arXiv:1402.5639v1  [cs.SY]  23 Feb 2014
Decentralized Rendezvous of Nonholonomic Robots with
Sensing and Connectivity Constraints
Zhen Kan a, Justin Klotz a, Eduardo L. Pasiliao Jr b, John M. Shea c,
Warren E. Dixon a,
aDepartment of Mechanical and Aerospace Engineering, University of Florida, Gainesville, USA
bMunitions Directorate, Air Force Research Laboratory, Eglin AFB, FL 32542, USA.
cDepartment of Electrical and Computer Engineering, University of Florida, Gainesville, USA
Abstract
A group of wheeled robots with nonholonomic constraints is considered to rendezvous at a common speciﬁed setpoint with
a desired orientation while maintaining network connectivity and ensuring collision avoidance within the robots. Given
communication and sensing constraints for each robot, only a subset of the robots are aware or informed of the global
destination, and the remaining robots must move within the network connectivity constraint so that the informed robots can
guide the group to the goal. The mobile robots are also required to avoid collisions with each other outside a neighborhood of the
common rendezvous point. To achieve the rendezvous control objective, decentralized time-varying controllers are developed
based on a navigation function framework to steer the robots to perform rendezvous while preserving network connectivity
and ensuring collision avoidance. Only local sensing feedback, which includes position feedback from immediate neighbors
and absolute orientation measurement, is used to navigate the robots and enables radio silence during navigation. Simulation
results demonstrate the performance of the developed approach.
1
Introduction
Distributed cooperative control of networked multi-
agent systems has attracted considerable interest. One
particular cooperative control problem is the rendezvous
problem, where a number of agents arrive at a predeﬁned
destination simultaneously, ideally using limited infor-
mation from the environment and team members. Some
example applications of the rendezvous problem are co-
operative strike and cooperative jamming in [1] and [2].
In the cooperative strike scenario, multiple strikes are
executed on a target simultaneously by ﬁring from dif-
ferent locations. In cooperative jamming of a wireless
communication network with eavesdroppers, noisy sig-
nals are transmitted to jam the eavesdroppers at the
Email addresses: kanzhen0322@ufl.edu (Zhen Kan),
jklotz@ufl.edu (Justin Klotz), pasiliao@eglin.af.mil
(Eduardo L. Pasiliao Jr), jshea@ece.ufl.edu (John M.
Shea), wdixon@ufl.edu (Warren E. Dixon).
1 This research is supported in part by NSF award numbers
1161260, 1217908, and a contract with the AFRL Mathe-
matical Modeling and Optimization Institute. Any opinions,
ﬁndings and conclusions or recommendations expressed in
this material are those of the authors and do not necessarily
reﬂect the views of the sponsoring agency.
same time when the source transmits the message sig-
nal. Spacecraft docking, air-to-air refueling, and the in-
terception of an incoming missile can also be considered
as rendezvous problems. In these applications, coordina-
tion and collaboration are crucial to performance, and
agents are required to communicate and coordinate their
movements with others to achieve rendezvous.
Several rendezvous results are reported in [3–5]. Conver-
gence to a common point for a group of autonomous mo-
bile agents is studied in [3]. In [4] and [5], synchronized
and unsynchronized strategies are developed to drive
mobile agents to a single unspeciﬁed location by using
only position feedback from its sensing regions. A com-
mon assumption in [3–5] is that the network remains con-
nected during the motion evolution, allowing constant
interaction between agents. However, the assumption of
network connectivity is not always practical. Typically,
each agent can only make decisions based on the local
information from immediate neighbors within a certain
region due to sensing and communication constraints.
Since communication/sensing links generally depend on
the distance between agents, agent motion may cause the
underlying network to disconnect. If the network discon-
nects, certain agents may no longer be able to commu-
Preprint submitted to Automatica
August 13, 2018
nicate and coordinate their motion, leading to a failure
of cooperative tasks.
Recent results such as [6–11] have focused on maintain-
ing network connectivity when performing rendezvous
tasks. A circumcenter algorithm is proposed in [6] to
avoid the loss of existing links between agents. In [7–9]
a potential ﬁeld-based distributed approach is devel-
oped to prevent partitioning in the underlying graph
by using local information from each agent’s immedi-
ate neighbors. The results in [10] provide a connectivity-
preserving protocol for rendezvous of a discrete-time
multi-agent system, and a hybrid dynamic rendezvous
protocol is designed in [11] to address ﬁnite-time ren-
dezvous problems while preserving network connectivity.
