Adaptive formation control of a fleet of mobile robots:
Application to autonomous field operations.
Roland Lenain∗, Johan Preynat∗, Benoit Thuilot†, Pierre Avanzini†, Philippe Martinet†
24, av. des Landais
63172 Aubi` ere Cedex France
24, av. des Landais
63177 Aubi` ere Cedex France
Abstract—The necessity of decreasing the environmental
impact of agricultural activities, while preserving in the same
time the level of production to satisfy the growing population
demand, requires to investigate new production tools. Mobile
robotic can constitute a promising solution, since autonomous
devices may permit to increase production level, while reducing
pollution thanks to a high accuracy. In this paper, the use
of several mobile robots for field treatment is investigated.
It is here considered that they can exchange data through
wireless communication, and a formation control law, accurate
despite typical off-road conditions (low grip, terrain irregu-
larities, etc), is designed relying on nonlinear observer-based
adaptive control. The algorithm proposed in this paper is tested
through advanced simulations in order to study separately its
capabilities, as well as experimentally validated.
The continuous advances in autonomous mobile robot
control (concerning both a single robot , as well as
multi-robots , ) offer new possibilities in terms of
applications for every-day life improvement. For instance,
the development of automated multi-robot fleets can benefit
to many applications requiring to cover large areas , such
as surveillance, cleaning, exploration, etc. It is particularly
interesting in environmental applications such as farming,
where the use of several light robots in the field may permit
to reduce environmental impact while preserving the level of
production. This constitutes a challenging problem as stated
in . Rather than considering numerous small robots, as
in swarm robotics , a cooperation framework with a
limited number of light machines seems preferable when
field treatment is addressed : on one hand, some farming
operations such as harvesting require quite large machines
to achieve tasks properly, and on the other hand, it appears
more tractable from a practical point of view (maintenance,
monitoring, acceptability, etc). As a consequence, this paper
is focused on formation control of several light robots exe-
cuting operations in field (as illustrated in figure 1), allowing
the use of several autonomous entities instead of driving a
sole huge vehicle.
In the considered applications, a reference path is defined
by the leader vehicle, controlled either manually or auto-
nomously. The shape of the formation is not considered as
fixed, since the area covering may require a varying forma-
tion (tank unload, maneuvers, etc). Several approaches have
been proposed for mobile robot formation control , ,
but they are mainly dedicated to structured environments.
In contrast, the context of the considered tasks requires a
high accurate relative positioning of the robots despite the
numerous perturbations encountered in natural environment
(skidding, terrain irregularities, etc). This is not addressed by
In this paper, an
adaptive algorithm for
formation control is
proposed, relying on
a reference trajectory
and lateral dynamics
with respect to the
advance of each robot along the reference path can be
addressed independently from the regulation of its lateral
deviation with respect to this path. Longitudinal control is
based on the regulation of curvilinear inter-vehicle distances,
while lateral regulation relies on an observer-based adaptive
control approach. The control of the possibly varying
formation gathers both control laws, enabling an accurate
formation regulation for field operations, independently
from the reference path shape and environment properties.
The paper is organized as follows : a model dedicated
to a mobile robot formation is firstly introduced, together
with the observation strategy allowing to reflect the bad
grip conditions encountered in natural environment. Based
on this model, the control of each mobile robot is then
detailed : longitudinal control is recalled from previous
work, while lateral control is designed with respect to a
varying set point. The validation of the proposed control is
finally achieved thanks to advanced simulations and actual
experiments (limited in this paper to urban mobile robots).
Fig. 1.Illustration of the application
II. MOBILE ROBOT MODELING
The autonomous control of a fleet of mobile robots is
considered with respect to a desired path, used as a reference
frame for both longitudinal and lateral positioning of each ro-
bot. The objective is to ensure an accurate overall motion of
the robots in a desired, but potentially varying, configuration
along this chosen trajectory. The control problem investigated
in this paper then derives from the path tracking one. The
following framework for formation control is consequently
A. Model of a robot formation
The overall control strategy for the robot formation is
based on the modeling proposed in figure 2 (two robots
among n are shown).
Fig. 2.Longitudinal model of a robot fleet
In this representation, each robot is viewed as a bicycle,
as in the celebrated Ackermann model, see  : an unique
wheel for the front axle and another one for the rear axle. The
classical rolling without sliding assumption is not satisfied in
a natural context. As they affect significantly robot dynamics,
low grip conditions depreciate the path tracking accuracy.
