###### Abstract

Tactile sensing is an essential capability for robots that carry out dexterous
manipulation tasks. While cameras, Lidars and other remote sensors can assess
a scene globally and instantly, tactile sensors can reduce their measurement
uncertainties and gain information about the local physical interactions
between the in-contact objects and the robot, that are often not accessible
via remote sensing. Tactile sensors can be grouped into two main categories:
electronic tactile skins and camera based optical tactile sensors. The former
are slim and can be fitted to different body parts, whereas the latter assume
a more prismatic shape and have much higher sensing resolutions, offering a
good advantage for being used as robotic fingers or fingertips. One of such
optical tactile sensors is our GelTip sensor that is shaped as a finger and
can sense contacts on any location of its surface. As such, the GelTip sensor
is able to detect contacts from all the directions, like a human finger. To
capture these contacts, it uses a camera installed at its base to track the
deformations of the opaque elastomer that covers its hollow, rigid and
transparent body. Thanks to this design, a gripper equipped with GelTip
sensors is capable of simultaneously monitoring contacts happening inside and
outside its grasp closure. Experiments carried out using this sensor
demonstrate how contacts can be localised, and more importantly, the
advantages, and even possibly a necessity, of leveraging all-around touch
sensing in dexterous manipulation tasks in clutter where contacts may happen
at any location of the finger. In particular, experiments carried out in a
Blocks World environment show that the detected contacts on the fingers can be
used to adapt planned actions during the different moments of the reach-to-
grasp motion. All the materials for the fabrication of the GelTip sensor can
be found at https://danfergo.github.io/geltip/

Tactile sensors,

Dexterous manipulation,

GelTip sensor,

###### keywords:

##  Chapter 0 GelTip Tactile Sensor for Dexterous Manipulation in Clutter

###  1 Introduction

As humans, robots need to make use of tactile sensing when performing
dexterous manipulation tasks in cluttered environments such as at home and in
warehouses. In such cases, the positions and shapes of objects are uncertain,
and it is of critical importance to sense and adapt to the cluttered scene.
With cameras, Lidars and other remote sensors, large areas can be assessed
instantly peel2018localisation . However, measurements obtained using such
sensors often suffer from large uncertainties, occlusions and variance of
factors like light conditions and shadows. Thanks to the direct interaction
with the object, tactile sensing can reduce the measurement uncertainties of
remote sensors and it is not affected by the changes of the aforementioned
surrounding conditions. Furthermore, tactile sensing gains information of the
physical interactions between the objects and the robot end-effector that is
often not accessible via remote sensors, e.g., incipient slip, collisions and
detailed geometry of the object. As dexterous manipulation requires precise
information of the interactions with the object, especially in moments of in-
contact or near-contact, it is of crucial importance to attain these accurate
measurements provided by tactile sensing. For instance, failing to estimate
the size of an object by $1\text{\,}\mathrm{mm}$, or its surface friction
coefficient, during (and also right before) a grasp might result in severely
damaging the tactile sensor or dropping the object. In contrast, failing to
estimate the object shape by a few centimeters farther away, will not make a
big impact on the manipulation. To this end, camera vision and other remote
sensors can be used to produce initial estimations of the object and plan
manipulation actions, whereas tactile sensing can be used to refine such
estimates and facilitate the in-hand manipulation luo2017robotic ;
luo2021vitac .

The usage of tactile sensors for manipulation tasks has been studied since
opticalSensorsDexterousManipulation  and in the past years a wide range of
tactile sensors and working principles have been studied in the literature
dahiya2013directions ; luo2017robotic ; luo2021vitac , ranging from flexible
electronic skins kaltenbrunner2013ultra , fiber optic based sensors
xie2013fiber , capacitive tactile sensors maiolino2013flexible , to camera
based optical tactile sensors TacTipFamily ; GelSight2017 , many of which have
been employed to aid robotic grasping kappassov2015tactile . Electronic
tactile skins and flexible capacitive tactile sensors can be adapted to
different body parts of the robot that have various curvatures and geometry
shapes. However, due to the necessary of dielectrics for each sensing element,
they produce considerably low resolution tactile readings. For example, a WTS
tactile sensor from Weiss Robotics used in luo2015novel ; luo2016iterative ;
luo2019iclap  has 14 $\times$ 6 taxels (tactile sensing elements). In
contrast, camera based optical tactile sensors provide higher-resolution
tactile images. However, on the other side, they usually have a bulkier shape
due to the requirement of hosting the camera and the gap between the camera
and the tactile membrane.

Figure 1: There are two distinct areas of contact highlighted in the robot
gripper during a manipulation task: (A) Outside contacts when the robot is
probing or steering the object to be grasped; (B) Inside contacts when the
object is within the grasp closure, which can guide the grasping.

Optical tactile sensors can be grouped in two main groups: marker-based and
image-based, with the former being pioneered by the TacTip sensors TacTip2009
and the latter by the GelSight sensors RetrographicSensing . As the name
suggests, marker-based sensors exploit the tracking of markers printed on a
soft domed membrane to perceive the membrane displacement and the resulted
contact forces. By contrast, image-based sensors directly perceive the raw
membrane with a variety of image recognition methods to recognise textures,
localise contacts and reconstruct the membrane deformations, etc. Because of
the different working mechanisms, marker-based sensors measure the surface on
a lower resolution grid of points, whereas image-based sensors make use of the
full resolution provided by the camera. Some GelSight sensors have also been
produced with markers printed on the sensing membrane GelSight2017GeometrySlip
, enabling marker-based and image-based methods to be used with the same
sensor. Both families of sensors have been produced with either flat sensing
surfaces or domed/finger-shaped surfaces.

In this chapter, we will first review existing optical tactile sensors in
Section 2, and then we’ll look in detail into one example of such image-based
tactile sensors, i.e., the GelTip geltip ; gomes2020geltip , in Section 3. The
GelTip is shaped as a finger, and thus it can be installed on traditional and
off-the-shelf grippers to replace its fingers, and enable contacts to be
sensed inside and outside the grasp closure that are shown in Figure 1. In
Section 4, we will look into experiments carried out using the GelTip sensor
that demonstrate how contacts can be localised, and more importantly, the
advantages, and possibly a necessity, of leveraging all-around touch sensing
in dexterous manipulation tasks in clutter. In particular, experiments carried
out in a Blocks World environment show that the detected contacts on the
fingers can be used to adapt planned actions during the different moments of
the reach-to-grasp motion.

###  2 An overview of the tactile sensors

Compared to remote sensors like cameras, tactile sensors are designed to
assess the properties of the objects via physical interactions, e.g.,
geometry, texture, humidity and temperature. A large range of working
principles have been actively proposed in the literature in the past decades
dahiya2013directions ; luo2017robotic ; dahiya2009tactile . An optical tactile
sensor uses a camera enclosed within its shell and pointing at its tactile
membrane (an opaque window membrane made of a soft material), to capture the
properties of the objects, from the deformations caused to its tactile
membrane by the in contact object. Such characteristics ensure that the
captured tactile images are not affected by the external illumination
variances. To perceive the elastomer deformations from the captured tactile
images, multiple working principles have been proposed. We group such
approaches in two categories: marker tracking and raw image analysis. Optical
tactile sensors contrast to electronic tactile skins that usually have lower
thickness and are less bulky. They are flexible and can adapt to different
body parts of the robot that have various curvatures and geometry shapes.
However, each sensing element of most of the tactile skins, e.g., a capacitive
transducer, has the size of a few square millimetres or even centimetres,
which results in a limited spatial resolution of the tactile skins. Here we do
not cover such skins as these are an extensive topic on its own, however, we
point the reader to two surveys that extensively cover these sensors
DexterousTactileSensorsSurvey ; electronicSkins .

