IEEE Robotics & Automation Magazine - December 2022 - 57

the external disturbance, the measurement errors, and so
on. Then, we sequentially update the particle importance
weight
w .()
i
k
hhk+1
+1
wp w
w
()
()
i
k+1
i
k ^h k
()i
=+e
=
N 1
-
hhk+1 N 1
i =0
k+1 = /
where iV
of the measurements, and Vstd
pe |V ()
^h
h =
VV ()i
- h
() /
2
2
-
2
std
Vk+1 hk+1
()i
|,
,
()i
wk+1
/ k+1
i =0
()i
is the importance weight
is the factor of the importance
weight.
We would like to mention that, according to the principle of
the electric sense, our proposed electric sense-based scheme can
provide not only the position (x, y) but also the orientation ψ of
the robot. However, specific to the electric sense module, the
robot's position is updated using the electric sense, while the
robot's orientation information is mainly contributed from the
IMU module, where the electric sense just serves as an auxiliary.
The reason is that the IMU sensor is a very cheap and established
way to estimate the orientation. Thus, we focus on estimating the
position of the robot based on the electric sense when designing
the localization algorithm of the electric sense module.
Multi-Information Fusion-Based
Localization Methods
By selectively fusing the previous three perception modules,
we propose three localization methods (Figure 4): 1) electric
sense + IMU + dynamic model, 2) electric sense + IMU, and
3) IMU + dynamic model.
Specifically, the IMU module is chosen as a necessary part
of all three localization methods. Since the emitters are fixed
in the underwater environment, their orientations are known
ahead of time by the robot. Thus, the IMU on the robot can
be calibrated before the localization starts, so that the robot
obtains a more accurate initial orientation. Then, for the
dynamic-model module and the electric sense module, we
choose one or both of them to integrate with the IMU module.
These designs enable us to evaluate the contributing factors
of the electric sense-based localization.
Our Robot and Experimental Platform
To verify and evaluate our proposed hardware solution to the
electric sense and the corresponding localization methods, we
design a robot equipped with an electric receiver for conducting
the experiments. Thus, in this section, we first introduce
the robot's design and its dynamic model and then present the
experimental platform including the physical parameters of
the electric sense hardware.
Design of the Robot
The small underwater robot that we have developed is shown
in Figure 6(a) and (c); its size is about 34 × 27 × 24 cm3
()i
(4)
()i
k
Third, we normalize the importance weights of
all of the particles and calculate the final value of the pose
estimation. That is,
,
(length × width × height). The robot has two propellers and a
caudal fin. Each propeller is steered by a servo motor so that
the direction of propulsion can be flexibly changed. The caudal
fin is designed to increase the steering resistance of the
robot, allowing it to swim straighter, so it is fixed on the robot.
Moreover, an electric receiver consisting of four electrodes is
equipped on the bottom of the robot to realize electric sense.
For our experiments in this article, we only need the robot to
swim in two dimensions because of the limitation of the
experimental pool, so we just change the thrust of the two
propellers but fix their thrust directions.
The Dynamic Model
Based on the design of the robot as well as its previous specific
actuation mode when used in our experiments, we construct
a 2D dynamic model of the robot. Following the notation
developed by Fossen [18], the dynamic model of an underwater
robot is generally described by the equation
Mv Cv vD vv x++ =
o
()
()
,
(5)
where v denotes the vehicle velocity in the body-fixed frame,
which contains the vehicle surge velocity, sway velocity, and
yaw velocity. M is the added mass and inertia matrix.
()
()
Cv R33
#
!
is the matrix of Coriolis and centripetal terms.
constitute the vector of body-frame forces, which
x==>
BF
!
1
1
aa
22
-
H=
fu
fu
11
22
()
()
G,
(6)
where a is the distance between the robot's two propellers,
and ,[ ,]uu 0112
are the normalized control outputs of the
left and right propellers, respectively. f1( · ), f2( · ) are the mapping
relationships between the control input u and the thrust
vector F, and F to x is a linear mapping.
Based on the robot's dynamic model, we use the model
predictive control (MPC) method as the control law for the
free-swimming robot. Specifically, we establish a rolling
optimization model and perform continuous convex optimization
[19]. We would like to emphasize that the control
law will not affect the performance of our electric sense.
The Experimental Platform
As shown in Figure 6(b), we conduct the experiments in a
pool of size 3 × 2 × 0.38 m3 (length × width × height). There
is an overhead camera to capture the position and orientation
of the robot, which is used as the ground truth of our localization
experiments.
For the electric sense-based hardware used in our experiments,
the physical parameters of the emitters and the receiver
are as follows: the emitter's voltage V0 is 5 V, the radius of
the emitter's electrode R is 0.75 cm, the distance between the
emitter's two electrodes de is 10 cm, and the distance between
the receiver's two electrodes dr is 20 cm.
DECEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
57
Dv is the drag matrix. The propulsion forces and moments
x ! R31#
can be written as
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