Distributed sliding mode control for leader‐follower formation flight of fixed‐wing unmanned aerial vehicles subject to velocity constraints

Abstract

This paper considers the leader‐follower formation flight of fixed‐wing unmanned aerial vehicles (UAVs) subject to velocity constraints. A novel distributed sliding mode control law is proposed for each UAV, whose kinematics is described by a unicycle model with a saturated angular velocity and a bounded linear velocity within an interval. The designed control law of each follower UAV only uses its own information and the information of its leader UAV. Driven by the designed control law, the desired formation is achieved with rigorous proof, while the follower UAVs' constraints of both the linear and angular velocities are satisfied. Moreover, the follower's speed adjustment range is relaxed and not required to be strictly larger than their leaders'. Finally, numerical simulations are presented to verify the results.

Full text

Received: 16 January 2020 Revised: 28 February 2020 Accepted: 16 April 2020
DOI: 10.1002/rnc.5030
SPECIAL ISSUE ARTICLE
Distributed sliding mode control for leader-follower
formation flight of fixed-wing unmanned aerial vehicles
subject to velocity constraints
Xiangke Wang1Yangguang Yu1Zhongkui Li2
1College of Intelligence Science and
Technology, National University of
Defense Technology, Changsha, China
2State Key Laboratory for Turbulence and
Complex Systems, Department of
Mechanics and Aerospace Engineering,
Peking University, Beijing, China
Correspondence
Xiangke Wang, College of Intelligence
Science and Technology, National
University of Defense Technology,
Changsha 410073, China.
Email: xkwang@nudt.edu.cn
Funding information
National Natural Science Foundation of
China, Grant/Award Number: 61973309,
61973006
Summary
This paper considers the leader-follower formation flight of fixed-wing
unmanned aerial vehicles (UAVs) subject to velocity constraints. A novel dis-
tributed sliding mode control law is proposed for each UAV, whose kinematics is
described by a unicycle model with a saturated angular velocity and a bounded
linear velocity within an interval. The designed control law of each follower UAV
only uses its own information and the information of its leader UAV. Driven by
the designed control law, the desired formation is achieved with rigorous proof,
while the follower UAVs' constraints of both the linear and angular velocities
are satisfied. Moreover, the follower's speed adjustment range is relaxed and not
required to be strictly larger than their leaders'. Finally, numerical simulations
are presented to verify the results.
KEYWORDS
distributed control, fixed-wing UAV, formation control, velocity constraints
1INTRODUCTION
Formation control of multiple autonomous vehicles, including unmanned aerial vehicles (UAVs),1unmanned ground
vehicles2and unmanned underwater vehicles,3has received many attentions due to its various potential applica-
tions, such as cooperative surveillance and searching, cooperative transport, et al.4,5 A number of control approaches
have been proposed for achieving the formation,6-8 and in general the conventional approaches include the
leader-follower method,9,10 behavior-based method,11,12 consensus-based method,13,14 and so on. Among these meth-
ods, the leader-follower approach was intensively studied in recent years due to its relatively simple and clear control
architecture.15-20 In the leader-follower approach, a single or multiple UAVs in the formation are assigned as the leaders
and fly along a predefined trajectory while the other UAVs, acting as the followers, are to maintain the desired distances
and orientations with respect to the leader.20
In the past decades, many theoretical achievements have been obtained on the leader-follower formation control
problems. In 2002, Das et al proposed a leader-follower local control law for the cooperative control of a group of nonholo-
nomic mobile robots.15 Based on the work of Das, the formation problem for multiple nonholonomic mobile robots was
further analyzed,16 in which the robots' control inputs were forced to satisfy some constraints that restrict the set of lead-
er's possible paths and admissible positions of the follower with respect to the leader. Further, in order to tackle the case of
multiple leaders, the leader-follower formation problem with multiple leaders was investigated and it was shown that the
Int J Robust Nonlinear Control. 2020;1–16. wileyonlinelibrary.com/journal/rnc © 2020 John Wiley & Sons, Ltd. 1