####  1 Marker-based optical tactile sensors

The first marker-based sensor proposal can be found in gelforce , however,
more recently an important family of marker-based tactile sensors is the
TacTip family of sensors described in TacTipFamily . Since its initial domed
shaped version TacTip2009 , different morphologies have been proposed,
including the TacTip-GR2 tactipGR2  of a smaller fingertip design, TacTip-M2
tactipM2  that mimicks a large thumb for in-hand linear manipulation
experiments, and TacCylinder to be used in capsule endoscopy applications.
Thanks to their miniaturised and adapted design, TacTip-M2 tactipM2  and
TacTip-GR2 tactipGR2  have been used as fingers (or finger tips) in robotic
grippers. Although each TacTip sensor introduces some manufacturing
improvements or novel surface geometries, the same working principle is
shared: white pins are imprinted onto a black membrane that can then be
tracked using computer vision methods.

As shown in Table 1, there are also other optical tactile sensors that track
the movements of markers. In FingerVision , an optical tactile sensor named
FingerVision is proposed to make use of a transparent membrane, with the
advantage of gaining proximity sensing. However, the usage of the transparent
membrane makes the sensor lack the robustness to external illumination
variance associated with touch sensing. In ColorMixingTactileSensor , semi-
opaque grids of magenta and yellow makers, painted on the top and bottom
surfaces of a transparent membrane are proposed, in which the mixture of the
two colours is used to detect horizontal displacements of the elastomer. In
greendots , green florescent particles are randomly distributed within the
soft elastomer with black opaque coating so that a higher number of markers
can be tracked and used to predict the interaction with the object, according
to the authors. In greenDotsMulti , a sensor with the same membrane
construction method, 4 Raspberry PI cameras and fisheye lenses has been
proposed for optical tactile skins.

Table 1: A summary of influential Marker-based optical tactile sensors |  Sensor Structure |  Illumination and Tactile Membrane
---|---|---
TacTip TacTip2009  |  The TacTip has a domed (finger) shape, $40\times 40\times 85$ mm, and tracks 127 pins. It uses the Microsoft LifeCam HD webcam. |  The membrane is black on the outside, with white pins and filled with transparent elastomer inside. Initially the membrane was cast from VytaFlex 60 silicone rubber, the pins painted by hand and the tip filled with the optically clear silicone gel (Techsil, RTV27905); however, currently the entire sensor can be 3d-printed using a multi-material printer (Stratasys Objet 260 Connex), with the rigid parts printed in Vero White material and the compliant skin in the rubber-like TangoBlack+.
TacTip-M2 tactipM2  |  It has a thumb-like or semi-cylindrical shape, with TacTip-M2 $32\times 102\times 95$ mm and it tracks 80 pins.
TacTip-GR2 tactipGR2  |  It has a cone shape with a flat sensing membrane, and is smaller than the TacTip, $40\times 40\times 44$ mm, tracks 127 pins and uses the Adafruit SPY PI camera. |
TacCylinder tacCylinder  |  A catadioptric mirror is used to track the 180 markers around the sensor cylindrical body. |
FingerVision FingerVision  |  It uses a ELP Co. USBFHD01M-L180 camera with an 180 degree fisheye lens. It has approximately $40\times 47\times 30$ mm. |  The membrane is transparent, made with Silicones Inc. XP-565, with $4\text{\,}\mathrm{mm}$ of thickness and markers spaced by $5\text{\,}\mathrm{mm}$. No internal illumination is used, as it the membrane transparent.
Subtractive Color Mixing ColorMixingTactileSensor  |  N/A |  Two layers of semi-opaque colored markers is proposed. SortaClear 12 from Smooth-On, clear and with Ignite pigment, is used to make the inner and outer sides.
Green Markers greendots  |  The sensor has a flat sensing surface, measures $50\times 50\times 37$ mm and is equipped with a ELP USBFHD06H RGB camera with a fisheye lens. |  It is composed of three layers: stiff elastomer, soft elastomer with randomly distributed green florescent particles in it and black opaque coating. The stiff layer is made of ELASTOSIL® RT 601 RTV-2 and is poured directly on top of the electronics, the soft layer is made of Ecoflex™ GEL (shore hardness 000-35) with the markers mixed in, and the final coat layer is made of ELASTOSIL® RT 601 RTV-2 (shore hardness 10A) black silicone. A custom board with an array of SMD white LEDs is mounted on the sensor base, around the camera.
Multi-camera Skin greenDotsMulti  |  It has a flat prismatic shape of $49\times 51\times 17.45$ mm. Four Pi cameras are assembled in a $2\times 2$ array and fish-eye lenses are used to enable its thin shape.

####  2 Image-based optical tactile sensors

On the other side of the spectrum, the GelSight sensors, initially proposed in
RetrographicSensing , exploit the entire resolution of the tactile images
captured by the sensor camera, instead of just tracking makers. Due to the
soft opaque tactile membrane, the captured images are robust to external light
variations, and capture information of the touched surface’s geometry
structure, unlike most conventional tactile sensors that measure the touching
force. Leveraging the high resolution of the captured tactile images, high
accuracy geometry reconstructions are produced in GelSightSmallParts ;
luo2018vitac ; lee2019touching ; cao2020spatio ; lu2019surface ;
jiang2021vision . In GelSightSmallParts , this sensor was used as fingers of a
robotic gripper to insert a USB cable into the correspondent port effectively.
However, the sensor only measures a small flat area oriented towards the grasp
closure. In gomes2019gelsight ; gomes2021generation , simulation models of the
GelSight sensors are also created.

Markers were also added to the membrane of the GelSight sensors, enabling
applying the same set of methods that were explored in the TacTip sensors.
There are some other sensor designs and adaptations for robotic fingers in
GelSight2017 ; GelSlim ; digit . In GelSight2017 , matte aluminium powder was
used for improved surface reconstruction, together with the LEDs being placed
next to the elastomer, and the elastomer being slightly curved on the
top/external side. In GelSlim , the GelSlim is proposed, a design wherein a
mirror is placed at a shallow and oblique angle for a slimmer design. The
camera was placed on the side of the tactile membrane, such that it captures
the tactile image reflected onto the mirror. A stretchy textured fabric was
also placed on top of the tactile membrane to prevent damages to the elastomer
and to improve tactile signal strength. Recently, an even more slim design has
been proposed $2\text{\,}\mathrm{mm}$ gelslim3 , wherein an hexagonal
prismatic shaping lens is used to ensure radially simetrically illumination.
In digit , DIGIT is also proposed with a USB “plug-and-play” port and an
easily replaceable elastomer secured with a single screw mount.

In these previous works on camera based optical tactile sensors, multiple
designs and two distinct working principles have been exploited. However, none
of these sensors has the capability of sensing the entire surface of a robotic
finger, i.e., both the sides and the tip of the finger. As a result, they are
highly constrained in object manipulation tasks, due to the fact that the
contacts can only be sensed when the manipulated object is within the grasp
closure GelSightSmallParts ; IncipientSlip ; RegraspVisionTouch . To address
this gap, we propose the finger-shaped sensor named GelTip that captures
tactile images by a camera placed in the center of a finger-shaped tactile
membrane. It has a large sensing area of approximately
$75\text{\,}{\mathrm{cm}}^{2}$ (vs. $4\text{\,}{\mathrm{cm}}^{2}$ of the
GelSight sensor) and a high resolution of 2.1 megapixels over both the sides
and the tip of the finger, with a small diameter of $3\text{\,}\mathrm{cm}$
(vs. $4\text{\,}\mathrm{cm}$ of the TacTip sensor). More details of the main
differences between the GelSight sensors, TacTip sensors and our GelTip sensor
are given in Table 2.

