test the prediction using repeatable experiments over a wide range of wing
opening and leg oscillation amplitudes. In addition, we modeled the escape
from the metastable state to self-right as a probabilistic barrier-crossing
transition on an evolving potential energy landscape of the self-deforming
robot/animal, facilitated by kinetic energy. The landscape is the
gravitational potential energy of the robot in its body pitch-roll space.
Because self-righting could in principle occur via both roll and pitch modes,
we analyzed the potential energy barriers on landscape and the kinetic energy
from wing opening (primary propulsion) and leg flailing (secondary
perturbation) along roll and pitch directions. Considering the effects of wing
opening and leg flailing separately gave new insight into the physical
mechanism of self-righting. Finally, we examined whether the observed
stereotypy of the animal’s body motion can be explained by the potential
energy landscape.

We designed and controlled our robotic physical model to achieve similar,
strenuous self-righting behavior as the animal’s, where both wing and leg use
are crucial (see discussion). The robot consisted of a head, two wings, a leg,
and motors to actuate the wings and leg (Figure 2B). To emulate the animal’s
wing opening, both robot wings opened by rolling and pitching about the body
by the same angle (defined as wing opening amplitude, $\theta_{wing}$; Figures
3.2B, 3.3). To simplify leg flailing of the animal, the robot used a pendulum
leg which oscillated in the coronal plane by the same angle to both sides
(defined as leg oscillation amplitude, $\theta_{leg}$; Figures 3.2B, 3.3 ). We
opened and closed the robot’s wings (hereafter referred to as an attempt)
repeatedly while oscillating its legs to generate repeated attempts observed
in the animal. The robot’s leg oscillation was feedforward-controlled,
considering that the animal’s leg flailing motion did not correlate with wing
opening motion (see Materials & Methods for detail). Sufficiently large or
sufficiently asymmetric wing opening alone guarantees self-righting (Li et
al., 2016, 2017)). Here, to study the effect of using both wings and legs
under the most strenuous condition, we chose to open both wings symmetrically
and only used sufficiently small $\theta_{wing}$ with which the robot did not
always self-right with wing opening alone. We emphasize that our goal was not
to simply achieve successful self-righting in a robot.

We chose to focus potential energy landscape modeling on the robotic physical
model because it offers two advantages. First, the animal’s complex 3-D motion
with many degrees of freedom is difficult to quantify. It would take 540 hours
( 12 working weeks) to track our animal dataset ( 5 seconds per trial at 200
frames/s, with 3 markers on the body, each wing, and each leg) to quantify 3-D
motion required for calculating the potential energy landscape. In addition,
wing motion is often impossible to quantify due to occlusion under the body.
By contrast, the robot’s simpler mechanical design, controlled actuation, and
an onboard inertial measurement unit (IMU) sensor allowed easier
reconstruction of its 3-D motion. Second, the animal’s wing opening and leg
flailing are highly variable (Xuan and Li, 2020a) and cannot be controlled.
This results in the potential energy landscape varying substantially from
trial to trial and makes it difficult to evaluate how the system behaved
probabilistically on the landscape. By contrast, the robot’s controlled
variation of wing opening and leg flailing allowed us to do so. Considering
that body rolling is induced by centrifugal force from leg flailing, we
compared the ratio of leg centrifugal force to leg gravitational force between
the animal and robot and verified they are dynamically similar (see Materials
and Methods for detail). In addition, because the animal and robot are
geometrically similar, their potential energy barriers also scale as expected
(Table 2). Thus, the physical principles discovered for the robot are
applicable to the animal.

###  3.5 Methods

### Animal experiments

###  3.6 Animals

We used 30 adult male Blaberus discoidalis cockroaches (Figure 3.2A) (Pinellas
County Reptiles, St Petersburg, FL, USA), as females were often gravid and
under different load-bearing conditions. Prior to experiments, we kept the
animals in individual plastic containers at room temperature (24 $\degree$C)
on a 12h:12h light: dark cycle and provided water and food (rabbit pellets) ad
libitum. Animals weighed 2.6 $\pm$ 0.2 g and measured 5.3 $\pm$ 0.2 cm in
length, 2.3 $\pm$ 0.1 cm in width, and 0.8 $\pm$ 0.1 cm in thickness. All data
are reported as mean $\pm$ s.d. unless otherwise specified.

####  3.6.1 Leg modification

To study the effect of leg flailing, we directly modified both hind legs of
the animal. We attached stainless steel spheres of diameter 0.32 cm and mass
0.14 g (5% of body weight, 180% of leg weight (McMaster-Carr, Elmhurst, IL,
USA) to the tibia-tarsus joint of both hind legs (Figure 3.2A) using
ultraviolet curing glue (BONDIC, Ontario, Canada). We verified that the added
mass increased the average kinetic energy during leg flailing (Figure 3.6, see
section ‘Kinetic energy measurement’).

####  3.6.2 Experiment protocol

We used a flat, wooden surface (60 cm $\times$ 60 cm) covered with cardstock
and walled with transparent acrylic sheets as the righting arena. Four 500 W
work lights (Coleman Cable, Waukegan, IL, USA) illuminated the arena for high-
speed imaging. We maintained the arena at an ambient temperature of $40\pm 2$C
during experiment. We used two synchronized cameras (Fastec IL5, Fastec
Imaging, San Diego, CA, USA) at 200 frames s-1 and 200 $\mu$s shutter time to
record the self-righting maneuver from top (1200 $\times$ 1080 pixels) and
side (1200 $\times$ 400 pixels) views, with a small lens aperture to maximize
the focal depth of field.

For each trial, we first started video recording, held the animal upside-down
by its pronotum, and gently released it from a height of $\approx$ 1 cm above
the center of the righting arena. The small drop was to ensure that the animal
did not begin leg searching, a common strategy used to self-right (Camhi,
1977), before it was released. The animal was given 10 seconds to attempt to
self-right during each trial. After it self-righted or 10 seconds elapsed, the
animal was picked up, and video recording was stopped. After each trial, we
returned the animal to its container and continued testing a different animal.
This way, each animal was allowed to rest for $\approx$ 30 minutes before its
next trial to minimize the effects of fatigue (Camhi, 1977).

We tested 30 animals, each with five trials with its hind legs intact and then
modified, resulting in a total of 300 accepted trials (N = 30 animals, n = 150
trials for each leg treatment). We excluded trials in which the animal
collided with the walls of the righting arena or moved out of both camera
views.

####  3.6.3 Self-righting performance

For each animal trial, we watched the videos to determine whether the animal
self-righted. Because the animal did not always immediately begin to self-
right when placed on the arena (Camhi, 1977; Li et al., 2019), we defined the
beginning of the self-righting attempt as the instant when the animal began
moving its body or appendages to self-right. We defined the animal to have
successfully self-righted if it attained an upright orientation with all six
legs on the ground within 10 s of starting its attempt. We identified the
trials in which animal succeeded in self-righting using the leg-assisted,
winged strategy. For each animal and each leg treatment, we defined and
measured self-righting probability as the number of trials that self-righted
using winged attempts divided by the total number of trials. We counted the
trials that used the legged strategy as failed. We then calculated average
self-righting probability for each leg treatment by averaging across all
animals.

####  3.6.4 Preference of self-righting strategies

We verified that the animal’s preference of winged and legged self-righting
strategies (see Footnote 1) did not change with leg modification. To compare
the animal’s preference of winged and legged self-righting strategies before
and after leg modification, for each trial, we examined the videos to identify
winged and legged self-righting attempts and measured the percentage of time
spent on each strategy. Then, for each leg treatment and each animal, we
averaged it across all the trials from that animal. For each treatment, we
then averaged across each animal to calculate the average percentage of time
spent on each strategy (Figure 3.5).

####  3.6.5 Kinetic energy measurement

To measure the animal’s pitch and roll kinetic energy during self-righting, in
a separate experiment, we used three high speed cameras (Photron FASTCAM Mini
UX-100) to record the animal self-righting at 2000 frames $s^{-1}$ and a
resolution of 1280 $\times$ 1024 pixels, first with its hind legs intact (N =
2 animals, n = 2 trials) and then modified (N = 2 animals, n = 2 trials).

We used DeepLabCut (Mathis et al., 2018) to track the tip and femur-tibia
joint of both hind legs, head anterior tip, abdomen posterior tip, and body
midpoint (Figure 3.6A, B). We then used Direct Linear Transformation software
DLTdv5 (Hedrick, 2008b) to reconstruct 3-D motion of the tracked points and
used a sixth order Butterworth filter with a cut-off frequency of 25 Hz to
filter their 3D positions. To calculate kinetic energy, we approximated the
animal body as an ellipsoid cut into two parts at 38% of total length from the
anterior end, connected by a hinge joint (thorax-abdomen joint, Figure 3.6A).
The smaller part represented the animal’s head and thorax, and the larger part
represented its abdomen. We assumed uniform mass distribution for both parts.
We used the geometric center of the body parts when their fore-aft axes are
aligned to approximate body center of mass . For both hind legs, we
approximated the coxa-femur and tibia-tarsus segment as rigid rods. One end of
the rod representing coxa-femur segment was connected to the body at the
midpoint of thorax-abdomen joint, and the other end connected to the rod
representing tibia-tarsus segment, both via spherical joints (Figure 3.6A, B,
thick black lines connected by blue dots). For modified hind legs, we
approximated the stainless steel spheres at the leg tip as a point mass
attached to the free end of the tibia-tarsus rod (Figure 3.6A, B). We defined
pitch and roll kinetic energy as the sum of kinetic energy from translational
and rotational velocity components from all body parts that contribute to
pitching and rolling motion, respectively. We obtained pitch and roll kinetic
energy by summing contributions from the body ellipsoid parts and the hind leg
segments. For each part, we measured its rotational velocity components about
the animal’s body fore-aft ($X_{body}$) and lateral ($Y_{body}$) principal
axes, and we measured the translational velocity components of its center of
mass along the fore-aft and lateral directions (Figure 3.6B, red vs. blue
arrows). For the sphere attached to modified leg, we measured its
translational velocities. Because vertical translational velocity and yaw
angular velocity did not contribute to motion along the pitch or roll
direction, we did not consider them. For each of the ellipsoid parts and rigid
rods, we calculated its pitch and roll kinetic energy as follows:

$KE_{pitch,j}=\frac{1}{2}I_{yy,j}\omega^{2}_{y,j}+\frac{1}{2}m_{j}v^{2}_{x,j}$
(3.1)
$KE_{roll,j}=\frac{1}{2}I_{xx,j}\omega^{2}_{x,j}+\frac{1}{2}m_{j}v^{2}_{y,j}$
(3.2)

where $I_{xx,j}$ and $I_{yy,j}$ are the moments of inertia the $j^{th}$ object
measured about the animal’s body fore-aft ($X_{body}$) and lateral
($Y_{body}$) principal axes, respectively, $m_{j}$ is the mass of $j^{th}$
object, $\omega_{x,j}$ and $\omega_{y,j}$ are the rotational velocities of the
$j^{th}$ object about body fore-aft and lateral principal axes, and $v_{x,j}$
and $v{y,j}$ are the translational velocity of the center of mass of the
$j^{th}$ object along fore-aft and lateral directions, respectively (Figure
3.6B). For both hind leg segments, we used the mass reported in ((Kram et al.,
1997)) (0.07 g for coxa-femur segments and 0.01 g for tibia-tarsus segment).
To calculate the mass of the two body parts, we assumed body density to be
uniform.

