# Visually Guided Swarm Motion Coordination via Insect-inspired Small Target
Motion Reactions

Md Arif Billah  and Imraan A. Faruque M.A. Billah is with Oklahoma State
University, Stillwater, OK 74078 USA (e-mail: arif.billah@okstate.edu). I. A.
Faruque is with Oklahoma State University, Stillwater, OK 74078 USA (e-mail:
i.faruque@okstate.edu).This work was supported in part by ONR Young
Investigator Award N00014-19-1-2216. Copyright by the authors June 2022; this
work may be under consideration for publication and copyright may be
transferred without notice, after which this version may no longer be
accessible.

###### Abstract

Despite progress developing experimentally-consistent models of insect in-
flight sensing and feedback for individual agents, a lack of systematic
understanding of the multi-agent and group performance of the resulting bio-
inspired sensing and feedback approaches remains a barrier to robotic swarm
implementations. This study introduces the small-target motion reactive (STMR)
swarming approach by designing a concise engineering model of the small target
motion detector (STMD) neurons found in insect lobula complexes. The STMD
neuron model identifies the bearing angle at which peak optic flow magnitude
occurs, and this angle is used to design an output feedback switched control
system. A theoretical stability analysis provides bi-agent stability and state
boundedness in group contexts. The approach is simulated and implemented on
ground vehicles for validation and behavioral studies. The results indicate
despite having the lowest connectivity of contemporary approaches (each agent
instantaneously regards only a single neighbor), collective group motion can
be achieved. STMR group level metric analysis also highlights continuously
varying polarization and decreasing heading variance.

##  1 Introduction

Insects robustly achieve the coordination required for collective group
flight, largely by leveraging a small collection of neurons in their
visuomotor system. Tethered insect studies have identified individual neurons
in the visual sensory pathway and their respective roles in environmental
navigation. Despite the progress in identifying the insect visuomotor neural
pathways, the resulting flight control functions of these neural processes are
not always known, particularly when stimulated by large numbers of neighboring
agents.

Some insects demonstrate high visual acuity during interactions with prey,
predators or conspecific neighbors. This interaction requires visual detection
of targets in unstructured environments [1]. Targets often subtend only a
small region of the visual field relative to sensor resolution. The small-
target motion detector (STMD) neurons found in the lobula complex of certain
insects such as dragonflies, blowflies and hoverflies are highly sensitive to
small, dark, fast-moving objects. Models predicting the behavior of STMDs
suggest that these neural models can be used for target detection and pursuit
in artificial vision-based robotic applications [2, 3, 4, 5]. Although recent
progress in multi-agent modeling has shown how a parallel wide-field visual
processing pathway can participate in swarm motion [6, 7], there is not a
clear understanding for how the STMDs behave in a multi-agent environment.
This study investigates whether the responses of STMD neurons can help insects
participate in swarm motion through theory, simulation, and experiment.

This paper builds on Billah and Faruque [8], which introduced the mathematical
feedback model form and provided the first simulations of various individual
feedback laws, and on Billah and Faruque [9], which suggests a theoretical
bound on trajectories under average dwell time constraints for various
individual feedback laws, without any further theoretical or trajectory
analysis or experimental implementation. This paper builds on the general
feedback approach identified in these papers by providing explicit theoretical
convergence analysis of the behavior, which is applied to design a controller
convergent to local formations. This study also formalizes the STMR approach’s
implicit visual interaction between agents through a graph theoretic
representation, which supports understanding of the implicit information flow
and instantaneous computational/attentional workload of this approach relative
to alternatives. Finally, this study develops an experimental robotic
implementation of the STMR approach, using ground robots to provide
experimental data that quantifies the real-world performance variation of
these visually-guided algorithms. This study also addresses a broader context
by comparing the performance of this controller against existing swarm
approaches. The addition of a stability proof, a graph-theoretic implicit
information framework, development of an implemented robotic experiment and
its data analysis, and contemporary swarm approach simulations and systematic
comparison analysis, this study provides a more comprehensive treatment that
motivated this paper.

The main contributions of this paper include (a) a concise retina-to-motor
neuron dynamic model of STMD neuronal feedback, which goes beyond the STMD
neuronal models as simply sensory mechanisms, (b) analysis and stability proof
of state boundedness over time, necessary for reliable robotic
implementations, (c) simulation and analysis of information flow showing low
levels of connectivity, indicating low attentional requirements relative to
contemporary approaches, and (d) experimental work using external recording of
onboard implementation in robotic hardware, used to provide an uncertainty
bound by showing repetitive behavior.

This paper is structured as follows: section 2 reviews previous work related
to the characteristics of small-target motion detector neurons, feedback laws
for pursuit flight in insects and stability of switched systems. Section 3
proposes a mathematical model of the STMD neurons, designs a continuous-time
multi-agent switched feedback control system using the STMD neuron model as a
sensory function and analyzes stability of the designed multi-agent system.
This section also discusses the simulation and experimental methods used for
validation and further analysis. Section 4 presents results and discusses the
findings and outcomes from this study and their implications.

##  2 Previous Work and Background

###  2.1 Small target motion detector neurons

The small-target motion detector (STMD) neurons were first identified in
dragonflies [10] by O’Carroll. STMDs have also been studied in dipteran flies
such as hoverflies [11] and blowflies [12]. Studies by Nordström et al. [11]
have shown the STMDs to have extremely high contrast sensitivity and
background motion rejection [11]. Experimental studies found STMDs to be
highly-sensitive to small moving targets [13], exhibit selective attention to
a particular target in the presence of multiple targets [14], and show
predictive gain modulation to target trajectories [15]. The STMD neurons also
perform gain modulation for low contrast targets [16], even amongst cluttered
dynamic backgrounds [17], and the gain modulation response can depend on
factors such as target angular size and velocity [18]. Lancer et al. [19]
further studied the selective attention property of STMDs that randomly tracks
a particular target in the presence of multiple targets and discovered the
“lock on” mechanism where the STMD neurons fixate on a low-contrast target to
reduce switching between targets even when a high-contrast distractor is
introduced [20], suggesting a varied mechanism for processing the visual
stimuli compared to spatial wide-field integration (WFI) of optic flow [21]
performed in the lobula complex of the insect visual ganglia. Although
individual insect targets may be isolated and tracked by STMD neurons, the
question of whether STMDs function to inhibit or enable conspecific swarm
flight is not yet fully addressed in literature.

Several quantitative models of STMDs have been proposed in recent years to
define their underlying mechanisms [4, 5, 9, 22, 23, 24]. These include
elementary STMD (ESTMD) [22], cascaded EMD-ESTMD [4], ESTMD with feedback [5],
directionally selective STMD (DSTMD) [23] and STMD+ [24]. Implementation of
some of these models on autonomous robots for target tracking has highlighted
the effectiveness of these models in artificial visual feedback systems [3,
2].

