Himani Sinhmar, Hadas Kress-Gazit 11institutetext: Cornell University, Ithaca
NY 14853, USA,
11email<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>
# Multi-Source Encapsulation With Guaranteed Convergence Using Minimalist
Robots

Himani Sinhmar  Hadas Kress-Gazit

###### Abstract

We present a decentralized control algorithm for a minimalist robotic swarm
lacking memory, explicit communication, or relative position information, to
encapsulate multiple diffusive target sources in a bounded environment. The
state-of-the-art approaches generally require either local communication or
relative localization to provide guarantees of convergence and safety. We
quantify trade-offs between task, control, and robot parameters for guaranteed
safe convergence to all the sources. Furthermore, our algorithm is robust to
occlusions and noise in the sensor measurements as we demonstrate in
simulation.

###### keywords:

Minimalist robots, Swarms, Multi-Source encapsulation, Obstacle avoidance,
Guaranteed convergence

##  1 Introduction

Signal source detection, localization, and encapsulation play a critical role
across biological systems, spanning from chemotaxis in microscopic organisms
[1] to predators’ olfactory tracking [2]. In robotics, researchers develop
sensor equipped robots for identifying signal sources for applications such as
nanomedicine [3], chemical hazard detection [4], nuclear radiation tracing
[5], and environmental monitoring [6]. Extensive research exists for source
localization using either individual agents or a network of agents. A common
approach involves gradient computation, either directly through sensors
capable of measuring gradients [7] or by robots that deduce the gradient by
sequentially measuring the field’s concentration and making incremental
movements. Using multiple robots as a network can enhance accuracy and
efficiency by sharing information for collective gradient estimation at each
time-step [8], requiring less memory and computational power. However, this
method depends on inter-robot communication and the ability to discern their
neighbors’ signal direction.

In this paper, we consider multi-source encapsulation using a swarm of
minimalist robots (no memory, no self-localization, no direct communication,
and inability to determine neighbors’ relative locations) equipped only with
omnidirectional sensors [9] to detect the concentration of environmental
signals. Here, each robot independently estimates signal gradients at each
time step without requiring information from its neighbors or memory for
storing past concentration values. This method is particularly useful in
contexts where traditional navigation and communication technologies are
unfeasible, such as nano-medicine for tumor encapsulation [3] or in underwater
environments [10] where restricted communication bandwidth challenges the
coordination of autonomous vehicles. Previously [9, 11], we established the
mathematical foundation for a swarm of minimalist robots to encapsulate
multiple targets with non-overlapping influence regions. In this paper, we
tackle multi-source encapsulation with overlapping influence regions. A target
is encapsulated if a user-specified number of robots are present
simultaneously in a user-specified encapsulation ring around it.

In contrast to source localization where the robots can follow a gradient,
encapsulation is more complex since it also involves accurately determining
the encirclement contour for each target. By treating each target as a point
source with no encapsulation ring, the multi-source encapsulation problem
boils down to the localization of all peaks of a spatiotemporal multi-modal
function.

Related Work: While the literature covers the detection of a single source by
its gradient extensively [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26], the simultaneous detection of multiple overlapping sources is not as
well explored [27]. Most existing source-seeking strategies either require
communication between the agents and/or relative position information of an
agent’s neighbors. In [28, 29, 30], the authors introduced a glowworm swarm
optimization algorithm (GSO) designed for multiple source localization. Each
robot can determine its position relative to others and avoid collisions. They
share signal information to select a leader robot, which guides the others.
The GSO algorithm cannot guarantee the complete localization of all sources
and its effectiveness depends on the initial distribution of the swarm. [31]
introduces a fish-school-inspired [32] distributed approach for single source
seeking using mobile sensor networks without the need for agents to estimate
gradients. However, for groups larger than two agents, communication among the
agents becomes necessary. In [33], authors developed an algorithm for a
robotic swarm to encapsulate diffusive targets among convex obstacles by using
relative position information to build an artificial gene network for
coordinated movement toward targets and used simulations for verification of
the algorithm. In [34], authors utilize communication between a mobile sensor
and uniformly distributed localized static sensor nodes to perform gradient
descent for contour following. [35] introduces a reactive control policy for
holonomic robot clusters for tasks like extrema finding, contour following,
ridge/trench tracking, and saddle point keeping. This strategy relies on the
cluster maintaining a fixed geometry optimized for scalar field navigation
effectively turning the cluster into a larger entity with distributed sensors
along its perimeter.

There is less work that provides guarantees of convergence for source
encapsulation using minimalist robots. One of the earliest efforts to
establish a theoretical basis for source localization is in [36, 37, 38, 39]
using projected gradients and artificial potential functions to maintain
formation, hence requiring all-to-all communication. In [40], the stability of
a system, wherein each robot estimates a second-order representation of an
unknown scalar field through a communication network, is validated using
Lyapunov analysis. In contrast, we do not estimate the scalar field, leading
to a control policy that requires less computational effort. There also exists
work on using maximum likelihood localization [41], optimization [42, 43, 44,
45, 46, 47, 48, 49, 50], and machine learning techniques [51, 52, 53, 50] for
multiple source localization. These approaches generally diverge from the use
of minimalist robots, which is central to our work.

Contribution: The novelty of this paper is threefold: (i) we introduce a
scalable, decentralized control law for a minimalist robotic swarm (each robot
is devoid of memory, self-localization, and explicit communication ability)
that can encapsulate multiple signal sources (targets) without the need for
accurate detection of relative positions of the targets, neighboring robots,
or obstacles; (ii) we present a derivative-free algorithm based on the simplex
gradient to identify multiple optima of a multimodal scalar spatiotemporal
field in an obstacle-cluttered environment using distributed sensing; (iii) we
offer theoretical guarantees for convergence and provide bounds on task,
control, and robot parameters to ensure all targets are encapsulated safely.
Furthermore, we analyze the robustness of the control algorithm to sensor
noise, occlusions, and asynchronous execution.

