# PeaTMOSS: Mining Pre-Trained Models in Open-Source Software

Wenxin Jiang**footnotemark: *${}^{1^{*}}$, Jason Jones††footnotemark:
†${}^{1^{*}}$, Jerin Yasmin‡‡footnotemark: ‡${}^{2^{*}}$, Nicholas Synovic3,
Rajeev Sashti1, Sophie Chen4,
George K. Thiruvathukal3, Yuan Tian2, James C. Davis§§footnotemark: §1 West
Lafayette, Indiana, USA1 ; Kingston, Ontario, CA2 ; Chicago, Illinois, USA3 ;
and Ann Arbor, Michigan, USA4 Purdue University1 ; Queen’s University2 ;
Loyola University–Chicago3 ; and University of Michigan–Ann Arbor4

###### Abstract

Developing and training deep learning models is expensive, so software
engineers have begun to reuse pre-trained deep learning models (PTMs) and
fine-tune them for downstream tasks. Despite the wide-spread use of PTMs, we
know little about the corresponding software engineering behaviors and
challenges.

To enable the study of software engineering with PTMs, we present the PeaTMOSS
dataset: Pre-Trained Models in Open-Source Software. _PeaTMOSS_ has three
parts: a snapshot of (1) 281,638 PTMs, (2) 27,270 open-source software
repositories that use PTMs, and (3) a mapping between PTMs and the projects
that use them. We challenge _PeaTMOSS_ miners to discover software engineering
practices around PTMs. A demo and link to the full dataset are available at:
https://github.com/PurdueDualityLab/PeaTMOSS-Demos.

11footnotetext: These authors contributed equally and are listed
alphabetically.

##  I High-Level Overview

Motivation: Deep Neural Networks (DNNs) have become a common component in
software systems over the past decade. While some software engineers develop
DNNs from scratch, many others integrate DNNs into their software following a
typical re-use pattern: (1) pre-trained DNN models are published to registries
such as Hugging Face, similar to traditional software package registries
(e.g., NPM, PyPI); and (2) other software depends on these Pre-Trained Models
(PTMs), accessed via libraries or web APIs.

_Despite wide-spread adoption of PTMs, we know relatively little about how
PTMs are integrated into software systems._

Challenge: We propose the _PeaTMOSS_ challenge to learn more about Pre-Trained
Models in Open-Source Software (Figure 1).

Figure 1: Data for Pre-Trained Models in Open-Source Software.

The _PeaTMOSS_ dataset comprises snapshots of PTMs and open-source
repositories utilizing PTMs, as well as a mapping of PTMs to projects.
_PeaTMOSS_ contains 281,638 PTM packages, 27,270 GitHub projects that use PTMs
as dependencies, and 44,337 links from these GitHub repositories to the PTMs
they depend on. For both PTMs and GitHub projects, _PeaTMOSS_ contains
metadata (commits, issues, pull requests) and data (e.g., model architecture
and weights; git repositories). A uniform schema for retrieving PTM and
project metadata is provided to assist in mining efforts. Most information is
indexed; some is stored as blobs.

The dataset can be accessed in two formats. The “metadata” version of
_PeaTMOSS_ is 7.12 GB and contains only the metadata of the PTM packages and a
subset of the GitHub project metadata. The “full” version is 48.2 TB, adding
(1) the PTM package contents in each published version, and (2) git history of
the main branches of the GitHub projects.

##  II PeaTMOSS Dataset Structure

The metadata version of _PeaTMOSS_ is stored in a SQLite database. The tables
include hyperlinks to tarred copies of the PTM package or GitHub repository.
Dataset schemas are described in Appendix A.

The mapping between GitHub projects and PTMs are cases where a GitHub
repository is known to depend on a particular PTM. Additional detail is given
in section B-B.

##  III Accessing and Working with PeaTMOSS

Access: The metadata and full versions of _PeaTMOSS_ are available through a
Globus share hosted at Purdue University. _PeaTMOSS_ can be downloaded via the
official Globus Connect application, which is available for all major
operating systems, e.g., Linux, MacOS, and Windows. We include a Python script
to download and configure an SQLite instance of the metadata version. For more
instructions, see https://github.com/PurdueDualityLab/PeaTMOSS-Demos.

Working with PeaTMOSS: To interact with _PeaTMOSS_ , we recommend ORM or SQL.
Examples are provided in Appendix C.

Required Skills: _PeaTMOSS_ is accessible to miners with a range of skills and
interests. _PeaTMOSS_ includes both standard mining artifacts from GitHub
(e.g., git repositories, issues, PRs) and unusual artifacts from PTMs (e.g.,
neural networks, weights, model cards). Miners interested in program analysis,
software repository mining, and natural language processing, etc. may apply
these techniques to GitHub data, PTM data, or both.

_Neither expertise in deep learning nor access to hardware such as GPUs is
necessary for use of _PeaTMOSS__. Of course, miners with deep learning
expertise or hardware resources could explore more advanced questions about
PTM usage on GitHub or delve deeper into the PTM data.

Data Samples: We offer a subset of samples from _PeaTMOSS_ at:
https://github.com/PurdueDualityLab/PeaTMOSS-Demos.

##  IV Possible Research Questions

Table I presents sample research questions for miners to investigate. This
table includes questions focused on the GitHub portion of the dataset, on the
PTM portion of the dataset, and on both parts. It notes some prior work as
starting points for miners.

TABLE I:  Example questions for miners to investigate. These questions are divided into three groups. The first group of questions makes use of the GitHub portion of the dataset (_GH_). The second group uses the Pre-Trained Model portion of the dataset (_PTM_). The third group asks questions that require Integrating both parts of the dataset (_I_).  Research questions |  Related work
---|---
GH1: What kinds of defects are opened related to PTM use in the GitHub projects? How do these defects differ from defects opened on other aspects of the GitHub projects? |  [1]
GH2: What do developers on GitHub discuss related to PTM use, e.g., in issues, and pull requests? What are developers’ sentiments regarding PTM use? Do the people do pull requests of PTMs have the right expertise? |  [2, 3]
GH3: How commonly do developers change the specific PTM they use to implement a feature? What factors influence these changes? |  [4]
GH4: Sometimes a PTM is introduced into a GitHub repository as part of the initial implementation of a software feature, but other times a PTM is introduced to replace part or all of an existing feature implementation. How common are these two modes of PTM adoption? In the second kind, how does the feature’s defect types or defect rates change after the PTM is introduced? |  [5, 6, 7, 8]
PTM1: What factors predict the popularity of a PTM, and what is their relative importance? Intuition suggests that performance aspects such as accuracy and latency may dominate; what is the role played by other factors such as engineering quality? |  [9, 10]
PTM2: Recent qualitative work determined that software engineers struggle to re-use PTMs because of their limited documentation. What are the typical characteristics of this documentation? Can natural-language model cards be automatically parsed into a structured schema? |  [10, 11, 12]
PTM3: One aspect of re-use is finding a candidate model. What naming conventions do PTMs follow? Are they consistent enough (within an architecture family? across families?) to support engineers looking for similar models? When do PTM maintainers release a model under a new name, and when do they simply bump the version number? |  [13]
PTM4: PTM authors may reuse each others’ work, e.g., building off of model checkpoints or incorporating architectural building blocks. This might be viewed as an extreme form of “forking” from open-source software, but it may also reflect a novel form of software exchange. What is the phylogeny, or perhaps the “supply chain”, of the major families of PTMs? |  [11]
PTM5: Many research papers describe techniques for identifying DNNs with unexpected behavior, e.g., hidden malicious behaviors. How common are such DNNs in the PTM dataset? |  [10, 14, 15, 16]
I1: It can be difficult to interpret model popularity numbers by download rates. To what extent does a PTM’s download rates correlate with the number of GitHub projects that rely on it, or the popularity of the GitHub projects? |  [17]
I2: What are the code smells related to PTM in the downstream projects, and how do they affect theses projects? |  [18, 19, 20, 21]
I3: When PTMs are used in GitHub repositories, what are engineers’ testing practices for the PTMs they add as dependencies? Is there any correlation between the tests of the PTM by its maintainers, and the tests of the PTM by the downstream users? Do practices vary based on the purpose of the PTM, e.g., computer vision vs. natural language processing? How do PTM downstream users deal with flakiness when testing a PTM? |  [22, 23, 8, 24]
I4: Updating dependencies is a core software engineering activity. Suppose a GitHub repository depends on a PTM. How often does the GitHub repository update the dependency when that PTM is changed, e.g., due to (1) PTM deprecation, (2) PTM improvement via a new version, or (3) PTM being made outdated by the release of a new model? What is the typical lag time for such updates? |  [25, 26]
I5: Software engineers often communicate through filing issue reports. What are the characteristics of issue reports on the PTM packages, e.g., in terms of the kinds of questions asked, responsiveness of maintainers, issue density, and issue staleness? How often does the topic of reproducibility come up (cf. the “ML reproducibility crisis”)? How do these attributes differ from the characteristics of issue reports in GitHub repositories? |  [27, 28]
I6: When engineers deploy software applications that make use of PTMs, they may prefer to use a deployment framework, e.g., the ONNX RunTime, rather than a development framework such as PyTorch. Which of the several competing deployment frameworks (ONNX RunTime, MM-DNN, NNET, TFLite, etc.) is the most popular, and is there any evidence of why? Do GitHub users make the transformation to deployment themselves or do the PTM authors provide the deployment-ready version? |  [29, 30, 31]

##  Appendix A Dataset schema

The detailed schema is shown in Figure 2. The definition in SQL is available
to miners.