We calculated the pitch and roll kinetic energy of the added spherical mass as
follows:

$KE_{pitch,sphere}=\frac{1}{2}m_{sphere}v^{2}_{x,sphere}$ (3.3)
$KE_{roll,sphere}=\frac{1}{2}m_{sphere}v^{2}_{y,sphere}$ (3.4)

where $m_{sphere}$ is the added spherical mass, and $v_{x}$, sphere and
$v_{y}$, sphere are the translational velocity components of the sphere along
fore-aft and lateral directions, respectively. We considered kinetic energy
from the added spherical mass only for animal with modified legs.

We obtained the pitch and roll kinetic energy of the intact animal from
equations (1) and (2) respectively. For the modified animal, we added
equations (1) and (3) to obtain total pitch kinetic energy and added (2) and
(4) to obtain total roll kinetic energy. For each trial, we first averaged the
measured kinetic energy along pitch and roll directions over the recorded
interval (2.5 s) for each trial. Then for each leg treatment, we further
averaged it across all the trials of that treatment (intact: N = 2 animals, n
= 2 trials; modified: N = 2 animals, n = 2 trials).

####  3.6.6 Relationship between wing opening and leg flailing

We examined whether the animal’s leg flailing during self-righting was more
feedforward-driven or more towards a feedback-controlled reflex coordinated
with wing opening. To do so, we measured the correlation between wing opening
and leg flailing motions as well as their self-correlations (Figure 3.7).
Because wing opening was difficult to measure due to occlusion of wings by the
body during self-righting, we used abdomen tip height as a proxy for wing
opening, considering that abdomen tip height typically increased as wings
opened. For each hind leg, we used its leg tip height as a proxy of the
flailing motion (Figure 3.6). To check whether the height of abdomen tip and
hind leg tips were correlated to each other and to themselves, we measured the
normalized cross-correlations between each pair of these variables and the
normalized autocorrelation of each of them (Figure 3.7). Normalized cross
correlation h between two signals f (t) and g(t) is defined as

$h(t)=\frac{\int_{-\infty}^{\infty}f^{*}(\tau-t)g(\tau)d\tau}{\sqrt{\int_{-\infty}^{\infty}|f(\tau)|^{2}d\tau.\int_{-\infty}^{\infty}|g(\tau)|^{2}d\tau}}$
(3.5)

where $t$ is the time lag between $f(t)$ and $g(t)$ and is a variable, $\tau$
is the variable of integration, and $f*(t)$ is the complex conjugate of
$f(t)$. When $f(t)=g(t)$, $h(t)$ is the normalized autocorrelation.

All normalized cross correlations plots lacked a prominent peak whose value
was close to 1, and all normalized autocorrelations plots had a prominent peak
only at zero lag. This showed that abdomen tip height did not correlate with
itself or with either of the two hind leg tips heights (Figure 3.7A, B, F).
This meant that wing opening and leg flailing motions were not correlated to
each other during self-righting. However, the normalized cross correlation
between both hind legs had recurring oscillations as the lag increased in
magnitude (Figure 3.7). This suggested that leg flailing had some rhythm,
despite a large temporal variation and difference between the two hind legs
((Delcomyn, 1987); (Sherman et al., 1977); (Zill, 1986)).

### Robotic physical modelling

####  3.6.7 Design and actuation

The robot consisted of a head, two wings, a leg, and four motors to actuate
the wings and one to actuate the leg (Table 1, Figures 3.2B, Figure 3.3). The
head and wings were cut from two halves of a thin ellipsoidal shell thermo-
formed (Formech 508FS, Middleton, WI, USA) from 0.16 cm thick polystyrene
plastic sheet (McMaster-Carr, Elmhurst, IL, USA). We connected different parts
using joints 3-D printed using PLA (Ultimaker 2+, Geldermalsen, Netherlands)
(Figure 3.2B). We used DC servo motors (Dynamixel XL-320, ROBOTIS, Lake
Forest, CA, USA) to actuate both the wings and the leg.

To measure the robot’s 3-D orientation (roll, pitch, and yaw angles), we
attached an inertial measurement unit (IMU) (BNO055, Adafruit, New York, NY,
USA) near its center of mass determined from the robot CAD model. We used the
Robot Operating System (Version: melodic) ((Quigley et al., 2009)) to send
actuation signals for the wing and leg motors and record IMU data. To ensure a
constant voltage for repeatable experiments, we used an external 8 V voltage
source (TP3005DM, TEK Power, Montclair, CA, USA) to power the robot. We used
fine flexible wires (30 AWG, 330-DFV, Vishay Sensor, USA) for powering robot
and sending/acquiring signals and ensured that they were loose and did not
interfere with robot motion.

Table 3.1:  Mass distribution of the robot. Component | Mass (g)
---|---
Head | 13.4
Leg rod | 4.3
Leg added mass | 51.5
Leg motor | 28.6
Two wings | 57.4
Two wing pitch motors | 56.0
Two wing roll motors | 48.8
Total | 260.0

####  3.6.8 Similarity to animal

To examine whether the robotic physical model was similar to the animal and
reasonably approximated its self-righting motion, we examined how well they
were geometrically similar and their leg flailing motions were dynamically
similar. To evaluate geometric similarity, we compared their dimensions. For
geometrically similar objects, length $l$ should scale with mass $m$ and
density $\rho$ as $l\propto(m/\rho)^{1/3}$ (Alexander, 2006). Following this,
potential energy should scale as $E\propto m\cdot(m/\rho)^{1/3}\propto
m^{4/3}\rho^{-1/3}$.The robot, which was 90 times as much heavy and 2.3 times
as much dense as the animal with modified legs (Table 2) was expected have
dimensions $(90/2.63)^{1/3}=3.4$ times those of the animal. For the animal,
$m$ includes the added mass from leg modification because we used the same for
calculating the potential energy landscape. Because gravitational potential
energy is proportional to mass and center of mass height, the potential energy
barriers should scale by a factor of $90^{4/3}{\times}2.63^{-1/3}=305$ (Table
2). We found that the robot’s length, thickness, and pitch potential energy
barriers scaled up roughly as expected (Table 2). The larger scaling factor
for robot’s width and roll potential energy barrier is due to the robot being
designed wider to make self-righting via rolling more strenuous.

To evaluate dynamic similarity between the robot and animal, we calculated
Froude Number for their leg flailing. Here, we used the following definition
of Froude number (Biewener, 2003):

$Fr=\frac{\text{Inertial force from leg flailing}}{\text{Gravitational force
of leg}}=\frac{mv^{2}/r}{mg}=\frac{v^{2}}{rg}$ (3.6)

where $m$ is the mass of the animal or robot leg(s), plus the added mass
attached it for the modified animal, $v$ is the leg translational velocity
along the body lateral principal axis, $g$ is gravitational acceleration, and
$r$ is leg length.

Table 3.2:  Comparison between animal and robot.

Parameter | Animal | Robot | Ratio
---|---|---|---
Body length 2a (mm) | 53 | 260 | 4.9
Body width 2b (mm) | 23 | 220 | 9.6
Body thickness 2c (mm) | 8 | 43 | 5.4
Mass attached to leg (g) | 0.14 | 51.5 | 368
Total mass m* (g) | 2.84 | 260 | 90
Density $\rho$ ($\times 10^{-3}g\cdot mm^{-3}$) | 0.88 | 2.05 | 2.3
Expected length scale factor $(m/\rho)^{1/3}$ | 1.47 | 5.06 | 3.4
Expected potential energy scale factor $m^{4/3}/\rho^{1/3}$ | 4.28 | 1306 | 305
Maximum pitch potential energy barrier (mJ) | 0.58 | 282 | 486
Maximum roll potential energy barrier (mJ) | 0.19 | 244 | 1284
Froude number for leg flailing Fr | Intact legs | 0.37 | 0.78 | 2.1
Modified legs | 1.27 | 0.61

* Includes mass attached to the legs.

We found that the Froude numbers for the robot and both the intact and
modified animals were similar (within a factor of two). This dynamic
similarity demonstrated that the robot provided a good physical model for
studying the animal’s self-righting.

####  3.6.9 Experiment protocol

For robot experiments, we used a level, flat, rigid wooden surface (60 cm
$\times$ 60 cm) covered with sandpaper as the righting arena. We used two
synchronized webcams (Logitech C920, Logitech, Newark, CA, USA) to record the
experiment from top and side views at 30 frames s-1 and a resolution of 960
$\times$ 720 pixels. Using the onboard IMU, we recorded the robot body
orientation relative to the lab coordinate system ($X$-$Y$-$Z$ in Figure 3.2B)
at $\approx$ 56 Hz and synchronized them with the motor actuation timings
angles (Figure 3.3, bottom right).