##  2 Environment and Robot Model

We adopt the environment, robot, and target models from [11], adding convex
obstacles in this work. We consider a $2D$ bounded environment
$E\subseteq\mathbb{R}^{2}$ with a fixed global frame $\mathcal{I}$. A target
$g=(\mathbf{c}_{g},r_{g})$ is a disk of radius $r_{g}\geq 0$ centered at
$\mathbf{c}_{g}\in\mathcal{I}$. The set $O=\\{O_{1}\cdots O_{w}\\}$ represents
closed convex obstacles in the environment. A robot,
$R=(\mathbf{c}_{r},\gamma_{r},r_{r},p,Z)$, is a disk of radius $r_{r}>0$
centered at $\mathbf{c}_{r}\in\mathcal{I}$ with heading
$\gamma_{r}\in\mathbb{S}$. The following set of equations gives the
differential drive kinematics of a robot:
$\gamma_{r,T}=\gamma_{r,T-1}+\theta_{r}$ and
$\textbf{c}_{r,T}=\textbf{c}_{r,T-1}+d_{r}[\mathrm{cos}\gamma_{r,T}\quad\mathrm{sin}\gamma_{r,T}]$.
At each time step T, the robot is controlled in a turn-then-move scheme with
control inputs $\theta_{r}\in\mathbb{S}$ and $d_{r}\in\mathbb{R}^{+}$ with the
step-size upper bounded as $d_{r}\leq d_{r}^{\mathrm{max}}$. Robots are
reactive, memoryless, with no knowledge of others’ locations or communication
ability, and equipped with $p$ isotropic sensors around their periphery.
Angles $\phi^{k}$ for $k\in\\{1\cdots p\\}$ denote each sensor’s orientation
relative to the robot’s heading. Sensor measurements are aggregated in the set
$Z$. The robot’s frame, $\mathcal{B}$, is centered at $\textbf{c}_{r}$, with
its $x-$axis always aligned with the heading direction $\gamma_{r}$.

We consider four types of signal-emitting sources detectable by a robot’s
sensor: point sources located at the center of each target ($s_{g}$), each
robot ($s_{r}$), each obstacle ($s_{o}$), and a line source spanning the
boundary of the environment ($s_{e}$). For simplicity, we refer to these
signal types as $\\{g,r,o,e\\}$ with a signal strength function
$f_{s}:\mathbb{R}^{2}\rightarrow\mathbb{R}$. The influence radius of a source
is limited to $\beta_{s}$, such that at a point $\textbf{x}\in\mathcal{I}$ and
a source located at $\textbf{S}\in\mathcal{I}$, $f_{s}(\textbf{x})=0\textrm{
}\mathrm{if}\left\lVert\textbf{x}-\textbf{S}\right\rVert\geq\beta_{s}$. For
brevity, we define $d=\left\lVert\textbf{x}-\textbf{S}\right\rVert$. For the
$k^{\text{th}}$ sensor, $N^{k}_{s}$ denotes the set of sources of type $s$
within its sensing range, with $d_{j}^{k}$ representing the distance to source
$j\in N^{k}_{s}$. The sensor’s reading, $z_{s}^{k}$, is the sum of signal
strengths from all $N^{k}_{s}$ sources, $z_{s}^{k}=\sum_{j\in
N^{k}_{s}}f_{s}(d^{k}_{j})$. For a line source within $\beta_{e}$, this sum
transforms into an integral along the boundary segment. The tuple
$(z_{g}^{k},z_{r}^{k},z_{o}^{k},z_{e}^{k})$ corresponds to the measurements of
the $k^{th}$ sensor and $Z=\cup_{k}(z_{g}^{k},z_{r}^{k},z_{o}^{k},z_{e}^{k})$.
We define user-specified safe distances $r^{\text{safe}}_{s}\textrm{
},s\in\\{g,r,o,e\\}$ as the minimum distance a robot is required to keep from
a source. An encapsulation ring is an annular region with an inner radius
$r^{\mathrm{safe}}_{g}$ and outer radius $r^{\mathrm{encap}}_{g}$ centered at
$\mathbf{c}_{g}$. A robot is considered to be in the encapsulation ring if,
$r^{\mathrm{safe}}_{g}<\left\lVert\textbf{c}_{r,T}-\textbf{c}_{g}\right\rVert\leq
r^{\mathrm{encap}}_{g}$. A target is considered encapsulated when $n_{g}$
robots, a user-specified number, occupy its encapsulation ring concurrently.

##  3 Problem Formulation

We consider $n$ robots each equipped with $p$ sensors, $m$ static targets, and
$w$ static obstacles in a bounded environment $E$ with a random initial
distribution and no information about their location and the locations of
targets, obstacles, and other robots. Given safe distances
$r^{\mathrm{safe}}_{s},s\in\\{g,r,o,e\\}$, and the encapsulation ring
parameters ($r^{\mathrm{encap}}_{g}$, $n_{g}$) such that
$\sum_{g\in\mathcal{G}}n_{g}\leq n$ we find a real-time decentralized control
law for each robot such that all targets are eventually encapsulated while
maintaining minimum safety distances at all times. We make the following
assumptions:

###### Assumption 1

The sensor placement is such that when a robot’s center is
$r_{s}^{\mathrm{safe}}$ away from a source $s$, at least one sensor is in the
influence region of the source.

###### Assumption 2

We constrain a target to maintain a minimum distance of
($r_{g}^{\mathrm{encap}}+r_{e}^{\mathrm{safe}}+d_{r}^{\mathrm{max}}$) from the
environment boundary. This ensures that robots can encapsulate the target
without colliding with the environment boundary.

###### Assumption 3

On encapsulation, a target emits a single burst of a shut-off signal and stops
emitting any signal after that. Robots within the encapsulation ring stop
moving ($d_{r}=0$) and start emitting an obstacle-type signal $s_{o}$.

###### Assumption 4

The function $f_{s}(d)$ is convex and decreases monotonically as the distance
$d$ from a source increases. It is continuously differentiable, with its
gradient being L-Lipschitz continuous. For a point $P$ at position
$\textbf{x}\in\mathcal{I}$ influenced by $m$ sources of type $s$ at positions
$\textbf{S}_{1},\textbf{S}_{2},\ldots,\textbf{S}_{m}\in\mathcal{I}$, the
aggregate signal strength at $P$ is defined by
$F_{s}(\textbf{x})=\sum_{i=1}^{m}f_{s}(\left\lVert\textbf{x}-\textbf{S}_{i}\right\rVert)$.
The function $F_{s}$ (sum of convex functions) adheres to the continuity and
differentiable properties of its constituent functions [54, Chapter 3].
Furthermore, the robots know the function $f_{s}(d)$ and its inverse $\forall
s\in\\{g,r,o,e\\}$. However, it does not know the aggregate signal function
$F_{s}$, hence it lacks comprehensive information about the combined effects
of multiple sources.

###### Assumption 5

There is no static obstacle in the encapsulation ring of a target.

###  3.1 Mathematical Preliminaries

To direct a robot toward the closest target without knowing the number or
positions of targets, we calculate gradients using its sensor readings. The
simplex gradient method approximates gradients where direct derivative
calculation is unfeasible, by constructing an affine approximation of the
function near a point of interest. It selects points within the function’s
domain to form a simplex, and the gradient is derived from the original
function’s evaluations at these points.