Figure 2:  The _PeaTMOSS_ data schema. The database has three regions: one set
of tables for PTMs, one set of tables for GitHub projects, and one table
linking the two. The underlying PTMs and GitHub repositories are stored in the
Globus share and can be fetched on demand. A more navigable version of the
schema is available in the demo repository.

##  Appendix B Data collection

###  B-A PTMs

####  B-A1 What is a PTM and PTM package?

Pre-trained deep learning models (PTMs) are often shared through deep learning
model registries, such as Hugging Face. Engineers can directly reuse these
PTMs, or fine-tune them for specific downstream tasks. A PTM package typically
includes a model card (a form of README), as well as the model’s architecture
and weights [10]. In recent years, the popularity of PTMs has been steadily
rising [32, 33]. As illustrated in Figure 3, the total number of Hugging Face
models has seen a consistent increase on a monthly basis. Recent work shows
increasing interest from the software engineering mining community in PTMs
[11, 10, 34, 35]. These works have identified the potential mining data that
the community can take advantage of. In the past year the first mining efforts
of PTMs and software engineering practices have begun [33, 32].

Figure 3: Evolution of the total number of Hugging Face models per month. This
figure is reused from [33].

####  B-A2 PTM Collection

Our PTM data collection includes three parts: (1) We saved 15,250 PTM
snapshots. This included the most popular PTM packages (i.e., with over 10K
downloads) on Hugging Face, which resemble a “git-like” structure, and all
PTMs on PyTorch. This part of the data can provide a comprehensive view of PTM
packages. (2) Among these “full” metadata, 44,337 links from the PTMs to the
downstream GitHub repositories have been identified. This part of the data can
be connected to downstream GitHub data and allow miners to analyze the
relationship between them. (3) For all PTMs hosted on Hugging Face and
PyTorch, we retrieved their metadata, resulting in a total number of 281,638
PTM package metadata being included in _PeaTMOSS_. The miners can answer
research questions based on metadata only, such as analyzing PTM naming
conventions.

####  B-A3 Soundness and Completeness

_PeaTMOSS_ is comprehensive in terms of popular PTM packages, as it includes
snapshots of those with over 10,000 downloads on Hugging Face. This provides a
full view of widely-used PTMs and their connections to downstream GitHub
projects, facilitating in-depth analysis. Additionally, the dataset includes
metadata from all other PTMs on Hugging Face, which can be used for metadata-
based analyses. _PeaTMOSS_ enhances the diversity of PTM data by incorporating
PTM packages from PyTorch Hub, including all available model repositories and
their associated pull requests and issues.

####  B-A4 Implementation

Metadata is collected using an Extract-Translate-Load (ETL) pipeline for each
model hub. The ETL pipeline can be generalized to the following steps:

  * •

Extract: Obtain metadata that is available from each model hub’s API.

  * •

Transform: Use metadata to collect more information about PTMs (i.e., examine
Github Metadata for a linked repository) as well as download PTM package of
GitHub repository snapshot. Transform the data into intermediate
representation to simplify Loading.

  * •

Load: Load the transformed data into a database for long-term storage and
retrieval

Each model hub has a unique implementation of the Extract stage, but the
functionality is the same.

  * •

Hugging Face: PTM package metadata is downloaded using the HuggingFace_hub
Python library.

  * •

PyTorch: The markdown files in the root of the PyTorch GitHub repository,
which correspond to each PTM package’s repository, are downloaded and
subsequently scraped for metadata.

During the Transform stage, data that matches the PeaTMOSS.db metadata schema
is transformed into an intermediate representation, while data that doesn’t
match the schema is transformed into a JSONB blob. This is to allow for both
consistency across model hubs, as well as maintaining hub specific metadata.

###  B-B Mapping GitHub projects to PTMs

To evaluate PTM usage within projects, a PTM must be mapped back to a GitHub
project. While it is possible to use Hugging Face and PyTorch Hub hosted
projects with libraries outside of the Python ecosystem, we filtered on GitHub
projects that utilize Python libraries to interface with PTMs as the vast
majority of projects we found utilized Python libraries. As Hugging Face and
PyTorch hub do not provide metadata or a method to retrieve PTM usage within
projects, by analyzing the source code of a GitHub project, it is possible to
map the two. The whole methodology has been described below:

####  B-B1 Signature Collection

For leveraging Hugging Face PTMs, dedicated functions/methods specific to each
library are available. These functions/methods allow loading PTMs by inputting
their names as arguments. In this step, we manually retrieve the libraries,
along with their corresponding import identifiers and the necessary
function/method names essential for PTM loading. This compilation process is
guided by Hugging Face’s official
documentation.¶¶¶https://github.com/HuggingFace/hub-docs/blob/main/js/src/lib
/interfaces/Libraries.ts The culmination of import identifiers and the
essential function/method names is referred to as a Signature. For example,
when accessing the PTMs provided by the Diffusers library, the corresponding
signature encompasses diffusers and from_pretrained. Note that some libraries
offer multiple methods to load PTMs, and we have taken all of these methods
into account. For instance, the transformers library presents two distinct
approaches: from_pretrained and pipeline, both of which can be utilized to
load PTMs. Figure 4 shows an example code snippet to load the PTMs provided by
Transformers library.

Throughout this process, we exclude libraries not visible on the website (such
as k2, doctr, mindspore), as well as those unsuitable for downstream projects.
An example of such incompatibility is when PTMs can only be used via command
line or download, or when they lack the "use in library"
feature∥∥∥https://HuggingFace.co/facebook/musicgen-large. After filtering, a
total of 23 libraries remain in the compilation.

To use PyTorch PTMs, there are two approaches: (1) torch.hub.load function
that provides a way to load PTMs by specifying their names as arguments. and
(2) Alternatively, one can utilize library-specific classes provided by
torchvision or instances provided by torchaudio and torchtext. These classes
and instances are tailored to represent individual PTMs. The first approach
and alternative approach provided by the torchvision library contain keyword
based parameters i.e., pretrained/weights to control the model usage with
pretrained weights or random initialization. The above mentioned two
approaches give us a total of 163 signatures. Figure 5 shows an example code
snippet to load the PTMs provided by the PyTorch library.

Figure 4: An Example Code Snippet to Use PTMs from Hugging Face Hub

Figure 5: An Example Code Snippet to Use PTMs from PyTorch Hub

####  B-B2 Preliminary repository collection based on Sourcegraph Search

The subsequent task involves locating these signatures within the code.
Although we attempted to use GitHub search, it does not facilitate a
comprehensive search as it provides the top 1000 results. Instead, we utilize
the Sourcegraph command line interface, src
******https://docs.sourcegraph.com/cli, to detect projects containing
pertinent files that employ the gathered signatures to interact with Hugging
Face and PyTorch Hub.

Our search pattern incorporates the signatures gathered earlier, and we match
them against the content of files within GitHub repositories. Our search
criteria encompass repositories that are not archived (default behavior), not
forked (default behavior), and publicly accessible, specifically focusing on
Python files. For example, a query for Diffusers is structured as ‘‘src
select:file visibility:public count:all lang:Python content:‘from diffusers’
AND from_pretrained(’’. Our search query accommodates both ‘from import’ and
‘import’ statements. Our search results include the corresponding code
snippets, file names, and repository names. The projects were identified
during July 5-12. Based on the count of the repositories, we select the top 5
Hugging Face libraries for our data collection including Transformers, Spacy,
Sentence-Transformers, Diffusers, and Timm. For PyTorch, we consider all of
the corresponding signatures. Our dataset comprises well-recognized GitHub
repositories with an average star count of 201.

We have obtained local copies of the GitHub repositories by using the git
clone command. This process took approximately 12 days to complete and
resulted in the download of 36,251 repositories, collectively amounting to a
total size of 3.5 TB.

####  B-B3 Extracting PTMs via Static Analysis

As Sourcegraph’s search feature relies on text-based patterns, the possibility
of encountering false positive results exists. To mitigate this concern, we
perform static analysis on GitHub repositories with the Scalpel framework
[36]. For each relevant source code file associated with a specific function
signature, we construct an abstract syntax tree and extract the function calls
contained within the file. Subsequently, we retrieve the complete and
qualified names of each identified function call and cross-reference them with
our predefined signatures which gives us a total of 28,575 repositories.
Additionally, we go a step further by extracting both the positional and
keyword arguments that are associated with the function calls that match our
target signatures. Our analysis is equipped to capture any argument that
possesses a static value. We then utilize the list of PTMs from PTM Torrent V2
to identify the repositories that statically call PTMs which gives us a total
of 15,129 repositories. We store the corresponding repositories and files for
each of the matched PTMs. It is important to note that a single repository can
utilize multiple PTMs, and similarly, a single PTM can be employed across
multiple repositories.

