaystyle V\geq\mathbf{z}^{\top}(\mathbf{g}-\mathbf{g}^{*})$
$\displaystyle\geq\frac{1}{2\max_{(i,j)\in\mathcal{E}}\|\mathbf{z}_{ij}\|}\mathbf{p}^{\top}\mathbf{L}_{b}^{*}\mathbf{p}$
$\displaystyle\geq\frac{\lambda_{\min}(\mathbf{L}_{ff}^{*})\|\boldsymbol{\delta}\|^{2}}{2\|\bar{\mathbf{H}}\|(\|\boldsymbol{\delta}\|+\|\mathbf{p}^{*}\|)},$

which shows that
$\frac{\lambda_{\min}(\mathbf{L}_{ff}^{*})\|\boldsymbol{\delta}\|^{2}}{2\|\bar{\mathbf{H}}\|(\|\boldsymbol{\delta}\|+\|\mathbf{p}^{*}\|)}$
is bounded. Suppose that the upper bound is $\alpha>0$, i.e.,

$\displaystyle\vartheta\frac{{\|\boldsymbol{\delta}\|^{2}}}{\|\boldsymbol{\delta}\|+\|\mathbf{p}^{*}\|}\leq\alpha,$

where
$\vartheta:=\frac{\lambda_{\min}(\mathbf{L}_{ff}^{*})}{2\|\bar{\mathbf{H}}\|}$.
Let $x=\|\boldsymbol{\delta}\|\geq 0$ and $\zeta=\frac{\alpha}{\vartheta}>0$,
the inequality

$x^{2}-\zeta x-\zeta\|\mathbf{p}^{*}\|\leq 0,$

has solution

$\displaystyle\|\boldsymbol{\delta}(t)\|\in\left[0,\frac{\zeta+\sqrt{\zeta^{2}+4\zeta\|\mathbf{p}^{*}\|}}{2}\right],~{}\forall
t\geq 0.$ (26)

Thus, $\boldsymbol{\delta}(t)$ is uniformly bounded. ∎

The following lemma gives a sufficient condition for guaranteeing collision
avoidance between neighboring agents.

######  Lemma 5.

Consider the Problem 2. Suppose that
$0<\eta:=\frac{\zeta+\sqrt{\zeta^{2}+4\zeta\|\mathbf{p}^{*}\|}}{2}<\frac{1}{\sqrt{n}}\min_{i,j\in\mathcal{V}}\|\mathbf{p}_{i}^{*}-\mathbf{p}_{j}^{*}\|$,
then
$\min_{i,j\in\mathcal{V}}\|\mathbf{p}_{i}-\mathbf{p}_{j}\|\geq\min_{i,j\in\mathcal{V}}\|\mathbf{p}_{i}^{*}-\mathbf{p}_{j}^{*}\|-\sqrt{n}\eta,~{}\forall
t\geq 0$.

###### Proof.

For each $i,j\in\mathcal{V}$, we can write
$\mathbf{p}_{i}-\mathbf{p}_{j}=(\mathbf{p}_{i}-\mathbf{p}_{i}^{*})-(\mathbf{p}_{j}-\mathbf{p}_{j}^{*})+(\mathbf{p}_{i}^{*}-\mathbf{p}_{j}^{*})$.
Thus,

$\displaystyle\|\mathbf{p}_{i}-\mathbf{p}_{j}\|$
$\displaystyle\geq\|\mathbf{p}^{*}_{i}-\mathbf{p}_{j}^{*}\|-\|\mathbf{p}_{j}-\mathbf{p}_{j}^{*}\|-\|\mathbf{p}_{i}-\mathbf{p}_{i}^{*}\|$
$\displaystyle\geq\|\mathbf{p}^{*}_{i}-\mathbf{p}_{j}^{*}\|-\sum_{i=1}^{n}\|\mathbf{p}_{i}-\mathbf{p}_{i}^{*}\|$
$\displaystyle\geq\|\mathbf{p}^{*}_{i}-\mathbf{p}_{j}^{*}\|-\sqrt{n}\|\boldsymbol{\delta}\|.$

