# Negative Shaping Order $K$ in Set Shaping Theory: A Comprehensive Analysis

Sochima Biereagu111Author<EMAIL_ADDRESS>
###### Abstract

This paper explores an innovative aspect of the Set Shaping Theory [1], the
use of a negative shaping order $K$. Traditionally, the theory utilizes a
positive $K$ [2] to extend the length of data strings, enhancing their
testability [3] and compressibility [4]. We propose a paradigm shift by
employing a negative $K$, which shortens data strings and potentially improves
compression efficiency. However, this approach sacrifices the local
testability of the data, a cornerstone in traditional Set Shaping Theory. We
examine the theoretical implications, practical benefits, and challenges of
this new methodology.

## Introduction

The Set Shaping Theory [1], since its inception, has revolved around the
utilization of a positive shaping order $K$ to transform data strings.
However, recent developments suggest that employing a negative $K$ can
significantly optimize compression, especially when coupling high entropy
sequences with longer strings. This innovative approach potentially marks a
paradigm shift in data compression methodologies, offering a deeper
understanding of the $K$ parameter and its implications.

#### Brief Overview of Set Shaping Theory

Fundamentally, the Set Shaping Theory investigates bijection functions that
transform one set of strings into another. The crux is the transformation of
string sets, such as $X^{N}$, into longer string sets, like $Y^{N+K}$, with
each string in the latter being $K$ symbols longer than its counterpart in the
former.

#### Definitions and Basic Concepts

  * •

Source Representation: A source, denoted as $X$, is characterized by the
triplet $X=(x;A;P)$, where:

    * –

$x$ is the value of a random variable.

    * –

$A$ represents the possible states or values of $x$.

    * –

$P$ is the probability distribution associated with the states.

    * –

We call $X^{N}$ the set of all possible strings
$x=x_{1},\ldots,x_{j},\ldots,x_{N}$ generated by $X$.

  * •

Entropy [4]: A measure of unpredictability or randomness of a set, $H(X)$, is
given by:

$H(X)=-\sum_{i}p_{i}\log_{b}p_{i}$

  * •

Information Content (zero-order empirical entropy): is a measure of
uncertainty or randomness associated with a discrete probability distribution,
For a specific sequence $x$ in $X^{N}$, its information content, $I(x)$, is:

$I(x)=-\sum_{j=1}^{N}\log_{2}p(x_{j})$

where $p(x_{j})$ refers to the probability of the occurrence of the symbol
$x_{j}$ in the sequence $x$.

Note: In this article, the information content always refers to the zero-order
empirical entropy.

####

The bijection function $f$ that performs the transform is defined as:

$f:X^{N}\rightarrow Y^{N+K}$

The shaping order, $K$ [2] acts as a determinant of how much length is added
to each string during the transformation. If you’re transforming a string from
the set $X^{N}$ using the function $f$, the resulting string in $Y^{N+K}$ will
be $K$ characters/symbols longer.

Transformation process:

  * •

Using the bijection function $f$, a string $x$ in $X^{N}$ is mapped to a new
string $y$ in $Y^{N+K}$

  * •

The key insight from set shaping theory is that the strings in $Y^{N+K}$ are
chosen based on their lower information content from the set $X^{N+K}$. This
means that, even though the strings in $Y^{N+K}$ are longer, they might be
more ”compressible” due to their reduced information content.

### Local testability

Local testability [5] [6] is a fundamental concept in data integrity and error
detection, playing a pivotal role in modern data processing and communication
systems. It refers to the ability to efficiently test the integrity of a piece
of data without the necessity to examine the entire data set. This capability
is crucial in large-scale data environments where scanning entire datasets for
errors is impractical.