####  B-B4 Soundness and Completeness of Collected Repositories

For the PTMs hosted on the Hugging Face hub, our dataset provides usage
considering the five libraries, i.e., Transformers, Spacy, Timm, Sentence-
Transformers, and Diffusers. These libraries were chosen because they comprise
the top five libraries used in GitHub projects as shown in Figure 6. For the
PTMs from the PyTorch hub, we did not filter by the library. Our dataset
comprises torchvision, torchaudio, torchtext, along with PyTorch hub.

Static analysis was carried out due to the limitations of the text search
conducted using Sourcegraph. We resolve the fully qualified names for each
function call to accurately identify True Positive results. This results in a
total of 28,575 repositories that genuinely contain the practical utilization
of the PTMs. Our dataset encompasses projects created up until July 10, 2023.

Figure 6: Number of projects that use a specific library as captured via
Sourcegraph code search

####  B-B5 Extracting GitHub Issues and Pull Requests

By analyzing the discussions that community members have about the project
within the project’s issue tracker, it is possible to identify not only PTMs
of interest w.r.t the project, but the potential future direction of a project
w.r.t the techniques implemented by the PTM.

To collect the issues and pull requests associated with the GitHub
repositories, we use GitHub’s official command line interface gh. We consider
all states (i.e., open and closed) while collecting the issues and pull
requests associated with each repository. Each of the issue and pull request
metadata responses contain all available relevant fields provided by GitHub
CLI. Specific response fields are listed in LABEL:tab:GHCLI_RequestFields.
Example commands to retrieve relevant data can be found in _PeaTMOSS_ GitHub
repository. Targeting 28,575 repositories, we retrieve the issues or pull
requests resulting in 19,507 repositories with issues and 12,159 repositories
with pull requests.

Altogether, our dataset encompasses a total of 27,270 repositories, which
involve occurrences of issues, pull requests or static utilization of PTMs.

TABLE II:  JSON response fields captured when collecting issues and pull requests for 28,575 GitHub repositories.  Request Type |  Response Fields
---|---
Issue Metadata |  assignees, author, body, closed, closedAt, comments, createdAt, id, labels, milestone, number, projectCards, reactionGroups, state, title, updatedAt, url
Pull Request Metadata |  additions, assignees, author, baseRefName, body, changedFiles, closed, closedAt, comments, commits, createdAt, deletions, files, headRefName, headRepository, headRepositoryOwner, id, isCrossRepository, isDraft, labels, maintainerCanModify, mergeCommit, mergeStateStatus, mergeable, mergedAt, mergedBy, milestone, number, potentialMergeCommit, projectCards, reactionGroups, reviewDecision, reviewRequests, reviews, state, statusCheckRollup, title, updatedAt, url

##  Appendix C Data Access Examples

To answer several of the proposed research questions in Table I, we have
released examples on how to interface with _PeaTMOSS_. ORM methods and SQL
examples for interfacing with the PeaTMOSS.db database are provided. Code
snippets for these examples are made available via the /Examples/ filepath in
our GitHub repository.

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# Optimised Design and Performance Comparison of Soft Robotic Manipulators

Arnau Garriga-Casanovas1, Shen Treratanakulchai1, Enrico Franco1, Emilia
Zari1, Varell Ferrandy2, Vani Virdyawan2, Ferdinando Rodriguez y Baena1
1Mechatronics In Medicine Laboratory, Department of Mechanical Engineering,
and Hamlyn Centre, Imperial College London, UK. a.garriga-
<EMAIL_ADDRESS>Engineering Department, Institut
Teknologi Bandung, Kota Bandung, Indonesia.

###### Abstract

Soft robotic manipulators are attractive for a range of applications such as
medical interventions or industrial inspections in confined environments. A
myriad of soft robotic manipulators have been proposed in the literature, but
their designs tend to be relatively similar, and generally offer a relatively
low force. This limits the payload they can carry and therefore their
usability. A comparison of force of the different designs is not available
under a common framework, and designs present different diameters and features
that make them hard to compare. In this paper, we present the design of a soft
robotic manipulator that is optimised to maximise its force while respecting
typical application constraints such as size, workspace, payload capability,
and maximum pressure. The design presented here has the advantage that it
morphs to an optimal design as it is pressurised to move in different
directions, and this leads to higher lateral force. The robot is designed
using a set of principles and thus can be adapted to other applications. We
also present a non-dimensional analysis for soft robotic manipulators, and we
apply it to compare the performance of the design proposed here with other
designs in the literature. We show that our design has a higher force than
other designs in the same category. Experimental results confirm the higher
force of our proposed design.

##  I Introduction

Soft robotic manipulators offer a range of attractive features such as low
cost, lack of moving parts which mean that they can be easily miniaturised,
flexible transmission lines, compatibility with magnetic-resonance imaging
(MRI) and with biological tissue, and inherent safety when interacting with
delicate environment, to name a few. This makes them well suited for
applications in medicine such as endoluminal surgery, specifically for
colorectal surgery or lung interventions [1]. They also show promise in the
field of industrial inspections to enter confined environments and perform in
situ inspections. In this work, we focus on soft robots actuated by a
pressurised fluid, typically air, since they represent one of the most common
designs.

A myriad of soft robotic manipulators with fluidic actuation have been
proposed in the literature, with notable examples [2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12], and minimally invasive surgery has been a common application for soft
robots for at least a decade [13, 14, 15, 16]. The majority of existing
designs tend to be relatively similar, and orbit around a robot design with a
tubular silicone structure and a set of internal chambers that can be
pressurised.

The main drawback of these robots is that they have low lateral force, which
limits significantly the payloads they can carry and the forces they can apply
[1]. In this work, we focus on lateral force because it is generally the
weakest feature of soft robots. Axial tension force in these robots is
generally related to the stiffness of their silicone and can be increased with
stiffer materials in the axial direction, so it is not relevant. Axial
compression force can be increased by pressurising all chambers of the
manipulator, but that is typically equivalent to a pressurized cylinder made
of an inextensible fabric, so it is easy to achieve and not relevant from a
design perspective. It should also be noted that, in this work, we primarily
consider soft robotic manipulators that offer the capability of bending in any
direction in 3D space. This is because soft robots with 1 degree of freedom
(DOF) exist and offer relatively high lateral force in the direction of
bending, but they offer only passive, low force in the other directions.

Figure 1: Cross-sections of the proposed morphing design in undeformed and
deformed configuration from applying pressure, obtained using finite element
simulation in Abaqus. The distance between the equivalent centre of pressures
and the centre of tension is shown with a yellow line for both undeformed and
deformed cross-section. The increase in that distance when the cross section
deforms implies a larger lever arm for the bending moment created by pressure.

A recent comparison of soft robot designs can be found in [1]. However, most
existing designs have different diameters, which makes it difficult to
extrapolate and compare them. In addition, in a number of cases, the
specifications and performance of the different designs is not fully reported.
It should also be noted that one design in [1] stands out as offering a high
force, which is Stiff-flop [3, 17]. However, Stiff-flop is 34 mm diameter, it
is difficult to miniaturise because of the minimum critical size to enable
granular jamming, and it only offers high force when it is rigid, but loses it
when it is moving. As a result, it cannot carry a significant payload nor
apply forces actively to the environment in the range of maximum force
reported.

In this paper, we present the design of a soft robotic manipulator optimised
to maximise its force while respecting typical application constraints such as
size and miniaturisation potential, mobility, payload carrying capability, and
limits on maximum usable pressure (typical in medicine). The design presented
here has the advantage that it exploits the internal structural deformation
that occurs when it is pressurised, morphing towards an optimal design as it
is pressurised to move in different directions, as illustrated in Figure 1.
This leads to higher lateral force, which is employed here as key performance
indicator. In order to compare our design with many other ones in the
literature, we develop a non-dimensional analysis. We apply this analysis and
we show that our design is the strongest in the category of soft robots with
pneumatic actuation and similar mobility in terms of DOFs and range. We also
present experiments to validate the analysis, and to confirm that our design
offers higher force.

The main contributions of the paper are the following: first, the design of a
soft robotic manipulator that morphs as it is pressurised, and that is
stronger than existing designs; second, the development of a non-dimensional
analysis for soft robotic manipulators relying on Buckingham’s Pi theorem;
third, the experimental comparison of our design and multiple other designs to
show which is the strongest and validate the work.

The paper is structured as follows. The design of the soft robotic manipulator
proposed in this work is presented in section II, together with a description
of its morphing features. Then the non-dimensional analysis for soft robotic
manipulators is developed in section III. The experiments to validate the work
are described in section IV, and their results are reported in section V.
Lastly, concluding remarks are described in section VI.

Figure 2: CAD render of the proposed morphing design showing its cross section
(left) and a perspective view (right).