It follows from (26) that $\|\boldsymbol{\delta}(t)\|\leq\eta,~{}\forall t\geq
0$. Thus, we have

$\min_{i,j\in\mathcal{V}}\|\mathbf{p}_{i}-\mathbf{p}_{j}\|\geq\theta:=\min_{i,j\in\mathcal{V}}\|\mathbf{p}_{i}^{*}-\mathbf{p}_{j}^{*}\|-\sqrt{n}\eta,~{}\forall
t\geq 0.$

∎

The main result of this section is given in the following theorem, where a
sufficient condition for achieving the desired target formation will be
derived.

######  Theorem 2.

Consider the Problem 2. Under the adaptive bearing-only control law (17),
there exists a positive constant $\alpha>0$ such that if the Lyapunov function
in (24) satisfies $V(0)<\alpha$, then
$\min_{(i,j)\in\mathcal{E}}\|\mathbf{p}_{i}-\mathbf{p}_{j}\|\geq\theta>0,$
$\forall i,j\in\mathcal{V},~{}\forall t\geq 0$,
$\lim_{t\to+\infty}\mathbf{p}(t)=\mathbf{p}^{*}$, and there exists
$\boldsymbol{\gamma}^{*}$ such that
$\lim_{t\to+\infty}\boldsymbol{\gamma}(t)=\boldsymbol{\gamma}^{*}$.

###### Proof:

Based on Lemma 5, as $\mathbf{z}^{\top}(\mathbf{g}-\mathbf{g}^{*})\leq
V(t)\leq V(0)\leq\alpha$, it follows that

$\displaystyle\frac{{\|\boldsymbol{\delta}\|^{2}}}{\|\boldsymbol{\delta}\|+\|\mathbf{p}^{*}\|}\leq\frac{\alpha}{\vartheta}=\zeta,$

and we can obtain

$\displaystyle\|\boldsymbol{\delta}(t)\|$
$\displaystyle\leq\frac{\zeta+\sqrt{\zeta^{2}+4\zeta\|\mathbf{p}^{*}\|}}{2}$
$\displaystyle=\frac{\alpha/\vartheta+\sqrt{\alpha^{2}/\vartheta^{2}+4(\alpha/\vartheta)\|\mathbf{p}^{*}\|}}{2},~{}\forall
t\geq 0.$ (27)

Thus, for $\alpha$ sufficiently small, the inequality
$0<\eta:=\frac{\zeta+\sqrt{\zeta^{2}+4\zeta\|\mathbf{p}^{*}\|}}{2}<\frac{1}{\sqrt{n}}\min_{i,j\in\mathcal{V}}\|\mathbf{p}_{i}^{*}-\mathbf{p}_{j}^{*}\|$
can always be satisfied. It follows from Lemma 4 that no collision can happen,
and
$\min_{i,j\in\mathcal{V}}\|\mathbf{p}_{i}-\mathbf{p}_{j}\|\geq\theta:=\min_{i,j\in\mathcal{V}}\|\mathbf{p}_{i}^{*}-\mathbf{p}_{j}^{*}\|-\sqrt{n}\eta,~{}\forall
t\geq 0.$

It follows that $\dot{V}$ is uniformly continuous. By Barbalat’s lemma,
$\lim_{t\to+\infty}\dot{V}=0$, or $\mathbf{g}\to\mathbf{g}^{*}$. Equivalently,
we conclude that $\lim_{t\to+\infty}\mathbf{p}(t)=\mathbf{p}^{*}$.