Table 2: A summary of influential flat and finger-shaped GelSight sensors |  Sensor Structure |  Illumination |  Tactile Membrane
---|---|---|---
GelSightGelSightSmallParts  |  It has a cubic design with a flat square surface. A Logitech C310 (1280 $\times$ 720) camera is placed at its base pointing at the top membrane. |  Four LEDs (RGB and white) are placed at the base. The emitted light is guided by the transparent hard surfaces on the sides, so that it enters the membrane tangentially. |  A soft elastomer layer is placed on top of a rigid, flat and transparent acrylic sheet. It is painted using semi-specular aluminum flake powder.
GelSight GelSight2017  |  It has a close-to hexagonal prism shape. The used webcam is also the Logitech C310. |  Three sets of RGB LEDs are positioned (close to) tangent to the elastomer, with a $120\text{\,}\mathrm{\SIUnitSymbolDegree}$ angle from each other. |  A matte aluminium powder is proposed for improved surface reconstruction. Its elastomer has a flat bottom and a curved top.
GelSlim GelSlim  |  A mirror placed at a shallow oblique angle and a Raspberry Pi Spy (640 $\times$ 480) camera is used to capture the tactile image reflected by the mirror. |  A single set of white LEDs is used. These are pointed at the mirror, so that the light is reflected directly onto the tactile membrane. |  A stretchy and textured fabric on the tactile membrane prevents damages to the elastomer and results in improved tactile signal strength.
GelSlim v3gelslim3  |  It is shaped similar to GelSightSmallParts ; GelSight2017  however slimmer $20\text{\,}\mathrm{mm}$ of thickness, and a round sensing surface. |  A custom hexagonal prism is constructed to ensure radially symmetric illumination. |  An elastomer with Lambertian reflectance is used, as proposed in GelSight2017 .
DIGIT digit  |  A prismatic design, with curved sides. An OmniVision OVM7692 (640 $\times$ 480) camera is embedded in the custom circuit board. |  Three RGB LEDs are soldered directly into the circuit board, illuminating directly the tactile membrane. |  The elastomer can be quickly replaced using a single screw mount.
Round Fingertip br2020soft  |  It has a round membrane, close to a quarter of sphere. A single $160\text{\,}\mathrm{\SIUnitSymbolDegree}$ FoV Raspberry Pi (640 $\times$ 480) is installed on its base. |  Two rings of LEDs are placed on the base of the sensor, with the light being guided through the elastomer. |  Both rigid and soft parts of the membrane are cast, using SLA 3D printed molds.
OmniTact padmanabha2020omnitact  |  It has a domed shape. Five endoscope cameras (400 $\times$ 400) are installed on a core mount, and placed orthogonally to each other: pointing at the tip and sides. |  RGB LEDs are soldered both onto the top and sides of the sensor. |  The elastomer gel is directly poured onto the core mount (and cameras) without any rigid surface or empty space in between.
GelTip geltip  |  It has a domed (finger) shape, similar to a human finger. A Microsoft Lifecam Studio webcam (1920 $\times$ 1080) is used. |  Three sets of LEDs, with a $120\text{\,}\mathrm{\SIUnitSymbolDegree}$ angle from each other, are placed at the sensor base, and the light is guided through the elastomer. |  An acrylic test tube is used as the rigid part of the membrane. The deformable elastomer is cast using a three-part SLA/FFF 3D printed mold.

With their compact design, the GelTip geltip  and other GelSight
GelSightSmallParts ; GelSlim ; gelslim3 ; digit ; cao2021touchroller  sensors
are candidate sensors to be mounted on robotic grippers. Recently, custom
grippers built using the GelSight working principle have also been proposed
wilson2020design ; she2019cable . Two recent works br2020soft ;
padmanabha2020omnitact  also address the issue of the flat surface of previous
GelSight sensors. However, their designs have large differences to ours. In
br2020soft , the proposed design has a tactile membrane with a surface
geometry close to a quarter of a sphere. As a consequence, a great portion of
contacts happening on the regions outside the grasp closure is undetectable.
In padmanabha2020omnitact , this issue is mitigated by the use of five
endoscope micro cameras looking at different regions of the finger. However,
this results in a significant increase of cost for the sensor, according to
the authors, approximately US$3200 (vs. only around US$100 for ours).

###  3 The GelTip sensor

####  1 Overview

Figure 2: (A) The working principle of the proposed GelTip sensor. The three-
layer tactile membrane (rigid body, elastomer and paint coating) is shown in
gray. The light rays emitted by the LEDs travel through the elastomer. As one
object, shown in green, presses the soft elastomer against the rigid body, an
imprint is generated. The resulted tactile image is captured by the camera
sensor, placed in the core of the tactile sensor. An opaque shell, enclosing
all the optical components, ensures the constant internal lighting of the
elastomer surface. (B) Two-dimensional representation of the geometrical model
of the GelTip sensor. The tactile membrane is modeled as a cylindrical surface
and a semi-sphere. An optical sensor of a focal-length $f$ is placed at the
referential origin of the sensor, which projects a point on the surface of the
sensor $P$ into a point $P^{\prime}$ in the image plane. The sensor has a
radius $r$ and its cylindrical body has a length $d$.

As illustrated in Figure 2 (A), the GelTip optical tactile sensor is shaped as
a finger, and its body consists of three layers, from the inside to the outer
surface: a rigid transparent body, a soft transparent membrane and a layer of
opaque elastic paint. In its base, a camera is installed, looking at inner
surface of the cylinder. When an object is pressed against the tactile
membrane, the elastomer distorts and indents according to the object shape.
The camera can then capture the obtained imprint into a digital image for
further processing. As the membrane is coated with opaque paint, the captured
tactile images are immune to external illumination variances, which is
characteristic of tactile sensing. To ensure that the imprint is perceptible
from the camera view, LED light sources are placed adjacent to the base of the
sensor, so that light rays are guided through the membrane.

####  2 The sensor projective model

For flat sensors, the relationship between the surface and the captured image
can be often be easily obtained, or simply substituted by a scaling factor
from single pixels to meters GelSightSmallParts ; GelSight2017 ; IncipientSlip
. However, when considering highly curved sensors, it is important to study a
more general projective function. In this subsection, we will look into how to
derive such projective function $m$. As for the case of GelTip sensor, $m$
maps pixels in the image space $(x^{\prime},y^{\prime})$ into points $(x,y,z)$
on the surface of the sensor. Obtaining the protective function for other
curved GelSight sensors should be similar, requiring only sensor-specific
adaptations. The camera is assumed to be placed at the referential origin,
looking in the direction of $z$ axis. The sensor space takes the center of its
base, which is also the center point of the camera, as the coordination origin
$(0,0,0)$; the image space takes the center of the image as the origin
$(0,0)$. Such a projection model is necessary for, among other applications,
detecting the position of contacts in the 3D sensor surface.