##  II Soft robotic manipulator design

###  II-A Design requirements

The soft robotic manipulator design presented in this work is aimed at a
prototypical medical application, since it is one of the most common use cases
for soft robots. A soft robotic manipulator for medicine may consist of a set
of segments stacked serially to create a robot with a high number of DOFs.
Here we focus on the design of one segment of robotic manipulator capable of 2
DOFs, since it is the fundamental building block for a manipulator. However,
the design of the robot segment needs to offer the possibility of stacking
multiple segments to create a manipulator with more DOFs.

The requirements for the robot segment are defined based on typical
requirements for colorectal interventions, and are summarised in Table I.

The design objective is to maximise the lateral force of the soft robotic
manipulator segment so that it can carry a payload as heavy as possible or
apply forces on the environment.

Max. OD | DOFs | Bending | Max. pressure | Payload
---|---|---|---|---
12 mm | 2 per segment | 90 deg | 3 bar | 2 mm channel
TABLE I: Design requirements for a segment of soft robotic manipulator.

###  II-B Design principles

The design is derived using the design principles in [18] to maximise its
force. The main principles are summarised below for completeness.

The total cross-section of the device should be maximised to occupy all
available room. The area of any internal chambers should be maximised, to
ensure that the region of the cross-section corresponding to the pressurised
fluid is maximal. The thickness of any internal partition walls separating
chambers should be minimised. The longitudinal stiffness in the part of the
robot structure that extends when the robot bends should be minimised. The
longitudinal stiffness in the part of the robot that does not extend to
achieve bending should be maximised. The longitudinal stiffness distribution
should be selected so that the maximum stiffness is concentrated near the edge
of the cross section, near side toward which the robot bends. The geometry of
the cross-section should be designed so that the distance between the centre
of pressures and the centre of structural tension forces in the cross section
is maximised.

###  II-C Proposed design

The proposed design was derived by applying the design principles in
subsection II-B to the design requirements in II-A. The resulting design is
shown in Figure 2. It consists of a tubular structure made of silicone
(Elastosil 4601M) with three chambers in the cross section in order to be able
to bend in any direction. Using more than three chambers would lead to a
larger part of the cross section being occupied by partition walls, which
reduces the area available for the chambers, and is thus undesirable as it
would leads to lower force. The centre of the design has an inextensible tube
that absorbs the majority of the axial tension when one or more chambers are
pressurised, and thereby acts as the counterbalance for the pressure force,
creating a bending moment. The inextensible central tube also offers a working
channel, which in this case is 2 mm ID. The partition walls have a minimal
thickness in order to maximise chamber area, and to facilitate cross-sectional
deformation. In this work, the partition walls are 0.8 mm thick which is the
limit that could be achieved in manufacturing without leakage. The outer wall
of the design has an inextensible fibre wrapped around it in a double helix
arrangement to prevent radial expansion while enabling longitudinal extension
and thereby bending, as in standard soft robots. The ideal design would
consist of a pleated outer wall that minimises bending resistance, but
manufacturing such an outer wall at the scale required here remains an
unsolved challenge, so a smooth outer wall with helical fibres is used
instead. A manufactured segment of the design together with its cross section
are shown in Figure 3.

Figure 3: Manufactured segment of the proposed showing its cross section
(left), and a side view of the segment (right).

The design proposed may resemble existing soft robots cosmetically since it
has a tubular silicone structure with three chambers. However, it differs
significantly since it is designed to morph as it is pressurised, and its
features, such as chamber geometry, deformable partition walls, inextensible
central tube that can bend and absorbs structural tension, are engineered to
tend to an optimal design. When a differential pressure is applied in the
chambers, the device is engineered to facilitate cross-sectional deformation.
The result is a significant cross-sectional deformation when pressure is
applied, as shown in Figure 1 (right), which was obtained by simulating the
pressurisation of the robot design shown in 1 (left) in two chambers using
finite element simulation software Abaqus, and following the simulation setup
described in [18]. In particular, the pressurised chambers occupy practically
all of the cross-section, and the inextensible central tube is fully displaced
to the side opposing the applied pressure, which leads to a desirable
configuration that offers significantly higher bending force. This is because
the bending moment generated by the pressure is roughly proportional to the
product of pressure times chamber area times the distance between the centre
of pressures and the centre of tension forces in the structure. In this case,
the main stiff element in the structure is the inextensible central tube,
which absorbs the majority of axial tension, and thus it acts as the centre of
tension forces in the structure. As a result, the distance between the centre
of pressures and centre of tension forces in the structure is nearly equal to
the radius, which is nearly equal to the maximum available. Similarly, the
area of the pressurised chambers occupies practically the entire cross-
section, which is also close to the maximum available. Thus, the configuration
is nearly optimal and maximises force.

This near optimal configuration remains near optimal when bending in any
direction. Since the cross-section deforms when a differential pressure is
applied, and the inextensible central tube is always pushed to the side
opposing the applied pressure, the robot always morphs to an optimal
configuration regardless of the direction in which it needs to bend.

It should also be noted that, at the two ends of the robot segment, the
inextensible central tube is attached to the centre of the segment. This means
that, when a differential pressure is applied and the tube is pushed to one
side of the robot, it would need to follow a longer path between the two ends
of the segment to remain attached to them, which creates additional tension in
the inextensible tube. This behavior is similar to that found in a bow when it
is tensioned, with the difference that in the soft robot the pressure applies
the force that displaces the inextensible tube, whereas in a bow it is a
person that pulls the string from the bow frame. The result however is
similar, as the tension in the inextensible central tube of the soft robotic
manipulator also contributes further to the bending force.

The improvement of the design proposed here with respect to designs that have
constant cross-section, such as Stiff-flop, is significant from a theoretical
perspective. The chamber area in this design can reach nearly double the size
when pressurised with respect to the original size, as shown in Figure 1. In
addition, the distance between the centre of pressures and the centre of
structural tension also nearly doubles when the cross-section is pressurised,
as also highlighted using a yellow line in Figure 1. Both of these elements
contribute to the bending moment, which is the product of both of them times
pressure, so the result can be expected as a bending force that could
theoretically reach four times the force of a similar design with fixed cross-
section. The experimental evaluation of force is presented in section V.

The improvement of the design proposed here with respect to traditional
designs such as the flexible microactuator (FMA) [2, 19] is also significant.
In general, soft robots can be considered to achieve bending either via an
asymmetry in geometry or in stiffness. In the case of the FMA, it is mainly
designed to achieve bending via an asymmetry in geometry (even if some cross-
sectional deformation may occur accidentally). In addition, the FMA can
”waste” some of the applied differential pressure in extending its structure
instead of bending because it has a relatively low axial stiffness throughout
its structure. However, the design proposed here employs both the asymmetry in
geometry and in stiffness thanks to the morphing in the cross-section when it
is pressurised, leading to higher performance. Moreover, the design proposed
here has the inextensible central tube which absorbs most of the axial
tension, and ensures that practically all of the applied pressure translates
into bending.

The design presented here consists of one segment with 2 DOF, and a
manufactured prototype is shown in Figure 3. Multiple segments can be stacked,
and a 6 DOF robot can be created with three segments, as in the prototype
shown in Figure 4. Miniature tubes of 0.3 mm OD can be used to pressurise the
distal segments, and these can be passed through the central working channel
or the small channels next to it shown in Figure 3 (left), so practically no
changes are needed to the design of the individual segments to create the full
robot shown in 4.

Figure 4: Full prototype of the soft robotic manipulator with 12 mm OD made of
three segments, which have 2 DOF each, resulting in a 6 DOF robotic
manipulator.

##  III Non-dimensional analysis

###  III-A Theoretical formulation

A non-dimensional analysis is developed in this section for soft robotic
manipulators actuated by a pressurised fluid, relying on Buckingham’s Pi
theorem. According to Buckingham’s Pi theorem, the deformation of the
manipulators considered in this work depends on a set of non-dimensional
groups, as long as they are geometrically equivalent, under equivalent
external load distribution, and pressurisation in one chamber or multiple
chambers with a given ratio of pressures between chambers. These non-
dimensional groups can be selected to be $\frac{p}{E_{r}}$,
$\frac{F}{E_{r}d^{2}}$, $\frac{E_{f}}{E_{r}}$, $\frac{E_{c}}{E_{r}}$,
$\frac{E_{p}}{E_{r}}$, as well as a parameters $\nu_{r}$, $\nu_{f}$,
$\nu_{c}$, and $\nu_{p}$, where $p$ is the reference magnitude of the pressure
applied, $F$ is the magnitude of the external wrenches applied for a given
wrench distribution that generally excludes gravitational forces since they
are typically negligible, $d$ is a reference length corresponding to the
diameter of the device, $E_{r}$ is a coefficient related to the stiffness of
the rubber (which can correspond to the Young’s modulus or the $c_{10}$
coefficient in a Neo-Hookean constitutive law), $E_{f}$ is the stiffness of
the outer fibres, $E_{c}$ is the stiffness of the central tube, $E_{p}$ is the
stiffness of the partition walls in designs where these are made of a
different material than the outer wall, and $\nu_{r}$, $\nu_{f}$, $\nu_{c}$,
and $\nu_{p}$ are the Poisson ratios of the rubber, outer fibres, central rod
and partition walls, respectively. The condition of geometric equivalence for
soft robotic manipulators in the non-dimensional analysis implies that the
aspect ratio between diameter and length is maintained, and thus only one
dimensional variable $d$ is necessary to specify size. It should be noted that
here the use of the magnitude $p$ in the non-dimensional groups imposes the
stress at the surfaces where pressure is applied, and is equivalent to the use
of a reference magnitude of stress in the structure of the device.

