# RASS: Risk-Aware Swarm Storage

Samuel Arseneault, David Vielfaure, Giovanni Beltrame

###### Abstract

In robotics, data acquisition often plays a key part in unknown environment
exploration. For example, storing information about the topography of the
explored terrain or the natural dangers in the environment can inform the
decision-making process of the robots. Therefore, it is crucial to store these
data safely and to make it available quickly to the operators of the robotic
system. In a decentralized system like a swarm of robots, this entails several
challenges. To address them, we propose RASS, a decentralized risk-aware swarm
storage and routing mechanism, which relies exclusively on local information
sharing between neighbours to establish storage and routing fitness. We test
our system through thorough experiments in a physics-based simulator and test
its real-world applicability with physical experiments. We obtain convincing
reliability, routing speeds, and swarm storage capacity results.

###### Index Terms:

Swarm Robotics, Information Sharing, Robot Safety

##  I Introduction

Using multi-robot systems for the exploration of unknown environments is very
appealing. Indeed, if robots do not overlap in their exploration task, the
amount of terrain covered increases proportionally with the number of robots
in the system [1]. This can, for example, be particularly useful for search
and rescue scenarios [2] where the speed at which the environment is covered
is of critical importance. However, as the number of robots increases so does
the amount of data collected, which puts pressure on the data storage
infrastructure. Unfortunately, multi-robot systems usually suffer from
unreliable connectivity [3], and directly sending the information to an
external storage system (e.g., the cloud) may not always be feasible. An
alternative to overcome this issue is using the swarm of robots as means of
distributed data storage. However, because the robots composing the swarm
often have very limited memory and storage capacities, saving large amounts of
data across the swarm can prove challenging. Furthermore, the robots in a
swarm are not necessarily reliable [4]: even in controlled environments, they
are usually meant to be easily replaced. This issue is aggravated when they
must face situations that decrease their reliability, such as exposure to
dangerous environments like search and rescue scenarios, forest fire
reckoning, or nuclear inspection and cleanup. Therefore, giving the robots a
way to eventually relay the information they acquired during their mission to
a control station for more permanent storage is a definite advantage: it not
only allows the information to be stored in a safe location accessible by
human operators, it also alleviates the memory usage of the robots and enables
continuous operation. Nonetheless, because in practice robots have a finite
communication range, the information collected at the periphery of the swarm
usually needs to be routed through other robots before reaching a base
station. Although forwarding the data through the shortest path towards the
base station enables prompt retrieval [5], in real-world scenarios, this
approach can be problematic for two reasons: first, such a path might not
exist at all, for example if the robots are in an area completely cut off from
external communication (e.g. a cave); second, if that path does exist, it
might be too dangerous to use. For example, a robot trying to relay
information by going through a highly radioactive area might cause a physical
failure or data corruption. Furthermore, the base station cannot request the
data, as communication with the robots is never guaranteed. Instead, data must
be autonomously forwarded to the base station by the robots. For the same
reason, and because of the overhead it entails, it is not practical to use
leaders to coordinate the storage and routing processes. In addition, the
system must take into account that the robots are constantly moving and
acquiring new data, meaning that storage and routing conditions are dynamic.
These constraints motivate the need for a completely decentralized, risk-aware
swarm storage system that can safely and efficiently store data and route it
towards a base station in a percolating fashion whenever possible. In this
paper, we make the following contributions to multi-robot storage:

  * •

A distributed, dynamic and risk-aware storage system

  * •

A fully decentralized risk-aware routing algorithm

We name the combination of these two contributions RASS: Risk-Aware Swarm
Storage. The rest of the article is structured as follows: in section II, we
present related work and useful concepts upon which we build RASS; in section
III we present our system model; in section IV and V we detail our simulations
and physical experiments along with their respective results; finally in
section VI we draw some conclusions.

##  II Related Work and Background

###  II-A Distributed Storage

Distributed hash tables (DHTs) are data structures specifically designed to
partition data across a network of storage nodes in a key-value format. They
come with a set of issues susceptible of hindering their performance,
consistency and partial connectivity being some of them [3]. The virtual
stigmergy presented in [6] tackles the problem using Conflict-free Replicated
Data Types (CRDTs) and allowing decentralized robust information sharing for
multi-robot systems. This virtual stigmergy is implemented directly into the
Buzz programming language [7]. However, in the virtual stigmergy, each node
stores a full copy of the data (in an eventually consistent way). This means
it has full redundancy and no partitioning, thus offering good reliability but
poor storage capacity optimization.