Before each trial, we placed the robot upside-down (Figure 3.9Ai) on the
arena, with its wings closed and leg aligned with the body midline and started
video recording. We then actuated the robot to repeatedly open and close its
wings at 2 Hz and oscillate its legs at 2.5 Hz to self-right. Because the
animal was likely to move its leg before wings at the start of self-righting
(59% of intact leg trials and 81% of modified leg trials), for non-zero robot
leg oscillation amplitudes, the first wing opening was started after
completing one cycle of leg oscillation (0.4 s). If the robot did not self-
right after five wing opening attempts (10 s), we powered down the robot,
stopped video recording, and reset the robot for the next trial. We tested
self-righting performance of the robot by systematically varying leg
oscillation amplitude $\theta_{leg}$ (0$\degree$, 15$\degree$, 30$\degree$,
45$\degree$) and wing opening amplitude $\theta_{wing}$ (60$\degree$, 72,
83$\degree$). We collected five trials for each combination of $\theta_{wing}$
and $\theta_{leg}$. This resulted in a resulted in a total of 60 trials with
134 attempts.

To reconstruct the robot’s 3-D motion, in a separate experiment, we
characterized how the wing and leg actuation angles changed over time during
an attempt (Figure 3.3). We attached BEETag markers (Crall et al., 2015) to
the body frame and to each link actuated by the motors and tracked their
positions using two calibrated high speed cameras (Fastec IL5, Fastec Imaging,
San Diego, CA, USA) at 500 frame $s{-1}$ and a resolution of 1080 $\times$
1080 pixels, as the robot actuated its wings and legs to self-right. We
obtained 3-D kinematics of the markers using the Direct Linear Transformation
method DLTdv5 (Hedrick, 2008b). We then measured the rotation of the link
actuated by each motor about its rotation axis as a function of time during an
attempt. Because the wings were controlled to roll and pitch by the same
angle, we used the average measured wing actuation profile of all the four
motors (two for wing pitching and two for wing rolling). The actual wing
opening and leg oscillation angles were smaller than the commanded (solid blue
and red) due to the inertia of robot body components attached to each motor.

####  3.6.10 Self-righting performance

We defined the beginning of the righting attempt as the instant when the robot
first started opening its wings and measured this instance from the commanded
motor actuation profile (Figure 3.3Bii). We defined the robot to have
successfully self-righted if it attained an upright orientation within 10
seconds (five attempts). We used the IMU to measure the projection of the
gravity acceleration vector "g" onto the body Z-axis "Z" body as a function of
time. This allowed us to determine when the robot became upright. We then
counted the number of successful and failed attempts for each trial. For each
trial, we defined self-righting probability as the ratio of the number of
successful attempts to the total number of attempts of that trial. At each
wing opening and leg oscillation amplitude, we then averaged it across all
trials of that treatment to obtain its average self-righting probability.
Among all the 134 attempts observed across all 60 trials, 44 attempts
succeeded (12, 15, and 17 attempts at $\theta_{wing}$ = 60$\degree$,
72$\degree$, and 83$\degree$, respectively), and 90 attempts failed (46, 27,
and 17 attempts at $\theta_{wing}$ = 60$\degree$, 72$\degree$, and
83$\degree$, respectively).

####  3.6.11 Robot 3-D motion reconstruction

For each robot trial, we measured the robot’s 3-D orientation in the lab frame
using Euler angles (yaw $\alpha$, pitch $\beta$, and roll $\gamma$, Z-Y’-X”
Tait-Bryan convention). We divided each trial temporally into 0.01 s intervals
and used the measured motor actuation angles and body 3-D orientation (Figure
3.3 B, C) at each interval to reconstruct the robot’s body shape and 3-D
orientation, respectively. Because the IMU measured only the 3-D orientation
of the robot, we constrained the robot’s center of mass to translate only
along the vertical direction (Figure 3.2B, Z-axis of lab frame) while
maintaining contact with the ground. We then used the reconstructed 3-D motion
of the robot to obtain the translational and rotational velocity components of
all robot parts.

####  3.6.12 Kinetic energy measurements

For each robot trial, we measured pitch and roll kinetic energy for all
attempts. We defined pitch and roll kinetic energy as the kinetic energy of
the entire robot due to translational and rotational velocities along body
fore-aft and lateral directions, respectively. Because vertical translation
and yawing do not contribute to body pitching or rolling towards self-
righting, we did not consider vertical velocities or rotational velocities
about the vertical axis. Considering that the five motors, leg, and mass added
to the leg could be approximated as regular, symmetric shapes with uniform
mass distribution (motors and leg as solid cuboids and added mass as a solid
sphere), the moment of inertia at the center of mass of each part could be
directly calculated. Then, we calculated the total pitch and roll kinetic
energy of the motors and leg with added mass as:

$KE_{pitch}=\sum_{j=1}^{k}(\frac{1}{2}I_{yy,j}\omega^{2}_{y,j}+\frac{1}{2}m_{j}v^{2}_{x,j})$
(3.7)
$KE_{roll}=\sum_{j=1}^{k}(\frac{1}{2}I_{xx,j}\omega^{2}_{x,j}+\frac{1}{2}m_{j}v^{2}_{y,j})$
(3.8)

where $j$ enumerates the five motors, leg, and mass added to the leg,
$I_{xx,j}$ and $I_{yy,j}$ are the moments of inertia of object $j$ about the
body fore-aft and lateral principal axes (measured at the part’s center of
mass), $m_{j}$ is the mass of object $j$, and $v_{x,j}$ and $v_{y,j}$ are
translational velocities of object j along fore-aft and lateral directions of
robot, and $\omega_{x}$ and $\omega_{y}$ are rotational velocities of object
$j$ about fore-aft and lateral directions of the robot, respectively. For both
the wings and head with complex shapes, we imported their CAD model and
approximated them with uniformly distributed point mass clouds and calculated
the pitch and roll kinetic energy of each part as:

$KE_{pitch,cloud}=\frac{m}{2k}\sum_{j=1}^{k}v^{2}_{x,j}$ (3.9)
$KE_{roll,cloud}=\frac{m}{2k}\sum_{j=1}^{k}v^{2}_{y,j}$ (3.10)

where $m$ is the total mass of the wing or head, $k$ is the number of point
masses in the point cloud, and $v_{x,i}$ and $v_{y,i}$ are the velocity
components of the $i_{th}$ point mass along the body fore-aft and lateral
principal axes.

To obtain total pitch and roll kinetic energy, we summed the pitch and roll
kinetic energy of all the parts. To compare pitch and roll kinetic energy at
each combination of wing opening and leg oscillation amplitudes, we first
averaged the total pitch and roll kinetic energy respectively over the phase
when wings were fully open in the first attempt of each trial to avoid bias
from the large rolling kinetic energy during successful self-righting in later
attempts. We then averaged these temporal averages across the five trials at
each combination of wing opening and leg oscillation amplitudes (Figure 3.8A,
B).

###  3.7 Potential energy landscape modelling

####  3.7.1 Model definition

The gravitational potential energy of the animal or robot is:

$E=mgz_{CoM}$ (3.11)

where $m$ is the total mass of the animal or robot, $g$ is gravitational
acceleration, $z_{CoM}$ is center of mass height from the ground. To determine
the robot’s center of mass, we used a CAD model of the robot (Figures 3.2A,
Figure 3.3) and measured the 3-D positions and orientations of all robot body
parts for a given body orientation and wing opening (see consideration of leg
oscillation below). We approximated the animal body as a rigid ellipsoid, with
the animal’s center of mass at the body geometric center, and its wings as
slices of an ellipsoidal shell. Because the animal or robot did not lift off
during self-righting, in the model we constrained the lowest point of the
animal or robot to be always in contact with the ground. The potential energy
depended on body pitch and roll, wing opening angle, and leg oscillation
angle. Because the effect of leg oscillation was modelled as a part of kinetic
energy, for simplicity, we set the leg to be held fixed in the middle when
calculating the potential energy landscape. We verified that potential energy
landscape did not change considerably (roll barrier changed only up to 13%)
when the leg moved. Because we used Euler angles for 3-D rotations, change in
body yaw did not affect center of mass height. Because the robot’s initial
wing opening was negative (-6$\degree$) due to body weight, in our model
calculations, we varied wing opening angle within the range [-10$\degree$,
90$\degree$] with a 0.5$\degree$ increment. For each wing opening angle, we
then varied both body pitch and roll within the range [-180$\degree$,
180$\degree$] with a 1$\degree$ increment and calculated zCoM to obtain the
system potential energy (Figure 3.3). Because the animal or robot did not
pitch backward significantly, in the figures we do not show landscape for body
pitch < -90$\degree$; the full landscape maybe visualized using data and code
provided (Othayoth and Li, 2021a).

####  3.7.2 System state trajectories on potential energy landscape

To visualize how the robot’s measured system state behaved on the landscape,
we first discretized each righting attempt into time intervals of 0.01 s. For
each interval, we used the measured the wing opening angle (Figure 3.3, dashed
blue curves) to calculate the potential energy landscape. We then projected
the measured body pitch and roll onto the landscape to obtain the system state
trajectory over time. Note that only the end point of the trajectory, which
represented the current state, showed the actual potential energy of the
system at the corresponding wing opening angle. The rest of the visualized
trajectory showed how body pitch and roll evolved but, for visualization
purpose, was simply projected on the landscape surface. The exact system state
trajectories are shown in Figure 3.11.

####  3.7.3 Potential energy barrier measurements

We measured the potential energy barrier that must be overcome to escape from
metastable basin to transition to an upright basin (Figure 3.9C, 3.13). For
each wing opening angle (Figure 3.13B, dashed blue), at each time interval, we
considered imaginary straight paths away from the metastable local minimum
(Figure 3.9B, white dot) in the body pitch-roll space, parameterized by the
polar angle $\psi$ from the positive pitch direction (body pitching up, Figure
3.9Ci). Along each path, we obtained a cross section of the landscape. Then,
we defined and measured the potential energy barrier along this path as the
maximal increase in potential energy in this cross section. Finally, we
plotted the potential energy barrier as a function of $\psi$ (Figure 3.9C). We
defined the roll barrier as the lowest potential energy barrier within $\psi$
= $\pm$ [45$\degree$, 135$\degree$], because both roll upright minima always
lay in this angular range. We defined the pitch barrier as the potential
energy barrier at $\psi$ = 0$\degree$ towards the pitch local minimum.
Finally, we measured both pitch and roll barriers as a function of wing
opening angle (Figure 3.13, Figure 3.14).

