#  : An Exemplar for
Self-Adaptive Underwater Vehicles ††thanks: This work was supported by the
European Union’s Horizon 2020 Framework Programme through the MSCA network
REMARO (Grant Agreement No 956200).

Gustavo Rezende Silva1, Juliane Päßler2, Jeroen Zwanepol1, Elvin Alberts31, S.
Lizeth Tapia Tarifa2,
Ilias Gerostathopoulos3, Einar Broch Johnsen2, Carlos Hernández Corbato1
<EMAIL_ADDRESS>1Technical University of Delft, Delft, The
Netherlands
Email<EMAIL_ADDRESS>2University of Oslo, Oslo,
Norway
Email<EMAIL_ADDRESS>3Vrije Universiteit Amsterdam,
Amsterdam, The Netherlands
Email<EMAIL_ADDRESS>
###### Abstract

Once deployed in the real world, autonomous underwater vehicles (AUVs) are out
of reach for human supervision yet need to take decisions to adapt to unstable
and unpredictable environments. To facilitate research on self-adaptive AUVs,
this paper presents , an exemplar for two-layered system-level adaptation of
AUVs, which clearly separates the application and self-adaptation concerns.
The exemplar focuses on a mission for underwater pipeline inspection by a
single AUV, implemented as a ROS2-based system. This mission must be completed
while simultaneously accounting for uncertainties such as thruster failures
and unfavorable environmental conditions. The paper discusses how can be used
with different self-adaptation frameworks, illustrated by an experiment using
the Metacontrol framework to compare AUV behavior with and without self-
adaptation. The experiment shows that the use of Metacontrol to adapt the AUV
during its mission improves its performance when measured by the overall time
taken to complete the mission or the length of the inspected pipeline.

###### Index Terms:

exemplar, self-adaptation, robotics, underwater robots, Metacontrol, SUAVE

##  I Introduction

Autonomous robots are an excellent case for applying self-adaptation
techniques [1, 2, 3, 4, 5, 7, 6]. These robots face uncertainty in their
operation stemming from both the system (e.g., sensor failures) and the
environment (e.g., different terrains). They need to complete their missions
despite such uncertainty [8] with minimal or no human supervision [9]. A
subclass of these robots, autonomous underwater vehicles (AUVs)[10] which are
used for, e.g., subsea observation, are particularly challenging: once they
have been deployed in the real world, they need to take both low-level (e.g.,
increase thruster power) and high-level (e.g., dive deeper) adaptive decisions
without _any_ human supervision.

Self-adaptive systems can be implemented as two-layered systems consisting of
a _managed_ and a _managing_ subsystem [11]. The managed subsystem handles the
domain concerns, while the managing subsystem implements the adaptation logic
and exploits functional alternatives of the managed subsystem to handle the
self-adaptation process.

This paper proposes the exemplar 111 Self-adaptive Underwater Autonomous
Vehicle Exemplar. to facilitate research in the challenging domain of self-
adaptive AUVs and to allow the comparison of different self-adaptation
strategies. is based on ROS2 – one of the most widely adopted robotics
software frameworks [12]. This ensures that the system built for can (i) run
directly on real robots and not only in simulation environments, (ii) serve as
a basis for other adaptive robotic missions, and (iii) be easily extended with
new functionalities and adaptation concerns. The exemplar is publicly
available at https://github.com/kas-lab/suave.

The exemplar focuses on the scenario of _pipeline inspection_ for a single
AUV. The AUV’s mission is to first search for a pipeline on the seabed, then
follow and inspect the pipeline. The functionalities required to accomplish
this mission are implemented in the _managed subsystem_ of . During the
execution, of the mission, two types of uncertainties are considered:
component failures in the form of thruster failures (e.g., due to debris
getting stuck in a thruster) and changes in the environmental conditions in
the form of changes in the water visibility (e.g., due to currents disturbing
sediment from the seabed). While the first uncertainty may impact the robot’s
motion by making it move unexpectedly, the second impacts the efficiency of
the pipeline search and detection by forcing the robot to be closer to the
seabed to detect the pipeline in case of poor water visibility, which results
in a smaller field of view while searching.

The exemplar enables the development of a _managing subsystem_ to address the
previous uncertainties. The managing subsystem should be able to monitor the
current runtime circumstances, recover the AUV’s thrusters in case of a
thruster failure, and adjust the AUV’s path generation algorithm to account
for changes in water visibility.

To illustrate the use of adaptation frameworks in , the managing subsystem was
implemented with Metacontrol [13, 14], a framework that enables self-
adaptation in robotic systems and promotes the reuse of the adaptation logic
by exploiting a model of the managed subsystem at runtime. Metacontrol’s
strength lies in the separation between the application and adaptation
concerns, i.e., in the separation between the robot’s operation and the logic
of when and how to adapt. This separation of concerns allows the adaptation
logic to be reused in a straightforward way in different applications.
However, it is important to highlight that even though is equipped with a
Metacontrol-based adaptation logic, the exemplar can also be used without
Metacontrol, which in addition allows for comparing other approaches to
Metacontrol-based ones.

In summary, the contributions of this paper are:

  * •

a _self-adaptation exemplar for AUVs using ROS2_ that can be equipped with
different adaptation logics, enables the comparison of different self-
adaptation strategies, forms a basis for other adaptive robotic missions, and
can run both on real robots and in simulation environments; and

  * •

a _Metacontol-based adaptation logic formulation_ that can serve as a baseline
for future research and as a benchmark for self-adaptation strategies, and is
easily reusable for other robotic and non-robotic applications.

_Paper outline._ Section II presents related work, after which Section III
further details the use case and the overall architecture. The managed
subsystem is described in Section IV, while Section V discusses the managing
subsystem and how Metacontrol is applied to the use case. Section VI briefly
explains how the exemplar can be reused and extended, and Section VII presents
and discusses the results of applying Metacontrol. Finally, Section VIII
concludes the paper.

##  II Related work

The UNDERSEA exemplar by Gerasimou _et al._ [7] provides an AUV simulation in
which the robot performs self-adaptation to deal with uncertainties such as
sensor failures and changing goals. is related to UNDERSEA as both address the
domain of self-adaptive AUVs. However, a key difference is the underlying
libraries used to develop software for the robot. UNDERSEA uses MOOS-IvP while
uses ROS2, a more widely used framework that is considered state of the art in
the robotics research community, which contributes to the reusability and
extensibility of .

There have been previous exemplars that do use ROS, in particular, the Body
Sensor Network by Gil _et al._ [15]. However, its application differs
significantly from as it concerns health monitoring through a series of
sensors rather than a robot vehicle fulfilling a mission autonomously.

Cheng _et al._ proposed AC-ROS [6], a framework which uses assurance cases to
endow a ROS-based system with self-adaptive capabilities. Specifically, it
concerns an ‘EvoRally’ vehicle, a terrestrial robot tasked with patrolling an
environment as its mission, while meeting requirements such as energy
efficiency. The authors do not provide the source code of the proposed system,
which means it does not serve as an exemplar as does.

