1
A Survey on Aerial Swarm Robotics
Soon-Jo Chung, Senior Member, IEEE, Aditya Paranjape, Philip Dames, Member, IEEE,
Shaojie Shen, Member, IEEE, and Vijay Kumar, Fellow, IEEE
Abstract—The use of aerial swarms to solve real-world prob-
lems has been increasing steadily, accompanied by falling prices
and improving performance of communication, sensing, and pro-
cessing hardware. The commoditization of hardware has reduced
unit costs, thereby lowering the barriers to entry to the field of
aerial swarm robotics. A key enabling technology for swarms is
the family of algorithms that allow the individual members of the
swarm to communicate and allocate tasks amongst themselves,
plan their trajectories, and coordinate their flight in such a way
that the overall objectives of the swarm are achieved efficiently.
These algorithms, often organized in a hierarchical fashion,
endow the swarm with autonomy at every level, and the role of
a human operator can be reduced, in principle, to interactions
at a higher level without direct intervention. This technology
depends on the clever and innovative application of theoretical
tools from control and estimation. This paper reviews the state of
the art of these theoretical tools, specifically focusing on how they
have been developed for, and applied to, aerial swarms. Aerial
swarms differ from swarms of ground-based vehicles in two
respects: they operate in a three-dimensional (3-D) space, and the
dynamics of individual vehicles adds an extra layer of complexity.
We review dynamic modeling and conditions for stability and
controllability that are essential in order to achieve cooperative
flight and distributed sensing. The main sections of the paper
focus on major results covering trajectory generation, task
allocation, adversarial control, distributed sensing, monitoring,
and mapping. Wherever possible, we indicate how the physics
and subsystem technologies of aerial robots are brought to bear
on these individual areas.
Index Terms—Aerial robotics, distributed robot systems, net-
worked robots.
I. INTRODUCTION
Aerial robotics has become an area of intense research
within the robotics and control community. Autonomous aerial
robots can capitalize on the three-dimensional (3-D) airspace
with aplomb, oftentimes equipped with vertical take-off and
landing capabilities using zero-emission distributed electric
fans. Swarms of such aerial robots or autonomous Unmanned
Aerial Vehicles (UAVs) are emerging as a disruptive technol-
ogy to enable highly-reconfigurable, on-demand, distributed
intelligent autonomous systems with high impact on many
S.-J. Chung is with the Graduate Aerospace Laboratories of California
Institute of Technology (GALCIT), Pasadena, CA 91125, USA (Email:
sjchung@caltech.edu).
A. A. Paranjape is with the Department of Aeronautics, Imperial
College London, South Kensington, London, United Kingdom (Email:
a.paranjape@imperial.ac.uk).
P. Dames is with the Department of Mechanical Engineering, Temple
University, Philadelphia, PA 19122, USA (Email: pdames@temple.edu).
S. Shen is with the Department of Electronic & Computer Engineering,
Hong Kong University of Science and Technology, Hong Kong (Email:
eeshaojie@ust.hk).
V. Kumar is with the School of Engineering & Applied Science,
University of Pennsylvania, Philadelphia, PA 19104, USA (Email: ku-
mar@seas.upenn.edu).
areas of science, technology, and society, including tracking,
inspection, and transporting systems. In any application, au-
tonomous aerial swarms are expected to be more capable
than a single large vehicle, offering significantly enhanced
flexibility (adaptability, scalability, and maintainability) and
robustness (reliability, survivability, and fault-tolerance) [1].
This survey article reflects on advances in aerial swarm
robotics and recognizes that a number of technological gaps
need to be bridged in order to achieve the aforementioned
benefits of swarms of aerial robots through autonomous and
safe operation. The papers included in this survey article rep-
resent the most important and promising approaches to mod-
eling, control, planning, sensing, design, and implementation
of aerial swarms, with an emphasis on enhanced flexibility,
robustness, and autonomy.
Swarming aerial robots must autonomously operate in a
complex 3-D world including urban canyons and an airspace
that is getting increasingly crowded with drones and com-
mercial airplanes. The success of aerial swarms flying in a
3-D world is predicated on the distributed and synergistic
capabilities of controlling individual and collective motions of
aerial robots with limited resources for on-board computation,
power, communication, sensing, and actuation (the so-called
size, weight and power, or SWaP, tradeoff). The goal is to
provide a unified framework within which to analyze the three-
way trade-off among computational efficiency for large-scale
swarms, stability and robustness of control and estimation
algorithms, and optimal system performance.