#### Set Shaping Theory and Local Testability

In Set Shaping Theory, local testability stands out as a significant feature
due to its efficient approach to maintaining data integrity [3]. In
traditional error correction techniques, redundancy is incorporated within the
codewords, Set Shaping Theory offers an alternate method. The essence of this
method revolves around the bijection functions $f(X^{N})=Y^{N+K}$ that
transform one set of strings into another of equal size but with strings of
greater length. The transformation process conditions the emission probability
of the dependent variable $y$ based on previously emitted variables. As a
consequence, if a decoder encounters a symbol with a conditional probability
of zero, it is an immediate indication of an error in the message,
facilitating local testability [3].

Several advantages arise from this Approach:

  * •

Efficient Local Testing: The reshaped strings in $Y^{N+K}$ are selected for
their minimized average information content, which streamlines the process of
local testing. This efficiency arises from the predictability and structured
nature of these strings, making anomalies more detectable.

  * •

Enhanced Error Detection through Reduced Entropy: The selection of strings
based on lower entropy not only facilitates better compression but also
contributes to more effective error detection. Lower entropy in strings means
fewer variations and unexpected elements, allowing for easier identification
of errors during local testing.

  * •

Intrinsic Error Detection Capability: The unique feature of the Set Shaping
Theory is its intrinsic error detection mechanism. By conditioning the
emission probabilities of each symbol in the string, the theory enables the
immediate identification of errors. A symbol with a conditional probability of
zero is a clear indicator of a fault in the message, thus providing a robust
method for error detection without the need for extensive data scanning.

### Rationale for exploring a negative $K$ in Set Shaping Theory.

Set Shaping Theory, originally designed for transforming and compressing data
with positive shaping orders, mainly extends data strings. This usual method
using a positive $K$ improves compressibility and keeps local testability.
Yet, always using positive $K$ values might miss out on more efficient data
compression. This has sparked interest in studying the effects of a negative
$K$ in Set Shaping Theory.

Some reasons to explore Negative $K$:

  * •

Pushing the Boundaries of Compression: The primary motivation behind
considering a negative $K$ is the pursuit of higher compression efficiency. By
shortening data strings, there is an inherent potential for reducing the
overall data footprint significantly.

  * •

Theoretical Novelty and Exploration: From a theoretical standpoint, extending
the Set Shaping Theory to include negative $K$ values opens new avenues of
research. It challenges the conventional wisdom of data transformation and
presents an opportunity to deepen our understanding of the interplay between
string length, information content, and compressibility.

  * •

Practical Implications in High-Entropy Contexts: In practical applications
involving high-entropy data sets, a negative $K$ could yield considerable
advantages. These contexts often involve managing large volumes of data where
conventional compression methods may fall short. Employing a negative $K$
could potentially unlock new methods of handling such data more efficiently.

  * •

Exploring Trade-offs: The exploration of a negative $K$ is not without its
trade-offs, primarily the loss of local testability. This exploration is
crucial for understanding the limits and potential drawbacks of the Set
Shaping Theory when applied in unconventional ways. It allows for a
comprehensive understanding of the theory’s applicability and limitations.

## Theoretical Foundation of Negative $K$

Set Shaping Theory traditionally explores bijection functions
$f:X^{N}\rightarrow Y^{N+K}$ with a single shaping order $K$, where
$N,K\in\mathbb{N}^{+}$, and $|X^{N}|=|Y^{N+K}|$. This implies a direct one-to-
one correspondence between the elements of the sets, preserving their
cardinality. Additionally, the subset relationship $Y^{N+K}\subset X^{N+K}$
holds, indicating that the transformed set is a part of the extended original
set.

When considering a negative shaping order $K$, the dynamics of this bijection
undergo a significant alteration. With $K$ being negative, the length of
strings in the transformed set $Y$ is reduced compared to their counterparts
in set $X$. This reduction in length introduces challenges in maintaining the
bijection, primarily due to the potential decrease in the total number of
unique strings that can be formed with a shorter length.