####  3.7.4 Comparison of kinetic energy and potential energy barriers

To understand how wing opening and leg oscillation together contribute to the
robot’s self-righting, we compared the measured kinetic energy and potential
energy barriers along both pitch and roll directions throughout each attempt.
For each attempt, we measured kinetic energy minus potential energy barrier
over time along both pitch and roll directions (Figure 3.15, 3.16 A-C). We
then examined whether there was a surplus or deficit of kinetic energy to
overcome the potential energy barrier in both pitch and roll directions,
comparing between successful and failed attempts (Figure Figure 3.15D, 3.16D).
To examine how maximal surplus varied with wing opening and leg oscillation
amplitudes, for each combination of the two, we recorded the maximal surplus
when the wings are held fully open in each attempt and averaged it across all
attempts (Figure 3.15E vs. 3.16E).

####  3.7.5 Data analysis and statistics

We tested whether the animal’s percentage of time spent on winged and legged
self-righting attempts and self-righting probability changed with leg
modification using a mixed-effects ANOVA, with leg treatment as the fixed
factor and individual as a random factor to account for individual
variability. We tested whether the animal’s pitch and roll kinetic energy
depended on leg modification using ANOVA with leg treatment a fixed-factor. We
tested whether the animal’s self-righting probability depended on leg
treatment using a mixed-effect ANOVA with leg treatment as a fixed factor and
individual as a random factor.

We tested whether the robot’s self-righting probability, number of attempts
required to self-right, pitch and roll kinetic energy depended on leg
oscillation amplitude at each wing opening amplitude using a chi-squared test
for probability and an ANOVA for the rest, with wing opening magnitude as a
fixed factor. We tested whether kinetic energy minus potential energy barrier
along the pitch and roll directions depended on leg oscillation amplitude at
each wing opening amplitude, using ANOVAs with leg oscillation amplitude as
the fixed factor. We also tested whether they depended on wing opening
amplitude at each leg oscillation amplitude, using ANOVAs with wing opening
amplitude as the fixed factor. To test whether kinetic energy minus potential
energy barrier differed between successful and failed attempts, we used an
ANOVA with the attempt outcome (success or failure) as the fixed factor. All
statistical tests were performed using JMP Pro 14 (SAS Institute Inc., NC,
USA).

###  3.8 Results

####  3.8.1 Leg flailing facilitates animal winged self-righting

As leg modification increased the animal’s average kinetic energy in both
pitch and roll directions (by 2 and 10 times, respectively; Figures 3.4A, 3.6;
P < 0.05, ANOVA), its probability of self-righting using wings increased
(Figure 3.4B; P < 0.0001, mixed-effect ANOVA). These observations supported
our hypothesis. Leg modification did not change the animal’s relative
preference of using winged and legged self-righting strategies (Figure 3.5).
In addition, wing opening and leg flailing did not show temporal correlation.
Furthermore, the approximate time period of leg flailing (100 ms) was
comparable to combined sensory feedback (6-40 ms (Ritzmann et al., 2012) and
neuromuscular (45 ms (Sponberg and Full, 2008)) delays. These, combined with
the fact that previous studies observed minimal proprioceptive sensory input
from legs during flailing (Camhi, 1977; Delcomyn, 1987; Zill, 1986), indicate
that leg flailing was more feedforward-driven than a feedback-controlled
reflex coordinated with wing opening (Figure 3.6). Moreover, large trial-to-
trial variations in the number of attempts required to self-right showed that
the animal’s self-righting was stochastic (Figure 3.7).

Figure 3.4: Animal’s kinetic energy and self-righting probability. Comparison
of (A) average pitch and roll kinetic energy and (B) self-righting probability
between intact animals and animals with modified hind legs. Error bars in show
$\pm$ s.d. Asterisk indicates a significant difference (P < 0.05) and n.s.
indicates none. Statistical tests: Pitch kinetic energy: P = 0.34, F1, 1 =
1.53, ANOVA. Roll kinetic energy: P = 0.02, F1, 1 = 50.35, ANOVA. Probability:
P < 0.0001, F1, 29 = 93.38, mixed-effect ANOVA. Sample size: (A) N = 2
animals, n = 2 trials. (B) Intact: N = 30 animals, n = 150 trials. Modified: N
= 30 animals, n = 150 trials. Figure 3.5: Comparison of average percentage of
time spent on winged and legged††ADDFOOTNOTE self-righting attempts between
animals with intact and modified legs. Error bars show $\pm$ s.d. n.s.
indicates no significant difference. Winged: P = 0.19, F1,269 =1.71; legged: P
= 0.78, F1,269 = 0.07 mixed-effect ANOVA. Sample Size: N = 30 animals, n = 150
trials for each treatment.  Figure 3.6: Animal kinetic energy calculation. (A)
Representative snapshot of body and appendage with definition of markers
tracked. (B) Multi-body model of animal for calculating pitch and roll kinetic
energy. Red and blue arrows show velocity components that contribute to pitch
and roll kinetic energy, respectively. (C) Pitch and roll kinetic energy as a
function of time for animal with (i) intact and (ii) modified hind legs from a
representative trial. Figure 3.7: Correlation between animal’s body and leg
motion. (A-C) Pair-wise normalized cross correlations between left hind leg
tip height, right hind leg tip height, and abdomen tip height, as a function
of lag between each pair of variables. (D-F) Normalized autocorrelation of
left hind leg tip height, right hind leg tip height, and abdomen tip height as
a function of lag between a variable and itself. N = 1 animal, n = 1 trial.

####  3.8.2 Wing opening and leg flailing together facilitate robot self-
righting

The robot’s self-righting performance increased with both wing opening
amplitude $\theta_{wing}$ and leg oscillation amplitude $\theta_{leg}$ (Figure
3.8). Similar to the animal, the robot’s self-righting was stochastic, with
large trial-to-trial variation in the number of attempts required to self-
right and body pitching and rolling motions (Figures 3.11, 3.12 ). For each
$\theta_{wing}$ tested, as $\theta_{leg}$ increased, average roll kinetic
energy increased (Figure 3.8B; P < 0.0001, ANOVA) and the robot’s self-
righting probability increased (Figure 3.8C; P < 0.0001, Nominal logistic
regression), reaching one at higher $\theta_{leg}$. Meanwhile, the number of
attempts required for self-righting decreased (Figure 3.8D; P < 0.05, ANOVA).
At the maximal $\theta_{leg}$ tested (45$\degree$), the robot always self-
righted (3.8C) and always did so in the first wing opening attempt (Figure
3.8D). Together, these results demonstrated that wing opening and leg flailing
together facilitate the robot’s self-righting performance over the wide range
of parameter space tested.

Figure 3.8: Robot’s kinetic energy and self-righting performance. (A, B)
Average pitch and roll kinetic energy during self-righting as a function of
leg oscillation amplitude $\theta_{leg}$ at different wing opening amplitudes
$\theta_{wing}$. (C, D) Self-righting probability and average number of
attempts required to self-right as a function of $\theta$leg at different
$\theta_{wing}$. Error bars in A, B, and D are $\pm$ s.d., and those in C are
confidence intervals of 95%. Asterisks indicate a significant dependence (P <
0.05) on $\theta_{leg}$ at a given $\theta_{wing}$ and n.s. indicates none.
See Figure 4 source data for detail of statistical tests. Sample size: Kinetic
energy: n = 20 attempts at each wing opening amplitude. Self-righting
probability and number of attempts: n = 58, 42, and 34 attempts at
$\theta_{wing}$ = 60$\degree$, 72$\degree$, and 83$\degree$. For kinetic
energy, only the first attempt from each trial is used to measure the average
to avoid bias from large pitching or rolling motion during subsequent attempts
that self-right. Figure 3.9: Robot’s self-righting motion and potential energy
landscape. (A) Snapshots of reconstructed robot upside down (i), in metastable
state (ii), self-righting by pitch (iii) and roll (iii’) modes, and upright
afterwards (iv, iv’). (B) Snapshots of potential energy landscape at different
wing opening angles corresponding to (A) i, ii, iii. Dashed curves are
boundary of upside-down/metastable basin. Green dots show saddles between
metastable basin and the three upright basins. Gray curves show constant
potential energy contours. Black, dashed blue, and red curves are
representative trajectories of being attracted to and trapped in metastable
basin, self-righting by pitch mode, and self-righting by roll mode,
respectively. i, ii, iii in (A, B) show upside-down (1), metastable (2), and
upright (3iii, iii’) states. (C) Polar plot of potential energy barrier to
escape from upside-down or metastable local minimum along all directions in
pitch-roll space. $\Psi$ is polar angle defining direction of escape in body
pitch-roll space. Green arrow in (i) shows direction of upright minima at
pitch = 180$\degree$ ($\Psi$ = 0$\degree$). Black circle shows scale of energy
barrier (100 mJ). Blue and red arrows in (ii) define pitch and roll potential
energy barriers. Blue and red error bars in (iii) show average maximal pitch
and roll kinetic energy, respectively.

Figure 3.10: Animal’s potential energy landscape. Snapshots of potential
energy landscape at different wing opening angles. Black curve is
representative trajectories of failed attempts and dashed blue and red curves
are for successful attempt by pitch mode and self-righting by roll mode,
respectively. Thin black curves on landscape are constant potential energy
contours. Dashed black curves show boundary of upside-down/metastable basins.
Green dots show saddles between metastable basin and the three upright basins.

####  3.8.3 Robot self-righting resembles animal’s

The robot’s winged self-righting behavior resembled that of the discoid
cockroach in multiple aspects (Figures 3.1, 3.9A, 3.10). First, it often took
the robot multiple attempts (Figure 3.8D) to self-right probabilistically
(Figure 3.8C). In addition, as the wings opened, the robot’s body pitched up
(Figure 3.9Ai), and the head and two opened wings formed a triangular base of
support in which the center of mass projection fell (metastable state, Figure
3.9Aii). In failed attempts, after the wings opened fully, the robot was
unable to escape this metastable state by either pitching over the head or
rolling sideways and fell back to the ground upside-down as the wings closed
(Figure 3.9A). In successful attempts, the robot escaped the metastable state
and always self-righted by rolling to either side (Figure 3.9Aiii’-iv’, red).
Moreover, the robot never lifted off the ground during self-righting. Finally,
the robot’s motion trajectories in the space of body pitch, roll, and center
of mass height were stereotyped for both failed and successful attempts
(Figures 3.11, 3.12), although they are also stochastic with trial-to-trial
variations in body pitch and roll.