Compared to prior survey articles focused on robotic
swarms [2], we emphasize swarms of aerial robots flying in
a 3-D world. Other related survey papers on swarm robotics
include [3], which focused on problems such as formation con-
trol, cooperative tasking, spatiotemporal planning, and consen-
sus for generic multi-robot system. Our survey paper addresses
the challenges associated with transitioning from 2-D to 3-D
with limited SWaP with applications to swarm coordination
or collaboration and distributed tracking and estimation. Our
survey paper also addresses the challenges of integrating
autonomous aerial swarm systems with other types of robots,
such as ground vehicles. From a technological standpoint, the
broader impacts of research in aerial swarm robotics include
scalability and down-compatibility with 2-D robotic networks
(e.g., ground robots) and other 3-D unmanned systems such
as spacecraft swarms [4], [5] and underwater swarms [6].
The distinguishing characteristics of aerial swarm robotics are
summarized as follows:
3-D Flow and Swarm Autonomy: Motion planning and
control methods for aerial swarms rely on autonomously-
generated 3-D traffic flows that do not have fixed edges or
roads. Real-time flight control and swarm operation must
2
also take into account high-fidelity six-degree-of-freedom (6-
DOF) flight dynamic models, traffic variations, weather, and
other time-varying operational conditions found in crowded
urban environments. These aspects stand in stark contrast
to those focused on 3-D air traffic flow control with much
longer time horizon [7]–[9] as well as 2-D road traffic flow
theory, bipartite matching, and transport operation theory that
assume fixed flight pathways and road/route topologies [10].
Furthermore, existing air traffic control systems require human
operators to perform real-time control of airport congestion
and prevention of mid-air collision [7]. We will describe
methods of simultaneous 6-DOF trajectory generation and
optimal swarm routing or control techniques for autonomous
aerial swarm system that require a minimal level of human
intervention.
Scalability Through Hierarchy and Multi-Modality: En-
abling large-scale swarm autonomy in complex environments
will require theoretically well-founded, computationally-
efficient, and scalable algorithms. This can be realized through
the use of hierarchical architectures for decentralized planning,
reasoning, learning, and perception that address scalability
and information management in the presence of uncertain-
ties. Hierarchical approaches are pervasive in both the ma-
chine learning and control fields for dealing with complex-
ity and high dimensionality (e.g., hierarchical task networks
(HTNs) [11], hierarchical tree or lattice networks employed
in Sequential Game Theory [12], and singular perturbation
theory [13], [14]). They are also especially well-suited for
aerial robots due to the inherent diversity of time scales in
the system. The inner-loop flight control, and especially the
attitude dynamics, must run faster than the timescales of the
rigid body dynamics of the aerial robot as well as the structural
dynamics of the wings or propellers to ensure stable flight.
Onboard perception algorithms must also run at a time scale
that is appropriately small to enable robots to avoid collisions
with dynamic, unexpected obstacles. Transient maneuvers of
aerials swarms are controlled at the same time scale as the
rigid-body flight dynamics, while outer-loop control (i.e.,
motion planning of swarms) and the cooperative estimation
and planning algorithms run an order of magnitude slower
than the flight dynamics. These outer-loop components must
be integrated closely with perception and reasoning of other
vehicles, environmental conditions, and scientific or customer
needs. This complexity in aerial swarms can be reduced by
exploiting hierarchical connections in spatial and temporal
scales of large-scale aerial swarm networks. In this survey
paper, we expand on the algorithms and technologies for aerial
robotics that depend on hierarchical architectures.
The organization of the present paper is shown in Fig. 1.
In each section, we attempt to provide elementary working
solutions, taken from the literature, for each subproblem.
We will then present refinements of these solutions, which
constitute the state of the art in the respective subject areas.
In Sec. 2, we review modeling the dynamics of a swarm
and nonlinear stability tools, in particular for hierarchical
decomposition, as well as issues of controllability for aerial
swarms. In Sec. 3, we review optimal control, motion plan-
ning, task assignment, and other control algorithms. In Sec.
Fig. 1. Major themes in swarm control and the organization of the paper.
4, we discuss distributed sensing and estimation using aerial
swarms, specifically addressing the problems of (multi-)target
tracking, distributed surveillance, and cooperative mapping.
In Sec. 5, we review essential system-level and component
technologies for aerial swarms. Finally, we conclude the paper
in Sec. 6 with a discussion of open problems in the area of
aerial swarms.
