IEEE Robotics & Automation Magazine - December 2022 - 82

FC AU u
FC AU u
norm =
long =
ii iw ii
2
1
ii iw i
2
1
long
where the relative velocity, U ,i
Cnormi and Clongi
t
norm
t
2
<
i
,
2
=
,
(2)
is decomposed into the normal
and longitudinal components of the feather's longitudinal
axis; Ai
is the characteristic area of the feather unit; and
are the hydrodynamic coefficients. These
coefficients were found to be affected by the flow incidence
angle. The normal component of velocity was used in this
model instead of the incidence angle because for our flow
regime, the streamlines of flow in the neighborhood of each
feather element are bent such that the position of the separation
point does not change for small angles of relative flow.
The hydrodynamic coefficients for each mass of the flexible
element were obtained from steady-state computational fluid
dynamics simulations carried out using ANSYS software for
Reynolds numbers ranging from 10 to 104, estimated to be
fitting for such biological systems [17].
Finally, the added mass force,
F ,ai must also be accounted
force is the added mass of the movement of fluid around the
accelerated body due to the action of pressure [18] and can be
defined as
FA , where Ai
aw anormiii
= t
is the cross-sectional
area of each element and wt is the water density. The hydrodynamic
added mass force is implemented on the normal
component of the acceleration,
anormi , to the feather's longitudinal
axis. The added mass force in the longitudinal direction
was ignored for this approximation. The handling of the
hydrodynamic forces captures a nonlinear feature of the
fluid-structure interactions as dynamic feedback resulting
from the motion of the individual elements. The actuation at
the base of the feather therefore drives the kinematics and
deformation dynamics.
Simulation Results
Using the hydrodynamic simulation, the design space
spanned by pdes
for in the total force formulation. The hydrodynamic mass tt thold
fall
50
45
40
35
30
25
20
15
80
90 100 110
Length, l (mm)
(a)
120 130 140
1
0.8
0.6
0.4
0.2
was explored. Specifically, we explored
all combinations of geometries (w = 15-50 mm, in 5-mm
increments, and l = 80-140 mm, in 20-mm increments)
and all combinations of the motion parameters
(. ,. ,. ,, and
rise 01 05 051005 1
== = , ., s). The geo -
metries were chosen to match what
could be used on the physical robotic
setup. Each simulation was run for 10
cycles of the periodic motion, and
the average thrust over one period at
the base of the feather was calculated
as
In the remainder of the article, when
referring to the " thrust, " we consider
this to be the average thrust Tr
r = #
TT .
n
tperiod
()
tdt . R T
1
nt
over
a period.
Figure 3 shows the simulation
Top-Six Controllers
Bottom-Six Controllers
Standard Deviation
results of the thrust produced by various
combinations of the parameters.
These results allow us to select
a smaller subspace to explore ex -
perimentally to find the optimum
feather design parameter, represented
as
p .des
*
Although the simulation
R:0.1
F:0.5
H:0
R:0.1
F:0.5
H:0.5
R:0.1
F:1
H:0.5
R:0.1
F:0.5
H:1
R:0.1
F:1
H:1
R:0.1
F:1
H:0
R:0.5
F:0.5
H:0
(b)
Figure 3. (a) A heat map showing the variation of the maximum Tr across all controllers
for every feather geometry combination. A subset of l and w is identified (the green box)
to be further investigated using physical experiments. (b) The simulation results for the
mean and standard deviation of Tr
controllers are identified. R, F, and H correspond to rise ,, and thold
82 * IEEE ROBOTICS & AUTOMATION MAGAZINE * DECEMBER 2022
across all feather geometries. The worst and best
ttfall
.
Experimental Exploration
To validate the simulation, we created
an experimental setup (Figure 4) that
R:0.5
F:0.5
H:1
R:0.5
F:0.5
H:0.5
R:0.5
F:1
H:1
R:0.5
F:1
H:0
R:0.5
F:1
H:0.5
allows general trends to be captured
and the search space to be reduced,
there remains a significant reality
gap such that a small number of
more costly real-world experiments
must be performed. We
explored five feather geometries
([ ,] [, ],[, ],[. ,],
=
40 12040 140 and seven
control motions for the top six
best-performing controllers and
one poorly performing controller.
wl 25 12025 140 32 5130
[, ], and [, ])
Average Thrust
(Arbitrary Units)
Width, w (mm)
IEEE Robotics & Automation Magazine - December 2022

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