As illustrated in Figure 2 (B), the sensor surface can be modeled as a joint
semi-sphere and an opened cylinder, both sharing the same radius $r$. The
cylinder surface center axis and the $z$-axis are collinear, therefore, the
center point of the semi-sphere can be set to $(0,0,d)$, where $d$ is the
distance from the center point of the base of the semi-sphere to the center
point of the base of the sensor. The location of any point on the sensor
surface $(x,y,z)$ can be represented as follows:

$\displaystyle x^{2}+y^{2}+(z-d)^{2}=r^{2}$ for $z>d$ (1) $\displaystyle
x^{2}+y^{2}=r^{2}$ for $z<=d$ (2)

By making the usual thin lens assumptions, the optical sensor is modeled as an
ideal pinhole camera. The projective transformation that maps a point in the
world space $P$ into a point in the tactile image $P^{\prime}$ can be defined
using the general camera model szeliski2011computer  as:

$\displaystyle P^{\prime}$ $\displaystyle=K[\boldsymbol{R}|\boldsymbol{t}]P$
(3) $\displaystyle K$
$\displaystyle=\begin{bmatrix}\begin{array}[]{llll}fk&0&c_{x}&0\\\
0&fl&c_{y}&0\\\ 0&0&1&0\\\ \end{array}\end{bmatrix}$ (4)

where $P^{\prime}=[x^{\prime}z,y^{\prime}z,z]^{T}$ is an image pixel and
$P=[x,y,z,1]^{T}$ is a point in space, both represented in homogeneous
coordinates here, $[R|t]$ is the camera’s extrinsic matrix that encodes the
rotation $R$ and translation $t$ of the camera, $K$ is the camera intrinsic
matrix ($f$ is the focal length; $k$ and $l$ are the pixel-to-meters ratios;
$c_{x}$ and $c_{y}$ are the offsets in the image frame). Assuming that the
used camera produces square pixels, i.e., $k=l$, $fk$ and $fl$ can be replaced
by $\alpha$, for mathematical convenience.

The orthogonal projections in the $XZ$ and $YZ$ of a generic projection ray
can be obtained by expanding the matrix multiplication given by Equation 3 and
solving it w.r.t. $x$ and $y$:

$\displaystyle\systeme*{x^{\prime}z=\alpha x+c_{x}z,y^{\prime}z=\alpha
y+c_{y}z,z=z}\Leftrightarrow\systeme*{\alpha x=x^{\prime}z-c_{x}z,\alpha
y=y^{\prime}z-c_{y}z}\Leftrightarrow\systeme*{x=(\frac{x^{\prime}-c_{x}}{\alpha})z,y=(\frac{y^{\prime}-c_{y}}{\alpha})z}$
(5)

The desired mapping function $m:(x^{\prime},y^{\prime})\rightarrow(x,y,z)$ can
then be obtained by constraining the $z$ coordinate through the intersection
of the generic projection ray with the sensor surface, described in Equation
6, where $\chi=x^{\prime}-c_{x}$ and $\gamma=y^{\prime}-c_{y}$ and
$\omega=\chi^{2}+\gamma^{2}$. The discontinuity region, i.e., a circumference,
is found by setting $z=d$ in Equation 5:

$m(x^{\prime},y^{\prime})=\left\\{\begin{array}[]{ll}x&=\frac{\chi}{\alpha}z\\\
y&=\frac{\gamma}{\alpha}z\\\
z&=\left\\{\begin{array}[]{lll}\sqrt{\frac{(r\alpha)^{2}}{\omega}}&\mbox{if
}\omega<(\frac{r\alpha}{d})^{2}\\\ \\\
\frac{{\alpha^{2}}2d+\sqrt{{(-{\alpha^{2}}2d)}^{2}-4\omega{{(d^{2}-r^{2})}{\alpha^{2}}}}}{2{(\omega+\alpha^{2})}}&otherwise\end{array}\right.\\\
\end{array}\right.$ (6)

The introduced sensor model is validated and visualised in Figure 3. Two
projection rays, corresponding to the spherical and cylindrical regions, are
depicted. Each ray intersects three relevant points: the frame of reference
origin, the point in the 3D sensor surface, and the corresponding projected
point in the image plane.

Figure 3: Two projection rays that correspond to the spherical (in red) and
cylindrical (in navy blue) regions are depicted in the figure. Each ray
intersects three relevant points: the frame of reference origin, a point in
the sensor surface and the corresponding projected point in the image plane.

####  3 Fabrication process

As for any other optical tactile sensors, the fabrication of the GelTip sensor
is the fabrication of the sensing membrane. It requires the fabrication of
three parts: the bottom rigid layer, the deformable elastomer, and the coat of
paint. For constructing the rigid layer, a flat sheet of transparent acrylic
can be used, in the case of flat sensors GelSight2017GeometrySlip . However,
for finger-shaped sensors, a curved rigid surface is necessary. In the case of
the GelTip, a simple off-the-shelf transparent test tube is used. These
commercially available tubes are made of plastic/acrylic or glass and one
disadvantage of using such test tubes, particularly the plastic ones, is that
they contain small imperfections that result from the manufacturing process.
An alternative approach is to print the rigid tube using a stereolithography
3D-printer and clear resin, however, proper polishing is necessary to ensure
its optical transparency br2020soft . To fabricate the elastomer in the
desired shape a mold can also be created, for instance 3D printed. Fused
Filament Fabrication (FFF) and Stereolithography (SLA) printers, yield
different textured surfaces and consecutive differently textured elastomers.
Example 3D-printed parts are shown in Figure 4-(C).

The soft elastomer is then created by mixing a two-components silicone, such
as XP-565. Mixing these two parts in different ratios yields elastomers with
different a elastic properties. Additional additives can also be considered,
such as, Slacker for increasing the silicone tackiness. A commonly used
mixture used for the GelTip would be 1 gram of XP-565 part-A, 22 grams of
XP-565 part-B and 22 grams of the Slacker.

For painting the transparent elastomer, off-the-shelf spray paints tend to
form a rigid coat and cracks will develop in the coat when the elastomer
deforms or stretches. To avoid these issues, custom paint can be fabricated
and applied using a paint gun or an airbrush GelSight2017 . Pigment powder is
mixed with a small portion of part-A and part-B of XP-565, with the same ratio
as used in the elastomer. The paint pigment commonly consists on aluminium
powder ($1\text{\,}\mathrm{\SIUnitSymbolMicro m}$), however, other options can
also be considered as well. After mixing them properly, the mixture is
dissolved using a silicone solvent until a watery liquid is achieved, that
then can be sprayed onto the elastomer.

Finally the three sets of LEDs can be soldere: either of different colors,
red, green and blue, or all white. Since different LEDs emit different light
intensities, each cluster is preceded by a independent resistor. The power
source can be either extracted from the camera or from an external source,
e.g., adding a secondary USB cable. At the core of any optical sensor, a
camera is installed. In the case of the GelTip, a wide angle lens is also
considered, enabling the recording of the internal surface of the entire
finger.

Figure 4: (A) Exploded view of the GelTip tactile sensor design. (B) A GelTip
sensor, next to a British one pound coin, for relative size comparison. The
sensor has a length of approximately $10\text{\,}\mathrm{cm}$; its shell has a
diameter of $2.8\text{\,}\mathrm{cm}$; and the tactile membrane has a length
of $4\text{\,}\mathrm{cm}$ and a diameter of $2\text{\,}\mathrm{cm}$. (C) The
three-part mold next to the remaining parts used in the GelTip construction.
(D) The plastic tube is inserted into the sleeve and then mounted onto the
mold, afterwards the tube is measured and trimmed and then the elastomer is
poured. (E) The tactile membrane after being de-molded and before being
painted.

###  4 Evaluation

In this section, we look into two sets of experiments carried out using a
GelTip sensor. The first set of experiments demonstrates how an image-based
tactile sensor can be used to localise contacts and; the second set of
experiments illustrates the advantages of leveraging all-around touch sensing
for dexterous manipulation. Video recordings of these experiments and the CAD
models for 3D printing the GelTip sensor can be found at
https://danfergo.github.io/geltip/.

