IEEE Robotics & Automation Magazine - March 2016 - 48
The forward velocity of the robot is denoted by yrt ! R and is
defined as the component of the CM velocity along the current heading of the snake, i.e.,
yrt
= po x cos ir + po y sin ir.
(4)
Hydrodynamic Modeling
As has been noted in the biorobotics community, underwater
snake (eel-like) robots bring a promising prospective to improve the efficiency and maneuverability of modern-day underwater vehicles. The dynamic modeling of the contact
forces however, is, quite complicated compared to the modeling of the overall rigid motion. The Navier-Stokes equations
are very difficult to solve and not suited for robotics control
design purposes. The hydrodynamic modeling approach
from [2] that is considered in this article takes into account
both the linear and the nonlinear drag forces (resistive fluid
forces), the added mass effect (reactive fluid forces), the fluid
moments, and current effects.
In [2], it is shown that the fluid forces on all links can be
expressed in vector form as
fx
fA x
f ID x
f IID x
f = ; E = ; E + = I G + = II G .
fy
fA y
f Dy
f Dy
(5)
The vectors fA x and fA y represent the effects from added mass
forces and are expressed as
fA x
n n ^S ih
- n n S i C i Xp
; E = -=
G; E
fA y
- n n S i C i n n ^C ih2 Yp
- n n S i C i - n n ^S ih2 V ax o
G= aG i,
-=
2
n n ^C ih
nn Si Ci V y
2
(6)
where V xa = diag ^Vx, 1, f, Vx, nh ! R n # n, V ay = diag (V y, 1,
f, V y, n) ! R n # n and [Vx, i, V y, i] T is the current velocity expressed in inertial frame coordinates. The drag forces on the
robot are given by
ct Ci
f ID
= I xG = - ;
ct Si
f Dy
II
ct Ci
fD
= IIxG = - ;
ct Si
f Dy
- c n S i V rx
E; E,
c n C i Vry
V rx V r 2
- cn Si
E sgn c; Em= x2G,
cn Ci
V ry V ry
(7)
(8)
where f ID x, f ID y and f IID x, f IID y are the linear and nonlinear
drag forces, respectively, and where the relative link velocities
Vrx and Vry are given by
V rx
C i S i Xo - Vx
E=
; E= ;
G.
(9)
- S i C i Yo - V y
V ry
In addition, the fluid torques on all links are
x
= - K 1 ip - K 2 io - K 3 io | io |,
(10)
where K 1 = m 1 I n, K 2 = m 2 I n and K 3 = m 3 I n . The coefficients c t, c n, m 2, m 3 represent the drag forces parameters due
to the pressure difference between the two sides of the body,
and the parameters n n, m 1 represent the added mass of the
48
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IEEE ROBOTICS & AUTOMATION MAGAZINE
*
march 2016
fluid carried by the moving body. Note that the added mass
parameter in the x-direction is considered equal to zero
^ n t = 0 h because the added mass of a slender body in the
longitudinal direction can be neglected compared to the
body mass [2].
Equations of Motion
This section presents the equations of motion for the underwater snake robot. In [2] and [37], it is shown that the acceleration of the CM may be expressed as
k 11 k 12 lK T (C i io 2 + S i ip )
pp x
E=
= G =-Mp ;
G
k 21 k 22 lK T (S i io 2 - C i ip )
pp y
k 12 - k 11 V ax
e T fDx
E= aG io + M p = T G, (11)
- Mp ;
k 22 - k 21 V y
e fDy
where the detailed derivation of the matrix M p and vectors
k 11, k 12, k 21, and k 22 are given in [2] and [37]. In addition, it
is shown that, under the influence of fluid forces (5) and
torques (10), the complete equations of motion of the underwater snake robot are obtained by (11) and
M i ip + W i io 2 + Vi io + K 3 | io | io
+ K Dx fDx + K Dy fDy = D T u,
(12)
with fDx = f IDx + f IIDx and fDy = f IDy + f IIDy representing the
drag forces in x- and y-directions, and u ! R n - 1 the control
input. For more details and the derivation of the matrices
M i, W i, Vi, K Dx and K Dy, see [37].
T
By introducing the state variable x = 6i T , p TCM, io T , po CMT@
2n + 4
!R
, we can rewrite the model of the robot compactly in
state-space form as
xo = 6io T , po TCM, ip T , pp CMT@T = F (x, u),
(13)
where the elements of F (x, u) are found by solving (11) and
(12) for pp CM and ip , respectively.
Remark 1
It is interesting to note that if, in the dynamic model (11)
and (12), we set the fluid parameters to zero and replace
the drag forces in the x- and y-direction with ground
friction models [3], then the model exactly reduces to the
dynamic model of a ground snake robot described in [3].
The underwater snake robot model is thus an extension of
the land snake robot model and may be used for amphibious snake robots moving both on land and in water.
LOS Path-Following Control
In this section, we present an LOS path-following control
scheme for underwater snake robots moving in a virtual horizontal plane [41], based on the general sinusoidal motion pattern proposed in [49]. In particular, a function to describe a
quite general class of sinusoidal motion patterns suitable for locomotion of underwater snake robots was derived in [49] and is
briefly presented in the "Outer-Loop Controller" section.
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