Adaptive Formation Control of Autonomous
Ground Vehicles in Leader-Follower Structure
György Max∗ , Béla Lantos†
∗ Budapest
University of Technology and Economics, Department of Control Engineering and Information Technology,
† MTA-BME Control Engineering Research Group, Budapest, Hungary
Abstract—In this paper an adaptive control law is designed
for formation control of underactuated autonomous ground
vehicles based on the leader-follower approach and kinematics
equations. The follower vehicles have limited knowledge about
the leader’s states. The unknown term containing the velocity
information of the leader is estimated using neural networks with
online adaptive weight tuning laws. The decentralized formation
control algorithm incorporates neural network estimation and
guarantees semi-global ultimate uniform boundedness of all
closed loop error signals. The stability is proven using Lyapunov
technique and showed that the tracking error can be made small
by appropriate choice of control parameters. The effectiveness
and high precision of the formation controller is investigated via
realistic, high-speed motion maneuvers.
Index Terms—Unmanned Ground Vehicles, Formation Control, Vehicle Kinematics, Adaptive Neural Network
I. I NTRODUCTION
Formation control of unmanned autonomous vehicles has
obtained great importance over the past few years. It is
due to the fact that many practical applications require a
group of autonomous vehicles to follow a prescribed trajectory
while maintaining a desired spatial configuration. Moving in
formation has many advantages over single-vehicle systems.
Increased robustness, more flexibility, efficient energy consumption and redundancy against failures can be achieved [1],
[2] and [3].
There are many issues needed to be solved when designing a
formation controller for autonomous ground vehicles, such as
the stability of the formation, controllability considerations due
to underactuation, handling safety and model uncertainties.
Many recent control approaches have been introduced to
solve the problems in formation control, for example, leaderfollower strategy [1], optimal control [4] and consensusbased method [5]. In practice, among these approaches, the
leader-follower strategy is preferred due to its simplicity and
comprehensiveness [6].
On the other hand, the existing leader-follower formation
control algorithms use the leader’s velocity information [1]
and [7]. In practice, however, it is not necessarily manageable
to equip all vehicles with appropriate velocity sensors. One
approach to tackle this problem is to employ feedforward
neural networks (NNs) for estimating the uncertainties [8]. By
incorporating neural networks into the formation controller, the
level of robustness can be increased. However, due to these
uncertainties, stability issues may arise. From a practical point
of view, the semi-global ultimate and uniform boundedness
(SGUUB) of all error signals is a sensible objective, since
asymptotic stability may be too strong to achieve in the
presence of uncertainties and disturbances.
In this paper, a nonlinear adaptive formation controller is
presented using the kinematics of the vehicle and taking into
consideration the model uncertainties based on earlier results
for surface ships [9]. The decentralized controller employs
single hidden layer NNs for compensating the uncertainties
whose parameters are tuned in real-time. Based on Lyapunov technique, the adaptive NN control scheme ensures
that the tracking errors are semi-globally uniformly ultimately
bounded. By appropriate choice of the control parameters, the
ultimate bound can be decreased and thus the tracking error
can be made smaller.
The main contributions of the paper are i) the extension
of earlier results for the kinematic control of autonomous
car-like vehicle formations, ii) the more practical choice of
relative orientation in the leader-follower scheme, iii) the
applicability of the algorithms for time-varying high-speed
velocity and overlapping time-varying orientation, iv) selfcontained derivation of the system parameters for SGUUB
stability guarantee.
The structure of the paper is as follows. In Section II
the kinematic model of the vehicle is introduced and the
formation control objective is formulated. In Section III a basic
introduction and properties of multi-layer neural networks are
presented. It is followed by the derivation of the adaptive
formation control law in Section IV. The tracking performance
of the developed controller are investigated via high-speed
maneuver in Section V by simulation. Finally, the paper is
concluded in Section VI.
II. P ROBLEM S TATEMENT
A. Leader-Follower Approach
Consider a group of i = 1 . . . n underactuated unmanned
ground vehicles (UGVs) following a leader (L) vehicle in the
horizontal plane described with kinematics equation of
cos ψi − sin ψi
R(ψi ) 0
ν̇ , R(ψi ) =
(1)
η̇i =
sin ψi
cos ψi
0
1 i
T
where ηi = [xi , yi , ψi ] ∈ R3 is the pose of the ith vehicle
T
expressed in the reference frame and νi = [ui , vi , ri ] ∈ R3
contains the body-fixed linear and angular velocities.