####  1 Contact localisation

For the contact localisation experiment, a set of seven small objects is 3D
printed, each with object with a maximum size of $1\times 1\times 2$ cm3. The
objects are shown in Table 4 . A 3D printed mount is also built, and placed on
top of a raised surface, ensuring that all the objects are kept in the same
position throughout the experiment. GelTip sensors are installed on a robotic
actuator, i.e., the 6-DoF Universal Robots UR5 arm with a Robotiq 2F-85
gripper. The actuator rotates and translates to tap objects at multiple known
positions of one of its fingers surface, as illustrated in Figure 5. The
actuator starts with its fingers pointing downwards, i.e., orientation 0,
being is visually aligned with the cone object. Contacts are then registered,
firstly on the sensor tip, by rotating the sensor, and then on the side, by
translating the sensor. In Figure 5 (B) markings show the location of such
contacts and in Figure 5 (C) the necessary ($\Delta x$, $\Delta z$)
translation to obtain contacts on the finger skin is also shown. To use the
projection model described in Section 3, five parameters are necessary to be
known: $r$, $d$, $c_{x}$, $c_{y}$ and $\alpha$. The first two are extracted
from the dimensions of the sensor design, however, the latter three are the
intrinsic parameters of the camera, which need to be calibrated. To this end,
we obtain such parameters from a known pair of corresponding
$(x^{\prime},y^{\prime})$ and $(x,y,z)$ points. We set the actuator to tap the
object in the $15\text{\,}\mathrm{mm}$ translation position. The center of the
sensor tip ($c_{x},c_{y}$) and the contacted point are manually annotated in
the image space. The $\alpha$ parameter can then be derived by fitting the
known information into Equation 5. After detecting the contact in the image
space and projecting it into $(x,y,z)$ coordinates, the Euclidean distance
between the predicted and the true contact positions are computed. For each of
the seven objects, a total of eight contacts are recorded, i.e., four
rotations ($\theta$): $0$, $\pi/6$, $\pi/4$, $\pi/3$; and four translations
($\tau$): $0\text{\,}\mathrm{mm}$, $5\text{\,}\mathrm{mm}$,
$10\text{\,}\mathrm{mm}$, $15\text{\,}\mathrm{mm}$.

The resulting localisation errors between the observed and true localisation,
expressed in millimeters, are summarised in Table 3. Overall, the variance of
the localisation errors is large: in some contacts the obtained errors are
lower than $1\text{\,}\mathrm{mm}$, while in others are over
$1\text{\,}\mathrm{cm}$. On the other hand, the localisation error, for each
object or position, is correlated with its variance. The largest localisation
errors happen on objects with large or rounded tops i.e., sphere, edge and
slab; contrariwise, the lowest errors are observed for objects with sharp
tops, i.e., cone, tube and cylinder. In terms of the localisation errors at
different positions, contacts happening near the sensor tip, i.e., the
rotations, present lower errors than contacts happening on the sensor side,
i.e., translations. In particular, contacts happening at $pi/4$ and $pi/6$
have the lowest errors.

From these experiments, we find three main challenges using a finger-shaped
sensor, such as the GelTip, for contact localisation: 1) weak imprints,
created by light contacts, may not be captured by the localisation algorithm;
2) forces perpendicular to the main axis of the sensor may flex the sensor
tip, resulting in localisation errors; and 3) imperfections in the sensor
modeling and calibration further contribute to these localisation
inaccuracies. Examples of captured tactile images and corresponding
predictions for the smallest (i.e., $<Cone,\theta=0>$) and largest
localisation errors (i.e., $<Slab,\tau=$15\text{\,}\mathrm{mm}$>$) are shown
in Figure 6. In the first case, due to the bright imprint provided by the
sharp cone top, the algorithm successfully locates the contact. In the second
case, due to the imperceptible contact imprint the algorithm incorrectly
predicts the contact in the sensor tip.

Figure 5:  (A) Two GelTip sensors are installed on a robotic actuator and a 3D printed mount that holds a small 3D printed shape (a cylinder here) placed on top of a wooden block. The actuator moves in small increments and collects tactile images annotated with the known contact positions. (B, C) Illustration of the motion of the sensor during the data collection. The sensor starts pointing downwards, as shown in (B). To obtain contacts on the sensor surface, while moving, the sensor is also translated by ($\Delta x$, $\Delta z$), as shown in (C). A total of eight contacts are collected per object: four rotations ($\theta$) on the sensor tip and four translations ($\tau$) on the sensor side, as highlighted in (B). Table 3: Contact errors per position, expressed in millimeters ROTATIONS | TRANSLATIONS
---|---
$0$ | $\pi/6$ | $\pi/4$ | $\pi/3$ | 0 | 5 | 10 | 15
$4.71$ $\pm 0.75$ |  $2.01$ $\pm 0.90$ |  $1.04$ $\pm 0.46$ |  $6.96$ $\pm 4.82$ |  $7.87$ $\pm 5.08$ |  $8.03$ $\pm 1.92$ |  $7.55$ $\pm 5.00$ |  $4.86$ $\pm 8.41$
|  |  |  |  |  |  |
Table 4: Contact errors per object, expressed in millimeters 0.093 |  |  |  |  |  |
---|---|---|---|---|---|---
Cone |  0.096 |  |  |  |  |
Sphere |  0.12 |  |  |  |  |
Irregular |  0.094 |  |  |  |  |
Cylinder |  0.093 |  |  |  |  |
Edge |  0.093 |  |  |  |  |
Tube |  0.093 |  |  |  |  |
Slab |  |  |  |  |  |
$3.63$ $\pm 3.26$ |  $6.79$ $\pm 5.38$ |  $5.61$ $\pm 4.08$ |  $4.57$ $\pm 4.30$ |  $7.47$ $\pm 6.29$ |  $3.33$ $\pm 1.90$ |  $6.27$ $\pm 8.17$
|  |  |  |  |  |

Figure 6: Reference and in contact frames for two evaluated contacts. The
expected contact region is highlighted with a dash circumference, the
predicted contact position with a yellow circle and the axis where the
contacts occur with dotted lines. The top row shows the contact for the
$<Cone,\theta=0>$ that results in the smallest error, and the bottom row shows
the $<Slab,\tau=$15\text{\,}\mathrm{mm}$>$ that results in the largest error.

####  2 Touch-guided grasping in a Blocks World environment

In the task of grasping objects, the initial motion of the gripper is often
planned using remote sensing, e.g., camera vision or Lidar. However, remote
sensing suffers from occlusions and inaccurate predictions about geometry and
distances. In such cases, the final grasp and re-grasp control policies have
to rely on inaccurate information of where and when contacts occur. In
contrast, touch sensing offers accurate feedback about such contacts.

We can clearly verify the importance of considering of touch sensing, by
studying two different policies in a simple grasp experiment, i.e., Random
grasp (Rg) and Random grasp + Touch informed re-grasp (RgTr), in a simple
grasp experiment. In this experiment, the robot is presented with a 4x4 board
with one wooden block placed in each row in unknown columns (to the robot).
The robot attempts to grasp each block, and if it fails to grasp one block
after 5 attempts, it is considered a failure, and it skips to the next one, as
shown in Figure 7. Here, the random grasp in Rg and RgTr mimics the
inaccuracies from remote sensing based grasping, by sampling the block
position randomly. The touch informed re-grasp in RgTr, mimics an adaptation
carried out by touch sensing, by sensing possible collisions, and moving the
gripper towards the column in which the contact is detected. A Control policy
(C) can also be implemented for reference. In this case, the agent always
knows the position of each block and consequently always moves directly
towards it. The results of this experiment are summarised in Table 5, after
executing these policies 5 times. As it can be seen, in all the measured
metrics, RgTr is a more successful policy than Rg. For instance, Rg fails to
grasp 20% of the blocks, i.e., on average one block is left on the board at
the end of each run. In contrast, with the RgTr policy all the blocks are
grasped, resulting in a failure rate of 0%. Similarly, both the average number
of attempts and the average number of collisions per block with the RgTr
policy is also lower than the Rg policy, i.e., $1.85$ and $0.55$ versus $3.30$
and $1.45$. This difference in performance is justified by the fact that in
the case of RgTr, once a collision occurs the regrasp policy ensures that the
grasp attempt is successful. If the grasp position is sampled randomly, it
will be a success chance of $1/4$ for each grasping attempt. In contrast, with
the touch feedback enabled, this chance jumps to $2.5/4$ on average. As a
consequence, the RgTr policy finds a successful grasp more quickly and thus
grasps more blocks within the maximum 5 attempts limit. This experiment shows
that sensing contacts outside the grasp closure offers an important feature to
improve the success chance of a given grasp attempt.