2WANG  .
controllability of a multiagent system can be uniquely determined by the interconnection graph.17,21-23 The controllability
of the leader-follower formation for the multi-agent systems was also discussed.24 Following this way, Hu and Feng fur-
ther considered the effect of noise and presented a distributed tracking control scheme for a leader-follower multi-agent
system with measurement noises.18 Afterwards, the cases of switching interconnection topologies were discussed in this
paper via both static and dynamic feedback.19 As the dynamics considered in most previous works are first-order integra-
tors, the leader-follower problem with agents governed by a second-order dynamics was considered in References 25-27.
The measurement delays in leader-follower formation control were considered in Chen's work,28 in which a low-level con-
troller for leader-follower formations of nonholonomic vehicles was designed and the stability of the closed-loop system
was proved.
Formation flight of multiple fixed-wing UAVs has attracted significant attentions due to its increasing demands in
civil and military domains. However, as a special type of robots, fixed-wing UAVs have some special properties owing
to its special dynamics. Firstly, the dynamics of the fixed-wing UAV is an under-actuated system and its kinematics can
be simplified into a unicycle system.29 In addition, the fixed-wing UAV can neither move backward directly nor slow
down the linear velocity lower than a certain positive value, namely, it is required to maintain a positive minimum air-
speed due to the stall conditions.30 Therefore, different from the traditional unicycle model, the velocity constraints of
each fixed-wing UAV are described by a saturated angular velocity and a linear velocity bounded by two positive con-
stants. The leader-follower formation problem for the unicycle-type vehicles was considered in References 31,32, in which
a cascaded approach was used to achieve the exponential stability of the closed-loop system. However, the approach
cannot be applied to track a leader along straight paths, and this problem was fixed by Loria.33 Further, an integral
sliding-mode control strategy was proposed to eliminate the need for measurement or estimation of the absolute veloc-
ity of the leader.34 Afterwards, a distributed controller for the leader-follower formation was proposed in Reference 35
with the aid of the small-gain method, for the networked unicycle dynamical systems with positive minimum linear
velocity constraints. Following this work, a distributed control law was proposed for the leader-follower formation of
UAVs subject to both linear and angular velocity constraints in Reference 36. However, the proposed control law in
Reference 36 required that the follower UAVs have a wider adjustable range of linear velocity than that of their lead-
ers. In the case where the mobility of the leader is very close to (even the same as) the follower, this method may be
inappropriate.
In this paper, we will investigate the leader-follower formation flight control problem of fixed-wing UAVs subject to
velocity constraints, that is, the velocity of each UAV is described by a saturated angular velocity and a bounded linear
velocity lying between two positive constants. Distributed sliding-mode control laws both for two and more UAVs under
directed communication graph are proposed. A nonlinear sliding mode surface is defined for each follower UAV. Using
the Lyapunov method, it is proven that the error dynamical system with designed control laws will converge to the slid-
ing mode surface in finite time, and then converge to the origin once it is reaching on the sliding mode surface. It is
worthy pointing out that the convergence of the overall system is guaranteed with the condition that the follower UAVs'
constraints of both the saturated angular velocity and the bounded linear velocity are satisfied.
The main contributions of the proposed distributed formation control law include:
•The proposed sliding-mode control for leader-follower formation is distributed in the sense that each follower only has
access to the information of its leader, and the overall formation is in a network modeled by a tree-like directed graph.
•The desired formation is achieved with the designed control law, and the input constraints of each UAV for both the
linear and angular velocities can be always satisfied.
•Compared with the existing works, the leader's speed adjustment range is relaxed in this paper. In the existing works,
such as the work in Reference 36, the adjustable range of the followers' linear velocity is required to be larger than
that of their leaders' linear velocity. While the speed adjustment range of followers in our method can be the same as
that of the leaders'. As a consequence, the whole leader-follower formation system will have a better maneuverability,
as its maneuverability is largely determined by that of the leaders. In addition, this relaxation is helpful to construct a
“deep” leader- follower system with multiple cascade leaders, because the linear speed adjustment range of the leader
can be the same as that of its followers.
The rest of the paper is organized as follows: Section 2 presents some mathematical preliminaries, including the nota-
tions problem and some basic control theories. Section 3 gives the problem statement for the distributed leader-follower
formation flight of UAVs. In Section 4, a distributed formation control law is proposed and the stability analysis of the

WANG  . 3
closed-loop system is proved. A numerical simulation as well as the the comparison with the algorithm proposed in
Reference 36 are presented in Section 5. Finally, the conclusion is drawn in Section 6.
2MATHEMATICAL PRELIMINARIES
We employ ⋅to denote the absolute value of a real number. For a vector x∈Rn, its Euclidean norm is repre-
sented by x=(
n
i=1xi2)1
2and xTis its transpose. For a matrix X∈Rm×n, its Euclidean norm is denoted as XF=
(m
i=1n
j=1xij2)1
2. For notation simplicity, the subscript Fis omitted and ⋅also denotes the Euclidean norm for a
matrix.
Afunction𝛾∶R≥0→R≥0is positive definite if 𝛾(s)>0 for all s>0and𝛾(0)=0. 𝛾∶R≥0→R≥0is a class function
if it is continuous, strictly increasing and 𝛾(0)=0; it is a class ∞function if it is a class function and also satisfies
𝛾(s)→∞as s→∞. Supposed b≤c, the saturation function z=sat(x,b,c)∶R3→Ris defined as
z=sat(x,b,c)=




b,if x <b,
x,if b <x<c,
c,if x >c.
(1)
The sign(s)is defined as:
sign(s)=




1,s>0,
0,s=0,
−1,s<0.
(2)
Definition 1 (37 Globally Asymptotically Stability). For a system with no inputs ̇
x=f(x), if there exists a function 𝛽of
class  such that:
x(t,xo)≤𝛽(xo,t),∀xo,∀t≥0,
then the system is globally asymptotically stable.
Lemma 1 (37). Consider an autonomous system
̇
x=f(x),(3)
where f ∶M→Rnis a locally Lipschitz map from a domain M ⊂Rninto Rn.Letx=0be an equilibrium point for ( 3). Let
V∶Rn→Rbe a continuously differentiable function such that:
V(0)=0and V(x)>0,∀x≠0,(4)
x→∞⇒V(x)→∞,(5)
̇
V(x)<0,∀x≠0,(6)
then x =0is globally asymptomatically stable.
3PROBLEM FORMULATION
Consider a group of Nfixed-wing UAVs with kinematic models:





̇
xi=vicos 𝜃i,
̇
yi=visin 𝜃i,
̇
𝜃i=𝜔i,i∈[1,N],
(7)

4WANG  .
where xiand yiare the position of UAV iin the inertial frame and 𝜃i∈(−𝜋,𝜋]is its heading angle, viand 𝜔irepre-
sent the linear velocity and the angular velocity, respectively. In addition, viand 𝜔iare determined by the control inputs
𝜇i=(𝜇v
i,𝜇
𝜃
i),whichare
vi=𝜇v
i,
𝜔i=𝜇𝜃
i.(8)
Further, the following velocity constraints are considered for UAV i:
0<v−
i<vi<v+
i,(9)
𝜔i≤𝜔+
i,𝜔
+
i>0,(10)
where v−
iand v+
iare the constants which determine the minimum and maximum velocities of the UAV i, respectively,
and 𝜔+
iis a constant that determines the maximum angular velocity of UAV i.
Remark 1. Compared with the traditional unicycle model, the model of the fixed-wing UAV is additionally constrained
by the velocity constraints (9) and (10). Namely, the velocity of each UAV is constrained by a saturated angular velocity
and a linear velocity bounded within a positive interval. These constraints make the formation control of UAVs a more
challenging problem.
In the leader-follower formation, there is one uncontrolled UAV labeled 0 and this UAV acts as the origin leader of the
formation. The other UAVs are followers whose desired trajectories are determined by the origin leader or other follower
UAVs. To describe the leader-follower architecture, an acyclic directed graph is utilized.
Definition 2 (Formation Control Graph38). A formation control graph =(V,E,S)is a directed acyclic graph
consisting of:
•A finite set V={𝜈1,…,𝜈
N}of Nvertices and a map assigning to each vertex 𝜈ia control system ̇𝜒i=fi(t,𝜒
i,ui)where
𝜒i∈Rn.
•An edge set E⊂V×Vencoding leader-follower relationships between vehicles. The ordered pair (𝜈i,𝜈
j)≜eij belongs
to Eif ujdepends on the state of vehicle i,𝜒i.
•A collection D={dij}of edge specifications, defining control objectives (setpoints) for each j∶(𝜈k,𝜈
j)∈Efor some
𝜈i∈V.
For UAV j, the tail of the incoming edge to vertex jrepresents the unique leader of UAV j, which is denoted by Lj.
Obviously, the origin leader of the whole formation labeled by 0 has no leader, thus L0=∅.Foravertexj(1 ≤j≤N), we
have the following assumption:
Assumption 1. For any vertex 𝜈j∈V(1 ≤j≤N) and its leader vertex Li, the conditions (9) and (10) always hold. In
addition, [v−
i,v+
i]⊆[v−
j,v+
j]and [𝜔−
i,𝜔
+
i]⊂[𝜔−
j,𝜔
+
j]. Meanwhile, there exists an appropriate time interval [T1,T2]that the
leader's linear speed is smaller than the follower's linear speed, that is, for ∀t∈[T1,T2],vi(t)<vj(t)holds.
Remark 2. Note that from Assumption 1, the adjustable interval of the linear velocity of the leader can be the same as
that of the followers. Actually, Assumption 1 is a relaxed condition compared with the work in Reference 36, which
requires the adjustable interval of the linear velocity of the leader should be smaller than that of the followers'. Therefore,
under Assumption 1, the maneuverability of the leader can be enhanced to a certain extent, and furthermore, the whole
leader-follower formation will have a better maneuverability, as its maneuverability is generally determined by that of its
leader.
Through the above definition, a team of UAVs can be represented by a tree-like control graph ={V,E,D}, shown
as in Figure 1. The root of the spanning graph is the uncontrolled UAV labeled 0 and is the first class leader of the
whole formation. The vertices 1 and 2 are the follower of the vertex 0 while being the leader of the vertices of 3, 4
and 5, 6, respectively. For an ordered pair (𝜈i,𝜈
j)≜eij belonging to E,theleaderUAVi's position pi, heading angle
𝜃i,velocityviand angular velocity 𝜔ican be accessed by the follower UAV jvia a perfect communication without
delays.