The geometry of the leader-follower configuration is presented in Fig. 1 where diL ∈ R+ is the distance and
i.e. bounded tracking of the leader vehicle is required by satisfying the prescribed formation specification with imperfect
knowledge of the leader’s actual states, and by guaranteeing
that the tracking error can be made small.
III. F EEDFORWARD N EURAL N ETWORKS
Fig. 1.
Leader-follower configuration on the x–y plane
φiL ∈ (−π, π] is the relative bearing angle between the leader
and the ith follower formulated by
p
diL = (xL − xi )2 + (yL − yi )2
(2)
yL − yi
− ψL = γiL − ψL
(3)
φiL = arctan
xL − xi
In this section the basic background and properties of
feedforward neural networks (NNs) are introduced using the
notations and results of [8].
Consider a single hidden layered (SHL) NN as depicted in
Fig. 2. The ith NN output for input x = [x1 , x2 , . . . , xN 1 ]T ∈
RN1 is defined by
"
!
#
N1
N2
X
X
yi =
wij σj
vjk xk + θvj + θwi
(7)
j=1
k=1
for i = 1, . . . , N3 with activation function σ(·) and vjk , wij
layer interconnection weights. The input and output bias terms
associated with the jth hidden layer and the ith output are
denoted with θvj and θwi , respectively.
T
Define φi = γiL − ψi and denote with fi = [diL , φiL ]
the formation specification vector, then its time derivative
can be expressed by the body-fixed velocities using (1) and
trigonometric identities as follows:
f˙i = Bi ci + ∆i ,
ci = RT (φi )
Bi = −diag(1, d1iL ),
(4a)
ui
u
0
, ∆i = −Bi RT (φiL ) L −
(4b)
vi
vL
rL
In order to facilitate the control objective and design, the
following assumptions are made.
Assumption 1. The vehicles are underactuated in the sense
that the lateral velocity vi cannot be controlled directly, i.e.
νi = [ui , 0, ri ]T .
Assumption 2. The leader tracks a smooth and bounded
path and each vehicle can perfectly measure its own pose.
The leader’s velocities, and thus the nonlinear term (∆i ), are
unknown to the followers.
B. Control Objective
The follower must maintain a desired position relative to the
leader while it only has knowledge about the leader’s position
and orientation. Before establishing the control objective,
introduce the tracking error as
ei = fid − fi
(5)
where fid ∈ R2 is the desired formation specification. Taking
its derivative with respect to time yields
ėi = f˙id − Bi ci − ∆i
(6)
The control objective is to find the control input for the
follower that for any ǫ > 0 ensures kei (t)k ≤ ǫ for ∀t > t0 ,
Fig. 2.
Structure of SHL neural network
Construct V T = [vjk ] and W T = [wij ] weight matrices
and define σ = [1, σ1 (·), . . . , σN2 (·)]T ∈ RN2 +1 and the new
input vector x = [1, xT ]T ∈ RN1 +1 to incorporate the bias
terms, then the NN equation (7) can be expressed as
y = W T σ(V T x)
(8)
The universal approximation theorem [10] states that given
a continuous real-valued function f : Ω → RN3 over the
compact set Ω ∈ RN1 and any ǫ⋆ > 0 precision, then for
some sufficiently large N2 there exist bounded V T and W T
weights such that the approximation
f (x) = W T σ(V T x) + ǫ(x)
(9)
satisfies ǫ(x) ≤ ǫ⋆ . Typically, non-decreasing and bounded
activation functions are chosen, such as the sigmoid function
whose range is [0, 1].
In order to employ NN approximators for control purposes,
the unknown weight matrices have to be estimated and also
adapted in real-time such that the estimation error does not
violate the stability of the closed-loop system. The weight
adaptation laws will be later introduced in Section IV, here
some properties of the weight approximation is presented [8].
