# Finite-time bearing-based maneuver of acyclic leader-follower formations

Minh Hoang Trinh, and Hyo-Sung Ahn The work of M.H. Trinh is funded by the
Hanoi University of Science and Technology (HUST) under project number
T2020-SAHEP-007. The work of H.-S. Ahn was supported by the National Research
Foundation of Korea (NRF) under the grant NRF- 2017R1A2B3007034M. H. Trinh is
with Department of Automatic Control, School of Electrical Engineering, Hanoi
University of Science and Technology (HUST), Hanoi 11615, Vietnam. E-mail:
<EMAIL_ADDRESS>Ahn is with School of Mechanical Engineering,
Gwangju Institute of Science and Technology (GIST), Gwangju 61005, Korea.
E-mail<EMAIL_ADDRESS>
###### Abstract

This letter proposes two finite-time bearing-based control laws for acyclic
leader-follower formations. The leaders in formation move with a bounded
continuous reference velocity and each follower controls its position with
regard to three agents in the formation. The first control law uses only
bearing vectors, and finite-time convergence is achieved by properly selecting
two state-dependent control gains. The second control law requires both
bearing vectors and communications between agents. Each agent simultaneously
localizes and follows a virtual target. Finite-time convergence of the desired
formation under both control laws is proved by mathematical induction and
supported by numerical simulations.

###### Index Terms:

bearing-only measurements, formation control, directed acyclic graph

##  I Introduction

For many years, much research efforts have been put on understanding the
mechanisms of collective behaviors displayed in nature and realizing them in
large-scale systems such as robotic-, sensor-, traffic-, and electrical
networks [1]. A notable application is formation control, where a team of
autonomous agents (UAVs, UUVs, mobile robots, etc) is required to achieve and
maintain a desired formation shape. Different solutions have been proposed to
the problem based on various assumptions on the
sensing/controlling/communication variables and topologies among the agents
[2].

In the bearing-based approach, the main sensing and controlling variables
among the agents are the bearing vectors (aka the directional information) or
the subtended angles which can be obtained from low-cost cameras [3, 4, 5].
Bearing-only formation control of a stationary formation with undirected [6,
7] and special directed topologies such as acyclic leader-follower [8, 9] or
directed cycle [10] have been studied in the literature. In flocking control,
the agents simultaneously form a desired formation and agree on their
velocities. To achieve flocking behavior, the agents sense the relative
geometric variables and the relative velocity with regard to a few followers
[11, 12, 13]. Formation tracking is more demanding since it requires the
agents to achieve a target formation and follow a few leaders, whose
velocities can be time-varying. The ability to maintain a moving target
formation shape is crucial for engineering applications such as search-and-
rescue, truck platooning, or flight maneuvering. The authors in [14] proposed
a control law for single and double-integrator agents using the relative
positions and relative velocities. Bearing-only formation tracking with
constant leaders’ velocity has been studied for single integrators [15, 16],
double integrators [15, 17], or robotics agents [18, 15, 19, 20]. However,
bearing-only formation tracking with time-varying leader’s velocity has not
been studied in the literature.

This work considers the bearing-only maneuver problem for directed acyclic
leader-follower formations where the leaders’ velocity is a bounded continuous
function, thus filling a gap in the literature. The directed acyclic leader-
follower structure can somehow describe the V-shape formation in immigrating
birds, where each bird only sees and follows several individuals (immediate
leaders) in its sight [21]. Thus, the interacting graph has a hierarchical
structure and the formation is led by a small number of leaders [22, 23]. To
ease the analysis, it is assumed that during the formation maneuver, the
positions of the immediate leaders are not collinear. Each follower, modeled
by a single-integrator, controls its position based on information obtained
from exactly three agents. First, a finite time bearing-only formation
maneuver control law is proposed. Finite-time convergence of the target
formation is achieved by appropriately choosing two control gains based on the
measured bearing vectors. Note that this method has not been introduced in
existing formation tracking laws in the literature [24, 25, 26]. Second, in
case the agents can communicate with each others, an estimation-based control
strategy is proposed. Each agent determines a virtual target point based on
the desired bearing vectors and the received position estimates of the
immediate leaders. Simultaneously, the agent tracks its target point and sends
this information to its followers. Although this control strategy requires
more information than just the sensed bearing vectors, the followers have some
freedom to choose their own trajectories to reach the target point. Thus,
collision avoidance between agents or obstacle can be included under this
approach. Finally, simulations are given to support the analysis.

The remainder of this paper is organized as follows. Section II contains
background and formulates the problem studied in this paper. Section III
proposes and studies the bearing-only formation tracking law. The target-point
based formation tracking strategy is investigated in Section IV. Section V
contains simulation results and Section VI concludes the letter.

##  II Preliminaries and problem formulation

###  II-A Preliminaries

####  II-A1 Notations

The set of real numbers is denoted by $\mathbb{R}$. Given $x\in\mathbb{R}$,
the signum function $\text{sign}(x)$ takes value 1, 0, -1 if $x>0$, $x=0$, and
$x<0$, respectively. For $\alpha>0$,
$\text{sig}(x)^{\alpha}=\text{sign}(x)|x|^{\alpha}$, where $|x|$ is the
absolute value of $x$. Also, let $\bm{x}\in\mathbb{R}^{d}$, one defines
$|\bm{x}|^{\alpha}=\sum_{i=1}^{d}|x_{i}|^{\alpha}$. The kernel and image of a
matrix $\bm{A}\in\mathbb{R}^{n\times n}$ are denoted by $\text{ker}(\bm{A})$
and ${\text{im}}(\bm{A})$. The vec operator is defined as
vec$(\bm{a}_{1},\ldots,\bm{a}_{n})=[\bm{a}_{1}^{\top},\ldots,\bm{a}_{n}^{\top}]^{\top}$.
Let $\bm{A}\in\mathbb{R}^{d\times d}$ be a symmetric matrix, one uses
$\lambda_{i}(\bm{A})$ to denote the $i$-th smallest eigenvalue of $\bm{A}$.

Let $\bm{x}$ be a nonzero vector in $\mathbb{R}^{d}$, defining the projection
matrix
$\bm{P}_{\bm{x}}=\bm{I}_{d}-\frac{\bm{x}\bm{x}^{\top}}{\|\bm{x}\|^{2}}$. For
any vector $\bm{y}\in\mathbb{R}^{d}$, $\bm{P}_{\bm{x}}\bm{y}$ is the
projection of $\bm{y}$ onto the orthogonal space of $\text{im}(\bm{x})$.
$\bm{P}_{\bm{x}}$ is symmetric, positive semidefinite,
$\text{ker}(\bm{P}_{\bm{x}})={\text{im}}(\bm{x})$, and in addition to a zero
eigenvalue, $\bm{P}_{\bm{x}}$ has $n-1$ eigenvalues 1.

####  II-A2 Graph theory

A directed graph $\mathcal{G}$ consists of a vertex set
$\mathcal{V}=\\{1,\ldots,n\\}$ of $|\mathcal{V}|=n$ vertices and an edge set
$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ of $|\mathcal{E}|=m$ edges.
The neighbor set of a vertex $i\in\mathcal{V}$ is defined as
$\mathcal{N}_{i}=\\{j\in\mathcal{V}|~{}(i,j)\in\mathcal{E}\\}$. A simple path
is a sequence of edges in $\mathcal{E}$ connecting vertices in $\mathcal{V}$
so that there is no repeated vertex (excepting for possibly the first and the
last vertices) and edges. If the first and the last vertices of a path
coincide, it is called a directed cycle. $\mathcal{G}$ is a directed acyclic
graph if it does not contain any directed cycle. Indexing the edges so that
$\mathcal{E}=\\{e_{1},\ldots,e_{m}\\}$, the incidence matrix
$\bm{H}=[h_{ki}]\in\mathbb{R}^{m\times n}$ has $h_{ki}=-1$ if $i$ is the
starting vertex of $e_{k}$, $h_{ki}=1$ if $i$ is the end vertex of $e_{k}$,
and $h_{ki}=0$, otherwise.