######  Lemma 3.1.

Simplex Gradient:[55] Let $F:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and
$\mathcal{X}=\\{\textbf{x}_{0},\textbf{x}_{1}\cdots\textbf{x}_{k}\\}$ be an
ordered set of $k+1$ affine independent points in $\mathbb{R}^{n}$ such that
$k\geq n$. Define,
$H(\mathcal{X})=[\textbf{x}_{1}-\textbf{x}_{0}\cdots\textbf{x}_{k}-\textbf{x}_{0}]\in\mathbb{R}^{n\times
k}$ and $\delta_{F}(\mathcal{X})=[F(\textbf{x}_{1})-F(\textbf{x}_{0})\cdots
F(\textbf{x}_{k})-F(\textbf{x}_{0})]^{T}\in\mathbb{R}^{k}$. Then for $k=n$,
$H(\mathcal{X})$ is invertible and the simplex gradient of $F$ at
$\textbf{x}_{0}$ over $H(\mathcal{X})$ is given by
$\nabla_{\mathrm{\mathrm{SG}}}F(\textbf{x}_{0};H)=H^{-\mathrm{T}}\delta_{F}(\mathcal{X})$.

######  Lemma 3.2.

Error bound for Simplex Gradient:[55] Let $F$ be a continuously differentiable
function in an open domain $\Omega$ containing the closed ball
$B(\textbf{x}_{0},\Delta_{H})=\\{\textbf{x}\mid\left\lVert\textbf{x}-\textbf{x}_{0}\right\rVert\leq\Delta_{H}\\}$,
where $\Delta_{H}=\mathrm{max}_{1\leq i\leq
k}\left\lVert\textbf{x}_{i}-\textbf{x}_{0}\right\rVert$. Further, assume that
$\nabla F$ is Lipschitz continuous in $\Omega$ with Lipschitz constant $L\geq
0$ and $\hat{H}(\mathcal{X})^{\mathrm{T}}=H(\mathcal{X})^{\mathrm{T}}/\Delta$.
Then, $\left\lVert\nabla_{\mathrm{\mathrm{SG}}}F(\textbf{x}_{0})-\nabla
F(\textbf{x}_{0})\right\rVert\leq k^{1/2}\frac{L}{2}\Delta_{H}$.

##  4 Approach

In our prior work [9, 11], we considered targets with non-overlapping
influence regions, i.e. a robot can only sense one target at a time. A robot
then estimates the position of the sensed target using the sensor receiving
the strongest signal. When considering overlapping influence regions, two
challenges emerge due to aggregated signals: first, the sensor with the
strongest signal no longer reliably indicates the direction to the closest
target; second, it is challenging to compute the distance to a target. In this
paper, we develop a control policy based on fusing distributed information
from a robot’s sensors to encapsulate all targets while maintaining a safety
distance from obstacles, other robots, and targets. In Section 4.1, we derive
bounds on $d_{r}$ and $\beta_{r}$ so that a robot maintains user-specified
safety distances. We relax the upper bound, with respect to [11], on the step
size a robot can take. In Section 4.2, we utilize simplex gradients to
estimate a robot’s heading for motion toward the closest target. In Section
4.3, we present the reactive control policy for a robot to encapsulate a
target.

###  4.1 Maintaining safety distances

Given a robot is equipped with isotropic sensors, the same measurement could
correspond to a single source nearby or a cluster of sources further away.
Therefore, for each sensor reading, $z_{s}^{k},s\in\\{r,o,e\\}$, we define a
virtual source on a circle centered at the sensor $k$ such that the closest
possible location of the virtual source with respect to the robot’s center is
given by,

$d_{s}=r_{r}\mathrm{cos}(\pi/p)+\sqrt{(d_{s}^{k})^{2}-r_{r}^{2}\mathrm{sin}^{2}(\pi/p)}$
(1)

Using Assumption 4, we can say that there cannot be an actual source closer to
the robot than the virtual source [9]. For asymmetric sensor placement,
$\pi/p$ is substituted with half the largest angle formed by the
$k^{\text{th}}$ sensor and either of its neighboring sensors. Likewise, for
robots with non-circular geometries, $r_{r}$ is adjusted to reflect the
distance from the $k^{\text{th}}$ sensor to the robot’s center.

At each time step, the robot estimates the relative distance between its
center and the nearby sources using Eq. (1) for the sensor with the maximum
sensor reading $z_{s}^{k}=\mathrm{max}(Z_{s})$. If this distance is less than
or equal to $(r_{s}^{\mathrm{safe}}+d_{s}^{\mathrm{max}})$, where
$d_{s}^{\mathrm{max}}$ is the maximum step-size of the source $s$, a collision
avoidance behavior is triggered.

Figure 1: Computing $d_{r}$ that ensures a safe distance from nearby obstacles

The step-size $d_{r}$ that a robot takes at time step $T$ in a given heading
direction $\gamma_{r,T+1}$ must be chosen such that, at next time step $T+1$,
the robot maintains at least a distance of $r_{s}^{\mathrm{safe}}$ from the
closest virtual source of type $s$. As shown in Fig. 1, let $k$ and $l$ be the
indices of the sensors closest to the intended heading direction
$\gamma_{r,T+1}$ with $d_{s}^{k}$ and $d_{s}^{\ell}$ being the radii of
virtual sources respectively such that $d_{s}^{k}>d_{s}^{\ell}$. If we
consider only the sensor reading from the $k^{\mathrm{th}}$ sensor, the
closest virtual source to the robot is located at
$\textbf{S}_{1}\in\mathcal{I}$ as depicted in Fig. 1. Then the step-size
$d_{r}$ that the robot can take should be such that
$\left\lVert\textbf{c}_{r,T+1}-\textbf{S}_{1}\right\rVert\geq
r_{s}^{\mathrm{safe}}+d_{s}^{\mathrm{max}}$ as given by Eq. (2).

$0\leq d_{r}\leq
r_{r}\mathrm{cos}(\phi^{k}-\theta_{r})+\sqrt{(d_{r}^{k}-r_{r}^{\mathrm{safe}}-d_{s}^{\mathrm{max}})^{2}-r_{r}^{2}\mathrm{sin}^{2}(\phi^{k}-\theta_{r})}$
(2)

Sources that lie outside the sensing radius of the $k^{\mathrm{th}}$ sensor
(shaded green region in Fig. 1) may create a virtual source that is closer
than the one described above. Therefore, the closest virtual source could also
be at the intersection of the virtual source circle of the
$\ell^{\mathrm{th}}$ sensor and the sensing circle of the $k^{\mathrm{th}}$
sensor at $\textbf{S}_{2}\in\mathcal{I}$. The robot should maintain a safe
distance from $\textbf{S}_{2}$ as well, i.e.
$\left\lVert\textbf{c}_{r,T+1}-\textbf{S}_{2}\right\rVert\geq
r_{s}^{\mathrm{safe}}+d_{s}^{\mathrm{max}}$. Consider the triangle
$\triangle\textbf{S}_{2}\textbf{c}_{r,T}\textbf{c}_{r,T+1}$, let
$b_{0}=\left\lVert\textbf{c}_{r,T}-\textbf{S}_{2}\right\rVert$ and
$\psi=\angle\textbf{S}_{2}\textbf{c}_{r,T}\textbf{c}_{r,T+1}$. Then,

