# Engineering AI Tools for Systematic and Scalable Quality Assessment in
Magnetic Resonance Imaging

Yukai Zou
University of Southampton
University Hospital Southampton NHS Foundation Trust
<EMAIL_ADDRESS>
Ikbeom Jang11footnotemark: 1
Massachusetts General Hospital
Harvard Medical School
<EMAIL_ADDRESS>
Equal Contribution

###### Abstract

A desire to achieve large medical imaging datasets keeps increasing as machine
learning algorithms, parallel computing, and hardware technology evolve.
Accordingly, there is a growing demand in pooling data from multiple clinical
and academic institutes to enable large-scale clinical or translational
research studies. Magnetic resonance imaging (MRI) is a frequently used, non-
invasive imaging modality. However, constructing a big MRI data repository has
multiple challenges related to privacy, data size, DICOM format, logistics,
and non-standardized images. Not only building the data repository is
difficult, but using data pooled from the repository is also challenging, due
to heterogeneity in image acquisition, reconstruction, and processing
pipelines across MRI vendors and imaging sites. This position paper describes
challenges in constructing a large MRI data repository and using data
downloaded from such data repositories in various aspects. To help address the
challenges, the paper proposes introducing a quality assessment pipeline, with
considerations and general design principles.

##  1 Background

A desire to achieve a large medical imaging dataset keeps increasing as
machine learning algorithms, parallel computing, and hardware evolve.
Accordingly, there is a growing desire for larger datasets by pooling
harmonized data from multiple institutions. One of the most frequently used
imaging modalities is Magnetic Resonance Imaging (MRI). MRI can non-invasively
monitor the physiological processes of the human body under health and disease
conditions. There is no ionizing radiation compared to X-ray Computed
Tomography (CT), allowing subjects to be monitored at multiple time points
without the concern of tissue damage due to radiation exposure.

In T1- and T2-weighted imaging, MR signals are stored in relatively arbitrary
values, and this provides anatomical contrasts that radiologist is
traditionally trained to read and make clinical decisions. With the
advancement of hardware and infrastructure for data collection, storage, and
computing, the development of quantitative MRI techniques becomes fast-
evolving and adds new dimensions to probing normal biological processes and
pathophysiology. In quantitative MRI, parameters are extracted from the MRI
signals through computational modeling, which can serve as an imaging-based
biomarker that is objectively measured and evaluated to correspond to a
specific physical or physiological property [5]. Therefore, quantitative MRI
opens up a new way of thinking about MRI signals, imaging processes,
optimization tools, and scientific computing, spanning from physiology to
tissue microstructure and morphology, and the research activities are highly
interdisciplinary [20].

However, the interpretation of imaging-based biomarkers still faces barriers.
Clinical translation of quantitative MRI has been slow, with cautions about
directly using the pooled data. The barriers of poor reproducibility and low
efficiency need to be overcome in order to translate quantitative MRI into
clinical practice. As such, AI tools hold the potential to fulfill the unmet
need so that high-quality MRI data can be guaranteed to be integrated into
routine clinical care.

##  2 Challenges

###  2.1 Challenges in constructing multi-site MRI data repositories

Constructing a large MRI data repository has multiple challenges. MRI data are
stored as “Digital Imaging and Communications in Medicine (DICOM)", a
standardized format for the storage, exchange, and transmission of medical
imaging data. A DICOM consists of a header and the actual image itself. The
header consists of metadata that describes the image, acquisition parameters,
study and participant information, institution, etc. Once raw MRI data are
acquired, the data are reconstructed into images that can be clinically
interpreted. To note, the raw data are typically not stored as DICOM, given
that they are often huge in file size.

DICOM files are typically large in size, which can be an issue because both
the data storage and transfer come at a price. For example, it takes about
1–1.5 GB of storage to host the data for one routine exam of a clinical brain
study. Often, each patient has multiple exams, i.e., data at multiple time
points. About 6–13.5 TB are required when building a data repository for a
longitudinal study that comprises 3 timepoints, about 1,000 patients, with 1–2
backups. This is even without adding 4-D MRI data, such as functional or
diffusion MRI. One potential solution can be to utilize image compression
technology. Recent technologies using generative adversarial networks make it
possible to compress images by a factor of up to 100 without significantly
compromising image quality [12].

DICOMs must be de-identified before uploads. Because DICOM files contain
metadata that include protected health information (PHI) and personally
identifiable information (PII), it is considered a violation to Health
Insurance Portability and Accountability Act (HIPAA) to access, disclose, or
share them with purposes other than treatment, healthcare operation, or
payment unless an authorization is obtained from the patient in advance. In
principle, PHI in a DICOM header may be de-identified in two ways [1]. The
anonymization removes information carried by header elements or replaces it
with random values so that the original identity cannot be restored at all.
The pseudonymization is a more frequently used method that replaces the real
identifiers within a data record using an artificial identifier and a list of
mappings, that could be used by authorized personnel to track down the
original identity of the patient [13, 14, 15]. Not only DICOM header entries
but the image itself is advised to be anonymized because facial features can
be rendered from the image. Defacing algorithms (e.g., [3]) have been
developed that use models of non-brain structures for removing facial features
that may potentially allow the identification of a participant from their MR
scan.

To build a big MRI data repository, hospitals and other institutes need to
collaborate and host centralized databases. This overhead can quickly become a
logistical challenge and usually requires time-consuming approval processes
due to data privacy and ethical concerns associated with data sharing in
healthcare [9]. Even when these challenges can be addressed, institutions
owning the valuable dataset may prefer not to make it publicly available [19].
One alternative to the data sharing hurdles is federated learning [11], where
only model weights are shared between participating institutions without
sharing the actual data.

Data retrieval from clinical systems also poses some challenges. Retrieving
large amounts of data may cause issues that lead to a breakdown of the Picture
Archiving and Communication System (PACS), which can be detrimental for
clinicians and technicians. Cloud services such as Amazon Web Services and
Microsoft Azure can provide solutions that ensure secure and scalable storage,
as well as fast and secure data transmission. However, when switching to a
cloud-based solution, the risk of data retrieval must be well managed. For
security purposes, the types of data that cannot be hosted on a public
repository must be identified beforehand; in addition, a strategy such as
prefetching [7] allows marking those data pulled from remote repositories to
be deleted after the analysis is done.

###  2.2 Challenges in using data pooled from MRI data repositories

Using data downloaded from such data repositories can also be challenging in
several aspects. First, directly using the pooled data is not advised, due to
heterogeneity of the MRI data in acquisition parameters, magnetic field
strength, receiver coil, reconstruction parameters, post-processing
parameters, and operators who acquired the data. The images acquired are not
standardized across the major vendors of clinical MRI machines, including GE,
Philips, Siemens, and Canon, which makes data harmonization particularly
challenging.

Image quality is another factor to consider. Adding data with poor quality
often harms the model, hence it is better to filter them out. For this reason,
many hospitals and institutes have a quality assessment pipeline that filters
out corrupted data or data with too much noise or artifacts.

Although DICOM has already been standardized to a large extent to enable
medical imaging data more accessible, there are numerous private DICOM tags
that are defined by different vendors. Those private tags often contain
important parameters, e.g., coil element, acquisition, and storage of the
data, etc. Unfortunately, the naming of those tags is often not descriptive,
causing difficulty in parsing and filtering the data. Several open-source
software provides functionality to handle these private tags – e.g.,
dicom_parser for Siemens tags [2].

Series selection from MRI exams can be extremely challenging. The most
informative DICOM tag related to series selection is “Series Description",
however, there is no standard form to describe a series and mixes information
of different nature such as workflow, acquisition, anatomy, and post-
processing, e.g., Ax T1 Pre whole brain, Sag MPRAGE RFMT. Also, a wide variety
of non-standard annotation formats are used and they are not inter-operable.
See Fig. 1 for a few more examples.

Another major difficulty in using a large MRI dataset is that it takes too
much memory, especially when housing large quantities of 3D data. Training a
model (e.g., deep learning model) with such large data can be troublesome
because one may not be able to fit the data into the limited graphic card
memory and would have to write a project-specific data loader.

Figure 1: Highly variable MRI ‘Series Description’s. Above are a few examples
from a clinical GE scanner.

##  3 Quality Assessment (QA): Considerations and General Design Principles

Introducing a QA pipeline provides a way to resolve at least some of the
challenges described above by ensuring important elements for a certain study.
Such elements not only include image quality but also can include acquisition
parameters, processing pipeline, sequence, and anatomy.

The ways “quality" is defined and assessed have been well established within
the world of business. Quality refers to the perception of the degree to which
a set of characteristics of the product or service meets the customer’s
expectations. QA, according to Mainz et al. [10], is “the data collection and
analysis through which the degree of conformity to predetermined standards and
criteria are exemplified." When the product or service has been found
unsatisfactory through QA, actions shall be made to identify the cause and to
make improvements [10].