To circumvent this challenge and preserve the bijection when employing
negative $K$ values in Set Shaping Theory, a nuanced strategy that involves
the integration of both negative and positive $K$ values is essential. This
approach ensures that the transformation maintains a one-to-one correspondence
and effectively utilizes residual sequences, which are pivotal in this context

#### Integrating Negative $K$ with Positive $K$

In the application of negative $K$ values, it is crucial to first apply only
positive $K$ values for the initial $|-K|$ values of $N$. This method ensures
that the zero-order empirical entropy is increased appropriately, setting a
foundation for the subsequent application of negative $K$. For instance, in a
scenario where $K=-6$, the transformation with positive $K$ is applied for the
first six values of $N$ ($1$, $2$, $3$, $4$, $5$, $6$). Once $N$ exceeds this
range, negative $K$ values are employed, utilizing the residual sequences left
from the transformations at smaller values of $N$.

#### Example of Applying Negative $K$

To illustrate this concept further, let’s examine a binary example
($A=\\{0,1\\}$) where $N=3$ and $K=1,-2$:

  * •

For $N=1$, we can only apply positive $K$. The transformation is as follows:

    * –

0 becomes 00

    * –

1 becomes 11

    * –

Residual sequences of length $N=1$ are $(0,1)$, and for $N=2$ are $(01,10)$

  * •

For $N=2$, only positive $K$ is applied again:

    * –

00 becomes 000

    * –

11 becomes 111

    * –

01 becomes 001

    * –

10 becomes 100

The residual sequences for $N=3$ include $(010,110,101,011)$. When $N$ reaches
3, which is greater than $|-K|$, the negative $K$ is also applied, using the
residual sequences from $N=1$ with $K=-2$.

  * •

For $N=3$, we can apply negative $K$

    * –

000 becomes 0

    * –

111 becomes 1 (Here we use the residual sequences from $N=1$ $(0,1)$)

    * –

001 becomes 0000

    * –

010 becomes 1111

    * –

100 becomes 0001

    * –

110 becomes 1110

    * –

101 becomes 1011

    * –

011 becomes 1101

Then with $N=4$ we will use negative $K=-2$ and the residual sequences of
length $N=2$ which are $(01,10)$.

The application of a negative shaping order $K$ within the Set Shaping Theory
necessitates a multi-valued approach, incorporating both a negative and a
positive value of $K$. This strategy is essential to maintain the theory’s
integrity and effectiveness in data compression. Specifically, when the length
of the data string $N$ is less than or equal to the absolute value of the
negative $K$ (i.e., $N\leq|K|$), the transformation process employs the
positive value of $K$. This initial phase ensures an increase in the zero-
order empirical entropy $NH_{0}(S))$, preparing the groundwork for more
effective data compression in subsequent steps.

Once $N$ exceeds the absolute value of the negative $K$ (i.e., $N>|K|$), the
negative $K$ is applied. This shift to negative $K$ allows for the utilization
of residual sequences, which are crucial in enhancing the compressibility of
the data. These residual sequences, left unused in the positive $K$ phase,
become integral to the transformation process under the negative $K$ regime,
contributing to a more efficient and comprehensive compression strategy.

This multi-valued approach to applying negative $K$ in Set Shaping Theory,
balancing both negative and positive values, is particularly significant from
a compression standpoint. As $N$ trends towards infinity, the initial phase
where only the positive $K$ is applied (affecting the first $|K|$ values of
$N$) has a minimal impact on the overall average compression. This is because
the vast majority of the data transformation process, occurring as $N$ becomes
large, leverages the full potential of both positive and negative $K$ values.

#### How Negative $K$ Affects Entropy and Compressibility

With a negative $K$, the length of some strings in the transformed set
$Y^{N+K}$ is reduced. This reduction in length can lead to a decrease in the
potential configurations of the strings, thereby potentially decreasing the
overall entropy of the set. The transformed strings, being shorter and
potentially having lower entropy, might be more amenable to efficient
encoding. Standard compression algorithms could potentially achieve better
compression ratios with these transformed strings.