In order to account for this specificity, two sideslip angles
are added : βFand βR, respectively for front and rear
axles. Their estimation is described in section II-B. These
variables are representative of the difference between the
tire orientation and the actual tire speed vector direction.
Longitudinal sliding has not been described, since in the
considered applications, longitudinal guidance accuracy is
not as critical as lateral one. Based on these assumptions,
the notations used in the sequel are depicted in figure 2 for
the ithrobot and hereafter listed :
• Γ is the common reference path for each robot defined
in an absolute frame (computed or recorded before-
• Oiis the center of the ithmobile robot rear axle. It is
the point to be controlled for each robot.
• si is the curvilinear coordinate of the closest point
from Oibelonging to Γ. It corresponds to the distance
covered along Γ by robot i.
• c(si) denotes the curvature of path Γ at si.
•˜θidenotes the angular deviation of robot i w.r.t. Γ.
• yiis the lateral deviation of robot i w.r.t. Γ.
• δiis the ithrobot front wheel steering angle.
• l is the robot wheelbase.
• viis the ithrobot linear velocity at point Oi.
i denote the sideslip angles (front and rear)
of the ithrobot.
i and βR
Using these notations, the motion equations for the ith
mobile robot can be expressed as (see  for details) :
Expression (1) does not exist if [1−c(si)yi] = 0 (i.e. point
Oiis superposed with the instantaneous reference path center
of curvature). Such a situation is not encountered in practice,
since robots are supposed to be properly initialized. The
state vector for robot i is then defined as Xi= [si yi˜θi]T,
and is supposed to be measured. As a result, model (1) is
entirely known as soon as sideslip angles βF
accessible. As these variables cannot be easily measured,
they are estimated thanks to an observer described below.
B. Sideslip angle estimation
As sideslip angles integrated into robot model (1) are
hardly measurable directly, their indirect estimation has to
be addressed. The observer-based approach detailed in 
is here implemented. It follows the algorithm described in
figure 3, taking benefit of the duality principle between
observation and control.
Fig. 3.Observer principle scheme
More precisely, model (1) is considered as a process
to be regulated thanks to sideslip angles. The observer
consists then in a control law designed for these sideslip
angles, with the aim to ensure the convergence of some
model outputs (Xobs
measured ones (¯ Xi= [¯ yi¯˜θi]T). Such a convergence ensures
that model (1) is representative of vehicle actual behavior,
whatever the grip conditions, and sideslip angle values can
then be reported into mobile robot control laws. The detailed
computation of this observer and proofs of stability are
available in .
]T) with the corresponding
C. Model exact linearization for control
Kinematic model (1) has been extended to account for
low grip conditions. Nevertheless, it is still consistent with
classical kinematic models, such as considered in . It can
consequently be turned into a chained form, enabling then
an exact linearization. This can be achieved by imposing the
following invertible state and control transformations :
[si,yi,(1 − c(si)yi)tan(˜θi+ βR
[m1i,m2i] = [vicos(˜θ+βR
which turn system (1) into system (3).
Let us now consider, in system (3), the derivative with respect
to curvilinear abscissa, instead of the derivative with respect
to time. This leads finally to system (4) :
which constitutes an exact linear form. Since a1i= si, then
1i= 1 and is consequently removed from the model, which
is then unchanged whatever the robot velocity. As a result,
these transformations permit to separate formally the robot
longitudinal behavior from its lateral motion with respect to
the path to be followed. Both longitudinal and lateral control
can then be addressed independently.
III. MOBILE ROBOT FORMATION CONTROL
To address the control of a fleet of mobile robots in a path
tracking context, the relative positioning of each robot with
respect to the reference trajectory is achieved and then shared
within the fleet via wireless communication. The control of
each robot aims then at ensuring the convergence to desired
set points in terms of curvilinear offset (longitudinal control)
and lateral deviation offset (lateral control). In the sequel,
the longitudinal control issued from previous work is briefly
described. Then, the lateral control law, constituting the main
contribution of this paper, is detailed.