Figure 2: Considerations and general design principles to build a quality
assessment pipeline.

QA for MRI needs to be both systematic and scalable (Fig. 2). The complexities
of MRI processing require a procedure that is capable of systematically
tracking and reporting the processing and outcomes. It needs to take into
account what steps are involved in the image processing, the input and output
of each step, the purpose of the QA, and whether the procedure spans multiple
institutes. The QA procedure should also be scalable, meaning that performing
hundreds and thousands of QA should be as easy and efficient as performing
only one QA. Addressing these factors will significantly make it faster for
the research community to use the datasets without concerning the quality.

General design principles of establishing a quality management system apply
for engineering AI tools for systematic and scalable QA for MRI. According to
the International Organization for Standardization (ISO9001:2015) [8], there
are four principles (Fig. 2):

  1. 1.

Goal-oriented. It is important to keep in mind who the customer is, and
depending on the needs of the customer, the set of characteristics for
evaluating image quality can be different, hence the goals would vary. For
radiologists who read images and identify regions with pathology, QA should
focus on the image quality such as contrast and artifacts. For researchers who
carry out big data analysis, the goal would be to provide a dataset that have
been properly filtered and harmonized.

  2. 2.

Monitor process. To develop reliable AI tools in the medical field, the
following items are recommended to be monitored: reliability of the reference
standards, whether the training dataset matches the intended use, independence
of hyperparameter tuning, and independence of model evaluation [6]. Because
the QA pipeline for MRI can comprise numerous steps, key elements within each
step need to be listed, with standardized operation procedures.

  3. 3.

Manage risk. According to Ropeik and Gray [18], there are four elements to
consider: hazard, exposure, consequence, and probability. What is the
probability that exposure to hazard will cause a negative consequence? This
question can be asked throughout the QA pipeline, from acquisition to
reconstruction and to processing, to ensure MRI data are reliable for accurate
patient diagnoses and real-world applications [4].

  4. 4.

Ensure outcomes. Addressing the issue of reproducibility [17] is the key to
ensuring reliable outcomes. As researchers that strive for best practices in
neuroscience, we ignore reproducibility at our own peril [16]. AI tools can
effectively eliminate the primary source of variability, which is fatigue from
manual labeling. For quantitative MRI, ensuring reproducible objective
measures require such AI tools can unfold the complexity in data acquisition
and post-processing.

## References

  * [1] KYE Aryanto, M Oudkerk and PMA Ooijen “Free DICOM de-identification tools in clinical research: functioning and safety of patient privacy” In _European radiology_ 25.12 Springer, 2015, pp. 3685–3695
  * [2] Zvi Baratz, Gal Ben-Zvi and Lily Cheng “dicom_parser” In _GitHub repository_ GitHub, https://github.com/open-dicom/dicom_parser, 2021
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  * [5] Biomarkers Definitions Working Group “Biomarkers and surrogate endpoints: preferred definitions and conceptual framework.” In _Clinical pharmacology and therapeutics_ 69.3, 2001, pp. 89–95 DOI: 10.1067/mcp.2001.113989
  * [6] Akifumi Hagiwara, Shohei Fujita, Yoshiharu Ohno and Shigeki Aoki “Variability and Standardization of Quantitative Imaging: Monoparametric to Multiparametric Quantification, Radiomics, and Artificial Intelligence.” In _Investigative radiology_ 55.9, 2020, pp. 601–616 DOI: 10.1097/RLI.0000000000000666
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  * [8] International Organization for Standardization “ISO 9001:2015 Quality management systems - Requirements” Genève, Switzerland: International Organization for Standardization
  * [9] David B Larson et al. “Ethics of using and sharing clinical imaging data for artificial intelligence: a proposed framework” In _Radiology_ 295.3 Radiological Society of North America, 2020, pp. 675–682
  * [10] Jan N. Mainz, E M Hammershøy, A M Worning and S Juul “Quality assessment, quality assurance and quality improvement in health care. Concepts and terminology.” In _Ugeskrift for laeger_ 154.17, 1992, pp. 1151–1156
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  * [14] Thomas Neubauer and Bernhard Riedl “Improving patients privacy with Pseudonymization” In _Studies in health technology and informatics_ 136 IOS Press; 1999, 2008, pp. 691
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  * [16] Russell A. Poldrack “The Costs of Reproducibility” In _Neuron_ 101.1, 2019, pp. 11–14 DOI: https://doi.org/10.1016/j.neuron.2018.11.030
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††institutetext: Center for Joint Quantum Studies and Department of Physics,
School of Science, Tianjin University, Yaguan Road 135, Jinnan District,
300350 Tianjin, P. R. China

# Inverse problem of correlation functions in holography and bulk
reconstruction

Run-Qiu Yang<EMAIL_ADDRESS>
###### Abstract

This paper shows that the exterior of a planar/spherically/hyperbolically
symmetric asymptotically anti-de Sitter static black brane/hole can be
reconstructed from its two arbitrary boundary frequency 2-point correlation
functions by solving a Gel’fand-Levitan-Marchenko integral equation. A method
about how to solve this integral equation is also presented by using BTZ black
hole as an example. Since the frequency correlation function is easily handled
in experiments and theories, this paper not only proposes an explicit method
to “directly measure” the corresponding holographic spacetime for a material
that has holographic dual but also provides a powerful approach to
experimentally check if a system has holographic dual.

##  1 Introduction

The mathematical expression of a physical law is usually a set of equations
determined by the parameters and its solutions stand for the results which can
be compared with observations. Deriving the “computed result” for a given set
of parameters is called “solving the direct problem”. Conversely, obtaining
the set of the parameters from a given set of results is called “solving the
inverse problem”. The “solving inverse problem” plays essential roles both in
theoretical studies and in practice of applied holography Maldacena:1997re ;
Gubser:1998bc ; Witten:1998qj ; Witten:1998zw ; Zaanen2015 ; Baggioli:2019rrs
. In theoretical viewpoint, bulk reconstruction is the central question in
understanding the emerging spacetime in holography and answer how the
gravitational degrees of freedom emerge from field theory. In the viewpoint of
application, it will help us to build holographic spacetime and theory more
efficiently by directly from the observable data when we try to use
holographic duality to deal with strongly coupling materials.

A few advances have been pushed forward towards this issue. When conformal
dimension $\Delta$ is much larger than spacetime dimensional $d$, boundary
2-point correlation functions can be expressed in terms of geodesics
Balasubramanian:1999zv  and so have been used to reconstruct the bulk geometry
Hubeny:2012ry . However, the requirement $\Delta\gg d$ restricts its
application in practice since materials in general will not satisfy this
requirement. There is also a method called “bulk-cone singularities”, which
uses the relationship between singular correlators and bulk null geodesics to
recover some special spherically symmetric static bulk geometries
Bilson:2008ab ; Hammersley:2006cp . In most situations, the materials we are
concerned about have planar structures. For those boundary systems, this
method cannot be applied since the bulk null geodesics of planar symmetric
static anti-de Sitter (AdS) black brane will no longer return to boundary
again if they start from the boundary. What’s more, locating all singularities
of correlator in boundary spacetime coordinates is also a huge challenge for
measurements. Another well-known method of reconstruction is called
“entanglement wedge reconstruction”, which uses the boundary entanglement
spectrum to reconstruct bulk geometry and local observables Dong:2016eik ;
Espindola:2018ozt ; Penington:2019npb . Such a reconstruction is given via
proving its “existence” rather than explicitly giving the method of
reconstruction. There are also some other approaches based on holographic
entanglement entropy Ryu:2006bv ; Nishioka:2009un , Wilson loop
Maldacena:1998im ; Rey:1998ik , tensor network Swingle:2009bg ;
Pastawski:2015qua  or complexity Susskind:2014rva ; Susskind:2014jwa ;
Brown:2015bva , e.g. see Refs. Hashimoto:2020mrx ; Hashimoto:2021umd ;
Xu:2023eof ; Bilson:2010ff ; Caputa:2017urj ; Milsted:2018san . In these
approaches, some boundary quantities are conjectured to dual to some bulk
geometric quantities and so give the chance to recover the bulk metric. It
needs to be noted that except for incredibly special systems, it is still a
challenge to experimentally measure the entanglement entropy or Wilson loop of
general materials. The complexity currently is a “theoretical quantity” and
the way to measure it is unclear. Taking all these into account, one may ask:
is there any universal method to reconstruct bulk geometry in an explicit
manner by using directly experimentally measurable quantities of materials?