Define the weight estimation errors as
Ṽ = V̂ − V,
W̃ = Ŵ − W,
T
T
σ̃ = σ − σ̂ , σ(V x) − σ(V̂ x)
(10a)
(10b)
where V̂ , Ŵ are the estimations of the ideal V and W weights
provided by the weight adaptation laws. The Taylor series
expansion of the hidden-layer output can be given by
σ(V T x) = σ(V̂ T x) + σ ′ (V̂ T x)Ṽ T x + O(Ṽ T x)2
(11)
T
2
where σ ′ (ẑ) = dσ
dz |z=ẑ and O(Ṽ x) denotes second order
error terms. Consequently, the function approximation error
can be written as
fˆ(x) − f (x) = Ŵ T σ(V̂ T x) − W T σ(V T x)
= W̃ T (σ̂ − σ̂ ′ V̂ T x) + Ŵ T σ̂ ′ Ṽ T x + ρ (12)
with higher-order residual terms of
ρ = W̃ T σ ′ V̂ T x − W T O(Ṽ T x)2
= −W T (σ − σ̂) − W T σ̂ ′ V̂ T x + Ŵ T σ̂ ′ V T x
where σ̂ ′ = σ ′ (V̂ T x)
term takes form of

0
σj′

σ̂ ′ =  .
 ..
0
(13)
is used for shorthand. The derivative
...
...
..
.
0
0
..
.
...
′
σN
2



 ∈ RN2 +1×N1

(14)
Definition 1. [11]. The solution x(t) ∈ Rn of ẋ = f (t, x(t)) is
semi-globally uniformly ultimately bounded (SGUUB) if, for
any compact (bounded and closed) set Ω and ∀x(t0 ) ∈ Ω,
there exists some µ > 0 and T (µ, x(t0 )) such that kx(t)k ≤ µ
for all t ≥ t0 + T .
Theorem 1. [9]. For vehicle {i} described by kinematics (1)
with Assumptions 1-2 satisfied, let the control law be
i
h
(18)
c̄i = Bi−1 f˙id + kiN ei − ŴiT σ(V̂iT x)
and define the adaptive NN weight tuning laws as
˙
V̂i = −Gi xeTi ŴiT σ̂ ′ + kV i V̂i
i
h
˙
Ŵi = −Fi (σ̂ − σ̂ ′ V̂iT x)eTi + kW i Ŵi
(19a)
(19b)
where Fi > 0, Gi > 0, kW i > 0, kV i > 0 and the error gain
is chosen as
kiN = k1i + k2i kxŴiT σ̂ ′ k2F + k3i kσ̂ ′ V̂iT xk2
where σj′ (·) is the derivative of the activation function evaluated at the corresponding input.
Lemma 1. [8]. The residual term (13) with sigmoid activation
function can be upper bounded by
kρk ≤ kV kF kxŴ T σ̂ ′ kF + kW kkσ̂ ′ V̂ T xk + αkW k
√
where α = N2 + 1.
A. Controller Design
The leader-follower formation control algorithm is derived
using Lyapunov-theory and employs adaptive neural network
with online weight tuning laws to approximate the unknown
nonlinear function ∆i (t) of the leader’s velocity. The decentralized controller design follows [9] and the results are given
by the following definition and theorem.
(15)
Proof: Using properties of norms yields
kρk ≤ kW T (σ − σ̂)k + kW T σ̂ ′ V̂ T xk + kŴ T σ̂ ′ V T xk
≤ kW kkσ − σ̂k + kW kkσ̂ ′ V̂ T xk + kŴ T σ̂ ′ V T xk
≤ αkW k + kW kkσ̂ ′ V̂ T xk + kV kF kxŴ T σ̂ ′ kF
each activation function σj (·) ∈
Since σ ∈ RN2 +1 and for √
[0, 1], therefore kσ − σ̂k ≤ N2 + 1.