WANG  . 5
FIGURE 1 A tree-like control graph
FIGURE 2 Leader-Follower unmanned aerial vehicle formation
geometry
As shown in Figure 2, the follower UAV jis expected to keep a preset relative position to its leader UAV iand maintain
the same heading angle with the UAV i. If the follower reaches its desired posture with respect to the leader, a desired
formation is formed. Let pi=(xi,yi)denote the position of UAV iand 𝜃irepresent its heading angle. Given a specification
dij =(dx
ij,dy
ij)on edge (𝜈i,𝜈
j)∈E, a setpoint for the follower UAV jcan be expressed as pd
j=pi−dij. Taking the consistence
of the heading angle into account, the desired state 𝜂d
j=(xd
j,yd
j,𝜃
d
j)for UAV jis





xd
j=xi+dx
ij,
yd
j=yi+dy
ij,
𝜃d
j=𝜃i,
where j∈Fi. Therefore, the state error ̃𝜂j=(
̃
xj,̃
yj,̃
𝜃j)of the follower UAV jis defined as





̃
xj=xj−xi−dx
ij,
̃
yj=yj−yi−dy
ij,
̃
𝜃j=𝜃j−𝜃i.
(11)
Extending the state error defined by (11) to the whole formation, the formation error vector is constructed by stacking
the errors of all followers:
̃𝜂 ≜…̃𝜂j…T,j∈[1,NE],(12)
where N|E|is the number of edges in graph . Then, the objective of this paper is to design a controller for each follower
UAV to make the stack error ̃𝜂 converge to the origin.
4MAIN RESULTS
To simplify the problem, the tracking control problem for only one pair of leader and follower is considered firstly, and
then it is extended to the formation tracking case in the general case.

6WANG  .
4.1 Leader-Follower Tracking Control
Consider a pair of fixed-wing UAVs in which one acts as the leader and the other acts as the follower. For the reading
convenience, let (xF,yF,𝜃
F)and (xL,yL,𝜃
L)denote the planar coordination and the heading angle of the follower and the
leader UAV in the inertial frame, respectively. Then (11) in this case becomes:





̃
xF=xF−xL−dx
LF,
̃
yF=yF−yL−dy
LF,
̃
𝜃F=𝜃F−𝜃L,
(13)
where (dx
LF,dy
LF)is the desired relative position between the leader UAV and the follower UAV. Meanwhile, the follower
UAV obeys the following velocity constraints:
0<vmin ≤vF≤vmax,(14)
0≤𝜔F≤𝜔max,𝜔
max >0,(15)
where 𝜔max is the maximum angular velocity of the follower UAV, vmin and vmax are, respectively, the minimum and
maximum linear velocities of the follower UAV.
A leader-aligned coordinate called -coordinate is firstly defined. As illustrated in Figure 3, the frame Ogxgygis the
inertial frame and the origin of -coordinate (ODxDyD) is fixed with the desired position of the follower UAV and its
x-coordinate is consistent with the heading direction of the leader UAV. Based on this, a coordinate transformation is
performed as follows:

xe
ye
𝜃e
=
cos 𝜃Lsin 𝜃L0
−sin 𝜃Lcos 𝜃L0
001

̃
xF
̃
yF
̃
𝜃F
,(16)
where (xe,ye,𝜃
e)is the tracking error in the -coordinate and 𝜃e∈(−2𝜋, 2𝜋). By performing the transformation (11) and
(16), the position of the follower is transferred from the world-coordinate into the -coordinate. In the new coordinate,
the error dynamics between the leader and the follower becomes





̇
xe=𝜔Lye−vL+vFcos 𝜃e,
̇
ye=−𝜔Lxe+vFsin 𝜃e,
̇
𝜃e=𝜔F−𝜔L.
(17)
FIGURE 3 The illustration of two coordinates:
world-coordinate and -coordinate [Colour figure can be viewed at
wileyonlinelibrary.com]

WANG  . 7
A control law 𝜇=(𝜇v
F,𝜇
𝜃
F)that enforces the trajectory of (17) to be globally stable is to be designed. In the following
part, we design a sliding mode controller which guarantees the global stabilization of the system (17). Firstly define the
sliding mode surface sas
s=𝜃e+k1
ye
k2+re
,(18)
where re=x2
e+y2
eand k1,k2are constants. Then a controller is designed as
𝜇v
F=




