IEEE Robotics & Automation Magazine - March 2016 - 50

Outer-Loop Controller
As noted previously, the outer-loop controller generates the
reference joint angles to achieve the desired sinusoidal gait
pattern and the desired orientation for the robot. Regarding
the sinusoidal gait pattern, previous approaches keep the parameters a and d fixed, while ~ and z 0 are used to control
the speed and the direction of the snake robot, respectively,
[3], [42], [50]. In this article, the same approach will be adopted. The orientation ir of the robot is given by (2). Moreover,
for the desired orientation, motivated by [51] and [1], we propose to define the reference orientation using the following
LOS guidance law

Force/Torque
Sensor
Water
Microcontroller Card
Sensor

Power-Supply Card

Servo Motor

irref

= - arctan c

Remark 3
The look-ahead distance T is a fundamental parameter for
the LOS guidance law since this parameter directly affects the
transient motion of the underwater snake robot. A large value
of T results in a well-damped transient motion, but the convergence to the desired path becomes slow (Figure 4). In contrast, a too small value of T forces the system to have a poor
or unstable performance. A rule of thumb is to define a value
for T larger than twice the length of the robot (see [1]).
Motivated by the effective application of LOS guidance
laws for path-following control of marine surface vessels [1],
[52] and especially in the corresponding case of ground snake
robots [3], we choose the joint angle offset z 0 as
(19)

Inner-Loop Controller
To make the joint angle z i follow its reference signal z )i , a PD
controller is used:
ui =

zp i

)

+ k d (zo )i -

zo i) +

k p (z )i -

z i),

i = 1, f, n - 1,
(20)

where k p > 0 and k d > 0 are the gains of the controller.
Note that, for the experimental and the simulation results
presented in the following sections, the values of the gait parameters a , ~ , and d in (17) and the controller gains, k p and
k d in (20) are chosen arbitrarily based on our experience on
undulatory motion of underwater snake robots. In the future,
optimization techniques may be used for choosing the optimal gait parameters, and, preferably, the controller gains
should be based on the model-based analysis.

(b)

Experimental Setup
This section describes the experimental setup employed for the
fluid parameter identification and the investigation of the performance of the LOS path-following controller proposed in [41].

(c)
Figure 5. The underwater snake robot Mamba implemented the
at Norwegian University of Science and Technology to support
our group's research activity about both ground and underwater
snake robot locomotion: (a) the internal components of the joint
module, (b) two modules connected with orthogonal joint axes,
and (c) motion inside a dummy underwater structure.

IEEE ROBOTICS & AUTOMATION MAGAZINE

= k i ^ir - irref h,

where k i > 0 is a control gain [37].

Joint ID
Temperature
Switch
Sensor
Accelerometer
(a)

*

(18)

where p y is the cross-track error (i.e., the position of the underwater snake robot along the global y-axis), while T is a
constant design parameter. In particular, T denotes the lookahead distance that influences the rate of convergence to the
desired path [1]. Note that LOS guidance laws are much used
in practice for path-following control of marine surface vessels [1], [52] and have been used for path-following control of
ground snake robots [3].

z0

50

py
m, D > 0,
D

*

march 2016

Underwater Snake Robot-Mamba
In this section, the underwater snake robot that was used in
our experiments is presented. A more detailed description of
the robot can be found in [17].
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2016