II. MODELS, STABILITY AND CONTROLLABILITY OF
SWARMS
A. Types of Multiagent Systems
Table I presents a classification of multiagent systems based
on the number of agents and their interaction. It has a direct
bearing on how the systems are modeled: the choice of
the governing equations, the assumptions made about the
underlying connectivity, and the nature of the control inputs
and information exchange.
In a team, the behavior and strategies of each individual
agent seek to explicitly maximize a local objective. In some
cases, this may cause the agents to compete against each other,
while in other cases the locally optimal behavior may also
(approximately) maximize the global reward. The latter is the
premise of game-theoretic methods and auction algorithms
[15], [16]. In auctions, for instance, maximizing the local
benefit also maximizes the net global benefit (defined as the
sum of individual benefits) and concurrently solves the dual
pricing problem [15]. In contrast to a team, a formation
almost always consists of cooperative interactions, and the
relationship between the states of the agents is well-defined
for objectives such as energy efficiency (e.g., flocks of birds
in an aerodynamically optimum V-formation [17]). A swarm
generally refers to a group of similar agents that displays
emergent behavior arising from local interactions among the
agents. The local interaction can be competitive or cooperative.
Although a swarm typically implies a large group of agents
(10s to 100s or more), this survey article uses “swarm” to also
include smaller groups as well (see Table I).
3
TABLE I
CLASSIFICATION OF MULTI-AGENT SYSTEMS
Type Scope Size
Team Typically small groups; each agent opti-
mizes individual objectives in a cooperative
or competitive manner
typically
≤ 10
Formation Each agent is typically assigned a specific
sub-task, role, or placement
typically
≤ 10s
Swarm Typically large groups of dispensable
agents; global capability arises from emer-
gent behavior
large
B. Models for Swarm Dynamical Systems
One of the earliest engineering models for flocking is from
Reynolds [18], who used it to generate a realistic visualization
of flocks for computer graphics. Reynolds rules cover basic
neighbor-to-neighbor interaction: a nonlinear function which
governs the steady state separation between the agents, and a
velocity feedback term which seeks to ensure that the velocity
of each agent tracks the average of its neighbors. Reynolds’
model is given as:
¨
x
i
=
˙
v
i
=
X
j∈N
i
(k
s
∇ W (x
j
− x
i
)+k
a
(v
j
− v
i
))+f
i
(1)
where x
i
and v
i
denote the position and the velocity of
the i
th
agent; W (x
j
− x
i
) is a coupling function; N
i
is
the neighborhood of i
th
agent; and f
i
denotes an external
influence on the agent, such as that of the leader or an intruder.
Another early work [19] studied a flock moving in two-
dimensional space and discrete time using the following
equations:
x
i
(t + 1) = x
i
(t) + v
i
(t)∆t
θ
i
(t + 1) =
1
card(N
i
)
X
j∈N
i
θ
j
(t) + ∆θ
i
(t) (2)
where the noise ∆θ
i
(t) is normally distributed in the set
[−η, η]. Importantly, the velocity v
i
is assumed to have a
constant magnitude for all i and t, with its heading given by
θ
i
(t). Despite the apparent simplicity of the model, it is able to
capture the possibility of long-range order, as explained later
in this section.
A generalized representation of the models in [18] and [19]
can be obtained by using partial difference equations (PdEs)
[20], [21]. The rules for obtaining PdEs permit a natural
association with continuum PDEs, and consequently, ways for
deriving flocking laws based on PDEs other than the wave
equation used in [20].
A unified, nonlinear continuum model, as against models
based on discretely defined agents on a graph, was proposed
in [22]:
∂v
∂t
+ (v · ∇)v
| {z }
convection
= αv − βkvk
2
v − ∇P (ρ)
+ D
L
∇(∇ · v) + D
1
∇
2
v + D
2
(v · ∇)
2
v
| {z }
diffusion
+f
∂ρ
∂t
+ ∇ · (ρv) = 0 (3)
This model was claimed to resemble that of bird flocks for two
spatial dimensions, although the model itself is not constrained
to any particular number of dimensions and could be applica-
ble to three-dimensional flocks as well. The constants β, D
{·}
are all positive; the term α > 0 corresponds to an ordered
velocity state (steady flight speed kvk =
p
α/β), while α < 0
gives rise to a disordered phase (e.g., a flock loitering around
a fixed point). The pressure term P =
P
k
σ
k
(ρ−ρ
0
)
k
, where
σ
k
’s are constants and ρ
0
is the mean local density, replaces
the potential-like term in Reynolds’ model. Finally, f denotes
disturbances, modeled as Gaussian noise.