Moreover, since $(\gamma_{i}(t)-\gamma_{0}),~{}\forall i=l+1,\ldots,n,$ are
bounded and nonincreasing, $\mathbf{z}^{\top}(\mathbf{g}-\mathbf{g}^{*})\to
0$, it follows that there exists $\boldsymbol{\gamma}^{*}$ such that
$\lim_{t\to+\infty}\boldsymbol{\gamma}(t)=\boldsymbol{\gamma}^{*}$. ∎

##  V Application on formation tracking

Let the leaders move with the same velocity $\mathbf{v}^{*}(t)$, which is
assumed to be a bounded, uniformly continuous function. The desired formation
$\mathbf{p}^{*}$ in Definition 1 is now time-varying, with
$\dot{\mathbf{p}}^{*}=\mathbf{1}_{n}\otimes\mathbf{v}^{*}$. Thus, it is
assumed that $(\mathcal{G},\mathbf{p}^{*}(0))$ is infinitesimally rigid in
$\mathbb{R}^{d}$. We will show that the adaptive formation control laws (3)
and (17) are still capable of stabilizing the desired leader-follower
formation.

The motion of the $n$-agent system under the control law (3) is now given in
matrix form as follows:

$\displaystyle\dot{\mathbf{p}}=\begin{bmatrix}\dot{\mathbf{p}}^{\rm L}\\\
\dot{\mathbf{p}}^{\rm F}\end{bmatrix}=\begin{bmatrix}\mathbf{0}_{dl}\\\
\dot{\mathbf{p}}^{\rm
F}-\mathbf{1}_{f}\otimes\mathbf{v}^{*}\end{bmatrix}+\mathbf{1}_{n}\otimes\mathbf{v}^{*}.$
(28)

Let
$\boldsymbol{\delta}=\mathbf{p}-\mathbf{p}^{*}=\begin{bmatrix}\mathbf{0}_{dl}\\\
{\mathbf{p}}^{\rm F}-{\mathbf{p}}^{\rm F*}\end{bmatrix}$, then

$\displaystyle\dot{\boldsymbol{\delta}}=\begin{bmatrix}\mathbf{0}_{dl}\\\
\dot{\mathbf{p}}^{\rm
F}-\mathbf{1}_{f}\otimes\mathbf{v}^{*}\end{bmatrix}=\bar{\mathbf{Z}}\left(\begin{bmatrix}\mathbf{0}_{dl}\\\
\dot{\mathbf{p}}^{\rm
F}\end{bmatrix}-\mathbf{1}_{n}\otimes\mathbf{v}^{*}\right).$

Suppose that the displacement-based control law (3) is adopted for followers,
we have

$\displaystyle\dot{\boldsymbol{\delta}}=-\bar{\mathbf{Z}}((\tilde{\mathbf{R}}_{b}^{*})^{\top}\bar{\boldsymbol{\Gamma}}{\rm
sign}({\rm
blkdiag}(\mathbf{P}_{\mathbf{g}^{*}_{k}})\bar{\mathbf{H}}\boldsymbol{\delta})-\mathbf{d}+\mathbf{1}_{n}\otimes\mathbf{v}^{*}),$
$\displaystyle\dot{\boldsymbol{\gamma}}=\kappa\left[\|\mathbf{q}_{1}\|_{1},\ldots,\|\mathbf{q}_{m}\|_{1}\right]^{\top},$
(29)

which is of the same form as (4), but having an additional disturbance term
$-\bar{\mathbf{Z}}(\mathbf{1}_{n}\otimes\mathbf{v}^{*})$. Thus, the following
theorem can be proved.

######  Theorem 3.

Consider the $n$-agent system (29) under the displacement-based control law
(3), the following statements hold:

  1. (i)

$\boldsymbol{\delta}(t)\to\mathbf{0}_{dn}$, as $t\to+\infty$,

  2. (ii)

There exists a constant vector
$\boldsymbol{\gamma}^{*}=[\ldots,\gamma_{ij}^{*},\ldots]^{\top}=[\gamma_{1}^{*},\ldots,\gamma_{m}^{*}]^{\top}$,
such that $\boldsymbol{\gamma}(t)\to\boldsymbol{\gamma}^{*}$, as
$t\to+\infty$,

  3. (iii)

Additionally, if
$\gamma_{k}^{*}>\gamma_{0}^{\prime}:=(\beta+\|\mathbf{v}^{*}\|_{\infty})\sqrt{\frac{dn}{\lambda_{\min}(\mathbf{L}_{ff}^{*})}},~{}\forall
k=1,\ldots,m$, and there exists a finite time $T$ such that
$|\gamma_{k}-\gamma_{k}^{*}|<\min_{k}|\gamma_{k}-\gamma_{0}|,~{}\forall
i=1,\ldots,m,$ then $\boldsymbol{\delta}(t)\to\mathbf{0}_{dn}$ in finite time.