This paper will give a positive answer. It will propose a new method about
bulk reconstruction for a static asymptotically AdS black brane/hole by
solving the inverse problem of 2-point correlation functions. This supplies a
possibility to “directly measure” the corresponding holographic spacetime. The
purpose of following parts is to present a proof of following theorem,

###### Theorem 1

For a stable planar symmetric static asymptotically AdS black brane, the bulk
geometry can be recovered up to a scaling transformation from the boundary
frequency 2-point correlation functions of arbitrary two scalar operators.

In addition, we will also give explicit steps about how to reconstruct the
bulk metric by using BTZ black hole as a concrete example. The generalizations
to spherically and hyperbolically symmetric cases are straightforward.

##  2 Overview on correlation function in holography

Before we present our proof on above theorem and the method of reconstructing
bulk metric, let us first briefly review the “solving direct problem” of
holographic 2-point correlation functions. We focus on the planar symmetric
asymptotically AdS black brane, of which the metric will have following form

$\textmd{d}s^{2}=\frac{1}{z(\rho)^{2}}[h(\rho)(-\textmd{d}t^{2}+\textmd{d}\rho^{2})+\textmd{d}\vec{x}^{2}_{d-1}],~{}~{}\rho\in\mathbb{R}^{+}\,.$
(1)

The functions $h(\rho)$ and $z(\rho)$ will satisfy following boundary
conditions

$z(0)=0,~{}~{}z^{\prime}(0)=h(0)=1\,$ (2)

and

$z(\infty)=z_{h},~{}~{}h(\infty)=0\,.$ (3)

The horizon then locates at $\rho\rightarrow\infty$ and the AdS boundary
locates at $\rho=0$.

We consider a real scalar field $\Psi$ which is described by Klein-Gorden
equation $\nabla^{2}\Psi+m^{2}\Psi=0$ in the bulk. Here $m^{2}$ is the mass-
square of the scalar field and is relative to the conformal dimension $\Delta$
of boundary scalar operator $\mathcal{O}$ such that
$\Delta=d/2+\sqrt{d^{2}/4+m^{2}}$. In general, the asymptotical expansion of
the scalar field near the AdS boundary has two independent branches,

$\Psi=\Phi^{(+)}_{\Delta}\rho^{\Delta}(1+\cdots)+\Phi^{(-)}_{\Delta}\rho^{d-\Delta}(1+\cdots)\,.$
(4)

Without loss of generality, we assume $d-\Delta\leq\Delta$ such that we have
$\Delta\geq d/2$. Here we choose that so called “standard quantization”, which
treats $\Phi^{(-)}_{\Delta}$ as the source term of boundary theory and
$\Phi^{(+)}_{\Delta}\sim\langle\mathcal{O}\rangle$ as the value of expectation
for boundary scalar operator. In some cases, we may have an alternative
quantization by choosing $\Phi^{(+)}_{\Delta}$ as the source, for which the
conformal dimension of dual operator is $d-\Delta$. In the following
discussion, we shall focus on the “standard quantization”. The Weyl anomaly
coming from gravity is zero Henningson:1998ey ; deHaro:2000vlm  due the
flatness of boundary. The Weyl anomaly of the matter sector will appear for
some fine-tuned values of mass-square, so we will do not consider it here.

To compute the frequency 2-point correlation function of dual boundary scalar
operator, we consider the bulk scalar field of a form
$\Psi=z^{(d-1)/2}\phi(\rho)\textmd{e}^{-i\omega t}$ on the fixed background
(1). The Klein-Gordon equation then reads

$-\frac{\textmd{d}^{2}\phi}{\textmd{d}\rho^{2}}+\left(V_{\Delta}+\frac{4m^{2}+d^{2}-1}{4\rho^{2}}\right)\phi=\omega^{2}\phi\,.$
(5)

Here

$\begin{split}V_{\Delta}&=-\frac{(d-1)z^{\prime\prime}}{2z}+\frac{(d^{2}-1)}{4}\left(\frac{z^{\prime
2}}{z^{2}}-\frac{1}{\rho^{2}}\right)+m^{2}\left(\frac{h}{z^{2}}-\frac{1}{\rho^{2}}\right)\,.\end{split}$
(6)

Eq. (5) has four kinds of solutions
$\\{\psi_{\omega}(\rho),\psi_{\omega}(\rho)^{*},\varphi^{(+)}_{\omega}(\rho),\varphi^{(-)}_{\omega}(\rho)\\}$.
The $\\{\psi_{\omega}(\rho),\psi_{\omega}(\rho)^{*}\\}$ are defined by
following boundary conditions

$\lim_{\rho\rightarrow\infty}\textmd{e}^{-i\omega\rho}\psi_{\omega}(\rho)=\lim_{\rho\rightarrow\infty}\textmd{e}^{i\omega\rho}\psi_{\omega}(\rho)^{*}=1\,.$
(7)

The solutions $\\{\varphi^{(\pm)}\\}$ are defined by following boundary
conditions when $\rho\rightarrow 0^{+}$

$\varphi^{(\pm)}_{\omega}(\rho)=\rho^{1/2\pm
v}[1+\mathcal{O}(\omega^{2}\rho^{2})]\,.$ (8)

Here $v=\sqrt{d^{2}/4+m^{2}}=\Delta-d/2$. Due to the fact that Eq. (5) is 2nd
order linear differential equation with real-valued coefficients, above 4
types of solutions are not all linear independent and any one of them can be
expressed in terms of linear combination of any other two. In the case that we
consider the retarded Green function of holography, we impose the ingoing wave
boundary condition $\Phi\propto\textmd{e}^{-i\omega(t-\rho)}$ near the
horizon. This leads to $\phi(\rho)\propto\psi_{\omega}(\rho)$. We then have

$\psi_{\omega}(\rho)=\Phi^{(-)}_{\Delta}(\omega)\varphi_{\omega}^{(-)}(\rho)+\Phi^{(+)}_{\Delta}(\omega)\varphi_{\omega}^{(+)}(\rho)\,.$
(9)

The freqeuncy 2-point correlation function $\mathcal{G}(\omega)$ is defined as
111Here we omit an irrelevant factor that is determined by scaling dimension.

$\mathcal{G}_{\Delta}(\omega)=\Phi^{(+)}_{\Delta}(\omega)/\Phi^{(-)}_{\Delta}(\omega)\,.$
(10)

In the “solving direct problem” of correlation functions, what we have known
is the metric functions $\\{z(\rho),h(\rho)\\}$ and the purpose is to find the
correlation function $\mathcal{G}_{\Delta}(\omega)$ for a give conformal
dimension $\Delta$.

For the purpose of proving Theorem 1 and following notations in Refs.
Chadan1989 ; Kravchenko2020 ; Kravchenko2021 , let us introduce a function
$F_{\Delta}(\omega)$ such that

$F_{\Delta}(\omega)=\lim_{\rho\rightarrow
0^{+}}\frac{\sqrt{2\pi}(-\omega\rho)^{v-1/2}}{2^{v}\Gamma(v)}\textmd{e}^{i\pi(v-1/2)/2}\psi_{\omega}(\rho)\,.$
(11)

This function plays an important role in discussions of Refs. Chadan1989 ;
Kravchenko2020 ; Kravchenko2021 , which is not determined by correlation
function. However, this paper proves following lemma to relate it to
correlation function,

###### Lemma 1

The modulus of $F_{\Delta}(\omega)$ is completely determined by the
correlation function

$|F_{\Delta}(\omega)|^{2}=\frac{\pi|\omega/2|^{2v}v}{\Gamma(v+1)^{2}\Im[\mathcal{G}_{\Delta}(\omega)]}\,.$
(12)

See appendix A for a proof.

##  3  “Solving the inverse problem” and bulk reconstruction

The “solving the inverse problem” of correlation functions then is to recover
the metric functions $\\{z(\rho),h(\rho)\\}$ from the correlation
$\mathcal{G}_{\Delta}(\omega)$. Though our reconstruction will be performed in
real-valued frequency, let us make a short discussion on mathematical property
of analytical continuation in complex frequency.

The function $\psi_{\omega}(\rho)$ is not square-integrable for a general
$\omega\in\mathbb{C}$. However, there may be some special complex frequencies
$\omega_{k}$ such that $\Phi^{(-)}(\omega_{k})=0$. Such modes are called
“quasi-normal modes (QNMs)”. The QNMs are related to the stability of the
holographic system. Particularly, if there is a QNM $\omega_{k}$ such that
$\Im(\omega_{k})>0$, then the system will be unstable under perturbations
since the scalar field
$\Psi=z^{(d-1)/2}\phi(\rho)\textmd{e}^{-i\omega_{k}t}\propto\textmd{e}^{\Im(\omega_{k})t}$
will grow exponentially. Instead, the system will be stable if no QNM has
positive imaginary part. For those unstable QNMs we have following lemma

###### Lemma 2

If $\omega_{k}\in\mathbb{C}$ is an unstable QNM, then $\psi_{\omega_{k}}$ is
square-integrable in the interval $(0,\infty)$. If $\psi_{\omega_{k}}$ is
square-integrable in the interval $(0,\infty)$, then $\omega_{k}$ is a QNM and
$\omega_{k}^{2}<0$.