The paper by Bozhinoski _et al._ [14] concerns an earlier iteration of using
MROS for runtime adaptation similar to this paper. Their work revolves around
two cases, a manipulator robot with a “pick and place” task and a mobile robot
navigating around obstacles on a factory floor. Both of the use cases show a
need to deal with uncertainties, e.g., with a safety concern by disabling one
of the pick and place arms. When compared to , the key differences are the
migration from ROS to ROS2, as well as the use case being an AUV rather than a
manipulator or mobile terrestrial robot.

##  III Pipeline inspection exemplar

This section describes the use case and system architecture, the two system
layers are detailed in Sections IV and V.

###  III-A Use case description

The use case in this exemplar is about an AUV inspecting pipelines located on
a seabed. Its mission consists of two sequential tasks, $(T1)$ searching for
the pipeline, then $(T2)$ simultaneously following and inspecting the
pipeline.

When performing its mission, the AUV is subject to two sources of uncertainty
that could trigger self-adaptation: $(U1)$ thruster failures and $(U2)$
changes in water visibility. $U1$ arises from the possibility of the AUV’s
thrusters failing at runtime, which may cause the AUV to move unexpectedly.
This is relevant for both $T1$ and $T2$. To overcome $U1$, the managed
subsystem of the AUV contains functional alternatives. When one or more
thrusters fail, it is possible to enter a recovery state in which the
thrusters are recovered. $U2$ influences the maximum distance at which the AUV
can visually perceive objects. This is relevant for $T1$, higher water
visibility allows the AUV to search for the pipeline at higher altitudes,
resulting in a larger field of view and the possibility of discovering the
pipeline faster. On the other hand, if the water visibility is low, the AUV
has to move closer to the seabed to search for the pipeline, which limits its
field of view and therefore may lead to a longer time to discover the
pipeline. Thus, changing the altitude of the AUV provides functional
alternatives for dealing with $U2$.

This exemplar focuses on the problem of overcoming $U1$ and $U2$ using a self-
adaptation logic, implemented by a managing subsystem, that can be extended
and reused for other sources of uncertainty. The managing subsystem shall
overcome $U1$ by recovering the failed thrusters at runtime, and $U2$ by
adapting the maximum altitude for the path generator algorithm according to
the measured water visibility. Thus, by reacting to $U1$ and $U2$, the
managing subsystem increases the reliability and performance of the system.

For the feasibility of the exemplar, the use case was simplified while still
allowing for a worthwhile application of self-adaptation to an AUV. It is
important to highlight that a realistic operation of an AUV used for pipeline
inspection would include steps that are related to pre-dive, launching and
recovery, human interaction, and intermediary missions that are necessary to
enable the inspection. Furthermore, there are several sources of uncertainty
not considered here, including ocean dynamics, sensor failures, and battery
duration.

###  III-B System Architecture

To accomplish the mission described in Section III-A, the managed subsystem
requires the functions represented in Fig. 1. $T1$ requires the functions
Control Motion, Maintain Motion, Localization, Detect Pipeline, Generate
Search Path, and Coordinate Mission, while $T2$ requires the functions Control
Motion, Maintain Motion, Localization, Detect Pipeline, Follow Pipeline,
Inspect Pipeline, and Coordinate Mission. During runtime, the functions must
be activated and deactivated according to the task being performed.

Figure 1: Managed Subsystem’s Functional Hierarchy

To overcome the uncertainties $U1$ and $U2$, a managing subsystem requires the
functionalities to _monitor_ the environment and the managed subsystem’s
internal state, _reason_ about it, and _execute_ the managed subsystem’s
reconfiguration.

Figure 2: System Architecture

The required functions of the managed and managing subsystems are realized as
depicted in Fig. 2. The managed subsystem is detailed in Section IV and the
managing subsystem in Section V. It is important to mention that managed
subsystem functions Control Motion and Localization are achieved by ArduSub,
and the function Inspect Pipeline is not realized since the actual inspection
of the pipeline is not the focus of this work. It is also important to
highlight that this exemplar implements the function to _reason_ about the
managing subsystem with Metacontrol to provide a baseline for future research.
However, it can be replaced with other solutions, as long as they are
compatible with the monitor and execute interfaces, as described in Section
VI.

##  IV Managed Subsystem

The managed subsystem is implemented as a ROS2-based system and is depicted in
Fig. 2. The only non-ROS2 component is ArduSub222https://www.ardusub.com/,
which is an open-source autopilot for underwater vehicles. In this application
it is used to solve the functions Control Motion and Localization333It is
assumed that the AUV has appropriate sensors for localization. The MAVROS
package works as a bridge between ArduSub and the ROS2 components. The Detect
Pipeline node detects the pipeline and informs Follow Pipeline and the
Coordinate Mission node about its position444A mock perception system is
used.. The Coordinate Mission node coordinates the tasks’ execution and sets
the adaptation goals. Note that the function Inspect Pipeline is not
implemented, since the actual inspection of the pipeline is not the focus of
this work. However, the exemplar can easily be extended with this
functionality by adding a new node that implements the pipeline inspection.

Follow Pipeline, Generate Search Path, and Maintain Motion are lifecycle
nodes, which means that they have internal states, such as _active_ and
_inactive_ , and it is possible to switch between these states at runtime.
Furthermore, the System Modes package [16] extends the state _active_ with
additional modes, e.g., _active.low_altitude_.

To adapt the managed subsystem, the managing subsystem adapts the lifecycle
nodes by changing their states. This is done by the Mode Manager node, which
is used off-the-shelf from the System Modes package. The states available for
Generate Search Path are deactivated, low altitude, medium altitude, and high
altitude. Subsequently, the states available for Follow Pipeline are
deactivated and activated, while the states for Maintain Motion are
deactivated, and recover thrusters.

To enable other developers of self-adaptive systems to use this exemplar and
compare different approaches, a Gazebo-based 555https://gazebosim.org/home
simulation of a pipeline inspection environment and a model of the AUV is
provided. The BlueROV2666https://bluerobotics.com/store/rov/bluerov2/ robot
was selected as the AUV for the exemplar because (i) it is compatible with
ArduSub; (ii) it is easily integrated with Gazebo via plugins; and (iii) the
robot has a low price compared to other available AUVs, making it more
accessible to researchers to reproduce the exemplar with a real robot.

##  V Managing Subsystem

The managing subsystem exploits functional alternatives of the managed
subsystem to enable adaptation and thereby increase system reliability.
Metacontrol is used as an example of how a managing subsystem can be
implemented. This section introduces Metacontrol and shows how the adaptation
problem can be formulated and implemented with Metacontrol.

###  V-A Metacontrol Background

_Metacontrol_ uses the MAPE-K feedback loop [17, 18] to implement self-
adaptation. It _Monitors_ the managed subsystem during runtime, _Analyzes_
whether the system meets its requirements, _Plans_ a new configuration if the
system does not meet the requirements, and then _Executes_ the reconfiguration
of the managed subsystem. All this is done using a shared _Knowledge Base_ to
which each step refers. In Metacontrol, the knowledge base conforms to the
TOMASys (Teleological and Ontological Metamodel for Autonomous Systems)
metamodel [13].