###### Proof:

The proof is similar to the proof of Theorem 1 and will be omitted. ∎

Finally, if the bearing-only control law (17) is adopted for followers, the
$n$-agent formation can be expressed in matrix form as

$\begin{split}\dot{\boldsymbol{\delta}}&=\bar{\mathbf{Z}}\left(-\bar{\boldsymbol{\Gamma}}\text{sign}\left(\bar{\mathbf{H}}^{\top}({\mathbf{g}_{k}}-\mathbf{g}^{*})\right)+\mathbf{d}-\mathbf{1}_{n}\otimes\mathbf{v}^{*}\right),\\\
\dot{\boldsymbol{\gamma}}&=\kappa\left[\ldots,\left|\left|\sum_{j\in\mathcal{N}_{i}}(\mathbf{g}_{ij}-\mathbf{g}_{ij}^{*})\right|\right|_{1},\ldots\right]^{\top},\end{split}$
(30)

which is of the same form as (18a)–(18b), but having an additional unknown
disturbance term $-\bar{\mathbf{Z}}(\mathbf{1}_{n}\otimes\mathbf{v}^{*})$. We
have the following theorem, whose proof is similar to the proof of Theorem 2
and will be omitted.

######  Theorem 4.

Consider the $n$-agent system (30) under the adaptive bearing-only based
control law (17). There exists a positive constant $\alpha>0$ such that if the
Lyapunov function in (24) satisfies $V(0)<\alpha$, then
$\lim_{t\to+\infty}(\mathbf{p}(t)-\mathbf{p}^{*})=\mathbf{0}_{dn}$, and
$\lim_{t\to+\infty}\boldsymbol{\gamma}(t)=\boldsymbol{\gamma}^{*}$, for some
constant vector $\boldsymbol{\gamma}^{*}$.

######  Remark 4.

In formation tracking, the leaders’ trajectories can be embedded into each
leader from beginning, or can be remotely regulated from a control center. The
leader agents are assumed to be equipped with better positioning system, so
that their positions are available for control and monitoring objectives.
Suppose that the leaders are also subjected to bounded unknown disturbances,
i.e.,

$\displaystyle\dot{\mathbf{p}}_{i}(t)$
$\displaystyle=\mathbf{u}_{i}(t)+\mathbf{d}_{i}(t),~{}\forall i=1,\ldots,l,$

where $\|\mathbf{d}_{i}\|<\beta$. To assure that the leaders track their
desired trajectories $\mathbf{p}_{i}^{*}(t)$, and thus, eventually acting as
moving references for follower agents, the following position tracking law is
respectively proposed

$\displaystyle\mathbf{u}_{i}(t)=-k_{p}(\mathbf{p}_{i}-\mathbf{p}_{i}^{*})-\beta_{1}{\rm
sign}(\mathbf{p}_{i}-\mathbf{p}_{i}^{*})$

where $\beta_{1}>\beta$. By considering the Lyapunov function
$V=\frac{1}{2}\|\mathbf{p}_{i}-\mathbf{p}_{i}^{*}\|^{2}$, we can proved that
$\mathbf{p}_{i}(t)\to\mathbf{p}_{i}^{*}$ in finite time.

##  VI Numerical examples

In this section, we provide a few numerical examples to demonstrate the
effectiveness of the formation control laws proposed in Sections III, IV, and
VI. In all simulations, the target formation is described by a graph
$\mathcal{G}$ of 20 vertices and 39 edges and a desired configuration
$\mathbf{p}^{*}$ (a dodecahedron) as depicted in Figure 1. It can be checked
that $(\mathcal{G},\mathbf{p}^{*})$ is infinitesimally bearing rigid in 3D. In
the simulations, there are $l=3$ leaders and $17$ followers.