A memory-efficient storage mechanism designed for multi-robot systems was
proposed in SwarmMesh [8]. It uses the available memory of an agent and its
connectivity with its peers to generate unique identifiers, which are the
basis of its partitioning scheme and which represent an agent’s fitness to
store data. Storage of larger items (such as binary files) in DHTs has been
studied in [9], where an auction process is used to partition data. Other
partitioning systems have been proposed, such as Locality-aware Distributed
Hash Tables [10] which posits that routing data through nodes that are close
in a network will reduce latency. Therefore, they use information from the
nodes’ Autonomous System Numbers (ASNs) to partition data. Geographic Hash
Tables [11], Cell Hash Routing [12] and [13] all use position-based
information in their partitioning systems, and the second specifically
addresses DHT implementation for ad-hoc networks of resource-constrained
nodes.

These systems show various metrics that can be used to devise a partitioning
system for a DHT. However, none of them takes into account the effects of
environmental dangers on the reliability of the distributed storage system.
RASS, our decentralized risk-aware storage and routing mechanism, aims to
address this issue by including a risk measurement in its partitioning scheme.

###  II-B Risk Assessment

Taking risk into account when designing autonomous systems is of high
importance as excessive exposure to risk can lead to system failures. In an
unknown environment, risks are usually tied to a certain location. Information
about these locations could therefore provide increased situational awareness
for robots to effectively perform a given mission. Belief maps and occupancy
maps provide this situational awareness by assigning values to cells of a
discretized environment. Whereas occupancy maps define the presence or absence
of a feature (e.g. fire), beliefs maps assign probabilities to the cells of
the discretized environment and usually offer better performance [14]. Belief
maps have been extensively used in the past for robotic exploration [15, 16,
17].

A risk-aware exploration algorithm leveraging a belief map of the risk
associated with the environment has been recently proposed in [18]. In this
work, robots combine efforts in building a shared belief map of the
environmental risks and use it to spatially avoid the dangerous pitfalls of
the environment. The belief map is used to provide risk awareness to an
exploration algorithm which ultimately results in fewer robot failures. In
RASS, we seek to use similar shared risk awareness to establish a reliable
distributed data storage and routing mechanism, while improving its memory
consumption by relying on more localized information.

###  II-C Routing

There exist many types of routing algorithms for wireless sensor networks. For
example, packets can be explicitly routed towards a known destination by
forwarding messages towards a neighbour which optimizes a given metric. One
simple approach is to send messages through the shortest path in terms of
Euclidean distance. However, this assumes a complete knowledge of the nodes’
Cartesian coordinates, which is far from realistic in swarm robotics. Indeed,
robots might be operating into GPS-denied environments, or they might not even
be equipped with GPS at all. A comparative study of metrics for multi-hop
wireless networks shows that hop-count performs the best in scenarios where
nodes are mobile [19] because it is robust in dynamic topologies [20], which
is the case for RASS’s nodes. Notable implementations using this metric are
given in [21, 22, 23], which are respectively greedy, power-adaptive, and
grid-based routing algorithms. Routing algorithms have also been inspired by
biology: some are based on slime molds [24, 25] and others on ant colonies
[26, 27]. However, the former category has applications mostly in static
topology networks, which makes it ill-suited to our needs. The latter category
specifically addresses data aggregation, which is particularly relevant to our
objective our percolating data towards a base station.

##  III System Model

We consider a fully decentralized multi-robot system tasked with the
exploration of a dangerous environment. The multi-robot system is denoted as
the collection of agents $a_{i}\in A$. We assume the robots to have limited
storage and communication capabilities. We also assume the presence of an
operator or base station who is interested in collecting the data generated by
the swarm. Note that this last assumption does not mean that our system is
centralized: it is reasonable to assume that the autonomous system has to
produce some value (i.e. data) for the humans deploying it, and the base
station does not influence the decision-making process of the swarm, but
rather act as an information sink. We consider the amount of data needed to be
stored to be greater than the individual storage capabilities of the robots.
It is therefore impossible for the individual agents to store a complete copy
of the system’s data, but the data can be fully stored by the base station.
Also, we consider that the robots are only able to communicate if they are
within a certain communication radius $R$. As a result, if a message needs to
be sent to a distant location (e.g. to the base station), it might need to be
routed through multiple nodes before reaching the desired destination.