IV. A DAPTIVE NN F ORMATION C ONTROLLER
Taking into consideration the underactuated property, it
yields
ūi
cos φ̄i
T
c̄i = R (φ̄i )
=
ū
(16)
0
− sin φ̄i i
hence, selecting an appropriate control law c̄i = [c̄i1 , c̄i2 ]T ,
then ūi and ψ̄i , i.e. the longitudinal velocity and orientation
of the ith vehicle, can be determined according to
p
ūi
c̄2i1 + c̄2i2
(17)
=
ψ̄i
atan2(c̄i2 , c̄i1 ) + γiL
These high-level signals ūi and ψ̄i can deal as reference signals
for the low-level control system (not discussed here).
(20)
with positive k constants, then all closed-loop signals are
semi-globally uniformly ultimately bounded (SGUUB).
Proof: The proof follows [9]. Construct the Lyapunovfunction candidate
Li = 12 eTi ei + 21 tr{W̃ T F −1 W̃ } + 12 tr{Ṽ T G−1 Ṽ } (21)
and suppress indices i onward for better readability. Differentiating (21) by time and using that the ideal W and V are
˙
˙
˙ = Ŵ
constant matrices, hence W̃
and Ṽ˙ = V̂ , it follows
L̇i = eTi (f˙id − Bi ci − ∆i )+
˙
˙
tr{W̃ T F −1 Ŵ } + tr{Ṽ T G−1 V̂ }
(22)
Each component of the unknown function ∆i can be
approximated by a SHL NN to arbitrary accuracy by
∆i = W T σ(V T x) + ǫi (x)
(23)
where ||ǫi (x)|| ≤ ǫib is the bounded NN reconstruction error.
Substituting (18), (19) and (23) into (22) and using tr{A +
B} = tr{A} + tr{B} yields
h
i
L̇i = eTi Ŵ T σ(V̂ T x) − W T σ(V T x) − ǫi (x)
−tr{W̃ T (σ̂ − σ̂ ′ V̂ T x)eTi } − tr{Ṽ T xeTi Ŵ T σ̂ ′ }
−kV tr{Ṽ T V̂ } − kW tr{W̃ T Ŵ } − kiN ||ei ||
2
(24)
Moreover, using tr{AB} = tr{BA} and tr{λ} = λ for any
λ ∈ R, it follows from (12) and (20) that
L̇i = eTi [ρ − ǫi (x)] − kV tr{Ṽ T V̂ } − kW tr{W̃ T Ŵ }
i
h
2
− ||ei || k1 + k2 kxŴ T σ̂ ′ k2F + k3 kσ̂ ′ V̂ T xk2 (25)
The derivative of the Lyapunov function can be upper
bounded by applying (15) as
L̇i ≤ |eTi ρ| + |eTi ǫi (x)| − kV tr{Ṽ T V̂ } − kW tr{W̃ T Ŵ }
i
h
2
− ||ei || k1 + k2 kxŴ T σ̂ ′ k2F + k3 kσ̂ ′ V̂ T xk2
o
n
≤ kei k kV kF kxŴ T σ̂ ′ kF + kW kkσ̂ ′ V̂ T xk + αkW k
+kei kkǫi (x)k − kV tr{Ṽ T V̂ } − kW tr{W̃ T Ŵ }
h
i
2
− ||ei || k1 + k2 kxŴ T σ̂ ′ k2F + k3 kσ̂ ′ V̂ T xk2
(26)
Upper bounds can be given on the following terms by using
kAk2F = tr{AT A} and completing the squares
kV
kV
−kV tr{Ṽ T V̂ } ≤
kV k2F −
kṼ k2F
2
2
kW
kW
kW k2F −
kW̃ k2F
−kW tr{W̃ T Ŵ } ≤
2
2
2
(27)
2
for any a, b ∈ R,
Furthermore, from the fact that ab ≤ a +b
2
1
it follows ka k1 b ≤ k2 kak2 + 2k
kbk2 for any k > 0, hence the
following inequalities hold
k4
1
kei k2 +
kǫi (x)k2
2
2k4
1
kV k2F + k2 kei k2 kxŴ T σ̂ ′ k2F
kei kkV kF kxŴ T σ̂ ′ kF ≤
4k2
1
kW k2 + k3 kei k2 kσ̂ ′ V̂ T xk2
kei kkW kkσ̂ ′ V̂ T xk ≤
4k3
k5
1
αkei kkW k ≤ α2 kei k2 +
kW k2
(28)
2
2k5
Plugging inequalities (27) and (28) back into (26) and
collecting the terms of different norms, one can obtain
kei kkǫi (x)k ≤
k5
kW
k4
kV
− )kei k2 −
kṼ k2F −
kW̃ k2F
2
2
2
2
1
kV
1
kV k2F
+
kǫi (x)k2 +
+
2k4
2
4k2
1
1
kW
2
kW k2
(29)
kW̃ kF +
+
+
2
4k3
2k5
L̇i ≤ −(k1 − α2
Introducing the following notations
ηi , min k1 − α2 k25 − k24 ; k2V ; k2W
µi , 2k14 kǫi (x)k2 + k2V + 4k12 kV k2F
+ k2W kW̃ k2F + 4k13 + 2k15 kW k2
then, the upper bound on the Lyapunov candidate function