(a)

(b)

(c)

(d)

Figure 2: Simulation 1a: the 20-agent system under the control law (3). (a)
Trajectories of agents from 0 to 40 seconds (leaders are marked with $\Delta$,
followers’ initial and final positions are marked with ${\rm x}$ and ${\rm
o}$, respectively); (b) Trajectories of agents from 40 to 80 seconds; (c)
Formation’s error versus time; (d) A subset of the adaptive gains
$\gamma_{ij}$ versus time.

(a)

(b)

(c)

(d)

Figure 3: Simulation 1b: the 20-agent system under the control law (11). (a)
Trajectories of agents from 0 to 30 seconds (leaders are marked with $\Delta$,
followers’ initial and final positions are marked with ${\rm x}$ and ${\rm
o}$, respectively); (b) Trajectories of agents from 30 to 90 seconds; (c)
Formation’s error versus time; (d) A subset of the adaptive gains
$\gamma_{ij}$ versus time.

###  VI-A Bearing-based formation control with disturbance rejection

First, we simulate the formation with the control law (3). Let each follower
$i$ be modeled by a single integrator with disturbance $\mathbf{d}_{i}$ given
as

$\displaystyle\mathbf{d}_{i}(t)=\begin{cases}0.1\mathbf{h}_{i}(t),&\mbox{if
}0\leq t\leq 40s,\\\ 0.15\mathbf{h}_{i}(t),&\mbox{if }t\geq 40s,\end{cases}$
(31)

where
$\mathbf{h}_{i}(t)=[\sin(it)+1,~{}\cos(it)+\tanh(t),~{}1-e^{-it}]^{\top}$.

The control law (3) is used with $\kappa=0.2$ and
$\gamma_{i}(0),~{}i=4,\ldots,20,$ are randomly generated on the interval
$[0,0.05]$. Simulation results are given as in Fig. 2.

According to Figs. 2(a), 2(c), and 2(d), for $0\leq t\leq 40$ seconds, the
desired formation is asymptotically achieved and the adaptive gains
$\gamma_{ij}$ increase until the corresponding bearing constraint is
stabilized. From $t=40$ seconds, the magnitude of the disturbance suddenly
increases, which drives the agents out of the desired formation. The errors
invoke the adaptive mechanism, $\gamma_{ij}$ increase again. It can be seen
from Figs. 2(b), 2(c), and 2(d) that followers are driven out from their
desired positions from 40 to 55 seconds, as the magnitudes of their formation
control laws are not big enough to counter the disturbance. From 55 to 80
seconds, when $\gamma_{ij}$ are sufficiently large, the agents are pulling
back to the desired positions, and the desired formation is eventually
achieved.

Second, we conduct a simulation of the formation under the adaptive control
law with increasing/decreasing gains (11). The disturbance acting on a
follower $i$ in this simulation is given as

$\displaystyle\mathbf{d}_{i}(t)=\begin{cases}0.15\mathbf{h}_{i}(t),&\mbox{if
}0\leq t\leq 30s,\\\ 0.3\mathbf{h}_{i}(t),&\mbox{if }30\leq t\leq 65s,\\\
0.1\mathbf{h}_{i}(t),&\mbox{if }65\leq t\leq 90s.\end{cases}$ (32)

With $\gamma_{ij}(0)=0.05,\forall(i,j)\in\mathcal{E}$, $k_{p}=0.5$
(proportional gain), $\kappa=1$ (rate of adaptation), $\alpha=0.05$ (leakage
coefficient), and $\mathbf{p}(0)$ chosen the same as the previous simulation,
we obtain the simulation results as depicted in Figure 3.