###  2.2 Insect individual feedback control

Several individual feedback guidance laws have been used to describe the
target tracking behavior of insects [25, 26, 27, 28, 29]. An engineering
history underlies the comparatively straightforward pure pursuit algorithm
[30], in which the error angle defined as the difference between the pursuer
heading and the bearing angle of the evader is reduced to zero. If the pursuer
regulates its heading to a certain non-zero value of error angle instead, that
feedback law is referred to as biased pursuit or deviated pure pursuit [30]. A
proportional navigation controller aims to maintain a constant bearing angle
of the evader [31], and a hybrid controller is a combination of both the pure
pursuit controller and proportional navigation controller. Motion camouflage
strategies minimize relative motion to reduce the visual signature of the
agent, and one method to achieve visual signature reduction is to regulate the
optic flow generated by the pursuer on the retina of the evader to zero [29].
This approach achieves target tracking while hiding pursuer motion from the
evader. The CDMC algorithm maintains constant distance to the evader while
concealing the pursuer’s motion, for simple linear trajectories of the evader
[26]. Effect of sensory noise and delay in the motion camouflage guidance law
have also been considered for a more biologically consistent approach [32].
Studies suggest each of these feedback control strategies have been found in
various insects [25]. Implementation of insect guidance laws for collision
avoidance and pursuit on robotic platforms have shown consistency with
theoretic formulations for individual agents [33, 34]. Bio-inspired insect
feedback laws are chosen as feedback for the multi-agent switched system
developed in this paper that utilizes the response of STMD neurons for
feedback control.

###  2.3 Switched Systems

A switched system is defined as a family of dynamical systems wherein a
switching signal activates only one subsystem at a time. Switched systems have
been widely used in various fields such as power electronics [35], automotive
control [36], robotics [37] and aerial systems [38]. Significant effort has
been directed toward developing stability and robustness criteria for switched
systems. Stability of switched linear systems was studied in terms of
construction of special classes of Lyapunov functions [39, 40] which decreased
over time despite switching to ensure stability. Hespanha and Morse [41]
showed that the stability properties of individual subsystems can be
translated to the switched system when switching is sufficiently slow, i.e.,
the switching signal satisfies an average dwell-time bound. The use of average
dwell time for stability was further investigated in [42] to study the input-
to-state stability (ISS) of switched continuous systems; in [43] to relax the
ISS requirement on each subsystem for switching under disturbances; in [44,
45] for switched systems with multiple equilibria; and, more recently, in [46]
for switched systems with multiple equilibria under disturbances. The
stability criteria based on average dwell time constraints from Veer and
Poulakakis [46] is used for stability analysis of the STMD-based multi-agent
switched control system developed in this study.

##  3 Methods and Approach

In this section, the modeling, analysis and validation methods for the multi-
agent feedback control system is described. A simplified model of the STMD
neuron function is proposed in 3.1, a multi-agent switched feedback control
system using the STMD function is developed in 3.2, stability analysis of the
designed control system is provided in 3.3 followed by implementation in
simulation (3.4) and on ground robots (3.5).

###  3.1 Small-Target Motion Detector (STMD) Neuronal Model

The magnitude of optic flow signal depends upon the size, relative velocity
and the distance between the observer and the target, along with the
observer’s own motion [7]. Literature shows the STMD neurons found in insects
are also highly responsive to a target’s size, velocity, contrast, relative
distance and direction of motion [17]. The STMD neurons also respond to a
single target at any particular time [14]. From the cascaded model in
Wiedermann and O’Carroll [4], it is evident that the neural pathways for both
optic flow and STMDs can be correlated, or even be assumed to be cascaded one
after the other in an EMD-ESTMD formation. STMD neurons process the optic flow
patterns to extract relevant information about the surroundings. This control-
oriented study uses a concise model of STMD neurons’ output, in particular as
mathematical maxima ($\max$) and argument of maxima
($\operatorname*{arg\,max}$) operators on the optic flow signal from its
preceding neural pathway [9]. The STMD-model thus identifies the maximum
magnitude of optic flow signal and its corresponding angular position from the
entire visual field of view as

$\displaystyle y_{\mathrm{stmd,1}}$
$\displaystyle=\max_{\gamma,\beta}(\dot{\bm{Q}})$ $\displaystyle
y_{\mathrm{stmd,2}}$
$\displaystyle=\operatorname*{arg\,max}_{\gamma,\beta}(\dot{\bm{Q}}),$ (1)

where $\dot{\bm{Q}}$ is the optic flow signal defined in a continuous, body-
frame-relative spherical coordinate system as developed in Humbert [47], and
$\gamma$ and $\beta$ are the respective azimuth and elevation angles (Fig. 1).
This proposed STMD-model captures the STMD neurons’ dependency on target’s
size, velocity, relative distance and direction of motion, along with the
single target selection property of the STMD neurons.

Figure 1: Geometry of the optic flow definition [47].

For initial analysis of the implications of Eqn. (3.1) in a multi-agent
system, the STMD-model is implemented in a group of homogeneous agents (in
simulation and ground robots experiments) constrained to horizontal planar
motion and the STMD neurons are assumed to be perfect sensors with no temporal
or frequency dependence, resulting in zero time delay and signal amplification
over a wide range of frequencies. Boundary walls or obstacles are not
considered.

For horizontal planar motion, $\gamma$ spans between $[-\pi,\pi]$, $\beta=0$
and the planar optic flow equation for multi-agent environment can be
expressed as [7]

$\displaystyle\dot{\bm{Q}}(\gamma,\bm{q}_{a},$
$\displaystyle\dot{\bm{q}}_{a},\bm{q}_{b},\dot{\bm{q}}_{b})=-\dot{\theta}_{a}-\mu(\gamma,\bm{q}_{a},\bm{q}_{b})\dot{\theta}_{b}$
$\displaystyle+$
$\displaystyle\mu(\gamma,\bm{q}_{a},\bm{q}_{b})\\{(\dot{x}_{b}-\dot{x}_{a})\sin\gamma-(\dot{y}_{b}-\dot{y}_{a})\cos\gamma)\\},$
(2)

where ($\bm{q}_{a},\dot{\bm{q}}_{a}$) and ($\bm{q}_{b},\dot{\bm{q}}_{b}$) are
the states of the viewer and the target agents respectively, such that
$\bm{q}_{a}=[x_{a},y_{a},\theta_{a}]$ and
$\bm{q}_{b}=[x_{b},y_{b},\theta_{b}]$. The nearness function $\mu$ is the
inverse of the magnitude of the vector $\bm{r}_{ab}$ connecting the viewer and
the target agents

$\displaystyle\mu(\gamma,\bm{q}_{a},\bm{q}_{b})=\frac{1}{|\bm{r}_{ab}|}.$ (3)

###  3.2 Bio-inspired Multi-Agent Feedback Control System

The STMD-model detects and outputs a selected target agent in terms of the
maximum optic flow magnitude ($y_{\mathrm{stmd,1}}$) and its corresponding
angular position ($y_{\mathrm{stmd,2}}$). A feedback law to close the loop
must then specify how the STMD-output is used to regulate the agent’s motion.
Dragonflies use STMD neurons during flight to pursue and encounter smaller
prey and conspecifics [17] and a feedback guidance law from insect pursuer-
evader models known as ‘pure pursuit’ strategy in literature [25] was chosen
to define the higher-order neural feedback following the STMD-output. The
feedback rule is given by

$\displaystyle
u_{i}=K[{y}_{\mathrm{stmd,2}}]_{i}=K[\operatorname*{arg\,max}_{\gamma,\beta}(\dot{\bm{Q}_{i}})],$
(4)

where $u_{i}$ is the feedback law for $i^{th}$ subsystem, $i=1,2,\ldots,N$,
and $K>0$ is an arbitrary scalar feedback gain. This proportional feedback
rule structure aims to maintain a constant bearing angle between the target
and the pursuer.

