IEEE Robotics & Automation Magazine - December 2022 - 81

identical shape and mass, which are joined by internal
springs and dampers, allowing bending [Figure 2(c)]. The
first mass element, a, of the first beam element (i = 1) is rigidly
fixed to an angular actuator at a feather's base, while
the second mass element, b, of the last beam unit (i = 10) is
a free end. All flexible elements are connected by rigid rotational
joints.
The influence of the rest of the feather is imposed
on each flexible element at the leading and trailing
rigid joint locations as a combination of the force,
and bending moment, M ,r
F ,r
acting on each joint, as
defined in [Figure 2(c)]. The bending deformation of
the feather is captured by imposing a structural bending
stiffness,
k ,i
and damping, d ,i
a and b, which results in moments, Mki
rotational stiffness, k ,i
was calculated as EI /,lii
between the masses,
and M .di The
where E
is the elastic modulus, I is the area moment of inertia,
and l is length of the section. The elastic modulus was
tuned by comparing the simulation's visual output
against a physical rectangular polypropylene feather in
water. The damping was approximated to be zero for all
simulated joints.
Each of the masses is subject to a set of lumped external
forces. Although, in the simulation, the forces are
solved for each mass, for this derivation, the formulations
will be expressed for the generalized flexible element,
i. For each flexible element, i, the total lumped
external force,
buoyancy force, F ;bi hydrodynamic force, F ;hydi
added mass force, F :ai
F ,exti consists of gravitational force, F ;gi
and
FF FF F .
exthydgb
ii ii i
=+ ++ a
tfi 1
- t
t
f
(1)
Since gravity and buoyancy always oppose each other,
they can be combined as
tional acceleration; and ug
direction of gravity.
The hydrodynamic
force, F ,hyd
is the total
force due to the viscous
interaction between the
fluid and the structure. In
some literature, the hydrodynamic
force is decomposed
into its lift and drag
components, where the
drag is in the direction of
the relative velocity
between the fluid and the
body. In our simulation,
we decompose the hydrodynamic
force into forces
in the normal and longitudinal
directions of each
feather element, Fnorm
and
F ,long where these individual elements can be approximated as
a rectangular prism.
The hydrodynamic opposing forces are approximated
using the following equations [16]:
By providing the robot
with the ability to alter, or
morph, its body, we want
to show how changes in
maneuverability can be
achieved through altering
the body structure.
Vu ,g^h where ft and
w
t w are the density of the polypropylene feather and
water, respectively; Vi
is the volume; g is the gravitais
the unit vector in the
Fhydi
w Simulated Feather Model
ω(t)
l
ω(t)
(a)
trise/2
40
20
-20
-40
0 0.51 1.5
Time (s)
(b)
2 2.5 3
thold
tfall
thold
trise
A
Fri,a
, Mri,a
Fnormi,a
Fgi,a
Flexible Element i
Frx
Mki,a
Mki,b
+ Mdi,a
Flongi,a
Fai,a
+ Fbi,a
Fgi,b
Flexible Element i
: External Forces
: Reaction Forces/Moments
: Internal Forces/Moments
(c)
Figure 2. The (a) simulation of an actuated parameterized feather in Simscape Multibody, (b) plot of the parameterized control
signal, and (c) underlying multibody model of a single feather and its interaction with water, with a close-up free-body diagram of
a flexible element.
DECEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
81
Fai,b
+ Fbi,bz
Fri,b
, Mri,b
+ Mdi,b
Fnormi,b
Flongi,b
T = Fry
Fai
Fgi
+ Fbi
Ten Flexible Elements
Amplitude (°)
IEEE Robotics & Automation Magazine - December 2022

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