###  II-B Problem formulation

Consider a formation of $n$ agents in the $d$-dimensional space ($d\geq 2$).
Each agent in the formation has a local coordinate system ${}^{i}\Sigma$, and
the axes of these local coordinate systems are aligned to each other. The
position of agent $i$, written in a global coordinate system ${}^{g}\Sigma$,
is denoted by $\bm{p}_{i}\in\mathbb{R}^{d}$.

A formation is characterized by $(\mathcal{G},\bm{p})$, where
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ represents both the sensing and
control interactions among agents and
$\bm{p}=\text{vec}(\bm{p}_{1},\ldots,\bm{p}_{n})\in\mathbb{R}^{dn}$ is a
configuration of the formation. If $\bm{p}_{i}\neq\bm{p}_{j}$, the bearing
vector between $i$ and $j$ is defined as
$\bm{g}_{ij}=\frac{\bm{p}_{j}-\bm{p}_{i}}{\|\bm{p}_{j}-\bm{p}_{i}\|}=\frac{\bm{z}_{ij}}{\|\bm{z}_{ij}\|}$,
where $\bm{z}_{ij}$ is the displacement between agents $i$ and $j$. If
$(i,j)\in\mathcal{E}$, agent $i$ can sense $\bm{g}_{ij}$ and control a desired
bearing vector $\bm{g}_{ij}^{*}$ with regard to agent $j$. The set of desired
bearing vectors in the formation is denoted by
${\Gamma}=\\{\bm{g}_{ij}^{*}\\}_{(i,j)\in\mathcal{E}}$. It is assumed that
$\Gamma$ is feasible, i.e., there exists a target configuration
$\bm{p}^{*}=\text{vec}(\bm{p}_{1}^{*},\ldots,\bm{p}_{n}^{*})\in\mathbb{R}^{dn}$
satisfying all the bearing vectors in $\Gamma$.

Following the definition in [6], $(\mathcal{G},\bm{p})$ and
$(\mathcal{G},\bm{q})$ are bearing equivalent if and only if
$\bm{P}_{(\bm{p}_{i}-\bm{p}_{j})}(\bm{q}_{i}-\bm{q}_{j})=\bm{0},\forall(i,j)\in\mathcal{E}$.
They are bearing congruent if and only if
$\bm{P}_{(\bm{p}_{i}-\bm{p}_{j})}(\bm{q}_{i}-\bm{q}_{j})=\bm{0},\forall
i,j\in\mathcal{V},i\neq j$. A formation $(\mathcal{G},\bm{p})$ is globally
bearing rigid if any formation $(\mathcal{G},\bm{q})$ bearing equivalent to
$(\mathcal{G},\bm{p})$ is also bearing congruent to it.

The augmented bearing rigidity matrix is defined as
$\tilde{\bm{R}}_{B}=\text{diag}(\bm{P}_{\bm{g}_{k}})(\bm{H}\otimes\bm{I}_{d})$.
The formation $(\mathcal{G},\bm{p})$ is infinitesimally bearing rigid if and
only if the kernel of $\tilde{\bm{R}}_{B}$ is only spanned by infinitesimal
bearing rigid motions, i.e.,

$\text{ker}(\tilde{\bm{R}}_{B})=\text{im}(\bm{1}_{n}\otimes\bm{I}_{d},\bm{p})=\text{im}(\bm{1}_{n}\otimes\bm{I}_{d},\bm{p}-\bm{1}_{n}\otimes\bar{\bm{p}}),$

where $\bar{\bm{p}}=(\bm{1}_{n}^{\top}\otimes\bm{I}_{d})\bm{p}/n$ is the
formation centroid.

Suppose that in the formation, there are $l\geq 3$ leaders moving under the
following equation

$\displaystyle\dot{\bm{p}}_{i}=\bm{v}_{i},~{}i\in\mathcal{V}_{l}=\\{1,\ldots,l\\},$
(1)

where $\bm{v}_{i}$ is the velocity of the leader $i$. Denote
$\bm{p}^{l}=\text{vec}(\bm{p}_{1},\ldots,\bm{p}_{l})$, and
$\bm{v}^{l}=\text{vec}(\bm{v}_{1},\ldots,\bm{v}_{l})$. The remaining agents
are followers, which are modeled by single-integrators

$\displaystyle\dot{\bm{p}}_{i}=\bm{u}_{i},~{}\forall
i\in\mathcal{V}_{f}=\\{l+1,\ldots,n\\},$ (2)

where $\bm{p}_{i},\bm{u}_{i}\in\mathbb{R}^{d}$ are respectively the position
and the control input of agent $i$.

The following assumptions are adopted in this paper:

###### Assumption 1

The leaders are not in collinear positions and they move with the same bounded
continuous reference velocity $\bm{v}^{*}(t)$ satisfying
$\|\bm{v}^{*}(t)\|<\beta$, $\forall t\geq 0$. Furthermore, no collision
happens between the agents.

###### Assumption 2

The followers cannot sense the leaders’ velocity but have information on the
upper bound $\beta$. The bearing sensing and controlling graph $\mathcal{G}$
is a directed graph generated by the following procedure:

  * •

Starting with $l$ vertices $1,\ldots,l$ where $l\geq 3$.

  * •

For each $l+1\leq i\leq n$, inserting vertex $i$ together with $r\geq 3$ new
edges $(i,j_{k})$, where $j_{k}\in\\{1,\ldots,i-1\\}$.

Obviously, any graph $\mathcal{G}$ satisfies Assumption 2 is directed acyclic
(see Fig. 3(a) for an example).

###### Assumption 3

The set of desired bearing vectors $\Gamma$ is feasible. The desired bearing
vectors of each agent $i\in\mathcal{V}_{f}$, given by $\bm{g}_{ij}^{*},\forall
j\in\mathcal{N}_{i}$, are not all parallel to each other.

In this paper, the following problems will be studied.

###### Problem 1

Let Assumptions 1–3 hold. Design control laws for the followers modeled by (2)
using only bearing vector measurements so that
$\bm{g}_{ij}\to\bm{g}_{ij}^{*},\forall(i,j)\in\mathcal{E}$ in finite time.

###### Problem 2

Suppose that Assumptions 1–3 hold. Further, suppose that if
$(i,j)\in\mathcal{E}$, agent $j$ sends its position estimate
$\hat{\bm{p}}_{j}$ to agent $i$ and the leaders have information on their true
positions. Design control laws for the followers modeled by (2) so that
$\bm{g}_{ij}\to\bm{g}_{ij}^{*},\forall(i,j)\in\mathcal{E}$ in finite time.

##  III Finite-time bearing-only formation maneuver

###  III-A Proposed control law

The following bearing-only control law is proposed to solve Problem 1:

$\displaystyle\bm{u}_{i}$
$\displaystyle=-k_{i1}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}\text{sig}(\bm{P}_{\bm{g}_{ij}}\bm{g}_{ij}^{*})^{\alpha}$
$\displaystyle\qquad\qquad\qquad-
k_{i2}\beta\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}\text{sign}(\bm{P}_{\bm{g}_{ij}}\bm{g}_{ij}^{*}),$
(3)

where $i\in\mathcal{V}_{f}$, $\alpha\in(0,1)$,
$\bm{M}_{i}=\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}$,
$k_{i1}=\lambda_{1}(\bm{M}_{i})^{-\frac{\alpha+1}{2}}$,
$k_{i2}=\lambda_{1}(\bm{M}_{i})^{-\frac{1}{2}}$. In (3), the functions
$\text{sig}(\cdot)$ and $\text{sign}(\cdot)$ are defined component-wise,
$\bm{P}_{\bm{g}_{ij}}=\bm{I}_{d}-\bm{g}_{ij}\bm{g}_{ij}^{\top}\in\mathbb{R}^{d\times
d}$ is the projection matrix associating with $\bm{g}_{ij}$.