The selection of one ‘target’ agent based on the STMD-model output results in
a continuous-time switching system, where the selection of target by a single
agent switches between different neighboring agents over time depending on the
highest magnitude of optic flow generated by the neighboring agents. The
resulting bio-inspired multi-agent feedback control system is illustrated in
Fig. 2.

Figure 2: Small-target motion reactive (STMR), a bio-inspired multi-agent
closed-loop feedback model based on STMD neurons as sensory mechanism.

###  3.3 Stability Analysis of the Multi-Agent System

####  3.3.1 Bi-agent stability

For stability analysis of the multi-agent feedback system, we consider
differential-drive ground vehicles set at a constant forward speed $v$,
following the non-holonomic constraint, whose kinematics can be defined for 2D
horizontal planar motion as

$\displaystyle\dot{x_{i}}$ $\displaystyle=v_{i}\cos\theta_{i}$
$\displaystyle\dot{y_{i}}$ $\displaystyle=v_{i}\sin\theta_{i}$
$\displaystyle\dot{v_{i}}$ $\displaystyle=0$ $\displaystyle\dot{\theta_{i}}$
$\displaystyle=u_{i},$ (5)

where $u_{i}$ is the feedback control input given in Eqn. 4 and $v_{i}>0$ is
the fixed forward speed.

Figure 3: Geometry of the bi-agent problem.

First consider a bi-agent system with only two agents interacting with each
other. Let the position error, defined as the distance between the agents, be
given as

$\displaystyle\tilde{x}=x_{a}-x_{b},$ $\displaystyle\tilde{y}=y_{a}-y_{b}.$
(6)

Let $\theta$ be the angle of the line-of-sight (LOS) of agent b w.r.t. agent
a, then from Fig. 3

$\displaystyle\tilde{x}$ $\displaystyle=r\cos\theta_{t},$
$\displaystyle\tilde{y}$ $\displaystyle=r\sin\theta_{t}.$ (7)

Differentiating error $\tilde{x}$ and $\tilde{y}$ with respect to time yields

$\displaystyle\dot{\tilde{x}}$
$\displaystyle=\dot{r}\cos\theta_{t}-r\sin\theta_{t}.\dot{\theta}_{t}=v_{a}\cos\theta_{a}-\dot{x}_{b},$
$\displaystyle\dot{\tilde{y}}$
$\displaystyle=\dot{r}\sin\theta_{t}+r\cos\theta_{t}.\dot{\theta}_{t}=v_{a}\sin\theta_{a}-\dot{y}_{b}.$
(8)

Rewriting the equations for $\dot{r}$ and $\dot{\theta}$ yields

$\displaystyle\dot{r}$
$\displaystyle=v_{a}\cos(\theta_{a}-\theta_{t})-(\dot{x}_{b}\cos(\theta_{t})+\dot{y}_{b}\sin{\theta_{t}}),$
$\displaystyle\dot{\theta}$
$\displaystyle=\frac{1}{r}(v_{a}\sin(\theta_{a}-\theta_{t})+(\dot{x}_{b}\sin(\theta_{t})-\dot{y}_{b}\cos{\theta_{t}})).$
(9)

Substituting $\dot{x}_{b}=v_{b}\cos\theta_{b}$ and
$\dot{y}_{b}=v_{b}\sin\theta_{b}$ in Eqn. (3.3.1) simplifies to

$\displaystyle\dot{r}$
$\displaystyle=v_{a}\cos(\theta_{a}-\theta_{t})-v_{b}\cos(\theta_{t}-\theta_{b}),$
$\displaystyle\dot{\theta}$
$\displaystyle=\frac{1}{r}(v_{a}\sin(\theta_{a}-\theta_{t})+v_{b}\sin(\theta_{t}-\theta_{b})).$
(10)

The feedback law for horizontal planar motion is given by

$\displaystyle\dot{\theta}_{i}=K[{y}_{\mathrm{stmd,2}}]_{i}=K_{i}\gamma_{i},$
(11)

where $K$ is the feedback gain constant and $\gamma$ is the azimuth angle.
Elevation angle, $\beta=0$ for horizontal planar motion.

From Fig. 3, the azimuth angles $\gamma_{a}$ and $\gamma_{b}$ can be defined
as

$\displaystyle\gamma_{a}$ $\displaystyle=\theta_{t}-\theta_{a},$
$\displaystyle\gamma_{b}$ $\displaystyle=\pi+\theta_{b}-\theta_{t}.$ (12)

Since both the agents have the same forward speed, $v_{a}=v_{b}=v$. Let the
nearness function be defined as $\mu=1/r$. For notational convenience, define
$\alpha=v\mu$.

Using Eqns. (3.3.1) and (11), and differentiating Eqn. (3.3.1), the error
dynamics with state variables $\gamma_{a}$ and $\gamma_{b}$ can be expressed
in the polar coordinate as

$\displaystyle\dot{\gamma_{a}}$
$\displaystyle=\alpha\sin\gamma_{a}+\alpha\sin\gamma_{b}-K_{a}\gamma_{a},$
$\displaystyle\dot{\gamma_{b}}$
$\displaystyle=K_{b}\gamma_{b}-\alpha\sin\gamma_{a}-\alpha\sin\gamma_{b}.$
(13)

Assume only one input to the system and let $K_{b}=0$, making the motion of
point B a straight line. Then, linearizing the above system about the origin
gives

$\displaystyle\begin{bmatrix}\dot{\gamma}_{a}\\\
\dot{\gamma}_{b}\end{bmatrix}=\begin{bmatrix}\alpha-K_{a}&\alpha\\\
-\alpha&-\alpha\end{bmatrix}\begin{bmatrix}\gamma_{a}\\\
\gamma_{b}\end{bmatrix}.$ (14)

From the characteristic equation of this system, it can easily be seen that
the origin is exponentially stable for $K_{a}>0$.

####  3.3.2 Multi-agent stability

Now, to analyze the multi-agent switching system with $N$ agents, Theorem 2
from Veer and Poulakakis [46] is considered which describes stability of
switched systems with multiple equilibria under disturbances. The theorem
predicts bounded trajectories for input-to-state stable (ISS) subsystems under
average dwell time bounds. The average dwell time $N_{a}$ is defined in terms
of the number of switches $N_{\sigma}(t,\underline{t})\in\mathbb{Z}_{+}$ over
the interval $[\underline{t},t)\subset\mathbb{R}_{+}$ as

$\displaystyle N_{\sigma}(t,\underline{t})\leq
N_{0}+\frac{t-\underline{t}}{N_{a}}\qquad\forall t\geq\underline{t}\geq 0,$
(15)

where $N_{0}>0$ is a finite constant.

The bounds for average dwell time is given by

$\displaystyle N_{0}\geq 1\quad\mbox{and}\quad N_{a}\geq\bar{N_{a}},$ (16)

where $\bar{N_{a}}>0$ represents the lower bound on the average dwell time of
the system. For our system, we define $\bar{N_{a}}$ for any constant
$\epsilon$ in the open interval $(0,\lambda)$ as

$\displaystyle\bar{N_{a}}=\frac{\ln\mu(k)}{\lambda-\epsilon},$

where $\lambda>0$ and $\mu(k)\geq 1$ are defined in Veer and Poulakakis [46].