$d_{r}\leq
b_{0}\mathrm{cos}(\psi)-\sqrt{(r_{r}^{\mathrm{safe}}+d_{s}^{\mathrm{max}})^{2}-b_{0}^{2}\mathrm{sin}^{2}(\psi)}\\\
\mathrm{OR}\quad
d_{r}\geq\;b_{0}\mathrm{cos}(\psi)+\sqrt{(r_{r}^{\mathrm{safe}}+d_{s}^{\mathrm{max}})^{2}-b_{0}^{2}\mathrm{sin}^{2}(\psi)}$
(3)

Furthermore, to ensure that two robots never deadlock, the bounds on
$d_{r}^{\mathrm{max}}$, and the influence region of a robot’s source
$\beta_{r}$, is given by Eq. (4) and Eq. (5), respectively. The proof is
detailed in Lemma V.3 of [11].

$d_{r}^{\mathrm{max}}<\frac{r_{r}^{\mathrm{safe}}+r_{r}\mathrm{cos}(\pi/p)}{2}-\frac{\sqrt{(r_{r}^{\mathrm{safe}})^{2}+r_{r}^{2}-2r_{r}r_{r}^{\mathrm{safe}}\mathrm{cos}(\pi/p)}}{2}$
(4)
$\sqrt{(r_{r}^{\mathrm{safe}})^{2}+r_{r}^{2}-2r_{r}r_{r}^{\mathrm{safe}}\mathrm{cos}(\pi/p)}+2d_{r}^{\mathrm{max}}<\beta_{r}<r_{r}^{\mathrm{safe}}+r_{r}\mathrm{cos}(\pi/p)$
(5)

Given a heading direction $\gamma_{r,T}$, at each time step we choose the
maximum possible step size $d_{r}$ such that it satisfies the bounds of Eq. 2,
Eq. 3, and Eq. 4 for a source $s\in\\{r,o,e\\}$. To maintain a safe distance
from the environment boundary, the robot’s heading direction $\theta_{r}$ must
be chosen from the angular range
$\Theta_{e}^{\mathrm{avo}}=[\phi^{k}+\pi/p+\pi/2,\textrm{
}\phi^{k}-\pi/p+3\pi/2]$ as derived in [9].

###  4.2 Target attraction

The primary objective of a robot’s control policy is to navigate toward the
closest target by calculating the appropriate heading ($\theta_{r}$) at every
time step and taking a step-size ($d_{r}$) that ensures safety. The collision
avoidance strategy above dictates the step-size. To determine the heading that
ensures navigation towards the closest target, we use simplex gradients. Once
the robot arrives at the encapsulation ring, it switches to an encirclement
behavior. The control policy requires determining the heading direction to
move towards a target, and identifying when the robot has entered an
encapsulation ring.

Direction Towards the Closest Target: We use a simplex gradient approach
(Lemma 3.1) to estimate the Line of Sight (LOS) vector, $\zeta$, to the
closest target. For a given sensor $\ell$ located at
$\textbf{x}^{\ell}\in\mathcal{B}$, let
$H(\mathcal{X})=[\textbf{x}^{\ell-1}-\textbf{x}^{\ell}\quad\textbf{x}^{\ell+1}-\textbf{x}^{\ell}]$
and $\delta{F_{g}}(\mathcal{X})=[z_{g}^{\ell-1}-z_{g}^{\ell}\quad
z_{g}^{\ell+1}-z_{g}^{\ell}]^{T}$, then the simplex gradient at
$\textbf{x}^{\ell}$ is computed as
$\nabla_{\mathrm{SG}}F_{g}(\textbf{x}^{\ell})=H^{-T}\delta{F_{g}}(\mathcal{X})$.
At each time step, a robot chooses the heading direction dictated by the
sensor with the maximum target signal reading.
Detecting the Encapsulation Ring: When targets’ influence regions overlap, a
robot’s sensor reading cannot be used to estimate the distance to a target.
Instead, we leverage a key observation: Assumption 4 suggests that the unique
signature of the aggregate signal ($F_{g}$) at any point
$\textbf{x}\in\mathcal{I}$ can be identified by analyzing the magnitude of the
aggregate signal gradient ($\nabla F_{g}(\textbf{x})$) and the expected
gradient of a single target’s signal ($\nabla f_{g}(\textbf{x})$). For
instance, when two targets lie in opposite directions with respect to a
sensor, the overall signal strength may be high, but the combined signal
gradient’s magnitude will be lower than if the signal came from one target
only.

Let the $k^{\mathrm{th}}$ sensor located at $\textbf{x}^{k}\in\mathcal{B}$ be
such that $z_{g}^{k}=\mathrm{max}(Z_{g})$ and $d_{g}$ be the robot center’s
distance to the closest target. As shown in Fig. 2, we define different zones
that a robot can be in: secondary zone if $d_{g}>r_{g}^{\mathrm{encap}}$,
encapsulation ring as defined in Section 2, encirclement contour
$O_{g}^{\mathrm{encap}}$, safety violation zone if
$d_{g}<r_{g}^{\mathrm{safe}}$, and dead zone if $z_{g}^{k}=0$.

Figure 2: Environment zones.

For encapsulation, a robot does not need to know the exact distance to the
closest target; it only needs to know if it has entered the encapsulation ring
to initiate the encirclement behavior. We use the known encapsulation ring
parameters and function $f_{g}(d)$. The expected gradient magnitude on the
encapsulation ring, assuming the robot’s $k^{\mathrm{th}}$ sensor perceives a
single target (i.e. if $F_{g}=f_{g}$,) is $\left\lVert\nabla
f_{g}(r_{g}^{\mathrm{encap}})\right\rVert$. A discrepancy between
$\left\lVert\nabla_{\mathrm{SG}}F_{g}(\textbf{x}^{k})\right\rVert$ and
$\left\lVert\nabla f_{g}(r_{g}^{\mathrm{encap}})\right\rVert$ indicates that
the robot is not in the encapsulation ring.

Specifically,
$\left\lVert\nabla_{\mathrm{SG}}F_{g}(\textbf{x}^{k})\right\rVert<\left\lVert\nabla
f_{g}(r_{g}^{\mathrm{encap}})\right\rVert$ suggests that the robot is in the
secondary zone, whereas $\left\lVert\nabla
f_{g}(r_{g}^{\mathrm{encap}})\right\rVert\leq\left\lVert\nabla_{\mathrm{SG}}F_{g}(\textbf{x}^{k})\right\rVert<\left\lVert\nabla
f_{g}(r_{g}^{\mathrm{safe}})\right\rVert$ implies the robot is within the
encapsulation ring. For this methodology to be valid, it is necessary that
$F_{g}\approx f_{g}$ within a target’s encapsulation ring. In Section 5, we
derive constraints on target separation to maintain this condition. Note that
a robot can only determine whether it is inside or outside the encapsulation
ring; it can neither asse.ss its distance from the encapsulation ring nor from
the target.