Figure 7: The Blocks World experimental setup. In each experiment, the robot actuator moves row by row, attempting to grasp each block. The experiment shows that, even with the initial uncertainty, the robot grasps all the blocks successfully, using the all-around touch feedback. Table 5: The table summarises the percentage of failing to grasp blocks (failure rate), and the average number of attempts and collisions per block in all the grasping attempts ($4\times 5$). It can be noted that the Random Grasp + Touch Informed Regrasp policy outperforms Random Grasp in all three metrics, i.e., obtains lower failure rate, less attempts and less collisions. Policies |  Failure rate |  # of attempts per block |  # of collisions per block
---|---|---|---
Control |  0% |  1 |  0
Random Grasp |  20% |  3.30 |  1.45
Random Grasp + Touch Informed Regrasp |  0% |  1.85 |  0.55

Figure 8: (A) Tactile images captured using our proposed GelTip sensor. From
left to right, top to bottom: a fingerprint pressed against the tip of the
sensor, two fingerprints on the sides, an open-cylinder shape being pressed
against an side of the sensor and the same object being pressed against the
corner of the tip, all highlighted in red circles. (B) A plastic strawberry
being grasped by a parallel gripper equipped with two GelTip sensors, with the
corresponding imprint highlighted in the obtained tactile image (in gray-
scale).

###  5 Conclusions and Discussions

In this chapter, we have reviewed the tactile sensors for robot grasping and
manipulation, highlighted our proposed GelTip sensor, which can detect
contacts around the robot finger. As illustrated in Figure 8, it can capture
fine ridges of human fingerprints and the fine texture of a plastic
strawberry. The grasping experiments in the Blocks World environment show the
potential of the all-around finger sensing in facilitating dynamic
manipulation tasks. In our future research, we will introduce imprinted
markers to the GelTip sensor to track the force fields. The use of the GelTip
sensor in the manipulation tasks, such as grasping in cluttered environments,
will also be of our interest.

Compared to the GelSight sensors GelSightSmallParts ; GelSight2017 , due to
the sensor design of a finger shape, the light distribution throughout the
sensor internal surface is no longer homogeneous. Specifically, a brightly
illuminated ring can be observed near the discontinuity region (see Figure 2
(B)). Shadows can also be observed in the bottom-left sample of Figure 8 (A)
when contacts of large pressure are applied, due to the placement of the
camera and light sources. It may pose a challenge to geometry reconstruction
using the Poisson reconstruction method GelSightSmallParts ; GelSight2017 ;
br2020soft  that builds a fixed mapping of pixel intensities to surface
orientations and requires carefully placed RGB LEDs. In future research,
Convolutional Neural Networks could be used for geometry reconstruction of the
GelTip sensor.

### ACKNOWLEDGMENT

This work was supported by the EPSRC project “ViTac: Visual-Tactile Synergy
for Handling Flexible Materials” (EP/T033517/1).