However, most of the aforementioned works only con-
sider linear motion models. Although agents with non-
holonomic kinematics are considered in [7], like other
results such as [3–5, 11], the agents can only converge
to a destination determined by the initial deployment.
A dipolar navigation function was proposed and a dis-
continuous time-invariant controller was developed for
a multi-robot system in [12] to perform nonholonomic
navigation for networked robots. The dipolar navigation
function is a particular class of potential functions, which
is developed from [13] and [14] such that the negative
gradient ﬁeld does not have local minima, and the closed-
loop navigation function guarantees convergence to the
global minimum. The result in [12] was then extended
to navigate a nonholonomic system in three dimensions
in [15]. Other recent results focused on nonholonomic
systems with various cooperative tasks such as formation
control and ﬂocking are reported in [16–19]. However,
network connectivity is not considered in [12,15–19].
The rendezvous problem for mobile robots with non-
holonomic constraints is studied in this work, and the
objective is to reach a common speciﬁed setpoint with a
desired orientation. Only a small subset of robots (i.e.,
informed agents) are assumed to be equipped with ad-
vanced sensors (e.g., GPS) and provided with global
knowledge of the destination, while the remaining robots
(i.e., followers) only have a range sensor (e.g., a passive
range sensor such as a camera, or active sensors such as
sonar, laser, or radar), which provides local feedback of
the relative trajectory of other robots within a limited
sensing region. Since the follower robots are not aware
of the global position of the destination, they have to
stay connected with the informed agents when perform-
ing rendezvous. To avoid collision among robots, the
workspace is divided into a collision-free region and a
rendezvous region. Particularly, the robots are required
to avoid collisions with other robots outside a neigh-
borhood of the common goal. Based on our prelimi-
nary eﬀorts in [20–22], a decentralized time-varying con-
troller, using only local sensing feedback from its im-
mediate neighbors, is designed to stabilize the robots at
the speciﬁed destination while preserving network con-
nectivity and ensuring collision avoidance. The devel-
oped decentralized controller only uses local sensing in-
formation and no inter-agent communication is required
(i.e., communication-free global decentralized group be-
havior). Although network connectivity is maintained so
that radio communication is available when required for
various tasks, communication is not required for nav-
igation. Using the navigation function framework, the
multi-robot system is guaranteed to rendezvous at a
common destination with a desired orientation without
being trapped by local minima from almost all initial
conditions, excluding a set of measure zero. Compared
to [22] where the formation control for a group of agents
with fully actuated dynamics is investigated, networked
mobile robots with nonholonomic constraints are con-
sidered in this work. Unlike our centralized result in [20]
or our preliminary result in [21] in which all the robots
are required to know the goal destination and only undi-
rected interaction between robots are considered, the
current result models the interaction among robots as a
digraph, and only requires a subset of the robots (i.e., one
or more) to have knowledge of the global position of the
destination and the desired orientation. This advance-
ment reduces required resources and sensor loads on the
remaining robots. Within this setting, the informed sub-
set of robots can perform a task-level controller, while
the remaining robots just execute a local interaction-
based strategy. Moreover, the developed controller al-
lows the robots to rendezvous at any desired destination,
versus an unspeciﬁed destination determined by their
initial deployment as in [3–5,7,11]. The result can also be
extended by replacing the objective function in the nav-
igation function to accommodate diﬀerent tasks, such as
formation control, ﬂocking, and other applications.
2
Problem Formulation
Consider N networked mobile robots operating in a
workspace F, where F is a bounded disk area with ra-
dius Rw. Each robot in F moves according to the follow-
ing nonholonomic kinematics:
˙qi =


cos θi 0
sin θi 0
0
1


"
vi (t)
ωi (t)
#
, i = 1, · · · , N
(1)
where qi (t) ≜
h
pT
i (t) θi (t)
iT
∈R3 denotes the states
of robot i, with pi (t) ≜
h
xi (t) yi (t)
iT
∈R2 denoting
the position of robot i, and θi (t) ∈(−π, π] denoting the
robot orientation with respect to the global coordinate
frame in F. In (1), vi (t) , ωi (t) ∈R are the control
inputs that represent the linear and angular velocity of
robot i, respectively.