###  4.3 Local Control Law

The reactive control policy for a robot is as follows: It first estimates its
distance to the environment boundary (Eq. (1)), if about to violate the safety
distance $r_{e}^{\mathrm{safe}}$, the heading direction is chosen such that
the robot moves away from the boundary
($\theta_{r}\in\Theta_{e}^{\mathrm{avo}}$) with a maximum possible step size
$d_{r}$ while avoiding nearby robots and obstacles as described in Section
4.1. Once the robot is at a safe distance from the boundary, it computes its
control parameters depending on which zone it is situated in (Fig. 2). When
the robot is present in the dead zone (i.e. $\mathrm{max}(Z_{g})=0$), it
performs a biased random walk, preferring directions dictated by the sensor
receiving minimum signal reading from nearby robots.

Secondary zone (
$\left\lVert\nabla_{\mathrm{SG}}F_{g}(\textbf{x}^{k})\right\rVert<\left\lVert\nabla
f_{g}(r_{g}^{\mathrm{encap}})\right\rVert$): the robot’s heading direction is
$\theta_{r}=\zeta$, where $\zeta$ is the LOS vector to the target. If the
robot cannot move towards the target due to safety constraints, it moves
tangentially to the LOS vector ($\theta_{r}=\zeta+\pi/2$ or
$\theta_{r}=\zeta+3\pi/2$ ) preferring the one that minimizes deviation from
its current heading. The step-size $d_{r}$ is chosen to ensure safety with
respect to other robots (Section 4.1) and obstacles (Section 4.3.1). If the
robot can neither move towards a target nor tangentially, it chooses a
direction of motion that maximizes the possible step size $d_{r}$. This
behavior prevents temporary deadlocks in the swarm due to local minima created
by other dynamic robots between the target and the robot.

Encapsulation ring (Section 4.3.2) : The robot moves tangentially to the LOS
vector $\zeta$ closely following the encirclement contour
$O_{g}^{\mathrm{encap}}$ (Fig. 2) with a step size $d_{r}$ such that it
maintains safe distances from nearby robots.

####  4.3.1 Behavior Close to an Obstacle in the Secondary Zone

(a)

(b)

Figure 3: (a) A robot circumnavigates an obstacle in the tangential direction
closest to the LOS vector, shown with a thicker teal arrow. (b) When targets
are closer to each other than their minimum separation threshold $D_{g}$
(Section 5.3), robots encapsulating them create an open obstacle. Robots at
locations A and B both assess the obstacle as being behind them relative to
the target’s LOS vector $\zeta$, prompting them to advance toward the target.
While this action is appropriate for the robot at B, it results in a livelock
for the robot at A.

We want the robots to avoid obstacles while still moving towards the target.
Thus, an efficient navigation strategy to avoid an obstacle is for a robot to
deviate as little as possible from the target’s LOS vector $\zeta$. To do so,
we encode a “wall-following” behavior whenever a robot encounters an obstacle.
A robot receiving obstacle signals ($Z_{o}\neq 0$) calculates
$\nabla_{\mathrm{\mathrm{SG}}}F_{o}(\textbf{x}^{\ell})$, where
$\textbf{x}^{\ell}$ is the $\ell^{\mathrm{th}}$ sensor’s location in
$\mathcal{B}$ with the highest obstacle signal intensity.

Let $\zeta_{o}$ represent the direction toward the obstacle. To circumnavigate
an obstacle, the robot moves tangentially with either
$\theta_{o}^{\text{tan,-1}}=\zeta_{o}+\pi/2$ or
$\theta_{o}^{\text{tan,+1}}=\zeta_{o}+3\pi/2$ (clockwise or counterclockwise
rotation, respectively). We choose the direction that minimizes the angular
deviation from the target’s LOS vector $\zeta$ (Fig. 3(a)). However, if the
chosen direction deviates by more than $\pi/2$ from the robot’s current
heading, the robot sets $\theta_{r}=0$, i.e. it continues with its current
heading. This is to minimize alternating (avoiding livelocks) between
clockwise ($\Theta_{o}^{\mathrm{tan},-1}$) and counterclockwise
($\Theta_{o}^{\mathrm{tan},+1}$), especially in scenarios where multiple
nearby targets and obstacles influence the robot. The robot executes its
obstacle circumnavigation strategy until the angle between $\zeta_{o}$ and
$\zeta$ exceeds $\pi/2$, indicating that the obstacle is now behind the robot
with respect to the closest target.

Following Assumption 3, robots that have encapsulated a target also emit an
obstacle signal $s_{o}$ and stop moving, essentially becoming stationary
obstacles. This allows a robot to distinguish encapsulating robots from other
dynamic robots nearby. Without this ability, the robot’s lack of memory could
result in livelocks around the “ridge and trench” formations created by
encapsulating robots blocking the line of sight to a target; the robot would
repeatedly advance toward the target, then retreat to avoid the encapsulating
robots.

####  4.3.2 Behavior in the Encapsulation Ring

A robot located at $\textbf{x}_{0}\in\mathcal{I}$ is in the encapsulation ring
of the closest target $g$, if $\left\lVert\nabla
f_{g}(r_{g}^{\mathrm{encap}})\right\rVert<\left\lVert\nabla_{\mathrm{SG}}F_{g}(\textbf{x}_{0})\right\rVert<\left\lVert\nabla
f_{g}(r_{g}^{\mathrm{safe}})\right\rVert$. To ensure that a robot remains in
the encapsulation ring, we define an encircling orbit
$O_{g}^{\mathrm{encap}}=r_{g}^{\mathrm{encap}}-d_{r}^{\mathrm{max}}$ as shown
in Fig. 2. We define the target attraction and avoidance angular ranges as:
$\Theta_{g}^{\mathrm{LOS,-1}}=[\zeta,\textrm{ }\zeta+\pi/2]\textrm{ and
}\Theta_{g}^{\mathrm{LOS,+1}}=[\zeta+3\pi/2,\textrm{ }\zeta]$,
$\Theta_{g}^{\mathrm{avo,-1}}=[\zeta+\pi/2,\textrm{ }\zeta+\pi]\textrm{ and
}\Theta_{g}^{\mathrm{avo,+1}}=[\zeta+\pi,\textrm{ }\zeta+3\pi/2].$ A robot’s
control then consists of the following behavior:
1\. If
$\left\lVert\nabla_{\mathrm{SG}}F_{g}(\textbf{x}^{k})\right\rVert\leq\left\lVert\nabla
f_{g}(O_{g}^{\mathrm{encap}})\right\rVert$, the robot chooses a heading
direction $\theta_{r}$ to move towards the target from the set
$\Theta_{g}^{\mathrm{LOS},-1}$. If it cannot move in any of the directions in
this set (i.e. $d_{r}=0$), it chooses a heading from the set
$\Theta_{g}^{\mathrm{LOS},+1}$.
2\. If
$\left\lVert\nabla_{\mathrm{SG}}F_{g}(\textbf{x}^{k})\right\rVert>\left\lVert\nabla
f_{g}(O_{g}^{\mathrm{encap}})\right\rVert$, the robot chooses the heading
direction to avoid the target from the set $\Theta_{g}^{\mathrm{avo},+1}$. If
it cannot move in any of the directions in this set (i.e. $d_{r}=0$), it
chooses a heading from the set $\Theta_{g}^{\mathrm{avo},-1}$.