As shown in Figs. 2(a), 2(c), and 2(d), for $0\leq t\leq 15$ seconds, the
adaptive gains $\gamma_{ij}$ increase and the control law drives the agents to
a neighborhood of the desired formation. Due to the existence of a leakage
term $-\epsilon\gamma_{ij}$ in (11)(c), once a desired bearing constraint is
sufficiently small, $\gamma_{ij}$ tends to reduce from 15 to 30 seconds. The
decrements of $\gamma_{ij}$ make the formation errors raise again, however,
$\mathbf{p}(t)$ remains on a small ball $\mathcal{B}_{R_{1}}(\mathbf{p}^{*})$
centered at $\mathbf{p}^{*}$, whose radius $r$ is jointly determined by the
controller’s parameters, the desired formation, and the magnitude of the
unknown disturbance.

From $t=30$ to $45$ seconds, as the magnitude of the disturbance is doubled,
the agents are out from $\mathcal{B}_{R_{1}}(\mathbf{p}^{*})$. As the errors
increase, the term $\|\mathbf{q}_{ij}\|$ dominates the leakage term in the
adaptive mechanism (11)(c), and thus $\gamma_{ij}$ increase again. It can be
seen from Figs. 2(b), 2(c), and 2(d) that followers are driven further from
their desired positions from 30 to about 38 seconds, and then being attracted
to a ball $\mathcal{B}_{R_{2}}(\mathbf{p}^{*})$ centered at $\mathbf{p}^{*}$,
with $R_{2}>R_{1}$, from 38 to 65 seconds. For $45\leq t\leq 65$, the bearing
constraints are sufficiently small, it can be seen that $\gamma_{ij}$ decrease
again due to the leakage term. For $t\geq 65$, as the disturbance magnitude
decreased to 0.1, as $\gamma_{ij}(t=65s)$ satisfy the requirement of Lemma 1,
$\mathbf{p}$ converges to $\mathbf{p}^{*}$ after a short time
($\mathbf{p}(t)=\mathbf{p}^{*}$ at $t=68$ s). However, from $t=68$s, because
the leakage term is the only active term in (11)(c), $\gamma_{ij}$ decreases.
Gradually, once the control law cannot fully reject the disturbance, the
disturbances make $\mathbf{p}$ out of $\mathbf{p}^{*}$. The control law will
still keep $\mathbf{p}$ inside a ball $\mathcal{B}_{R_{3}}(\mathbf{p}^{*})$
centered at $\mathbf{p}^{*}$, with $R_{3}<R_{1}$.

###  VI-B Bearing-only formation control with disturbance rejection

In this subsection, we simulate the adaptive bearing-only control law (17) for
the 20-agent system. The simulation’s parameters are $\kappa_{i}=2$,
$\gamma_{ij}(0)=0.5$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Simulation 2: the 20-agent system under the bearing-only control law
(17). (a) Trajectories of agents from 0 to 5 seconds (leaders are marked with
$\Delta$, followers’ initial and final positions are marked with ${\rm x}$ and
${\rm o}$, respectively); (b) Trajectories of agents from 5 to 15 seconds; (c)
Trajectories of agents from 15 to 25 seconds; (d) Formation’s error versus
time; (e) A subset of the adaptive gains $\gamma_{i}$ versus time. (f)
Magnitude of control input versus time

The disturbance acting on a follower $i$ in this simulation is given as

$\displaystyle\mathbf{d}_{i}(t)=\begin{cases}\mathbf{0}_{3},&\mbox{if }0\leq
t\leq 5s,\\\ 1.5\mathbf{h}_{i}(t),&\mbox{if }5\leq t\leq 15s\\\
3\mathbf{h}_{i}(t),&\mbox{if }15\leq t\leq 25s.\end{cases}$ (33)