###  3.4 Simulation Study

Simulation of the multi-agent switched system in Fig. 2 under average dwell
time bound from Eqn. (16) is implemented in Matlab. A total of $N$ homogeneous
agents are considered in the 2D planar motion simulation environment and
assumed to be following the kinematics of ground vehicles described in Eqn.
(3.3.1). For each agent, this framework results in $N-1$ identical subsystems
that the STMD output can switch among. The simplified simulation assumes
sparse environment with no background or obstacles, and no explicit collision
avoidance algorithm in place (outside of any given by the feedback law). All
of the agents are set at the same fixed forward speed $v_{i}$ with arbitrary
initial positions and orientations. Measurement of optic flow and STMD output
uses the idealized equations from Eqn. (3.1) and Eqn. (3.1) respectively.
Target switching for each agent is allowed only if the average dwell time
bound is satisfied.

###  3.5 Experimental Study

For real-time experimental implementation on ground robots, five Turtlebots
[48] equipped with panoramic cameras and running the decentralized feedback
controller onboard are used (Fig. 4, top), starting from arbitrary initial
conditions that were kept constant for all the experiments. The experiment
arena uses white boundary walls for low visual texture to ensure the swarm
agents’ motions are the primary sources of optic flow (Fig. 4, bottom). The
Turtlebots are operated by Raspberry Pi 3b running Raspbian OS. Python, C++,
and ROS Kinetic are used to implement the onboard feedback system processes,
including onboard image acquisition and processing, feedback control
calculation and implementation. The agents have a maximum linear velocity of
22 cm/s and maximum rotational velocity of 2.84 rad/s. No explicit collision
avoidance algorithm is implemented on the Turtlebots. The experimental arena
spans 8 x 12 ft. The camera used is Picam360 with a vertical FOV of
$220^{\circ}$ and a horizontal FOV of $360^{\circ}$. It captures 640 x 480
pixel images at 20 fps. The raw, circular image is then distortion corrected
into a panoramic image using image processing. The Picam360 is mounted face
down to ensure the target agents occupied a large FOV in the dewarped images.
A single horizontal line from the input image contains all the target agents
and is used to perform onboard optic flow calculation. Optic flow calculation
from the camera image input is performed using the Gunnar Farnebäck algorithm
using the opencv_apps ROS package. The Gunnar Farnebäck algorithm calculates
dense optic flow at fixed points in the visual input based on pixel intensity
changes between successive image frames. Refer to [49] for more details of the
experimental setup and optic flow measurement.

Figure 4: Turtlebot (top), the ground robot used for implementation of the
feedback system in Fig. 2, equipped with an inverted $360^{\circ}$ camera for
optic flow measurement, and experimental setup (bottom) showing five
Turtlebots in the experiment arena.

A remote desktop running Linux-based Ubuntu 16.04 initializes all the agents
at the start of the experiment. Each experimental trial consisted of 15
seconds.

Two cases have been explored in this setup for comparison: (a) pure pursuit
STMR, which uses the feedback rule defined in Eqn. (4), and (b) motion
camouflage STMR, which uses the feedback rule for motion camouflage defined in
Justh and Krishnaprasad [29]. Both cases use STMD-based sensory model for
optic flow processing, given in Eqn. 3.1. The scalar feedback gains $K_{i}$
were chosen arbitrarily, and the average dwell time was chosen to be 1.5s,
based on the stability analysis in Sec. 3.3. All the Turtlebots were set with
a fixed forward speed of $v_{i}=11$ cm/s. Gains $K_{i}$ and average-dwell time
were also the same for all agents. The initial conditions for all agents were
chosen arbitrarily and kept constant for all trials of both the cases.

###  3.6 Comparison Study

For comparison of the STMR framework with contemporary swarm approaches, three
other multi-agent systems are considered: the Vicsek model [50], the Cucker-
Smale (C-S) model [51] and the multi-agent optomotor response (MA-WFI) model
developed previously by the authors [7] that uses wide-field integration (WFI)
of optic flow signals for regulating an agent’s motion relative to its
neighbors. The Vicsek model updates agent orientation based on the average
orientation of agents in its neighborhood, defined by a radius of interaction.
The C-S model assumes inter-agent interaction occurs between all the agents in
the environment, regardless of their inter-agent distances, and the agents
update velocity based on a weighted function of the average velocity of all
the other agents. The Vicsek model achieves orientation-matching whereas C-S
model does velocity-matching. The MA-WFI model is designed to be able to do
both orientation-matching and velocity-matching for a controlled-agent in an
established formation of other agents based on the choice of basis functions.

Two different environments are considered for comparison of feedback systems.
For the first case, the MA-WFI simulation environment is used, where only one
agent is controlled among other agents in an established formation. The
controlled agent has an initial orientation offset, while the rest starts with
no offset. The simulation is run four times, once for each of the different
feedback control systems. For the second case of comparison, all agents are
being controlled by the four different control systems starting from the same
initial conditions as in the first case. The C-S agents have varying initial
velocity, the other models have the same fixed forward speed for all agents
during the entire simulation.

For further comparison analysis of the different frameworks, we study the
connectivity of a system of 50 agents governed by the four different multi-
agent systems mentioned earlier: STMR, MA-WFI, C-S and Vicsek. For this study,
the agents are converted to nodes ($V$) of a weighted graph
$\mathcal{G}=(V,E,W)$, where an edge ($E$) exists between a pair of nodes only
if the two agents interact with each other. Edge weights ($W$) are determined
based on the feedback rules that the agents use. Algebraic connectivity (also
called the Fiedler eigenvalue) is defined as the second-smallest eigenvalue of
the Laplacian matrix of $\mathcal{G}$ [52] and is a measure of how well
connected the agents are among each other in a multi-agent system. Three swarm
analysis metrics (swarm polarisation [53], average swarm heading and heading
variance) are analyzed to provide a systematic performance comparison across
the four frameworks.

##  4 Results and Discussion

###  4.1 Results

####  4.1.1 Simulation results

Simulation results of $N=20$ agents for six different initial conditions are
presented in Fig. 5. Feedback gains and agents’ forward speeds are set to
$0.1$ and $10$ cm/s respectively. Parameters to determine the minimum average
dwell time are set to: $\mu(k)=10$, $\lambda=1$ and $\epsilon=0.3$. Each
simulation is run for 50 seconds.

The markers in Fig. 5 represent the final position of the agents while the
solid lines represent their trajectories. Collision avoidance is not designed
into the feedback rule, so collision between agents is observed in some
instances. Fig. 6 shows all the agents’ orientations $\theta_{i}$ over the
simulation period. The agents show general directional consensus and bounded
$\theta_{i}$ for each of the different initial conditions.

Fig. 7 (top) presents the switching signal plot with and without the average
dwell time bounds. The plot indicates which agents are ‘tracked’ over time by
an arbitrary agent. Each agent ‘tracks’ one other agent based on the STMD-
output (Eqn. (3.1)). Fig. 7 (bottom) shows the average dwell time plot of an
agent. The minimum average dwell time based on Eqn. (16) is also plotted. It
is clear that switching between ‘tracked’ agents occurs only when the average
dwell time for that agent is above the minimum specified. After each switch,
the average dwell time decreases as expected from Eqn. (15), resulting in the
sawtooth-like shape of the plots.