Increasing the value of the noise (i.e., η) in (2) causes the
flock to spontaneously choose an ordered state [19], where
the critical value of the noise is correlated with the number
of agents in the flock. This is conjectured to be due to
the diffusive flow of information in the flock; i.e., agents
interacting with a time-varying set of neighbors and, in the
long run, this causes diffusion of information throughout the
flock. This conjecture was borne out in [22] for a two-
dimensional flock, wherein the nonlinear convection terms in
(3) were found to be responsible for stabilizing the ordered
state across large length scales.
In the context of swarms, one is interested in the questions
of stability and convergence of the states of the individual
agents. For such analysis, it is common to use a system of
linear(ized) equations, the simplest of which is the system
˙
x
i
=
X
j∈N
i
w
ij
(x
j
− x
i
), i = 1, . . . , n (4)
⇔
˙
x = −(L ⊗ I
p
)x, L
ij
=
w
ij
∃ edge from node j
to node i
0 otherwise
The matrix L or (L ⊗ I
p
) is called a Laplacian matrix
and satisfies L1
n
= 0, where 1
n
∈ R
n
is a vector of
ones. It is evident that a constant L corresponds to a fixed
communication topology; when the communication topology
evolves with time, a time-varying L(t) is used. This is identical
to the diffusive coupling term one would find from (2).
For problems involving assignment or routing, it helps to
model the environment as a collection of “functional bins,”
together with accessibility conditions which restrict the agents’
transition between the bins. The end objective is to assign n
agents to a set of m bins, where each bin can accommodate
up to p
i
≥ 1 (m < n; i = {1, 2, . . . , m}) agents. For
each agent i and a bin j, the accessible set E
ij
implicitly
accounts for the dynamics of the agent as well as the geometric
constraints imposed by the environment. Such models have
been used to control swarm shape with probabilistic transition
maps between the bins [23] and quadrotor formation control
with deterministic transition laws [24], [25].
C. Physics-Based Models for Robotic Agents
General linear systems similar to (4) can be constructed
readily in a double integrator setting (e.g., attitude dynamics
on SO(3) or rigid body motions on SE(3)), or by replacing the
4
dynamics with a nonlinear version. Of particular interest here
are swarm systems comprising Euler-Lagrange equations:
M
i
(q
i
)
¨
q
i
+C
i
(q
i
,
˙
q
i
)
˙
q
i
+g
i
(q
i
)=τ
i
(q
i
,
˙
q
i
, q
d
, q
j∈N
i
,
˙
q
j∈N
i
)
(5)
where q
i
∈ R
p
are the generalized states of the i
th
agent; q
d
(t)
is the desired trajectory or a virtual leader for a target collective
motion; and τ
i
are the external forces/torques, which are
the source of coupling between agents. If a linear diffusive
coupling is used, τ
i
would produce L similar to (4). The Euler-
Lagrange equations appear routinely robotics in the study of
rigid body motions of manipulators [26], [27] and spacecraft or
aircraft (SE(3)), which have attitude dynamics on SO(3) [28]–
[30] and often times have articulated wings [14], [31], [32],
appendages, or manipulators attached:
In [33], the full 6-DOF aircraft model is used with actuator
time delays to compute optimal motion primitives and 3-D
path planning for fast flight through a forest. It shows that
a conventional 2-D Dubin’s vehicle model, often times used
for 2-D aircraft motion planning and swarm control, is not
appropriate for aerial robots moving in 3-D. For the purpose
of studying swarms of fixed- or flapping-wing aerial robots, it
may suffice to model the aerial robots as point-masses (mass
m) with velocity dynamics (speed V , climb angle γ, and
heading χ) described by
[ ˙x, ˙y,
˙
h] = V [cos γ cos χ, cos γ sin χ, sin γ]
m
˙
V = T cos α − D(V, α) − mg sin γ
mV ˙γ =
1
mV
(L(V, α) + T sin α) cos µ − W cos γ
mV ˙χ = (L + T sin α)
sin µ
cos γ
(6)
where L, D, and T are the lift, drag, and thrust, respectively.
