IEEE Robotics & Automation Magazine - September 2019 - 33

rule extracted from the cownose ray, each chordwise cross
section on the pectoral foil employed movement consistent
with a sinusoidal discipline, i.e., flapping frequency and
wavelength. Only the amplitudes changed with different
spanwise positions; therefore, the flapping foil can be
taken as a 1-D flexible foil that may be used to calculate
the influences of the key parameters on the propulsionforce performance.
Considering the value of the Reynolds number of the
bionic fish at this dimension and for these swim conditions,
we omitted the influence of the viscous force in our calculations. The coordinate system illustrated in Figure 3(d) was
employed once again. An instantaneous position of the 1-D
pectoral foil was taken as the analysis target, [indicated by
the yellow line shown in Figure 6(a)]. According to the
ongoing analysis of the cownose ray, all parts of the pectoral foil oscillated with the same sinusoidal discipline, and
the simplified pectoral foil obeyed the same law.
The flapping movements and the induced velocity generated by the pectoral foil can be stated by
h (t) = h 0 cos (~t)

(1)

v (t) = ho (t) = h 0 ~ sin (~t),

(2)

and
where h represents the instantaneous longitudinal motion
position of the pectoral foil, h 0 represents the flapping
amplitude of the pectoral foil, and ~ represents the angular
frequency. Then, as shown in Figure 6(a), the velocity relative to the coming flow can be deducted, which is the vector sum of the movement velocity v and the incoming flow
velocity u:
v i = u + v.

(5)

Then, based on the momentum theorem, the force perpendicular to the infinitesimal element can be calculated as
dF = Tmv in - Tmv on ,
Tt

# tl sin a (v cos a - u sin a)

2

dl.

(7)

In an entire flapping cycle, (7) can be simplified to
Fx = t c sin a (h 0 ~ sin (~t) cos a - u sin a ) 2,

(8)

where a represents the average pitching angle of all the
infinitesimal elements subdivided by the simplified 1-D foil,
and cr represents the average chord length of the pectoral
foil. We can use (8) to discuss conditions that produced the
propulsion forces. According to the movement rules extracted from the cownose ray, whether the pectoral foil flaps
upstroke or downstroke, the value of the average pitching
angle is always between 0 and r/2. This means that the pectoral foil can produce positive propulsion force in both flapping directions.
Some general regularities can be deduced accordingly.
First, both the flapping amplitude and pitching angle have
obvious positive effects on the generation of the propulsion
force. This means that if we apply the larger of these two
parameters, it will produce a higher propulsion force within

Flow-Velocity Analysis
vi
The Schematic of the Simplified
1-D Flapping Foil
During Downstroke Cycle
(a)

(4)

where l represents the length of the infinitesimal element, i is
the included angle between the movement velocity and the
incoming flow, and a is the angle of attack. Therefore, the liquid mass swept by the infinitesimal element is
Tm = tTS = tlTt (v cos a - u sin a) .

Fx =

(3)

The force condition of each infinitesimal element on the
simplified pectoral foil is shown in Figure 6(b). The infinitesimal element is sufficiently small so that it can be treated as a
linear segment. For a short variable time Tt, the infinitesimal
element sweeps out an area of
TS = v i Ttl sin (i - a),

where v in represents the normal component of the inflow
velocity and von represents the normal component of the outflow velocity. The liquid flow cannot cross the infinitesimal
element. After interacting with the element, its velocity vo
translates to the tangential direction parallel to the element.
Therefore, von = 0 in (6). Then, the force perpendicular to
the element generated by the inflow velocity can be obtained.
The propulsion force generated by the entire simplified pectoral foil is

(6)

α

vo

v

θ
u

dFy

dF

B
Force Analysis dFx
A

α

C

d

The Infinitesimal
Element of the 1-D Foil
D

vo

l

vi

(b)
Figure 6. A schematic of the 1-D analysis method of a linearforward swimming cownose ray. (a) A model of the 1-D flapping
foil and (b) a force model of the infinitesimal element; the red
box on the 1-D foil identifies the infinitesimal element.

SEPTEMBER 2019

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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IEEE Robotics & Automation Magazine - September 2019

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