The subsequent development is based on the assumption
that all robots have equal actuation capabilities and each
2
robot has sensing and communication limitations en-
coded by a disk area with radius R, which indicates that
two moving robots can sense and communicate with each
other as long as they stay within a distance of R. We also
assume that only a subset of the robots, called informed
robots, are provided with knowledge of the destination,
while the other robots can only use local state feedback
(i.e., position feedback from immediate neighbors and
absolute orientation measurement). Furthermore, while
multiple informed robots may be used for rendezvous,
the analysis and results of this work are focused on a
single informed robot. The techniques proposed in this
work could be extended to the case of multiple informed
robots by using containment control, as explained in Re-
mark 2. The interaction among the robots is modeled as
a directed graph G (t) = (V, E(t)), where the node set
V = {1, · · · , N} represents the group of robots, and the
edge set E(t) denotes time-varying edges. The set of in-
formed robots and followers are denoted as VL and VF ,
respectively, such that VL ∪VF = V and VL ∩VF = ∅.
Let VL = {1} and VF = {2, · · · , N}. A directed edge
(i, j) ∈E in G (t) exists between node i and j if their
relative distance dij ≜∥pi −pj∥∈R+ is less than R.
The directed edge (i, j) indicates that node i is able to
access the states (i.e., position and orientation) of node
j through local sensing, but not vice versa. Accordingly,
node j is a neighbor of node i (also called the parent
of node i), and the neighbor set of node i is denoted as
Ni = {j | (i, j) ∈E}, which includes the nodes that can
be sensed. A directed spanning tree is a directed graph,
where every node has one parent except for one node,
called the root, and the root node has directed paths to
every other node in the graph. Since the follower robots
are not aware of the destination, they have to stay con-
nected with the informed robot either directly or indi-
rectly through concatenated paths, such that the knowl-
edge of the destination can be delivered to all the nodes
through the connected network. Hence, to complete the
desired tasks, maintaining connectivity of the underly-
ing graph is necessary.
Collision avoidance among robots has not been consid-
ered for rendezvous problems in existing literature (e.g.,
[4–9]), since it conﬂicts with the objective of meeting at a
common goal. To enable collision avoidance in this work,
the workspace F is divided into a collision-free region Ωc
and a rendezvous region Ωr, such that Ωc ∪Ωr = F. The
rendezvous region Ωr is a bounded disk area with radius
Rr centered at the common destination p∗, while the re-
maining area in F is the collision-free region Ωc. Assume
that the workspace F and the rendezvous region Ωr sat-
isfy that Rw ≫Rr and Rr > R (N −1). The classical
rendezvous problem enables the robots to rendezvous at
p∗with a desired orientation θ∗in Ωr, we additionally
constraint this model by requiring collision avoidance
among robots outside the neighborhood of common p∗
(i.e., Ωc). The main contribution of this work is to derive
a set of distributed controllers using only local informa-
tion (i.e., position feedback from immediate neighbors
and the absolute orientation measurement) to perform
rendezvous, ensure network connectivity, and avoid col-
lisions. To achieve these goals, the following assumptions
are required in the subsequent development.
Assumption 1 The initial graph G (0) has a directed
spanning tree with the informed node as the root.
Assumption 2 The destination p∗and desired orienta-
tion θ∗are achievable, which implies that p∗and θ∗do
not coincide with some unstable equilibria (i.e., saddle
points).
3
Control Design
3.1
Dipolar Navigation Function
Artiﬁcial potential ﬁeld-based methods that use attrac-
tive and repulsive potentials have been widely used to
control multi-robot systems. Due to the existence of lo-
cal minima when attractive and repulsive force are com-
bined, robots can be trapped by local minima and are
not guaranteed to reach the global minimum of the po-
tential ﬁeld. A navigation function is a particular cate-
gory of potential functions where the potential ﬁeld does
not have local minima and the negative gradient vector
ﬁeld of the potential ﬁeld guarantees almost global con-
vergence to a desired destination, along with (guaran-
teed) collision avoidance, if the initial conditions do not
lie within the sets of measure zero. Formally, a naviga-
tion function is deﬁned as follows:
Deﬁnition 1
[13] [14] Let F be a compact connected
analytic manifold with a boundary ∂F, and qd be a goal
point in the interior of F. A mapping ϕ : F →[0, 1] is
a Navigation Function, if it satisﬁes the following condi-
tions: 1) smooth on F (at least a C2 function); 2) admis-
sible on F, (uniformly maximal on the manifold bound-
ary ∂F and constraint boundary); 3) polar on F, (qd is
a unique minimum); and 4) a Morse function, (critical
points of the navigation function are non-degenerate).
The second condition in Deﬁnition 1 establishes that the
generated trajectories are collision-free, since the result-
ing vector ﬁeld is transverse to the boundary of F. The
third point indicates that, using a polar function on a
compact connected manifold with a boundary, all initial
conditions are either brought to a saddle point or to the
unique minimum qd. The requirement that the naviga-
tion function is a Morse function ensures that the initial
conditions that bring the system to saddle points are
sets of measure zero [13]. Given this property, all initial
conditions not within sets of measure zero are brought
to the unique minimum.