## References

## References

  * (1) H. Peel, S. Luo, A. Cohn, R. Fuentes, Localisation of a mobile robot for bridge bearing inspection, Automation in Construction 94 (2018) 244–256.
  * (2) S. Luo, J. Bimbo, R. Dahiya, H. Liu, Robotic tactile perception of object properties: A review, Mechatronics 48 (2017) 54–67.
  * (3) S. Luo, N. F. Lepora, U. Martinez-Hernandez, J. Bimbo, H. Liu, Vitac: Integrating vision and touch for multimodal and cross-modal perception, Frontiers in Robotics and AI 8 (2021).
  * (4) P. Dario, D. De Rossi, Tactile sensors and the gripping challenge: Increasing the performance of sensors over a wide range of force is a first step toward robotry that can hold and manipulate objects as humans do, IEEE Spectrum 22 (8) (1985) 46–53.
  * (5) R. S. Dahiya, P. Mittendorfer, M. Valle, G. Cheng, V. J. Lumelsky, Directions toward effective utilization of tactile skin: A review, IEEE Sensors Journal 13 (11) (2013) 4121–4138.
  * (6) M. Kaltenbrunner, T. Sekitani, J. Reeder, T. Yokota, K. Kuribara, T. Tokuhara, M. Drack, R. Schwödiauer, I. Graz, S. Bauer-Gogonea, et al., An ultra-lightweight design for imperceptible plastic electronics, Nature 499 (7459) (2013) 458–463.
  * (7) H. Xie, H. Liu, S. Luo, L. D. Seneviratne, K. Althoefer, Fiber optics tactile array probe for tissue palpation during minimally invasive surgery, in: 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, 2013, pp. 2539–2544.
  * (8) P. Maiolino, M. Maggiali, G. Cannata, G. Metta, L. Natale, A flexible and robust large scale capacitive tactile system for robots, IEEE Sensors Journal 13 (10) (2013) 3910–3917.
  * (9) B. Ward-Cherrier, N. Pestell, L. Cramphorn, B. Winstone, M. E. Giannaccini, J. Rossiter, N. F. Lepora, The TacTip Family: Soft Optical Tactile Sensors with 3D-Printed Biomimetic Morphologies, Soft Robotics 5 (2) (2018) 216–227.
  * (10) W. Yuan, S. Dong, E. H. Adelson, GelSight: High-Resolution Robot Tactile Sensors for Estimating Geometry and Force., Sensors (Basel, Switzerland) 17 (12) (11 2017).
  * (11) Z. Kappassov, J.-A. Corrales, V. Perdereau, Tactile sensing in dexterous robot hands, Robotics and Autonomous Systems 74 (2015) 195–220.
  * (12) S. Luo, W. Mou, K. Althoefer, H. Liu, Novel tactile-sift descriptor for object shape recognition, IEEE Sensors Journal 15 (9) (2015) 5001–5009.
  * (13) S. Luo, W. Mou, K. Althoefer, H. Liu, Iterative closest labeled point for tactile object shape recognition, in: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2016, pp. 3137–3142.
  * (14) S. Luo, W. Mou, K. Althoefer, H. Liu, iclap: Shape recognition by combining proprioception and touch sensing, Autonomous Robots 43 (4) (2019) 993–1004.
  * (15) C. Chorley, C. Melhuish, T. Pipe, J. Rossiter, Development of a tactile sensor based on biologically inspired edge encoding, International Conference on Advanced Robotics (ICAR) (2009).
  * (16) M. K. Johnson, E. H. Adelson, Retrographic sensing for the measurement of surface texture and shape Retrographic sensing for the measurement of surface texture and shape, in: IEEE Conference on Computer Vision and Pattern Recognition, 2009.
  * (17) S. Dong, W. Yuan, E. H. Adelson, Improved gelsight tactile sensor for measuring geometry and slip, in: 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2017, pp. 137–144.
  * (18) D. F. Gomes, Z. Lin, S. Luo, Blocks world of touch: Exploiting the advantages of all-around finger sensing in robot grasping, Frontiers in Robotics and AI 7 (2020) 127.
  * (19) D. F. Gomes, Z. Lin, S. Luo, Geltip: A finger-shaped optical tactile sensor for robotic manipulation, in: 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, 2020, pp. 9903–9909.
  * (20) R. S. Dahiya, G. Metta, M. Valle, G. Sandini, Tactile sensing—from humans to humanoids, IEEE transactions on robotics 26 (1) (2009) 1–20.
  * (21) H. Yousef, M. Boukallel, K. Althoefer, Tactile sensing for dexterous in-hand manipulation in robotics - A review, Sensors and Actuators, A: Physical 167 (2) (2011) 171–187.
  * (22) B. Shih, D. Shah, J. Li, T. G. Thuruthel, Y.-L. Park, F. Iida, B. Zhenan, R. Kramer-Bottiglio, M. T. Tolley, Electronic skins and machine learning for intelligent soft robots, Science robotics 41 (5) (2020).
  * (23) K. Vlack, K. Kamiyama, T. Mizota, H. Kajimoto, N. Kawakami, S. Tachi, Gelforce: A traction field tactile sensor for rich human-computer interaction, in: IEEE Conference on Robotics and Automation, 2004. TExCRA Technical Exhibition Based., 2004, pp. 11–12.
  * (24) B. Ward-Cherrier, N. Rojas, N. F. Lepora, Model-free precise in-hand manipulation with a 3d-printed tactile gripper, IEEE Robotics and Automation Letters 2 (4) (2017) 2056–2063.
  * (25) B. Ward-Cherrier, L. Cramphorn, N. F. Lepora, Tactile manipulation with a tacthumb integrated on the open-hand m2 gripper, IEEE Robotics and Automation Letters 1 (1) (2016) 169–175.
  * (26) A. Yamaguchi, C. G. Atkeson, Combining finger vision and optical tactile sensing: Reducing and handling errors while cutting vegetables, in: IEEE-RAS International Conference on Humanoid Robots, 2016, pp. 1045–1051.
  * (27) X. Lin, M. Wiertlewski, Sensing the Frictional State of a Robotic Skin via Subtractive Color Mixing, IEEE Robotics and Automation Letters 4 (3) (2019) 2386–2392.
  * (28) C. Sferrazza, R. D’Andrea, Design, motivation and evaluation of a full-resolution optical tactile sensor, Sensors 19 (4) (2019).
  * (29) C. Trueeb, C. Sferrazza, R. D’Andrea, Towards vision-based robotic skins: a data-driven, multi-camera tactile sensor, in: 2020 3rd IEEE International Conference on Soft Robotics (RoboSoft), 2020, pp. 333–338.
  * (30) B. Winstone, C. Melhuish, T. Pipe, M. Callaway, S. Dogramadzi, Toward bio-inspired tactile sensing capsule endoscopy for detection of submucosal tumors, IEEE Sensors Journal 17 (3) (2017) 848–857.
  * (31) R. Li, R. Platt Jr, W. Yuan, A. Pas, N. Roscup, M. A. Srinivasan, E. H. Adelson, Localization and Manipulation of Small Parts Using GelSight Tactile Sensing, IEEE International Conference on Intelligent Robots and Systems (2014).
  * (32) S. Luo, W. Yuan, E. Adelson, A. G. Cohn, R. Fuentes, Vitac: Feature sharing between vision and tactile sensing for cloth texture recognition, in: IEEE International Conference on Robotics and Automation (ICRA), IEEE, 2018, pp. 2722–2727.
  * (33) J.-T. Lee, D. Bollegala, S. Luo, “touching to see” and “seeing to feel”: Robotic cross-modal sensory data generation for visual-tactile perception, in: 2019 International Conference on Robotics and Automation (ICRA), IEEE, 2019, pp. 4276–4282.
  * (34) G. Cao, Y. Zhou, D. Bollegala, S. Luo, Spatio-temporal attention model for tactile texture recognition, 2020.
  * (35) C. Lu, J. Wang, S. Luo, Surface following using deep reinforcement learning and a gelsighttactile sensor, arXiv preprint arXiv:1912.00745 (2019).
  * (36) J. Jiang, G. Cao, D. F. Gomes, S. Luo, Vision-guided active tactile perception for crack detection and reconstruction, in: The 29th Mediterranean Conference on Control and Automation (MED 2021), 2021.
  * (37) D. F. Gomes, A. Wilson, S. Luo, Gelsight simulation for sim2real learning, in: ICRA ViTac Workshop, 2019.
  * (38) D. F. Gomes, P. Paoletti, S. Luo, Generation of gelsight tactile images for sim2real learning, IEEE Robotics and Automation Letters 6 (2) (2021) 4177–4184.
  * (39) E. Donlon, S. Dong, M. Liu, J. Li, E. Adelson, A. Rodriguez, Gelslim: A high-resolution, compact, robust, and calibrated tactile-sensing finger, in: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, 2018, pp. 1927–1934.
  * (40) M. Lambeta, P.-W. Chou, S. Tian, B. Yang, B. Maloon, V. R. Most, D. Stroud, R. Santos, A. Byagowi, G. Kammerer, et al., Digit: A novel design for a low-cost compact high-resolution tactile sensor with application to in-hand manipulation, IEEE Robotics and Automation Letters 5 (3) (2020) 3838–3845.
  * (41) I. Taylor, S. Dong, A. Rodriguez, Gelslim3. 0: High-resolution measurement of shape, force and slip in a compact tactile-sensing finger, arXiv preprint arXiv:2103.12269 (2021).
  * (42) S. Dong, D. Ma, E. Donlon, A. Rodriguez, Maintaining grasps within slipping bounds by monitoring incipient slip, in: IEEE International Conference on Robotics and Automation (ICRA), 2019, pp. 3818–3824.
  * (43) R. Calandra, A. Owens, D. Jayaraman, J. Lin, W. Yuan, J. Malik, E. H. Adelson, S. Levine, More than a feeling: Learning to grasp and regrasp using vision and touch, IEEE Robotics and Automation Letters 3 (4) (2018) 3300–3307.
  * (44) B. Romero, F. Veiga, E. Adelson, Soft, Round, High Resolution Tactile Fingertip Sensors for Dexterous Robotic Manipulation, in: IEEE International Conference on Robotics and Automation, 2020.
  * (45) A. Padmanabha, F. Ebert, S. Tian, R. Calandra, C. Finn, S. Levine, Omnitact: A multi-directional high resolution touch sensor, in: IEEE International Conference on Robotics and Automation, 2020.
  * (46) G. Cao, J. Jiang, C. Lu, D. F. Gomes, S. Luo, Touchroller: A rolling optical tactile sensor for rapid assessment of large surfaces, arXiv preprint arXiv:2103.00595 (2021).
  * (47) A. Wilson, S. Wang, B. Romero, E. Adelson, Design of a fully actuated robotic hand with multiple gelsight tactile sensors, arXiv preprint arXiv:2002.02474 (2020).
  * (48) Y. She, S. Wang, S. Dong, N. Sunil, A. Rodriguez, E. Adelson, Cable manipulation with a tactile-reactive gripper, arXiv preprint arXiv:1910.02860 (2019).
  * (49) R. Szeliski, Computer vision algorithms and applications, Springer Science & Business Media, 2010.