Figure 3: A simplified representation of the TOMASys elements

A simplified version of the TOMASys metamodel is displayed in Fig. 3. TOMASys
uses _functions_ $F$ to represent the functionalities of the system, e.g.,
generating a search path for the AUV. The architectural variants that
implement these requirements are captured by _function designs_
$FD(F,\mathcal{C},\operatorname{\mathcal{QA}}^{exp})$. To distinguish during
runtime which function design is most suited in a given situation, a set
$\operatorname{\mathcal{QA}}^{exp}$ of _expected quality attributes_ is
associated with it. An expected quality attribute value reflects how well a
function design is supposed to fulfill the function $F$ it solves.
Furthermore, a function design requires a set $\mathcal{C}$ of _components_ of
the managed subsystem to solve $F$. A component $C(S_{C})$ is a piece of
hardware or software, e.g., a sensor or a path-planning algorithm,
respectively. The status $S_{C}$ of a component indicates its availability,
i.e., whether it is functioning or not.

An _objective_ $O(F,S_{O},\operatorname{\mathcal{QA}}^{req})$ is a runtime
instantiation of a function $F$, e.g., generating a search path with a minimum
required water visibility, whose status $S_{O}$ reflects whether the objective
is currently achieved. Furthermore, the set
$\operatorname{\mathcal{QA}}^{req}$ of _required quality attributes_ specifies
which quality attribute value the objective requires in order to work
properly.

An objective $O$ is solved by a _function grounding_
$FG(O,FD,S_{FG},\operatorname{\mathcal{QA}}^{meas})$, which represents the
function design $FD$ that is currently used to solve the objective. Its status
$S_{FG}$ reflects whether the function grounding is currently able to achieve
the objective. The set $\operatorname{\mathcal{QA}}^{meas}$ of _measured
quality attributes_ reflects how well the function grounding currently
fulfills $O$ and is computed using sensor data.

###  V-B Metacontrol Formulation

The functions, architectural variants, and quality attributes required to
solve the tasks $(T1)$ Search Pipeline and $(T2)$ Inspect Pipeline, described
in Section III, are modeled conforming to the TOMASys metamodel. Table III
specifies the functions ($F_{1}$) maintain_motion, ($F_{2}$)
generate_search_path, and ($F_{3}$) follow_pipeline, while Table III describes
the quality attributes ($QA_{1}$) water_visibility, and ($QA_{2}$)
performance. Functions $F_{1}$ and $F_{2}$ are required to achieve $T1$,
whereas $T2$ is achieved by $F_{1}$ and $F_{3}$. The function designs that
solve these functions are specified in Table III. The set of required
components is empty for function designs $FD_{2}-FD_{6}$ because they do not
require any components that are susceptible to adaptation, or used in the
reasoning process.

Since the objectives and function groundings are instantiated during runtime,
they are not specified here. An objective for function $F_{2}$ is for example
to generate a search path with no required quality attribute, which is defined
as $O_{2}(F_{2},\text{\emph{ok}},\varnothing)$ in the notation introduced
above. A possible function grounding for this objective is
$FG_{2}(O_{2},FD_{4},\text{\emph{ok}},\\{QA_{1}^{meas}=1.1\\})$.

TABLE I: The TOMASys functions used for the exemplar

Function | Name |  Requirement
---|---|---
$F_{1}$ | maintain_motion |  Maintain the motion of the robot
$F_{2}$ | generate_search_path |  Generate a path to search for the pipeline
$F_{3}$ | follow_pipeline |  Follow and inspect the pipeline

TABLE II: The TOMASys quality attributes used for the exemplar

QA | Name | Unit |  Description
---|---|---|---
$QA_{1}$ | water_visibility | [$0,\infty)$ |  Reflects the maximum altitude (in meters) from which the AUV can perceive the seabed
$QA_{2}$ | performance | $[0,1]$ |  Reflects how efficient the current search strategy is

TABLE III: The TOMASys function designs used for the exemplar

Function Design |  Name |  Description
---|---|---
$FD_{1}(F_{1},\\{\text{thruster}\\_x\mid x=1,\dots,6\\},\\{QA_{2}^{exp}=1\\})$ |  all_thrusters |  Uses all thrusters
$FD_{2}(F_{1},\varnothing,\\{QA_{2}^{exp}=0.5)$ |  recover_thrusters |  Recovers the thrusters that are in failure
$FD_{3}(F_{2},\varnothing,\\{QA_{1}^{exp}=0.5,$ $QA_{2}^{exp}=0.25\\})$ |  spiral_low |  Produces a spiral search path with low altitude
$FD_{4}(F_{2},\varnothing,\\{QA_{1}^{exp}=1,$ $QA_{2}^{exp}=0.5\\})$ |  spiral_medium |  Produces a spiral search path with medium altitude
$FD_{5}(F_{2},\varnothing,\\{QA_{1}^{exp}=2,$ $QA_{2}^{exp}=1\\})$ |  spiral_high |  Produces a spiral search path with high altitude
$FD_{6}(F_{3},\varnothing,\varnothing)$ |  follow_pipeline |  Follows the pipeline

The MAPE-K loop steps in this exemplar are formulated as follows. The monitor
step is responsible for measuring $QA_{1}^{meas}$ and for monitoring the state
of the six thrusters.

The analyze step uses Horn rules to reason about the knowledge base. One
example rule that analyzes whether the measured water visibility
$QA_{1}^{meas}$ still satisfies the expected water visibility $QA_{1}^{exp}$
of the grounded function design is displayed in Fig. 4. Note that it is
written in terms of the notation introduced in Section V-A. Line 1 expresses
that the rule reasons about a function grounding $FG$ that solves an objective
$O$, is of type $FD$, has a status $S_{FG}$ and an associated set of measured
quality attributes $\operatorname{\mathcal{QA}}^{meas}$. Furthermore, the
function design $FD$ solves the function $F$, has a set of required components
$\mathcal{C}$ and an associated set of expected quality attributes
$\operatorname{\mathcal{QA}}^{exp}$. Note that it is implicitly assumed that
$FG$ is well-formed, i.e., that the function of which $O$ is a type of is the
same as the function that $FD$ solves. Since this rule should analyze the
water visibility, Line 2 ensures that $QA_{1}^{meas}$ is an element of the set
$\operatorname{\mathcal{QA}}^{meas}$ and that $QA_{1}^{exp}$ is an element of
the set $\operatorname{\mathcal{QA}}^{exp}$, i.e., that both $FG$ and $FD$ are
related to water visibility. Finally, if the measured value of $QA_{1}$ is
less than its expected value associated with the grounded function design, see
Line 3, then the status of the function grounding is set to _error_ , see Line
4.

$\displaystyle
FG\bigl{(}O,FD,S_{FG},\operatorname{\mathcal{QA}}^{meas}\bigr{)}\boldsymbol{\wedge}~{}FD(F,\mathcal{C},\operatorname{\mathcal{QA}}^{exp})$
(1)
$\displaystyle\boldsymbol{\wedge}~{}QA_{1}^{meas}\in\operatorname{\mathcal{QA}}^{meas}~{}\boldsymbol{\wedge}~{}QA_{1}^{exp}\in\operatorname{\mathcal{QA}}^{exp}$
(2) $\displaystyle\boldsymbol{\wedge}~{}QA_{1}^{meas}<QA_{1}^{exp}$ (3)
$\displaystyle\boldsymbol{\Rightarrow}~{}S_{FG}=\text{\emph{error}}$ (4)
Figure 4: Rule to analyze whether the measured water visibility
$QA_{1}^{meas}$ still satisfies the expected water visibility $QA_{1}^{exp}$
of the grounded function design

In the planning step, the function designs with $QA_{1}^{exp}$ higher than
$QA_{1}^{meas}$ are filtered out as the visibility they would expect is not
measured, afterward the remaining function design with the highest expected
search performance ($QA_{2}^{exp}$) is selected as the desired configuration.
The selected configuration is then carried out in the execute step.