A proof will be found in appendix A. From this lemma we see that the system is
stable if and only if there is no square-integrable solution for Eq. (5). In
practice, the correlation function is measured according to linear response of
the system to external perturbations. Then the system should be stable against
the small perturbations otherwise the correlation function can not be obtained
experimentally. Thus, it is natural to assume that Eq. (5) has no square-
integrable solution. Under such an assumption, we will have following
mathematical results to reconstruct effective potential (see the chapters 1-3
of Ref. Chadan1989  for more details and explanations on its mathematical
aspects)

$V_{\Delta}(\rho)=2\frac{\textmd{d}}{\textmd{d}\rho}K_{\Delta}(\rho,\rho)\,.$
(13)

The transmutation kernel $K_{\Delta}(\rho,\sigma)$ is determined by the
Gel’fand-Levitan-Marchenko integral equation

$K_{\Delta}(\rho,\sigma)+\Omega_{\Delta}(\rho,\sigma)+\int_{0}^{\rho}K_{\Delta}(\rho,y)\Omega_{\Delta}(y,\sigma)\textmd{d}y=0\,.$
(14)

The function $\Omega_{\Delta}(\rho,\sigma)$ then is given by function
$F_{\Delta}(\omega)$ according

$\Omega_{\Delta}(\rho,\sigma)=\int_{0}^{\infty}\hat{J}_{v}(\omega\rho)\hat{J}_{v}(\omega\sigma)\left(\frac{1}{|F_{\Delta}(\omega)|^{2}}-1\right)\textmd{d}\omega\,.$
(15)

Here $\hat{J}_{v}(x)=\sqrt{x}J_{v}(x)$ and $J_{v}(x)$ is the $v$-th order
Bessel function of the first kind. When the system is not stable, i.e.
correlation function has poles at upper complex plane, we still have similar
results but the function $\Omega_{\Delta}(\rho,\sigma)$ will be more
complicated, see Refs. Chadan1989 ; Kravchenko2020 ; Kravchenko2021 . This
method of reconstruction originally was not related to correlation function
directly. However, thanks to the Lemma 1, the effective potential can be
recovered from the correlation function.

Take $\mathcal{G}_{\Delta_{1}}(\omega)$ and $\mathcal{G}_{\Delta_{2}}(\omega)$
to be correlation functions of arbitrary two different real scalar operators.
After we reconstruct their corresponding effective potentials $V_{\Delta_{1}}$
and $V_{\Delta_{2}}$, the function $z(\rho)$ will be determined by

$\begin{split}\frac{m_{2}^{2}V_{\Delta_{1}}-m_{1}^{2}V_{\Delta_{2}}}{m_{2}^{2}-m_{1}^{2}}&=\frac{(d^{2}-1)}{4}\left(\frac{z^{\prime
2}}{z^{2}}-\frac{1}{\rho^{2}}\right)-\frac{(d-1)z^{\prime\prime}}{2z}\,.\end{split}$
(16)

The function $h(\rho)$ then is given by

$h=\frac{V_{\Delta_{1}}-V_{\Delta_{2}}}{m_{1}^{2}-m_{2}^{2}}z^{2}+\frac{z^{2}}{\rho^{2}}\,.$
(17)

Naively, one may think that we can solve Eq. (16) by using initial condition
(2), i.e. $\\{z(0)=0,z^{\prime}(0)=1\\}$. However, such a Cauchy problem is
ill-defined due to that the Eq. (16) is singular at $z=0$. From the physical
viewpoint, this is because that the solution of above system is not unique but
can be different up to a scaling transformation. Instead of solving Eq. (16)
as a Cauchy problem, we solve this problem as a boundary-value-problem by
requiring

$\lim_{\rho\rightarrow 0^{+}}z/\rho=1,~{}~{}\text{and}~{}z(\infty)=z_{h}\,.$
(18)

The first boundary condition insures metric is asymptotically AdS when
$\rho\rightarrow 0^{+}$. Without losing generality, we can set “inverse
horizon radius” $z_{h}=1$ by using the scaling transformation. This setup
fixes the scaling symmetry. The boundary-value-problem now is well defined and
so we finished the proof of Theorem 1.

##  4 Example of BTZ black hole

Let us consider the BTZ black hole as an example and exhibit how to recover
the metric from its 2-point correlation functions. By setting the inverse
horizon $z_{h}=1$, the BTZ black hole is given by $z(\rho)=\tanh\rho$ and
$h(\rho)=1/\cosh^{2}\rho$. The boundary 2-point correlation function is give
by

$\mathcal{G}_{\Delta}(\omega)=\frac{\Gamma(-v)\Gamma(\frac{1-i\omega+v}{2})^{2}}{\Gamma(v)\Gamma(\frac{1-i\omega-v}{2})^{2}}\,.$
(19)

The correlation function has pole when $\omega_{k}=-i(1+v+2k)$ with
$k\in\mathbb{N}^{+}$. Thus there is no QNMs at upper complex plane and so no
unstable QNM.

In following we will applied the “Fourier-Jacobi series representation” to
solve Eq. (14). Since it may not be familiar to the potential readers of this
paper, we here first briefly explain this method. For more mathematical
aspects, one can refer to Refs. Kravchenko2020 ; Kravchenko2021  and
references therein.

Step 1) We write the kernel into following series expansion

$K_{\Delta}(\rho,\sigma)=\sum_{n=0}^{\infty}\frac{\beta_{n}(\rho)}{\rho^{v+3/2}}\sigma^{v+1/2}P_{n}^{(v,0)}(1-2\sigma^{2}/\rho^{2})\,,$
(20)

where $P_{n}^{(\alpha,\beta)}$ stands for the Jacobi polynomial. Introduce a
matrix $A_{ln}$ and an vector $B_{n}$ such that

$A_{ln}(\rho)=\int_{0}^{\infty}\frac{J_{v+1+2l}(\omega\rho)J_{v+1+2n}(\omega\rho)}{\omega\rho}\left(\frac{1}{|F_{\Delta}(\omega)|^{2}}-1\right)\textmd{d}\omega$
(21)

and

$B_{n}(\rho)=-\int_{0}^{\infty}J_{v}(\omega\rho)J_{v+1+2n}(\omega\rho)\left(\frac{1}{|F_{\Delta}(\omega)|^{2}}-1\right)\textmd{d}\omega\,.$
(22)

Then Eq. (14) becomes a discrete form

$\frac{\beta_{l}(\rho)}{2(2l+v+1)\rho}+\sum_{n=0}^{M}A_{ln}(\rho)\beta_{n}(\rho)=B_{l}(\rho)\,.$
(23)

Here $l=0,1,2\cdots,M$ with $M\rightarrow\infty$. To perform numerical
computation, we can truncate system into a finite system by setting a finite
$M$.
Step 2) Discrete the coordinate $\rho$ to be $\rho_{k}=k\delta\rho$. For every
$\rho_{k}$, compute $B_{m}(\rho_{k})$ and $A_{mn}(\rho_{k})$ according to the
formulas (21) and (22). One then solves the truncated system (23) to find
$\beta_{0}(\rho_{k})$ for every $\rho_{k}$.
Step 3) Use the finite differential methods to obtain the
$\beta^{\prime}_{0}(\rho_{k})$ and $\beta^{\prime\prime}_{0}(\rho_{k})$ based
on the values of $\beta_{0}(\rho_{k})$. The effective potential at the $k$-th
grid can be recovered according to

$V_{\Delta}(\rho_{k})=\frac{\rho\beta_{0}^{\prime\prime}(\rho_{k})+(2v+1)\beta^{\prime}_{0}(\rho_{k})}{\rho_{k}[\beta_{0}(\rho_{k})+2(v+1)]}\,.$
(24)

Step 4) Repeat the steps (1)-(3) to obtain effective potentials
$\\{V_{\Delta_{1}},V_{\Delta_{2}}\\}$ for two different scaling dimension and
then solve the differential equation (16) to obtain function $z(\rho_{k})$.
Step 5) Take the solution $z(\rho_{k})$ into Eq. (17) to obtain the function
$h(\rho_{k})$.
Though here we discuss the BTZ black hole, above 5 steps are valid for general
planar symmetric static asymptotically AdS spacetime.

In Fig. 1 we show the reconstructed metric components and their exact values.
Here we choose the scaling dimensions of dual boundary theory to be
$\Delta=5/3$ and $\Delta=3/2$ and only use 5 truncated equations (i.e. setting
$M=4$). We see that the reconstructed method gives excellent results.