becomes
h
i
L̇i ≤ −ηi kei k2 + kṼ k2F + kW̃ k2F + µi
(30)
Consider the compact set
µi
(31)
Γi = (ei , V̂ , Ŵ ) : kei k2 + kṼ k2F + kW̃ k2F ≤
ηi
Clearly, if kei k2 + kṼ k2F + kW̃ k2F > µηii , then
µi
<0
(32)
L̇i ≤ −ηi kei k2 + kṼ k2F + kW̃ k2F −
ηi
This means that Li is negative outside the compact set
(31), e.g. any closed-loop trajectory starting outside Γi will
eventually reach and remain inside this set. In fact, ei (t), Ṽi (t)
and W̃i (t) are semi-globally uniformly ultimately bounded.
Remark 1. The so-called σ-modification adaptation law (19)
may be changed to e-modification term for better trajectory
tracking by using
˙
(33)
V̂i = −Gi xeTi ŴiT σ̂ ′ + kV i kei kV̂i
h
i
˙
Ŵi = −Fi (σ̂ − σ̂ ′ V̂iT x)eTi + kW i kei kŴi
(34)
For further details see [12] and [13].
Remark 2. For practical systems operating in the presence
of uncertainties, SGUUB is a satisfactory objective since
asymptotic stability may be too strong to achieve. The tracking
performance can be improved by increasing the number of
neurons in the NN hidden layer which results smaller approximation error and thus smaller Γ. Moreover, increasing the
control parameters results larger η which means that Γ is
reduced. On the other hand, larger µ increases the domain of
attraction (semi-globally).
B. Implementation Strategies
The actual control input ūi to system (1) is computed by
(17). The angular velocity r̄i can be acquired by using filtered
differentiation with time constant Td as
r̄i , ψ̇F i = (ψ̄i − ψF i )/Td .
(35)
For practical reasons, it is important to take into account the
limitations of the control efforts. Usually, the vehicles do not
necessarily start close to the desired formation, thus large
control signals may appear initially. A common approach is
to use a command filter to influence the trajectories of the
vehicles at the initial stage [14]. The following guidance law
was chosen
f˙id = Tc (fic − fid ),
fid (0) = fid0
(36)
where Tc is the time constant of the command filter and
fic is the constant command of desired relative distance and
bearing angle. The output of the filter determines a timevarying desired formation specification.
Another important consideration is the input choice and the
weight initialization of the NNs. Generally, the input vector
x can be constructed from the actual and previous control
signals and system outputs [13]. For the formation controller
presented in Section IV, the NN inputs of vehicle {i} are
selected using simulation step size Ts as follows:
x = [1, fiT (t), fiT (t − Ts ), ψi (t) − ψL (t), c̄Ti (t − Ts )]T
Initializing the NN weights to zeros will not decrease the
approximation performance [12]. However, from a practical
point of view, it is preferred if the weight adaptation starts
close to the accurate values in order to avoid large control
signals at the initial stage. Thus, if feasible, a pre-initialization
may be required before the actual control takes place.
1.
2.
3.
0
y (m)
1.
3.
2.
-100
4.
trajectory
reference
leader
desired
-200
0
100
5.
6.
200
300
400
500
600
700
t = 5.00s
t = 0.00s
x (m)
ψ (rad)
0.5
t = 10.00s
4.
5.