The navigation function introduced in [13] and [14] en-
sures global convergence of the closed-loop system; how-
3
−5
0
5
−5
0
5
0
0.5
1
X axis
Y axis
Potential Value
Figure 1. An example of a dipolar navigation function with
a workspace of Rw = 5 and destination located at the origin
with a desired orientation θ∗= 0.
ever, the approach is not suitable for nonholonomic sys-
tems, since the feedback law generated from the gradient
of the navigation function can lead to undesirable be-
haviors, which may be overcome by extending the orig-
inal navigation function to a Dipolar Navigation Func-
tion in [23] and [24]. The ﬂow lines created in the poten-
tial ﬁeld resemble a dipole, so that the ﬂow lines are all
tangent to the desired orientation at the origin and uti-
lized by the vehicle to achieve the desired orientation. An
example of the dipolar navigation is shown in Fig. 3.1,
where the potential ﬁeld has a unique minimum at the
destination (i.e., p∗= [0, 0]T and θ∗= 0), and achieves
the maxima at the workspace boundary of Rw = 5. Note
that the surface x = 0 divides the workspace into two
parts and forces all the ﬂow lines to approach the desti-
nation parallel to the y-axis.
Inspired by the work in [23] and [24], the control strat-
egy here is to develop a dipolar navigation function for
the informed robot, which creates a feasible nonholo-
nomic trajectory for the nonholonomic robot and guar-
antees the achievement of the speciﬁed destination with
a desired orientation, while other follower robots aim to
achieve consensus with the informed robot and main-
tain network connectivity by using only local interaction
with neighboring robots. Following this idea, the dipo-
lar navigation function is designed for the informed node
i ∈VL as ϕd
i (t) : F →[0, 1],
ϕd
i =
γd
(γα
d + Hd · βd)1/α ,
(2)
where α ∈R+ is a tuning parameter. The goal func-
tion γd (t) : R2 →R+ in (2) encodes the control objec-
tive of achieving the desired destination, which is spec-
iﬁed by the distance from pi (t) ∈R2 to the destination
p∗∈R2, and is designed as γd = ∥pi (t) −p∗∥2. The
factor Hd (t) ∈R+ in (2) creates a repulsive potential
to align the trajectory of node i at the destination with
the desired orientation. The repulsive potential factor is
designed as
Hd = εnh +

(pi −p∗)T · nd
2
,
(3)
where εnh is a small positive constant, and nd
=
h
cos (θ∗) sin (θ∗)
iT
∈R2. A small disk area with radius
δ1 < R centered at node i is denoted as a collision re-
gion. To prevent a potential collision between node i and
the workspace boundary, the function βd : R2 →[0, 1]
in (2) is designed as
βd =
1
1 + e−2
δ1 log( 1−ǫ
ǫ )(di0−1
2 δ1)
(4)
where 0 < ǫ ≪1 is a positive constant, and di0 ≜
Rw −∥pi∥∈R is the relative distance of node i to the
workspace boundary.
Since γd and βd in (2) are guaranteed to not be zero
simultaneously by Assumption 2, the navigation func-
tion candidate in (2) achieves its minimum of 0 when
γd = 0 and its maximum of 1 when βd = 0. Our previous
work in [22] proves that the original navigation function
with the form of ϕi =
γi
(γα
i +βi)
1/α is a qualiﬁed naviga-
tion function. It is also shown in [12] that the naviga-
tion properties are not aﬀected by the modiﬁcation to a
dipolar navigation with the design of (3), as long as the
workspace is bounded, Hd in (2) can be bounded in the
workspace, and εnh is a small positive constant. As a re-
sult, the decentralized navigation function ϕd
i proposed
in (2) can be proven to be a qualiﬁed navigation function
by following a similar procedure in [12] and [22]. From
the properties of the navigation function, it is known
that almost all initial positions (except for a set of mea-
sure zero points) asymptotically approach the desired
destination.
To achieve consensus with the informed node while en-
suring network connectivity and collision avoidance, a
local interaction rule is designed for each follower node
i ∈VF as ϕf
i (t) : F →[0, 1],
ϕf
i =
γi
(γα
i + βi)1/α ,
(5)
where α ∈R+ is a tuning parameter. The goal function
γi (t) : R2 →R+ in (5) encodes the control objective of
achieving consensus on the position between node i and
neighboring nodes j ∈Ni, which is designed as
γi =
X
j∈Ni
∥pi (t) −pj (t)∥2 .