###  III-A Risk Modelling

Radiation is known to cause performance loss and failures in robots [28, 29].
We therefore adapt the risk modelling from [18], which is based on a set of
independent point radiation sources $S$ with individual intensity
$I_{j}\sim\mathcal{U}(0,1)$. Note that we use radiation as a _model_ of risk,
but our method could be applied to other sources of danger: vertical air
currents, high temperature areas, etc.

Risk is assumed to be dynamic, meaning that the position of the point
radiation sources $\bm{s}_{j}(t)\in E$ can vary across time. For example,
radiation can spread to new areas if radioactive particles are transported by
wind. It is important to account for the dynamic nature of risk as it gives
the system the capability to adapt to changing environments. We achieve
adaptability by having each agent sense the radiation at every time step at
its current position. The perceived intensity decays exponentially (with
$\lambda$ as a decay parameter) as the Euclidean distance $\rho(\bm{x}_{i})$
between $\bm{s}_{j}$ and $\bm{x}_{i}$ increases. The total perceived radiation
level by a robot $a_{i}$ at position $\bm{x}_{i}\in E$ is given by:

$r(\bm{x}_{i})=b+\sum_{\bm{s_{j}}\in
S}\frac{I_{\bm{s_{j}}}}{1+\lambda\rho(\bm{x}_{i})^{2}}$ (1)

and is measured by an on-board sensor with Gaussian measurement noise
$b\sim\mathcal{N}(0,0.05)$. We posit the radiation’s effect on the system is
to cause data corruption (which is one of its possible effects [29]). Let the
probability of the event of a datum $d$ getting corrupted while stored on
robot $a_{i}$ due to an individual radiation source $\bm{s}_{j}$ be
$\mathbb{P}(c_{i}=1|\bm{s}_{j})\sim\mathcal{B}(r_{\bm{s}_{j}}(\bm{x}_{i}))$,
which follows a Bernoulli distribution. Because we assume that the sources of
radiation affect the robots (and thus the stored data) independently, the
probability of a datum being corrupted due to the combined effect of radiation
sources follows a Bernoulli distribution given by:

$\mathbb{P}(c_{i}=1|S)\sim\mathcal{B}(r(\bm{x}_{i}))=1-\prod_{\bm{s_{j}}\in
S}1-\mathbb{P}(c_{i}=1|\bm{s}_{j})$ (2)

###  III-B Distributed Risk-Aware Storage

Our risk-aware storage system is built upon three assumptions:

  1. 1.

Because nodes exposed to a higher level of risk also have a higher failure
probability, they should be used less, thus maximizing overall storage
reliability

  2. 2.

Efficiently moving data away from the periphery of the swarm and towards the
base station will increase the storage capacity of the system and the
persistence potential of the data, because the base station usually has more
storage and reliability than the swarm

  3. 3.

Percolating data from edge nodes to the base station should be done by
choosing routes devoid of risk to minimize data loss

From these assumptions, we derive RASS’ two principal mechanisms: the routing
table and the fitness-based percolation. We combine them to obtain a high-
level algorithm described in Alg. 1, in which $\mathcal{N}$ is the set of
neighbours of a given agent. Because of the distributed nature of the
algorithm and because all robots execute the same code (except the base
station), we forego the indices notation to simplify the algorithms.

while  _True_ do

$update\\_routing\\_table()$

$update\\_fitness()$

if  _not $is\\_fit()$ and $|\mathcal{N}|>0$_ then

$evict\\_data()$

end if

$store\\_measurements()$

end while

Algorithm 1 RASS Execution Loop

####  III-B1 Routing Table

We assume that the swarm can be represented by a connected graph $\mathcal{G}$
with nodes $\mathcal{A}$ and edges $\mathcal{L}$ respectively representing
agents and their wireless communication links. In practice, this connectivity
cannot be maintained at all times, because of the quality of the links in
$\mathcal{L}$ and because of the movements of nodes $\mathcal{A}$. However,
this assumption can be mostly maintained through mechanisms for connectivity
maintenance [30] if necessary. Furthermore, the need for this assumption can
be relaxed because RASS does not need a constant link with the base station;
it can opportunistically move data towards it when possible, and store them
locally in the meantime.