### The Trade-off: Compressibility vs. Testability

#### Impact of Negative $K$ on Local Testability

The incorporation of negative $K$ values alongside positive $K$ in SST
presents a nuanced shift in the theory’s application, particularly affecting
the interplay between compressibility and testability. This integration
method, while optimizing for compression, has significant implications on the
testability of the transformed sequences. By employing both negative and
positive $K$ values, SST leverages all possible sequences, leaving no
sequences with a probability of zero. This comprehensive usage of sequences
ensures maximal exploitation of the available data for compression purposes.
However, it also implies the elimination of sequences that would traditionally
be used to maintain testability. In standard SST applications using only
positive $K$, certain sequences are inherently left with zero probability,
serving as indicators for error detection. The use of negative $K$ eliminates
these zero-probability sequences, thereby removing a key mechanism for
testability within the framework.

## Practical Implementation and Results

To demonstrate the practicality and implications of employing a negative
shaping order $K$ in Set Shaping Theory (SST), we conducted an empirical study
with a source defined by an ensemble $X=(x;A;P)$ having a uniform probability
distribution. Our goal was to assess the impact of different $K$ values,
ranging from negative to positive, on the compressibility and overall
characteristics of the transformed data sets.

### Experimental Setup

We defined a source ensemble $X$ as follows:

  * -

$A=\\{0,1,2\\}$: The set of possible states or values of $x$.

  * -

$N=10$: The length of each string in the source set $X^{N}$.

  * -

This configuration implies that $X^{N}$ contains all possible strings of
length $10$ produced by the source $X$, amounting to $3^{10}=59,049$ strings.

The experiment was structured to apply SST with values of $K$ ranging from -3
to 3. This range was chosen to explore how these transformations impact the
string’s information content.

### Methodology and Data Transformation

For our empirical investigation, the transformation of data strings was
conducted using the bijection function $f:X^{N}\rightarrow Y^{N+K}$, with the
following key methodologies:

  * •

For Positive $K$:

    * -

The function $f:X^{N}\rightarrow Y^{N+K}$ was applied, wherein each string in
the resulting set $Y^{N+K}$ was longer by $K$ symbols compared to its
counterpart in $X^{N}$.

    * -

The set $Y^{N+K}$ is a carefully chosen subset of $X^{N+K}$ characterized by
having lower information content, thereby optimizing for compressibility while
preserving data integrity.

    * -

The cardinality equality $|X^{N}|=|Y^{N+K}|$ was maintained, ensuring a one-
to-one correspondence between the elements of the sets.

  * •

For Negative $K$:

    * -

In our experiment, we employed a multi-valued strategy for negative $K$
values. Each negative $K$ was paired with a corresponding positive $K$ value,
specifically chosen as $K=1$.

    * -

Given that our chosen string length $N$ was 10, and since $N\geq|K|$ for all
tested negative $K$ values, we were able to apply both negative and positive
$K$ values. This application was made possible by utilizing the residual
sequences, which would typically emerge from transformations with smaller $N$
values than our chosen length, (i.e $N<|K|$).

    * -

This integrated approach allowed for the utilization of all possible
sequences, enhancing compressibility by encompassing a broader spectrum of
string lengths and information content.

    * -

Similar to the positive $K$ scenario, the cardinality condition
$|X^{N}|=|Y^{N+K}|$ was upheld, ensuring that the transformation remained
bijective.

This approach allowed us to systematically analyze the effects of both
extending and shortening the data strings under the Set Shaping Theory
framework, providing insights into the compressibility and other
characteristics of the transformed data sets across a spectrum of $K$ values.

### Results and Analysis

#### Some Definitions:

  * •

The probability $P(x_{i})$ that the source $X$ generates the sequence $x_{i}$
is given as:

$P(x_{i})=\prod_{j=1}^{N}p(x_{j})$

  * •

We call the average information content of a sequence generated by a source
$X=(x;A;P)$ the summation of the product between the information content of
the sequences belonging to $X^{N}$ is their probability:

$I(x)=\sum_{i=1}^{|X|^{N}}P(x_{i})I(x_{i})$

  * •

Because our bijection function $f$ transforms the strings $x\in X^{N}$ into
the strings $y\in Y^{N+K}$ consequently, the average information content
changes as follows.