(6)
4
Assume each node i has a collision region deﬁned as a
small disk with radius δ1 < R, and an escape region
deﬁned as the outer ring of the sensing area centered at
the node with radius r, R−δ2 < r < R, where δ2 ∈R+ is
a predetermined buﬀer distance. Any node j ∈Ni inside
the collision region has the potential to collide with node
i, and each edge formed by node i and j ∈Ni in the
escape region has the potential to break connectivity. To
ensure collision avoidance and network connectivity, the
constraint function βi : R2N →[0, 1] in (5) is designed as
βi =
Y
j∈Ni bijBij,
(7)
by only accounting for nodes within its sensing area.
Particularly, bij (pi, pj) : R2 →[0, 1] in (7) is a continu-
ously diﬀerentiable sigmoid function, designed as
bij =
1
1 + e−2
δ2 log( 1−ǫ
ǫ )(R−1
2 δ2−dij) ,
(8)
where 0 < ǫ ≪1 is a positive constant. The designed
bij ensures connectivity of nodes i and its neighboring
nodes j ∈Ni (i.e., nodes j ∈Ni will never leave the
sensing and communication zone of node i if node j is
initially connected to node i).
Since collision avoidance among robots are only required
in Ωc, Bij(pi, pj) : R2 →[0, 1] in (7) is designed as
Bij =
1
1 + e−2
δ1 log( 1−ǫ
ǫ )(dij−1
2 δ1)
(9)
which indicates that collision avoidance is activated if
the robots are in Ωc, i.e., node i is repulsed from other
nodes to prevent a collision in Ωc. If the robots are in Ωr,
the collision avoidance is deactivated by removing Bij
from βi in (7). Since Ωr is deﬁned by the distance to the
destination and only the leader in the group is informed
about the destination, the collision avoidance scheme
designed in (9) is deactivated only when the leader is
close enough to the destination in Ωr 2 .
That is βi in (5) for ∀i ∈VF switches from βi =
Q
j∈Ni bijBij to βi = Q
j∈Ni bij if the leader is close
enough to the destination in Ωr, so that collision avoid-
ance among robots is not considered any more. The bij
and Bij are illustrated in Fig. 2. Note that the constraint
function in (7) is designed to vanish whenever node i in-
tersects with one of the constraints in the environment,
2 The network will be proven to be connected all the time
in the subsequent analysis, which implies that the distance
between any two nodes is bounded by R (N −1). To ensure
all followers are in Ωr when the collision avoidance scheme is
deactivated, the leader is required to deactivate the collision
avoidance when its distance to the destination is less than
Rr −R (N −1).
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
dij
bij
(a) Network Connectivity
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
dij
Bij
(b) Collision Avoidance
Figure 2. The plot of bij and Bij with δ1 = 0.5, δ2 = 1,
R = 2, and ǫ = 0.01.
(i.e., if node i touches another node in Ωc, or separates
from adjacent nodes j ∈Ni by distance of Rc). Since γi
and βi in (5) will not be zero simultaneously from their
deﬁnitions, it is clear that ϕf
i achieves its minimum of 0
if γi = 0 (i.e., consensus is reached between node i and
its immediate neighbors), and ϕf
i achieves its maximum
of 1 if βi = 0 (i.e., either the network connectivity or
collision constraint is met).