Figure 1: Comparison between the reconstructed metric components and their
exact values by using 5 equations in the truncated system. Here it uses
correlation functions of $\Delta=5/3$ and $\Delta=3/2$ to recover the metric.

##  5 Summary

To summary, this paper considers the bulk reconstruction of metric from the
viewpoint the “solving inverse problem” of correlation functions. It shows
that a planar symmetric static asymptotically anti-de Sitter black brane can
be reconstructed from 2-point correlation functions of arbitrary two different
real scalar operators. By using BTZ black hole as a concrete example, we give
a step-by-step method to recover the metric. Though the Theorem 1 and above
discussion focused on real scalar field and planar symmetric case, the
generalizations to other kinds of operators and spherical/hyperbolic symmetric
case are straightforward. Thus, the planar/spherical/hyperbolic symmetric
static bulk geometry can be reconstructed from arbitrary two different
frequency 2-point correlation functions of boundary.

It has been shown that the high point correlation functions can be computed
holographically according to Witten diagram Witten:1998qj ; Liu:1998ty ;
Freedman:1998tz ; Bzowski:2022rlz . If its holographic dual in the bulk
couples only with gravity, its Witten diagram is completely determined by bulk
metric. In this case, the result in this paper implies that its all
correlation functions are determined from the 2-point correlation functions of
arbitrary two different operators. In addition, the entanglement entropy of
large-$c$ limit in holography is completely determined by bulk metric
Ryu:2006bv ; Nishioka:2009un , so it can also be determined by two different
2-point correlation functions. These seem to uncover that there may be a deep
underlying unknown structure hidden in the holography, which is worth of being
understood further. On the other hand, one can use different operators to
probe the material and use these correlation functions to reconstruct
holographic metrics. To have a holographic dual, the metrics reconstructed
from these different correlation functions must be the same. The result in
this paper then not only proposes an explicit method to build a holographic
spacetime for a material that has holographic dual but also provides a
powerful approach to check if a material has holographic dual. In fact, the
correlation functions belong to the scattering data of bulk theory. For the
case of less symmetries, it is complicated to recover the bulk geometry from
its scattering data. This is still an active topic (see Refs. Barreto ;
Joshi2000 ; doi:10.1142/e040 ; Hora2015  and Isozaki2020  for a recent
review), and we hope that some useful progresses will be made in the future.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
under Grant No. 12005155 and No. 12375051.

##  Appendix A Proof on Lemmas 1 and 2

### Proof of Lemma 1

Let us define Wronskian determinate $W[f,g]=fg^{\prime}-f^{\prime}g$. For any
two solutions of Eq. (5), their Wronskian is constant. We now compute
$W[\psi_{\omega},\psi_{-\omega}]$ at the infinity and
$W[\varphi^{(+)}_{\omega},\varphi^{(-)}_{\omega}]$ at AdS boundary. The
results then read

$W[\psi_{\omega},\psi_{-\omega}]=-2i\omega,~{}~{}W[\varphi^{(+)}_{\omega},\varphi^{(-)}_{\omega}]=-2v\,.$
(25)

The function $C_{\Delta}^{(\pm)}(\omega)$ can also be determined according to
Wronskian determinate

$C_{\Delta}^{(+)}(\omega)=W[\psi_{\omega},\varphi_{\omega}^{(+)}],~{}~{}C_{\Delta}^{(-)}(\omega)=iW[\psi_{\omega},\varphi_{\omega}^{(-)}]\,.$
(26)

Using the asymptotically behavior Eq. (8), we can find

$C_{\Delta}^{(+)}(\omega)=2v\lim_{\rho\rightarrow
0^{+}}\rho^{v-1/2}\psi_{\omega}(\rho)$ (27)

if $v>0.$ From boundary conditions (7) we can find

$\psi_{\omega}(\rho)^{*}=\psi_{-\omega}(\rho)\,.$ (28)

for $\omega\in\mathbb{R}$. From the fact that Eq. (5) is invariant under
transformation $\omega\rightarrow-\omega$, we will see that

$\varphi^{(\pm)}_{\omega}(\rho)=\varphi^{(\pm)}_{-\omega}(\rho)=\varphi^{(\pm)}_{\omega}(\rho)^{*}\,$
(29)

for $\omega\in\mathbb{R}$. Two functions $C_{\Delta}^{(-)}(\omega)$ and
$C_{\Delta}^{(+)}(\omega)$ satisfy

$\varphi_{\omega}^{(+)}(\rho)=\frac{i}{2\omega}\left\\{C_{\Delta}^{(+)}(\omega)\psi_{\omega}(\rho)^{*}-C_{\Delta}^{(+)*}(\omega)\psi_{\omega}(\rho)\right\\}\,$
(30)

and

$\varphi_{\omega}^{(-)}(\rho)=\frac{1}{2\omega}\left\\{C_{\Delta}^{(-)}(\omega)\psi_{\omega}(\rho)^{*}+C_{\Delta}^{(-)*}(\omega)\psi_{\omega}(\rho)\right\\}\,.$
(31)

We can solve the $\psi_{\omega}$ from Eqs. (30) and (31) and obtain

$\psi_{\omega}(\rho)=\Phi^{(-)}_{\Delta}(\omega)\varphi_{\omega}^{(-)}(\rho)+\Phi^{(+)}_{\Delta}(\omega)\varphi_{\omega}^{(+)}(\rho)\,.$
(32)

Here

$\begin{split}\Phi^{(+)}_{\Delta}&=\frac{-2i\omega
C_{\Delta}^{(-)*}}{C_{\Delta}^{(-)}C_{\Delta}^{(+)*}+C_{\Delta}^{(-)*}C_{\Delta}^{(+)}}\\\
\Phi^{(-)}_{\Delta}&=\frac{2\omega
C_{\Delta}^{(+)*}}{C_{\Delta}^{(-)}C_{\Delta}^{(+)*}+C_{\Delta}^{(-)*}C_{\Delta}^{(+)}}\,.\end{split}$
(33)

In the case that we consider the retarded Green function of holography, we
impose the ingoing wave boundary condition
$\Phi\propto\textmd{e}^{-i\omega(t-\rho)}$. This leads to
$\Phi(\rho)\propto\psi_{\omega}(\rho)$. The Green function
$\mathcal{G}(\omega)$ then is given by

$\mathcal{G}_{\Delta}(\omega)=\frac{\Phi^{(+)}_{\Delta}(\omega)}{\Phi^{(-)}_{\Delta}(\omega)}=-i\frac{C_{\Delta}^{(-)}(\omega)^{*}}{C_{\Delta}^{(+)}(\omega)^{*}}\,.$
(34)

We note from Eqs. (30) and (31) that

$\begin{split}W[\varphi^{(+)}_{\omega},\varphi^{(-)}_{\omega}]&=\frac{i}{4\omega^{2}}W[\psi_{-\omega},\psi_{\omega}]\left(C_{\Delta}^{(-)}(\omega)C_{\Delta}^{(+)}(\omega)^{*}+C_{\Delta}^{(-)}(\omega)^{*}C_{\Delta}^{(+)}(\omega)\right)\,.\end{split}$
(35)

Thus, we obtain

$C_{\Delta}^{(-)}(\omega)C_{\Delta}^{(+)}(\omega)^{*}+C_{\Delta}^{(-)}(\omega)^{*}C_{\Delta}^{(+)}(\omega)=4v\omega\,.$
(36)

On the other hand, Eq. (34) shows

$C_{\Delta}^{(-)}(\omega)=i\mathcal{G}(\omega)^{*}C_{\Delta}^{(+)}(\omega)\,.$
(37)

Take it into Eq. (36) and we will find

$|C_{\Delta}^{(+)}(\omega)|^{2}=2v\omega/\Im[\mathcal{G}_{\Delta}(\omega)]\,.$
(38)

The function $F_{\Delta}(\omega)$ is defined as

$F_{\Delta}(\omega)=\lim_{\rho\rightarrow
0^{+}}\frac{\sqrt{2\pi}(-\omega\rho)^{v-1/2}}{2^{v}\Gamma(v)}\textmd{e}^{i\pi(v-1/2)/2}\psi_{\omega}(\rho)\,.$
(39)

Then we see that the function $F_{\Delta}(\omega)$ will is related to the
coefficient $C^{(+)}_{\Delta}$ such that

$F_{\Delta}(\omega)=\frac{\sqrt{\pi}(-\omega/2)^{v-1/2}}{2\Gamma(v+1)}\textmd{e}^{i\pi(v-1/2)/2}C^{(+)}_{\Delta}\,.$
(40)

so we find the modulus of $|F_{\Delta}(\omega)|$ is completely determined by
the correlation function

$|F_{\Delta}(\omega)|^{2}=\frac{\pi|\omega/2|^{2v}v}{\Gamma(v+1)^{2}\Im[\mathcal{G}_{\Delta}(\omega)]}\,.$
(41)