$I(y)=\sum_{i=1}^{|X|^{N}}P(x_{i})I(y_{i})$

The first column shows the value of $K$, the second column shows the average
information content of the strings in set $Y^{N+K}$.

$K$ | I(y)
---|---
1,-3 | $14.001$
1,-2 | $13.851$
1,-1 | $13.636$
0 | $14.263$
1 | $14.136$
2 | $14.006$
3 | $13.694$
Table 1: The average information content (in bits) calculated for strings in
the set $Y^{N+K}$ for varying values of $K$

#### Key Observations:

  * -

For negative values of $K$, the incorporation of shorter string lengths in the
sets led to a decrease in the average information content, suggesting an
increase in compressibility.

  * -

For positive values of $K$ the information content trends were consistent with
traditional SST expectations. [2]

###

We also tested for the binary alphabet $A=\\{0,1\\}$ and found an interesting
result:

$K$ | I(y)
---|---
1,-3 | $9.375$
1,-2 | $9.274$
1,-1 | $9.222$
0 | $9.235$

It is noteworthy to highlight that with the use of negative $K$, SST can also
be applied in the case of a binary alphabet $A=\\{0,1\\}$. In fact, in the
case of the binary alphabet, the inefficiency is very low, so testability must
be sacrificed.

### Implications and Future Directions

The exploration of negative shaping order $K$ in SST opens new avenues in the
field of data compression and information theory. This section discusses the
broader implications of this novel approach, highlights potential challenges
and limitations, and suggests future research directions.

#### Enhanced Compressibility with Negative $K$

The use of negative $K$ values in SST demonstrates a significant increase in
compressibility, particularly for high-entropy data sets. This method, by
reducing string lengths, can lead to more efficient storage and transmission
of data.

#### Exploring the Entropy Coding Limit with Negative $K$

The concept of negative $K$ values in SST offers a groundbreaking perspective
in the area of data compression, particularly in the context of approaching or
potentially surpassing the entropy coding limit. The entropy coding limit,
often associated with the parameter $NH_{0}(S)$ (the zero-order empirical
entropy multiplied by the length of the sequence), is traditionally considered
the average coding limit of the symbols of a sequence $S$ using a uniquely
decipherable and instantaneous code [7].

Critical Analysis of $NH_{0}(S)$ as the Entropy Coding Limit: Recent
developments and experimental results challenge the conventional understanding
that $NH_{0}(S)$ represents the absolute limit for data compression [8]. By
applying negative $K$ values in SST, the transformed sequences, despite being
shorter, exhibit reduced self-information per symbol on average. This
contradicts the long-held belief that $NH_{0}(S)$ is the minimum self-
information per symbol achievable in any compression scheme.

The Role of Negative $K$ in Set Shaping Theory: Negative $K$ values in SST
facilitate the creation of transformed sequences that, while shorter in
length, contain symbols from the same alphabet and are characterized by lower
empirical entropy. The key lies in the innovative approach of SST, which,
unlike traditional methods, does not focus solely on string length but on the
information content of the strings. This strategy opens up the possibility
that the transformed sequences, when subjected to efficient encoding
techniques like Huffman coding [9], might result in a compressed form that is,
on average, shorter than the $NH_{0}(S)$ limit of the original sequence [8].

Implications for Information Theory: This revelation that negative $K$ values
in SST can potentially reduce the average self-information per symbol below
the supposed $NH_{0}(S)$ limit challenges the foundational assumptions of
information theory. It suggests that there might exist transformations, as
demonstrated by SST, that can compress data beyond the presumed entropy coding
limits without changing the alphabet. This finding prompts a re-evaluation of
the theoretical limits of data compression and the efficiency of current
compression algorithms.