6.
ψL
ψ1
ψ2
ψ3
0
-0.5
-1
10
15
20
25
t (s)
Fig. 3.
The number of hidden layer neurons are chosen experimentally, for satisfactory approximation N2 = 10 is sufficient.
The NN weights are initially set to zeros. Before enabling the
actual control to the vehicle, the neural network adaptation
process is activated for 0.5s in order to decrease the weight
errors and thus the initial control effort.
The control parameters are selected for each vehicle as
follows:
−2
Fi = 10
,
t = 25.00s
Output trajectories: positions (to scale) and orientations of three UGVs moving in V-shape formation
V. S IMULATION R ESULTS
In this section the simulation results of the adaptive formation control system are presented and its effectiveness is
investigated through a realistic high-speed maneuver.
A group of three follower vehicles and one leader is
considered moving in the x–y plane. The reference trajectories
for the leader are generated by a 16 degree of freedom fullvehicle dynamic model using a prescribed steering rate and
longitudinal velocity profile [15]. The resulting motion consists of an overlapping acceleration/deceleration and turning
phase at high speed ranging between 25-35m/s with small side
slip angle (less than 3 degrees).
For each vehicle, the neural network based adaptive formation controller are implemented separately using sigmoid
activation functions in form of
1
σ(x) =
1 + exp(−x)
Gi = 10−2 ,
t = 20.00s
t = 15.00s
kV i = 0.3,
kW i = 0.3,
k2i = 10−3 ,
k1i = 2,
−3
k3i = 10
For high-speed maneuvering, simulation time Ts = 0.01s is
chosen and the filter time constants are set to Td = 0.1s and
Tc = 0.5s satisfying Shannon’s rule.
A V-shape formation is commanded such that the first
follower travels 10m behind the leader and the rest of the
vehicles are aligned in 45 degree angle with 7m distance
relatively to the first follower.
d1L
20
(m)
5
d2L
d3L
did
15
10
0
5
10
15
20
25
0.5
0
(rad)
0
-0.5
φ1L
φ2L
φ3L
φid
-1
0
5
10
15
20
25
t (s)
Fig. 4.
Convergence of formation specifications
The output positions and orientation along with a few
snapshots of the motion for different times are shown in
Fig. 3. It can be observed that the followers are efficiently
estimated the leader’s unknown velocity and the formation
is well-established during the motion maneuver. This can be
verified by inspecting Fig. 4 where the formation specifications
converge to the desired values after a short transient.
Also, note that small lateral deviation may occur from the
(real) reference trajectory obtained from the vehicle dynamics
(Fig. 3), which can be explained by the spatial configuration
of the formation and the presence of small lateral velocity due
to the side slip angle.
Fig. 5 shows the control signals for each vehicle and Fig. 6
demonstrates the adaptation of the unknown terms ∆i ∈ R2
and their true values. Clearly, the leader’s uncertainties are
efficiently estimated and compensated by the NN controller.
In summary, the developed adaptive NN-based controller is
40
u (m/s)
35
30
25
ū1
ū2
ū3
uL
20
0
5
10
15
20
25
r (rad/s)
0.5
0
ACKNOWLEDGEMENTS
r̄1
r̄2
r̄3
rL
-0.5
0
5
10
15
20
25
t (s)
Fig. 5.
Control signals
(m/s)
30
NN11
NN21
NN31
∆i1
25
20
0
5
10
0
5
10
15
1
NN12
NN22
20 NN32
∆i2
25
0.5
0
-0.5
-1
15
20
25
t (s)
Fig. 6.
The research of B. Lantos was supported by the MTABME Control Engineering Research Group. The research of
G. Max was supported by Hungarian National Grant UNKP1-13: "Advanced nonlinear control of electric vehicles".
R EFERENCES
35
(rad/s)
The developed control algorithm can be considered as a part
of a higher level system which generates velocity reference
signals for a lower level controller where the actual control
takes place.
The effectiveness and high precision of the formation
controller was investigated via realistic, high-speed motion
maneuvers. Simulation results demonstrated the effectiveness
of the proposed algorithm.
Future research will concentrate on the extension of the
formation controller to incorporate realistic vehicle dynamics.