sat vL−k3xe
cos 𝜃e
,vmin,vmax ,if 𝜃e≠𝜋
2,3𝜋
2,
vmax,if 𝜃e=𝜋
2,3𝜋
2,
(19)
𝜇𝜃
F=−k1
(vLxe−vFcos 𝜃exe−vFsin 𝜃eye)ye
re(k2+re)2+k1
vFsin 𝜃e−𝜔Lxe
k2+re
+k4sign(s)+𝜔L,(20)
where k3and k4are the coefficient constants and the functions sat(⋅)and sign(⋅)are defined in (1) and (2), respectively.
Let v+
Land w+
Ldenote the maximum linear velocity and maximum angular velocity of the leader, respectively, and the
stability proof of the system (17) with controller (19) and (20) is provided in the following theorem:
Theorem 1. Consider the system (17 ) under the controller (19 ) and (20 ). If the parameters k1,k
2,k
3,andk
4satisfy the
following conditions:
0<k1<4
𝜋2,(21)
k2>k1
v+
L+2vmax
𝜔max −k4−(k1+1)w+
L
>0,(22)
k3>0,and k4>0,(23)
then the closed-loop system (17) will converge to the origin and the velocity constraints (14) and (15) are satisfied.
Proof. Firstly, we will prove that the system will globally converge to the slide mode surface s=0 defined in (18) in finite
time from any initial states, driven by control law (19) and (20). Define a Lyapunov function candidate as
Vs=1
2s2.
Then, the time derivative of Vsis
̇
Vs=ṡ
s=s̇
𝜃e−k1
̇
reye
(k2+re)2+k1
̇
ye
k2+re
=s̇
𝜃e−k1
ye
(k2+re)2⋅−vLxe+vFcos 𝜃exe+vFsin 𝜃eye
re
+k1
−𝜔Lxe+vFsin 𝜃e
k2+re.(24)
Note that here ̇
𝜃e=𝜔F−𝜔L=𝜇𝜃
F−𝜔L,where𝜇𝜃
Fis the designed controller defined in (20). As a consequence, we have
̇
Vs=−k4s⋅sign(s)=−k4s≤0.(25)
The equality is achieved if and only if s=0.
Denote W=2Vs=s, and the right differential D+Wof Wsatisfies the differential inequality:
D+W≤−k4.

8WANG  .
Then the comparison lemma37 shows that
W(s(t)) ≤W(s(0)) − k4t.(26)
Therefore, the trajectory reaches the manifold s=0 in finite time, and, once on the manifold s=0, it cannot leave it,
as seen from the inequality ̇
Vs≤−k4s.
Next the convergence of the system on the sliding mode surface s=0 will be analyzed. Recall the sliding mode
surface (18). When s=0, it yields
𝜃e=− k1
k2+re
ye.(27)
Obviously, 𝜃ewillalso converge to 0 when ye→0. Thus the convergence of 𝜃eis guaranteed as long as the convergence
of yeis ensured.
Define a Lyapunov function candidate as
V=1
2x2
e+1
2y2
e.(28)
When s=0, it can be obtained from (27) that
ye=−
k2+re
k1
𝜃e.(29)
Then combined with (17) and (29), the time derivative of Vyields
̇
V=xė
xe+yė
ye
=(vFcos 𝜃e−vL)xe+vFsin 𝜃eye
=(vFcos 𝜃e−vL)xe+vFsin 𝜃e−k2+re
k1
𝜃e
≤(vFcos 𝜃e−vL)xe−k2+re
k1
vF𝜃2
e.(30)
Firstly, when cos 𝜃e=0, that is,𝜃e=𝜋∕2or𝜃e=3𝜋∕2, we have 𝜃2
e≥𝜋2∕4 and can further deduce the following
inequality:
̇
V≤−k2
k1
vF𝜃2
e+vmax xe−𝜋2re
4k1
≤−k2
k1
vF𝜃2
e+vmaxxe1−𝜋2
4k1.
As k1<4∕𝜋2revealed by the condition (21), it holds that (1−𝜋2∕(4k1)) <0, which yields
̇
V<−k2
k1
vF𝜃e2−vmax 1−𝜋2
4k1xe<0.(31)
Then, when cos 𝜃e≠0 and the desired velocity vFis within the allowed range, that is, vmin ≤vF=vL−k3xe
cos 𝜃e
≤vmax,
(30) becomes
̇
V≤−k3x2
e−k2+re
k1
vF𝜃2
e≤0.(32)
The equality in (32) holds if and only if xe=0and𝜃e=0, which further leads to the global stability of the system in
this case.
On the other hand, when cos 𝜃e≠0 and the desired velocity vFis limited by the velocity saturation, there are 8 cases
listed as follows:

WANG  . 9
(1) vL−k3xe
cos 𝜃e
>vmax,cos𝜃e>0, xe≥0, vF=vmax;
(2) vL−k3xe
cos 𝜃e
>vmax,cos𝜃e>0, xe<0, vF=vmax;
(3) vL−k3xe
cos 𝜃e
>vmax,cos𝜃e<0, xe≥0, vF=vmax;
(4) vL−k3xe
cos 𝜃e
>vmax,cos𝜃e<0, xe<0, vF=vmax;
(5) vL−k3xe
cos 𝜃e
<vmin,cos𝜃e>0, xe≥0, vF=vmin;
(6) vL−k3xe
cos 𝜃e
<vmin,cos𝜃e>0, xe<0, vF=vmin;
(7) vL−k3xe
cos 𝜃e
<vmin,cos𝜃e<0, xe≥0, vF=vmin;
(8) vL−k3xe
cos 𝜃e
<vmin,cos𝜃e>0, xe<0, vF=vmin.
It is easy to verify that cases (4) and (8) are impossible. Therefore, we will discuss the rest cases in the following part.
•For cases (1) and (6), we have
−vLxe<−k3x2
e−vFcos 𝜃exe.(33)
Substituting (33) into (30) gives(32).
•For cases (3), (5), and (7), it is clear that xe>0 holds in these cases. Considering (vFcos 𝜃e−vL)<0, it yields
̇
V≤−vFcos 𝜃e−vLxe−k2+re
k1
vF𝜃2
e≤0.(34)
Obviously, the equality in (34) holds also only when xe=0and𝜃e=0.
•For case (2), substituting vF=vmax into (30) yields
̇
V≤−k2
k1
vmax𝜃2
e+(vL−vmax cos 𝜃e)xe−re
k1
vmax𝜃2
e
≤−k2
k1
vmax𝜃2
e+vmax −vmax cos 𝜃e−vmax
k1
𝜃2
exe
=−
k2
k1
vmax𝜃2
e+1−cos 𝜃e−1
k1
𝜃2
exevmax.(35)
Note that
1−cos 𝜃e=2sin2𝜃e
2≤𝜃2
e
2.(36)
Substituting (36) into (35) yields
̇
V≤−k2
k1
vmax𝜃2
e−1
k1
−1
2vmax𝜃2
exe.(37)
Note that k1<4∕𝜋2<2, thus 1∕k1−1∕2>0. Further from (37), it can be observed that ̇
V<0aslongas𝜃e≠0. Accord-
ing to Assumption 1, vL≤v+
L≤vmax,wherev+
Lis the maximum velocity of the leader. Further when 𝜃e=0, (30) can
be written as
̇
V=(vmax −vL)xe=−vmax −vLxe.(38)

10 WANG  .
From (38), it can be induced that ̇
V≤0 holds as well and the equality holds only if xe=0. Thus combining (37) with
(38), it can be concluded that ̇
V<0aslongas𝜃e≠0andye≠0 for the case 2).
From (27), it is known that 𝜃e≠0ifye≠0. Therefore, when s=0, it always holds that ̇
V≤0 and the equality holds
if and only if xe=0andye=0 through the discussions above. Following that, it can be concluded from Lemma 1 that xe
and yewill converge to the origin when s=0. Together with the result that the manifold s=0 can be reached in finite
time, we can conclude that the closed-loop system (17) under the controller (19) and (20) is globally stable. ▪
The boundedness of the input will be analyzed. Firstly, the velocity constraint that vmin ≤vF≤vmax holds obviously.
Then, for the angular velocity input 𝜇𝜃
F,define𝜙=atan 2(ye,xe)and we have
𝜇𝜃
F≤k1
−vLxe+vFcos 𝜃exe+vFsin 𝜃eye
re(re+k2)+k1
−𝜔Lxe+vFsin 𝜃e
re+k2+𝜇𝜃
L+k4
≤k1
re+k2
(vL+vFcos 𝜃ecos 𝜙+vFsin 𝜃esin 𝜙)+k1𝜔L+vFsin 𝜃e
re+k2+𝜇𝜃
L
≤k1
re+k2
(vL+vFcos(𝜃e−𝜙)+vF)+k1𝜔L+𝜇𝜃
L+k4
≤k1
k2
(v+
L+2vmax)+(k1+1)w+
L+k4.(39)
By substituting the condition (22) into (39), we obtain 𝜇𝜃
F≤𝜔max.
Remark 3. The parameter k3is related with the convergence rate of xe. In general, a larger k3will bring a faster convergence
of xe. The parameter k1is related with the convergence rate of yeand an excessively small k1will incur a slow convergence
of ye. In order to keep k1not to be excessively small whilethe condition (22) is satisfied, the parameter k2could be relatively
large.
Remark 4. It should be noted that the chattering may happen at the neighborhood of the equilibrium s=0 because of the
signum function sign(s). In order to avoid the chattering in practice, the function sign(s)defined in (2) can be replaced
with other “softer” functions, such as the tangent function tanh(s∕𝜀)or a continuous function s
s+𝜀,where𝜀is a positive
number selected to reduce the chattering problem. For more details, please refer to Reference 37.
Remark 5. From Theorem 1, the formation control law composed of (18), (19), and (20) guarantees the stability of the
closed-loop leader-follower system while the velocity constraints are satisfied. Namely, the proposed control law can still
work well even when the adjustable range of the leader's linear velocity is the same as that of the followers', which is
different from the previous work. In order to guarantee the stability of the overall system, the parameters in Reference 36
is quite conservative when the adjustable range of the leader's linear velocity approaches that of the follower's, and the
performance may be not so good in this case. From the proof of Theorem 1, it can be inferred that the proposed algorithm
can still work well as long as there exists a time interval that the leader's speed is smaller than that of the followers.
4.2 Leader-Follower Formation Control
Now, we extend the previous result to the case of formation tracking control. For a given ordered pair (𝜈j,𝜈
i)≜eji belonging
to E, define the error vector 𝜂ei =(xei ,yei,𝜃
ei), which is the extension of (16) as follows:
xei
yei
𝜃ei=cos 𝜃jsin 𝜃j0
−sin 𝜃jcos 𝜃j0
001
̃
xi
̃
yi
̃
𝜃i.(40)
Then similar to (17), the dynamics of error 𝜂ei becomes