Some remarks on the proposed control law (3) are given as follows. First,
$\bm{u}_{i}$ consists of two terms: the first term is for controlling the
bearing-vectors of the agents to the desired ones; the second term is included
to reject the disturbances resulting from leaders’ motions which vary the
bearing vectors. Second, as the leaders are not collinear, $\bm{M}_{i}$ is
symmetric positive definite [9]. One has
$\lambda_{1}(\bm{M}_{i})\leq\lambda_{n}(\bm{M}_{i})=\|\bm{M}_{i}\|=\|\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}\|\leq\sum_{j\in\mathcal{N}_{i}}\|\bm{P}_{\bm{g}_{ij}}\|=|\mathcal{N}_{i}|$.
Thus $k_{i1}\in(|\mathcal{N}_{i}|^{-\frac{\alpha+1}{2}},\infty)$ and
$k_{i2}\in\left(|\mathcal{N}_{i}|^{-\frac{1}{2}},\infty\right)$. The number of
neighbors for each follower $i$ is at least three, so that if its neighbors
are in a non-collinear configuration and no collision happens between agents,
$\lambda_{1}(\bm{M}_{i})>0$.

###  III-B Convergence Analysis

Since $\mathcal{G}$ has a directed acyclic structure, the convergence analysis
begins from the first follower. It will be shown that the first follower can
achieve the desired bearing vectors and follow the leaders in finite time.
Then, finite-time convergence of the overall desired formation will be
established based on mathematical induction.

####  III-B1 The first follower

Consider agent $i=l+1$ moving under the control law (3). Define the
subformation $(\mathcal{K}_{l},\bm{p}^{l})$, where $\mathcal{K}_{l}$ is the
complete graph of $l$ vertices. From Assumption 1,
$\bm{v}^{l}=\bm{1}_{l}\otimes\bm{v}^{*}$ belongs to the space of infinitesimal
bearing rigid motions of $(\mathcal{K}_{l},\bm{p}^{l})$. Thus, the relative
bearing vector between the leaders are maintained during maneuver.

The desired position of follower $i$ is determined by equations
$\bm{P}_{\bm{g}_{ij}^{*}}(\bm{p}_{i}^{*}-\bm{p}_{j})=\bm{0}$, $\forall
j\in\mathcal{N}_{i}$. By summing up these equations, it follows that
$\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}(\bm{p}_{i}^{*}-\bm{p}_{j})=\bm{0}$
or equivalently
$\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\bm{p}_{i}^{*}=\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\bm{p}_{j}$.
According to the Assumption 1, the leaders are not colinear, which implies
that the matrix $\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}$ is
positive definite. It follows that
$\bm{p}_{i}^{*}=\big{(}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\big{)}^{-1}\big{(}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\bm{p}_{j}\big{)}$
and thus,
$\dot{\bm{p}}_{i}^{*}={\bm{v}}_{i}^{*}=\big{(}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\big{)}^{-1}\big{(}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\dot{\bm{p}}_{j}\big{)}=\bm{v}^{*}$.

###### Lemma 1

Suppose that Assumptions 1–3 hold. Under the control law (3), for $i=l+1$,
$\bm{p}_{i}\to\bm{p}_{i}^{*}$ in finite time.

###### Proof:

Consider the Lyapunov function
$V_{i}=\frac{1}{2}\|\bm{p}_{i}-\bm{p}_{i}^{*}\|^{2}$, which is positive
definite and radially unbounded. As the control law (3) is discontinuous,
solution of (2) is understood in Filippov’s sense [27]. One has
$\dot{V}_{i}\in^{a.e.}\dot{\tilde{V}}_{i}=\bigcap_{\bm{\xi}\in\nabla{V}_{i}(\bm{p}_{i})}\bm{\xi}^{\top}K[\dot{\bm{p}}_{i}]$.
Then, $\nabla{V}_{i}(\bm{p}_{i})=\bm{p}_{i}-\bm{p}_{i}^{*}$, and

$\displaystyle\dot{\tilde{V}}_{i}$
$\displaystyle=(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\Big{(}-k_{i1}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}\text{sig}(\bm{P}_{\bm{g}_{ij}}\bm{g}_{ij}^{*})^{\alpha}$
$\displaystyle\qquad\qquad-
k_{i2}\beta\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}K[\text{sign}](\bm{P}_{\bm{g}_{ij}}\bm{g}_{ij}^{*})-{\bm{v}}^{*}\Big{)}$
$\displaystyle=(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\Big{(}-k_{i1}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}\text{sig}\Big{(}\bm{P}_{\bm{g}_{ij}}\frac{\bm{p}_{j}-\bm{p}_{i}^{*}}{\|\bm{z}_{ij}^{*}\|}\Big{)}^{\alpha}$
$\displaystyle\qquad\qquad-
k_{i2}\beta\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}K[\text{sign}](\bm{P}_{\bm{g}_{ij}}\bm{g}_{ij}^{*})-{\bm{v}}^{*}\Big{)}.$

Because
$\bm{P}_{\bm{g}_{ij}}(\bm{p}_{j}-\bm{p}_{i}^{*})=\bm{P}_{\bm{g}_{ij}}(\bm{p}_{j}-\bm{p}_{i}+\bm{p}_{i}-\bm{p}_{i}^{*})=\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*}),$
it follows that

$\displaystyle\dot{\tilde{V}}$
$\displaystyle=-k_{i1}(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}\text{sig}\left(\frac{\bm{P}_{\bm{g}_{ij}}}{\|\bm{z}_{ij}^{*}\|}(\bm{p}_{i}-\bm{p}_{i}^{*})\right)^{\alpha}$
$\displaystyle\qquad-
k_{i2}\beta(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}K[\text{sign}](\bm{P}_{\bm{g}_{ij}}\bm{g}_{ij}^{*})$
$\displaystyle\qquad\qquad-(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}{\bm{v}}^{*}$
$\displaystyle\leq-
k_{i1}(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}\text{sig}\Big{(}\frac{1}{\|\bm{z}_{ij}^{*}\|}\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*})\Big{)}^{\alpha}$
$\displaystyle\qquad-
k_{i2}\beta(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}K[\text{sign}](\bm{P}_{\bm{g}_{ij}}\bm{g}_{ij}^{*})$
$\displaystyle\qquad\qquad+\|\bm{p}_{i}-\bm{p}_{i}^{*}\|\|{\bm{v}}^{*}\|$ (4)

From [27] that for any $x\in\mathbb{R}$, $xK[\text{sign}](x)=\\{|x|\\}$, one
has

$\displaystyle(\bm{p}_{i}$
$\displaystyle-\bm{p}_{i}^{*})^{\top}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}\text{sig}\Big{(}\frac{1}{\|\bm{z}_{ij}^{*}\|}\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*})\Big{)}^{\alpha}$
$\displaystyle=\sum_{j\in\mathcal{N}_{i}}\frac{1}{\|\bm{z}_{ij}^{*}\|^{\alpha}}|\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*})|^{\alpha+1}$
$\displaystyle=\sum_{j\in\mathcal{N}_{i}}\frac{1}{\|\bm{z}_{ij}^{*}\|^{\alpha}}|(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*})|^{\frac{\alpha+1}{2}}$
$\displaystyle\geq\frac{1}{\max_{j\in\mathcal{N}_{i}}(\|\bm{z}_{ij}^{*}\|)^{\alpha}}\left|(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\bm{M}_{i}(\bm{p}_{i}-\bm{p}_{i}^{*})\right|^{\frac{\alpha+1}{2}}$
$\displaystyle\geq\frac{1}{\max_{j\in\mathcal{N}_{i}}(\|\bm{z}_{ij}^{*}\|)^{\alpha}}\big{(}\lambda_{1}(\bm{M}_{i})\big{)}^{\frac{\alpha+1}{2}}\|\bm{p}_{i}-\bm{p}_{i}^{*}\|^{\alpha+1},$

where the first inequality follows from [28][Lemma 2], and

$\displaystyle(\bm{p}_{i}$
$\displaystyle-\bm{p}_{i}^{*})^{\top}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}K[\text{sign}](\bm{P}_{\bm{g}_{ij}}\bm{g}_{ij}^{*})$
$\displaystyle=\sum_{j\in\mathcal{N}_{i}}(\bm{p}_{i}-\bm{p}_{i}^{*})^{\top}\bm{P}_{\bm{g}_{ij}}K[\text{sign}]\Big{(}\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*})\Big{)}$
$\displaystyle=\sum_{j\in\mathcal{N}_{i}}\|\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*})\|_{1}\geq\left|\left|\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*})\right|\right|_{1}$
$\displaystyle\geq\left|\left|\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}(\bm{p}_{i}-\bm{p}_{i}^{*})\right|\right|\geq{\lambda_{1}(\bm{M}_{i})}^{\frac{1}{2}}\|\bm{p}_{i}-\bm{p}_{i}^{*}\|.$