Given the existence of at least one (multi-hop) path between any node and the
base station, we can establish a routing table based on hop count. Because
peer communication is not the focus of this work, the routing table held by
each node only contains the minimal hop count between its neighbours and the
base station and can therefore be implemented with a hash table. The process
to construct and periodically update this table involves exchanging messages
between the base station and the nodes as suggested by [31], as we implement
in Alg. 2. Building a routing table with the aim of prioritizing higher
capacity nodes is similar to the Gateway Optimization [32] implemented in the
B.A.T.M.A.N mesh networking protocol [33], where in our case the base station
acts as a gateway. Assuming a message can only be forwarded to an immediate
neighbour within a given time step, the required time to build the routing
table in a connected graph is bounded by $\Omega(1)$ and $\mathcal{O}(|A|)$ as
it takes at most $|A|$ steps to send a message from the base station to the
furthest robot in a pathological topology (a line) and $1$ step if the network
is fully connected. However, in more realistic scenarios such as tree networks
or scale-free topologies, building the table can be expected to take on
average $\mathcal{O}(log|A|)$ steps because of the network’s depth. We
implement message forwarding through a gossip algorithm, which allows local
broadcasting between robots.

$routing\\_table\longleftarrow listen\\_neighbor\\_hop\\_count()$

if  _$id=0$_ then

$min\\_hops\longleftarrow 0$

else

$min\\_hops\longleftarrow min(routing\\_table)$

end if

$broadcast(min\\_hops+1)$

Algorithm 2 Building/Updating the Routing Table

####  III-B2 Potential-Based Percolation

Our risk-aware storage system draws inspiration from SwarmMesh [8], in that
each node periodically assigns itself a potential $\phi_{i}$ based on its
fitness to store data, given by:

$\phi_{i}=\left\\{\begin{array}[]{ll}\frac{1}{\alpha h_{i}+\beta
r({\bm{x}_{i}})}&\text{if }m_{i}>0\\\ 0&\text{otherwise}\end{array}\right.$
(3)

where $m_{i}$ is the memory available on node $i$, $r_{i}$ is the risk
associated with the current node’s location (stored in the distributed belief
map) and $h_{i}$ refers to the minimum hop count required to reach the base
station from $i$ as specified in the routing table. Parameters $\alpha$ and
$\beta$ are respectively the routing weights and risk weights, which allow
adapting the policy based on the relative importance of the routing time and
the environmental, risk with respect to each other. Similarly to [8], a node
which becomes unfit to store data will evict such data by moving it into its
routing queue. The condition for “unfitness” is simply:

$T\phi_{i}<\max_{j\in\mathcal{N}}\phi_{j}$ (4)

where $\mathcal{N}$ and $T$ are the set of $i$’s neighbours and the fitness
threshold, respectively. The latter’s purpose is to ensure data is transferred
only when the neighbours’ max fitness is significantly higher than $\phi_{i}$
to avoid instability and overhead which would result from frequent and
inefficient transfers. This fitness policy causes data to naturally percolate
along the edges towards the base station because nodes with a higher potential
are both closer to it and located in safer areas. When necessary, data is
evicted using a Least Recently Used (LRU) policy and transmitted to the
fittest neighbour.

##  IV Simulations

###  IV-A Experimental Setup

We ran extensive simulations in a physics-based simulator, ARGoS [34] with
models of Khepera IV [35] robots to eliminate the effect of potential hardware
issues on the conceptual validity of our system and also to verify that it
scales well to large swarm sizes. We executed 30 simulation runs with 100
robots for each type of experiment to reach results with low uncertainty. The
robots are placed within a 20m by 20m arena and their communication radius $R$
is set to 3m. Three (3) radiation sources are randomly distributed in the
environment around the origin. The base station is located in a corner of the
arena and its storage capacity is assumed to be infinite.

Figure 1: $400\text{m}^{2}$ environment in the ARGoS simulator with 100
KheperaIV robots distributed in a grid-like pattern. Figure 2:
$400\text{m}^{2}$ environment in the ARGoS simulator with 100 KheperaIV robots
distributed in a scale-free pattern. Figure 3: $400\text{m}^{2}$ environment
in the ARGoS simulator with 100 KheperaIV robots in a formation obtained
through Lennard-Jones potential interactions. Figure 4: $400\text{m}^{2}$
environment in the ARGoS simulator with 100 KheperaIV robots in a formation
obtained through random walk motions.