###  V-C Metacontrol Implementation

As depicted in Fig. 2, the monitor step is implemented with the Water
Visibility Observer and the Thruster Monitor nodes. They are used for
measuring $QA_{1}$ and monitoring the status of the six thrusters
$\text{thruster}\\_x$ where $x\in\\{1,\dots,6\\}$, respectively. To simplify
the system and avoid the addition of unnecessary nodes, instead of adding
water visibility to the Gazebo simulator, the Water Visibility Observer
simulates water visibility measurements with a sine function, and instead of
probing the managed subsystem to identify thruster failures the Thruster
Monitor simulates the thruster failures events. Since the monitor step is
mocked up, and its probes and intermediary nodes that would be required to
provide the probes are not implemented, they are not included in Fig. 2. Both
nodes publish their data into the /diagnostics topic with the ROS2 default
DiagnosticArray message type.

The knowledge base (KB), the analyze and plan step are implemented using MROS2
777https://github.com/meta-control/mc_mros_reasoner [19], a ROS2-based
Metacontrol implementation, as the MROS Reasoner node. The KB is implemented
with the Ontology Web Language (OWL) [20], the Horn rules used for the analyze
step with the Semantic Web Rule Language (SWRL) [21], and the reasoning is
done with Pellet888https://github.com/stardog-union/pellet. The MROS Reasoner
receives water visibility measurements ($QA_{1}^{meas}$) and thruster status
information from the monitor step, then decides whether adaptation is
required, and, in this case, selects a desired configuration which it sends to
the execute step (see Section V-B for more details). The MROS Reasoner
initially does not have objectives, so it does not perform adaptation. The
adaptation reasoning only starts when the Coordinate Mission node sends new
objectives, such as $O_{2}(F_{2},null,\varnothing)$, via the Adaptation Goal
Bridge. New objectives do not have a status yet.

The execute step uses the System Modes’ Mode Manager to adapt the managed
subsystem, and the System Modes Bridge bridges the Mode Manager with the MROS
Reasoner. When a reconfiguration is needed, the MROS Reasoner requests the new
configuration via the /mros/request_configuration service to the System Modes
Bridge.Then the System Modes Bridge forwards the request to the Mode Manager
using the correct service names, depending on the lifecycle node being
adapted. The services used by the Mode Manager are listed in Table V, and the
available modes are listed in Table V.

TABLE IV: Available System Modes’ services

Node |  Service
---|---
f_generate_search_path |  /f_generate_search_path/change_mode
f_follow_pipeline |  /f_follow_pipeline/change_mode
f_maintain_motion |  /f_maintain_motion/change_mode

TABLE V: Available modes

Node |  Mode | Lifecycle state
---|---|---
f_generate_search_path |  fd_spiral_high | active
f_generate_search_path |  fd_spiral_medium | active
f_generate_search_path |  fd_spiral_low | active
f_generate_search_path |  fd_unground | inactive
f_follow_pipeline |  fd_follow_pipeline | active
f_follow_pipeline |  fd_unground | inactive
f_maintain_motion |  fd_all_thrusters | inactive
f_maintain_motion |  fd_recover_thrusters | active

##  VI Extending and connecting managing subsystems

With the described system implementation, the only Metacontrol-specific nodes
of the system are the MROS Reasoner, the System Modes Bridge, and the
Adaptation Goal Bridge. All other nodes of the system can be reused with
different managing subsystems. The only requirement to connect a managing
subsystem to the managed subsystem is to ensure that the managing subsystem
adheres to the provided monitor and execute ROS2 interfaces. As described in
the previous section, the monitor interface is the ROS2 topic /diagnostics,
and the execute interfaces are listed in Table V. To show that changing the
managing subsystem is possible, a managing subsystem that randomly picks a
configuration was also implemented.

Since the system is implemented with a modular design, it can be extended with
additional functionalities and adaptation scenarios by adding new lifecycle
nodes and updating the system modes’ configuration file accordingly. The
implemented functionalities can be replaced with different implementations as
long as they adhere to the same interfaces; e.g., the Pipeline Detection node
could be replaced by a node that actually performs perception instead of a
mock-up.

##  VII Evaluation

To evaluate the performance of different managing subsystems using this
exemplar, the mission described in Section III was implemented. The mission
consists of the AUV performing $T1$ and $T2$ while subject to $U1$ and $U2$
until a user-provided time limit is reached. To evaluate the mission, the
following metrics were used: the _search time_ , the amount of time elapsed
from the beginning of the search until the pipeline is found, and the total
_distance inspected_ of the pipeline.

To provide a baseline for the exemplar, the mission is performed with two
different managing subsystems and with no managing subsystem, using a fixed
configuration. The managing subsystems are the Metacontrol-based
implementation detailed in Section V and a random managing subsystem that
selects configurations arbitrarily.

Since the system is non-deterministic due to characteristics of Gazebo,
ArduSub, and the interaction between them, no run of the simulation is exactly
the same. Thus, the mission execution and metrics collection are automated
with a runner to allow multiple runs to be easily performed.

This section briefly describes how to configure the exemplar, and the results
of running the exemplar. Further details may be found in the exemplar
repository.

###  VII-A Configuring the exemplar

In , the AUV’s mission execution can be varied by changing the parameters of
the system. In the Water Visibility Observer, the available parameters are the
water visibility minimum and maximum values, periodicity, and initial phase
shift. In the Thrusters Monitor, the available parameter is a list with
thruster events indicating which thruster fails and when. In the Coordinate
Mission, the mission time limit can be set. In the random manager, the
adaptation periodicity can be set, and when using no manager, the default
states for the lifecycle nodes can be set. In addition, the runner is
parametrized with the number of runs to execute, and which managing subsystem
to select. All parameters are adjusted using configuration files packaged in
the exemplar.

###  VII-B Results

The mission was executed with a time limit of 300 seconds, water visibility
periodicity of 80 seconds, minimum and maximum values of 1.25 and 3.75, no
phase shift, and thruster 1 failing after 35 seconds from the start of the
mission. The results are shown in Table VI. It can be noticed that with the
Metacontrol managing subsystem both mean _search time_ 999When the pipeline is
not found, the time limit is used as the _search time_ is lower, and the
_distance inspected_ is higher. This indicates that, in this exemplar,
Metacontrol improves the performance of the system, and outperforms the random
managing subsystem and the system without a managing subsystem. In addition,
the standard deviation (Std) of the _search time_ is lower for Metacontrol,
indicating that it is more consistent when searching for the pipeline. The Std
of the random manager for the _distance inspected_ is lower, however, its mean
value is also low, indicating that the random manager is consistent in not
inspecting the pipeline. The results shown can be used as a baseline for
comparing different managing subsystems.