In the deformation of the soft robotic manipulator, it can be assumed that the
outer fibres are inextensible. In addition, the effect of $\nu_{f}$,
$\nu_{c}$, and $\nu_{p}$ on the behaviour of the complete device can be
assumed to be negligible in general. Moreover, rubbers used as soft material
are generally incompressible, so $\nu_{r}$ is generally constant. Thus, the
non-dimensional groups of $\frac{E_{f}}{E_{r}}$, $\nu_{f}$, $\nu_{c}$,
$\nu_{r}$ and $\nu_{p}$ are considered to have a negligible effect or be
irrelevant, and the dominant non-dimensional groups are reduced to
$\frac{p}{E_{r}}$, $\frac{F}{E_{r}d^{2}}$, $\frac{E_{c}}{E_{r}}$, and
$\frac{E_{p}}{E_{r}}$. Also, in some designs such as the one described in
section II, the central rod can be assumed to be practically inextensible, and
in other devices such as the FMA the central rod is not present. In these
cases, the group $\frac{E_{c}}{E_{r}}$ is not relevant. Finally, in designs
where the partition walls must also be made of the same material as the outer
wall due to fabrication constraints, such as the design in section II, the
non-dimensional groups are reduced to $\frac{p}{E_{r}}$,
$\frac{F}{E_{r}d^{2}}$.

Using this non-dimensional analysis, the behaviour of geometrically equivalent
devices with either different size, different material, different magnitude of
external wrenches, or different applied pressure can be expected to be
equivalent if the non-dimensional groups are equal. In addition, the number of
variables influencing the behaviour of a soft robotic manipulator in the non-
dimensional case is reduced by two relative to the dimensional case, which
simplifies the study of these devices. This analysis can also be generalised
to designs with additional regions or fibres with different stiffness by
adding non-dimensional groups corresponding to the ratio of stiffnesses in the
different regions.

###  III-B Application

This non-dimensional analysis enables the extrapolation of the behaviour of a
soft robotic manipulator to different sizes, pressures, loads, and materials.
This leads to a set of relevant insights.

Material | Ultimate stress [MPa] | Ultimate strain | Shore hardness |  $c_{10}$ coeff. [kPa]
---|---|---|---|---
DragonSkin 10 | 2.75 | 663$\%$ | 10A | 42.5
DragonSkin 20 | 3.8 | 620$\%$ | 20A | n/a
DragonSkin 30 | 3.45 | 384$\%$ | 30A | n/a
Elastosil M4601 | 6.5 | 700$\%$ | 28A | 131.2
Ecoflex OO-30 | 1.38 | 900$\%$ | OO30 | 12.7
Ecoflex OO-50 | 2.17 | 980$\%$ | OO50 | 25
TABLE II: Material properties of available hyperelastic rubbers relevant to
create deformable structures in soft robotic manipulators.

A first insight of interest is related to the non-dimensional group
$\frac{F}{E_{r}d^{2}}$. This elucidates that the external wrenches that can be
supported by a device with specified material properties and a given
deformation varies with the square of the diameter of the device. This implies
that scaling up a soft robotic manipulator increases the capability of
supporting forces as the square of the diameter.

Another relevant insight is that, for devices with equal material properties
and no external loads, the relation between applied pressure and deformation
is constant regardless of the device scale. This implies that the pressure
required for operation is determined by the design and desired deflection, but
not by the scale of the robot, which is particularly relevant when considering
medical applications.

In terms of materials, the non-dimensional analysis indicates that an equal
deformation is achieved in designs made of materials with different stiffness
if the pressure and external wrenches are varied proportionally to maintain
the non-dimensional groups constant. This suggests that stiffer materials lead
to robots capable of supporting higher wrenches with an equal deformation,
which is generally desirable, provided that pressure is also increased.

The stress in a stiffer material also increases proportionally in the case of
equivalent deformation. Thus, the use of stiff materials with a relatively low
ultimate stress can be undesirable. However, for materials with similar
ultimate strain, which is the case of some of the hyperelastic rubbers used as
soft materials in robotic manipulators with fluidic actuation, the non-
dimensional analysis indicates that the use of stiffer materials is desirable
as it leads to devices that can support higher wrenches, provided that the
assumptions in the previous subsection are satisfied. This guides the
selection of an appropriate silicone for the robot, as described in the next
subsection.

###  III-C Material selection

The hyperelastic material of which the structure of the soft robot is made
affects the performance, and needs to be selected carefully. The main
alternatives for hyperelastic rubbers in the literature and offered by
suppliers are summarised in Table II. The material properties were obtained
from the literature. There are two potential values in the literature for the
elastic coefficient $c_{10}$ for Elastosil 4601M, from [20] and [21]. In [20]
it is implied that $c_{10}=131.2\pm 24.3kPa$. Instead, in [21] they
characterise Elastosil 4601M using a Yeoh law with $c_{10}=110kPa$ and
$c_{20}=20kPa$, which can be used to approximate a Neo-Hookean law with
$c_{10}=110kPa$ for relatively low strains. In this work, we use the average
value from [20] since it is the average from a set of tests, but it should be
noted that some variation of up to $18\%$ is possible in practice.

The materials in Table II present significant differences in their stiffness
and ultimate strain. The selection of material can be made with aid from the
non-dimensional analysis presented in the previous subsection. As can be seen
in Table II, the ultimate strain of Elastosil M4601, by Wacker Chemie AG
(Munich, Germany), is higher than that of DragonSkin 10, 20, and 30, by
Smooth-On Inc. (Macungie, US). In addition, the stiffness of Elastosil M4601
is also higher or comparable to that of these other materials. Hence,
according to the non-dimensional analysis in the previous section III,
Elastosil M4601 is preferable over DragonSkin 10, 20 or 30, since it can reach
equivalent deformations to DragonSkin 10 or 20 while supporting higher
wrenches, and at the deformation where DragonSkin 30 reaches its ultimate
strain, Elastosil M4601 can still increase its strain by further increasing
pressure, leading to higher support of wrenches.

When comparing the ultimate strain of Elastosil M4601 with that of Ecoflex
OO-30 and OO-50, by Smooth-On Inc., it can be seen that the ultimate strain of
the former is somewhat lower. This means that a device made of Ecoflex OO-30
or OO-50 can reach an equal deformation as a device made of Elastosil M4601 at
the point where the latter reaches its ultimate strain, and then keep
increasing deformation and thus resistance to wrenches by increasing pressure.
However, the stiffness of Ecoflex OO-30 or OO-50 is at least three times lower
that of Elastosil M4601. According to the non-dimensional analysis in section
III, this implies that at the maximum strain of Elastosil M4601, with an
equivalent deformation, the wrenches that a device made of Elastosil M4601 can
support are at least three times those that a design made of Ecoflex OO-30 or
OO-50 can support. At the point of a strain of 700$\%$, devices made of
Ecoflex OO-30 or OO-50 can be further pressurised to increase strain up to
900$\%$, and thus increase somewhat the wrenches they can support. However,
the additional increase in wrenches that can be support when increasing
pressure and strain up to 900$\%$ is not expected to be triple the wrenches
that can be supported at 700$\%$ strain. Thus, the wrenches that a device made
of Elastosil M4601 can support at its maximum strain of 700$\%$ are expected
to be higher than those that a device made of Ecoflex OO-30 or OO-50 can
support at its maximum strain of 900$\%$. As a result, Elastosil M4601 is the
most suitable material to support the highest wrenches.

##  IV Experiments

Experiments were conducted to validate the non-dimensional analysis and to
measure the force of the design proposed in this paper.

Figure 5: Experimental setup used to measure the lateral force of the soft
robotic manipulator.

###  IV-A Experiment description

The experimental set-up is shown in Figure 5. It consists of a holder to keep
the base of the robot horizontal, a robot segment attached to it, and a scale
under the segment to measure the lateral force that it applies. The chambers
of the robot are pressurised using supply tubes, which are connected to a set
of pressure regulators (PRE1-U08, AirCom Pneumatic GmbH,Ratingen, Germany),
and controlled from a laptop via a microcontroller (mbed nxp lpc1768) via a
serial link. This set-up was used to measure the force of two robot segments
made of two different silicone rubbers with different stiffnesses.

###  IV-B Results and non-dimensional analysis validation

The results of force measured as a function of pressure for a segment made of
Elastosil 4601M with 12 mm OD are shown in Figure 6. As can be seen, the
segment can reach forces of 2.9N at 3 bar, at which pressure the segment
begins to fail. The tests were repeated using two equal segments made of the
same material to confirm the repeatability of the experiment, both reported in
Figure 6. The variation in force between the two equal segments was less than
10$\%$.

