IEEE Robotics & Automation Magazine - March 2016 - 54

V rf =

(p x (t end) - p x (t 0)) 2 + (p y (t end) - p y (t 0)) 2
, (28)
t end - t 0

respectively. Note that t end and t 0 indicate the beginning and
the end of the time horizon, respectively. The amplitudes of
the achieved forward steady-state velocities for the lateral undulation calculated by (27) and (28) were 0.1275 m/s and
0.1288 m/s for the simulated robot and the real robot, respectively. The error between these velocities was 1.01%, which indicates that there is quite good agreement between the
simulated dynamics of the robot and the real experiments. In
addition, the steady-state velocities for the eel-like motion
were 0.0897 and 0.0894 m/s for the simulated robot and the
real robot, respectively. The error between the velocities for
the case of eel-like motion was 0.33%.
In addition, by keeping the chosen values for the fluid coefficients constant, we obtained comparable results between
the simulation and experimental results both for the lateral
undulation and eel-like motion pattern. In Tables 2 and 3, we
can see the achieved forward velocities both for the simulated and the physical system for different values of the gait parameters. In particular, the first three columns of the tables
present the values of the gait parameters, while the last three
columns present the forward velocity of the simulated robot,
the velocity of the physical robot and the relative error between the forward velocities, respectively. From Tables 2 and
3, we can see that the maximum error between the simulated
and the physical robot was 10.03% and 12.92% for the lateral

undulation and eel-like motion pattern, respectively. These
preliminary results show that the fluid coefficients are quite
sensitive to variations of the gait parameters. However, in
[14], it was also shown that, for the swimming speed, the discrepancies between modeling and reality do not exceed 16%.
Remark 5
Note that another, more accurate method for the fluid coefficients identification should be investigated in the future for
more precise identified values of the drag and added mass coefficients since, as we can see from Tables 2 and 3, the identified fluid coefficients are quite sensitive to variations of the
gait parameters. In this study, preliminary results are obtained
for the model presented in the "Mathematical Model of Underwater Snake Robot" section, which mainly will be used to
investigate the efficacy of the path-following controller presented in the "LOS Path-Following Control" section by comparing the experimental results with the simulated ones. In
the future, the force/torque sensor installed inside the modules of the snake robot may be used to obtain more general
results for the fluid coefficients, avoiding the calculation of
these coefficients by fitting the simulated motion with the
motion of the physical robot.
Experimental Investigation
of LOS Path-Following Controller
In this section, the experimental results will be presented to
investigate the efficacy of the LOS path-following controller

Controller Implementation in External Computer
D
py

Kinematic
Equations

Underwater Snake Robot-Mamba

a, ~, d

LOS

iref
i

Heading
Control

{0 Gait-Pattern z*
Generator

Inner Loop Controller

Camera-Positioning System
z

LabVIEW Code for Position
and Orientation Measurements

z
Underwater
Reflective
Markers Attached
on the Snake

x, y, Yaw
x, y, z
Roll,
Pitch,Yaw
Qualisys

Figure 9. An illustration of the controller structure used in the experiments, with the markers attached to the tail of the robot for
position measurements.

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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march 2016
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2016
