IEEE Robotics & Automation Magazine - March 2014 - 32

(such as grip conditions, terrain geometry, and inertial
effects), or changes in the shape of the formation to match the
application expectations, or just because the reference path
exhibits high curvature variations. In such cases, the actuators
settling time may decrease the relative positioning accuracy. If
several variations are unpredictable (such as changes in leader
velocity or in grip conditions), the changes origiBecause of the adaptive
nating from reference
path properties can be
and predictive approaches, anticipated to prevent the
robots formation from
highly accurate relative
transient mismatches. As
a result, both longitudinal
positioning may be
and lateral control laws
are decomposed into two
obtained.
parts: 1) a reactive part,
dealing with unpredictable effects and 2) a predictive part, taking reference trajectory properties (mainly the curvature) into consideration. The
two control parts are hereafter detailed.
Reactive Control
Longitudinal Control
The objective of longitudinal control for the ith robot is to
maintain a desired curvilinear distance with respect to some
other robot within the formation. Depending on the chosen
reference robot (1 to n except i), the formation behavior can
be slightly different: if the regulation of the ith robot is
referred to the immediate preceding (i - 1)th robot, servoing
errors propagation may lead to an oscillating behavior. On
the contrary, if the longitudinal errors of all the robots are
defined with respect to the first one, there is no servoing
errors propagation, but a failure in one of the preceding
robots [i.e., the (i - 1)th robot] can lead to a collision. As a
result, to achieve a nonoscillating as well as safe behavior, the
longitudinal control of the ith robot with respect to all the
other ones is first evaluated and from the collection of n - 1
velocities v ki thus obtained, a mixed control is inferred by
means of a linear combination
vi =

k = 1, k ! i

v ki

e with

k = 1, k ! i

= 1 o.

(4)

The design of the nonlinear longitudinal control law v ki
when k = 1 can be found in [18]. The proposed control
expression is generalized below to achieve the regulation of
the ith robot with respect to the kth one
v ki =

1 - c (s i) y i v k cos (iuk + b Rk )
e
+ l ki e ki + do ki o
1 - c (s k) y k
cos (iui + b Ri )

IEEE ROBOTICS & AUTOMATION MAGAZINE

MARCH 2014

Lateral Control
Since the extended kinematic model (1) can be converted into
a chained form, lateral control can be designed independently
from longitudinal control. The objective can be described as a
generalized path tracking task: the ith robot has to follow the
reference path C, but at some given lateral distance y di . This
lateral distance may of course be null [e.g., platoon formation;
see Figure 4 (a)], but not necessarily [e.g., wing-shaped or line
formations; see Figures 3 or 4(b)], and moreover may also vary
in case online reconfiguration of the formation is expected
and/or required for safety reasons. More precisely, planned
variations in the formation shape (e.g., transition from a wingshaped formation to a platoon formation to cross a narrow
area) can easily be specified by designing y di as a function of
s i . In addition, y di should also depend on the lateral deviations
y k of the immediate neighbors of the ith robot, so that its
nominal lateral set point could be immediately altered if one of
its neighbors, for any reason, comes abnormally close to it and
therefore poses a collision risk. The path following control law
designed in [16] is generalized below to meet such requirements, i.e., to allow the tracking of the reference path C at
some given potentially varying lateral distance y di

(5)

with l ki , a negative scalar specifying the longitudinal settling
time, and e ki = s i - s k - d ki as the longitudinal error of the
ith robot with respect to the kth one. Finally, the set of coeffi32

cients v ki allows us to specify the expected longitudinal
behavior of the formation. These coefficients can be adapted
online to modify this behavior or to reflect the communication availability. They also provide a convenient way to
shorten or enlarge the formation: v ki is then set to zero or to a
nonnull value when the kth robot, leaves or enters the formation, respectively. As a generic control is presented here, the
computation or the online modification of these coefficients
are not further detailed. It can be computed by a dedicated
supervision algorithm, so that the control architecture could
fit specific end-user tasks.
Since the sideslip angles b Rk appear in longitudinal law
(4), variations in grip conditions, misestimations in robot
parameters or biases in measurements are explicitly taken
into account. This allows us to preserve a high level of accuracy in steady-state phases (i.e., slow-varying velocity, curvature, and terrain conditions). Nevertheless, when the reference path or the leader curvature are quickly varying, the
velocity of the ith robot has to quickly increase or decrease
depending on its lateral deviation. However, due to the actuator settling time, the speed modifications computed by longitudinal law (4) are not applied instantaneously, leading to
transient overshoots. The same phenomenon is also encountered in lateral control. A common predictive algorithm,
detailed in the "Predictive Control" section, has been developed to address this problem.

= arctan =tan b iR +
+

where

c (s i) cos c i
li
c
R
1
- c (s i) y i
cos b i

A i cos 3 c i
oG - b Fi ,
(1 - c (s i) y i) 2

(6)

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2014