IEEE Robotics & Automation Magazine - September 2023 - 48

directly relies on the derivative gain). This is
a theoretical settling time imposed by the
control law (1), which is satisfied if the robot's
dynamics allows such a settling time, which
depends on the robot's properties (the actuator
and inertia) and on the grip conditions.
SETTLING TIME FOR THE ROBOT'S
YAW RATE
To check whether the settling time expected
to be imposed by the controller gains is
achievable in practice, let us consider the
dynamic model of the mobile robot with
the linear tyre model described in (2). One
can write
X (, )( ,) ,
o=+
where
R 22
F F
A (, )CC =
FR
T
S
S
S
S
CC === G
o
B(, ), .
FR
R
T
S
S
S
S
LC
vm
Iz
C
2
FF
F
V
X
W
W
W
W
is obtained:
11
X
i
b
2 z
2
2
-- R R
vI
- LC LCFF RR
-
LC LC
vm
- 1
-+
-
Iz
+
2
LC LC
vm
CC
V
FF RR
FR
X
W
W
W
W
(6)
By deriving (5) over time, and by substituting bo with the
one in (2), the following second-order differential equation
over ~i= o
~~ ~b-- =
po
AA AA A12 22
,, ,, ,
12 21
.
(7)
The 5% response time is deduced from this second-order
differential equation (7) on yaw rate .~ Adding = steering
actuator response time
yaw rate of the robot as follows:
T
=+x 22+
~d
z
F F
LC LC
4vI
.
R R
Intuitively, (8) shows that CR and CF are inversely proportional
to the settling time for angular velocity ^hT .~
This
is logical as the worse the grip conditions are, the more
time it takes for a vehicle to establish a constant yaw rate
with respect to the set point imposed by a constant steering
angle
dF .
GAIN ADAPTATION
To prevent oscillations and due to the nature of the controller,
system response time Ty is expected to be larger than settling
time
tion TTy 22 ~
T .~ As such, using Shannon's sampling theory, condihas
to be met. In practice, one can define the
following constraint between the settling time Ty imposed by
control gain Kd and Kd to ensure the convergence of y, and
48 IEEE ROBOTICS & AUTOMATION MAGAZINE SEPTEMBER 2023
SELECTION OF A MACHINE LEARNING STRATEGY
To online optimize the parameter of the path tracking control
law, an approach based on machine learning is offered.
Three classes of techniques can be distinguished [11]: supervised,
unsupervised, and RL-based methods. The supervised
processes require examples of the desired optimal output at
each desired time, which is not known in our case. The
unsupervised method, by definition, is unguided, and as
(8)
x ,d one can derive settling time for the
AB
CCX CC
FR FR Fd
"
TO BE STABLE, THE
SETTLING TIME IMPOSED
BY (4) HAS
TO BE IN LINE WITH
THE ROBOT'S CAPABILITIES
DESCRIBED
IN THE TY = NT~
CONSTRAINT.
„
Using T~
(5)
the settling time T~
to converge as TN ,~Ty =
N ).8=
required for the yaw rate
with the aforementioned
N a natural number to be determined
experimentally (in the experiments, N was
determined as
To be stable, the settling time imposed by
(4) has to be in line with the robot's capabilities
described in the TNTy =
~ constraint. Therefore,
a condition for stability can be defined by
introducing (4) to obtain tuning gain Kd, satisfying
the robot's constraints:
Kd =
vNT
8
.
~
and Kp described in (8), a deterministic
means of adapting gains Kp and Kd can be achieved.
The prediction horizon of the controller is defined as
T ;~
however, due to discretization of the control loop at 10 Hz,
the prediction horizon fluctuates very little using this method,
implying an almost constant prediction horizon.
The proposed expression (9) relies mainly on the velocity,
grip conditions, and robot properties (inertia and actuator
delays), and does not account for perception and observer
uncertainty, which play roles in the robot's stability. Moreover,
immediate errors are not accounted for, and it could
be interesting to adapt reactivity of the robot (defined by the
desired settling distance imposed by Kp and Kd) with respect
to the robot's position to the reference path. Thus, a second
approach of gain adaptation is recommended, based on neural
networks.
NEURAL GAIN
A set of " optimal " controller parameters depends on the
objective as well as many factors in the model. Such dependencies
are difficult to determine in the framework of offroad
mobile robot control as it requires knowledge of many
parameters. As an example, deterministic gain tuning
(defined in the " Deterministic Gain " section) depends on the
speed and cornering stiffness of the system. Whereas tracking
(i.e., lateral error) and measurement accuracy (i.e., measurement
covariance) are important parameters for
determining the ideal reactivity of the controller but are not
taken into account by the method. As such, to improve the
adaptation of controller gains, a data-driven technique is proposed
to take into account more properties than the tactic
suggested in the previous section.
(9)
IEEE Robotics & Automation Magazine - September 2023

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