In order to replicate realistic operation scenarios, we chose to artificially
introduce bandwidth limitations. In all experiments, robots can only exchange
up to 10 data items at every time step. For the same reason, our simulated
robots have a limited storage capacity of 50 data items of at most 50 bytes
each, as the data are simple items such as small tables or floating-point
numbers. This gives a total storage capacity of 2500kb per robot. Because the
robots can only exchange up to 20% of their stored data at a given time step,
it amounts to a bandwidth of 0.5kb/s. Data are generated by each robot at
fixed out of phase intervals.

To evaluate the performance of our system in different scenarios, we tested it
with static topologies: a grid-like formation, and a scale-free network; as
well as with dynamic topologies: a formation obtained through Lennard-Jones
potential interactions and a formation evolving from random walk motions.
Testing with static topologies allows us to verify applicability with fixed
wireless sensor networks relevant to IoT applications, while experiments with
dynamic topologies are more relevant for mobile robotics applications.

To setup RASS, we used values of $\alpha=10$ and $beta=1$ in Eq. 3 because
risk measurement values are normalized between 0 and 1 while hop-count values
are typically upper-bounded to 10 (for 100 robots). Our first benchmark
algorithm is to use a fitness policy based purely on hop count, i.e. if
required, data is sent only to neighbours closer to the base station. Our
second comparison baseline is to store data in a virtual stigmergy (a CRDT)
[6]. Using the virtual stigmergy practically ensures that no data can be lost
due to corruption because it is fully replicated across the system. The first
metric we used in our performance evaluation is reliability, expressed as
$\frac{n_{g}-n_{l}}{n_{g}}$, where $n_{g}$ and $n_{l}$ are respectively the
amount of data generated and lost at every time step. The second metric is the
average data transfer speed, measured as the delay between the creation of a
given datum and its arrival by percolation to the base station. We excluded
results of this metric for the virtual stigmergy, as the stigmergy cannot
include the concept of a base station (since all nodes are peers), and
stigmergy propagation speeds are detailed in [6]. The third metric is the
evolution of the system’s total storage capacity over time, i.e. the amount of
data stored by the agents and the base station combined.

###  IV-B Results

The results obtained in the 30 simulation runs for the static topologies
(grid-like and scale-free) as well as for the dynamic topologies (Lennard-
Jones potential and random walk) are presented in Figs. 5, 6, 7 and 8
respectively.

(a) Evolution of reliability over time

(b) Distribution of transfer speeds

(c) Evolution of total swarm storage capacity over time

Figure 5: Performance comparison of RASS, hop count and stigmergy in a static
grid-like topology.

(a) Evolution of reliability over time

(b) Distribution of transfer speeds

(c) Evolution of total swarm storage capacity over time

Figure 6: Performance comparison of RASS, hop count and stigmergy in a static
Scale Free topology.

(a) Evolution of reliability over time

(b) Distribution of transfer speeds

(c) Evolution of total swarm storage capacity over time

Figure 7: Performance comparison of RASS, hop count and stigmergy in a dynamic
Lennard-Jones topology.

(a) Evolution of reliability over time

(b) Distribution of transfer speeds

(c) Evolution of total swarm storage capacity over time

Figure 8: Performance comparison of RASS, hop count and stigmergy in a dynamic
random search topology.

Results show that RASS outperforms the hop-count algorithm in terms of
reliability. Because of the risk awareness component included in its fitness
policy as detailed in Eq. 3, robots do not always route the data through the
shortest path to the base station. RASS avoids the dangerous storage nodes of
the system when routing data which explains the higher reliability levels
displayed in Fig. 5(a), Fig. 6(a), Fig. 7(a) and Fig. 8(a) . This is why, on
average, RASS takes 54.89% more time to route the data to the base station
when compared to the hop-count algorithm. The shortest path might not always
be the safest one; RASS will take an alternate route if the risk associated
with the shortest one is too high. On the other hand, the hop-count algorithm
always takes the shortest path towards the base station regardless of the risk
associated with it. This leads to a higher number of data losses due to
corruption and an overall lower reliability. However, hop count can yield
faster transfer speeds as shown in Fig. 5(b), Fig. 6(b), Fig. 7(b), Fig. 8(b)
and table I.

