IEEE Robotics & Automation Magazine - March 2016 - 49
The structure of the LOS
a, ~, d Inner-Loop Controller
Outer-Loop Controller
path-following controller, as
shown in Figure 3, consists of
two parts, the inner-loop proiref Heading {0 Gait-Pattern {)
u Underwater
D
Joint Control
LOS
Snake-like
portional-derivative (PD)
Control
Generator
Robot
controller that is used to con{
trol the joint angles z and the
i
py
outer-loop controller that is
used for generating the reference joint angles to achieve
Figure 3. The structure of the LOS path-following controller.
the desired sinusoidal gait
pattern and the desired heading irref . The latter controller is composed of three separate
components: 1) the gait-pattern generator, which extracts the
sinusoidal motion pattern to propel the robot forward; 2) the
heading controller, which steers the robot toward and, subse(px, py)
i
quently, along the desired path; and 3) the LOS guidance law
y
iref
(Figure 4), which generates the desired heading angle to
reach and follow the desired path. These three components
of the path-following controller will be presented in the folx
lowing sections.
D
Control Objective
The main control objective is the convergence of the robot to
the desired straight-line path. The forward velocity yrt of the
robot, defined in (4), does not require accurate control but
only yrt > 0 to ensure a nonzero forward velocity for the
robot. Regarding the position of the robot in the two-dimensional (2-D) plane, the desired path is aligned with the global
x-axis for simplicity, and thus the cross-track error along the
desired path coincides with the robot's position over the global y-axis. Note that the controller can easily be generalized to
follow a straight line in any direction by redefining the global
x-axis with a proper rotational transformation. Furthermore,
the heading of the robot (2) corresponds to the angle formed
between the robot's body and the desired straight-line path
(Figure 4). Considering these objectives, the control system
can be formalized as
lim
py = 0
t"3
(14)
lim
ir = 0
t"3
(15)
lim
yrt > 0.
t"3
(16)
Note that, since underwater snake robots have an oscillatory
gait pattern, the control objectives imply that p y and ir
should have steady-state oscillations of about zero.
Remark 2
As we have already mentioned, in this article, forward speed
control has not been considered. However, in [47], based on
extensive simulation results, we showed how it is possible to
achieve a desired forward velocity for underwater snake robots by simply choosing a proper set for the gait parameters
a, ~ and d. In the future, a formal control approach for speed
control should be investigated.
Figure 4. An illustration of the LOS guidance law.
Motion Pattern
Previous studies on swimming snake robots have focused on
two motion patterns: lateral undulation and eel-like motion. In
the present study, the adopted motion pattern is a more general sinusoidal motion pattern, which represents a broader class,
including the aforementioned ones. Lateral undulation [3]
constitutes the fastest and most common type of ground snake
locomotion. It is achieved by means of body waves with a constant amplitude, propagated from head to tail, while the snake
robot is commanded to follow the serpenoid curve [7]. On the
other hand, an eel-like motion can be achieved by propagating
lateral axial undulations with increasing amplitude from head
to tail [29]. To achieve the general sinusoidal motion pattern,
each joint i ! " 1, f, n - 1 , of the underwater snake robot is
commanded to track the reference signal
z i (t)
)
= ag (i, n) sin (~t + (i - 1) d) +
z 0,
(17)
where a and ~ are the maximum amplitude and the frequency, respectively; d determines the phase shift between the
joints; and the function g (i, n) is a scaling function for the
amplitude of joint i [49]. This scaling function allows (17) to
describe a quite general class of sinusoidal functions, including several different snake motion patterns. For instance,
g (i, n) = 1 gives lateral undulation, while g (i, n) = (n i) / (n + 1) gives eel-like motion [2]. Finally, the parameter z 0
is a joint offset coordinate that is shown to affect the direction
of locomotion in the case of land-based snake robots [3] and
fish robots [42] as well. In this article, the joint offset will be
used to control the direction of the locomotion of underwater
snake robots.
march 2016
*
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
49
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