3.2
Control Development
For brevity, ϕi is used to represent the potential function
designed for each node i, where particularly ϕi = ϕd
i in
(2) if i ∈VL, and ϕi = ϕf
i in (5) if i ∈VF . The de-
sired orientation for any robot i ∈V, denoted by θdi (t) ,
is deﬁned as a function of the negative gradient of the
decentralized function ϕi as,
θdi ≜arctan 2

−∂ϕi
∂yi , −∂ϕi
∂xi

,
(10)
where the mapping arctan 2 (·) : R2 →R denotes the
four quadrant inverse tangent function, and θdi (t) is
conﬁned to the region of (−π, π]. By deﬁning θdi |p∗=
arctan2 (0, 0) = θi |p∗, θdi remains continuous along any
approaching direction to the goal position. Based on the
deﬁnition of θdi in (10)
∇iϕi = −∥∇iϕi∥
h
cos (θdi) sin (θdi)
iT
,
(11)
where ∇iϕi =
h
∂ϕi
∂xi
∂ϕi
∂yi
iT
denotes the partial deriva-
tive of ϕi with respect to pi, and ∥∇iϕi∥denotes the
Euclidean norm of ∇iϕi. The diﬀerence between the cur-
rent orientation and the desired orientation for robot i
at each time instant is deﬁned as
˜θi (t) = θi (t) −θdi (t) ,
(12)
5
where θdi (t) is generated from the decentralized naviga-
tion function ϕi and (10). Based on the open-loop sys-
tem in (1), the controller for each robot (i.e., the linear
and angular velocity of robot i) is designed as
vi = kv,i ∥∇iϕi∥cos ˜θi,
(13)
ωi = −kw,i˜θi + ˙θdi,
(14)
where kv,i, kw,i∈R+ denote the control gains for robot
i. The term ˙θdi in (14) is determined as
˙θdi = kv,i cos(˜θi)
"
sin (θdi)
−cos(θdi)
#T
∇2
i ϕi
"
cos (θi)
sin (θi)
#
,
(15)
where ∇2
i ϕi denotes the Hessian matrix of ϕi with re-
spect to pi. Note that the computation of ∇iϕi, ˜θi, and
˙θdi only requires local position feedback and does not
depend on communication with any neighbors, which
highlights the decentralized nature of the controllers in
(13) and (14). Although the switch of βi will result in a
discontinuity of the controller in (13) and (14) when the
leader enters Ωr from Ωc, the controller remains contin-
uous within Ωr and Ωc respectively. Substituting (13)
into (1), the closed-loop system for robot i can be ob-
tained as
˙pi =
"
˙xi
˙yi
#
= kv,i ∥∇iϕi∥cos ˜θi
"
cos θi
sin θi
#
.
(16)
After
using
the
fact
that
h
cos θi sin θi
i
∇iϕi
=
−∥∇iϕi∥cos ˜θi from (11), the closed-loop error systems
can be expressed as
˙pi = −kv,i∇iϕi, i ∈V.
(17)
4
Connectivity and Convergence Analysis
4.1
Connectivity Analysis
Theorem 1 The controller in (13) and (14) ensures
that the initially connected spanning tree structure is pre-
served when performing rendezvous for nodes with kine-
matics given by (1), as well as collision avoidance among
robots in Ωc.
PROOF. The spanning tree structure in Assumption 1
ensures that there exists a path from the informed node
to every follower node in G (0). To show every existing
edge in the directed spanning tree in G (0) is preserved,
consider a follower i ∈VF located at a position p0 ∈
F that causes βi = 0, which will be true when either
only one node j is about to disconnect from node i or
when multiple nodes are about to disconnect with node
i simultaneously. If βi = 0, the navigation function ϕi
designed in (5) will achieve its maximum value. Driven
by the negative gradient of ϕi in (17), no open set of
initial conditions can be attracted to the maxima of the
navigation function [14]. Therefore, every edge in G is
maintained and the directed spanning tree structure is
preserved for all time.
Similar to the proof of the preservation of each link, if
two nodes i and j are about to collide in Ωc, that is
Bij(pi, pj) = 0 from (9), then the potential function ϕi
in (5) will reach its maximum. Based on the properties
of a navigation function driven by the vector ﬁeld in
(17), the system will not achieve its maximum. Hence,
collision among nodes is avoided.
✷
4.2
Convergence Analysis
Lemma 1
[25, 26] Let G be a directed graph of or-
der n and L ∈Rn×n be the associated (non-symmetric)
Laplacian matrix. Consider a linear system ˙x (t) =
−L (t) x (t) , where x (t) = [x1, . . . , xn]T ∈Rn. If the
time-varying matrix L (t) ∈Rn×n is a piecewise contin-
uous function of time with bounded elements, and G has
a directed spanning tree for all t ≥0, then consensus is
exponentially achieved, i.e., x1 = · · · = xn.
Theorem 2 Provided that G has a spanning tree with
the informed node as the root, the controller in (13) and
(14) ensures that all robots with kinematics given by (1)
converge to a common point with a desired orientation,
in the sense that ∥pi (t) −p∗∥→0 and
˜θi (t)
 →0 as
t →∞∀i ∈V .