The simulation results are depicted in Figure (4). For the first 5 seconds,
there is no disturbance acting on the formation, the control law stabilizes
$\mathbf{p}(t)$ to $\mathbf{p}^{*}$ after about 2 seconds. The adaptive gains
$\gamma_{i}$ increase correspondingly in $0\leq t\leq 2$ and remain unchanging
until $t=5s$, when there are disturbances acting on the agents. Due to the
disturbance, $\mathbf{p}$ leaves the target configuration $\mathbf{p}^{*}$,
bearing errors make $\gamma_{i}$ increase. In turn, the control law’s
magnitude increases, and is eventually capable of suppressing the disturbance
from $7$s. For $t\geq 7$s, $\mathbf{p}$ approaches to $\mathbf{p}^{*}$.
Approximately, $\mathbf{p}$ reached to $\mathbf{p}^{*}$ after $13$ seconds,
and $\gamma_{i}$ cease to increase as the bearing constraints were almost
satisfied. For $t\geq 15$s, as the disturbances increase their magnitudes,
$\mathbf{p}$ leaves $\mathbf{p}^{*}$ again. Then, adaptive gains $\gamma_{i}$
increase correspondingly, and eventually pull $\mathbf{p}$ back to
$\mathbf{p}^{*}$. It can be seen that the increment of $\gamma_{20}$ is
relatively slower than other displayed adaptive gains for $20\leq t\leq 35$s.
Chattering phenomenon can also be seen due to the disturbances (for $11\leq
t\leq 15$ and $20\leq t\leq 35$s), which cause significant fluctuations of
$\mathbf{p}$ around $\mathbf{p}^{*}$.

###  VI-C Bearing-based formation tracking

In this subsection, we simulate the formation (29) with moving leaders. The
leaders’ velocities are chosen as

$\mathbf{v}^{*}=\left[\sin\left(\frac{t}{2}\right),~{}1,~{}0\right]^{\top},\,t\geq
0.$

The simulation’s parameters are $\kappa=2$, $\gamma_{i}(0)=1$. The initial
positions of the agents are the same as in the previous simulation.
Disturbances are not included in the simulation.

(a)

(b)

(c)

(d)

Figure 5: Simulation 3: the 20-agent system with moving leaders under the
control law (3). (a) Trajectories of agents (leaders are marked with $\Delta$,
followers’ initial and final positions are marked with ${\rm x}$ and ${\rm
o}$, respectively); (b) Formation’s error versus time; (c) A subset of the
adaptive gains $\gamma_{ij}$ versus time; (d) The magnitude of the control
input versus time.

Simulation results are shown in Figure 5. It can be seen from Fig. 5(b) that
for $t\leq 6$ seconds, the formation’s error increases because the adaptive
control gains $\gamma_{i}(t)$, which specify magnitude of the control input,
is still quite small. For $t\geq 6$ second, the formation’s error
$\boldsymbol{\delta}$ decreases to 0. Fig. 5(c) shows that the adaptive gains
tend to increase for $0\leq t\leq 17$ second, and after the desired formation
has been achieved (approximately at $t=17$ second), $\gamma_{i}(t)$ remain
unchanged. The magnitude of the control input $\|\mathbf{u}(t)\|$ versus time
is correspondingly shown in Fig. 5(d), which vary accordingly to the adaptive
gains and the leaders’ velocity.

###  VI-D Bearing-only formation tracking

In this subsection, we simulate the formation with moving leaders (30). The
initial positions of the agents and the leaders’ velocities are chosen the
same as the previous simulation in section VII.C. The simulation’s parameters
are $\kappa=2$, $\gamma_{i}(0)=0.5$. The disturbances acting on agents $i$ are
chosen as

$\displaystyle\mathbf{d}_{i}(t)=\begin{cases}\mathbf{0}_{3},&\mbox{if }0\leq
t\leq 15s,\\\ 3\mathbf{h}_{i}(t),&\mbox{if }t\geq 15s.\end{cases}$ (34)

Simulation results are shown in Figure 6. For $0\leq t\leq 15$s, no
disturbances acting on agents, and the desired moving formation is tracked
after about 11 seconds. $\gamma_{i}$ are increasing during this time period.
The behavior of the system is quite similar to the previous simulation,
however, it is interesting to observed that the bearing-only control law (17)
somehow gives a relatively faster convergence rate then the displacement-based
control law (3). This can be explained by the fact that in (3), the
displacement $(\mathbf{p}_{i}-\mathbf{p}_{j})$ are projected into
im$(\mathbf{P}_{\mathbf{g}_{ij}^{*}})$. This makes $\mathbf{q}_{ij}$ becoming
relatively small, especially when the angles between
$(\mathbf{p}_{i}-\mathbf{p}_{j})$ and $\mathbf{p}_{i}^{*}-\mathbf{p}_{j}^{*}$
is small.