The robot first tries to choose a heading in the clockwise orbit rotation
($\Theta_{g}^{\mathrm{LOS,-1}}$) if moving towards the target, and
counterclockwise orbit rotation ($\Theta_{g}^{\mathrm{avo,+1}}$) if moving
away from the target to ensure that robots moving towards and away from the
target avoid oscillatory movements.

##  5 Convergence Guarantees

In this section, we assume noiseless sensors and derive constraints on various
parameters that, if satisfied, guarantee that safety distances are maintained,
the robots are never stuck in a deadlock or livelock, and all targets in the
environment are eventually encapsulated.

###  5.1 Target Attraction

Using the Lyapunov stability theorem [56, 9], we can show that a robot moves
towards the target when the angular deviation ($\vartheta$) of its estimated
LOS vector ($\zeta$) remains within $\pm\pi/2$ of the true LOS vector to the
closest target. We determine constraints on the sensor placement, the number
of sensors $p$, the safety distance $r_{g}^{\mathrm{safe}}$, and the source
function $f_{g}$ that will guarantee motion toward the target.

Let the maximum separation angle between any consecutive sensors on a robot be
$\Phi$, the largest linear distance between them be $\Delta_{H}$ calculated as
$2r_{r}\sin(\Phi/2)$, and the Lipschitz constant of $\nabla F_{g}$ in a ball
of radius $\Delta_{H}$ centered at $\textbf{x}^{k}$ be given by
$L=\mathrm{max}(\left\lVert\nabla^{2}F_{g}(\textbf{x}^{k})\right\rVert)$. Let
the maximum gradient error be $\bm{\epsilon}$, then from Lemma 3.2 we have,
$\left\lVert\nabla_{\mathrm{SG}}F_{g}(\textbf{x}^{k})-\nabla
F_{g}(\textbf{x}^{k})\right\rVert\leq
L\frac{\Delta_{H}}{\sqrt{2}}\coloneq\left\lVert\bm{\epsilon}\right\rVert$. We
can show that the maximum possible angular deviation between the simplex
gradient $\nabla_{\mathrm{SG}}F_{g}$ and true gradient $\nabla F_{g}$ is
$\vartheta=\mathrm{sin}^{-1}\frac{\left\lVert\bm{\epsilon}\right\rVert}{\left\lVert\nabla
F_{g}(\textbf{x}^{k})\right\rVert}$. The sufficient condition to ensure
$\vartheta\in[-\pi/2,\pi/2]$ is
$\left\lVert\bm{\epsilon}\right\rVert<\left\lVert\nabla
F_{g}(\textbf{x}^{k})\right\rVert$, which is satisfied if the following two
conditions are met:

Condition 1:  the total number of sensors $p$ on a robot and the corresponding
sensor placement is chosen such that
$\frac{\Delta_{H}}{\sqrt{2}}\coloneq\sqrt{2}r_{r}\mathrm{sin}(\Phi/2)<1$.

Condition 2:  the Lipschitz constant of $\nabla f_{g}$ in the ball
$B(\textbf{x}^{k},\Delta_{H})$ satisfies $L<\left\lVert\nabla
f_{g}(\textbf{x}^{k})\right\rVert$. Intuitively, this means that the gradient
of the source signal function $f_{g}$ varies smoothly throughout its domain,
despite potentially high gradient magnitudes at specific locations.

We examine these conditions for two possible cases: (I) regions with the
influence of multiple targets ($F_{g}\neq f_{g}$), and (II) regions with the
influence of a single target ($F_{g}=f_{g}$). For (I), let $\textbf{x}^{k}$ be
such that there are no critical points of $F_{g}$ in the ball
$B(\textbf{x}^{k},\Delta_{H})$, and let the closest target be at
$\textbf{x}_{g}\in\mathcal{I}$ such that
$d=\left\lVert\textbf{x}^{k}-\textbf{x}_{g}\right\rVert$. Since $F_{g}$
inherits the continuity and differentiability properties of $f_{g}$
(Assumption 4), $\left\lVert\nabla^{2}F_{g}(d)\right\rVert<\left\lVert\nabla
F_{g}(d)\right\rVert$, implying
$\left\lVert\bm{\epsilon}\right\rVert\coloneq\left\lVert\nabla^{2}F_{g}(d-\Delta_{H})\right\rVert\frac{\Delta_{H}}{\sqrt{2}}\leq\left\lVert\nabla
F_{g}(d)\right\rVert$ in the ball $B$, hence $\vartheta\in[-\pi/2,\pi/2]$.
Ensuring a constantly bounded angular error becomes challenging if one or more
critical points exist inside the ball $B$ (Fig. 4). By leveraging the
embodiment of the robot ($r_{r}>0$) and its distributed sensing, we set the
robot heading in the direction indicated by the highest-intensity sensor
reading ($\mathrm{max}(Z_{g})$), so that the robot is guaranteed to get out of
the local critical regions. For (II), given the signal function $f_{g}$ and
the number of sensors $p$, we set constraints on $r_{g}^{\mathrm{safe}}$ such
that the following inequality is satisfied,

$\displaystyle\left\lVert\nabla^{2}f_{g}(r_{g}^{\mathrm{safe}})\right\rVert\frac{\Delta_{H}}{\sqrt{2}}<\left\lVert\nabla
f_{g}(r_{g}^{\mathrm{safe}})\right\rVert$ (6)

Fig. 4 shows how, for a given number of sensors, gradient estimate
deteriorates in the critical regions of scalar field $F_{g}$.