####  3.8.4 Robot and animal have similar evolving potential energy landscapes

For both the animal and robot, the potential energy landscape over body pitch-
roll space were similar in shape, and both changed in a similar fashion as the
wings opened (Figures 3.9, 3.10). This is expected because the animal and
robot were geometrically similar (Table 2). When the wings were fully closed,
the potential energy landscape had a local minimum at near zero body pitch and
roll (Figure 3.9Bi). This is because either pitching or rolling of the body
from being upside-down increases center of mass height and thus gravitational
potential energy. Hereafter, we refer to this local minimum basin as the
upside-down basin. The landscape also had three other local minima
corresponding to the body being upright . One local minimum at (body pitch,
roll) = (180$\degree$, 0$\degree$) could be reached from the upside-down basin
by pitching forward (Figure 3.9A, blue dot). Two local minima at (body pitch,
roll) = (0$\degree$, $\pm$180$\degree$) could be reached by rolling left or
right (Figure 3.9Aiii’-iv’, red and blue curves are for roll and pitch modes
respectively). Hereafter, we refer to these basins as pitch and roll upright
basins, respectively . Transition from one basin to another required
overcoming the potential energy barrier separating them (Figure 3.9B, dashed
black curve). As the wings opened, both the robot’s and animal’s potential
energy landscape and its equilibria changed (Figure 3.9B, 3.10). The upside-
down basin evolved into a metastable basin around a local minimum with a
positive pitch and zero roll (Figures 3.9Bii, 3.10Aii, white dot). This local
minimum corresponded to the metastable state with the triangular base of
support (Figures 3.9Aii, 3.1B). The more the wings opened, the higher the
pitch of this local minimum was. To self-right via either the pitch (Figures
3.9Aiii-iv, 3.1Aiii-iv) or roll (Figures 3.9Aiii’-iv’, 3.1Aiii’-iv’) mode, the
system state must escape from the metastable basin to reach either the pitch
or a roll upright basin (e.g., Figure 3.9Biii, blue and red curves).

####  3.8.5 Self-righting transitions are destabilizing,
barrier-crossing transitions on landscape

Reconstruction of the robot’s 3-D motion on the potential energy landscape
revealed that its self-righting transitions are probabilistic barrier-crossing
transitions (Figure 3.11). Except when the robot was upright, upside-down, or
metastable, it was always statically unstable and its system state was
strongly attracted to one of these three local minima basins. At the beginning
of each attempt, the system state was in the upside-down basin. As the wings
opened, it was attracted towards the metastable basin that emerged. In failed
attempts, the system state was trapped in the metastable basin and unable to
escape it (Figure 3.11, black curves). In successful attempts, it crossed a
potential energy barrier (Figure 3.9B, dashed black curve) to escape the
metastable basin and reach a roll upright basin (Figure 3.11, white curves).
These observations are in accord with the animal’s center of mass height
measurements at the beginning, maximal pitch, and end of each attempt from the
previous study (Li et al., 2019) projected onto the animal’s potential energy
landscape (Figure 3.2 C, D).

Figure 3.11: Robot state trajectories on potential energy landscape. (A)
$\theta_{wing}$ = 60$\degree$. (B) $\theta_{wing}$ = 72$\degree$. (C)
$\theta_{wing}$ = 83$\degree$. Columns i and ii show successful (white) and
failed (black) self-righting attempts, respectively. n is the number of
successful/failed attempts at each $\theta_{wing}$. Note that only the end
point of the trajectory, which represented the current state, showed the
actual potential energy of the system at the corresponding wing opening angle.
The rest of the visualized trajectory showed how body pitch and roll evolved
but, for visualization purpose, was simply projected on the landscape surface.
Gray lines show energy contours. Green dots show saddles between metastable
basin and the three upright basins.

Figure 3.12: Robot’s stereotyped body motion during self-righting. State
trajectories in body pitch, body roll, and center of mass height space. (A)
$\theta_{wing}$ = 60$\degree$. (B) $\theta_{wing}$ = 72$\degree$. (C)
$\theta_{wing}$ = 83$\degree$. Columns i and ii show successful and failed
self-righting attempts, respectively. n is the number of successful or failed
attempts at each $\theta_{wing}$.

####  3.8.6 Self-righting via rolling overcomes smaller barrier than via
pitching

For both the animal and robot, the potential energy landscape model allowed us
to quantify the potential energy barrier for self-righting via the pitch and
roll modes. The barrier to escape the metastable state to self-right varied
with the direction along which the system moved in the body pitch-roll space
(Figures 3.9C, 3.13C, 3.14). We defined the pitch and roll barriers as the
minimal barriers to escape from the metastable local minimum towards the pitch
and roll upright basins (Figure 3.9C, blue and red arrow). At all wing opening
angles up to 90$\degree$, the roll barrier was always lower than the pitch
barrier (Figures 3.9C, 3.13C, 3.14C).

####  3.8.7 Barrier reduction by wing opening facilitates self-righting via
rolling

For both the animal and robot, as wing opening angle increased, both the pitch
and roll barrier decreased monotonically (Figure 3.13C). As the wings opened
to the range of $\theta_{wing}$ tested (Figure 3.13C, gray band), the pitch
barrier was still much greater than the average pitch kinetic energy (Figure
3.13C, 3.14C, solid curve vs. dashed blue line). By contrast, the roll barrier
was lowered to a similar level as the average roll kinetic energy (Figure
3.13C, solid curve vs. dashed red line). This explained why the modified
animal, with its higher average kinetic energy, self-righted at a higher
probability than the intact animal (Figure 3.14 solid vs. dashed lines). These
findings demonstrated that, even though wing opening did not generate
sufficient kinetic energy to self-right by pitching (Figure 3.13C), it reduced
the roll barrier so that self-righting became possible using small, perturbing
roll kinetic energy from leg flailing.

To further confirm this, we compared the robot’s kinetic energy with potential
energy barrier along the pitch and roll directions respectively during each
attempt (Figures 3.15, 3.16). The robot’s pitch kinetic energy was
insufficient to overcome even the reduced pitch barrier in both failed and
successful attempts (Figure 3.16). By contrast, as wing opening and leg
flailing amplitudes increased, the robot’s roll kinetic energy more
substantially exceeded the roll barrier during successful attempts (3.15; P <
0.001, nominal logistic regression), and the surplus enabled it to self-right
via rolling.

Figure 3.13: Robot’s potential energy barriers for self-righting via pitch and
roll modes. (A) Potential energy during self-righting via pitch mode as a
function body pitch and wing opening angle. (B) Potential energy during self-
righting via roll mode as a function of body roll and wing opening angle.
Dashed white curves in A and B show energy of metastable state. Dashed black
curve in A and B shows maximal energy when pitching forward or rolling from
metastable state, respectively. Vertical upward arrows define pitch (A) and
roll (B) barriers at a few representative wing opening angles. (C) Pitch
(blue) and roll (red) barrier as a function of wing opening angle. Blue and
red dashed lines show average maximal pitch and roll kinetic energy,
respectively. Gray band shows range of wing opening amplitudes tested. Inset
shows the same data magnified to better show kinetic energy. Figure 3.14:
Animal’s potential energy barriers for self-righting via pitch and roll modes.
(A) Potential energy of self-righting via pitch mode as a function body pitch
and wing opening amplitude. (B) Potential energy of self-righting via roll
mode as a function of body roll and wing opening amplitude. Dashed white
curves in A and B show energy of metastable state. Dashed black curve in A and
B shows maximal energy when pitching forward or rolling from metastable state,
respectively. Vertical arrows define pitch (A) and roll (B) barriers at a few
representative wing opening angles. (C) Pitch (blue) and roll (red) barrier as
a function of wing opening angle. Dashed and solid horizontal lines show the
intact (dashed) and modified (solid) animal’s average pitch kinetic energy
(blue) and average roll kinetic energy (red), respectively. Inset shows the
same data magnified to better show kinetic energy. Figure 3.15: Comparison
between robot’s kinetic energy and potential energy barrier along roll
direction. (A) Roll kinetic energy, (B) roll potential energy barrier, and (C)
roll kinetic energy minus potential energy barrier along roll direction over
time for a representative successful and failed attempt. Between two vertical
dashed lines is when wings are held fully open. (D) Surplus along roll
direction over time for all attempts from all trials. The attempts are grouped
along vertical axis, based on increasing leg oscillation amplitude
$\theta_{leg}$. For each $\theta_{leg}$, the attempts are further grouped
along by different wing opening amplitudes $\theta_{wing}$ (increasing along
upward direction). Columns (i) and (ii) are successful and failed attempts.
Asterisk indicates significant difference in roll kinetic energy minus
potential energy barrier between successful and failed attempts. (E) Average
of maximal roll kinetic energy minus potential energy barrier, as a function
of wing opening amplitude and leg oscillation amplitude. Red and blue show
surplus and deficit of roll kinetic energy minus potential energy barrier,
respectively. n = 134 attempts. See Figure 7—source data 1 for results of
statistical tests.

Figure 3.16: Comparison between robot’s pitch kinetic energy and pitch
potential energy barrier. (A) Kinetic energy, (B) potential energy barrier,
and (C) kinetic energy minus potential energy barrier as a function of time
for a representative successful (i) and failed (ii) attempt. Between two
vertical dashed lines is when wings are held fully open. (D) Pitch kinetic
energy minus potential energy barrier as a function of time of all attempts
from all trials. Along vertical axis, attempts are grouped into increasing leg
oscillation amplitude $\theta_{leg}$. For each $\theta_{leg}$, attempts are
further grouped into increasing wing opening amplitudes $\theta_{wing}$.
Columns (i) and (ii) are successful and failed attempts. Asterisk indicates
significant difference in pitch kinetic energy minus potential energy barrier
between successful and failed attempts. (E) Average of maximal pitch kinetic
energy minus potential energy barrier when wings are fully open as a function
of $\theta_{wing}$ and $\theta_{leg}$. Red and blue show surplus and deficit
of pitch kinetic energy minus potential energy barrier, respectively. n = 134
attempts. See Figure 7—source data 1 for results of statistical tests.