TABLE VI: Mission results

Managing subsystem |  Number of runs | Search time (s) | Distance inspected (m)
---|---|---|---
Mean | Std | Mean | Std
None | 20 | 187.85 | 42.40 | 31.40 | 13.00
Random | 20 | 180.15 | 82.05 | 3.88 | 3.19
Metacontrol | 20 | 106.95 | 39.46 | 51.15 | 12.01

##  VIII Conclusion

This work describes , a ROS2-based exemplar for self-adaptive underwater
vehicles used for pipeline inspection. Due to its modular design, enables
different managing subsystems to be applied to the system without the need to
modify the managed subsystem, the monitor nodes, and the executing mechanism.
In addition, the system can be easily extended with new functionalities and
adaptation scenarios by adding new nodes. Furthermore, this paper provides a
baseline for comparing the performance of different managing subsystems, and
it shows that the addition of a Metacontrol-based managing subsystem increases
the performance of the system in comparison to not using any managing
subsystem or one that chooses configurations arbitrary.

In future work, can be extended with: more metrics for a more in-depth
evaluation; more tasks (e.g., docking), functionalities (e.g. a de facto
perception system), and components (e.g. sonars) for more realistic missions;
more adaptation scenarios, e.g., adapting to changes in the water currents,
and adapting the thruster configuration matrix when a thruster can not be
recovered.

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# Absence of fractal quantum criticality in the quantum Newman-Moore model

Raymond Wiedmann  Department of Physics, Staudtstraße 7, Friedrich-Alexander-
Universität Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany  Max-Planck-
Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany
Lea Lenke  Department of Physics, Staudtstraße 7, Friedrich-Alexander-
Universität Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany  Matthias
Mühlhauser  Department of Physics, Staudtstraße 7, Friedrich-Alexander-
Universität Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany  Kai Phillip
Schmidt  Department of Physics, Staudtstraße 7, Friedrich-Alexander-
Universität Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany

###### Abstract

The quantum phase transition between the low-field fracton phase with type-II
fracton excitations and the high-field polarized phase is investigated in the
two-dimensional self-dual quantum Newman-Moore model. We apply perturbative
and numerical linked-cluster expansions to calculate the ground-state energy
per site in the thermodynamic limit revealing a level crossing at the self-
dual point. In addition, high-order series expansions of the relevant low-
energy gaps are determined using perturbative continuous unitary
transformations indicating no gap-closing. Our results therefore predict a
first-order phase transition between the low-field fracton and the high-field
polarized phase at the self-dual point.

Fracton phases of matter are connected to intriguing phenomena such as
topological order, glassy dynamics, and spin liquids [1]. Due to this and
their potential applications in quantum memories [2], there has been
increasing interest in models exhibiting such phases in the last years [3, 4,
5, 6]. One of the most important properties of systems with fracton order is
the restricted mobility of their low-energy elementary excitations, dubbed
fractons [1], which can be of type I or II. In type-I fracton phases,
topologically non-trivial composites of the low-energy excitations are still
mobile on the lattice, but usually restricted to lower dimensions (sub-
manifolds), i.e., in 3D they can only move on planes or along lines [7]. In
contrast, type-II fractons are fully immobile on the lattice.

Well known examples for models exhibiting fracton properties in 3D are the
X-Cube model, as introduced in [7], with type-I fractons, and Haah’s code [8]
as an example for a type-II fracton order. The quantum robustness of the
fracton order in these systems has been studied using quantum Monte Carlo
simulations [9] and perturbative linked-cluster expansions [10]. It is found
that in both models the fracton phase breaks down by a first-order quantum
phase transition when applying an external field. The same behavior is found
when studying the competition of fracton order in the X-Cube model and
intrinsic topological order in the 3D toric code [11]. The occurrence of only
first-order transitions can be traced back to the (partial) immobility of the
fracton excitations since the low-energy excitations can not lower their
energy sufficiently by quantum fluctuations to induce a second-order phase
transition.

Fascinatingly, the two-dimensional quantum version of the Newman-Moore model
[12], exhibiting type-II fracton order, has been recently suggested to display
a continuous quantum phase transition with an exotic fractal criticality based
on quantum Monte Carlo simulations and field-theoretic arguments [13]. Here
the low-energy quantum dynamics in the low-field fracton phase is intimately
linked to 2D fractal structures of Sierpinksi triangles giving rise to glassy
dynamics at finite temperature in the classical Newman-Moore Model.
Furthermore, the quantum Newman-Moore model (qNM) features an exact self-
duality relating the energy spectra of the low-field fracton and the high-
field polarized phase. The zero-temperature phase transition between these
phases is therefore located at the self-dual point [13, 14]. However, the
quantum Monte Carlo simulation performed in [13] is challenging due to the
glassy dynamics as well as finite-size effects and, in addition, earlier
numerical calculations in [14] based on transition path sampling indicate a
first-order transition. In this letter, we clarify this situation by applying
perturbative and numerical linked-cluster expansions to investigate this phase
transition directly in the thermodynamic limit and at zero temperature. Our
results for the ground-state energy and the relevant low-energy excitation
energies predict a first-order phase transition between the low-field fracton
and the high-field polarized phase at the self-dual point in the qNM.

Figure 1: Low-field limit $\Gamma=0$ [high-field limit $J=0$] of the qNM on
the triangular lattice with spins-1/2 indicated as black arrows and three-spin
interactions acting on downward-pointing triangles $\triangledown$ is
illustrated in a) [in b)]. a) In the low-field limit $\Gamma=0$ the fully
polarized state with spins pointing in $z$-direction is one ground state so
that all pseudo-spins on downward-pointing triangles point also upwards. A
single spin-flip changes the eigenvalue of the three attached pseudo-spins on
downward-pointing triangles which is illustrated in red. The three spin-flip
excitation on the next larger Sierpinski triangle is shown in blue. b) In the
high-field limit $J=0$ the fully polarized state with spins pointing in
$x$-direction is the unique ground state. Fully mobile excitations in the
high-field limit are associated with flipping the three spins on a downward-
pointing triangle (shown in red) as well as on the first and second non-
trivial Sierpinski triangle illustrated in blue and orange, respectively.

#### Quantum Newman-Moore model:

The qNM is the spin-1/2 Newman-Moore model [12] in a transverse field

$\displaystyle\mathcal{H}_{\mathrm{qNM}}=-J\sum_{\triangledown}\sigma_{i}^{z}\sigma_{j}^{z}\sigma_{k}^{z}-\Gamma\sum_{i}\sigma_{i}^{x}\,.$
(1)

Here the sum over $\triangledown$ ($i$) runs over the downward-pointing
triangles (sites) of a triangular lattice as illustrated in Fig. 1. The first
term of the Hamiltonian is a three-spin Ising interaction with $J>0$ acting
only on downward-pointing triangles $\triangledown$ while the second term is
the transverse magnetic field with field strength $\Gamma>0$. We note that the
number of sites $N$ is equal to the number of downward-pointing triangles.

