IEEE Robotics & Automation Magazine - September 2023 - 47

These lateral forces can be determined using
tyre models such as the Pacejka formula [14] or
the TMEASY model. However, these models
have many parameters that need to be identified,
and as such, they create models that can
be very biased with respect to reality. This, in
turn, increases the reality gap for data-driven
methods. To minimize the number of parameters
that need to be identified, a linear tyre
model is used where the sliding dynamics are
modeled using a linearization of the forces
with respect to tyre slip angle
and FC () ,RR Rb
= $
where C ()F $ and C ()R $
C ()F $
and C ()R $
FC ()FF Fb
are
= $
the front and rear cornering stiffnesses, respectively.
A higher
stronger force while cornering, which implies
better grip conditions.
Because the robot moves off-road, these
two cornering stiffnesses are considered variable,
and their adaptations are ensured thanks to a backstepping
observer. The first step is a tyre slip angle observer based
on the kinematic model, and the second step consists of adapting
cornering stiffnesses using the aforementioned dynamic
model. Details of the backstepping approach are given in [8]
and [13]. In the following section, both sideslip angles Fb and
bR , as well as the cornering stiffnesses CF and CR, are then
assumed to be known.
CONTROL PARAMETERS' ADAPTATION APPROACHES
The proposed method for online gain tuning, is illustrated in
Figure 2. It consists of using current outputs from observers
and the state estimator to return control parameters to the
steering controller before it calculates the steering angle to be
sent to the robotic platform.
The control parameters adaptation strategies (denoted by
the gain estimator in Figure 2) will be one of the following:
■ Constant gain method: This technique returns the same
value regardless of the input (typically .,K 0 0225
KH .).
p =
d== This setup is chosen experimentally
(i.e. it decreases a high gain until the oscillation stops)
in the worse conditions for path tracking (achieved using a
harsh curve on low-grip terrain at the maximal velocity of
03 and
.,
04
Ty =
implies a
"
AS THE OBJECTIVE
OF THIS ARTICLE IS
TO ACHIEVE ACCURATE
PATH TRACKING,
THE MOTION
OF THE ROBOT IS
DESCRIBED WITH
RESPECT TO THE
FOLLOWED TRAJECTORY
(D).
„
4m ) . This then leads to stable behavior,
.s
-1
regardless of the velocity and grip conditions,
but is not optimal when moving at low speeds
or on terrain with good grip conditions.
■ Deterministic gain method: This approach
is discussed in the " Deterministic Gain "
section for online gain adaptation and is
based on the robotic model and controller
described in the " Mobile Robot Path
Tracking " section.
■ Neural gain method: This technique is
explained in the " Neural Gain " section for
online gain adaptation based on data-driven
training.
DETERMINISTIC GAIN
The following section describes a gain adaptation
process that uses Kp and Kd to set up the
theoretical convergence distance of the robot
with the controller described in (1), and adapt it with respect to
the robot's capabilities derived from the dynamic model.
SYSTEM RESPONSE TIME
The controller described in (1) imposes the following dynamics
of the lateral error with respect to curvature (described in [7]):
2
2
is p dp2
2
y
s
2 dp .
++ =Ky 0
K
KK .
2
2
s
y
(3)
The relationship between gains and the damping factor p
=`j Then, using a damping factor of p 1 ,=
a quadratic relationship, (),KK 4
p =
d
2
between two gains, Kp
and Kd, is derived. From those considerations, one can determine
that the settling distance for a 5% tracking error can be
approximated by ().DK8yd=
As a result, one can derive a
settling time for the convergence of y according to the speed
of the robot as follows:
vK
8
,
d
which describes the system's response time in a closed loop
with the steering controller, depending on gain Kd (as Kp
(4)
1∗
Tracking
Set Points
Gain Estimator
2∗
Errors
IEEE Robotics & Automation Magazine - September 2023

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