#### Future Research Directions

Algorithmic Development for Negative $K$: Future research should focus on
developing sophisticated algorithms capable of efficiently applying negative
$K$ transformations to large datasets.

Integration with Existing Compression Techniques: Another research avenue is
exploring how SST with negative $K$ can be integrated with existing data
compression algorithms. This integration could potentially lead to hybrid
compression methods that leverage the strengths of both SST and conventional
techniques.

Potential Challenges and Limitations: Employing negative $K$ values in Set
Shaping Theory presents challenges such as the loss of local testability,
potentially impacting error detection and correction capabilities in
transformed data sets, along with the increasing complexity of maintaining
bijection during data transformations, particularly as string length
significantly reduces, which could lead to computational and practical
implications that need careful consideration.

Determining Optimal Positive/Negative $K$ Value Pairs: Exploring the optimal
pairing of positive and negative $K$ values is crucial for maximizing SST’s
compression efficiency. The simplest case $K=1,-1$ is not necessarily the most
effective, indicating that finding the optimal pair could also approach the
theoretical limits of entropic coding. This endeavor not only aims to enhance
SST’s practical utility but also potentially extends our understanding of the
fundamental principles of data compression.

## Conclusion

This paper has introduced and thoroughly explored the concept of a negative
shaping order $K$ within the framework of Set Shaping Theory (SST). Our
investigation reveals that employing a negative $K$ can significantly enhance
data compressibility, marking a substantial shift from traditional SST
applications, which predominantly utilize a positive $K$ to extend data
strings, thereby maintaining both compressibility and local testability.

The empirical analysis conducted with varying $K$ values demonstrates a clear
trend: when negative $K$ is utilized, there is a notable decrease in the
average information content of the strings. This reduction suggests an
improved potential for compression. However, this comes at a cost – the
reduction in string length inherent to a negative $K$ leads to challenges in
maintaining the bijection essential for SST and compromises local testability,
a cornerstone of data integrity in the original framework.

Given the profound implications of these findings, further research is
required to explore the full potential of negative $K$ values in SST. This
includes developing more sophisticated algorithms to optimize the
transformation process, exploring the integration of SST with other
compression techniques, and understanding the theoretical underpinnings of why
and how negative $K$ values can achieve such results. The scientific
community’s engagement in this exploration is crucial, given the enormous
impact that surpassing the $NH_{0}(S)$ limit could have on information theory
and data compression practices.

The exploration of a negative $K$ within SST opens new perspectives in the
field of data compression and information theory. The journey into uncharted
territories of SST, marked by the introduction of negative $K$, not only
enriches our understanding of the parameter $K$ but also sets the stage for
innovative advancements in the field of data compression.

## References

  * [1] Solomon Kozlov. (2021). Introduction to Set Shaping Theory. ArXiv, abs/2111.08369.
  * [2] Biereagu, Sochima. (2023). Introducing the Role of Shaping Order K in Set Shaping Theory. AfricArxiv, doi:10.31730/osf.io/ywmsr.
  * [3] Solomon Kozlov. (2022). Use of Set Shaping theory in the development of locally testable codes. ArXiv, abs/2202.13152.
  * [4] C.E. Shannon, ”A Mathematical Theory of Communication”, Bell System Technical Journal, vol. 27, pp. 379– 423, 623-656, July, October, 1948.
  * [5] Oded Goldreich. Introduction to Property Testing. Cambridge University Press, 2017. p.8.
  * [6] Cheraghchi, Mahdi. “Locally Testable Codes.” (2005).
  * [7] Cover, Thomas M. (2006). Elements of Information Theory. John Wiley & Sons. ISBN 0-471-24195-4.
  * [8] Koch, A., Petit, A., Schmidt, C., Vdberg, A., & Lewis, L. (2023). Overcoming the encoding limit NH0(S) using Set Shaping Theory. ArXiv, abs/2310.12732.
  * [9] Huffman, D. (1952). A Method for the Construction of Minimum-Redundancy Codes. Proceedings of the I.R.E. 40 (9):1098 1101. doi: 10.1109/JRPROC.1952.273898