Fig. 8 shows the agent trajectories for simulation of the system without the
average dwell time bounds, allowing the agents to switch instantaneously to
whichever neighboring agent has the highest magnitude of optic flow. Initial
conditions for the six cases were kept the same as those in Fig. 5 for
comparison. The agents still show general directional consensus over time, but
one observes fragmentation into smaller groups of agents in some cases (Fig.
8, top right).

Figure 5: Simulation results for the multi-agent switched system in Fig. 2
with the average dwell time constraints, showing trajectories of 20 agents for
six different initial conditions.

Figure 6: Simulation results for the multi-agent switched system in Fig. 2
with the average dwell time constraints, showing orientation of 20 agents for
the six initial conditions from Fig. 5.

Figure 7: Simulation results. Switching signal plot (top) showing the agents
“tracked” over time by a single agent (fifth agent from top in Fig. 5 bottom
center axes) in the multi-agent switched system with and without the dwell
time constraints, and the average dwell time plot (bottom) for the same agent
with the minimum average dwell time plotted.

Figure 8: Simulation results without the average dwell time constraints,
showing trajectories of 20 agents for initial conditions from Fig. 5.

####  4.1.2 Experimental results

Experimental results for the two feedback rules are shown in Fig. 9. Blue
lines indicate mean agent trajectories, and grey areas represent the variance
over 15 trials for the same initial condition. The agents converge towards
each other for pure pursuit STMR feedback in Fig. 9 (top). Collision occurs
for a minority of agents since no collision avoidance algorithm is being
implemented. Unlike the simulation, collision has a greater effect in the
experiments as ground vehicles continue to interfere after collision.
Increasing initial agent separation and reducing the experiment duration to a
relatively shorter time (15s) compared to the simulation (50s) allowed the
multi-agent system to show convergence behaviors while avoiding artifacts from
collision-induced behavior. Analyzing the effects and consequences of
collision among agents is not within the scope of this study.

For the case of using motion camouflage STMR feedback, results shown in Fig. 9
(bottom), the agents do not converge on the same point or collide, and
consistently demonstrate low variance in their trajectories across the 15
trials. The agents’ group motion demonstrates higher polarisation and more
consistent spatial separation.

Figure 9: Experimental results for the implementation of multi-agent STMR
feedback control system with two different feedback rules considered (15
trials each): (top) the pure pursuit feedback rule from Eqn. (4) and (bottom)
motion camouflage from [29].

####  4.1.3 Comparison study results

Comparison study results are shown in Figs. 10-13. Fig. 10 shows the output of
the first case where only one agent is controlled among an established swarm
motion. MA-WFI agent reaches an equilibrium condition of $\theta^{*}=0$ from
its initial orientation offset, as predicted. The Vicsek agent also reaches
equilibrium, using the average orientation of its neighbors as feedback, with
a response time faster than MA-WFI. The STMR agent initially continues in its
initial orientation, before returning towards the other agents. It does not
reach $\theta^{*}=0$ during the simulation but shows it can successfully
participate in an established group formation. The C-S model does not
participate in the motion; given how it is meant to velocity-match all agents,
the model may be less applicable when only one agent is being controlled.

Fig. 11 shows the trajectory of agents for the second case where all agents
are being controlled by the different swarm models. STMR agents show cohesion
and group motion, as expected. MA-WFI (Fig. 11, top) was designed for a single
feedback-controlled agent, and does not show convergence when all agents are
feedback-controlled. Vicsek agents (Fig. 11, middle) converge to the average
orientation of the group, whereas the C-S agents (Fig. 11, bottom) converge to
an equilibrium velocity. Once the agents converge to their equilibrium
conditions, the formation of motion does not change.

Fig. 12 shows the connectivity plots for the four multi-agent approaches
considered, with 50 agents in each simulation. Fig. 12(a) shows the
instantaneous connectivity of the underlying undirected graph of each system,
and Fig. 12(b) shows the connectivity of the union of underlying graphs over
time. At each instant in Fig. 12(a), STMR has the lowest connectivity (zero
connectivity) since each agent is only connected to one other agent, whereas
the other systems are either fully connected (C-S and MA-WFI) or a subset of
agents are fully connected (Vicsek). In Fig. 12(b), the area under the union
of underlying graphs gives a measure of the total information shared (or “work
done”) by the connected agents. One can quantitatively compare this effect by
defining the area under the union of underlying graphs to be attentional work.
The STMR approach achieves similar attentional work as the other approaches
with very low instantaneous connectivity.

Fig. 13 shows the swarm polarisation (Fig. 13, top left), average swarm
heading (Fig. 13, top right ) and heading variance plots (Fig. 13, bottom) for
the simulation results from Fig. 12. The Vicsek and C-S agents reach
convergence to an arbitrary general swarm heading, resulting in swarm
polarisation reaching 1 and orientation variance reaching close to 0. The MA-
WFI agents show low polarisation and high orientation variance over time,
indicating they do not reach convergence, which may result from all agents
being controlled. The STMR agents’ swarm polarisation values are continuously
variable due to their ‘milling about’ motions, and have a gradually decreasing
heading variance, which indicates the agents’ convergence towards a time-
varying average swarm heading (bulk convection) with bounded individual
orientation.

Figure 10: A single feedback-controlled agent surrounded by non-reactive
neighbors with established formation; comparison of STMR feedback control
system with the optomotor feedback control system from [7], the Vicsek model
[50] and the Cucker-Smale model [51].

Figure 11: A collection of feedback-controlled agents all reactive to each
other; comparison of the STMR feedback control system with the optomotor
feedback control system from [7], the Vicsek model [50] and the Cucker-Smale
model [51].

(a) Connectivity of the underlying undirected graph.

(b) Connectivity of the union of underlying graphs.

Figure 12: Comparison of algebraic connectivity (Fiedler eigenvalue) among the
STMR feedback control system proposed in this paper, the optomotor feedback
control system from [7], the Vicsek model [50] and Cucker-Smale model [51].

Figure 13: Comparison of swarm polarisation (top left), average swarm heading
(top right) and heading variance (bottom) over time among the STMR feedback
control system proposed in this paper, the optomotor feedback control system
from [7], the Vicsek model [50] and Cucker-Smale model [51].

###  4.2 Discussion

The implementation results of the feedback control system proposed in Fig. 2
suggest that group motion can be achieved using the small-target motion
reactive (STMR) framework. The agents’ states are bounded over time (Fig. 5,
6) and group motion is achieved. The average dwell time constraint ensures
switching between agents is sufficiently slow to ensure boundedness, but the
group shows general directional consensus without the constraint as well (Fig.
8). In this framework, each agent tracks only one other agent in the
environment, instead of tracking all or a large subset of agents as in other
traditional multi-agent systems [50, 51]. Tracking only one neighboring agent
allows the system to be easily scalable to large groups of agents without the
need for increased sensory capabilities or communication bandwidth.