###  3.9 Discussion

We integrated animal experiments, robotic physical modeling, and potential
energy landscape modeling to discover the physical principles of how the
discoid cockroach use propelling and perturbing appendages (wings and legs,
respectively) together to achieve strenuous ground self-righting. Ground self-
righting transitions are stochastic, destabilizing barrier-crossing
transitions on a potential energy landscape. Even though propelling appendages
cannot generate sufficient kinetic energy to cross the high potential energy
barrier of this strenuous locomotor task, they modify the landscape and lower
the barriers in other directions sufficiently so that kinetic energy from
perturbing appendages can help cross them probabilistically to self-right.
Compared to only using propelling or perturbing appendages alone, using them
together makes self-righting more probable and reduces the number of attempts
required, increasing the chance of survival.

Although the intact animal’s average kinetic energy from hind leg flailing was
not sufficient to overcome the potential barrier at the range of wing opening
observed, it still self-righted at a small but finite probability (Figure
3.4B). This was likely because of the additional kinetic energy from flailing
of fore and mid legs, small forces from legs scraping the ground, as well as
abdominal flexion and twisting and passive wing deformation under load ((Li et
al., 2019)), both of which induce lateral asymmetry and tilts the potential
energy landscape towards one side and lowers the roll barrier. This
consideration further demonstrates the usefulness of co-opting a variety of
appendages for propulsion and perturbation simultaneously to achieve strenuous
ground self-righting. Such exaptation ((Gould and Vrba, 1982)) of multiple
types of appendages that evolved primarily for other locomotor functions for
self-righting is likely a general behavioral adaptation and should be adopted
by terrestrial robots.

####  3.9.1 Stereotyped motion emerges from physical interaction constraint

Our landscape modeling demonstrated that the stereotyped body motion during
strenuous leg-assisted, winged self-righting in both the animal and robot is
strongly constrained by physical interaction of the body and appendages with
the environment. The stereotyped repeated body pitching up and down during
failed attempts and rolling during successful attempts directly result from
the strong attraction of the system state to the landscape basins, which
directly arise from physical interaction of body/appendages with the ground.
This finding suggested that potential energy landscape modeling can be used to
understand stereotyped ground self-righting strategies of other species
((Ashe, 1970); (Domokos and Várkonyi, 2008); (Golubović et al., 2013); (Li et
al., 2019); O’Donnel, 2018)and even infer those of extinct species (analogous
to ((Gatesy et al., 2009))). Similarly, it will inform the design and control
of self-righting robots (e.g., Caporale et al. (2020); Kessens et al. (2012))/

Although only demonstrated in a model system, the potential energy landscape
approach can in principle be applied to more complex and different self-
righting behaviors, as well as on ground of different properties ((Sasaki and
Nonaka, 2016)), to understand how propelling and perturbing effects work
together. For example, as the ground becomes more rugged with larger
asperities, the landscape becomes more rugged with more attractive basins
(Figure 3.17). In addition, for leg-assisted, winged self-righting, we can add
degrees of freedom for fore and mid leg flailing, abdomen flexion and
twisting, and even passive wing deformation due to load ((Li et al., 2019)) to
create fine-grained potential energy landscapes to understand how these
motions may emerge from physical interaction constraints. We can also
understand legged self-righting by modeling how the legs and deformable
abdomen ((Li et al., 2019)) affect the potential energy landscape when wings
are not used. This broad applicability will be useful for comparative studies
across species, strategies, and even environments, such as understanding why
some cockroach species’ self-righting is more dynamic than others ((Li et al.,
2019)). However, this approach does not apply to highly dynamic self-righting
strategies, such as those using jumping ((Bolmin et al., 2017); (Kovac et al.,
2008)) where kinetic energy far exceeds the potential energy barrier.

Figure 3.17: Dependence of potential energy landscape on ground geometry. (A)
Grounds of different geometry. (i) Flat ground. (ii, iii) Uneven ground with
small (ii) and large (iii) asperities compared to animal/robot size. (B)
potential energy landscapes for self-righting on corresponding ground. In ii
and iii, landscape is not invariant to robot body translation as in i.
Landscape is show for robot at the geometric center of the terrain with wings
closed. Robot shown for scale.

####  3.9.2 Towards potential energy landscape theory of self-righting
transitions

The potential energy landscape model here does not describe self-righting
dynamics. Recent dynamic modeling using multi-body dynamics simulations ((Xuan
and Li, 2020a)) and dynamical templates ((Xuan and Li, 2020b)) in our lab
revealed that wing-leg coordination affects self-righting by changing the
mechanical energy budget ((Xuan and Li, 2020b)) and that the randomness in the
animal’s motion helps it self-right ((Xuan and Li, 2020a)). However, these
approaches have their limitations: multi-body dynamic simulations are
effectively experiments on a computer; dynamical templates are increasingly
challenging to develop as system degrees of freedom increases. Further
development of a potential energy landscape theory that adds stochastic, non-
conservative forces to predict how the system “diffuses” across landscape
barriers (analogous to (Socci et al., 1996)) may be a relatively simple yet
intuitive way to model probabilistic barrier-crossing dynamics.

##  Chapter 4 Tracking and reconstructing large obstacle interaction of small
animals using a terrain treadmill

††footnotetext: This chapter is a paper by Ratan Othayoth*, Blake Strebel*,
Yuanfeng Han, Evains Francois, and Chen Li (*equal contributions) published in
The Journal of Experimental Biology (2022) (Othayoth et al., 2022)

###  4.1 Summary statement

A device keeps a small animal on top of a sphere while it traverses obstacles,
creating a “terrain treadmill” to study locomotion over a long time and
distance at high-resolution.

###  4.2 Author Contributions

Ratan Othayoth implemented 2-D tracking and 3-D reconstruction, analyzed data,
created visualizations, and wrote the paper; Blake Strebel designed and
constructed the treadmill, implemented the treadmill control system, and wrote
an early draft; Yuanfeng Han designed the treadmill and assisted construction;
Evains Francois collected animal data for testing treadmill performance; Chen
Li. oversaw the study, designed the treadmill, created visualizations, and
wrote the paper.

###  4.3 Introduction

In nature, terrestrial animals often move through spatially complex, three-
dimensional terrain ((Dickinson et al., 2000). Small animals are particularly
challenged to traverse many obstacles comparable to or even larger than
themselves (Kaspari et al., 1999). By contrast, the majority of laboratory
studies of terrestrial locomotion have been performed on flat surfaces
(Alexander and Jayes, 1983; Blickhan and Full, 1993; Cavagna et al., 1976;
Diederich et al., 2002; Ferris et al., 1998; Full and Tu, 1990; Koditschek et
al., 2004; Li et al., 2012; Minetti et al., 2002; Moritz and Farley, 2003;
Spagna et al., 2007; Spence et al., 2010), either rigid or with various
surface properties (friction, slope, solid area fraction, stiffness, damping,
ability to deform and flow, etc.).

Recent laboratory studies have begun to advance our understanding of animal
locomotion in complex terrain with obstacles (Birn-Jeffery and Daley, 2012;
Blaesing, 2004; Collins et al., 2013; Daley and Biewener, 2006; Dürr and
Schilling, 2018; Gart et al., 2018; Gart and Li, 2018; Harley et al., 2009;
Kohlsdorf and Biewener, 2006; Li et al., 2015; Olberding et al., 2012; Parker
and McBrayer, 2016; Sponberg and Full, 2008; Theunissen et al., 2014; Tucker
and Mcbrayer, 2012). Because of typical laboratory space constraints, the
terrain arenas used in these studies are usually no larger than a few dozen
body lengths in each dimension. Thus, they only allow experiments at
relatively small spatiotemporal scales beyond 10 body lengths and 10 movement
cycles. It remains a challenge to study animal locomotion in complex 3-D
terrain with large obstacles at larger spatiotemporal scales.

Experiments at large spatiotemporal scales are usually realized by treadmills
to keep the animal (including humans) stationary relative to the laboratory
(Buchner et al., 1994; Darken et al., 1997; Full, 1987; Herreid and Full,
1984; Jayakumar et al., 2019; Kram et al., 1998; Stolze et al., 1997; Watson
et al., 2002; Weinstein and Full, 1999). However, only small obstacles can be
directly mounted on such treadmills (Voloshina et al., 2013); larger obstacles
have to be dropped onto the treadmill during locomotion (Park et al., 2015;
Snijders et al., 2010). Furthermore, such linear treadmills allow only
untethered movement along one direction. Alternatively, spherical treadmills
use lightweight spheres of low inertia suspended on air bearing (kugels) to
allow small animals to rotate the spheres as they freely change their movement
speed and direction, (Bailey, 2004; Okada and Toh, 2000; Ye et al., 1995).
However, the animal is tethered . However, the animal is tethered, and
obstacles cannot be used.

Figure 4.1: Terrain treadmill. (A) Design of terrain treadmill. Colored
elements show example modular terrain that can be used. (B, C) Terrain
treadmill, with (B) sparsely and (C) densely spaced vertical pillars as
example terrain modules. ArUCo markers attached on the inner sphere are also
shown in (B).

Here, we used a terrain treadmill (Figure 4.1A, B) that enabled large
spatiotemporal scale, high-resolution observations of small animal locomotion
in complex terrain with large obstacles. Our terrain treadmill design was
inspired by a celestial globe model. The terrain treadmill consists of a
transparent, smooth, hollow, outer sphere rigidly attached to a concentric,
solid, inner sphere using a connecting rod (Figure 4.1A). Terrain modules can
be attached to the inner sphere (Figure 4.1) to simulate obstacles that small
animals encounter in natural terrain (Othayoth et al., 2021). The outer sphere
is placed on an actuator system consisting of three actuated omni-directional
wheels (Figure 4.1A). An overhead camera captures videos of the animal moving
on top of the inner sphere, with an ArUCo (Garrido-Jurado et al., 2014) marker
attached on its body. The animal’s position estimated from tracking the marker
is used by a feedback controller to actuate the connected spheres with the
opposite velocity to keep the animal on top (Figure 4.3D, E) as it moves
through the obstacle field (Figures 4.3A-C, 4.4A, B,). Finally, the
reconstructed 3-D motion can be used to estimated different metrics such as
body velocities and antennal planar orientation relative to the body heading
(Figures 4.4, 4.2,).