Thus, one has

$\displaystyle\dot{V}_{i}$ $\displaystyle\leq-\eta
V_{i}^{\frac{\alpha+1}{2}}-\big{(}\beta-\|{\bm{v}}^{*}\|\big{)}\|\bm{p}_{i}-\bm{p}_{i}^{*}\|,$
(5)

where
$\eta=\frac{2^{\frac{\alpha+1}{2}}}{\max_{j\in\mathcal{N}_{i}}(\|\bm{z}_{ij}^{*}\|)^{\alpha}}$.
Based on [29], $\bm{p}_{i}\to\bm{p}_{i}^{*}$ in a finite time upper bounded by
$T_{l+1}=\frac{2V_{l+1}(0)^{\frac{1-\alpha}{2}}}{\eta(1-\alpha)}$. ∎

####  III-B2 The n-agent formation

For each agent $i\geq l+1$, the velocities of agents
$j\in\mathcal{N}_{i}\subset\\{1,\ldots,i-1\\}$ are considered as external
inputs to the dynamics of agent $i$. The convergence of the overall formation
is given in the following theorem.

###### Theorem 1

Suppose that Assumptions 1–3 hold. Further, suppose that each agent and its
neighbors are always non-collinear. Under control law (3),
$\bm{p}_{i}(t)\to\bm{p}_{i}^{*}$ in finite time upper bounded by
$T_{i}=T_{i-1}+\frac{2{V}_{i}(T_{i-1})^{\frac{1-\alpha}{2}}}{\eta(1-\alpha)}$,
$i=l+1,\ldots,n$ and $T_{l}=0$, and $\bm{u}_{i}$ is bounded $\forall
i=l+1,\ldots,n$.

###### Proof:

We show this theorem by mathematical induction on $i$. From Lemma 1, the claim
is true for $i=l+1$. Next, suppose that the claim holds until $i~{}(l+1\leq
i\leq n)$, then $\forall j\in\mathcal{N}_{i}$, during $[0,T_{j-1}]$,
$\bm{u}_{j}$ is bounded and $\bm{p}_{j}=\bm{p}_{j}^{*}$ for $t\geq T_{j}$,
where
$T_{k}=T_{k-1}+\frac{2{V}_{i}(T_{k-1})^{\frac{1-\alpha}{2}}}{\eta(1-\alpha)}$,
$\forall k=l+1,\ldots,i-1$ and $T_{l}=0$. Since agent $i$ and its neighbor
agents are not collinear and no collisions happen,
$\bm{M}_{i}=\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}}$ is always positive
definite and $k_{i1},k_{i2}<\infty$ in $t\in[0,T_{i-1}]$. It follows that
$\dot{\bm{p}}_{i}(t)$ is bounded and thus
$\|\bm{p}_{i}(T_{i-1})-\bm{p}_{i}^{*}(T_{i-1})\|<\infty$.

Next, for $t\geq T_{i-1}$, consider the Lyapunov function
$V_{i}=\frac{1}{2}\|\bm{p}_{i}-\bm{p}_{i}^{*}\|^{2}$. Similar to Lemma 1, one
has

$\displaystyle\dot{V}_{i}$
$\displaystyle\leq-\frac{\|\bm{p}_{i}-\bm{p}_{i}^{*}\|^{\alpha+1}}{\max_{j\in\mathcal{N}_{i}}\|\bm{z}_{ij}^{*}\|^{\alpha}}-\|\bm{p}_{i}-\bm{p}_{i}^{*}\|(\beta-\|\bm{v}^{*}\|).$
(6)

This means $\bm{p}_{i}(t)\to\bm{p}_{i}^{*}$ in a finite time upper bounded by
$\sum_{k=l+1}^{i}T_{k}$, where
$T_{i}=\frac{2V_{i}(T_{i-1})^{\frac{1-\alpha}{2}}}{\eta(1-\alpha)}$.

Therefore, the claim is also true for $i=n$ by mathematical induction. ∎

###### Remark 1

After the target formation was achieved, the leaders can rescale the formation
by adopting the reference velocity
$\bm{v}^{l}(t)\in\text{im}(\bm{1}_{l}\otimes\bm{I}_{d},\bm{p}^{l}(t))$,
$\bm{v}^{l}(t)=\bm{v}^{l}_{\text{trans}}(t)+k_{\text{scale}}(t)\bm{v}^{l}_{\text{scale}}(t)$,
where $\bm{v}^{l}_{\text{trans}}(t)=\bm{1}_{l}\otimes\bm{v}^{*}$ and
$\bm{v}^{l}_{\text{scale}}(t)=\frac{\bm{p}^{l}-\bar{\bm{p}}^{l}\otimes\bm{1}_{l}}{\|\bm{p}^{l}-\bar{\bm{p}}^{l}\otimes\bm{1}_{l}\|}$,
$\bar{\bm{p}}^{l}=(\bm{1}_{l}\otimes\bm{I}_{d})\bm{p}^{l}/l$, are the
translational and scaling motions of the formation
$(\mathcal{K}_{l},\bm{p}^{l})$, respectively. Suppose that
$\bm{v}^{l}_{\text{trans}}$ is bounded and $k_{\text{scale}}(t)=\xi>0$,
$t\in[t_{1},t_{2}]$, $0\leq t_{1}<t_{2}<\infty,$ and $k_{\text{scale}}(t)=0$,
otherwise. This assumption describes that the formation mainly moves forward
and the rescaling process is occasionally performed during the maneuver, e.g.,
before and after traversing a narrow alley. For $t\geq t_{2}$, convergence of
the desired formation can be shown as in Thm. 1.

###### Remark 2

Since collision between agents and collinearity of
$\\{\bm{p}_{j}\\}_{j\in\mathcal{N}_{i}},\forall i=l+1,\ldots,n$ are excluded,
one cannot conclude in Thm. 1 about globally asymptotic stability of the
target formation. Also, the mathematical induction cannot be established
without finite time convergence of each follower since boundedness of
$\|\bm{p}_{i}-\bm{p}_{i}^{*}\|,\forall t$ will not be guaranteed.

##  IV Finite-time formation maneuver via target point localization

###  IV-A Proposed control law

In this section, we propose an control strategy to solve Problem 2. Instead of
directly controlling the position based on the bearing errors, each follower
estimates its desired position with regard to the leaders and track that
point. The following finite-time control law is proposed:

$\displaystyle\dot{\hat{\bm{p}}}_{i}$
$\displaystyle=\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\text{sig}\big{(}\bm{P}_{\bm{g}_{ij}^{*}}(\hat{\bm{p}}_{j}-\hat{\bm{p}}_{i})\big{)}^{\alpha}+\gamma_{i}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\text{sign}\big{(}\bm{P}_{\bm{g}_{ij}^{*}}(\hat{\bm{p}}_{j}-\hat{\bm{p}}_{i})\big{)},$
(7) $\displaystyle\dot{\bm{p}}_{i}$
$\displaystyle=-\text{sig}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}^{\alpha}-\beta\text{sign}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)},$
(8)

where $\alpha\in(0,1)$ and
$\gamma_{i}\geq\beta\left|\left|\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\right|\right|^{-1}$.
Due to the directed acyclic structure of the graph $\mathcal{G}$, agent $i$
can also calculate the target point directly from the information by
$\bm{p}_{i}^{*}=(\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}})^{-1}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\bm{p}_{j}^{*}$,
and track that point under the control law (8). In case the beacons are
stationary, a fixed-time network localization law was proposed in [30].