The simulation results of the multi-agent system verifies that the addition of
average dwell time bounds on the switching system ensures stability in terms
of bounded states and prevents high frequency switching between the
subsystems. Trajectories without the average-dwell time bound (Fig. 8) suggest
very high frequency switching can lead to factions splitting from the group.
Parameters such as $\epsilon$ and $\mu(k)$ can be used to tune the average
dwell time bound to allow higher switching frequency at the cost of the
trajectories converging to a bigger compact set of solutions. The trade-off is
between switching frequency and robustness of the system as explained in [46].
Several mechanisms for attention-like processes and selective attentional
capabilities have been discovered from studies of insect attention [54]. The
STMD neurons in insects such as dragonflies shows similar behavior by tracking
only one target when multiple targets are presented [14], even “locking onto”
a low-contrast target in spite of the introduction of a high-contrast
distractor [20]. This “lock-on” mechanism reduces the number of switches
between targets, but does not eliminate them completely [17]. While the “lock-
on” phenomenon is claimed to reduce confusion during prey selection in a swarm
[20], the results from this study may be used to show that it may also be
applicable to cohesive motion coordination.

From the comparison study in Fig. 11, the STMR agent trajectories show a more
unstructured behavior over time compared to the other models, due to the fact
that the stability analysis only guarantees boundedness for the STMR agents
instead of convergence. Factions of agents can split and rejoin (Fig. 5,
bottom right), or agents show “milling” motion [55] that does not eventually
settle down to any particular group formation. This propagation of ‘erratic’
behavior is different than traditional swarm models found in literature that
use attraction, repulsion and/or state alignment for stability. However, the
lower group polarisation levels observed in STMR agent trajectories relative
to C-S and Vicsek simulations (Fig. 13, top left) are consistent with insect
swarms (e.g., honeybees, dragonflies) that have less obviously structured
coordination [56].

Conventional swarming approaches often assume continuous large bandwidth of
information sharing among agents (Fig. 12), and low-error communication
systems which are not easily scalable or implementable in large groups. These
approaches do not account for the neural/computational limit of the biological
animals that it models, nor of the hardware that will be used for
implementation. Since STMR agents only need information from the largest optic
flow signal producer, individual agents can be disconnected from the rest of
the $N-2$ agents and still have group motion, as this study suggests (Fig.
12). Since the information shared in conventional approaches is to provide
convergence, the connectivity is much higher at the start of the simulation
and decreases over time as agents converge to an alignment consensus (Fig.
12(a)). Instead of having very high connectivity at the start of the
simulation to prioritize convergence, STMR spreads the attentional workload
over time to reach bounded trajectories instead, and shows consensus over
group directionality (Fig. 12(b)). Low connectivity could sometimes lead to
factions splitting off, but results suggest that these factions often rejoin,
given the average dwell time bounds are enforced (Fig. 5, bottom left, bottom
right).

The STMD neurons in predatory insects such as dragonflies predominantly aid in
pursuer-evader scenarios, STMD neurons have been found in non-predatory
insects as well [57]. This study proposes a method of using pairwise
interaction between agents for collective motion in multi-agent systems and
there is precedent to believe that this might also be the case in animals.
Flocking starlings interact with 6-7 neighbors [58], while honeybees in
convective group milling interact with 1-2 neighbors [59]. A similar behavior
as observed from STMR agents, such as faction splitting and rejoining
resulting in robust cohesion, is also observed in flocking starlings to avoid
predators, as the swarm constantly changes its density and structure and yet
no agent is isolated from the group [58].

###  4.3 Discussion of Assumptions and Implications

This model assumes the small-target motion detectors can reliably suppress
background motion, which is generally slow-moving and further away, generating
low frequency optic flow signals, similar to how STMD neurons in dragonflies
respond to background motion [17]. In contrast, the fast-moving small-target
agents closer to the observer are assumed to provide high frequency optic flow
signals that are used as input for the system. The model also assumes STMD
neurons are ideal sensors with no temporal or frequency dependence, resulting
in zero time delay and signal amplification over a wide range of frequencies.
This assumption allows for investigating the effect of concise STMD-type
neural sensing mechanisms in multi-agent systems in idealized conditions
before further analysis of the sensory model including temporal dynamics.

No exclusive collision avoidance algorithm is implemented for the STMR system,
and the results suggest the STMR feedback rule does not provide repulsion in
close range. In simulation, the agents were allowed to overlap and continue in
their trajectories, while in experiment the agents collided and moved together
after collision. Separating the effect of collision avoidance ensured the
effect of STMD-based sensing in group cohesion could be analyzed in depth.

The Veer and Poulakakis [46] theorem used for stability analysis of the
switched system provides a bound on the state trajectories for multiple
equilibrium of the sub-systems. The behaviors discussed thus require some
switch rate limit provided by the average dwell time constraint. Similar
behavior can be seen without the constraint (Fig. 8) but the agents are then
more prone to splitting into multiple groups. Attention in insects have been
well studied and several mechanisms identified for attention-like processes
[54]. Insects such as Drosophila have also been found to have attention-like
mechanisms where they can shift the focus of attention (FoA) internally, with
the time-span between each shift of FoA being 1s-4s [60], values comparable to
the average dwell time bounds used in the bio-inspired feedback model designed
in this study. In addition, insect attention has multiple definitions and
unifying them is outside the scope of this study. Instead, we use the ad-hoc
term attentional work to describe the area under the union of graphs (Fig.
12(b)), to provide a quantitative comparison of the total information shared
(or “informational work done”) by the connected agents.

The comparison and connectivity plots among different multi-agent systems in
Figures 10 \- 12 are one instance of simulation amid numerous other possible
trajectories based on the gains and tunable parameters chosen for each system.
Instantaneous connectivity values are dependent on the gains used for
convergence, but the trend remains similar to that shown in Fig. 12. Different
gains might affect the time it takes for attentional work to reach equality
for STMR and the other multi-agent systems, but does not necessarily affect
the existence of such a finite time.

###  4.4 Summary and Conclusion

This study introduces the small-target motion reactive (STMR) approach to
insect swarming by considering an engineering model of the STMD neurons found
in the lobula of insects such as hoverflies and dragonflies. The two major
outputs from this group of motion detection neurons are the highest magnitude
of optic flow and the bearing angle corresponding to the moving agent
generating the highest magnitude of optic flow. Assuming an idealized case of
the STMD neuron model, a multi-agent switched feedback system framework (STMR)
is developed. Stability analysis of the multi-agent system involves bi-agent
stability of each subsystem, which extends to state boundedness via the Veer-
Poulakakis theorem for switched systems. An average dwell time constraint
ensures the agent states are bounded over time by ensuring that switching
between alternating agents is sufficiently slow. The feedback control system
is implemented in simulation and on ground vehicles for validation, resulting
in agent trajectories that demonstrate group cohesion and general directional
consensus. Comparison with other multi-agent systems such as Vicsek and
Cucker-Smale models suggest that the low connectivity network with an agent
only tracking one other agent may still result in group motion comparable to
conventional swarm algorithms but with consistently very low sensory
computational requirements. Overall, STMR represents a bio-inspired strategy
that could provide a similar function to traditional engineered swarm
approaches while accounting for low computational capabilities of insect
visuomotor systems by spreading the processing workload over time and limiting
the amount of instantaneous individual “attention” requirements to the
surrounding.

## Acknowledgment

This work was supported in part by ONR Young Investigator Award
N00014-19-1-2216.

## Research Data

The data that support the findings of this study are available from the
authors upon reasonable request.