A. Longitudinal control law
The objective of longitudinal control is to maintain a
desired distance (denoted d) between curvilinear abscissas
of successive vehicles. Preferentially, each robot is controlled
with respect to the curvilinear abscissa s1of the leader (1st
vehicle). This enables to avoid an oscillating behavior due
to error propagation along the fleet. However, for obvious
safety reasons, the distance to the previous vehicle has also
to be considered. Therefore, as proposed in , a composite
error xi equal to the distance to the leader vehicle e1
the nominal case, and smoothly commuting to the distance
to the preceding vehicle ei−1
approached, is here regulated, see figure 4. The auxiliary
control m1i (and therefore vi) ensuring that xi converges
with zero can easily be designed from the first equation in
model (3), so that each vehicle can be accurately and safely
controlled longitudinally, whatever the velocity of the leader.
when the security distance is
B. Lateral control law
1) Lateral desired set point: Once longitudinal control has
been achieved, the control of the lateral position of each robot
can be addressed. In contrast to the classical path tracking
problem, where the tracking error is expected to be null ,
the lateral deviation of each robot in a formation has to
Fig. 4. Longitudinal control scheme
Fig. 5.Lateral control model
converge to a non-null desired set point. To this aim, the
model is extended with a new variable yd
of the desired lateral deviation of robot i, and permitting
the relative positioning of mobile robots in 2D space, see
2) Control law design: Relying on linear system (4)
derived from state transformations (2), the goal of lateral
control consists in regulating a2i= yito a desired set point :
yi → yd
i. This objective can be achieved by imposing the
virtual control law (5) for m3i:
This indeed leads to the following second order differential
equation satisfied by the regulation error of the ithrobot,
i= a2i− yd
which ensures the convergence of ǫy
The steering control law of robot i can then be deduced
from the virtual control m3ithanks to the inverse transfor-
mations (2). It leads (if ǫ′′y
is neglected) to :
ito zero (i.e : yi→ yd
• the longitudinal acceleration can be neglected (˙ vi= 0).
The gains of the longitudinal control law can be tuned
˜θ + βR
1 − c(si)yi
1 + tan2(˜θ2i) −
i− Kdαiη + c(si)αiηtan˜θ2i
Control law (8) exists under the following assumptions :
to meet this assumption, while keeping a satisfactory
• 1 − c(si)yi ?= 0 : model existence condition, already
perpendicular to the path to be followed. It is satisfied
when the formation is properly initialized.
In the same way that d permits to define the distance between
robots within the fleet and then their relative longitudinal
positions, the variable yd
iin (7) permits to define their lateral
positions with respect to the global formation motion. Lon-
gitudinal and lateral relative positions of each robot can then
be specified in the reference trajectory frame independently.
The set point yd
ihas now to be constructed to regulate a
desired formation, in order to achieve a multi-robot task.
2[π], i.e. the rear robot speed vector is not
3) Generation of desired set point: When achieving a field
treatment with several machines, the desired lateral distance
between the tracks of each vehicle is chosen as the implement
width, usually reduced with 15% in order to have a small
overlapping margin to ensure a proper field covering. Just
as in the longitudinal case, in order to avoid an oscillating
behavior due to error propagation along the fleet, each
robot must preferentially be controlled with respect to the
leader trajectory. This can easily be achieved by specifying a
iin (7). This first mode is completely satisfactory
as long as vehicles are never side-by-side (e.g. when field
treatment is achieved according to a winger configuration).
Mode 1 (fixed inter-track) : yd
tant chosen w.r.t. implement widths. The lateral position
of a robot is independent from previous robot behaviors
and does not repeat their possible deviations.
i(si) = dy
i, with dy
In contrast, when robots have to work side-by-side (e.g. a
farm tractor moving alongside a combine harvester to unload
it), robot i must reproduce robot i − 1 deviation in order
to enable joint work (e.g. unloading) and avoid collision.
Ideally, as long as robot i−1 deviation does not exceed some
pre-specified threshold, robot i should be controlled with
respect to the leader trajectory, in order to avoid the above
mentioned oscillating behavior due to error propagation, and
when the threshold is exceeded, then robot i should be
controlled with respect to robot i − 1 trajectory. Such a
behavior can actually be imposed, since the desired lateral
set point yd
i(si), introduced in figure 5, may be varying. For
this second mode, it is here proposed to design yd
i(si) as :
i+ σ(yi−1)[yi−1− dy
where σ(yi−1) ∈ [0; 1] is the smooth commutation function
shown in figure 6 : if yi−1is small, then σ(yi−1) = 0, so
ias in the first mode. In contrast, if yi−1is
large, then σ(yi−1) = 1, so that robot i lateral objective is
to reproduce robot i − 1 lateral deviation.