Figure 4: Gradient estimation deteriorates in critical region of $F_{g}$

###  5.2 Guaranteed Encirclement Behavior

Next, we establish additional constraints on the signal function $f_{g}$ and
the minimum distance between any two targets, so that a robot can detect when
it is within a target’s encapsulation ring and start the encirclement
behavior. Consider an environment with $m\geq 1$ targets and a robot in a
configuration such that target $g_{1}$ located at
$\textbf{x}_{g_{1}}\in\mathcal{I}$ is closest to the robot at any time $T$.
Let the $k^{\mathrm{th}}$ sensor receive maximum target intensity
($z_{g}^{k}=\mathrm{max}(Z_{g})$) and $r_{g_{1}}^{\mathrm{safe}}$,
$r_{g_{1}}^{\mathrm{encap}}$ be the inner and outer radius of the
encapsulation ring of the closest target $g_{1}$, respectively (bounds on
encapsulation ring radii provided in [9]). For a robot to enter the
encapsulation ring without breaching the safety violation zone, the following
conditions must be satisfied:

Condition 3: The gradient of the aggregate signal strength $F_{g}$ at the
inner radius, $\left\lVert\nabla
F_{g}(r_{g_{1}}^{\mathrm{safe}})\right\rVert$, must exceed the gradient of
single target influence at the outer radius $\left\lVert\nabla
f_{g}(r_{g_{1}}^{\mathrm{encap}})\right\rVert$, that is $\left\lVert\nabla
F_{g}(r_{g_{1}}^{\mathrm{safe}})\right\rVert>\left\lVert\nabla
f_{g}(r_{g_{1}}^{\mathrm{encap}})\right\rVert$

Condition 4: Conversely, the gradient of the aggregate signal strength $F_{g}$
at the outer radius, $\left\lVert\nabla
F_{g}(r_{g_{1}}^{\mathrm{encap}})\right\rVert$, must be less than the gradient
of single target influence at the inner radius $\left\lVert\nabla
f_{g}(r_{g_{1}}^{\mathrm{safe}})\right\rVert$, that is $\left\lVert\nabla
F_{g}(r_{g_{1}}^{\mathrm{encap}})\right\rVert<\left\lVert\nabla
f_{g}(r_{g_{1}}^{\mathrm{safe}})\right\rVert$

These are the necessary conditions to ensure that the encirclement behavior is
triggered only if a robot is in the encapsulation ring as detailed in Section
4.3.2. Furthermore, for successful encapsulation of a target $g$, it is
necessary that within the encapsulation ring, the target’s influence
dominates, i.e. $F_{g}\approx f_{g}$ for
$\left\lVert\textbf{x}^{k}-\textbf{x}_{g}\right\rVert\leq
r_{g}^{\text{encap}}$. Therefore, we require a minimum separation
$D_{g}\in\mathbb{R}^{+}$ between any two targets in the environment. The lower
bound on $D_{g}$ is a function of both the number of sensors and their
placement, and $f_{g}$. We elaborate on two specific scenarios to compute
$D_{g}$. These cases depict the extreme target configurations with all other
scenarios being variations of these extremes.
Case I: All other $m-1$ targets are located on the opposite side of the robot
with the next closest target located at $\textbf{x}_{g_{2}}$ such that
$D_{g}=\left\lVert\textbf{x}_{g_{1}}-\textbf{x}_{g_{2}}\right\rVert$. In this
case, $\left\lVert\nabla F_{g}(\textbf{x}^{k})\right\rVert<\left\lVert\nabla
f_{g}(\textbf{x}^{k})\right\rVert$. If the $k^{\mathrm{th}}$ sensor is located
at $\textbf{x}^{k}\in\mathcal{I}$ such that it is at the inner radius
$r_{g_{1}}^{\mathrm{safe}}$ then Condition 3 is satisfied if,

$\displaystyle\inf\left\lVert\nabla
F_{g}(\textbf{x}^{k})\right\rVert\\!\coloneq\\!\left\lVert\nabla
f_{g}(r_{g_{1}}^{\mathrm{safe}})\right\rVert\\!-\\!(m\\!-\\!1)\left\lVert\nabla
f_{g}(D_{g}-r_{g_{1}}^{\mathrm{safe}})\right\rVert\\!$
$\displaystyle>\\!\left\lVert\nabla
f_{g}(r_{g_{1}}^{\mathrm{encap}})\right\rVert$ (7)

Case II: All the other $m-1$ targets are located on the same side of the robot
with the second closest target located at $\textbf{x}_{g_{2}}$ such that
$D_{g}=\left\lVert\textbf{x}_{g_{1}}-\textbf{x}_{g_{2}}\right\rVert$. In this
case, $\left\lVert\nabla F_{g}(\textbf{x}^{k})\right\rVert>\left\lVert\nabla
f_{g}(\textbf{x}^{k})\right\rVert$. If the $k^{\mathrm{th}}$ sensor is located
at $\textbf{x}^{k}\in\mathcal{I}$ at the outer radius
$r_{g_{1}}^{\mathrm{encap}}$ then Condition 4 is satisfied if,

$\displaystyle\sup\left\lVert\nabla
F_{g}(\textbf{x}^{k})\right\rVert\\!\coloneq\\!\left\lVert\nabla
f_{g}(r_{g_{1}}^{\mathrm{encap}})\right\rVert\\!+\\!(m\\!-\\!1)\left\lVert\nabla
f_{g}(D_{g}-r_{g_{1}}^{\mathrm{encap}})\right\rVert\\!$
$\displaystyle<\\!\left\lVert\nabla
f_{g}(r_{g_{1}}^{\mathrm{safe}})\right\rVert$ (8)

Note that Condition 4 is satisfied for all $D_{g}$ for Case I, and Condition 3
is satisfied for all $D_{g}$ for Case II.

Example: One example of a target source signal function that satisfies
Conditions 1 and 2 is the radially decreasing function $f_{g}=C/d^{2}$ where
$C\in\mathbb{R}^{+}$ and $d$ is the distance from the source. For a circular
robot with $p$ sensors placed symmetrically on its periphery, the constraints
on the safety distance $r_{g}^{\mathrm{safe}}$ and the minimum separation
between targets $D_{g}$ are given below such that Condition 3 and Condition 4
are satisfied. From Eq. (6) we have,

$\frac{6C}{(r_{g}^{\mathrm{safe}})^{4}}\sqrt{2}r_{r}\mathrm{sin}(\pi/p)<\frac{2C}{(r_{g}^{\mathrm{safe}})^{3}}\implies
r_{g}^{\mathrm{safe}}\geq 3\sqrt{2}r_{r}\mathrm{sin}(\pi/p)$