PROOF. For the follower robots i ∈VF , the term ∇iϕi
in (17) is computed from (5) as
∇iϕi = αβi∇iγi −γi∇iβi
α(γα
i + βi)
1
α +1 ,
(18)
where ∇iγi and ∇iβi are bounded in the workspace F
from (6) and (7), and ∇iγi and ∇iβi in (18) can be
determined as
∇iγi = 2
X
j∈Ni (pi −pj) ,
(19)
and
∇iβi = 2
X
j∈Ni
 ∂bij
∂dij

¯bij
∥pi−pj∥(pi −pj) ,
(20)
6
respectively, where ¯bij ≜Q
l∈Ni,l̸=j bil. In (20), ∂bij
∂dij is
∂bij
∂dij
= −
2
δ2 log( 1−ǫ
ǫ )e
−2
δ2
log( 1−ǫ
ǫ )(R−1
2 δ2−dij)

1+e
−2
δ2
log( 1−ǫ
ǫ )(R−1
2 δ2−dij)2
,
(21)
which
is
negative,
since
δ2,
2
δ2 log
  1−ǫ
ǫ

,
and
e−2
δ2 log( 1−ǫ
ǫ )(R−1
2 δ2−dij) are all positive terms. Substi-
tuting (19) and (20) into (18), ∇iϕi is rewritten as
∇iϕi =
X
j∈Ni mij (pi −pj) ,
(22)
where
mij =
2αβi −2

∂bij
∂dij

¯bij
∥pi−pj∥γi
α(γα
i + βi)
1
α +1
(23)
is non-negative, based on the deﬁnitions of γi, βi, α, ¯bij,
and ∂bij
∂dij in (21). Using (17) and (22) yields the closed-
loop system for each node i as:



˙pi(t) = −kv,i∇iϕd
i ,
i ∈VL
˙pi(t) = −P
j∈Ni
kv,imij (pi −pj) , i ∈VF ,
(24)
which can be rewritten in a compact form as
˙p (t) = −(π (t) ⊗I2) p (t) + Fd,
(25)
where p (t) =
h
pT
1 , · · · , pT
N
iT
∈R2N denotes the
stacked vector of pi, Fd =
h
−kv,i∇T
i ϕd
i , 0, · · · , 0
iT
∈
R2N for i = 1 (recall that VL = {1} from Section 2), I2 is
a 2×2 identity matrix, and the elements of π (t) ∈RN×N
are deﬁned as
πik (t) =







P
j∈Ni kv,imij,
i = k
−kv,imik,
k ∈Ni, i ̸= k
0,
k /∈Ni, i ̸= k.
(26)
Using the fact that mij is non-negative from (23), and
kv,i is a positive constant gain in (13), the oﬀ-diagonal
elements of π (t) are negative or zero, and its row sums
are zero. Hence, π (t) is a Laplacian matrix. Since the in-
formed node acts as the root in the spanning tree struc-
ture in G, the ﬁrst row of π (t) is composed of all ze-
ros, which indicates that the motion of the informed
node is not dependent upon the motion of the followers.
From Lemma 1 and the properties of the dipolar nav-
igation function in (2), the ﬁrst term in (25) indicates
that p1 = · · · = pN, and the second term implies that
p1 →p∗, and hence, pi →p∗∀i ∈V.
Note that the properties of the developed dipolar nav-
igation function in (2) ensure that the informed node
achieves the speciﬁed destination with the desired ori-
entation. If the informed node always tracks its desired
orientation θdi and all the followers move along with the
informed node, the group will achieve the destination
with desired orientation. To show that
˜θi
 →0 , we
take the time derivative of ˜θi (t) in (12) and use (1) to
develop the open-loop orientation tracking error system
as
·
˜θi = ωi −˙θdi. Using (14), the closed-loop orientation
tracking error is
·
˜θi = −kw˜θi,
(27)
which has the exponentially decaying solution ˜θi (t) =
˜θi (0) e−kwt.
✷
Remark 1 Since the network is proven to be connected
in Theorem 1, the distance between any two nodes is
bounded by R (N −1), where R is the sensing radius and
N is the number of nodes. By requiring the informed robot
to move within a region with a distance to the workspace
boundary greater than R (N −1), the followers are guar-
anteed to avoid collision with the workspace boundary by
achieving consensus with the informed robot.
Remark 2 The previous analysis is based on the simpli-
ﬁcation that only one informed node is considered. The
result can be generalized to multiple informed nodes by
using containment control theory. Containment control
is a particular class of consensus problems in which all
nodes are grouped into followers and leaders, and the fol-
lowers, under the inﬂuence of leaders through local in-
formation exchange, converge to a desired region (i.e., a
convex hull) formed by the leaders’ states. Some recent
results are reported in [27–29] for containment control. In
our recent work in [30], a decentralized method is devel-
oped to inﬂuence followers in a social network to reach a
common desired state (i.e., within a convex hull spanned
by the leaders), while maintaining interaction among the
followers and leaders. As a special case of [30], if each
leader is assigned the same destination, the convex hull
formed by leaders will shrink to the common destination,
and the followers will converge to this desired destina-
tion. Therefore, following a similar approach in [30], all
nodes can be proven to converge to the common destina-
tion, if multiple informed nodes are considered.