Each three-spin Ising interaction has eigenvalues $\pm 1$. One can therefore
introduce a pseudo-spin-1/2 $\tau_{\mu}$ at the center $\mu$ of each
$\triangledown$ so that $-J\sum_{\mu}\tau_{\mu}^{x}$ yields the correct energy
of the three-spin Ising interactions. The local field operator
$\sigma_{i}^{x}$ flips the eigenvalues of the three Ising interactions
(pseudo-spins) containing site $i$ so that the dual qNM reads

$\displaystyle\mathcal{H}^{\rm
dual}_{\mathrm{qNM}}=-J\sum_{\mu}\tau_{\mu}^{x}-\Gamma\sum_{\vartriangle_{\rm
dual}}\tau_{\mu}^{z}\tau_{\nu}^{z}\tau_{\kappa}^{z}$ (2)

which is defined on the triangular lattice formed by the centers $\mu$ of each
$\triangledown$ [see Fig. 1a)]. The qNM is therefore self-dual and the energy
spectra of the low- and high-field phase are isospectral. Consistently, the
quantum phase transition is found at the self-dual point $J=\Gamma$ [13, 14].

The self-duality does not hold for degeneracies and states. In the low-field
phase $\Gamma<J$, the qNM displays type-II fracton order. For the limiting
case $\Gamma=0$ ground states have all eigenvalues of three-spin interactions
$+1$. This is either realized when all spins point in positive $z$-direction
[see Fig. 1a)] or when two of the three spins per triangle are flipped.
Therefore, the ground-state manifold is highly degenerate. Within the fracton
phase $0<\Gamma<J$, one finds that the degeneracy scales sub-extensively with
$N$ [13] which is one hallmark of fracton order. In contrast, in the high-
field phase with $\Gamma>J$, the qNM is in a featureless polarized phase with
a non-degenerate ground state. For $J=0$ this ground state is the product
state where all spins point in positive $x$-direction [see Fig. 1b)].

Elementary quasiparticle (QP) excitations in the high-field phase correspond
to dressed spin-flip excitations, while the elementary fracton excitation in
the low-field phase is related to a negative eigenvalue of a three-spin Ising
interaction which becomes dressed at finite fields (see Fig. 1 for both
phases). Due to the self-duality, it is sufficient to determine the energetic
properties of the trivial high-field phase. The associated energies in the
low-field fracton phase can be deduced directly by interchanging $J$ and
$\Gamma$.

#### Linked-cluster expansions:

We use a variant of linked-cluster expansions [15, 16] designed for multi-spin
interactions as perturbation [17] to calculate the ground-state energy per
site and relevant excitation energies of the high-field polarized phase in the
thermodynamic limit. For this phase the three-spin interactions link always
the three sites of a $\triangledown$. One therefore performs a full graph
decomposition in terms of connected $\triangledown$s. The calculation on the
linked, i.e., contributing, graphs can be done either perturbatively or
numerically using exact diagonalization. After embedding all graph
contributions into the thermodynamic limit, one then obtains either a high-
order series expansion in $J/\Gamma$ or a numerical data sequence for fixed
ratio $J/\Gamma$. While the perturbative linked-cluster expansion is exact up
the calculated perturbative order, the numerical linked-cluster expansion
(NLCE) [18] includes all quantum fluctuations in the thermodynamic limit up to
the length scale set by the maximally extended graph.

Figure 2: Series expansion of the ground-state energy per site $E_{0}/(\Gamma
N)$ up to order 26 in $J/\Gamma$ for the low- and high-field phase. The
individual orders of the series are shown with increasing opacity. By
construction, the series intersect in all orders at the self-dual point
$J/\Gamma=1$ marked by the vertical dashed line (left inset zoom close to the
self-dual point). The right inset shows the angle $\beta$ between the low- and
high-field energy expression from the perturbative (empty circles) and
numerical linked-cluster expansions (filled circles) as a function of
$1/\mathrm{order}$.

Concretely, for the ground-state energy per site $E_{0}/(\Gamma N)$, we apply
perturbative and numerical linked-cluster expansions. We consider all 186061
non-isomorphic connected clusters from 1 to 13 triangles. This allows us to
calculate the perturbative series of $E_{0}/(\Gamma N)$ up to order 26 in
$J/\Gamma$ in the thermodynamic limit using matrix perturbation theory [16].
Here the perturbation always has to act on each $\triangledown$ an even number
of times so that graphs up to 13 triangles are sufficient in order 26 and only
even orders contribute in the series expansion. Within the NLCE we define the
term order to be twice the maximal number of triangles of the involved
clusters which again yields a maximal order 26.

To derive high-order series expansions of relevant excitation gaps in the
high-field phase, we use perturbative continuous unitary transformations
(pCUTs) [19, 20]. The pCUT method maps Hamiltonian (1) perturbatively in
$J/\Gamma$ to an effective Hamiltonian, which conserves the number of
(dressed) spin-flip excitations corresponding to the elementary QP of the
high-field phase. The effective Hamilton is therefore block-diagonal and each
QP block can be treated separately. In this work we calculated the low-energy
excitation energies of the 1QP, 2QP, and 3QP sector (see Supplementary
Materials [21]).

Due to the exact self-duality of the qNM, the energetic properties of the QP
are identical to the fracton excitations in the low-field phase. As a
consequence, individual QPs are strictly local while pairs of QPs are always
only linked to a finite number of 2QP configurations so that one is left with
diagonalizing finite matrices in the 2QP sector. We denote the lowest
excitation gaps in the 1QP and 2QP sector by $\Delta_{1}$ and $\Delta_{2}$,
which we have calculated perturbatively up to order $14$ and $12$ in
$J/\Gamma$, respectively. The 3QP sector contains low-energy excitations which
are fully mobile. Indeed, considering three spin flips on the same
$\triangledown$, this configuration can either hop to other $\triangledown$ or
can be deformed to other 3QP configurations where the three spin-flips are
located at the corners of Sierpinski triangles of arbitrary size. These 3QP
configurations on Sierpinski triangles can again hop. In the calculated
perturbation order $12$ this part of the 3QP sector contains three different
types of Sierpinski triangle 3QP configurations with side length one, two, and
four [see Fig. 1b)]. Exploiting translational invariance, this infinitely
large sector can then be reduced to a $3\times 3$-matrix for fixed momentum.
We find that the energy gap in the 3QP sector is located at zero momentum and
we denote the three eigenvalues as $\Delta_{3}^{(n)}$ with $n\in\\{1,2,3\\}$.
The $\Delta_{3}^{(n)}$ can then be extracted as series expansions in order
$12$ by diagonalizing the $3\times 3$-matrix order by order.

For our analysis, we can extract information about the behaviour of the system
directly from the bare perturbative and numerical series and, additionally,
from extrapolations of the perturbative series which can increase the radius
of convergence and give access to potential critical exponents. Here we use
DlogPadé extrapolation techniques which are described in the Supplementary
Materials [21]. This includes biased DlogPadé extrapolations where we
incorporate that the phase transition takes place at the self-dual point
$J=\Gamma$.