## References

  * Lingenfelter et al. [2021] B. Lingenfelter, A. Nag, and F. van Breugel, “Insect inspired vision-based velocity estimation through spatial pooling of optic flow during linear motion,” _Bioinspiration & biomimetics_, vol. 16, no. 6, p. 066004, 2021.
  * Bagheri et al. [2017a] Z. M. Bagheri, B. S. Cazzolato, S. Grainger, D. C. O’Carroll, and S. D. Wiederman, “An autonomous robot inspired by insect neurophysiology pursues moving features in natural environments,” _Journal of neural engineering_ , vol. 14, no. 4, p. 046030, 2017.
  * Bagheri et al. [2017b] Z. M. Bagheri, S. D. Wiederman, B. S. Cazzolato, S. Grainger, and D. C. O’Carroll, “Performance of an insect-inspired target tracker in natural conditions,” _Bioinspiration & biomimetics_, vol. 12, no. 2, p. 025006, 2017.
  * Wiedermann and O’Carroll [2013] S. Wiedermann and D. C. O’Carroll, “Biologically inspired feature detection using cascaded correlations of off and on channels,” _Journal of Artificial Intelligence and Soft Computing Research_ , vol. 3, 2013.
  * Wang et al. [2021] H. Wang, H. Wang, J. Zhao, C. Hu, J. Peng, and S. Yue, “A time-delay feedback neural network for discriminating small, fast-moving targets in complex dynamic environments,” _IEEE Transactions on Neural Networks and Learning Systems_ , 2021.
  * Humbert et al. [2005] J. S. Humbert, R. M. Murray, and M. H. Dickinson, “Sensorimotor convergence in visual navigation and flight control systems,” _IFAC Proceedings Volumes_ , vol. 38, no. 1, pp. 253–258, 2005.
  * Billah and Faruque [2021] M. A. Billah and I. A. Faruque, “Bioinspired visuomotor feedback in a multiagent group/swarm context,” _IEEE Transactions on Robotics_ , vol. 37, no. 2, pp. 603–614, 2021.
  * Billah and Faruque [2022a] M. A. Billah and I. Faruque, “The multi-agent group motions generated by models of insect small target detector neurons and feedback,” in _AIAA SCITECH 2022 Forum_ , 2022, p. 0962.
  * Billah and Faruque [2022b] M. A. Billah and I. A. Faruque, “Modeling small-target motion detector neurons as switched systems with dwell time constraints,” in _2022 American Control Conference (ACC)_. IEEE, 2022, pp. 3192–3197.
  * O’Carroll [1993] D. O’Carroll, “Feature-detecting neurons in dragonflies,” _Nature_ , vol. 362, no. 6420, pp. 541–543, 1993.
  * Nordström et al. [2006] K. Nordström, P. D. Barnett, and D. C. O’Carroll, “Insect detection of small targets moving in visual clutter,” _PLoS biology_ , vol. 4, no. 3, p. e54, 2006.
  * O’Carroll and Wiederman [2014] D. C. O’Carroll and S. D. Wiederman, “Contrast sensitivity and the detection of moving patterns and features,” _Philosophical Transactions of the Royal Society B: Biological Sciences_ , vol. 369, no. 1636, p. 20130043, 2014.
  * Wiederman and O’Carroll [2011] S. D. Wiederman and D. C. O’Carroll, “Discrimination of features in natural scenes by a dragonfly neuron,” _Journal of Neuroscience_ , vol. 31, no. 19, pp. 7141–7144, 2011.
  * Wiederman and O’Carroll [2013] S. D. Wiederman and D. C. O’Carroll, “Selective attention in an insect visual neuron,” _Current Biology_ , vol. 23, no. 2, pp. 156–161, 2013.
  * Wiederman et al. [2017] S. D. Wiederman, J. M. Fabian, J. R. Dunbier, and D. C. O’Carroll, “A predictive focus of gain modulation encodes target trajectories in insect vision,” _Elife_ , vol. 6, p. e26478, 2017.
  * Geurten et al. [2007] B. R. Geurten, K. Nordström, J. D. Sprayberry, D. M. Bolzon, and D. C. O’Carroll, “Neural mechanisms underlying target detection in a dragonfly centrifugal neuron,” _Journal of Experimental Biology_ , vol. 210, no. 18, pp. 3277–3284, 2007.
  * Evans et al. [2022] B. J. E. Evans, D. C. O’Carroll, J. M. Fabian, and S. D. Wiederman, “Dragonfly neurons selectively attend to targets within natural scenes,” _Frontiers in Cellular Neuroscience_ , p. 151, 2022.
  * Fabian et al. [2019] J. M. Fabian, J. R. Dunbier, D. C. O’Carroll, and S. D. Wiederman, “Properties of predictive gain modulation in a dragonfly visual neuron,” _Journal of Experimental Biology_ , vol. 222, no. 17, p. jeb207316, 2019.
  * Lancer et al. [2022] B. H. Lancer, B. J. Evans, J. M. Fabian, D. C. O’Carroll, and S. D. Wiederman, “Preattentive facilitation of target trajectories in a dragonfly visual neuron,” _Communications biology_ , vol. 5, no. 1, pp. 1–13, 2022.
  * Lancer et al. [2019] B. H. Lancer, B. J. Evans, J. M. Fabian, D. C. O’Carroll, and S. D. Wiederman, “A target-detecting visual neuron in the dragonfly locks on to selectively attended targets,” _Journal of Neuroscience_ , vol. 39, no. 43, pp. 8497–8509, 2019.
  * Krapp [2022] H. G. Krapp, “Optic flow processing,” in _Encyclopedia of Computational Neuroscience_. Springer, 2022, pp. 2539–2558.
  * Wiederman et al. [2008] S. D. Wiederman, P. A. Shoemaker, and D. C. O’Carroll, “A model for the detection of moving targets in visual clutter inspired by insect physiology,” _PloS one_ , vol. 3, no. 7, p. e2784, 2008.
  * Wang et al. [2018] H. Wang, J. Peng, and S. Yue, “A directionally selective small target motion detecting visual neural network in cluttered backgrounds,” _IEEE transactions on cybernetics_ , vol. 50, no. 4, pp. 1541–1555, 2018.
  * Wang et al. [2019] H. Wang, J. Peng, X. Zheng, and S. Yue, “A robust visual system for small target motion detection against cluttered moving backgrounds,” _IEEE transactions on neural networks and learning systems_ , vol. 31, no. 3, pp. 839–853, 2019.
  * Varennes et al. [2020] L. Varennes, H. G. Krapp, and S. Viollet, “Two pursuit strategies for a single sensorimotor control task in blowfly,” _Scientific reports_ , vol. 10, no. 1, pp. 1–13, 2020.
  * Strydom and Srinivasan [2017] R. Strydom and M. V. Srinivasan, “Uas stealth: target pursuit at constant distance using a bio-inspired motion camouflage guidance law,” _Bioinspiration & biomimetics_, vol. 12, no. 5, p. 055002, 2017.
  * Mischiati et al. [2015] M. Mischiati, H.-T. Lin, P. Herold, E. Imler, R. Olberg, and A. Leonardo, “Internal models direct dragonfly interception steering,” _Nature_ , vol. 517, no. 7534, pp. 333–338, 2015.
  * Srinivasan [2011] M. V. Srinivasan, “Visual control of navigation in insects and its relevance for robotics,” _Current opinion in neurobiology_ , vol. 21, no. 4, pp. 535–543, 2011.
  * Justh and Krishnaprasad [2006] E. W. Justh and P. Krishnaprasad, “Steering laws for motion camouflage,” _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , vol. 462, no. 2076, pp. 3629–3643, 2006.