$\square$

### Proof of Lemma 2

Since $\Im(\omega_{k})>0$, we then find that $\psi_{\omega_{k}}$ will decay
exponentially when $\rho\rightarrow\infty$. On the other hand,
$\Phi^{(-)}(\omega)=0$ leads that $\psi_{\omega_{k}}\rightarrow 0$ when
$\rho\rightarrow 0$. Thus we can conclude that $\psi_{\omega_{k}}$ is square-
integrable in the interval $(0,\infty)$. On the other hand, if
$\psi_{\omega_{k}}$ is square-integrable in the interval $(0,\infty)$, we
first should have
$\lim_{\rho\rightarrow\infty}\psi_{\omega_{k}}=\textmd{e}^{i\omega_{k}\rho}\rightarrow
0$ and $\Phi^{(-)}=0$, so $\omega_{k}$ is a QNM. In addition, under these two
boundary conditions, Eq. (5) in fact forms a “Sturm-Liouville eigenvalue
problem” and the $\omega^{2}_{k}$ is one of eigenvalue, so the
$\omega^{2}_{k}$ must be real. The
$\lim_{\rho\rightarrow\infty}\psi_{\omega_{k}}=\textmd{e}^{i\omega_{k}\rho}\rightarrow
0$ means $\omega_{k}^{2}<0$. $\square$

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# Design and Mechanics of Cable-Driven Rolling Diaphragm Transmission for
High-Transparency Robotic Motion

Hoi Man Lam1, W. Jared Walker1, Lucas Jonasch1, Dimitri Schreiber2, ,
and Michael C. Yip2 1Hoi Man Lam, W. Jared Walker, and Lucas Jonasch are with
the Department of Mechanical and Aerospace Engineering, University of
California San Diego, La Jolla, CA 92093 USA<EMAIL_ADDRESS><EMAIL_ADDRESS>Schreiber and Michael C. Yip are
with the Department of Electrical and Computer Engineering, University of
California San Diego, La Jolla, CA 92093 USA. {dschreib<EMAIL_ADDRESS>
###### Abstract

Applications of rolling diaphragm transmissions for medical and teleoperated
robotics are of great interest, due to the low friction of rolling diaphragms
combined with the power density and stiffness of hydraulic transmissions.
However, the stiffness-enabling pressure preloads can form a tradeoff against
bearing loading in some rolling diaphragm layouts, and transmission setup can
be difficult. Utilization of cable drives compliment the rolling diaphragm
transmission’s advantages, but maintaining cable tension is crucial for
optimal and consistent performance. In this paper, a coaxial opposed rolling
diaphragm layout with cable drive and an electronic transmission control
system are investigated, with a focus on system reliability and scalability.
Mechanical features are proposed which enable force balancing, decoupling of
transmission pressure from bearing loads, and maintenance of cable tension.
Key considerations and procedures for automation of transmission setup,
phasing, and operation are also presented. We also present an analysis of
system stiffness to identify key compliance contributors, and conduct
experiments to validate prototype design performance.

##  I Introduction

Transmissions are essential in systems where placing actuators at the joints
is infeasible or dangerous. Examples of such systems include surgical robots
or wearable robots, where high inertias on distal joints can pose dangers to
the patient or user. Another example is MR-compatible surgical robots, where
actuators are incompatible and can disturb the magnetic fields used for
imaging.

Hydraulic transmissions are a viable solution to these problems, offering high
power density, routing flexibility, and high stiffness. Recent usage of
rolling diaphragms in hydraulic transmissions provide high force transparency
and bandwidth [1], which are important advantages in medical robotic
applications.

However, current research applications of rolling diaphragm transmissions
usually introduce a tradeoff between beneficial transmission pressures versus
detrimental frictional forces and torque loading. Furthermore, the fluid
transmission’s complex setup procedure is a detriment to the adoption of
rolling diaphragm transmissions beyond research.

[width=0.96trim=0 0 0 0 mm, clip=true]Figures/Cover/cover.jpg

Figure 1: Image of one prototype transmission unit with a motor attached to
the input/output shaft. Two of these units connected through fluid lines form
a full transmission.

###  I-A Related Works

Rolling diaphragms have been applied to many applications, such as in medical
or teleoperated robotics. In medical applications, rolling diaphragm
transmissions have found great use in MR-safe robots, which make surgery
during live scans possible by allowing the MR-incompatible actuator to power
the robot from outside the MR field [2], [3], [4], [5]. The usage of rolling
diaphragm transmissions also provides smooth force transmission in a compact
package, which is important for patient comfort, and accommodation of a
variety of patient sizes.

The hydrostatic rolling diaphragm transmission itself was investigated in
various works, where some works modelled the transmission as a second-order
spring model [2], [6], [7]. John Peter Whitney proposed a N+1 hybrid
hydrostatic transmission setup, utilizing N hydraulic lines and 1 common
pneumatic preload line, which minimizes transmission complexity without
hampering performance [8], [9], [10].

Whitney’s designs featured a belt-driven design for rotary actuation, and a
linkage design for directly actuating a finger, to convert the transmission’s
translational motion into the desired mechanical work. In the linkage hand
design, the load is attached to the diaphragm piston through the air-side
diaphragm, introducing a dynamic seal, which can be tolerance-intensive and
introduces constant pressure leakage.

In a hydrostatic transmission used for wearable robotic limbs, a ball-screw
design is used on the actuation side, whilst the manipulator side uses a
rolling diaphragm [11], [12]. A cable drive provides reduced friction and
backlash to accentuate the rolling diaphragm transmission’s transparency and
backdrivability, while the floating-cylinder layout makes use of the self-
aligning features of the rolling diaphragm.

###  I-B Contributions

In this work, we investigate an alternative coaxial rolling diaphragm
transmission design, which retains proven features such as the N+1 hybrid
transmission layout, and a cable drive for its ability to minimize backlash
and friction forces. Building on top of those elements, we introduce the
following:

  1. 1.

A translating inner core inline with the rolling diaphragm, coupled with a
force-balanced angled cable drive design, which decouples transmission
pressure from cable tension and friction, reduces bearing load thanks to cable
preload tension balancing, and provides a constant mechanical advantage over a
range of motion beyond a full rotation.

  2. 2.

A method for individual transmission component stiffness analysis to predict
full system stiffness and key compliance contributors.

  3. 3.

An electronic fluid transmission control system for automated pressure and
phasing regulation and control.

##  II Method

###  II-A Overview

The transmission consists of two identical paired units, where each unit uses
two rolling diaphragms to interface with the two air or water transmission
lines. In each transmission unit (Fig. 2(a)), a cable drive converts the
fluid-driven linear rolling diaphragm motion to rotary input-output motions
for robotic joints. A linearly translating core actuates that cable drive,
while isolating the transmission pressure forces from the cable tension and
central pillar bearings. Lastly, a stationary frame provides structural
support and alignment between rolling diaphragms and translating core.

[width=0.96trim=0 0 0 0 mm,clip=true]Figures/Method/labelledOverview.png

(a) Section view of one actuator unit.

[width=0.96trim=0 0 0 0 mm, clip=true]Figures/Method/core.png

(b) Translating capstan pillar relative to the core.

Figure 2: Mechanical design of one actuator unit. (a) shows the section view.
Fluid pressure difference between the two rolling diaphragms (red) actuate the
translating core (green), which drives the input/output shaft (cyan) through
the cable drive (purple). The coaxial rolling diaphragms are fully sealed, and
the translating core maintains cable tension even when the system is
depressurized. (b) shows how the capstan pillar rolls relative to the core.
Fixed to the housing via bearings, the pillar no longer translates, and is
limited to rotation as the core translates. Cable drive fleet angle is kept
constant across the full range of motion through the geometry of the capstan
pillar and the cable wrap-around walls, maintaining constant cable tension for
controllability. Fluid line pressure preload forces are coaxially balanced
through the translating core.

###  II-B Cable Drive and Translating Core

A cable drive was chosen due to its inherent advantages of low friction, low
backlash, and high stiffness [13], [14]. In the prototype transmission, a
1/16” diameter 302 stainless steel cable with 7x19 construction was chosen for
its high flexibility, and sufficient breaking strength of $2100N$ to match the
maximum supportable load of $600N$. The maximum supportable force is a
function of maximum rated diaphragm pressure $P_{max}=1.7MPa$, and diaphragm
piston radius $r_{piston}=15mm$ in Eqn. 1. The corresponding maximum
supportable torque is $6Nm$ for a capstan radius of $r_{capstan}=10mm$.

$F_{max}=\frac{P_{max}}{2}(\pi r_{piston}^{2})$ (1)

Due to the limited travel of the rolling diaphragm, there is a direct tradeoff
between the diameter of a flat cable pulley and the angular range of motion.
However, wrapping the cable helically up a capstan pillar allows for increased
capstan effect and more than one rotation of motion despite limited diaphragm
travel.