#### Ground-state energy:

The ground-state energy per site $E_{0}/(\Gamma N)$, as obtained by
perturbative and numerical linked-cluster expansions in order 26, is shown for
both phases in Fig. 2. All displayed results are well behaved and well
converged. The low- and high-field energies intersect by construction exactly
at the self-dual point $J/\Gamma=1$ with a clearly visible kink. This kink can
be quantified by the angle

$\beta=\left(\lim_{J/\Gamma\downarrow 1}-\lim_{J/\Gamma\uparrow
1}\right)\Bigg{|}\arctan\bigg(\frac{\partial}{\partial(J/\Gamma)}\frac{E_{0}}{\Gamma
N}\bigg{missing})\Bigg{|}$ (3)

which is shown in the right inset of Fig. 2 as a function of
$1/\mathrm{order}$ using the perturbative or numerical linked-cluster
expansions. One finds that both agree in high orders yielding a finite angle
$\beta\approx$15\text{\,}\mathrm{\SIUnitSymbolDegree}$$ in the infinite-order
limit. A rough lower bound for the angle at infinite order can be obtained by
performing a linear fit for the three values of highest order, resulting in
$\beta\approx$15.23(1)\text{\,}\mathrm{\SIUnitSymbolDegree}$$
[$\beta\approx$15.19(3)\text{\,}\mathrm{\SIUnitSymbolDegree}$$] for the
perturbative [numerical] data sequence. This finite angle corresponding to the
presence of a kink in the ground-state energy at the self-dual point is a
quantitative evidence that the quantum phase transition in the qNM between the
low-field fracton phase and the high-field polarized phase is first order.

Figure 3: Critical exponent $\alpha$ from various biased DLog Padé
approximants of the second derivative of $E_{0}/(\Gamma N)$ with respect to
$J/\Gamma$ displayed as a function of the order of the DLog Padé approximant.
The results are grouped in families ([L,L+C]) with order 2L+C) and shown for
$|C|\leq 5$. The inset plot shows the average value for each order with the
sample standard deviation. For the lowest order value only a single value is
available, marked by a circle.

Next, we check the reliability of these considerations by applying DLog Padé
approximation to the perturbative series of the second derivative of
$E_{0}/(\Gamma N)$ with respect to $J/\Gamma$. We bias the DLog Padé
approximation to have a pole at the self-dual point $J/\Gamma=1$. This allows
then to extract the critical exponent $\alpha$. A first-order phase transition
implies $\alpha=0$. Our results for the critical exponent $\alpha$ are shown
in Fig. 3 which are again well converged for the high-order DLog Padé
approximants. We obtain a critical exponent of $\alpha_{\mathrm{bias}}\approx
0.007(12)$ taking the average of the highest order value for every family.
This finding is obviously fully consistent with a first-order phase
transition.

#### Excitations gaps:

Figure 4: Low-energy excitation gaps $\Delta_{1}$ of the 1QP, $\Delta_{2}$ of
the 2QP, and $\Delta_{3}^{(1)}$ and $\Delta_{3}^{(2)}$ in the 3QP sector are
shown as a function of $J/\Gamma$ using average and standard deviation of non-
defective DLog Padé approximants. The vertical dashed line indicates the self-
dual point.

Our analysis of the ground-state energy revealed a first-order phase
transition at the self-dual point $J/\Gamma=1$. As a consequence, one expects
that all low-energy excitation energies of the high-field phase remain finite
for $J/\Gamma\leq 1$. This we investigate next by analyzing the series of the
low-energy excitation energies in the 1QP, 2QP, and 3QP sector. Due to the
restricted mobility of single and pairs of QP excitations, we do not expect
the associated gaps in these sectors to close. In contrast, the 3QP sector
contains fully mobile excitations which at the same time are superpositions of
Sierpinski triangle 3QP configurations. Here one expects quantum fluctuations
to be most important.

In Fig. 4, we show averaged DLog Padé approximations of the 1QP and 2QP gaps
$\Delta_{1}$ and $\Delta_{2}$ as well as the two lowest excitation energies
$\Delta_{3}^{(1)}$ and $\Delta_{3}^{(2)}$ with zero momentum in the 3QP
sector. Note that DLog Padé approximants with poles at values $J/\Gamma<1$ are
excluded in the averages. Not surprisingly, standard deviations increase for
all excitation energies when approaching the self-dual point $J/\Gamma=1$.
Nevertheless, they are small enough to conclude the absence of a gap-closing
in all three QP sectors which further confirms the presence of a first-order
phase transition.

#### Conclusions:

In this work we applied perturbative and numerical linked-cluster expansions
to investigate the quantum phase transition in the qNM. Our results for the
ground-state energy revealed the presence of a finite kink at the self-dual
point $J/\Gamma=1$. This finding is further consistent with the analysis of
the second derivative of the ground-state energy as well as with the absence
of any gap-closing in the 1QP, 2QP, and 3QP sector. Our results therefore
demonstrate the presence of a first-order phase transition in the qNM which is
in contradiction to the scenario of fractal quantum criticality proposed in
[13]. In particular, their critical value $\alpha=0.28(2)$ is not consistent
with our finding $\alpha_{\mathrm{bias}}=0.007(12)$. One might speculate that
the results in [13] from quantum Monte Carlo simulations suffer from finite-
size effects or from the glassy dynamics present at finite temperatures.
Indeed, in [22] it has been shown that boundary conditions and finite-size
effects are non-trivial in the qNM and that the approach to the thermodynamic
limit is different across different system sizes and geometries. This is
different in our approach using linked-cluster expansions which have the
benefit to work directly in the thermodynamic limit. Furthermore, our results
are in accordance with the numerical results by Vasiloiu et al. [14] who also
found a first-order transition in the qNM. Another attractive route for a
better understanding of the qNM are experimental investigations which have
been proposed for Rydberg atoms arrays in [23].

#### Acknowledgments:

We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 429529648 – TRR 306 QuCoLiMa (“Quantum
Cooperativity of Light and Matter”). KPS acknowledges the support by the
Munich Quantum Valley, which is supported by the Bavarian state government
with funds from the Hightech Agenda Bayern Plus.

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  * Sfairopoulos  _et al._ [2023] K. Sfairopoulos, L. Causer, J. F. Mair, and J. P. Garrahan, Boundary conditions dependence of the phase transition in the quantum Newman-Moore model (2023), arXiv:2301.02826 .
  * Myerson-Jain  _et al._ [2022] N. E. Myerson-Jain, S. Yan, D. Weld, and C. Xu, Phys. Rev. Lett. 128, 017601 (2022).
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## Supplementary Materials

In the supplementary Information we first briefly introduce DlogPadé
approximants. Second, we give the explicit series expansions for the ground-
state energy per site as well as low-energy excitation energies analyzed in
the main body of the article.

###  .1 DlogPadé extrapolations

To extract the quantum-critical point including critical exponents from the
pCUT method well beyond the radius of convergence of the pure high-order
series we use DlogPadé extrapolations. For a detailed description on DlogPadés
and its application to critical phenomena we refer to Refs. [24, 25]. The Padé
extrapolant of a physical quantity $\kappa$ given as a perturbative series is
defined as

$P[L,M]_{\kappa}=\frac{P_{L}(\lambda)}{Q_{M}(\lambda)}=\frac{p_{0}+p_{1}\lambda+\cdots+p_{L}\lambda^{L}}{1+q_{1}\lambda+\cdots+q_{M}\lambda^{M}}$
(4)

with $p_{i},q_{i}\in\mathbb{R}$ and the degrees $L$, $M$ of $P_{L}(x)$ and
$Q_{M}(x)$ with $r\equiv L+M$, i.e., the Taylor expansion of Eq. (4) about
$\lambda=0$ up to order $r$ must recover the quantity $\kappa$ up to the same
order. For DlogPadé extrapolants we introduce