̇
xei =𝜔jyei −vj+vicos 𝜃ei,
̇
yei =−𝜔jxei +visin 𝜃ei ,
̇
𝜃ei =𝜔i−𝜔j.
(41)

WANG  . 11
A sliding mode surface for UAV iis defined as
si=𝜃ei +k1
yei
k2+rei
,(42)
where rei =x2
ei +y2
ei. Then, the distributed sliding mode controller for UAV iis proposed as
𝜇v
i=




sat vj−k3xei
cos 𝜃ei
,v−
i,v+
i,if 𝜃ei≠𝜋
2,3𝜋
2,
v+
i,if 𝜃ei=𝜋
2,3𝜋
2,
(43)
𝜇𝜃
i=−k1
(vjxei −vicos 𝜃eixei −visin 𝜃ei yei)yei
rei(k2+rei )2+k1
visin 𝜃ei −𝜔jxei
k2+rei
+k4sign(si)+𝜔j,(44)
where j=Li,andk1,k2,k3,k4are constants and satisfies
0<k1<4
𝜋2,(45)
k2>k1
v+
j+2v+
i
𝜔+
i−k4−(k1+1)w+
j
>0,(46)
k3>0,k4>0.(47)
Now it is the position to present the formation controller.
Theorem 2. Consider the formation of N UAVs whose kinematic model is given by (7 ) and the communication relationship
is determined by a tree-like digraph , the error dynamics for any pair of two connected UAVs is given by (41). Given the
distributed sliding mode control law (42), (43), and (44), the convergence of the formation error ̃𝜂 defined by (11) and ( 12)
to the origin can be guaranteed.
Proof. Define a Lyapunov function Vsi for each UAV ias
Vsi =1
2s2
i.(48)
It can be concluded using the method similar to the proof of Theorem 1 that
Vsi =−k4si≤0,(49)
and the trajectory of siwill reach the manifold si=0 in finite time. By using a similar method in Theorem 1, it can be
proved that the system (41) driven by the control law (43) and (44) will asymptotically converge to the origin in finite
time if si=0. Let Tidenote the time when sireaches the manifold si=0. Combining with the definition of the globally
asymptotically stability in Definition 1 yields
𝜂ei(t)≤𝛽i(𝜂ei(Ti),t),∀t≥Ti,(50)
where 𝛽iis a class  function. Since ̃𝜂i(t)=𝜂ei(t),wehave
̃𝜂i(t)≤𝛽i(̃𝜂i(Ti),t),∀t≥Ti.(51)
Denoting Tmax =max
1≤i≤N{Ti}and summing over all 1 ≤i≤N, we obtain that
̃𝜂(t)≤
N

i=1̃𝜂i(t)≤
N

i=1
𝛽i(̃𝜂i(Tmax),t)
≤
N

i=1
𝛽i(̃𝜂(Tmax),t),∀t≥Tmax.(52)

