IEEE Robotics & Automation Magazine - December 2022 - 85

where I represents the inertia of the body and ,and
mmw are
the mass of the robot and the displaced water respectively.
Each feather is actuated following a periodic oscillatory
trajectory, producing an average thrust Tr directed upward in
the plane of motion, as described in the " Modeling and Optimization
of Bioinspired Feathers " section. Since all feathers
on the same layer are moved in synchrony, the thrust produced
by a generic feather on the jth layer can be described
as
T .j
r
on the environment can be written as
,
n
i
n
n
xpitchj jj jjj j
j
Tc T
dT cc cccc
dT cccc
toti,j rj
=
xroll
=- +- -+ +
=- ++ -
//
/
2 / r 1346
3
j
j
where d represents the distance between the center of mass
and the location of the resulting force for each feather. In
the following, the vector of inputs [, ,]uT
1tot pitchrollxx=
rr
uT T21 ns
= [,..., ]
<
will be used to simplify the physical intuition. However,
the vector u1 can be mapped back to the motor activation
input
Mu / u12
= 22
< through the mapping
=
^h such that it holds that uM .u12 Thanks
to this mapping, we can write the resulting forces on the
robot, given the configuration C and the activation state of
the motors.
We can then rewrite the system in state-space form. The
state of the system is given by the position and velocity of the
center of mass and the orientation and the angular velocity,
[, ,, ,, ,, ,, ,, ][ ,,,,,,,,
==<
x , ,,] ,xxx
9101112
xo =
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
L
x
x
x
7
8
9
xxyz zi } uvwpsv xxxxxxxx
< while the control input is [,...,].uT T1ns
12345678
= rr
Therefore, the complete nonlinear system becomes
J
x
x
x
10
11
12
xx xx gx x -
II
11 7108 45
12 11
coscos
-
yz
Ix
xx +
10 12
Ix
II xx +
-
zx
Iy
II
-
xy
Iz
xx
10 11
xpitch
Iy
xx xx gx x
-+ () ()
xroll
10 9127 45
xx xx gx-- sin ()
-+ sincos() ()
12 811105
T
tot
m
N
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
P
Using the nonlinear state-space representation, we define a
linearization around the hovering configuration, and we
study the controllability of the structure as a function of C. In
the linearized state around the hovering condition, θ = 0,
r
n
(( ),
(),
25 1346
2
1
jj jjj
(5)
z== -
0a Tm mg
, () holds. Therefore, the equand
/
w
tions of motion can be written as
o =+ =+
xAxBuAxBMu .
12
(7)
Given A and B, we can study the degrees of controllability of
the configuration by evaluating the rank of the controllability
Gramian
rank(gram (, ))ABM of the system [19]. A robot
Consequently, the forces produced by the whole robot
with a degree of controllability of n is able to exert forces
along n independent directions. This metric is then ultimately
linked to the robot's behavior, as it informs us of how
many independent directions the robot can accelerate or
react to disturbances in. Maximizing such a metric leads to a
robot that can react to any disturbance and that is fully controllable
[20].
Optimal Detachment
In this section, we present how the detachment of feathers
can be optimized to increase the performance of the
robot in one specific task, such as escaping a predator. In
this scenario, we want our robot to move as quickly as
possible in one direction. Considering that, the robot
configuration can be optimized to produce the maximum
directional thrust by solving the discrete optimization
problem
argmax kT kk
s.t. TT T0
c ! 01
i,j
{, }
jj j
-
where [, ,]kk k
totrollpitch
describe the desired behavior and T ,, and
T xx ! are weight parameters that
totrollpitchxx are
!{, ,},
R
3
defined as in (5). Note that the optimization is solved by evaluating
a combinatorial problem on the subset of binary variables
ci,j
that have value one in the current configuration.
On the other hand, a feather can be detached to restore a
higher degree of controllability. In this case, the optimal feather
to detach is found by solving the optimization problem
argmax
{, }
c ! 01
i,j
rank gram
s.t
.
(6)
(( ,)),
.
AB
TT
j = j
(9)
Through the evaluation of the controllability Gramian matrix,
we can distinguish among designs with the same (),Mrank C
where MC
represents the controllability matrix. In Figure
6(b) and (c), we show how, through the mathematical
model of the design, we can optimize the configuration of the
robot for a desired behavior. The results from the presented
algorithm are used as a framework for the experiments presented
in " Experimental Results " section.
Robotic Hardware Design
The robot is a multilayered system. Each layer has six feathers
arranged regularly in a hexagonal pattern [shown in
Figure 7(a)]. A servo motor is located in the center of the
layer and actuates two feather holders on opposite sides
through a linkage mechanism. On either side of the holder
DECEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
85
(8)
T ++totrollpitch
xx
|| || ||,
totrollpitch
xx
IEEE Robotics & Automation Magazine - December 2022

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