$\mathcal{D}(\lambda)=\derivative{\lambda}\ln(\kappa)\equiv
P[L,M]_{\mathcal{D}}$ (5)

the Padé extrapolant of the logarithmic derivative $\mathcal{D}$ with
$r-1=L+M$. Thus the DlogPadé extrapolant of $\kappa$ is given by

$\mathrm{d}P[L,M]_{\kappa}=\exp\left(\int_{0}^{\lambda}P[L,M]_{\mathcal{D}}\,\mathrm{d}\lambda^{\prime}\right).$
(6)

Given a dominant power-law behavior
$\kappa\sim|\lambda-\lambda_{c}|^{-\theta}$, an estimate for the critical
point $\lambda_{c}$ can be determined by excluding spurious extrapolants and
analyzing the physical pole of $P[L,M]_{\mathcal{D}}$. If $\lambda_{c}$ is
known, we can define biased DlogPadés by the Padé extrapolant

$\theta^{*}=(\lambda_{c}-\lambda)\derivative{\lambda}\ln(\kappa)\equiv
P[L,M]_{\theta^{*}}$ (7)

In the unbiased as well as the biased case we can extract estimates for the
critical exponent $\theta$ by calculating the residua

$\begin{split}\theta_{\rm unbiased}=\Res
P[L,M]_{\mathcal{D}}|_{\lambda=\lambda_{c}},\\\ \theta_{\rm biased}=\Res
P[L,M]_{\theta^{*}}|_{\lambda=\lambda_{c}}.\end{split}$ (8)

###  .2 Series expansions

The series for the ground-state energy per site in units of
$\Gamma=\frac{1}{2}$ is calculated up to order 26 and reads

$\displaystyle\frac{E_{0}}{N}=$
$\displaystyle-\frac{1}{2}-\frac{1}{3}J^{2}-\frac{2}{27}J^{4}-\frac{694}{8505}J^{6}-\frac{87917}{714420}J^{8}-\frac{1163156201}{5063451750}J^{10}-\frac{37554176289949}{76559390460000}J^{12}$
(9)
$\displaystyle-\frac{11683996058218949671}{10345853229812100000}J^{14}-\frac{61969629185820865703757811}{22369390019370530136000000}J^{16}$
$\displaystyle-\frac{366126366318767391658088809316147}{51389087844549822028782120000000}J^{18}$
$\displaystyle-\frac{7180291157644353011347796827422626494913}{377778862591504266872355265169280000000}J^{20}$
$\displaystyle-\frac{344371159846708207005245975888183886739063098907}{6595808140242338440210408155617664341760000000}J^{22}$
$\displaystyle-\frac{8458783982889010392413115078905896242564195637263595308619}{57579564833844486970612044494666792375213448960000000000}J^{24}$
$\displaystyle-\frac{6094307456345282285613515218215816652048109273373916121215182638251}{14451289168650119698445377495743278705265545839970529600000000000}J^{26}\;.$

Since the 1QP configuration is completely immobile, the block in the effective
Hamiltonian is one-dimensional and the series obtained up to order 14 in units
of $\Gamma=\frac{1}{2}$ reads

$\displaystyle\Delta_{1}=1$
$\displaystyle-2J^{2}+\frac{10}{9}J^{4}-\frac{398}{45}J^{6}+\frac{21286241}{893025}J^{8}-\frac{34989373088}{281302875}J^{10}$
(10)
$\displaystyle+\frac{2125889739834149}{4253299470000}J^{12}-\frac{683607613238271693151}{265278287943900000}J^{14}\;.$

The mobility of all 2QP configurations is highly restricted and the
corresponding block in the effective Hamiltonian is of finite size. In order
12 perturbation theory, the block is of size $15\times 15$, therefore yielding
15 energy eigenvalues. The 2QP energy eigenvalue decreasing strongest in
leading order and, therefore, considered as the 2QP gap reads in units of
$\Gamma=\frac{1}{2}$

$\displaystyle\Delta_{2}=2-8J^{2}+\frac{1336}{27}J^{4}-\frac{6264848}{8505}J^{6}+\frac{427973156}{33075}J^{8}$
$\displaystyle-\frac{2595333024547577}{10126903500}J^{10}$ (11)
$\displaystyle+\frac{1543912949140866037}{283553298000}J^{12}\;.$

The series for the first eigenvalue of the 3QP sector in units of
$\Gamma=\frac{1}{2}$ for $\vec{k}=0$ up to order 12 reads

$\displaystyle\Delta_{3}^{(1)}=$
$\displaystyle\,\,3-\frac{2}{3}(10+\sqrt{82})J^{2}+\frac{(109054+12907\sqrt{82})}{27(82+\sqrt{82})}J^{4}$
(12)
$\displaystyle-\frac{4(1436807939437+154950111676\sqrt{82})}{8505(82+\sqrt{82})^{3}}J^{6}$
$\displaystyle+\frac{(346835035620091968700+38457664526290637983\sqrt{82})}{2679075(82+\sqrt{82})^{5}}J^{8}$
$\displaystyle-\frac{(143882555062725026268280765837+15878921718665450600555048341\sqrt{82})}{5063451750(82+\sqrt{82})^{7}}J^{10}$
$\displaystyle+\left(\frac{63875776145970660095856118712991441467}{9569923807500(82+\sqrt{82})^{9}}\right.$
$\displaystyle\qquad+\left.\frac{7054556156002646215050241827786693722\sqrt{82}}{9569923807500(82+\sqrt{82})^{9}}\right)J^{12}\;.$

The series for the second eigenvalue of the 3QP sector in units of
$\Gamma=\frac{1}{2}$ for $\vec{k}=0$ up to order 12 reads

$\displaystyle\Delta_{3}^{(2)}=3-6J^{2}+\frac{10}{3}J^{4}-\frac{398}{15}J^{6}+\frac{20510669}{297675}$
$\displaystyle J^{8}-\frac{170229112967}{337563450}J^{10}$ (13)
$\displaystyle+\frac{117051872376412993}{22967817138000}J^{12}\;.$

The series for the third eigenvalue of the 3QP sector in units of
$\Gamma=\frac{1}{2}$ for $\vec{k}=0$ up to order 12 reads

$\displaystyle\Delta_{3}^{(3)}=$
$\displaystyle\,\,3+\frac{2}{3}(-10+\sqrt{82})J^{2}+\frac{(-109054+12907\sqrt{82})}{27(-82+\sqrt{82})}J^{4}$
(14)
$\displaystyle-\frac{4(-1436807939437+154950111676\sqrt{82})}{8505(-82+\sqrt{82})^{3}}J^{6}$
$\displaystyle+\frac{(-346835035620091968700+38457664526290637983\sqrt{82})}{2679075(-82+\sqrt{82})^{5}}J^{8}$
$\displaystyle-\frac{(-143882555062725026268280765837+15878921718665450600555048341\sqrt{82})}{5063451750(-82+\sqrt{82})^{7}}J^{10}$
$\displaystyle+\left(\frac{-63875776145970660095856118712991441467}{9569923807500(-82+\sqrt{82})^{9}}\right.$
$\displaystyle\qquad+\left.\frac{7054556156002646215050241827786693722\sqrt{82}}{9569923807500(-82+\sqrt{82})^{9}}\right)J^{12}\;.$