12 WANG  .
As 𝛽i(⋅)is a class  function, 𝛽i(̃𝜂(Tmax),t)→0whent→∞. Thus, it can be induced from (52) that ̃𝜂(t)→0
when t→∞. The proof is completed. ▪
Remark 6. It is clear that the proposed control law composing of (42), (43) and (44) only uses its own information and the
information of its leader UAV, which is distributed and scalable. In addition, the control law is of significance for a “deep”'
leader-follower systems with multiple cascade leaders, as shown in Figure 1, because the follower's speed adjustment
range can be the same as that of the leader. If the adjustable range of the followers' linear velocity is required to be strictly
larger than their leaders', the velocity adjustable range of the followers in the “deeper” class has to be larger and larger.
This is the reason why it is hard to construct a “deep” leader- follower system.
5SIMULATION RESULTS
In this section, two simulation scenarios are presented. In the first simulation, the case of only a pair of leader and follower
is considered. and in the second simulation, a formation consisting of a leader UAV and six follower UAVs is investigated.
In the first case, the reference path of the leader UAV is a sinusoidal-like curve. The follower is expected to keep a
relative position (40,40)from the leader. The initial positions of the leader and the follower are given by [30,40]and
[−300,−300],respectively. The velocity constraint for the follower UAV in the simulation is given as vF∈[12,20].The
maximum angular velocity for the follower is 1.2 rad/s. The linear velocity and the angular velocity of the leader robot are
given as vL(t)=(15 −cos 0.2t)m∕sand𝜔L(t)=0.1cos0.2t. Therefore, the leader's velocity is constrained by vL⊆[14,16].
The parameters in the designed control law (19) and (20) are given by k1=0.4, k2=30, k3=1andk4=0.2. Further,
we compare the proposed algorithm with the algorithm in,36 which is explicitly described as follows:
vF=vL+̂
k1̂
xe
1+̂
x2
e+̂
y2
e
,
𝜔F=𝜔L+
̂
k3sin ̂
𝜃ei
2
1+̂
x2
ei +̂
y2
ei
+
̂
k2sat vF,v+
L,v−
L̂
yei cos ̂
𝜃ei
2−̂
xei sin ̂
𝜃ei
2
1+̂
x2
ei +̂
y2
ei
,(53)
where ̂
xe,̂
ye,̂
𝜃eare given by
̂
xe
̂
ye
̂
𝜃e=−cos 𝜃Lsin 𝜃L0
sin 𝜃L−cos 𝜃L0
00−1̃
xF
̃
yF
̃
𝜃F,(54)
̂
k1,̂
k2,and̂
k3are coefficient constants and satisfy
̂
k1≤min vmax −v+
L,v−
L−vmin,
2̂
k2v+
L+̂
k3≤𝜔max −𝜔+
L.(55)
Constrained by (55), the parameter ̂
k1in (53) should satisfy ̂
k1≤2. Thus choose the parameters in (53) as ̂
k1=2,
̂
k2=0.008, ̂
k3=0.25.
The paths of the follower controlled by our algorithm and the algorithm in36 as well as the leader's path are illustrated
in Figure 4A. The tracking errors xe,ye,𝜃eare compared in Figure 4B, Figure 4C, and Figure 4D, respectively. The angu-
lar speed and velocity of the UAVs controlled by two methods are shown in Figure 5. It can be observed from Figure 5B
that the velocity of the UAV controlled by our algorithm can reach the maximum speed 20m∕s at the beginning of the
simulation. On the contrary, the UAV controlled by the algorithm in Reference 36 cannot reach the maximum perfor-
mance speed of airplane because the parameter k1is limited by k1≤v−
L−vmin =2. Therefore, if the UAV is controlled
by the algorithm in Reference 36, the velocity of the follower UAV is constrained by vF≤vL+̂
k1=18m∕s. As a conse-
quence, the tracking error xeof the algorithm in Reference 36 converges much slower than that of our algorithm, which
is illustrated in Figure 4B. Figure 5A shows that the angular speed of both algorithm is within the allowed range.
In the second simulation, we consider a formation consisting of a leader UAV (labeled 0) and six follower UAVs (labeled
1−6) with kinematics (1). The desired formation shape is a triangle. The communication topology as well as the desired
relative position inside the formation is shown in Figure 6. The velocity of the leader for the formation is given by v0=
16 −4sin(0.1𝜋t)and thus v0∈[12,20]. The adjustable range of the followers' velocity is just the same as that of the leader

WANG  . 13
FIGURE 4 Described paths and the comparison of tracking errors controlled by two methods [Colour figure can be viewed at
wileyonlinelibrary.com]
FIGURE 5 The illustration of the follower's speed and angular speed controlled by two methods [Colour figure can be viewed at
wileyonlinelibrary.com]

14 WANG  .
FIGURE 6 The illustration of the desired formation in the simulation
FIGURE 7 Trajectories of the leader unmanned aerial
vehicle (UAV) and all follower UAVs [Colour figure can be
viewed at wileyonlinelibrary.com]
FIGURE 8 Formation tracking
errors of each follower unmanned aerial
vehicle [Colour figure can be viewed at
wileyonlinelibrary.com]

WANG  . 15
vehicle, i.e., vi∈[12,20],i=1,2,…,6. The control parameters in this case are the same as those in the first simulation.
Figure 7 shows the trajectories of all UAVs during the simulation of 200s. The positions of all the UAVs at certain moments
during the simulation are marked by solid balls. It can be observed that the UAV formation converges to the desired
triangle shape. The tracking errors xe,ye,𝜃eof all follower UAVs are shown in Figure 8, which shows that the formation
tracking errors will converge to 0. The simulation results verify the effectiveness of the control law proposed in this paper
and prove that our algorithm can still work well even when the the adjustable range of the followers' velocity is just the
same as that of the leader's.
6CONCLUSION
In this paper, a distributed sliding-mode control law has been proposed for fixed-wing UAVs formation flight subject to
velocity constraints. Driven by the proposed control law, the linear velocity of each follower is guaranteed to lie between
two positive constants while the desired leader-follower formation is achieved. In addition, the adjustable range of the
followers' linear velocity is not required to be larger than that of the leader's, which is of significance in the leader-follower
formation flight for a large scale of UAVs.
Although we relax the limit on the adjustable range of the follower's linear velocity, the maximum angular velocity of
the follower is still required to be larger than its leader. This may limit its applications to the formation flight for a large
scale of UAVs. Besides, the collision between the UAVs will be considered in future.
ORCID
Xiangke Wang https://orcid.org/0000-0002-5074-7052
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SUPPORTING INFORMATION
Additional supporting information may be found online in the Supporting Information section at the end of this article.
How to cite this article: Wang X, Yu Y, Li Z. Distributed sliding mode control for leader-follower formation
flight of fixed-wing unmanned aerial vehicles subject to velocity constraints. Int J Robust Nonlinear Control.
2020;1–16. https://doi.org/10.1002/rnc.5030