Mode 2 (adapted inter-track) : yd
Robot i reproduces robot i − 1 deviation, if the latter
exceeds a pre-specified threshold.
i(si) = dy
i(si) is defined by (9)
Previous robot tracking error (m)
Fig. 6. Shape of commutation function
IV. SIMULATION RESULTS
In order to validate theoretically the proposed algorithm
and investigate its robustness with respect to sliding pheno-
mena, simulated results obtained with MATLAB/Simulink
coupled to multi-body dynamic simulation software Adams
are here reported. Such a simulation testbed permits to
render accurately the behavior of actual mechanical sys-
tems. In particular, the specificities of mobile robot motion
in natural environment can be emulated (grip conditions,
ground irregularities, low-level settling time, etc). In order
to investigate the capabilities of the proposed algorithm,
three mobile robots have been designed (as depicted in
figure 7(a)) and a soil with low grip conditions (equivalent to
wet grass) has been parameterized. Delays and settling times
of the actuators are also accounted : 500ms for the steering
actuator and 700ms for the velocity actuator, corresponding
to the values measured on available experimental robots. A
reference path consisting in an U-turn has been computed,
such as depicted in figure 7(b) in black plain line. It has to be
followed by the leader at a 3m.s−1velocity, and the desired
distance between robots has been set to d = 5.5 m.
(a) Virtual mobile robots
(b) Reference trajectory
Fig. 7.Simulation testbed
The objective of the three simulated robots, as well as their
control parameters, are described below :
• Leader : no lateral deviation is specified (i.e. yd
and the control law accounting for sliding is used.
• Robot 2 (first follower) : mode 1 has been considered,
i.e. a constant desired lateral deviation yd
has been specified. Moreover, in order to demonstrate
the influence of sliding, the sideslip angle estimation
has been frozen, i.e. (βF
• Robot 3 (second follower) : mode 2 has been consi-
dered, i.e. a composite objective has been specified
3 = −2m. The control law accounting for
sliding is used, in order to show how robot 3 reacts
to robot 2 large tracking error (following from sliding
phenomenon encountered during the curve).
2) = (0,0).
Global trajectories achieved by each robot are superposed
in figure 7(b). Next, the lateral deviations of the three robots
recorded during this test are reported in figure 8.
Lateral deviation (m)
Robot 3 lateral deviation
Robot 2 lateral deviation
Leader lateral deviation
Curve for rob.1
Curve for rob.2
Curve for rob.3
Fig. 8.Lateral deviations recorded in simulation
The leader tracking error, depicted in blue plain line, is
satisfactorily regulated to zero with an accuracy inferior to
10cm, despite the bad grip conditions. A 20cm overshoot
can nevertheless be observed at the beginning and at the end
of the curve (times 13s and 23s), due to the settling time of
the simulated actuator.
On the contrary, the second mobile robot is unable to
achieve its objective with such an accuracy : during the
straight line parts of the trajectory (before 14s and after
24s), robot 2 accurately reaches the expected -1m devia-
tion. However, during the curve, it converges with a -1.2m
constant lateral deviation (i.e. ǫy
grip conditions : sideslip angle estimated values are indeed
around 3◦, which is consistent with what is recorded in actual
experiments, see . This proves that sliding phenomenon,
when not reported in control laws as it is the case for robot 2,
significantly depreciates lateral guidance accuracy.
Finally, since sliding phenomenon is accounted in robot 3
control laws, this robot could have accurately met its objec-
tive. However, since mode 2 has been considered and robot 2
lateral deviation is large during the curve, robot 3 lateral
objective has been adapted : ǫy
center of the transition part in the commutation function,
see figure 6. Consequently, yd
3is shifted to -2.1m instead of
-2m, and it can be observed that robot 3 lateral deviation
satisfactorily converges with this adapted objective during
the curve. Of course, when the curve is over (after time
27s), robots no longer undergo sliding phenomenon : robot 2
lateral deviation then goes back to its objective, and so does
robot 3 lateral deviation.
2=20cm) because of the bad
2=20cm corresponds to the
These results demonstrate the capabilities of the algorithm
in accurately controlling a formation, despite sliding pheno-
menon. It also shows the ability of mode 2 (adapted inter-
track) to manage on-line formation reconfiguration.
051015 20 25 30
Robots interdistance (m)
(a) Distances between robots
05 10 15202530
Robot velocity (m/s)
Robot 3 velocity
Robot 2 velocity
(b) Mobile robot velocities
Fig. 9.Simulation results on longitudinal servoing during simulation
The longitudinal performances are investigated in figure 9.