###  IV-B Convergence analysis

Consider the first follower $i=l+1$. The following lemma will be proved.

###### Lemma 2

Let the assumptions of Problem 2 hold, for $i=l+1$,
${\hat{\bm{p}}}_{i}(t)\to{{\bm{p}}}_{i}^{*}$ and
${\bm{p}}_{i}(t)\to{\bm{p}}_{i}^{*}$ in finite time.

###### Proof:

First, consider the estimation dynamics (7). Using the Lyapunov function
$V=\frac{1}{2}\|\hat{\bm{p}}_{i}-\bm{p}^{*}_{i}\|^{2}$ and keeping in mind
that $\hat{\bm{p}}_{j}=\bm{p}_{j}^{*},\forall j\in\mathcal{N}_{i}$, one has

$\displaystyle\dot{V}$
$\displaystyle=(\hat{\bm{p}}_{i}-\bm{p}^{*}_{i})^{\top}\Big{(}\bm{P}_{\bm{g}_{ij}^{*}}\sum_{j\in\mathcal{N}_{i}}\text{sig}\big{(}\bm{P}_{\bm{g}_{ij}^{*}}({\bm{p}}_{j}^{*}-{\bm{p}}_{i}^{*}+{\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})\big{)}^{\alpha}+\gamma_{i}\bm{P}_{\bm{g}_{ij}^{*}}\sum_{j\in\mathcal{N}_{i}}\text{sign}\big{(}\bm{P}_{\bm{g}_{ij}^{*}}({\bm{p}}_{j}^{*}-{\bm{p}}_{i}^{*}+{\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})\big{)}-\bm{v}^{*}\Big{)}$
$\displaystyle=-\sum_{j\in\mathcal{N}_{i}}|\bm{P}_{\bm{g}_{ij}^{*}}({\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})|^{\alpha+1}-\gamma_{i}\sum_{j\in\mathcal{N}_{i}}\|\bm{P}_{\bm{g}_{ij}^{*}}({\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})\|_{1}-(\hat{\bm{p}}_{i}-\bm{p}^{*}_{i})^{\top}\bm{v}^{*}$
$\displaystyle\leq-\left|({\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})^{\top}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}({\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})\right|^{\frac{\alpha+1}{2}}-\gamma_{i}\left|\left|\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}({\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})\right|\right|_{1}+\|\bm{v}^{*}\|\|{\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i}\|$
$\displaystyle\leq-\eta_{i}|({\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})^{\top}({\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i})|^{\frac{\alpha+1}{2}}-\gamma_{i}\left|\left|\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\right|\right|\|{\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i}\|+\|\bm{v}^{*}\|\|{\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i}\|$
$\displaystyle\leq-{\chi_{i}}V^{\frac{\alpha+1}{2}}-\Big{(}\gamma_{i}\|\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\|-\|\bm{v}^{*}\|\Big{)}\|{\bm{p}}_{i}^{*}-\hat{\bm{p}}_{i}\|$
(9)

where
$\eta_{i}=\lambda_{1}\big{(}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\big{)}^{\frac{\alpha+1}{2}}$
and $\chi_{i}={\eta_{i}}2^{\frac{\alpha+1}{2}}$. It follows that
${\hat{\bm{p}}}_{i}(t)\to{{\bm{p}}}_{i}^{*}$ in finite time upper bounded by
$T_{i}$, and $\hat{\bm{v}}_{i}\to{\bm{v}}_{i}^{*}$ in finite time.

Next, consider the position tracking control law (8) with the Lyapunov
function $W=\frac{1}{2}\|\bm{p}_{i}-{\hat{\bm{p}}}_{i}\|^{2}$. For $t\leq
T_{i}$,

$\displaystyle\dot{W}$
$\displaystyle=-(\bm{p}_{i}-{\hat{\bm{p}}}_{i})^{\top}\text{sig}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}^{\alpha}-\beta(\bm{p}_{i}-{\hat{\bm{p}}}_{i})^{\top}\text{sign}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}-(\bm{p}_{i}-{\hat{\bm{p}}}_{i})^{\top}{\hat{\bm{v}}}_{i}$
$\displaystyle\leq-|\bm{p}_{i}-\hat{\bm{p}}_{i}|^{\alpha+1}-\beta\|\bm{p}_{i}-{\hat{\bm{p}}}_{i}\|_{1}+\|\bm{p}_{i}-{\hat{\bm{p}}}_{i}\|\|{\hat{\bm{v}}}_{i}\|,$
(10)

which shows that $\|\bm{p}_{i}-{\hat{\bm{p}}}_{i}\|$ is globally ultimately
bounded. Together with the boundedness of
$\|{\hat{\bm{p}}}_{i}-\bm{p}_{i}^{*}\|$, it follows from the triangle
inequality that
$\|\bm{p}_{i}-\bm{p}_{i}^{*}\|\leq\|\bm{p}_{i}-{\hat{\bm{p}}}_{i}\|+\|{\hat{\bm{p}}}_{i}-\bm{p}_{i}^{*}\|$,
i.e., $\|\bm{p}_{i}-\bm{p}_{i}^{*}\|$ is also bounded.

Now, for $t\geq T_{i}$, $\hat{\bm{p}}_{i}=\bm{p}^{*}_{i}$ and
$\|\hat{\bm{v}}_{i}\|=\|{\bm{v}}^{*}\|\leq\beta$, consider the function
$W_{1}=\frac{1}{2}\|\bm{p}_{i}-{\hat{\bm{p}}}_{i}^{*}\|^{2}$. Similar to (10),
one finds that
$\dot{W}_{1}\leq-|\bm{p}_{i}-\hat{\bm{p}}_{i}|^{\alpha+1}-\|\bm{p}_{i}-{\hat{\bm{p}}}_{i}\|(\beta-\|{\hat{\bm{v}}}_{i}\|)\leq-\zeta_{i}W_{1}^{\frac{\alpha+1}{2}}$,
where $\zeta_{i}=2^{\frac{\alpha+1}{2}}>0$ is a positive constant. Thus,
$\bm{p}_{i}\to\bm{p}_{i}^{*}$ in finite time. ∎

###### Theorem 2

Under the control laws (7)–(8), the desired moving formation is achieved in
finite time.

###### Proof:

The proof follows from Lemma 2 and mathematical induction on $i=l+1$ to $n$. ∎

###### Remark 3

Observe that (8) steers the agent to the virtual target along a curve. Let
$d=2$ and assume that the agents are equipped with proximity sensors which can
sense the distance to an obstacle located at $\bm{p}_{o}$ within a small range
$d_{max}$. To avoid collision with an obstacle, for $0<d<d_{\max}$, (8) can be
modified as follows:

$\displaystyle{\bm{u}}_{i}$
$\displaystyle=-(1-\zeta)\Big{(}\text{sig}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}^{\alpha}+\beta\text{sign}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}\Big{)}+\zeta\big{(}k(\bm{p}_{i}-{\bm{p}}_{o})+\bm{g}_{io}^{\perp}\big{)},$
(11)

where $\zeta=1$ if $\|\bm{p}_{i}-\bm{p}_{0}\|<d$ and $\zeta=0$ if
$\|\bm{p}_{i}-\bm{p}_{o}\|\geq d$,
$\bm{g}_{io}^{\perp}=\bm{J}\frac{\bm{p}_{o}-\bm{p}_{i}}{\|\bm{p}_{o}-\bm{p}_{i}\|}$,
$\bm{J}=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}$, $k>1$ is a sufficiently
large control gain.