The results of force of the same segment design and same dimension (12 mm OD)
but made of Ecoflex 0050 are shown in Figure 7. The experiment was conducted
up to a pressure where the robot began to experience failure. The segment can
reach a force of 0.52N at 0.6 bar, which was observed to be the failure
pressure for this material.

The ratio between the Neo-hookean stiffness coefficient of the materials used,
Elastosil 4601M and Ecoflex 0050, is 5.2 according to Table II. Comparing the
results in Figures 6 and 7, there is a scaling factor in terms of force and
pressure between the results corresponding to the segments made of the two
different materials of approximately 5. This is similar to the ratio of Neo-
hookean stiffness coefficient of the two materials.

The theoretical lateral force of a device made of Ecoflex 0050 scaled using
the non-dimensional analysis to a theoretical device made of Elastosil 4601M
using the ratio of Neo-hookean stiffness coefficient between the two devices
of 5.2 is plotted in Figure 8. The same Figure also shows the experimental
measurement of force for the devices made of Elastosil 4601M for comparison.
The both plots in Figure 8 show a relatively good agreement. This confirms
that the results from one segment made of one material can be extrapolated to
an equal segment made of a different material by using the scaling
corresponding to the ratio of stiffnesses and maintaining the non-dimensional
groups descried in section III, and matches the behavior predicted by the non-
dimensional analysis.

Figure 6: Lateral force from a robot segment with the optimal design, 12 mm
OD, and made of Elastosil 4601M.

It should be noted that the scaling is not perfect, and the trends in the
graphs in Figure 8 differ to some extent. This can be attributed both to
experimental inaccuracies in manufacturing the segments and to the fact that
the inextensible central tube used in all the segments made of different
materials was the same and was not scaled according to the non-dimensional
groups in section III. This means that the tube bending stiffness is more
dominant for the softer material, Ecoflex 0050, particularly at lower
pressures where the force of the robot is lower. This effect is less important
at higher pressures where the bending stiffness of the central tube is less
dominant compared to the bending force generated by pressure. Thus, the
comparison of the scaling between both graphs in Figure 8 is more relevant in
the region corresponding to higher pressures used in each test.

###  IV-C Force measurement of proposed design

The experiments show the force of the proposed design as a function of
pressure for different materials. Given that the maximum pressure for the
medical application considered in this work is 3 bar, the material Elastosil
4601M is confirmed as a suitable choice, since other materials exhibit failure
at lower force and pressures.

The maximum force of one segment of the design proposed here is approximately
2.9N, as can be seen in Figure 6. This implies that the force of a full
prototype made of three segments such as that shown in Figure 4, which is
three times the length and thus any lateral force at the tip applies a triple
moment at the base, is a third of that of a single segment, or approximately
0.97N.

Figure 7: Lateral force from a robot segment with the optimal design, 12 mm
OD, and made of Ecoflex 0050. Figure 8: Comparison of lateral force predicted
by non-dimensional analysis scaling of Ecoflex 0050 to Elastosil 4601M, and
experimental measurement of lateral force of device made of Elastosil 4601M.

##  V Performance comparison

The performance of the design proposed in this work is compared in this
section with that of other relevant existing designs reported in the
literature in the same category of soft robotic manipulators actuated by a
pressurised fluid that could meet the design requirements in section II-A. It
should be noted that the literature in various cases does not include values
of lateral force, as reported in [1]. Here we include the designs where
lateral force is reported, following [1] with the addition of recent published
work.

As previously mentioned, the designs in the literature all have different
diameters. The non-dimensional analysis is first used to extrapolate their
force to a chosen outer diameter of 12 mm. Table III shows the maximum force
of the main existing designs in the literature when scaled to a 12 mm OD.

Robot design | Max. lateral force at 12 mm OD [N]
---|---
Tsinghua University [22] | 0.384
University of Hong Kong [23] | 0.82
Stiff-flop design in [24] | 0.09
Stiff-flop jamming in [3, 17] | 0.84
Our design | 2.9
TABLE III: Maximum lateral force for one 2DOF segment of main designs in the
literature when scaled to a outer diameter of 12mm, and maximum force of the
design proposed in this work at the same diameter.

It should be noted that the force values for the stiff-flop design reported in
[3, 17] correspond to lateral force when the robot is locked in a rigid
configuration using granular jamming, and an external force is applied on it.
However, the robot cannot actively move, and thus this force cannot be used to
carry payloads or actively apply forces on the environment. In [3, 17], it is
also reported that the stiff-flop module can reach forces of 47.1N in a 33 mm
OD robot (equivalent to 6.2N in a 12 mm OD). However, the authors in [3, 17]
report that these forces are measured by placing a load cell on top of a
vertical robot and pressurising it, and thus are understood to correspond to
extension forces rather than lateral forces. The values for extension forces
are not used here since this paper focuses on lateral force. Extension forces
are typically significantly higher than lateral forces. For example, in [22],
extension forces are nearly an order of magnitude higher than lateral forces.

As can be seen in Table III, the lateral force of all designs reported in the
literature is below 1N when scaled to a 12 mm OD. The design proposed in this
work offers a higher force of 2.9N. This is 200$\%$ above the existing highest
lateral force.

One factor that affects lateral force is the length of the robot, as already
mentioned previously. As such, shorter designs with a smaller aspect ratio
will typically have comparatively higher lateral force. This could in some
cases bias the comparison of force to some extent. However, shortening a
design typically means reducing the bending angle, which is typically
proportional to the length of the design. Thus, to achieve the desired
performance of 90 degree bending selected in this work, which is common across
the literature, robots cannot be designed to be as short as possible, and
instead a design compromise needs to be struck. As a result, the comparison of
designs based on lateral force as in Table III for designs that can typically
reach bending of 90 degrees remains valid for most cases.

##  VI Conclusions

A soft robotic manipulator design optimised to maximise its force was
presented in this work. A non-dimensional analysis was also developed to
compare the performance of the proposed design with other designs in the
literature. The non-dimensional analysis was validated with experiments, and
insights were extracted from it. In particular, the non-dimensional analysis
was used to extrapolate the force of different designs in the literature to a
given diameter. The force of existing designs was compared with the design
proposed in this work, and it was shown that the design proposed here offers
the highest force.

The design proposed in this work offers a force, size, mobility, and payload
capabilities that are suitable for medical applications. The design exploits
internal structural deformation to maximise force by morphing its internal
features into an optimal configuration when pressurised to bend in each
desired direction. In addition, it is derived using a set of design principles
that make it possible to easily adapt it to the requirements of each
application. In future work, we plan to apply this design on a fully
functioning 6DOF robotic manipulator for MIS.

##  VII Acknowledgements

This work was supported by the British Council under the Newton Fund
Institutional Links, grant number 623531377.

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# Fermi Surface Nesting Driving the RKKY Interaction
in the Centrosymmetric Skyrmion Magnet Gd2PdSi3

Yuyang Dong  Yosuke Arai  Institute for Solid State Physics, The University of
Tokyo, Kashiwa, Chiba 277-8581, Japan  Kenta Kuroda  Institute for Solid State
Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Graduate
School of Advanced Science and Engineering, Hiroshima University, Higashi-
hiroshima, Hiroshima 739-8526, Japan  International Institute for
Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2), Hiroshima
University, Higashi-hiroshima, Hiroshima 739-8526, Japan  Masayuki Ochi
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
Forefront Research Center, Osaka University, Toyonaka, Osaka 560-0043, Japan
Natsumi Tanaka  Department of Physics, Tokyo Metropolitan University, Tokyo
192-0397, Japan  Yuxuan Wan  Institute for Solid State Physics, The University
of Tokyo, Kashiwa, Chiba 277-8581, Japan  Matthew D. Watson  Diamond Light
Source Ltd, Harwell Science and Innovation Campus, Didcot, OX11 0DE, United
Kingdom  Timur K. Kim  Diamond Light Source Ltd, Harwell Science and
Innovation Campus, Didcot, OX11 0DE, United Kingdom  Cephise Cacho  Diamond
Light Source Ltd, Harwell Science and Innovation Campus, Didcot, OX11 0DE,
United Kingdom  Makoto Hashimoto  Donghui Lu  Stanford Synchrotron Radiation
Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
Yuji Aoki  Tatsuma D. Matsuda  Department of Physics, Tokyo Metropolitan
University, Tokyo 192-0397, Japan  Takeshi Kondo<EMAIL_ADDRESS>Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba
277-8581, Japan  Trans-scale Quantum Science Institute, The University of
Tokyo, Tokyo 113-0033, Japan

###### Abstract

The magnetic skyrmions generated in a centrosymmetric crystal were recently
first discovered in Gd2PdSi3. In light of this, we observe the electronic
structure by angle-resolved photoemission spectroscopy (ARPES) and unveil its
direct relationship with the magnetism in this compound. The Fermi surface and
band dispersions are demonstrated to have a good agreement with the density
functional theory (DFT) calculations carried out with careful consideration of
the crystal superstructure. Most importantly, we find that the three-
dimensional Fermi surface has extended nesting which matches well the _q_
-vector of the magnetic order detected by recent scattering measurements. The
consistency we find among ARPES, DFT, and the scattering measurements suggests
the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction involving itinerant
electrons to be the formation mechanism of skyrmions in Gd2PdSi3.

