IEEE Robotics & Automation Magazine - March 2016 - 56

py (m)

2

px (m)

Path of CM of Robot
Reference Path

1
0
-1

-4 −2

0 2 4
px (m)

6

6
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py (m)

3

50
100
Time (s)

-50
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150

0.15
0.1
0.05
0

0

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(c)

z0 (°)

yt (m/s)

i (°)

0

0

Time (s)

0.2

i
iref

50

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(b)

(a)
100

3
2.5
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Time (s)

Time (s)

Time (s)

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Figure 12. A straight-line path following with the physical snake initially headed along the desired path, and with the initial distance
from the CM being p y = 2.75 m for eel-like motion with gait parameters a = 40c, ~ = 120c/s, and d = 40c: (a) the path of the CM,
(b) the position along the path p x, (c) the cross-track error p y, (d) the heading angle i, (e) the forward Velocity yrt , and (f) the joint
angle offset z 0 .

absolute link angles, i, of the underwater snake robot. The
LOS path-following controller of the underwater snake robot
was implemented on an external computer according to
(17)-(19), i.e., for the lateral undulation and eel-like motion
gait patterns. Specifically, the reference joint angles, computed by (17), were sent to each joint module of the robot via
the CAN bus running through the robot. A proportional
controller implemented in the microcontroller of each joint
module controlled the joint angle according to the received
reference angle. The joint torque controller given by (20) was
not implemented, since the servo motors installed in the
snake robot do not require torque control input as the servos
have built in angle regulation. The robot's orientation was
calculated according to (2), i.e., as the average of the individual link angles.
The LOS guidance law angle given by (18) was calculated
with a look-ahead distance equal to half the length of the
robot, i.e., D = 0.9 m [1] for fast convergence, due to the limited working area covered by the camera system. Furthermore, the control gain in (19) was k i = 0.4 and k i = 0.6 for
lateral undulation and eel-like motion, respectively. The joint
angle offset was saturated according to z 0 = [- 20c, 20c] to
keep the joint reference angles within reasonable bounds and
taking into account the physical robot's joint angle constraints. Moreover, the reference angles were calculated by
(17) for n = 9 choosing g (i, n) = 1 and g (i, n) = (n - i) /
(n + 1) in the case of lateral undulation or eel-like motion, respectively, while the rest of the gait parameters were a = 35c
for lateral undulation and a = 40c for eel-like motion,
a = 40c and ~ = 120c/s. The initial joint angles were zero in
all the trials, while the initial heading and position of the
robot will be specified for each trial.
56

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

march 2016

Experimental Results
The straight-line-path-following controller was experimentally investigated for both lateral undulation and eel-like
motion patterns. In particular, experimental results for two
different sets of initial conditions are presented here, both
for lateral undulation and eel-like motion patterns. In the
first four trials of the experiments, the robot was initially
headed along the desired path (the x-axis), and the initial
distance from the CM to the desired path was 1.89 m and
2.81 m for lateral undulation (Figures 10-11) and 2.75 m
and 2.98 m for eel-like motion (Figures 12-13). In the last
two trials, the robot was initially headed toward the desired
path (the x-axis) with initial heading ir (0) = - 91.3c and
ir (0) = - 88.3c for lateral undulation (Figure 14) and eellike motion (Figure 15), and the initial distance from the
CM to the desired path was 1.59 m and 1.97 m, respectively. The xy-plots of the experimental results for the different
trials are presented in Figures 10(a), 11(a), and 14(a) for lateral undulation and Figures 12(a), 13(a), and 15(a) for eellike motion pattern, where it is easily seen that the robot
converged nicely toward and moved along the desired path
during all trials both for lateral undulation and eel-like motion patterns. In particular, we can see that the CM of the
underwater snake robot converged to the desired path for
all the trials.
In Figures 10(d)-15(d), we can see that (19) made the
heading angle converge to and oscillate around zero for both
lateral undulation and eel-like motion patterns. Moreover,
Figures 10(c)-15(c) show that the cross-track error converged to and oscillated around zero. Furthermore, the
forward velocity of the robot is shown in Figures 10(e)-15(e)
and the joint angle offset is shown in Figures 10(f)-15(f).



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

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