  * Shneydor [1998] N. A. Shneydor, _Missile guidance and pursuit: kinematics, dynamics and control_. Elsevier, 1998.
  * Peppas [1992] L. Peppas, “D. proportional navigation and command to line of sight of a command guided missile for a point defence system,” Ph.D. dissertation, Ph. D. thesis, Naval postgraduate school Monterey, California, 1992.
  * Raju and Krishnaprasad [2016] V. Raju and P. Krishnaprasad, “Motion camouflage in the presence of sensory noise and delay,” in _2016 IEEE 55th Conference on Decision and Control (CDC)_. IEEE, 2016, pp. 2846–2852.
  * Huang et al. [2019] J. V. Huang, Y. Wei, and H. G. Krapp, “A biohybrid fly-robot interface system that performs active collision avoidance,” _Bioinspiration & Biomimetics_, vol. 14, no. 6, p. 065001, 2019.
  * Colonnier et al. [2019] F. Colonnier, S. Ramirez-Martinez, S. Viollet, and F. Ruffier, “A bio-inspired sighted robot chases like a hoverfly,” _Bioinspiration & biomimetics_, vol. 14, no. 3, p. 036002, 2019.
  * Vasca and Iannelli [2012] F. Vasca and L. Iannelli, _Dynamics and control of switched electronic systems: advanced perspectives for modeling, simulation and control of power converters_. Springer, 2012.
  * Wu et al. [2018] X. Wu, K. Zhang, and M. Cheng, “Optimal control of constrained switched systems and application to electrical vehicle energy management,” _Nonlinear Analysis: Hybrid Systems_ , vol. 30, pp. 171–188, 2018.
  * Namaki-Shoushtari et al. [2012] O. Namaki-Shoushtari, A. Pedro Aguiar, and A. Khaki-Sedigh, “Target tracking of autonomous robotic vehicles using range-only measurements: a switched logic-based control strategy,” _International Journal of Robust and Nonlinear Control_ , vol. 22, no. 17, pp. 1983–1998, 2012.
  * Carrillo et al. [2013] L. R. G. Carrillo, G. R. F. Colunga, G. Sanahuja, and R. Lozano, “Quad rotorcraft switching control: An application for the task of path following,” _IEEE Transactions on Control Systems Technology_ , vol. 22, no. 4, pp. 1255–1267, 2013.
  * Narendra and Balakrishnan [1994] K. S. Narendra and J. Balakrishnan, “A common lyapunov function for stable lti systems with commuting a-matrices,” _IEEE Transactions on automatic control_ , vol. 39, no. 12, pp. 2469–2471, 1994.
  * Mancilla-Aguilar and Haimovich [2020] J. L. Mancilla-Aguilar and H. Haimovich, “Uniform input-to-state stability for switched and time-varying impulsive systems,” _IEEE Transactions on Automatic Control_ , vol. 65, no. 12, pp. 5028–5042, 2020.
  * Hespanha and Morse [1999] J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” in _Proceedings of the 38th IEEE conference on decision and control (Cat. No. 99CH36304)_ , vol. 3. IEEE, 1999, pp. 2655–2660.
  * Vu et al. [2007] L. Vu, D. Chatterjee, and D. Liberzon, “Input-to-state stability of switched systems and switching adaptive control,” _Automatica_ , vol. 43, no. 4, pp. 639–646, 2007.
  * Chen and Yang [2016] G. Chen and Y. Yang, “Relaxed conditions for the input-to-state stability of switched nonlinear time-varying systems,” _IEEE Transactions on Automatic Control_ , vol. 62, no. 9, pp. 4706–4712, 2016.
  * Alpcan and Basar [2010] T. Alpcan and T. Basar, “A stability result for switched systems with multiple equilibria,” _Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis_ , vol. 17, no. 4, pp. 949–958, 2010.
  * Dorothy and Chung [2016] M. Dorothy and S.-J. Chung, “Switched systems with multiple invariant sets,” _Systems & Control Letters_, vol. 96, pp. 103–109, 2016.
  * Veer and Poulakakis [2019] S. Veer and I. Poulakakis, “Switched systems with multiple equilibria under disturbances: Boundedness and practical stability,” _IEEE Transactions on Automatic Control_ , vol. 65, no. 6, pp. 2371–2386, 2019.
  * Humbert [2005] J. Humbert, “Bio-inspired visuomotor convergence in navigation and flight control systems,” Doctoral dissertation, California Institute of Technology, 2005.
  * Amsters and Slaets [2020] R. Amsters and P. Slaets, “Turtlebot 3 as a robotics education platform,” in _Robotics in Education: Current Research and Innovations 10_. Springer, 2020, pp. 170–181.
  * Billah and Faruque [2023] M. A. Billah and I. A. Faruque, “Robustness in bio-inspired visually guided multi-agent flight and the gain modulation hypothesis,” _International Journal of Robust and Nonlinear Control_ , vol. 33, no. 2, pp. 1316–1334, 2023.
  * Vicsek et al. [1995] T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, and O. Schochet, “Novel Type of Phase Transition In a System of Self-Driven Particles,” _Physical Review Letters_ , vol. 75, no. 4, pp. 729–732, 1995.
  * Cucker and Smale [2007] F. Cucker and S. Smale, “Emergent Behavior in Flocks,” _IEEE Transactions on Automatic Control_ , vol. 52, no. May, pp. 1–20, 2007.
  * Godsil and Royle [2001] C. Godsil and G. F. Royle, _Algebraic graph theory_. Springer Science & Business Media, 2001, vol. 207.
  * Armbruster et al. [2017] D. Armbruster, S. Martin, and A. Thatcher, “Elastic and inelastic collisions of swarms,” _Physica D: Nonlinear Phenomena_ , vol. 344, pp. 45–57, 2017.
  * Nityananda [2016] V. Nityananda, “Attention-like processes in insects,” _Proceedings of the Royal Society B: Biological Sciences_ , vol. 283, no. 1842, p. 20161986, 2016.
  * Hindes et al. [2021] J. Hindes, V. Edwards, M. A. Hsieh, and I. B. Schwartz, “Critical transition for colliding swarms,” _Physical Review E_ , vol. 103, no. 6, p. 062602, 2021.
  * Schultz et al. [2008] K. M. Schultz, K. M. Passino, and T. D. Seeley, “The mechanism of flight guidance in honeybee swarms: subtle guides or streaker bees?” _Journal of Experimental Biology_ , vol. 211, no. 20, pp. 3287–3295, 2008.
  * Nordström and O’Carroll [2006] K. Nordström and D. C. O’Carroll, “Small object detection neurons in female hoverflies,” _Proceedings of the Royal Society B: Biological Sciences_ , vol. 273, no. 1591, pp. 1211–1216, 2006.
  * Ballerini et al. [2008] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini _et al._ , “Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study,” _Proceedings of the national academy of sciences_ , vol. 105, no. 4, pp. 1232–1237, 2008.
  * [59] S. Islam and I. A. Faruque, “Metric, topological, and temporal elements of tracking and deliberative decision-making in flying insect groups,” in review.
  * Koenig et al. [2016] S. Koenig, R. Wolf, and M. Heisenberg, “Vision in flies: measuring the attention span,” _PLoS One_ , vol. 11, no. 2, p. e0148208, 2016.