###### Proof:

Denote ${C}=\\{\bm{q}\in\mathbb{R}^{2}:\|\bm{q}-\bm{p}_{o}\|<d\\}$ and
$\partial C=\\{\bm{q}\in\mathbb{R}^{2}:\|\bm{q}-\bm{p}_{o}\|=d\\}$. Consider
the following scenarios:

  * •

Case 1: The solution of
$\dot{\bm{p}}_{i}=-\Big{(}\text{sig}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}^{\alpha}+\beta\text{sign}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}\Big{)},{\bm{p}}_{i}(t_{0})={\bm{p}}_{i0}$,
which is called $S$, does not intersect $C$. Then, $\zeta=0,\forall t$ and the
agent moves along the curve to the target under the control law:
${\bm{u}}_{i}=-\Big{(}\text{sig}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}^{\alpha}+\beta\text{sign}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}\Big{)}$.

Figure 1: Illustration of the control law (1): the blue dashed-line is the
trajectory without the obstacle. The blue line approximates the real
trajectory of the agent toward the target point.

  * •

Case 2: The solution of
$\dot{\bm{p}}_{i}=-\Big{(}\text{sig}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}^{\alpha}+\beta\text{sign}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}\Big{)},{\bm{p}}_{i}(t_{0})={\bm{p}}_{i0}$
intersects $C$.

    * –

Case 2 (a): $\bm{p}_{i}\in C$. In this case, $\zeta=1$ and the motion of agent
$i$ is governed by the control law
$\bm{u}_{i}=\big{(}\bm{p}_{i}-{\bm{p}}_{o}\big{)}+\bm{g}_{io}^{\perp}$. This
control law consists of two terms: the first term steers the agent toward the
boundary of $C$ in a finite-time. The second one steers the agent along the
orthogonal space of $\bm{g}_{io}$. Note that inside $C$, the first control
term dominates the second control term (because $k>>1$) and thus it prevents
possible collision. The second term changes the bearing vector $\bm{g}_{io}$.
Thus, the combination of two control laws eventually steers the agent to a
position in the boundary of $C$, denoted by $\bm{p}_{i}^{\prime}$. Two cases
may happen here: (i) the solution $S^{\prime}$ of
$\dot{\bm{p}}_{i}=-\Big{(}\text{sig}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}^{\alpha}+\beta\text{sign}\big{(}\bm{p}_{i}-\hat{\bm{p}}_{i}\big{)}\Big{)},{\bm{p}}_{i}(0)=\bm{p}_{i}^{\prime}$
does not intersect the region $C$, then we return to Case 1; (ii) $S^{\prime}$
intersects the region $C$, then we have Case 2(b). These arguments are
illustrated in Figure 1. To show these arguments, consider:

$\displaystyle\frac{d}{dt}d_{io}^{2}$
$\displaystyle=2(\bm{p}_{i}-\bm{p}_{o})^{\top}\dot{\bm{p}_{i}}$
$\displaystyle=2k(\bm{p}_{i}-\bm{p}_{o})^{\top}\big{(}\bm{p}_{i}-{\bm{p}}_{o}\big{)}+2(\bm{p}_{i}-\bm{p}_{o})^{\top}\bm{g}_{io}^{\perp}$
$\displaystyle=2k\|\bm{p}_{i}-\bm{p}_{o}\|^{2}>0,\forall\bm{p}_{o}\neq\bm{p}_{i}\in
C$

This implies $d_{io}\to d$ and $\bm{p}_{i}\to\partial C$ in a finite time.
Moreover,

$\displaystyle\frac{d}{dt}\bm{g}_{io}$
$\displaystyle=-\frac{\bm{P}_{\bm{g}_{io}}}{d_{io}}\dot{\bm{p}_{i}}=-\frac{\bm{P}_{\bm{g}_{io}}}{d_{io}}\big{(}\bm{p}_{i}-{\bm{p}}_{o}\big{)}-\frac{\bm{P}_{\bm{g}_{io}}}{d_{io}}\bm{g}_{io}^{\perp}=-\frac{1}{d_{io}}\bm{g}_{io}^{\perp}(\bm{g}_{io}^{\perp})^{\top}\bm{g}_{io}^{\perp}=-\frac{1}{d_{io}}\bm{g}_{io}^{\perp},$

and thus, the bearing vector $\bm{g}_{io}$ changes its direction if
$\bm{p}_{i}$ enters the set $C$.

    * –

Case 2 (b) $\bm{p}_{i}\notin C$. Then, the agent $i$ keeps moving toward the
target until it enters $C$. After $\bm{p}_{i}\in C$, we analyze as in Case
2(a).

∎

Figure 2: Simulation of the control law (1): the agent tracks the target point
after finite time.

We simulate the control law (11) and demonstrate the result as in Fig. 2. The
virtual target point is initially located at $[0,0]^{\top}$ and it moves with
velocity $\hat{\bm{v}}_{i}=[0.2*\cos(t),-0.2]^{\top}$. The obstacle is located
at $[0.5,2]^{\top}$. The control law’s parameters: $d=0.5$ and $k=5$. For
three different initial positions of the agent, we have three corresponding
trajectories (blue, green, magenta). It can be seen that the agent can track
the target point after finite time and achieve collision avoidance.

However, it is noted that the proposed collision scheme will not work if
multiple obstacles are presented in the system.

###### Remark 4

A fixed-time position estimation dynamics can be designed based on (7) and
[30, 31] as follows:

$\displaystyle\dot{\hat{\bm{p}}}_{i}$
$\displaystyle=\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\Big{(}\text{sig}(\bm{P}_{\bm{g}_{ij}^{*}}(\hat{\bm{p}}_{j}-\hat{\bm{p}}_{i}))^{\alpha}+\text{sig}(\bm{P}_{\bm{g}_{ij}^{*}}(\hat{\bm{p}}_{j}-\hat{\bm{p}}_{i}))^{\rho}\Big{)}+\gamma_{i}\sum_{j\in\mathcal{N}_{i}}\bm{P}_{\bm{g}_{ij}^{*}}\text{sign}\big{(}\bm{P}_{\bm{g}_{ij}^{*}}(\hat{\bm{p}}_{j}-\hat{\bm{p}}_{i})\big{)},$
(12)

where $0<\alpha<1$ and $\rho>1$. Note that (12) is also applicable for
infinitesimally bearing rigid leader-follower formations.

###### Proof:

Let the leaders be indexed by $1,\ldots,l$ and the followers be indexed by
$l+1,\ldots,n$. By noting that the desired moving formation is moving with the
same velocity, one may write

$\displaystyle\dot{\bm{p}}^{*}$ $\displaystyle=\bm{1}_{n}\otimes\bm{v}^{*}$

where
$\bm{p}^{*}=\text{vec}(\bm{p}^{l*},\bm{p}^{f*})=\text{vec}(\bm{p}_{1}^{*},\ldots,\bm{p}_{n}^{*})$.
Let $\hat{\bm{p}}_{i}$ denote the estimate of the target position of agent
$i$, and define
$\hat{\bm{p}}=\text{vec}(\hat{\bm{p}}^{l},\hat{\bm{p}}^{f})=\text{vec}(\hat{\bm{p}}_{1},\ldots,\hat{\bm{p}}_{n})$.
Since the leaders know their positions and has already been at the desired
position,
$\hat{\bm{p}}^{l}=\bm{p}^{l*}=\text{vec}(\bm{p}_{1}^{*},\ldots,\bm{p}_{l}^{*})$.
Defining the matrix $\bm{Z}=\begin{bmatrix}\bm{0}_{dl\times
dl}&\bm{0}_{dl\times df}\\\ \bm{0}_{df\times dl}&\bm{I}_{df}\end{bmatrix}$,
the equation governing the dynamic of the estimated desired configuration is
given as follows:

$\displaystyle\dot{\hat{\bm{p}}}=-\bm{Z}\bar{\bm{H}}^{\top}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\Big{(}\text{sig}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}^{\alpha}+\text{sig}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}^{\rho}+(\bm{\Gamma}\otimes\bm{I}_{d})\text{sign}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}\Big{)}+\text{vec}(\bm{1}_{l}\otimes\bm{v}^{*},\bm{0}_{df}),$
(13)

where $\bm{\Gamma}=\text{diag}(0,\ldots,0,\gamma_{l+1},\ldots,\gamma_{n})$.
The solution of (13) is understood in Fillipov sense.