The distances between robots (reported in figure 9(a)) show
that the 5.5m objective is satisfactorily obtained, despite
skidding and variations in the curvature (robots 2 and 3 have
a larger targeted curvature since they are on the external part
of the curve, as pointed out in figure 7(b)). In the same way
than for lateral behavior, some overshoots can be recorded
when each robot enters into the curve, due to the low-level
settling time. Since mobile robots 2 and 3 are on the external
part of the curve, they have to increase their speed to preserve
the desired inter-distance. Their velocities are compared in
figure 9(b), and it can be noticed that the velocities of robot
2 and 3 present the expected behavior.
V. EXPERIMENTAL RESULTS
A. Experimental mobile robots and equipment
As a first step, prior to
full-scale experiments with
agricultural vehicles opera-
ting off-road, the proposed
approach has been implemen-
ted on the electric vehicles
Cycab shown in figure 10.
They are equipped with an
RTK-GPS receiver supplying
an absolute position informa-
tion, accurate to within 2cm.
Coupled with a Kalman filter, this sensor permits also to
access to vehicle heading (and consequently to the orien-
tation error). The communication between robots is ensured
thanks to a WiFi access point enabling to transfer all the data
required to feed both longitudinal and lateral control laws.
These robots are able to move up to 10km.h−1.
A reference trajectory has been learnt beforehand, with
the leader robot manually driven. This trajectory, reported in
figure 11, is composed of a straight line, followed by a curve
and another part of straight line. This allows to investigate the
robustness of control algorithms with respect to path shape.
This path has been followed by a fleet composed of two
mobile robots. The complete control law, including sideslip
angle estimation, has been used. A 1m.s−1desired velocity
Fig. 10.Mobile robots
Fig. 11.Reference path Download full-text
has been imposed to the leader, and an 8m longitudinal
desired distance between the two robots has been specified.
For the second robot, the lateral set point has been chosen
as a sinusoidal curve, with a 0.35m variation range and a
10.5m period w.r.t. the curvilinear abscissa. This objective
is consistent with the vehicle steering angle and steering
rate limits. Such an experiment is not representative for a
specific agricultural task, but it permits to clearly investigate
the capabilities of the proposed algorithm when the lateral
set point is varying.
B. Validation with a varying lateral set point
The results regarding lateral regulation are reported in
figure 12. The desired sinusoidal error is reported in green
dotted line, while the actual follower’s error during the test
is depicted in red dashed line1.
Follower curvilinear abscissa (m)
Lateral deviation, yi (m)
Fig. 12.Lateral deviations obtained during the experiment
It can be noticed that, even when the lateral set point is
always varying, the difference between desired and actual
deviations stay quite small (below 10cm) all along the
trajectory tracking and whatever the trajectory shape. The
accuracy level is indeed not altered in the curve : the lateral
deviation error is unchanged between curvilinear abscissas
65m to 78m. Finally, the tracking error of the leader is
reported in black plain line in the same figure. It can be
seen that its lateral error stays below 10cm, even during the
curve, showing the relevance of the path tracking algorithm.
VI. CONCLUSION AND FUTURE WORKS
This paper proposes an algorithm for the accurate control
of a mobile robot formation moving off-road. This approach
considers the formation control as the combination of (i) a
ofthis test andanother
site lateraldeviation arethe:
platooning control and (ii) an extension of the path tracking
problem to a non-null lateral deviation regulation. As a result,
the control of each vehicle is decomposed into longitudinal
and lateral control with respect to a reference path. An
adaptive control strategy allows to take into account for low
grip conditions, as well as other phenomena encountered off-
road and depreciating the accuracy of classical algorithms.
The relative positioning of each robot with respect to a
possibly varying formation can then be regulated with a few
centimeter accuracy, whatever the shape of the reference
trajectory and the grip conditions. The efficiency of the
approach has been tested through actual experiments with
two urban mobile robots.
The efficiency of the control algorithm with respect to
sliding phenomena has been checked in advanced simulations
and must now be validated by experimental tests with off-
road mobile platforms. In addition, the proposed strategy is
focused on the regulation of a formation with respect to a
reference trajectory supplied beforehand. Such an algorithm
has now to be extended in order to manage automatically
the formation (modification of the formation, mobile robot
entering/leaving the fleet, obstacle avoidance, etc).
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