# Deformable Tip Mount for Soft Growing Eversion Robots

Cem Suulker, Sophie Skach, Danyaal Kaleel, Taqi Abrar, Zain Murtaza,
Dilara Suulker, and Kaspar Althoefer, Senior Member, IEEE  This work was
supported in part by the EPSRC National Centre for Nuclear Robotics project
(EP/R02572X/1), the Innovate UK GrowBot project (96566) and the Innovate UK
LianaBot project (89911).Authors are with the Centre for Advanced Robotics @
Queen Mary, School of Engineering and Materials Science, Queen Mary University
of London, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Here we present a flexible tip mount for eversion (vine) robots. This soft cap
allows attaching a payload to an eversion robot while allowing moving through
narrow openings, as well as the eversion of protruding objects, and expanded
surfaces.

##  I Introduction

Growing robots based on the eversion principle are known for being capable of
immensely extending along their longitudinal axis and reaching far into
inaccessible, remote spaces.

One major challenge has been to attach a meaningful payload, such as sensors,
at the tip which is never steady but rather evolving because of the way an
eversion robot is constantly moving its inside to the outside to progress
forward. Various types of mechanisms have been proposed by the robotics
community, caps that engulf the tip section of the eversion robot and are
often linked to the everting material of the eversion robot via complex sets
of rollers or magnets. As a disadvantage, these mechanisms do not work
reliably, especially when the eversion robot is integrated with functional
elements that protrude from the robot’s skin such as navigation pouches,
sensors or electronics. Another shortcoming of existing caps is that they are
made from rigid materials, preventing the robot from moving through narrow
openings and reducing its navigation capabilities.

We present here the soft cap for eversion robots. Our soft cap made from
fabric and with a diameter similar to that of the eversion robot is easily
slipped over the tip like a beanie. Our experimental study shows that the soft
cap sits on the eversion robot in a robust fashion and is capable of
transporting a payload (here: a camera) across long distances. We also show
that the robot’s capability to move through narrow openings or to bend its
body is virtually unhindered by the slipped-on cap.

Figure 1: Left, an inflated eversion robot. Middle, a soft cap made from
flexible fabric slipped over the tip of eversion robot. Right, a soft cap made
using non-flexible fabric and elastic bands, slipped over the tip of eversion
robot. A tool, here a camera, is attached to the caps.

##  II Soft Cap Design

Here, we take on the challenge to develop such soft cap and design around the
shortcomings of its rigid counterparts, taking advantage of the given
structural properties of an increasingly popular material for soft robotics:
textiles.

We exploit the friction force between the cap and the robot body as a way to
keep the cap in place without restricting robot movement. With the help of
elastic fabrics [1], the cap is slipped onto the tip of the everting robot,
like a beanie, able to slightly adapt its diameter through its stretch
character. This encapsulation of the body is still firm, yet flexible.

Figure 2: Pattern constructions a) using elastic finely knitted rib fabric
material, and b) using non-elastic woven fabric and elastic band.

There are various cutting and sewing patterns that could be explored, this
extended abstract focuses on the patterns shown in Fig. 2. In the first design
(Fig. 2 a) one-way elastic fabric is used to expand the cap diameter. Second
design (Fig. 2 b) the cap is made from non-stretch fabric but by using elastic
bands, the cap’s bottom section will squeeze onto the eversion robot body. An
elastic band is commonly used in clothing to create ruffles or elastic
waistbands. Being a standard method in the textile industry, it has rarely
been employed for the development of textile robotics [2]. Using this method
leads to a smaller effective contact surface. The thickness of the elastic
band gives the effective contact length (Red arrows in Fig. 2 b).

##  III Case Study & Results

In this section, we showcase the effectiveness of our soft caps by subjecting
them to challenging scenarios that would typically pose difficulties for rigid
caps. These challenges were designed to closely resemble realistic scenarios.