Within the cable drive, the cables are run at an angle enforced by wall and
capstan geometry throughout the full range of motion (Fig. 2(b)). The fleet
angle is kept constant by terminating the cables with the same angle as the
helix pitch. Constant fleet angle is necessary to maintain constant cable
tension for a linear performance, which is useful in direct teleoperation
applications.

The cable drive’s forces are balanced by the symmetrical placement of cable
angle and departure locations, which decouples cable preload tension from
bearing loading. The free body diagram moment balance of Fig. 3 sums to 0,
indicating full cancellation:

$\Sigma M_{y}=T_{L}r_{L}-T_{L}r_{L}+T_{R}r_{R}-T_{R}r_{R}=0$ (2)

Where $T_{L}$ and $r_{L}$ are the cable preload tension force and moment arm
on the left side of the capstan, and $T_{R}$ and $r_{R}$ on the right side.
High cable preload tension is preferred to improve axial stiffness, which
decreases with load range to preload ratio [15].

[width=0.7trim=0 0 0 0 mm, clip=true]Figures/Method/capstan_FBD.png

Figure 3: Cable tension forces are symmetrically balanced, thereby eliminating
moment bearing loads in the y-axis (out of page) and decoupling cable tension
from axial bearing loads in x/y directions.

The translating core structure fixes the cable terminations and absorbs the
transmission pressure between the two rolling diaphragms. One drawback to
cable drives can be the difficulty in tuning and maintaining cable tension,
which is a prerequisite for consistent performance. By fixing the cable
terminations within the rigid core structure, the cables can be preloaded at
high tension, allowing for low slack and high stiffness [13]. Cable tension is
held even when the transmission is depressurized, which is advantageous in
medical settings, where surgeons might not have the expertise or tools to
troubleshoot and repair cable slack issues. The prototype translating core
comprises of solid 3D printed termination walls in Markforged Onyx filament
and carbon fiber rods.

###  II-C Coaxial Enclosed Rolling Diaphragm Layout

Interacting with the fluid transmission lines, rolling diaphragms interface
with coaxial pistons on either side of the translating core. Rolling
diaphragms are flexible seals that extend and retract via a rolling action,
eliminating the sliding friction that typical piston O-Ring seals have [16].
The rolling diaphragms used in the prototype are DM3-35-35 rolling diaphragms
manufactured by IER Fujikura, with stroke of 46mm, diameter of 35mm, and
maximum pressure rating of $1.7MPa$.

Both fluid transmission lines have a pressure preload, which fills out the
rolling diaphragm’s convolution to prevent jamming, enables high fluid
stiffness, and determines transmission load capacity. This transmission
preload pressure has a much higher magnitude of force relative to the
magnitude of input/output force transferred.

In the proposed transmission, the two rolling diaphragms are aligned in a
coaxial opposed layout, such that preload pressures are balanced against one
another (Fig. 2(b)). These preload pressure forces compress the translating
core, but are isolated from bearings by the cable drive, decoupling
transmission pressure from bearing friction. The cable drive is situated
between the diaphragm pistons, keeping both fluid chambers fully enclosed to
avoid pressure leakage.

###  II-D Hydrostatic Transmission

Two identical actuators connected by a hydraulic line form one transmission
system, detailed in Fig. 4. The hydraulic line acts as an incompressible link
between the actuator, while an opposing pneumatic line provides a preload
pressure on the water line. High preload pressures help dissolve remaining air
into the water to achieve a stiff and responsive system.

[width=.99trim=0 0 0 0 mm, clip=true]Figures/Method/TransmissionOverview.png

Figure 4: Transmission system setup. Stiffness is maximized along the
hydraulic transmission distance by using a copper tube, while small sections
of flexible plastic tube at each end allow for some flexibility in actuator
placement. The incoming water line is maintained at $700kPa$ by a water pump.
The outgoing line releases into a depressurized reservoir which feeds into the
pump. The preload pressure is controlled through a proportional electric
pressure regulator, which allows for precise pressure control anywhere from 0
to $860kPa$. The design is intended for an N+1 configuration (introduced in
[8]), such that multiple transmissions are preloaded by one single pneumatic
line, simplifying scalability.

The volume of water in the hydraulic line determines the phase offset between
the input and output shafts. Manual alignment is time consuming and difficult,
and misaligned actuators may hit their endstops prematurely resulting in
reduced range of motion and control. Automation of this phasing process
removes a large hurdle for system adoption, and effectively negates long-term
issues such as minuscule leaks and settling. It also reduces misalignment
factors like tube and cable flex by allowing phasing at a high pressure, close
to the standard operating preload pressure.

###  II-E Predictive Phasing

The microcontroller phases the actuators efficiently through a proportional
controller, calculating the adjustment that is needed to rotationally align
the actuators. The flow rate $Q$ through a solenoid valve is given by:

$Q=K_{v}\sqrt{\Delta P}$ (3)

where $\Delta P$ is the pressure drop and $K_{v}$ the flow factor of the
valve. The relationship between water volume and phase offset was determined
empirically, resulting in the following relationship:

$V_{W}=\frac{|\Delta\phi|}{9.594}$ (4)

where $V_{W}$ is water volume in mL and $\Delta\phi$ is the phase offset in
degrees. By combining Eq. 3 and 4, the time the solenoid valve should be
opened is found. The sign of the phase offset $\Delta\phi$ determines if the
intake or outlet valve should be used.

$t=\frac{V_{W}}{Q}=\frac{|\Delta\phi|}{9.594K_{v}\sqrt{\Delta P}}$ (5)

[width=0.7trim=0 0 0 0 mm, clip=true]Figures/Method/AutoPhaseLoop.png

Figure 5: Automatic phasing algorithm. $\Delta P$ of $15kPa$, along with
potentiometer accuracy and solenoid valve response time, resulted in a 0.4°
acceptable maximum phase offset for our setup.

Fig. 5 shows the actuator phasing algorithm. The alignment resolution is
limited by the $\Delta P$ from the water refill system to the transmission
line. To achieve finer angle adjustments, the transmission pressure is brought
to just below the water injection pressure to decrease $\Delta P$. This
increases $\Delta P$ for water ejection, but it is inconsequential since any
overshoot can be corrected by water injection.

###  II-F Automated Operation

The system operating procedure is outlined in Fig. 6. Users operate the system
through a text-based interface with a microcontroller. Plain language
instructions and commands allow even a non-technical user to set up and adjust
a transmission. An air bleed mode removes air bubbles that entered the water
line during assembly. When not in operation, the system is stored in a
hibernation state rather than completely removing all water and air pressure.
Maintaining the system at a low pressure (for example, $100kPa$) removes the
need to bleed air from the water lines or reseat the rolling diaphragms.
Completely depressurizing the actuator is used to disassemble or move the
transmission setup.

[width=0.96trim=0 0 0 0 mm, clip=true]Figures/Method/OperationLoop.png

Figure 6: Operating procedure for the transmission system. Ovals represent
transitory steps that lead to steps intended to be used for extended periods
of time, represented by rectangles.

##  III Theoretical Transmission Stiffness

[width=0.96trim=0 0 0 0 mm, clip=true]Figures/modeling/stiffnessSystem.png

Figure 7: Transmission stiffness system model, assuming locked output shaft
and torqued input shaft applying a force into the system. The model assumes
some proportion of undissolved air left in the system, which acts as an
additional spring in series in the fluid transmission.

To predict the transmission stiffness and identify the main compliance
contributors in the system, a simple spring system, as illustrated in Fig. 7,
was used to model the transmission. Fluid stiffness and cable stiffness are
determined analytically, while more complex components such as the translating
core and diaphragm stiffnesses are determined respectively using FEA and
empirical methods. The system assumes that the change in force is applied by
the input side capstan pillar, while the output side capstan pillar is locked.

The fluid transmission stiffness is separated into a water stiffness
($K_{water}$) and air stiffness ($K_{air}$), assuming there is some proportion
of undissolved air $p_{air}$ in the system. These fluid stiffnesses were found
using Eq. 6 and the variables in Table I.