Without loss of generalization, we will only consider the case
$\gamma_{i}=\gamma_{j}=\gamma,\forall i=l+1,\ldots,n$ in the following
analysis. Consider the Lyapunov function:
$V=\frac{1}{2}\|\hat{\bm{p}}-\bm{p}^{*}\|^{2}$, one has
$\dot{V}\in^{a.e.}\dot{\tilde{V}}$, and

$\displaystyle\dot{\tilde{V}}=$
$\displaystyle-(\hat{\bm{p}}-\bm{p}^{*})^{\top}\bm{Z}\bar{\bm{H}}^{\top}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\Big{(}\text{sig}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}^{\alpha}+\text{sig}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}^{\rho}$
$\displaystyle\qquad+\gamma
K[\text{sign}]\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}\Big{)}-(\hat{\bm{p}}-\bm{p}^{*})^{\top}[(\bm{1}_{l}\otimes\bm{v}^{*},\bm{0}_{df})-\bm{1}_{n}\otimes\bm{v}^{*}]$
$\displaystyle=$
$\displaystyle-\text{vec}(\bm{0}_{dl},\hat{\bm{p}}^{f}-\bm{p}^{f*})^{\top}\bar{\bm{H}}^{\top}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\Big{(}\text{sig}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}^{\alpha}+\text{sig}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}^{\rho}$
$\displaystyle\qquad+\gamma
K[\text{sign}]\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bm{H}\hat{\bm{p}}\big{)}\Big{)}-(\hat{\bm{p}}^{f}-\bm{p}^{f*})^{\top}(\bm{1}_{f}\otimes\bm{v}^{*})$
$\displaystyle=$
$\displaystyle-(\hat{\bm{p}}-\bm{p}^{*})^{\top}\bar{\bm{H}}^{\top}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\Big{(}\text{sig}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}^{\alpha}+\text{sig}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}^{\rho}$
$\displaystyle\qquad+\gamma\text{sign}\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}\hat{\bm{p}}\big{)}\Big{)}-(\hat{\bm{p}}^{f}-\bm{p}^{f*})^{\top}(\bm{1}_{f}\otimes\bm{v}^{*})$
$\displaystyle=$
$\displaystyle-|\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}(\hat{\bm{p}}-\bm{p}^{*})|^{\alpha+1}-|\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}(\hat{\bm{p}}-\bm{p}^{*})|^{\rho+1}$
$\displaystyle\qquad-\gamma\|\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}(\hat{\bm{p}}-\bm{p}^{*})\|_{1}-(\hat{\bm{p}}^{f}-\bm{p}^{f*})^{\top}(\bm{1}_{f}\otimes\bm{v}^{*})$
$\displaystyle\leq$
$\displaystyle-\|(\hat{\bm{p}}-\bm{p}^{*})^{\top}\bm{L}_{b}(\bm{p}^{*})(\hat{\bm{p}}-\bm{p}^{*})\|^{\frac{\alpha+1}{2}}-\|(\hat{\bm{p}}-\bm{p}^{*})^{\top}\bm{L}_{b}(\bm{p}^{*})(\hat{\bm{p}}-\bm{p}^{*})\|^{\frac{\rho+1}{2}}$
$\displaystyle\qquad-\gamma\|\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}(\hat{\bm{p}}-\bm{p}^{*})\|_{1}+\|\hat{\bm{p}}-\bm{p}^{*}\|\|\bm{v}^{*}\|$

Note that
$\|\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}(\hat{\bm{p}}-\bm{p}^{*})\|_{1}\geq\|\big{(}\text{diag}(\bm{P}_{\bm{g}_{k}^{*}})\bar{\bm{H}}(\hat{\bm{p}}-\bm{p}^{*})\|=\Big{(}(\hat{\bm{p}}-\bm{p}^{*})^{\top}\bm{L}_{b}(\bm{p}^{*})(\hat{\bm{p}}-\bm{p}^{*})\Big{)}^{\frac{1}{2}}=(\hat{\bm{p}}^{f}-\bm{p}^{f*})^{\top}\bm{L}_{bff}(\bm{p}^{*})(\hat{\bm{p}}^{f}-\bm{p}^{f*})\geq\sqrt{\lambda_{\min}(\bm{L}_{bff}(\bm{p}^{*}))}\|\hat{\bm{p}}-\bm{p}^{*}\|$
.

Thus, if $\gamma$ is chosen such that
$\gamma\geq\frac{\|\bm{v}^{*}\|}{\sqrt{\lambda_{\min}(\bm{L}_{bff}(\bm{p}^{*}))}}$,
then it is not hard to show that

$\displaystyle\dot{V}\leq-\chi_{1}V^{\frac{\alpha+1}{2}}-\chi_{2}V^{\frac{\rho+1}{2}},$

which implies the fixed-time convergence of the estimated configuration to the
desired configuration $\bm{p}^{*}$ based on [31].

The fixed-time stability analysis in case of directed acylic leader-follower
graphs can be shown by a similar approach as in [26, 30] and will be omitted.
∎

##  V Simulation results

###  V-A Simulation 1: Bearing-only control law

Consider a formation of 12 agents with graph as shown in Fig. 3(a). Agents 1–4
are leaders, which move with the reference velocity $\bm{v}^{l}$ given as
follows

  * •

For $t\in[0,10]$, $\bm{v}^{l}=\bm{1}_{4}\otimes\bm{f}_{1}$, where
$\bm{f}_{1}=[1.9-0.14t,0]^{\top}$. Leaders move in straight lines along the
$x$-axis. At $t=10$, $\bm{v}_{1}(10)=[0.5,0]^{\top}$.

  * •

For $t\in[10,15]$,
$\bm{v}^{l}=\bm{1}_{4}\otimes\bm{f}_{2}-\frac{\bm{h}}{5\|\bm{h}\|}$, where
$\bm{f}_{2}=[0.5,0]^{\top}$,
$\bm{h}=\bm{p}^{l}-\bm{1}_{4}\otimes\bar{\bm{p}}^{l}$, and
$\bar{\bm{p}}^{l}=\frac{1}{4}(\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3}+\bm{p}_{4})$ is
the geometric center of four leaders. Leaders go along the $x$-axis and
downscale the formation’s size to fit the alley. At $t=15$s,
$\bm{v}_{1}(15)=[0.5,0]^{\top}$.

  * •

For $t\in[15,25]$, $\bm{v}^{l}=\bm{1}_{4}\otimes\bm{f}_{3}$, where
$\bm{f}_{3}=[0.5+0.05(t-15),0]^{\top}$. Leaders move through the alley. At
$t=25$s, $\bm{v}_{1}(25)=[1,0]^{\top}$.

  * •

For $t\in[25,30]$,
$\bm{v}^{l}=\bm{1}_{4}\otimes\bm{f}_{4}+\frac{\bm{h}}{5\|\bm{h}\|}$, where
$\bm{f}_{4}=[1,0]^{\top}$. The formation has passed the alley. Leaders go
along the $x$-axis and upscale the formation’s size back to normal. At
$t=30$s, $\bm{v}_{1}(30)=[1,0]^{\top}$.