TABLE I: Average transfer speed and average individual memory usage with different topologies Topology | Algorithm |  | Transfer speed
---
(hops)
| Memory used
---
(%)
| RASS | 11.45 | 1.35
| Hop-Count | 9.11 | 0.61
Grid-like | Stigmergy | N.A. | 100.00
| RASS | 11.44 | 1.95
| Hop-Count | 6.85 | 0.50
Scale-Free | Stigmergy | N.A. | 100.00
| RASS | 12.51 | 1.69
| Hop-Count | 7.32 | 0.51
Lennard-Jones | Stigmergy | N.A. | 100.00
| RASS | 12.68 | 1.67
| Hop-Count | 7.76 | 0.57
Random Search | Stigmergy | N.A. | 100.00

For the virtual stigmergy, most of the data losses can be attributed to the
storage having reached its maximum capacity. Indeed, because of the fully
replicated nature of the stigmergy, the memory of the agents is quickly
saturated. Table I shows that on average, across the 500 steps of the
simulations runs, the virtual stigmergy has 100% of the individual memories
used. This means that the nodes of the system are simply full and cannot store
data anymore. In comparison, RASS uses between 1% and 2% of the local memories
of the nodes, and hop-count is even lower at values around 0.5%. This full
redundancy prevents losing data from corruptions. However it entails a very
inefficient use of the memory of the robots and ultimately leads to data
losses due to insufficient memory capacity. The result is an unchanging
storage capacity over time and poor reliability as shown in Fig. 5(a) for the
virtual stigmergy strategy.

The low values of local memory usage shown in Table I for RASS and hop-count
were obtained because the topologies used to test the algorithms were usually
well connected in accordance with the connected graph assumption we made in
III-B1. For the most part, multiple routes were connecting the nodes to the
base station and as a result, the collected data was routed towards the base
station instead of being kept locally. Such a result implies that the system,
by maintaining a low individual storage occupancy, allows the robot to adapt
to situations in which they would be temporarily stranded: if their storage
were to be mostly full, they would not be able to generate new data without
promptly losing it. This storage buffer thus allows them to continue
collecting data while being temporarily disconnected from the rest of the
swarm.

##  V Physical experiments

###  V-A Experimental setup

We evaluated RASS’ performance with the same metrics on physical robots to
confirm the real-world applicability of our system. We used 5 small drones in
a controlled indoor environment. These drones use standard Raspberry Pi Zeros
as their main computer, meaning they have relatively low capabilities, and are
therefore well suited to verify our algorithm. We designed a static topology
with one drone acting as a base station and 4 others acting as agents. The
radiation source (red cone) in Fig. 9 is positioned to make one of the paths
more dangerous, allowing us to verify if data is routed in the longer but
safer path. The communication range is set to 1.5m. The topology is
illustrated in Fig. 9. To assess RASS’ performance, we compared it with a hop-
count algorithm over 3 runs.

Figure 9: Topology of the 3x3m environment with 4 drones, a base station and a
radiation source used in the physical experiments.

###  V-B Results

The reliability results of the physical experiments conducted on the drones
are presented in Fig. 10. They show that RASS outperforms the hop-count
algorithm in terms of reliability which lead to overall greater swarm storage.
Even if the topology used to assess the performance of our algorithm was
simple and the number of agents in the system was limited, the physical
experiments confirm the real-world applicability of RASS. Using only local
interactions, it was able to choose safer paths for the data to be routed
through which resulted in fewer data corruptions.

Figure 10: Evolution of reliability over time on the the physical experiments

##  VI Conclusions

We presented RASS, a Risk-Aware Swarm Storage system in which a swarm of
robots can collectively store data on strategically chosen members. This
choice is made without central coordination and is purely based on local
information shared between the robots. This information is simply composed of
risk measurements and topological distance from a robot to a base station, and
used to determine a robot’s fitness to store data as well as to establish the
most reliable and fast route towards the base station.

We show in our experiments that RASS largely outperforms a hop-count-based
solution as well as a virtual stigmergy in terms of reliability and total
swarm storage capacity, while only being slightly slower in terms of
percolation speed compared to the hop-count-based algorithm. RASS showed good
scalability in physics-based experiments as it repeatedly performed well with
a large number of robots. It performed well in experiments on physical robots.

An interesting direction for future work could be to conduct experiments in
more diverse scenarios, for example in search and rescue applications where
image storage and processing is required, therefore increasing the system’s
workload.