For $t\geq 15$s, due to the presence of the disturbances (which may be
originated from the wind or rain), $\mathbf{p}$ temporally cannot track
$\mathbf{p}^{*}$ (Figs. 6(a)–6(b)). Correspondingly, as depicted in Fig.
6(c)–6(d), the adaptive gains $\gamma_{i}$ and the control magnitude
$\|\mathbf{u}(t)\|$ increase again. As $\gamma_{i}$ is large enough, the
control law simultaneously rejects the disturbance and drives the agents to
its desired moving target point (approximately after 27 seconds).

(a)

(b)

(c)

(d)

Figure 6: Simulation 4: the 20-agent system under the control law (17) with
moving leaders. (a) Trajectories of agents (leaders are marked with $\Delta$,
followers’ initial and final positions are marked with ${\rm x}$ and ${\rm
o}$, respectively); (b) Formation’s error versus time; (c) A subset of the
adaptive gains $\gamma_{i}$ versus time; (d) The magnitude of the control
input versus time.

##  VII Conclusions

The bearing-constrained formation control with unknown bounded disturbances
has been studied for two types of measurements: displacements and bearing
vectors. The proposed control laws can adapt the control magnitudes separately
for each bearing constraint whenever the desired constraint has not been
satisfied. Once the control magnitudes have exceeded the magnitude of the
disturbances, it is possible to stabilize the desired configuration. Since the
disturbance’s magnitude may increase after the desired formation has been
achieved, it may temporarily make the agents leave the desired configuration.
The magnitude of the control laws will then increase accordingly to cope with
the disturbances and eventually stabilize the target formation again. This
process can be repeated as long as there is disturbance and control gains
which always depend on the constraints’ errors. Several modifications of the
proposed control laws with regard to the upper bounds of the matched
disturbance and the error’s bound have been also discussed. Notably, the
formation maneuver problem can be also solved with the proposed control
framework.

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|  Thanh Truong Nguyen received the Degree of Engineer (2020) in control and
automation engineering from Hanoi University of Science and Technology (HUST),
Hanoi, Vietnam. From 2020 to 2022, he worked as an automatic control engineer
with Flight Instrument Center, Viettel High Technology Industries Corporation,
Hanoi, Vietnam. He is currently an engineer at Bosch Global Software
Technologies, Hanoi, Vietnam. His research interests include control theory
and networked systems.
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|  Dung Van Vu received the Degree of Engineer (2020) in control and
automation engineering from Hanoi University of Science and Technology (HUST),
Hanoi, Vietnam. Since 2020, he has been with Viettel High Technology
Industries Corporation, Hanoi, Vietnam, where he is an automatic control
engineer at the Flight Instrument Center. His research interests include
control of multi-agent systems, optimal control, and control of unmanned
aerial vehicles.
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|  Tuynh Van Pham is a lecturer at Department of Automation Engineering,
School of Electrical and Electronic Engineering, Hanoi University of Science
and Technology (HUST), Vietnam. He received the Ph.D degree (2022) in control
and automation engineering from Chulalongkorn University, Thailand. His
research interests include control of network systems, robust control and
graph theory.
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|  Minh Hoang Trinh is a lecturer/researcher at the AI Department, FPT
University, Quy Nhon City, Binh Dinh, Vietnam. He received the B.S. degree
(2013) in electrical engineering from Hanoi University of Science and
Technology (HUST), Vietnam, the M.S. degree (2015) in mechatronics, and the
Ph.D degree (2018) in mechanical engineering both from Gwangju Institute of
Science and Technology (GIST), Republic of Korea. In 2016, he was at Technion
- Israel Institute of Technology, for a six-month research visit. From 2019 to
2023, he was a lecturer at Department of Automatic Control, School of
Electrical Engineering, Hanoi University of Science and Technology, Hanoi,
Vietnam. His research interests include distributed control of multi-agent
systems, graph rigidity theory, and network algorithms.
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