###  III-A Quality 1: Eversion of protruding solid objects

This scenario simulates the placement of sensors, electrical components, air
tubes, and rigid connectors on the body of the eversion robot. These rigid
components can cause jamming in caps with complex mechanisms, entirely
blocking the movement of the robot.

Figure 3: a) An eversion robot with a soft cap with a camera attached to it b)
The cap successfully allows eversion of 1.7 cm long protruding material.

To simulate this scenario, we roughly taped two 1 cm wide, 1.3 cm long and 1.6
cm wide, 1.7 cm long pipe connectors to the robot body in random places. In
the starting condition, the robot is inflated and protruding solid objects are
not everted yet (Fig. 3 a). The robot with a cap grows despite the protruding
objects, and the objects should be fully visible leaving the boundaries of the
cap (Fig. 3 b).

###  III-B Quality 2: Squeezability

Eversion robots are often marketed for their ability to easily navigate
through narrow openings [3]. However, when a rigid cap is attached to the tip,
this function is significantly compromised. Our soft caps ensure that this
property is maintained while also allowing the robot to carry a payload at its
tip.

Figure 4: An eversion robot with a soft cap and camera attached to it,
squeezing through a narrow opening. Diameter of the robot and width of the
opening are indicated.

In this particular scenario, we constructed a gate with a width of 9 cm for a
10 cm diameter eversion robot. Using our soft cap, the robot was able to
successfully pass through the narrow opening and continue its intended path
while successfully delivering the payload (Fig. 4).

###  III-C Quality 3: Navigability

When a navigation actuation takes place, materials often expand, which limits
the mobility of a rigid cap within a confined space. This problem can be
avoided by using a soft cap, which can withstand great expansion without
compromising tip mobility.

Figure 5: An eversion robot with a soft cap reorienting its tip.

In this scenario, we therefore activate our eversion robot with cap in situ,
the challenge being to retain the cap at the tip while allowing requisite
motion. At starting position the cap (Fig. 5 a) is attached to the eversion
robot which is the halfway through eversion of the navigation layers (pouches
[4]), and then first the navigation layers then the main chamber of the robot
are being inflated. The tip of the robot embraces the orientation change (Fig.
5 b) due to the navigation mechanism and keeps growing until it passes the
navigation layers.

##  IV Conclusions

This extended abstract presents our caps for eversion robots made from elastic
fabric parts. Exploiting textile properties, the cap adjusts to the everting
robot and sits firmly, yet flexible at the tip of the robot’s body throughout
the eversion process. This design preserves squeezability and navigability of
the robot, and the eversion of protruding elements is shown to be achievable.
More available at [5].

## References

  * [1] C. Suulker, A. Hassan, S. Skach, and K. Althoefer, “A comparison of silicone and fabric inflatable actuators for soft hand exoskeletons,” in _2022 IEEE 5th International Conference on Soft Robotics (RoboSoft)_. IEEE, 2022, pp. 735–740.
  * [2] C. Suulker, S. Skach, and K. Althoefer, “Soft robotic fabric actuator with elastic bands for high force & bending performance in hand exoskeletons,” _IEEE Robotics and Automation Letters_ , 2022.
  * [3] E. W. Hawkes, L. H. Blumenschein, J. D. Greer, and A. M. Okamura, “A soft robot that navigates its environment through growth,” _Science Robotics_ , vol. 2, no. 8, p. eaan3028, 2017.
  * [4] T. Abrar, F. Putzu, A. Ataka, H. Godaba, and K. Althoefer, “Highly manoeuvrable eversion robot based on fusion of function with structure,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2021, pp. 12 089–12 096.
  * [5] C. Suulker, S. Skach, D. Kaleel, T. Abrar, Z. Murtaza, D. Suulker, and K. Althoefer, “Soft cap for vine robots,” in _2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2023, pp. 6462–6468.