###  4.4 Methods

####  4.4.1 Experiment and data collection

We used discoid cockroaches (Blaberus discoidalis) to test the treadmill’s
ability to elicit free locomotion and measure animal-terrain interaction over
large spatiotemporal scales. We put the animal inside the outer sphere and
then sealed it. To pick and place the animal onto the inner sphere, we
attached a square magnet (16mm side length, 3.5g) on the animal’s dorsal side,
with an ArUCo marker attached to it for tracking (4.4A, B). We used a larger
magnet to pick up and move the animal to the top of the treadmill and dropped
it onto the inner sphere.

We then started the control program to keep the animal on top. The images
recorded by the camera were then sent to the ROS program, which first saved
each frame in its native format (a bagfile) and then processed the image to
track the marker position. Based on the tracked and then filtered marker
position, which were used to calculate the velocity of the animal through
forward kinematics, motor velocities required to keep the animal centered on
were calculated and commanded to the motors. After each experiment, the
bagfiles were retrieved and processed using custom MATLAB code to extract the
saved images for post processing.

Figure 4.2: Measuring motion of animal exploring sparse pillar fields. (A)
Coordinate frame transformation to measure animal motion relative to sphere.
Solid black arrows are relative 3-D poses ($T_{1}$, $T_{2}$ and $T_{3}$) that
are known or measured directly from acquired images. Dashed arrows are the two
relative 3-D poses ($T_{4}$ and $T_{5}$) that are calculated from measurements
to obtain animal motion relative to the sphere. Yellow squares with red and
green lines show the markers attached to the sphere and their x and y axes,
respectively. Thick green and blue lines show the y and z axes of the frame
attached to the inner sphere. (B) Histogram of left ($\theta_{left}$, blue)
and right ($\theta_{right}$, red) antenna planar orientation relative to body
heading (see schematic on right for definition). (C) Accuracy of antenna-
pillar contact detection outcomes. N = 3 animals, n = 3 trials.

####  4.4.2 Measuring animal movement in obstacle field

To measure the animal’s movement relative to the pillar obstacle field, we
first measured the movement of the pillar obstacle field (i.e., treadmill
rotation) relative to the camera. We attached 31 ArUCo markers to the inner
sphere, with one each at the center of hexagonal and pentagonal regions of the
soccer ball pattern projected on the sphere (Figure 4.2A). We then separately
created a map of all markers attached on the inner sphere (referred to as
marker map) using ArUCo Marker-mapper application. Because each marker and its
four corners were fixed relative to the coordinate frame attached to the inner
sphere (i.e., $T_{3}$ is known, Figure 4.2A), when one of the markers on
sphere is tracked (i.e., $T_{1}$ can be measured, Figure 4.2A), the relative
pose between sphere body frame and the camera (Figure 4.2A, $T_{4}$) can be
computed. When more than one marker on the sphere is detected, relative pose
of sphere and camera can be computed by solving the Perspective-n-points
problem (Lepetit et al., 2009), which estimates camera pose from a known set
of 3D points (marker corners) and the corresponding 2D coordinates in the
image. The ‘solvePnP’ program in in image processing toolboxes in MATLAB or
OpenCV may be used to for this purpose. Because the animal’s movement relative
to the camera (Figure 4.2A, $T_{2}$) is directly available from tracking via
the calibrated camera, the animal’s pose relative to the sphere body frame and
hence relative to the terrain obstacle field can be calculated (Figure 4.2A,
$T_{5}$). Because the ArUCo marker attached to the animal is not necessarily
at its center of mass, a constant position and orientation offset must be
manually determined and added.

####  4.4.3 Unwrapped 2-D trajectory

Considering that the sphere diameter is $\approx 9\times$ that of animal body
length, we approximated the immediate region surrounding the animal to be flat
and estimated the animal’s equivalent 2-D planar trajectory. To obtain the 2-D
trajectory, we integrated the body forward and lateral translational
velocities and body yaw angular velocity (Figure 4.4F-G) over time, with the
initial position at origin and body forward axis along x axis. Because during
portions of a trial the animal body marker was not tracked for a long
duration, we did not consider those video frames. As a result, each trial was
assumed to be composed of multiple segments, and each of their equivalent 2-D
trajectories were assumed to have the same initial conditions as described
above (Figure 4.4E).

###  4.5 Results

####  4.5.1 Free locomotion at large spatiotemporal scales

We tested the terrain treadmill’s performance in eliciting sustained
locomotion of discoid cockroaches (N = 5 animals, n = 12 trials, sparse
obstacles) through both sparse (Figure 4.1B) and cluttered (Figure 4.1C)
pillar obstacles (see ‘Experimental validation using pillar obstacle field’).
Even with cluttered obstacles, where gaps between obstacles were smaller than
animal body width, we were able to elicit continuous trials, in which the
animal moved through pillars for 25 minutes ($\approx$ 2500 stride cycles)
over 67 m ($\approx$ 1500 body lengths) (Video 2). For 83% of the experiment
duration, the terrain treadmill contained the animal within a circle of radius
4 cm (0.9 body length) centered about the image center (Figure 4.2D, E) even
at locomotion speeds of up to 10 body length/s (peak speed of 50 cm/s). We
implemented a Kalman filter (Harvey (1990)) to estimate the position of animal
and reduce the noise and error in marker tracking . The Kalman filter
continued to estimate the animal’s position even when the marker was obscured
from body rolling (Figure 4.3A) or the outer sphere’s seam. In addition, over
the course of 12 trials, the animal freely explored and visited almost the
entire obstacle field (Figure 4.4C, D). Finally, the animal’s motion relative
to the treadmill was used to estimate metrics such as body velocity components
(Figure 4.2F-H), antenna planar orientation relative to the body heading
(Figure 4.2B), and unwrapped 2-D trajectories (Figure 4.4E).

Figure 4.3: Animal behavior and performance of the treadmill. (A-C)
Representative snapshots of behaviors including (A) body rolling, (B) body
pitching and pillar climbing, and (C) antennal sensing observed during free
exploration of terrain. (D) Probability of animal’s detected location in the
image. Red circle of radius 2 animal body lengths is centered at the image
center. (E) Cumulative histogram of animal’s radial position (in body lengths)
from the center of the image. Vertical and horizontal red lines show a radius
of red circle in (A) and the percentage of frames in which animal’s position
was maintained within this circle. N = 5 animals, n = 12 trials.

####  4.5.2 Animal-obstacle interaction

We measured and reconstructed the animal-terrain interaction for 12 trials in
which the animal freely explored the sparse obstacle field (Figure 4.4). The
ArUCo markers attached on the animal and the inner sphere, allowed measuring
and reconstructing animal motion relative to obstacle field (see ‘Measuring
animal movement in obstacle field’ in Methods). Because lighting was not
optimized, the pillar shadow resulted in substantial variation of the
background, and because the left and right antenna are visually similar and
often moved rapidly, automated antenna tracking was accurate in only $\approx$
40% of frames after rejecting inaccurately tracked data (see ‘Automated animal
tracking’ in Methods). However, this can be improved with refinement of our
experimental setup in future (see Discussion).

We then detected which pillar the animal’s antennae contacted (Figure 4.4A, B)
by measuring the minimum distance from each antenna to all nearby pillars. To
determine which pillar the antenna interacted with, we determined whether any
pillars where within 3 cm from both antennae and which among them were closest
to both antennae. We also manually identified the antenna pillar contact,
which served as the ground truth. The antenna-pillar contact detected
automatically was accurate in over 70% of the contact instances (Figure 4.2C).

Figure 4.4: Representative metrics and 3-D reconstruction of animal exploring
sparse pillar obstacle field. (A, B) Representative snapshot and
reconstruction of animal moving through sparse pillar obstacle field.
Transparent green ellipsoid in (A) and brown ellipsoid in (B) show
approximated animal body. Red and blue dots show antenna tips. Yellow dot
shows the tracked point on animal’s head. Dashed cyan circle is the base of
the two pillars with which the animal’s antenna is interacting. (C, D)
Ensemble of trajectories (D) and probability density distribution of animal
center of mass (E) during free exploration of sparse pillar obstacle field, N
= 5 animals, n = 12 trials. (E) Unwrapped 2-D trajectories of animal, obtained
by integrating forward and lateral translational velocities and yaw angular
velocity over duration of trial. (F-H) Histogram of animal’s (F) forward and
(G) lateral translational velocities and (H) yaw angular velocity.

####  4.5.3 Multiple behaviors and behavioral transitions

In addition to walking or running while freely exploring the obstacle field,
the animal displayed other behaviors during interaction with the terrain. For
example, when moving in dense obstacle field, the animal often rolled its body
in to the narrow gap between the pillars (Figure 4.3A) to traverse and
occasionally climbed up the pillars Figure 4.3B). In sparse obstacle field,
the animal often swept its antennae during free exploration (Figure 4.3C). The
animal also transitioned between these behaviors and occasionally stopped
moving.

###  4.6 Discussion

We created a reconfigurable laboratory platform for large spatiotemporal scale
measurements of small animal locomotion through complex terrain with large
obstacles. Compared to existing locomotion arenas, our device increased the
limits of experiment duration by $\sim 100\times$ and traversable distance by
100$\times$. Such large spatiotemporal scales may be useful for studying
spatial navigation and memory (Collett et al., 2013; Varga et al., 2017) in
terrain with large obstacles, and the larger spatial resolution may be useful
for studying interaction of the animal (body, appendages, sensors) with the
terrain in detail (Cowan et al., 2006; Dürr and Schilling, 2018; Okada and
Toh, 2006). There may also be opportunities to advance neuromechanics of large
obstacle traversal by combining the terrain treadmill with miniature wireless
data backpacks (Hammond et al., 2016) for studying muscle activation (Sponberg
and Full, 2008) and neural control (Mongeau et al., 2015; Watson et al.,
2002). The treadmill design may be scaled down or up to suit animals (or
robots) of different sizes. Our treadmill enables large spatiotemporal scale
studies of how locomotor behavior emerges from neuromechanical interaction
with terrain with large obstacles.