Approximation performance of NNs
capable of high-speed formation tracking considering underactuation and the unknown velocity information of the leader
vehicle.
VI. C ONCLUSION
In this paper an adaptive control law is designed for the
formation control of underactuated autonomous ground vehicles based on the leader-follower approach and the kinematics equations. The follower vehicles have limited knowledge
about the leader’s states, only the position and orientation
information can be accessed. The unknown term containing
the velocity information of the leader is estimated using
neural networks with online adaptive weight tuning laws.
The decentralized formation control algorithm incorporates the
neural network estimation and guarantees semi-global ultimate
uniform boundedness of all closed loop error signals. The
stability is proven using Lyapunov techinque and showed that
the tracking error can be made small by appropriate choice of
the control parameters.
[1] C. B. Low, “A flexible leader-follower formation tracking control design
for nonholonomic tracked mobile robots with low-level velocities control
systems,” in 2015 IEEE 18th International Conference on Intelligent
Transportation Systems, 2015, pp. 2424–2431.
[2] S. E. Li, S. Xu, G. Li, and B. Cheng, “Periodicity based cruising
control of passenger cars for optimized fuel consumption,” in 2014 IEEE
Intelligent Vehicles Symposium Proceedings. IEEE, 2014, pp. 1097–
1102.
[3] M. A. Kamel, K. A. Ghamry, and Y. Zhang, “Fault tolerant cooperative
control of multiple uavs-ugvs under actuator faults,” in Unmanned
Aircraft Systems (ICUAS), 2015 International Conference on, 2015, pp.
644–649.
[4] M. Ono, G. Droge, H. Grip, O. Toupet, C. Scrapper, and A. Rahmani,
“Road-following formation control of autonomous ground vehicles,” in
2015 54th IEEE Conference on Decision and Control (CDC), 2015, pp.
4714–4721.
[5] A. Loria, J. Dasdemir, and N. A. Jarquin, “Leader–follower formation
and tracking control of mobile robots along straight paths,” IEEE
Transactions on Control Systems Technology, vol. 24, no. 2, pp. 727–
732, 2016.
[6] L. Consolini, F. Morbidi, D. Prattichizzo, and M. Tosques, “Leader–
follower formation control of nonholonomic mobile robots with input
constraints,” Automatica, vol. 44, no. 5, pp. 1343–1349, 2008.
[7] Z. Yao, Y. Song, and W. Cai, “Neuro-adaptive virtual leader based
formation control of multi-unmanned ground vehicles,” in Control
Automation Robotics & Vision (ICARCV), 2010 11th International
Conference on, 2010, pp. 615–620.
[8] F. L. Lewis, A. Yesildirek, and K. Liu, “Multilayer neural-net robot
controller with guaranteed tracking performance,” IEEE Transactions
on Neural Networks, vol. 7, no. 2, pp. 388–399, 1996.
[9] Z. Peng, D. Wang, Y. Yao, W. Lan, X. Li, and G. Sun, “Robust adaptive
formation control with autonomous surface vehicles,” in Proceedings of
the 29th Chinese Control Conference, 2010, pp. 2115–2120.
[10] K. Hornik, “Approximation capabilities of multilayer feedforward networks,” Neural networks, vol. 4, no. 2, pp. 251–257, 1991.
[11] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable adaptive neural
network control. Springer Science & Business Media, 2013, vol. 13.
[12] F. Lewis, S. Jagannathan, and A. Yesildirak, Neural network control of
robot manipulators and non-linear systems. CRC Press, 1998.
[13] Y. H. Kim and F. L. Lewis, High-level feedback control with neural
networks. World Scientific, 1998.
[14] N. Hovakimyan, F. Nardi, A. Calise, and N. Kim, “Adaptive output
feedback control of uncertain nonlinear systems using single-hiddenlayer neural networks,” IEEE Transactions on Neural Networks, vol. 13,
no. 6, pp. 1420–1431, 2002.
[15] G. Max and B. Lantos, “Active suspension, speed and steering control
of vehicles using robotic formalism,” in 2015 16th IEEE International
Symposium on Computational Intelligence and Informatics, Nov 2015,
pp. 53–58.