Magnetic skyrmions are topologically non-trivial particles with swirling spin
texture in real space, recently interested as a next-generation physical
medium leading toward future spintronic device applications [1, 2, 3, 4].
Magnetic skyrmions were first discovered in non-centrosymmetric magnets and
the consensus has been reached that the Dzyaloshinskii-Moriya interaction is
the key mechanism [5, 6]. However, the skyrmion size in this mechanism tends
to become rather large (10 nm - 200 nm), which is viewed as the major drawback
for applications [7].

Very recently, it was revealed from a study of Gd2PdSi3 that skyrmions can be
generated even in centrosymmetric crystals where the Dzyaloshinski-Moriya
interaction should not exist [8]. Interestingly, the skyrmion size of this new
type is extremely tiny, less than 4 nm. Its skyrmion formation mechanism is,
however, still controversial among many different ideas, such as the orbital
frustration [9], the geometrical frustration [10, 11], the magnetic dipolar
interaction [12], and the RKKY interaction induced by the Fermi surface (FS)
nesting [13, 14, 15, 16, 17, 18].

To solve this situation, it is crucial to reveal the electronic structure of
host materials [8, 19, 20, 21, 22]. Importantly, the zero-field ground state
is known to share the same magnetic modulations (_q_ -vectors) as the skyrmion
lattice emerging under the external magnetic field [8, 23]. Hence, pinning
down the direct relationship between the magnetic modulation and the
electronic structure of the ground state is the most fundamental issue in
elucidating the skyrmion mechanisms in centrosymmetric magnets.

The previous ARPES research of Gd2PdSi3 was conducted 14 years ago before the
discovery of the skyrmion [24]. The paper suggested that the magnetism in
Gd2PdSi3 and Tb2PdSi3 were driven by the FS nesting. However, the nesting
vector suggested [24] was later turned out different in direction and length
from the magnetic _q_ -vector of Gd2PdSi3 detected by resonant X-ray
scattering (RXS) [8] and neutron scattering [23]. There are two reasons behind
the confusion: Firstly, since the magnetic structures of Gd2PdSi3 were unknown
at that time, that of Tb2PdSi3 they determined was used on behalf of both
compounds for the discussion. Secondly, the nesting wave vector was determined
from a tight-binding fit to the ARPES data; however, the fitting was rather
rough due to the limited quality of data. Hence, the previous claim [24] is
only correct for Tb2PdSi3, but not for Gd2PdSi3.

Based on the current knowledge of the magnetic order in Gd2PdSi3, a recent
theory [18] newly suggested that the FS nesting which drives the RKKY
interaction exists in the barrel-shaped FS at the Brillouin zone center.
Another important aspect discovered after the previous ARPES study is that
Gd2PdSi3 crystals have complex superstructures [25] which should be taken into
account. In light of these circumstances, it is vital to revisit the band
structure of Gd2PdSi3 by the high-quality ARPES data and refined band
calculations.

In this letter, we present the first ARPES results for the thorough band
structure of Gd2PdSi3 including the $k_{\text{z}}$ dispersion. The DFT
calculations of the $2a\times 2a\times 8c$ superstructure [25] are also
conducted for the first time. Our ARPES data shows a good agreement with the
calculations, including the superstructure-induced band folding. Most
importantly, we find the extended FS nesting with the same direction and
length as the magnetic order revealed by the previous scattering experiments
[8, 23]; yet the FS location for the nesting differs from the prediction of
the most recent theory [18]. Our results indicate that the RKKY-interaction is
the formation mechanism of skyrmions in Gd2PdSi3.

Single crystals of Gd2PdSi3 were grown by the Czochralski pulling method in a
tetra-arc furnace. The raw materials used were 3N5 (99.95%-pure) Gd, 4N Pd,
and 5N Ge. ARPES with the vacuum ultraviolet (VUV-ARPES) was performed at
beamline 5-2 of Stanford Synchrotron Radiation Lightsource (SSRL) and beamline
I05 of Diamond Light Source in the photon energy range from 100 eV to 200 eV.
Soft X-ray ARPES (SX-ARPES) was performed at BL25SU of SPring-8 [26] in the
photon energy range from 380 eV to 650 eV. The (001) surface prepared by a
crystal cleavage $in$ $situ$ was measured at 10 K below the Néel temperature
($T_{\text{N}}$ = 21 K). The details of band calculations are described in
Supplemental Material.

SX-ARPES is commonly used for bulk-sensitive measurements. Figure 1(b) show
SX-ARPES intensities at ($k_{\text{x}}$,$k_{\text{y}}$)=(0,0) obtained by
sweeping photon energy (or $k_{\text{z}}$ value). A clear $k_{\text{z}}$
dispersion with a periodicity of $2\pi/c$ is observed, as traced by a yellow
dotted line, and it determines 435 eV and 500 eV as the $\Gamma$ and A points,
respectively. In Figs. 1(c) and 1(e), we plot the FS mappings over a wide
$k_{\text{x}}$-$k_{\text{y}}$ sheet measured at these two photon energies. The
FS sheet at $\Gamma$ [Fig. 1(c)] lies across the Brillouin zone (BZ) boundary,
whereas that at the A point [Fig. 1(e)] shows only circles at the zone center.
These are reproduced by our DFT calculations [Fig. 1(d),(f)].

Figure 1:  (a),(b) $k_{\text{z}}$ dispersion obtained by photon-energy-
dependent ARPES measurements with VUV and SX, respectively. The dotted yellow
lines are the guides to the eye, showing a periodicity along $k_{\text{z}}$.
(c),(d) Fermi surface mapping across the $\Gamma$ point by SX-ARPES and DFT
calculations, respectively. (e),(f) The same as (c) and (d), respectively, but
across the A point. The dashed lines indicate the Brillouin zones. (g),(h)
Band dispersions along H-A-H and K-$\Gamma$-K, respectively, measured by VUV-
ARPES.

We find that the VUV-ARPES data also show the $k_{\text{z}}$ dispersion [Fig.
1(a)] with the same periodicity as the SX-ARPES data under the same inner
potential $V_{0}$ = 17.9 eV, thus capturing the bulk information. This is
further examined in Figs. 1(g) and 1(h) by plotting the band dispersion maps
along the high-symmetry cuts near the A point (104 eV) and at the $\Gamma$
point (130 eV), respectively. Overall band dispersions, including the bottom
(yellow arrows) which corresponds to the yellow wavy line in Figs. 1(a), shift
energetically upward as going from A to $\Gamma$, due to the $k_{\text{z}}$
dispersion. Around 148 eV, which is the Gd 4$d$ \- 4$f$ resonant photoemission
photon energy [27, 28], high-intensity signals appear at $E_{\text{F}}$ [see
Fig. 1(a)]; this feature is advantageous to investigate the detailed
Fermiology of this compound.

To fully understand the band structure of Gd2PdSi3, the superstructure needs
to be taken into account. Without distinguishing the Pd and Si atoms, the
crystal structure is hexagonal as in Fig. 2(b). The real material has
$2a\times 2a\times 8c$ superstructure [Fig. 2(c)] due to the ordering of the
Pd and Si atoms and their systematic variation along the $c$-axis [25, 8].
Different stacking sequences of the Pd-Si layers lead to distinct
superstructure domains, but all these have equivalent centrosymmetry. The
superstructure can affect the band structure in two holds: One is that it
becomes a 2-fold symmetry. The DFT calculations carried out for one of the
superstructure domains result in the 2-fold symmetric Fermi surface [Figs.
1(d) and 1(f)]. The ARPES data, in contrast, show 6-fold symmetry, since
averaging signals from different domains. Second is that the superstructure
reduces the Brillouin zone (BZ), as represented in Fig. 2(a) with blue lines.
This effect is indeed observed as presented below.

In Fig. 2(d), we examine the band dispersions along the high-symmetry path of
$\Gamma$-K-M for the primitive BZ [red lines in Fig. 2(a)] which is obtained
by VUV-ARPES with 130 eV photons. Here the used light polarization (linear
vertical polarization) is different from that (linear horizontal polarization)
of Fig. 1(g). Our data show the bands folded about the high-symmetry point
denoted as (M) in the superstructure BZ; one of those is marked by the dashed
yellow line, which is symmetric to the main band (the solid yellow line) about
the (M) point.

The superstructure would also reduce the BZ along $k_{\text{z}}$ to
$\frac{1}{8}$ times the primitive one, as represented in Fig. 1(a) by blue
lines. We found that the resonant photon energy of 148 eV corresponds to the
($\Gamma$) point in the reduced BZ. This allows one to investigate the
detailed FS at $k_{\text{z}}$ = 0, as an alternative to measurements at 130 eV
where the intensities near $E_{\text{F}}$ are extremely weak due to the matrix
element effect. Figure 2(e) displays the band dispersion at 148 eV side by
side with that at 130 eV [Fig. 2(d)] along the in-plane momentum cut of
(M)-($\Gamma$)-(M) and M-$\Gamma$-M, respectively. The overall feature,
including the band folding due to the superstructure, is almost identical
between the two, except that the intensities near $E_{\text{F}}$ are much
higher at 148 eV. In Fig. 2(f), the corresponding DFT bands at $k_{\text{z}}$
= 0 are overlayed on the enlarged image of Fig. 2(e). Although not all bands
are visible in the data, a good agreement is seen between the calculations and
ARPES results.