In flapping-wing aerial robots, T is additionally a function of
V and α. This model is accurate under the assumption that
the rotational dynamics (α and µ) are stable and converge
rapidly to the commanded value. The 3-D aerial robot model
can be used effectively to reduce the computational burden on
a motion planning system and generate trajectories that are
optimal, stable, and safe (i.e., with collision avoidance) [14],
[33]. In [34], model-based control laws were derived, together
with a collision-avoiding system, for a swarm of parafoil-
payload systems. A model similar to (6) was employed, and
feedback about the position of the neighboring agents was
used to command the desired value of the turn rate ( ˙χ in (6))
of each agent.
Although the terms L, D, and T have been presented in
the spirit of control inputs in (6), it is important to note that
their values could be affected significantly in a swarm of aerial
robots by flow induced by neighboring aircraft. When an aerial
robot experience failures and is unable to hold its position
accurately, it could have a detrimental effect on the efficiency
of the formation due to the adverse disruption in the flow
field experienced by the faulty aircraft’s neighbors. This sort
of physics-based interaction is unique to atmospheric flight
vehicles.
Fig. 2. A swarm of heterogeneous rigid bodies converging to the desired
shape (ellipsoid), whose center can be viewed as a virtual leader for the
swarm [28]. The angular separation between the vehicles is synchronized
actively and shows smaller synchronization error than tracking errors. Dotted
lines show diffusive couplings via communication or relative sensing.
D. Synchronization with Leader Following
To control a swarm, it is useful at times to define a physical
or virtual leader that the rest of swarm agents then follow
(see Fig. 2). The motion of the leader can be given a priori
or controlled directly by separate dynamics. Alternatively, a
desired trajectory (i.e., the path of a virtual leader) can be
computed using optimal control or motion planning algorithms
(see Sec. 3). The remaining agents are controlled indirectly
through interaction between neighbors [35] or through inter-
action with the leader [36], [37]. The problem of tracking
the trajectory of the virtual leader or the desired collective
behavior for agents with highly nonlinear dynamics (e.g.,
swarms rigid bodies with dynamics on SE(3) or agents with
multi-DOF manipulators) can be addressed simultaneously
with the problem of synchronization with neighboring agents
[27]. Based on time-scale separation, this unified framework
integrates trajectory tracking with an exponentially-stabilizing
consensus controller that synchronizes the relative motions of
swarms faster than following a common leader or a desired
trajectory. This yields a smaller synchronization error than an
uncoupled tracking control law in the presence of bounded
disturbances and modeling errors [28] (see Fig. 2). This time-
scale separation can be interpreted as a hierarchical connection
of faster and slower dynamics as discussed in Sec. II-F. Other
works follow the same problem formulation of synchronizing
coupled nonlinear dynamical systems concurrently with tra-
jectory tracking for various multi-robot/multi-vehicle applica-
tions. We can leverage concurrent synchronization of mixing
multiple virtual leaders with many synchronized groups to
create a complex time-varying swarm comprised of numerous
heterogeneous systems [27], [28], [30], [38]. One needs to
determine how many (virtual) leaders need to be chosen,
and which agents to nominate as leaders. This question is
analogous to that of controllability, while the dual observabil-
ity problem corresponds to sensor placement for distributed
estimation.
E. Leader Selection and Sensor Placement
When the dynamics of the aerial swarm agents are identical,
controllability from a given set of leader nodes (equivalently,
observability from a given set of sensors) depends on the
topology of the graph as well as the individual edge weights. A
5
system defined on a graph is said to be structurally controllable
when it is controllable for almost all edge weights, and
strongly structural controllable when it is controllable for all
edge weights. The existence of a rooted tree is necessary
and sufficient for structural controllability with a single leader
node [39]–[42]. In [43]–[45], conditions and algorithms are
derived to determine whether a set of input nodes permit
strong structural controllability. Formulas linking the number
of driver nodes needed for a large network and its aggregate
properties (number of nodes, mean degree and the degree
exponent) are presented in [46]. It was observed that driver
nodes with the highest degree of controllability tend not to be
nodes of the largest degree.
In practical problems, we are generally interested in con-
trollability for a given set of edge weights, and especially for
identical edge weights (i.e., the system is described by the
Laplacian matrix –L in (4)). For this problem, there exist
necessary conditions based on symmetry and equipartition
[47]–[49]. While these conditions are not sufficient, a set of
sufficient conditions have been presented for path and cycle
graphs [50], and for a class of weakly-connected digraphs [51].