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###### List of Figures

  1. 1 Optical layout for our LUVOIR-B baseline mission parameters. We do not explicitly show dual parallel polarization channels for each wavelength channel, which we assume for the baseline coronagraph design.
  2. 2 Wavelength dependent optical throughput for our LUVOIR-B baseline mission. The optical throughput of the UV and VIS imagers are shown as solid blue and black lines, respectively, while the IFS is shown as a black dashed line.
  3. 3 Continuum SNR required for a strong detection of water vapor on an Earth-like exoplanet as a function of the long-wavelength edge of the bandpass for 20% (black) and 40% (red) bandwidth[20]. A broader bandwidth covers more water lines, allowing for detection at a lower continuum SNR. We adopt 20% bandwidth for our baseline mission and examine doubling the number of visible wavelength coronagraph channels in Section 4.2.2.
  4. 4 EEC yield distribution of our baseline mission considering only “luck of the draw” exoplanet sampling uncertainty. Without prior knowledge of which stars host EECs, there is no way to reduce this fundamental uncertainty. This distribution assumes Poisson draws ignoring exoplanet multiplicity and should be regarded as a lower limit on the amount of uncertainty.
  5. 5 _Left:_ EEC yield distribution of our baseline mission with exoplanet sampling and albedo uncertainties for the extreme scenarios in which all EECs turn out to be water-covered cloudless planets (dotted line) or cloud-covered water worlds (dashed line), compared to Earth-twins (solid line). _Right:_ Geometric albedo distributions of injected planets (dotted and dashed) compared to detected planets (solid). The detection rate for dark water worlds is $<$20%, explaining the shift in the yield distribution to lower values.
  6. 6 _Left:_ EEC yield distribution of our baseline mission with exoplanet sampling and albedo uncertainties assuming a distribution of albedo values. Yields are shown for a uniform distribution of water worlds (blue), a uniform distribution of rocky worlds (gray), a uniform distribution of water worlds with $\langle A_{G}\rangle=0.2$ (green), and Earth-twins for comparison (black). _Right:_ Geometric albedo distributions of injected planets (dotted) compared to detected planets (solid) for each of the three albedo distributions assumed. Observations are “tuned” to $A_{G}=0.2$, resulting in lower detection rates for $A_{G}<0.2$. We adopt the green curves as our fiducial albedo distribution.
  7. 7 EEC yield distribution of our baseline mission with exoplanet sampling, albedo, and exozodi sampling uncertainties (orange), compared to the green yield distribution calculated in Section 3.2. Drawing exozodi values from a distribution (as opposed to assigning all stars the same median value) shifts the yield distribution to lower values, as some high priority targets are assigned higher exozodi values.
  8. 8 EEC yield distribution of our baseline mission with exoplanet sampling, albedo, exozodi sampling, and exozodi distribution uncertainties (red), compared to the orange yield distribution calculated in Section 3.3. Drawing exozodi values from all distributions consistent with the LBTI HOSTS data negligibly impacts yield uncertainty.
  9. 9 _Left:_ 10k randomly drawn exozodi levels from 500 randomly drawn distributions when fitting the LBTI HOSTS data with maximum likelihood (solid red), log-normal (dotted black), non-parametric with $\alpha=0.1$ (dashed black), and non-parametric with $\alpha=0.25$ (solid black) approaches. _Right:_ Yield distributions including exozodi distribution uncertainty for each of these approaches; the red curve is identical to the red curve in Figure 8. Adopting a multi-modal non-parametric fit ($\alpha=0.25$) produces similar yields to the LBTI HOSTS maximum likelihood approach[19] because the fraction of distributions $\lesssim 3$ zodis is similar.
  10. 10 HZ completeness of selected targets for one representative run of the LUVOIR-B baseline scenario. Star-to-star variation in completeness is due to random assignment of exozodi levels. Red dashed lines roughly mark the boundaries of the observable targets, while gray dots indicate the range of the input target list.
  11. 11 EEC yield distribution of our baseline mission with all known sources of astrophysical uncertainty: exoplanet sampling, albedo, exozodi sampling, exozodi distribution, and $\eta_{\Earth}$ uncertainties (purple). The red yield distribution is the same as was calculated in Section 3.4, which excludes uncertainty in $\eta_{\Earth}$. The dotted purple line marks the median of the purple distribution. Uncertainties in $\eta_{\Earth}$ substantially broaden the EEC yield distribution.