  * •

For $t\in[30,35]$, $\bm{v}^{l}=\bm{1}_{4}\otimes\bm{f}_{5}$, where
$\bm{f}_{5}=[1+0.1(t-30),0]^{\top}$. The formation accelerates and continues
to move forward along the $x$-axis.

The followers adopt the control law (3) with $\beta=2$, $\alpha=0.5$. The
simulation results are depicted in Fig. 3(b),(c). The target formation shape
is achieved in less than 1 second (see Fig. 3(b)) and maintained except when
the formation rescales its size. Thus, simulation result is consistent with
Thm. 1.

(a)

(b)

(c)

Figure 3: Simulation 1: (a) The acyclic leader-follower graph $\mathcal{G}$;
(b) Bearing errors vs time [s]; (c) Trajectory of the agents.

###  V-B Simulation 2: Target point localization-based control law

The same 12-agent formation is simulated under the control law (7)–(8) for 5
seconds. The initial estimates are randomly selected. The trajectories of the
agents are depicted in Fig. 4(a). The desired formation is achieved in less
than 1 second, which is consistent with Thm. 2.

(a)

(b)

Figure 4: Simulation 2: (a) Trajectory of the agents. (b) Bearing errors vs
time [s].

##  VI Conclusions

In this letter, two finite-time bearing-based tracking control laws for
acyclic leader-follower formations have been proposed. The analysis partially
explains how individuals can follow leaders, who are moving at a time-varying
velocity, in collective behaviors such as bird immigration. As suggested in
[23], it will be interesting to consider the problem with delay and switching
in sensing/communication, or when agents can measure only the subtended
bearing angles.

## References

  * [1] B. D. O. Anderson _et al._ , “Rigid graph control architectures for autonomous formations,” _Control Systems Magazine_ , vol. 28, no. 6, pp. 48–63, 2008.
  * [2] K.-K. Oh, M.-C. Park, and H.-S. Ahn, “A survey of multi-agent formation control,” _Automatica_ , vol. 53, pp. 424–440, 2015.
  * [3] T. Eren, W. Whiteley, and P. N. Belhumeur, “Using angle of arrival (bearing) information in network localization,” in _45th IEEE Conf Decision Control, San Diego, CA, USA_ , 2006, pp. 4676–4681.
  * [4] S. Zhao and D. Zelazo, “Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions,” _Automatica_ , vol. 69, pp. 334–341, 2016.
  * [5] S. Zhao and D. Zelazo, “Bearing rigidity theory and its applications for control and estimation of network systems: Life beyond distance rigidity,” _IEEE Control Systems Magazine_ , vol. 99, no. 3, pp. 1–30, 2018.
  * [6] S. Zhao and D. Zelazo, “Bearing rigidity and almost global bearing-only formation stabilization,” _IEEE Transactions on Automatic Control_ , vol. 61, no. 5, pp. 1255–1268, 2016.
  * [7] Q. V. Tran _et al._ , “Finite-time bearing-only formation control via distributed global orientation estimation,” _IEEE Transactions on Control of Network Systems_ , vol. 6, no. 2, pp. 702 – 712, 2019.
  * [8] T. Eren, “Formation shape control based on bearing rigidity,” _International Journal of Control_ , vol. 85, no. 9, pp. 1361–1379, 2012\.
  * [9] M. H. Trinh _et al._ , “Bearing based formation control of a group of agents with leader-first follower structure,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 2, pp. 598 – 613, 2 2019.
  * [10] G.-H. Ko, M. H. Trinh, and H.-S. Ahn, “Bearing-only control of directed cycle formations: Almost global convergence and hardware implementation,” _Internat Journal of Robust and Nonlinear Control_ , vol. 30, no. 12, pp. 4789–4804, 2019.
  * [11] N. Moshtagh, A. Jadbabaie, and K. Daniilidis, “Vision-based control laws for distributed flocking of nonholonomic agents,” in _IEEE Int Conf Robotics Automat (ICRA)_. IEEE, 2006, pp. 2769–2774.
  * [12] Q. V. Tran, S.-H. Park, and H.-S. Ahn, “Bearing-based formation control via distributed position estimation,” in _IEEE Conf Control Technol Appl (CCTA)_. IEEE, 2018, pp. 658–663.
  * [13] G. Jing and L. Wang, “Multiagent flocking with angle-based formation shape control,” _IEEE Transactions on Automatic Control_ , vol. 65, no. 2, pp. 817–823, 2019.
  * [14] S. Zhao and D. Zelazo, “Translational and scaling formation maneuver control via a bearing-based approach,” _IEEE Transactions on Control of Network Systems_ , vol. 4, no. 3, pp. 429–438, 2015.
  * [15] S. Zhao, Z. Li, and Z. Ding, “Bearing-only formation tracking control of multiagent systems,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 11, pp. 4541–4554, 2019.
  * [16] J. Zhao _et al._ , “Bearing-only formation tracking control of multi-agent systems with local reference frames and constant-velocity leaders,” _IEEE Control Systems Letters_ , vol. 30, no. 1, pp. 1–6, 2021.
  * [17] M. H. Trinh _et al._ , “Robust tracking control of bearing-constrained leader-follower formation,” _Automatica_ , vol. 131, no. 109733, 2021.
  * [18] X. Li _et al._ , “Bearing-based formation control of networked robotic systems with parametric uncertainties,” _Neurocomputing_ , vol. 306, pp. 234–245, 2018.
  * [19] X. Li, C. Wen, and C. Chen, “Adaptive formation control of networked robotic systems with bearing-only measurements,” _IEEE Transactions on Cybernetics_ , vol. 51, no. 1, pp. 199–209, 2021.
  * [20] Y. Huang and Z. Meng, “Bearing-based distributed formation control of multiple vertical take-off and landing UAVs,” _IEEE Transactions on Control Network Systems, Early Access_ , 2021.
  * [21] I. L. Bajec and F. H. Heppner, “Organized flight in birds,” _Animal Behaviour_ , vol. 78, no. 4, pp. 777–789, 2009.
  * [22] M. Ballerini _et al._ , “Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study,” _Proceedings of the National Academy Sciences (PNAS)_ , vol. 105, no. 4, pp. 1232–1237, 2008.
  * [23] M. Nagy _et al._ , “Hierarchical group dynamics in pigeon flocks,” _Nature_ , vol. 464, no. 7290, pp. 890–893, 2010.
  * [24] Y. Cao and W. Ren, “Distributed coordinated tracking with reduced interaction via a variable structure approach,” _IEEE Transactions on Automatic Control_ , vol. 57, no. 1, pp. 33–48, 2011.
  * [25] D. V. Vu, M. H. Trinh, and H.-S. Ahn, “Distance-based formation tracking with unknown bounded reference velocities,” in _Internat Conf Control, Automat Syst (ICCAS), Busan_ , 2020, pp. 524–529.
  * [26] D. V. Vu _et al._ , “Tracking control of directed acyclic formation via target point localization,” in _3rd Internat Conf Eng Research Appl (ICERA)_ , 2020, pp. 839–845.
  * [27] D. Shevitz and B. Paden, “Lyapunov stability theory of nonsmooth systems,” _IEEE Transactions on Automatic Control_ , vol. 39, no. 9, pp. 1910–1914, 1994.
  * [28] L. Wang and F. Xiao, “Finite-time consensus problems for networks of dynamic agents,” _IEEE Transactions on Automatic Control_ , vol. 55, no. 4, pp. 950–955, 2010.
  * [29] S. P. Bhat and D. S. Bernstein, “Finite-time stability of continuous autonomous systems,” _SIAM Journal of Control and Optimization_ , vol. 38, no. 3, pp. 751–766, 1998.
  * [30] M. H. Trinh _et al._ , “Fixed-time network localization based on bearing measurements,” in _Proc Amer Control Conf (ACC)_. IEEE, 2020, pp. 903–908.
  * [31] A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” _IEEE Transactions on Automatic Control_ , vol. 57, no. 8, pp. 2106–2110, 2011.