Our study is only a first step and the terrain treadmill can use several
improvements in the future to realize its potential. First, we will add more
cameras from different views to minimize occlusions and diffused lighting from
different directions to minimize shadows, as well as increase camera frame
rate to accommodate rapid antenna and body movement, to achieve more reliable
tracking of the animal body and antenna through cluttered obstacles during
which 3-D body rotations are frequent. Second, feedback control of the sphere
can be improved to use not only position but also velocity of the animal to
better maintain it on top. This will be particularly useful if the animal
suddenly accelerates or decelerates when traversing obstacles. Furthermore,
for longer duration experiments, animal could be perturbed when at rest to
elicit movement by automatically moving the treadmill. Finally, we need to
take into account how locomotion on the spherical treadmill may affect the
animal’s sensory cues as compared to moving on stationary ground (Buchner et
al., 1994; Stolze et al., 1997)

##  Chapter 5 Conclusion

###  5.1 General remarks

In this dissertation, we integrated biological, robotic, and physics studies
of beam obstacle traversal and ground self-righting to demonstrate our
hypotheses of physical interaction-mediated locomotor transitions. We made the
following discoveries.

####  5.1.1 Reasoning about terradynamic favorability during physical
interaction

Physical interaction is easier via modes that are terradynamically favorable.
Due to terradynamic favorability, the system state is more likely to be
attracted to a more favorable mode (landscape basin) or transition to one when
it exists. When traversing flimsy beams, it is physically easier to move
forward by pushing down the beams (Figure 2.1). By contrast, stiff beams are
difficult to push against and deflect whereas transitioning rolling the body
into the beam gaps enables easy traversal. While this was not explicitly
demonstrated for ground self-righting, we can infer that both roll and pitch
upright modes are equally favorable as they have the same system potential
energy (Figure 3.9). However, transition to pitch mode does not occur due to
the large pitch barrier to be overcome compared to the pitch kinetic energy
fluctuation (Figure 3.13; see below).

####  5.1.2 Criteria for barrier-crossing locomotor transitions

We demonstrated that locomotor transitions of animals and robots during beam
obstacle traversal and ground self-righting are probabilistic barrier-crossing
transitions on their potential energy landscape. Our observations from robotic
physical modelling showed that transitions mediated by purely mechanical
interactions occur stochastically from one mode (landscape basin) to a more
favorable mode when system kinetic energy fluctuation is comparable to the
barrier to be overcome. In beam traversal, the kinetic energy fluctuation was
generated by oscillatory leg propulsion whereas in self-righting, vigorous leg
flailing generated them.

Alternatively, when kinetic energy fluctuation is not enough to overcome
potential energy barrier along a direction, potential energy landscape maybe
modified by changing relevant parameters of locomotor-terrain system, to
reduce the barrier along other directions and enable smaller kinetic energy
fluctuation to probabilistically induce transitions. For example, in self-
righting, wing opening reduces the barrier for the roll mode and enabled small
kinetic energy fluctuation along roll direction (from leg flailing) to
probabilistically induce body rolling.

In general, transitions in animal beam traversal and self-righting followed
the similar criteria as in passive mechanical robots. However, observations
from the beam traversal studies provided evidence that they can often
transition even when kinetic energy fluctuation is insufficient. This strongly
suggests that transitions maybe facilitated by sensory feedback (see Future
Directions). Because self-righting is an emergency response, it is likely that
the self-righting occurs in a feedforward manner with minimal sensory
feedback.

###  5.2 Stereotypy and Variations

Locomotor modes in both beam traversal (pitch and roll) and self-righting
(metastable and roll) are strongly attracted to energy basins on potential
energy landscapes, leading to stereotypy in locomotor modes, transitions, and
system state trajectories on landscape. Stereotypy emerges directly from
animals’ and robots’ mechanical (and likely neural) systems directly
interacting with the physical environment under constraints. In traversal of
stiff beams, the animal and the robot are constrained to pitch-up into the gap
due to the large restoring forces from beam deflection; because the robot’s
other degrees of freedom are constrained except pitching and rolling,
variations in movement will induce favorable rolling mode with no restoring
forces. Similarly in self-righting, the metastable triangular base of support
that emerges from wing opening constrains the robot from pitching or rolling,
but sufficient kinetic energy can induce body rolling. Variations in movement
that create kinetic energy fluctuation in both model systems lead to
stochastic transitions resulting in beam traversal and self-righting and can
be advantageous—in our case, for traversal and self-righting—when locomotor
behavior is separated into distinct modes.

###  5.3 Implications for animal and robot locomotion

  * •

Methods developed to measure and analyze animal movement across large
spatiotemporal scales in the existing treadmill will help begin to understand
how transitions occur during slow, free exploration where sensory feedback is
dominant (Chapter 4).

  * •

Variations such as kinetic energy fluctuation are usually considered as
adverse to the robot performance. Our discovery of kinetic energy fluctuation-
drive locomotor transitions suggest that embracing the variations from
oscillatory self-propulsion may prove useful (to a certain degree) when moving
in unpredictable natural environments with beam-like obstacles or self-
righting (of course, too large a variation or fluctuation may begin to affect
robot control). This also suggests that robots must destabilize themselves, at
least momentarily, in order to transition to a different mode (Chapters 2 and
3).

  * •

Simultaneous use of multiple appendages can modulate kinetic energy
fluctuation relative to potential energy barriers and facilitate transitions
(Chapter 3).

###  5.4 Future directions

While the investigation in this dissertation have addressed a few of the
knowledge gaps, more questions have been raised than those answered. Here I
briefly mention some of the related directions that I find intriguing and are
possible extensions of the studies in this dissertation.

####  5.4.1 Neuromechanical interaction with during locomotor transitions

The studies in this dissertation (Othayoth and Li (2021); Othayoth et al.
(2020)) focused on locomotor transitions in the feedforward regime of
locomotion (Dickinson et al. (2000); Nishikawa et al. (2007)), by eliciting
transitions during escape responses in which the role of sensory feedback is
diminished due to the inherent neuromechanical delays (Sponberg and Full
(2008)). However, an interesting observation from cockroach beam obstacle
traversal was that the animal can make a barrier-crossing transition even if
kinetic energy was not sufficient to overcome potential energy barriers
(Chapter 2, Figure 2.17) (Othayoth et al. (2020)). Following this, a recent
study led by Yaqing Wang (Wang et al. (2022)) demonstrated that during beam
traversal, cockroaches actively adjust body and appendages, which facilitates
locomotor transitions. An interesting question that is yet to be answered is
how different neural and mechanical sensory streams (Mongeau et al. (2021);
Roth et al. (2016))—for example, antennal sensing (Cowan et al. (2006);
Mongeau et al. (2014); Mongeau et al. (2015)), proprioception from legs
(Agrawal et al. (2020); Tuthill and Azim (2018)) and strain sensors (Tuthill
and Wilson (2016)) distributed across body—are integrated (Dickinson et al.
(2000); Mongeau et al. (2021)) to elicit or avoid locomotor transitions (Xuan
et al., 2021).

####  5.4.2 Decision making in natural environments

In addition to the combined interactions of the animal’s internal states
(Calhoun et al. (2019)) (such as hunger and fear) and external cues perceived
by other modalities (Kennedy et al. (2014)) (such as such as incoming
predators and escaping prey), decision making, and more broadly behavior in
natural environments, emerges also from neuromechanical interaction with
terrain (Dickinson et al. (2000)). The locomotor modes that the animal may
choose and the necessary or desired physical interaction possibly depends on
its higher-level goals as well (Cisek and Kalaska (2010))—for example, during
escape responses, transitioning to modes by that requires overcoming higher
energy barrier maybe terradynamically unfavorable, but beneficial in the sense
that it reduces the risk of predation. By contrast, during free exploration of
environment during foraging, animal may choose to make transitions to access
ecological patches with more resources (Fryxell et al. (2008)). A challenge to
studying decision making during locomotor transitions is that while physical
interaction may be quantified concretely using first principles (e.g.,
biomechanical templates and anchors (Full and Koditschek (1999)), potential
energy landscapes (Othayoth and Li (2021)), modelling the effects of risk,
rewards, and resources perceived by the animal is not yet clear. Recent data-
driven and optimization-based approaches have begun to represent internal
states (Calhoun et al. (2019)) and biomechanical risk (Hackett et al. (2020))
and holds promise towards developing comprehensive models that would inform
how animals make decisions.

####  5.4.3 Detecting and leveraging terrain affordances

For animals moving in complex natural environments, physical interaction with
terrain components is the rule rather than an exception. However, physical
interaction in cluttered, obstacle-laden terrain have been traditionally
deemed as unfavorable or hostile for robots. As a result, obstacle avoidance
(Khatib (1986)) has been the focus of various robot motion planning approaches
(Latombe (2012)). However, recent opinions in robotics (Koditschek (2021);
Roberts et al. (2019); Roberts et al. (2020)) inspired by James Gibson’s
theory of affordances (Gibson (2014)) posit that that obstacles in terrain,
which are otherwise avoided, present an opportunity to apply and direct forces
to elicit motion that is otherwise not possible on flat, featureless
environments. For example, bump- or gap-like terrain elements (Gart and Li
(2018); Gart et al. (2018)) present surfaces against which the animal or robot
can exert forces to climb or push-off(Hunt et al. (2021))—such contact
interactions are often impossible on flat, featureless terrain. In other
words, obstacles afford opportunities in locomotion by providing a physical
element against which effort can be directed to generate desired movement,
with the requirement that the predictive models of interaction physics must be
available to predict the desired forces. Development tractable physics models
(Astley et al. (2020); Holmes et al. (2006); Li et al. (2013)) combined with
intelligent terrain detection and planning algorithms will enable robots to
leverage the affordances provided by their environment to move better.

###  5.5 Closing remarks

Working towards the projects explored in this dissertation have helped changed
my outlook on doing science and engineering, and to a certain extent, life.
These studies gave me an opportunity to acquaint myself with fascinating
topics in fields both related and unrelated to locomotion. More importantly,
it gave me an opportunity to meet and learn from amazing people of diverse
backgrounds. In hindsight, some plans never materialized; some were “not
great, but not terrible”; but I’d like to think that most of them worked out
better than I anticipated. Nevertheless, it has been a reasonably constructive
endeavor for me! I wrap up with a quote from the Twelfth Doctor:

“Things end. That’s all. Everything ends, and it’s always sad. But everything
begins again too, and that’s… always happy. Be happy.”

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