$K_{fluid}=[p(2\frac{L_{cyl}}{A_{cyl}E_{fluid}}+\frac{L_{hose}}{A_{hose}E_{fluid}})]^{-1}$
(6)

TABLE I: Theoretical fluid line stiffness variables Variable | Value | Description
---|---|---
$E_{water}$ | 2.20 [$GPa$] | Bulk modulus of water
$E_{air}$ | 1.42 [$GPa$] | Bulk modulus of air
$A_{cyl}$ | 9.62E-4 [$m^{2}$] | Cross section of diaphragm cylinder
$A_{hose}$ | 3.17E-5 [$m^{2}$] | Cross section of hose
$L_{cyl}$ | 3.80E-2 [$m$] | Length of diaphragm cylinder
$L_{hose}$ | 4.26E-2 [$m$] | Length of hose
$p_{water}$ | 0.99 [%] | Proportion of water in transmission
$p_{air}$ | 0.01 [%] | Proportion of air in transmission
$K_{water}$ | 1.58E6 [$N/m$] | Est. K of fluid line water
$K_{air}$ | 3.99E3 [$N/m$] | Est. K of fluid line undissolved air

Cable stiffness ($K_{cable}$) was calculated with Eq. 7 Where $A_{cable}$ is
the cable’s cross-section, $L_{cable}$ is the cable length between the capstan
and the wrap-around wall, and $E_{cable}$ is $200GPa$ for 304 stainless steel.

$K_{cable}=\frac{E_{cable}A_{cable}}{L_{cable}}$ (7)

For both rolling diaphragm stiffness ($K_{RD}$) and translating core stiffness
($K_{core}$), the stiffness was estimated via $K=\frac{\Delta F}{\Delta x}$,
where $\Delta F$ and $\Delta x$ are the change in force and deflection
respectively. For $K_{RD}$, the rolling diaphragm stretch under force
application was measured on a material test system. And for $K_{core}$, the
deflection of the cable termination point under a unit applied force was
estimated via FEA (Fig. 8).

[width=0.96trim=0 0 0 0 mm, clip=true]Figures/modeling/coreFEA.png

Figure 8: FEA estimated cable termination deflection from application of $1N$
from cables.

The resultant transmission stiffness estimate is a combination of the
individual component stiffness estimates:

$K_{tot}=(\frac{1}{K_{cable}}+\frac{1}{2K_{core}}+\frac{2}{K_{RD}}+\frac{1}{K_{water}}+\frac{1}{K_{air}})^{-1}$
(8)

Where the individual estimates and overall stiffness estimate are featured in
Table II. Of the individual stiffnesses, the rolling diaphragm and undissolved
air contribute the most to overall compliance. The equivalent estimated
angular stiffness over a $20mm$ diameter capstan pillar is $23.54Nm/rad$,
using $K_{rot}=K_{lin}r^{2}$. However, the stiffness value can vary greatly
with $p_{air}$, as shown in Fig. 9.

[width=0.96trim=0 0 0 0 mm, clip=true]Figures/modeling/airVstiffness.png

Figure 9: Theoretical transmission stiffness as a function of undissolved air % remaining in fluid line. TABLE II: Theoretical component stiffness values, and resultant total transmission stiffness assuming $p_{air}$ of 0.01 %. Source | Variable |  Stiffness [$N/m$] |  % Compliance
---|---|---|---
Water in fluid line | $K_{water}$ |  1.54E6 |  15.2%
Undissolved air in fluid line | $K_{air}$ |  9.97E5 |  23.6%
Single cable | $K_{cable}$ |  8.98E6 |  2.60%
Translating core | $K_{core}$ |  3.80E6 |  12.4%
Rolling diaphragm | $K_{RD}$ |  1.02E6 |  46.1%
Full Transmission | $K_{total}$ |  2.35E5 |

##  IV Experiments and Results

###  IV-A Experimental Setup

To characterize the system, experiments were conducted to fit a second-order
spring model, and to understand the hysteresis and friction of the system. On
the input side, the input shaft is actuated either by hand or by a motor
(Maxon EC393023). Torque inputs are measured through a torque sensor (Futek
TRS600) in the hand-actuated case, and estimated via the motor current draw in
the motor actuated case. On the output side, a second torque sensor at the
output shaft measures the torque output. Encoders (US Digital E5-2000) are
fitted to both shafts to measure the angular deflection both into and out of
the transmission.

The effects of hose length and diameter were minimized in this experiment by
using a minimum hose length of $42.6mm$, to focus on the characteristics of
the rolling diaphragm and cable drive. The system is bled of air bubbles until
no more air can be visually seen in the system. The stiffness of shaft couples
and torque sensors facilitating measurement are stiff enough to be disregarded
in the stiffness estimation.

[width=0.96trim=0 0 0 0 mm, clip=true]Figures/Results/ExperimentSetup.png

Figure 10: Experimental setup for hand-driven configuration. Shaft torques and
positions are measured via torque sensors (1) and encoders (2). The input
shaft is driven by a handle (3), or a motor can be substituted for motor-
driven inputs.

To estimate the second-order spring model, the output shaft is clamped, and
the input shaft is driven with a consecutive torque step signal. A tfest is
performed with measured torque and position as input and output (Eq. 9),
starting from an initial model based on parameters from prior theoretical
calculations (Table III).

$H(s)=\frac{\theta(s)}{\tau(s)}=\frac{1}{Js^{2}+Bs+K}$ (9)

The resultant fit (Table III) estimates a stiffness of $18.71Nm/rad$, damping
of $2.1mNm/s$, and inertia of $0.52\mu kgm^{2}$. This lines up with the
theoretical stiffness value corresponding to 0.02% undissolved air in the
fluid line (see Fig. 9), which is reasonable given that visually no air
bubbles were left in the transmission, but some imperceptible undissolved air
might still remain. Additionally, hand-actuated datasets were collected for
model validation, measuring input torque applied and input shaft deflection.
The model predicted position tracks the measured position closely in Fig. 11,
indicating model accuracy.

TABLE III: Fit results of mass-spring-damper model on step dataset Initial Model Coeffs |  Step Fitted Model Coeffs
---|---
J = 6.54E-5 $[kgm^{2}]$ |  J = 5.20E-5 $[kgm^{2}]$
B = 0.005 $[Nm/s]$ |  B = 0.0021 $[Nm/s]$
K = 23.54 $[Nm/rad]$ |  K = 18.71 $[Nm/rad]$

[width=0.95trim=6mm 1mm 20mm 0mm, clip=true]Figures/Results/modelTracking.png

Figure 11: Verification of fitted model against the hand-driven dataset, where
the model predicted position from measured input torque is compared against
the measured position, exhibiting accurate tracking and good model fit.

[width=0.95trim=6mm 0 50 5mm, clip=true]Figures/Results/Hysteresis.png

Figure 12: Hysteresis plot of one torque sin wave input starting from rest,
maximum hysteresis measured is $0.0760Nm$ (1.27% of $6Nm$ full torque range),
and maximum static friction measured is $0.0272Nm$ (0.45% of $6Nm$ full torque
range).

From a hysteresis plot of one torque sin wave motion from rest (Fig. 12),
there is an approximate maximum hysteresis value of $0.076Nm$, corresponding
to 1.25% of the full $6Nm$ torque range. This hysteresis is likely caused by
hose flexibility or air line pressure regulation inaccuracy, which affects
shaft phasing, absorbs input energy, and has directional behaviour under
positive and negative pressures. The static friction, identified by the change
in torque without change in angular position at either end of the hysteresis
curve, is $0.025Nm$, corresponding to 0.45% of the full torque range. Though
the rolling diaphragm has little static friction, some other elements such as
the cable drive and bearings still contribute to static friction.

To compare the tracking accuracy between the input and output shafts, the
transmission was actuated across a large range of motion by hand via a handle
on the input shaft, while a load with inertia $0.0387kgm^{2}$ is attached to
the output shaft (Fig. 13). The results show good tracking over the entire
motion, with slight tracking error at the position and torque extremities that
are likely caused by hysteresis and energy losses. The close tracking between
the input and output torques validate the transmission’s constant mechanical
advantage across a large range of motion.

[width=0.95trim=20 0 50 20 mm, clip=true]Figures/Results/PosTrqTracking3.png

Figure 13: Transmission input/output shaft angular position and torque over
time, where input shaft is actuated by hand and output shaft is loaded with an
inertia of $0.0387kgm^{2}$.

##  V Conclusion

In this work, a rolling diaphragm transmission featuring a coaxial opposed
rolling diaphragm layout, and a translating core enclosed cable drive, was
prototyped and tested. The prototype displays low hysteresis, low friction,
and good position and torque transparency. An automated transmission
pressurization and phasing system was also detailed, improving the ease of
system setup and maintenance.

According to the theoretical stiffness model, minuscule amounts of undissolved
air can have a significant impact on system stiffness. Apart from undissolved
air, the rolling diaphragm stiffness contributes the most to system
compliance, forming a ’bottom line’ on transmission stiffness. Further
investigation into methods to thoroughly dissolve and bleed air in the
transmission, as well as stiffer diaphragm choice, can greatly improve system
stiffness. Further understanding the rolling diaphragm in isolation, such as
it’s damping and rolling friction, may also help identify other design
limitations for rolling diaphragm based transmissions.

## Acknowledgements

The authors would like to thank Professor Raymond de Callafon for his advice
on system identification, and Alexander Luke for his work on prototype
mechanical design.

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