Remark 3 The switch of the controller (13) and (14)
from Ωc to Ωr will not aﬀect the stability of the system.
Theorem 3.2 in [31] states that a switched nonlinear sys-
tem is stable if the associated Lyapunov-like function Vi
in each region Ωi is nonincreasing, and Vi is also nonin-
creasing when switching occurs. It is proven that PN
i ϕi
is a qualiﬁed Lyapunov function in [22], and following a
similar approach as [22], PN
i ϕiis nonincreasing in Ωc
and Ωr, respectively. To show that the Lyapunov func-
tion PN
i ϕi is nonincreasing when switching occurs, note
that the denominator of ϕf
i in (5) is nondecreasing when
7
switching from Ωc to Ωr due to the fact that Bij ∈[0, 1],
which results in a nonincreasing ϕf
i . By invoking Theo-
rem 3.2 in [31], the system remains stable when the switch
occurs from Ωc to Ωr.
5
Simulation
The following numerical simulation demonstrates the
performance of the controller developed in (13) and (14)
in a scenario in which a group of six mobile robots with
the kinematics in (1) are navigated to the common desti-
nation p∗=
h
0 0
iT
with the desired orientation θ∗= 0.
The limited communication and sensing zone for each
robot is assumed as R = 2 m and δ1 = δ2 = 0.4 m The
tuning parameter α in (2) is selected to be α = 1.2. The
workspace F is a disk area centered at the origin with
radius Rw = 50, and the rendezvous region is a disk area
with radius Rr = 11.5. The collison avoidance scheme of
the group is deactivated only when the informed robot
is within a distance less than 1.5 to the common des-
tination. The group of mobile robots is arbitrarily de-
ployed in the workspace and forms a connected graph.
The interaction between nodes is described by the net-
work topology with a spanning tree as shown in Fig. 5,
where the light dots denote the follower robots and the
dark dot denotes the informed node. The informed node
is randomly selected from the group, and is the only node
aware of the desired destination p∗and orientation θ∗.
The control laws in (13) and (14) yield the simulation re-
sults shown in Fig. 5 and Fig. 5. Fig. 5 shows the trajec-
tory for each robot, where the associated arrows indicate
the initial or ﬁnal orientation. To demonstrate the colli-
sion avoidance among robots and connectivity of existing
links, the evolution of the inter-robot distance is plotted
in Fig. 5. As shown in Fig. 5, the inter-robot distance de-
creases signiﬁcantly for the ﬁrst few seconds. Since the
robots are moving in the collision free region initially,
where collision avoidance is activated, the inter-robot
distance stops to decrease when two robots are close to
each other. Once the robots enter the rendezvous re-
gion, where collision avoidance is deactivated, the inter-
robot distance decreases again to perform the desired
rendezvous. Note that inter-robot distance is maintained
less than the radius R = 2 m, which indicates that con-
nectivity of the underlying graph is preserved.
6
Conclusion
A decentralized dipolar navigation function-based time-
varying controller is developed to navigate a network of
mobile robots to a common destination with a desired
orientation while ensuring network connectivity and col-
lision avoidance, using only local sensing information
Figure 3. Network topology with directed edges indicating
the directed interaction between two nodes.
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x−axis (m)
y−axis (m)
 
 
Trajectory of IR
Trajectory of FR
Figure 4. Plot of robot trajectories with solid line and dot–
dash line indicating the trajectory of the informed robot (IR)
and the follower robot (FR), respectively.
0
5
10
15
20
25
30
35
40
45
50
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
time (sec)
inter−robot distance (m)
Figure 5. The evolution of inter-robot distance.
from one-hop neighbors. A distinguishing feature of the
developed decentralized approach is that no inter-agent
communication is required to complete the network con-
sensus objective. Another distinguishing feature is that
the more general problem of directed networks is con-
sidered, where only one robot is informed of the global
objective while other robots coordinate their motions
to perform the cooperative task by using local informa-
8
tion feedback from immediate neighbors. Although con-
sensus among robots is ensured in this work, the rate
of convergence is not considered. Generally, the rate of
convergence depends on the network topology, which is
a function of the roles of nodes (i.e., informed nodes or
followers) and their interactions. A diﬀerent set of in-
formed nodes may lead to diﬀerent convergence rates.
Additional methods could be developed to optimize per-
formance metrics such as the degree of connectivity and
the convergence rate of the network in scenarios where
the set of informed nodes can be determined and/or po-
sitioned a prior.
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9