IEEE Robotics & Automation Magazine - March 2016 - 58

5

3

0

-1
-4

-2

0
2
px (m)

4

0

-5

6

py (m)

Reference Path

0

i
iref

i (°)

50
0
- 50

- 100
0

20 40 60 80 100 120
Time (s)
(d)

yt (m/s)

(a)
100

0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

20 40 60 80 100 120
Time (s)
(b)

3
2.5
2
1.5
1
0.5
0
- 0.5
-1

0

20 40 60 80 100 120
Time (s)
(c)

0

20 40 60 80 100 120
Time (s)
(f)

15
10
z0 (°)

1

px (m)

py (m)

2

Path of CM of Robot

5
0
-5

0 20 40 60 80 100 120
Time (s)
(e)

- 10

Figure 15. A straight-line path following with the physical snake initially headed toward the desired path, and with the initial heading
and the distance from the CM being ir (0) = - 88.3c and p y = 1.97 m respectively, 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 .

t= 0s

t = 30 s

t= 0s

t = 60 s

t = 91 s

t = 90 s

*

t = 150 s

Figure 16. The motion of the underwater snake robot during path
following for the experimental results presented in Figure 10. The
yellow line indicates the desired path, i.e., the global x-axis.

Figure 17. The motion of the underwater snake robot during
path following for the experimental results presented in
Figure 12. The yellow line indicates the desired path, i.e., the
global x -axis.

and 12(c), which shows that the cross-track error converged
to and oscillated about zero.

section. The hydrodynamic parameters c t, c n, n n, m 1, m 2 and
m 3 were calculated for the fluid coefficients C f , C d , and
C A as identified in the "Model Identification" section. The
joint PD controller (20) was used for each joint with parameters k p = 20, k d = 5, and the reference angles corresponding to the horizontal joint motion of the robot were
calculated according to (17) with n = 9 by choosing
g (i, n) = 1 and g (i, n) = (n - i) / (n + 1) for lateral undulation and eel-like motion respectively, with the same gait parameters as presented in the "Fluid Parameter Identification"
section. Furthermore, the control gain in (19) is k i = 0.4
and k i = 0.6 for lateral undulation and eel-like motion, respectively, while the guidance law parameter in (18)

Simulation Results
To perform a back-to-back comparison of real experimental
and ideal simulation results, we simulated the model presented in the "Mathematical Model of Underwater Snake Robot"
section with the LOS path-following controller proposed in
the "LOS Path-Following Control" section using similar parameters as in the experiments. In particular, we considered
an underwater snake robot with n = 9 links, each one having length 2l = 0.18 m and mass m = 0.8 kg, i.e., identical
to the physical robot presented in the "Experimental Setup"
58

t = 30 s

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