IEEE Robotics & Automation Magazine - March 2014 - 30
additional measured variables. More precisely, the proposed
model differs from classical kinematic models in that two sideslip angles are considered, namely b F and b R for front and
rear axles, respectively. These added variables are representative of the difference
between the tire orientation and actual tire speed
The mission objectives,
vector direction. Their
estimation is discussed in
with the changes in the
the "Sideslip Angles Estimation" section. Longituformation shape, have
dinal sliding has been
neglected in this article,
been achieved.
because of the low speed
of the robots (around 2
m/s). This modeling approach offers two main advantages: 1)
it allows us to avoid the use of complete dynamical models
(such as those presented in [15]), hardly tractable since they
require the knowledge of numerous parameters and 2) control
design can still be derived by using the approaches proposed
when rolling without sliding assumptions are valid.
Based on these assumptions, two robot models are derived:
1) in a relative frame and 2) in an absolute one. In the "Robot
Control" section, the parameters describing the formation shape
are defined with respect to a common reference path. This path,
made of a sequence of GPS points, can be previously planned or
defined online by a first robot viewed as a leader.
Therefore, it is convenient to express the robot motion
equations with respect to the reference path. The following
notations, also depicted in Figure 2, are then introduced:
● C is the reference path used to specify the desired motion
of the formation.
● O i is the center of the ith mobile robot rear axle.
It is the point to be controlled for each robot.
● s i is the curvilinear coordinate of the closest point from O i
belonging to C. It corresponds to the distance covered
along C by the ith robot.
● c ^ s i h denotes the curvature of path C at s i .
ui denotes the angular deviation of the ith robot with
● i
respect to C.
● y i is the lateral deviation of the ith robot with respect to C.
● d i is the ith robot front wheel steering angle.
● l i is the ith robot wheelbase.
● v i is the ith robot linear velocity at point O i .
F
R
● b i and b i denote the sideslip angles (front and rear) of the
ith robot.
The motion equations for the ith mobile robot can then
be expressed as (see [16] for details)
Z
R
] so i = v i cos (iui + b i )
]]
1 - c (s i) y i
[ yo i = v i sin ^iui + b Ri h
]
F
R
u
]] iuoi = v i e cos (b Ri ) tan (d i + b i ) - tan (b i ) - c (s i) cos i i o .
li
1 - c (s i) y i
\
(1)
30
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IEEE ROBOTICS & AUTOMATION MAGAZINE
*
MARCH 2014
Equation (1) does not exist if 61 - c (s i) y i@ = 0 (i.e., if
point O i is superposed with the instantaneous center of curvature of C). This situation is not encountered in practice
however, since robots are supposed to be properly initialized.
The state vector 6s i, y i, iui@T of this first model is supposed
to be measurable. Nevertheless, for a control law to be
designed from model (1), the sideslip angles b Fi and b Ri have
also to be available. As these variables cannot be easily measured, they have to be estimated by means of an observer. For
reasons detailed below, the observer should preferably be
designed from a robot model expressed in an absolute frame.
The notations listed above are therefore supplemented with
the following ones, also depicted in Figure 2:
v v @ is an absolute reference frame
● 6C, A, B
^
●
a i, b ih are the coordinates in the absolute frame of O i ,
center of the ith mobile robot rear axle
v
● i i denotes the heading of the ith robot with respect to A .
The motion equations for the ith mobile robot expressed
v ,B
v @ are derived from basic geoin the absolute frame 6C, A
metric considerations
Z ao = v cos (i + b R)
i
i
i
] i
] bo = v
R
(
i
+
b
i sin
i
i)
[ i
tan (d i + b Fi ) - tan (b Ri )
]o
] i i = v i cos ^ b Ri h
.
li
\
(2)
Sideslip Angles Estimation
Both extended kinematic models (1) and (2) can be used to
design a sideslip angles observer. In [17], model (1) is considered and the observer is built relying on the duality principle
between observation and control. More precisely, this model
is regarded as a process whose inputs are the sideslip angles
F
R
b i and b i , and a control law is designed for these two variables to impose that the lateral and angular deviations yt i and
t
iui, computed from model (1), converge to the corresponding
measurements yr i and irui . Such a convergence ensures that
model (1) is representative of the actual behavior of the vehicle whatever the grip conditions and sensor biases.
However, this observer presents two limitations. First,
since the robot velocity v i appears as a factor in the three
equations in model (1), the observer is necessarily singular
when v i = 0. As a consequence, from a practical point of
view, it has to be frozen when v i is lower than an arbitrary
threshold. When v i increases and crosses the threshold, the
observer is restarted, but transient inaccuracies in the sideslip
angles estimation are likely to occur and may be detrimental
to guidance accuracy. Next, if the reference trajectory is not
an admissible path for the robots in some places (this may
happen, due to noisy measurements, when C is built online
from the data recorded by the leader robot), then the deviations yr i and irui measured at these places by the ith robot
present abrupt variations. These are erroneously interpreted
as a sudden sliding phenomenon by the observer and an inaccurate sideslip angle estimation, detrimental to the guidance
accuracy, may be returned.
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