From Eq. (7) and Eq. (8),

$D_{g}>\mathrm{max}\Bigg{\\{}r_{g}^{\mathrm{safe}}\bigg{[}1+\frac{m-1}{\big{[}1-\big{(}\frac{r_{g}^{\mathrm{safe}}}{r_{g}^{\mathrm{encap}}}\big{)}^{3}\big{]}}\bigg{]}^{1/3},r_{g}^{\mathrm{encap}}\bigg{[}1+\frac{m-1}{\big{[}\big{(}\frac{r_{g}^{\mathrm{encap}}}{r_{g}^{\mathrm{safe}}}-1\big{)}^{3}\big{]}}\bigg{]}^{1/3}\Bigg{\\}}$

###  5.3 Absence of Livelocks: Minimum Separation Between Obstacles

To ensure no livelocks, the minimum separation $D_{o}$ required between any
two obstacles for a robot to navigate between them is given by

$\displaystyle D_{o}$
$\displaystyle>2r_{\mathrm{o}}+2(r_{r}+r_{o}^{\mathrm{safe}\mid\mathrm{sensor}})+d_{r}^{\mathrm{max}}$
(9) $\displaystyle r_{o}^{\mathrm{safe}\mid\mathrm{sensor}}$
$\displaystyle=\sqrt{(r_{o}^{\mathrm{safe}})^{2}+r_{r}^{2}-2r_{r}r_{o}^{\mathrm{safe}}\mathrm{cos}(\pi/p)}$

where $r_{o}$ is the obstacle’s circumscribed radius,
$r_{o}^{\mathrm{safe}\mid\mathrm{sensor}}$ is the safe distance
$r_{o}^{\mathrm{safe}}$ with respect to a robot’s sensor. $D_{o}$ is a
function of the number of sensors $p$: a robot with more sensors can navigate
through more closely spaced obstacles.

Additionally, since encapsulating robots emit obstacle signals, we need to
ensure that multiple sets of encapsulated targets do not form a non-convex
open region in the environment, as shown in Fig. 3(b). Therefore, we put
additional constraints on the minimum separation between any two targets,

$\displaystyle
D_{g}>2r_{g}^{\mathrm{encap}}+2(r_{r}+r_{g}^{\mathrm{safe}\mid\mathrm{sensor}})+d_{r}^{\mathrm{max}}$
(10)

Satisfaction of Conditions 1-4, Eq. (6)-(10) and Lyapunov stability guarantee
that the robot moves toward the closest target and eventually reaches its
encapsulation ring. When the robot encounters an obstacle, it moves away to
successfully circumnavigate the obstacle while maintaining its overall motion
toward the target, ensuring a control policy that combines obstacle avoidance
with target attraction.

##  6 Simulations

We examine how the number of sensors ($p$), sensor noise, occlusions,
deviations from the determined bounds on $f_{g}$, and minimum target
separation $D_{g}$ affect the swarm’s emergent behavior. The environment
comprises 3 targets, 6 circular obstacles, and 18 robots. The total simulation
time is capped at 1500 time steps. We run 50 simulations for each data point
with the same initial conditions to study the effect of the considered
parameter on the emergent behavior by keeping other conditions constant.
Stochasticity within each simulation originates from the random selection of
$\theta_{r}$. All robots in the swarm move at different rotational and
translational speeds.

Effect of the total number of sensors $(p)$: Fig. 5 shows how increasing the
number of sensors reduces the time required to encapsulate all targets while
ensuring safety. Here $n_{g}=6$ and the control is synchronous. This is
because when $p$ increases, the error in computing the simplex gradient
decreases, resulting in a better impression of the local neighborhood.

Figure 5: The total time taken for task completion as a function of $p$.

Effect of noise: We add Gaussian noise to each sensor reading,
$z_{g}^{k}=(1-n_{g}^{k})\sum_{j\in N_{g}^{k}}f_{g}(d^{k}_{g}),$
$\,n_{g}^{k}\sim\mathcal{N}(0,\sigma^{2})$ and $n^{k}_{g}\leq 1$. Fig. 6(b)
shows the total time taken by the swarm for task completion with $p=5$. As the
noise increases, the deviation of the simplex gradient from the true gradient
increases, leading to inaccurate estimations of the closest target’s
encapsulation ring. Consequently, the success rate decreases considerably for
noise levels exceeding 70% and drops to zero when the noise threshold
surpasses 85%. Unlike [11], where task parameters (e.g.,
$r_{g}^{\mathrm{encap}}$,$n_{g}$, $r_{g}^{\mathrm{safe}}$) had to be adjusted
to ensure successful encapsulation, here we keep all parameters fixed and show
that the use of simplex gradients, which leverage distributed sensing on a
robot, enhances robustness to noise. Contrary to [9], the addition of noise to
sensor readings for robot sources did not result in noise filtering, and we
observed safety violations. This difference stems from our relaxation of the
robot’s step size upper bound (Section 4.1).

(a)

(b)

(c)

Figure 6: For $p=5$ (a) probability of success as a function of noise levels
(b) time taken for task completion (c) number of $r_{g}^{\mathrm{safe}}$
violations across all simulations.

Effect of occlusions: We adjusted the sensor measurement $z_{s}^{k}$ by
applying a coefficient $\eta\in[0,1]$ to quantify the obstruction level
between the source and the sensor. This adjustment accounts for scenarios
where obstacles or other robots obstruct the line of sight from the source to
the $k^{\text{th}}$ sensor. In this scale, 0 indicates complete obstruction,
and 1 indicates full visibility. We investigate the effects of varying the
total number of sensors ($p$) on a robot, considering different levels of
occlusion. Figure 7(a) illustrates that the success probability rises from 65%
to 100% for $\eta\geq 0.75$ with $p=3$. Increasing the sensor count to $7$
boosts success rates, with a minimum of 94% at $\eta=0$ and reaching 100% for
$\eta\geq 0.45$, indicating improved robustness with higher sensor resolution.

(a)

(b)

(c)

Figure 7: (a) probability of success as a function of occlusion levels (b)
total time taken for task completion. (c) number of safety violations across
all simulations

Supplementary Video [57]: We include simulations that show the effect of
constraint violations, asynchronous execution, and a demonstration of the
algorithm’s scalability through a large-scale simulation involving 100 robots,
15 targets, and 10 obstacles.

##  7 Conclusion

In this paper, we show how robots equipped with omnidirectional sensors and an
isotropic signal emitter can successfully encapsulate multiple diffusive
targets while maintaining safety distances from other robots and obstacles.
Our decentralized controller is agnostic to the number of robots, targets, and
obstacles in the environment. We determined bounds and constraints on task,
control, and robot parameters for guaranteed convergence to all sources with
user-specified safety constraints. Additionally, our algorithm can be adapted
for robots with non-isotropic sensors, provided the measurement error bounds
are known. In future research, we will explore relaxing the assumption that a
target source stops signal emission upon encapsulation, which currently
directs non-encapsulating robots to search for other targets. One strategy
could be to extend the influence region of encapsulating robots, prompting
dynamic robots to avoid these regions and explore remaining active target
sources.

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