Figure 2:  (a) Brillouin zones of Gd2PdSi3 for the primitive cell (red) and
the superstructure (blue), respectively. (b) A conventional cell without
distinguishing the Pd and Si atoms. The red lines indicate the primitive cell.
(c) $2a\times 2a\times 8c$ superstructure with the modulated Pd-Si
arrangement. (d) Band dispersion along $\Gamma$-K-M
[($\Gamma$)-(M)-($\Gamma$)] measured with 130 eV photons. The blue lines
indicate the symmetry axis at (M). (e) Band dispersion along
($\Gamma$)-(M)-($\Gamma$) measured at 148 eV (resonant photon energy). (f)
Magnified band dispersion of (e), overlayed by the corresponding DFT band
dispersions of the $2a\times 2a\times 8c$ superstructure.

The ARPES mapping at 148 eV, corresponding to $k_{\text{z}}$ = 0, is presented
in Fig. 3(a). It is similar to the data of SX-ARPES at 435 eV [Fig. 1(c)], but
with much higher quality. The FS is shaped like a windmill with six wings
preferable for the nesting. This is consistent with our DFT calculations [Fig.
3(b)]; Note that the 2-fold symmetry due to the superstructure is absent in
the ARPES data, which integrates signals from superstructure domains aligned
in different directions.

Figure 3:  (a) Fermi surface mapping by ARPES with 148 eV photons. (b) DFT
calculations [same as Fig. 1(d)]. (c) Band dispersion along the black line in
(a). Nesting vector lengths (green and blue arrows) are estimated from the
peak positions of MDC at $E_{\text{F}}$, which are determined by fitting to
Lonetzian functions (red curves) with a polynomial background. (d) Nesting
vectors obtained from ARPES results and DFT calculation are compared with the
magnetic _q_ -vector detected by scattering measurements [8, 23]. (e),(f)
Sketch demonstrating that the nesting condition is better in the case that the
vector is angled than perpendicular to the Fermi surface. Figure 4:  (a)-(c)
Band dispersions along the in-plane momentum cut same as in Fig. 3(c), but at
different photon energies (141 eV, 143 eV, and 147 eV, respectively). The
nesting vector lengths (arrows) were estimated from the MDC peak positions,
similarly in Fig. 3(c). (d) DFT Fermi surface along $k_{\text{z}}$ and
$k_{\text{x}}$ [the purple dotted line in Fig. 2(b)]. (e) Schematic drawing of
the magnified area within a black dimension arrow in (d). The blue solid and
dashed lines indicate the main and folded bands due to the superstructure,
respectively. (f) Nesting vectors at different $k_{\text{z}}$s determined by
ARPES and DFT calculations, and the magnetic _q_ -vector detected by
scattering measurements [8, 23]. The $k_{\text{z}}$ range of the horizontal
axis (or photon energy range of 138-151 eV) is shown by the arrow in (d). For
the ARPES results, the lengths of left and right vectors [see (a-c)] are
averaged. The red arrows mark the photon energies used in (a)-(c). For
simplicity, the $\Gamma$ and ($\Gamma$) points are treated as the same in this
panel.

The nesting condition is better when the vector is directed not perpendicular
to the Fermi surface (FS) [Fig. 3(e)] but angled by 30° [Fig. 3(f)]. When the
nesting wave vector is perpendicular to the FS, only two sheets out of six FS
sheets are connected [Fig. 3(e)]. In contrast, four FS sheets can be connected
by a wave vector when it is angled by 30∘ [Fig. 3(f)]. This is further
justified in Supplemental Material, where we calculate the nesting function
similar to Lindhard function and demonstrate that a peak indeed appears at the
angled nesting vector. Importantly, this vector direction is the same as that
of the magnetic order detected by the previous scattering measurements [8,
23]. [Note that the nesting vector direction suggested 14 years ago [24] was
neither cases of Fig. 3(e) nor Fig. 3(f).] Figure 3(c) exhibits the ARPES
dispersion along this direction represented by the solid line in Fig. 3(a).
Two V-shaped bands are observed. We estimate the nesting vector length [green
and blue arrows in Fig. 3(c)] by Lorenzian fitting to the momentum
distribution curve (MDC) at $E_{\text{F}}$. The vector lengths similarly
determined for various momentum cuts [arrows in Fig. 3(a)] are summarized in
Fig. 3(d). We find that the length is almost constant at different
$k_{\text{y}}$s, thus the FS is extensively nested. Most importantly, the
nesting length matches well with that of the magnetic _q_ -vector [the pink
line in Fig. 3(d)] obtained by the scattering measurements [8, 23]. In Fig.
3(d), we also plot the nesting vector length extracted from the DFT Fermi
surface. The length for the lower-right part of the FS (circled by purple) is
consistent with our ARPES results and the magnetic _q_ -vector, further
supporting our conclusion. For the lower-left part (circled by orange), there
is some mismatch, implying that the superstructure sacrifices the nesting
condition to some degree.

We also investigate the FS nesting along $k_{\text{z}}$ by changing photon
energy. Figures 4(a)-(c) show the band dispersions measured at different
photon energies (141 eV, 143 eV, and 147 eV, respectively) along a momentum
cut corresponding to the horizontal line in Fig. 3(a). The photon energies are
close to the resonance photon energy (148 eV), providing relatively intense
ARPES signals near $E_{\text{F}}$. Similar V-shaped bands are observed for all
those photon energies. As summarized in Fig. 4(f), the nesting vector length
is estimated to be nearly constant within a certain range of $k_{\text{z}}$.
Interestingly, however, we find that the length around 143 eV is longer than
the others. To understand this, we plot in Fig. 4(d) the DFT Fermi surface
against $k_{\text{z}}$ along the horizontal dashed line in Fig. 3(b). The
complex Fermi surface with the superstructure-induced band folding is seen
along $k_{\text{z}}$. The black arrow in Fig. 4(f) indicates the photon energy
range of 138-151 eV, at which we could estimate the FS nesting length by
ARPES. Figure 4(e) sketches the main and folded FSs (solid and dashed lines,
respectively) within this arrow region. This explains that the length of the
nesting vector (red arrows) peaks at the crossing point of these Fermi
surfaces. This behavior is reproduced in Fig. 4(f), which plots the DFT
calculations on top of the ARPES data. In Fig. 4(f), the magnetic _q_ -vector
[8, 23] is also overlayed, showing a good agreement with the nesting vectors,
except for the region with a peak. The extensive nesting observed in both in-
plane [Fig. 3(d)] and out-of-plane [Fig. 4(f)] indicates that the RKKY
interaction drives the magnetism in this compound.

In conclusion, the intrinsic electronic structure of the centrosymmetric
skyrmion magnet Gd2PdSi3 was revealed by ARPES and DFT calculations for the
first time. We demonstrated the extensive Fermi surface nesting with the
direction and length same as the _q_ -vector of the magnetic order previously
detected by scattering measurements. These results indicate that the RKKY
interaction, mediated by itinerant electrons, is the mechanism for the
magnetism in Gd2PdSi3. Since the zero-field ground state is known to share the
same magnetic modulations as those of the skyrmion lattice, the FS nesting-
driven RKKY interaction is implied to be the formation mechanism of the
skyrmions with small size ($<$ 4 nm). Our results will provide essential
guidance in the material design for centrosymmetric systems yielding small
skyrmions that are advantageous for device applications.

Finally, we emphasize that our findings differ from the previous results in
Gd2PdSi3 [24, 18]. The ARPES study [24] (14 years ago) claimed the FS nesting
in a different direction and length from our results and from the magnetic _q_
-vector [8, 23]. While the recent theory [18] suggests the nesting wave vector
same as our result in direction and length, it is claimed to be located in the
Fermi surface pocket around the Brillouin zone center. Such a structure is,
however, not identified by either our ARPES measurements or our DFT
calculations. Our data, instead, revealed that the FS nesting driving the RKKY
interaction exists near the boundaries of the Brillouin zone.

Acknowledgements: Use of the Synchrotron Radiation Lightsource, SLAC National
Accelerator Laboratory, is supported by the U.S. Department of Energy, Office
of Science, Office of Basic Energy Sciences under Contract No. DE-
AC02-76SF00515. We thank Diamond Light Source for access to beamline I05 under
proposals SI30646, SI28930, and SI25416 that contributed to our results. This
work was supported by the JSPS KAKENHI (Grants Numbers. JP21H04439,
JP22K03517, and JP23H04870), by the Asahi Glass Foundation, by MEXT Q-LEAP
(Grant No. JPMXS0118068681), by The Murata Science Foundation, and by Tokyo
Metropolitan Government Advanced Research (Grant Number. H31-1).

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