It must be noted that the selection of a leader (equivalently,
sensor placement) need not be optimal by the virtue of its con-
trollability (respectively, observability) properties alone, and it
is therefore necessary to measure the influence of the candidate
driver nodes on the actual control/estimation objectives [52].
If the objective is to optimize an objective function, as shall
be seen in (7), one can solve the problem of determining
the leader/sensor nodes computationally using techniques from
sub-modular optimization [53]–[55]. Maximization of sub-
modular functions is NP-hard; however, greedy algorithms can
yield approximate solutions with guaranteed sub-optimality
using at most O(n
2
) computations of the objective function.
Sub-modular optimization can also be viewed from the hier-
archical organization standpoint emphasized in this paper.
F. Synchronization and Hierarchical Stability for Swarms
Consider (4) with diffusive couplings. It is well-known that
the matrix L gives rise to a stable system under the following
conditions on the underlying graph:
1) Undirected time-invariant graph: the graph is con-
nected [56].
2) Directed time-invariant graph: consensus to the average
value if and only if the graph is balanced and weakly
connected [57]. Existence of a rooted tree guarantees
consensus, though not necessarily to the average value
[58].
3) Time-varying undirected/directed graph: satisfies a gen-
eralized strong connectivity condition [58, Propositions
1 and 2], [59].
The Laplacian matrix (L) captures the effect of diffusive
coupling terms on swarm or synchronization stability. The
spectral characteristics of Laplacian matrices have been used
to prove the stability of flocks obeying Reynolds’ rules [19],
[59]–[61], the stability under a distance-based communication
topology [62], and the exponential stabilization of networked,
nonlinear Euler-Lagrange systems [27], [28], [31], [63]. [64]
illustrate the effect of nonlinearities on the stability of net-
worked systems through bifurcations. Alternate methods for
stability analysis include tools from renormalization groups
[22] and the theory of normally hyperbolic invariant manifolds
[65]. The Laplacian matrix defined above can be replaced by
its variant, the edge Laplacian matrix, to solve for stability as
well as robustness and optimality [66].
The aforementioned conditions are conclusive in the ab-
sence of other dynamical terms like (4). The passivity of the
input-output dynamics [67]–[69] is commonly used to analyze
the stability of networked nonlinear systems that have both
the Laplacian matrix (L in (4)) and the nonlinear dynamical
terms (e.g., convection terms of (3) or the Lagrangian form in
(5)). Input-to-State Stability (ISS) is used to study stability of
swarm systems with bounded uncertainties [70], [71]. Contrac-
tion analysis [72] is used to study global exponential stability
of multiple solution trajectories, and hence forms a basis of
incremental stability analysis. Contraction-based incremental
stability analysis represents an important departure from tra-
ditional passivity-based methods using Lyapunov functions,
which are concerned primarily with stability of equilibrium
points.
Such an exponentially-safe and robust synchronization
framework can also be used to study the synchronization
stability and robustness of networked nonlinear dynamics
connected by a synchronization controller or by diffusive
communication couplings [27], [73]. One major advantage
of incremental stability in a synchronization framework [27],
[28], [73] over the passivity formalism is that a hierarchically-
combined structure of dynamic systems, emphasized in this
paper, can be handled more easily because of differential con-
traction analysis without using some implicit motion integrals.
Further, it can be shown that contraction-based exponential
incremental stability using a Riemannian metric possesses
superior robustness related to input-to-state stability (ISS),
output passivity, and finite-gain L
p
stability [28]. Many types
of model uncertainty can be cast into a bounded pertur-
bation term, including constant unknown time delays [27],
[72] and errors arising from heterogeneous dynamics [27],
[63]. Recently, incremental stability has been extended to
synchronization stability of multiple It
¯
o stochastic nonlinear
differential equations [38], [74] with unbounded stochastic
disturbances.
An extension of some of the aforementioned results arises
in the form of event-triggered information exchange. Instead
of exchanging communication continuously or over finite
intervals of time, as in the previous cases, it is sufficient
for stability to exchange information between neighboring
agents at discrete instants of time. Conditions for stability in
such cases have been found for single integrator dynamics on
undirected graphs [75], consensus on balanced digraphs [76],
convergence to a trajectory on time-varying graphs [77], and
synchronization of general nonlinear dynamics on balanced
graphs [28], [73], [78]. These conditions typically depend
on the underlying dynamics and also help determine the
conditions under which communication must be triggered.