  12. 12 EEC yield distributions for four different telescope diameters when adopting our baseline mission parameters including $\eta_{\Earth}$ uncertainty (purple) and excluding it (red). Solid, dotted, dashed, and dot-dashed lines correspond to IDs of 6, 7, 8, and 9 m, respectively.
  13. 13 Targets selected for observation for one simulation of the 7, 8, and 9 m ID telescope scenarios, color-coded by total completeness, assuming the same single representative exozodi draw as in Fig. 10. The red dashed lines indicate the boundaries of the accessible targets—the horizontal line roughly marks the assumed astrophysical noise floor while the curved line roughly indicates the working angle limit. The black dashed line indicates the working angle limit of a 6 m ID telescope for reference. Larger telescopes expand the range of accessible targets.
  14. 14 Possible spectral characterization time distributions of the first 18 EECS for our baseline mission parameters with a 6, 7, 8, and 9 m ID telescope shown in solid, dotted, dashed, and dash-dotted lines, respectively. A 9 m ID telescope reduces spectral characterization times by a factor of 6.
  15. 15 Fraction of yield distribution $>$25 EECs, $P_{25}$, when budgeting for detection and water vapor characterization, as a a function of telescope diameter for our baseline mission parameters. The purple line includes $\eta_{\Earth}$ uncertainty and the red line excludes it.
  16. 16 Optical layout for a Cassegrain design with single VIS channel. We do not explicitly show dual parallel polarization channels, which we assume for the baseline coronagraph design. See Figure 1 for legend.
  17. 17 Wavelength dependent optical throughput for a Cassegrain design with single VIS channel imager and IFS (red), an energy-resolving detector (green), and the LUVOIR-B baseline (black). Minimizing aluminum reflections can increase the throughput by 50% at 1000 nm and nearly double it at $\sim$700 nm. Opting for an energy resolving detector can increase the throughput by an additional 40% compared to an IFS.
  18. 18 _Left:_ Yield distributions for all design change scenarios considered, with dotted and solid lines including and excluding $\eta_{\Earth}$ uncertainty, respectively. Table 4 summarizes the design changes included in each scenario. _Right:_ Fraction of yield distribution $>$25 EECs, $P_{25}$, when budgeting for detection and water vapor characterization. Dotted and solid bar plots correspond to including and excluding $\eta_{\Earth}$ uncertainty, respectively.
  19. 19 Spectral characterization time distributions for all design change scenarios considered when excluding $\eta_{\Earth}$ uncertainty. Table 4 summarizes the design changes included in each scenario.
  20. 20 The targets selected for each scenario color-coded by total completeness. The scenarios are cumulative, in that we compile multiple design improvements. The dashed lines indicate the boundaries of the accessible targets—the horizontal line roughly marks the assumed astrophysical noise floor while the curved line roughly indicates the working angle limit.
  21. 21 The incremental change in completeness between successive scenarios. Each design improvement allows the mission to access more distant stars.
  22. 22 Optical layout for a Cassegrain design with two parallel VIS channels. We do not explicitly show dual parallel polarization channels for each wavelength channel, which we assume for the baseline coronagraph design. See Figure 1 for legend.
  23. 23 Skipper CCD raw QE (solid line) and the raw QE values adopted for the yield code when integrating over a 20% bandpass for detection (blue dashed) and spectral characterization (red dashed). The raw QE for the LUVOIR-B baseline EMCCD and energy-resolving detector (Scenario E) is shown for comparison (dotted line).
  24. 24 Optical layout for a Cassegrain design with two parallel VIS channels with energy-resolving detectors. See Figure 1 for legend.
  25. 25 Azimuthally averaged contrast for a star with diameter $0.1$ $\lambda/D$ (dotted line) and core throughput (solid line) for the two coronagraphs included in this study. The DMVC6 and PIAA-FPM2.5 are shown in black and red, respectively.
  26. 26 Fraction of yield distributions $>$25 EECS, $P_{\rm 25}$, as a function of expected EEC yield excluding (solid red) and including (solid purple) uncertainty in $\eta_{\Earth}$. Filled circles indicate yields obtained by changes to telescope diameter while filled triangles with a connecting line indicate those due to design changes. The agreement between lines and circles suggests $P_{\rm 25}$ is independent of the means used to obtain higher yields. The unfilled symbols and dashed lines show the expectation value of the benchmark yield, $Y^{\prime}$, which can be used to estimate $P_{25}$ without calculating a yield distribution.

###### List of Tables

  1. 1 Baseline Astrophysical Parameters
  2. 2 Coronagraph-based Mission Parameters
  3. 3 Detector parameters
  4. 4 Summary of design change impacts