## The system of the vortex-like structures:
the viewpoint on a turbulence modelling

S.V. Talalov

Dept. of Applied Mathematics, Togliatti State University,
14 Belorusskaya str., Tolyatti, 445020 Russia.
<EMAIL_ADDRESS>
###### Abstract

In this study we suggest new approach to turbulence modeling. To develop this
approach, we construct the set of the vortex-like dynamical systems evolving
in the space $E_{3}$. These systems are constructed using the AKNS hierarchy
so that they generalize thin closed vortex filament described in the local
induction approximation. We construct and investigate the Hamiltonian dynamics
of these systems in terms of the non - standard variables. We formulate the
principles of quantization these systems focusing on the quantum description
of the filaments with linearized dynamics. To describe the energy of vortex-
like structures, we apply a new method based on the use of the central
extended Galilei group. In a special case, we calculate the energy levels of a
system of quantized vortices. The proposed approach makes it possible to
describe both the interaction of such vortex - like filaments and the
processes of their creation and annihilation in a natural way. As a result, we
have constructed a quantum turbulent flow model that demonstrates the
relationship between stochastic and integrable dynamics.

keywords: vortex filament quantization; AKNS hierarchy; extended Galilei
group; turbulence modelling.

###

##  1 Introduction

The turbulence, both classical and quantum, continues to be a phenomenon that
is not fully understood. A common approach to describing this complex
phenomenon is to consider a turbulent flow as a tangle of vortex structures. A
large number of works are devoted to this issue. For the first time, the
description of a turbulent medium as a tangle of vortices was proposed in
Feynman’s work [1]. The achievements in this matter, which followed over
several decades, were summarized, for example, in the books [2, 3]. Despite
many years of studying turbulence, this complex phenomenon is still of
interest to researchers. Indeed, ”…the understanding of turbulent flows is one
of the biggest current challenges in physics” [4]. Quantum effects add
complexity to the study of this problem. So, the differences between the
classical and quantum turbulence were considered in the paper [5]. Along with
the theoretical understanding of the problem, numerical studies were also
carried out [6] (see also [7, 8] for resent results here).

As usual, the theoretical study of the phenomenon involves building some kind
of model. The Gross-Pitaevskii model is the most used theory in modeling
quantum turbulence. So, one of the predictions of this theory is the values of
quantum circulation $\Gamma_{n}\propto\hbar n$, where the numbers $n$ are
integers. The study of the statistics of the distribution of the $\Gamma_{n}$
value in a turbulent flow was investigated in work [9] by numerical
simulation. The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model is also widely
used for numerically study quantum turbulence. For example, this approach was
used for this purpose in the paper [11].

Currently, it is considered an established fact that turbulent flow is
characterized by a wide range of length and time scales. This question was
discussed, for example, in the paper [10]. Topological and geometrical aspects
of a quantum turbulent flow was studied in the work [12]. More complicated
problems were also considered: quantum turbulence in a channel with an
counterflow of superfluid turbulent helium was studied in the paper [13].

Despite the large number of works in the field of quantum turbulence, this
phenomenon cannot be considered fully investigated. In this study, the author
develops new approach to the turbulent flow modeling. This approach
demonstrates the relationship between the regular and stochastic dynamics of
vortex-like filaments in a tangle. Moreover, it makes it possible in principle
to calculate a partition function for a turbulent flow in certain cases.

##  2 The hierarchy of the vortex-like dynamical systems

Nevertheless, the following question would be appropriate in a framework of
any approach: how to describe a single vortex in a tangle? Corresponding
description will be necessary, for example, when constructing a quantum theory
of such complex structures. Approaches that have been developed to describe of
a single thin vortex filament ${\boldsymbol{r}}(t,s)$ are well known. One of
the most common ways of such a description is to use the Local Induction
Approximation. Corresponding equation (LIE) is written as follows::

$\frac{\partial{\boldsymbol{r}}({t^{\ast}},s)}{\partial{t^{\ast}}}~{}=~{}A\,\frac{\partial{\boldsymbol{r}}({t^{\ast}},s)}{\partial
s}\times\frac{\partial^{\,2}{\boldsymbol{r}}({t^{\ast}},s)}{\partial
s^{2}}\,,$ (1)

We use notation $s$ for the natural curve parameter and notation
$t^{\ast}={t\Gamma}/{4\pi}$ for the ”time” parameter. Symbols $\Gamma$ and $t$
mean the circulation and real time correspondingly. After certain assumptions
[14], a vortex filament with a nonzero flow inside the core111We assume that
the core radius is small but non-zero. is described by the equation:

$\displaystyle\frac{\partial{\boldsymbol{r}}({t^{\ast}},s)}{\partial{t^{\ast}}}$
$\displaystyle=$ $\displaystyle
A\,\frac{\partial{\boldsymbol{r}}({t^{\ast}},s)}{\partial
s}\times\frac{\partial^{\,2}{\boldsymbol{r}}({t^{\ast}},s)}{\partial s^{2}}+$
$\displaystyle+$ $\displaystyle
B\left(\frac{\partial^{\,3}{\boldsymbol{r}}({t^{\ast}},s)}{\partial
s^{3}}+\frac{3}{2}\,\biggl{|}\,\frac{\partial^{\,2}{\boldsymbol{r}}({t^{\ast}},s)}{\partial
s^{2}}\biggr{|}^{\,2}\frac{\partial{\boldsymbol{r}}({t^{\ast}},s)}{\partial
s}\right)\,.$ (2)

Of course, the application of equations (1) and (2) to describe individual
vortex filaments in a tangle is impossible. Indeed, the influence of other
vortices in the vortex-tangle cannot be ignored here. On the other hand, it is
also impossible to take into account the real physical influence a lot of
nearby random vortices on a particular filament in a turbulent flow. Most
probably, this problem is no less difficult than solving the Navier-Stokes
equations in a general case. In this regard, the following question arises:
can we modify equations (1) and (2) so that the dynamics of the filament is
described by a sufficiently large number of parameters? In this case, the
influence of neighboring vortices can be taken into account indirectly,
through one or another distribution of these parameters. With this approach,
we save the opportunity to describe the dynamics (classical and quantum) of a
separate vortex filament.

The purpose of our study is as follows. We develop a theory that can be
considered as a mathematical model of a tangle of vortex-like filaments. This
model will include the following items:

  1. 1.

We construct the dynamical theory of the separate vortex-like structure in
Hamiltonian form. This theory generalizes dynamics (1) and (2) in some way. To
solve this problem, we define the Hamiltonian structure of such systems in
terms of non-standard fundamental variables. So, these variables are divided
into two groups: internal and external ones;

  2. 2.

To quantize these systems, we linearize the dynamics of considered objects. We
will also discuss the relevance of this procedure in our model;

  3. 3.

We construct the quantum theory of introduced vortex-like filaments. The
constructed model makes it possible to describe the interaction of the vortex-
like structures at quantum level by means of the many-body theory methods;

  4. 4.

In a sense, the theory combines elements of integrability and stochasticity in
the behavior of the objects under consideration.

For our further studies,, we use a representation for smooth closed filament
of length $S$ in the form

${\boldsymbol{r}}(\tau,\xi)~{}=~{}{\boldsymbol{q}}+{R}\int\limits_{0}^{2\pi}\left[\,{\xi-\eta}\,\right]{\boldsymbol{j}}(\tau,\eta)d\eta\,,\qquad
R=\frac{S}{2\pi}\,,\qquad\xi=\frac{s}{R}\,,$ (3)

where dimensionless evolution parameter $\tau={t\Gamma}/{4\pi R^{2}}$ was
introduced. The vector ${\boldsymbol{q}}={\boldsymbol{q}}(\tau)$ defines the
position of the filament in the coordinate system (for example, the center of
the circular vortex ring). The notation $[\,x\,]$ means the integer part of
the number $x/{2\pi}$:

$[\,0\,]=0\,,\qquad[\,x+2\pi\,]=[\,x\,]+1\,,\qquad\forall\,x\,.$ (4)

Vector ${\boldsymbol{j}}(\tau,\eta)$ is the unit tangent (affine) vector for
this filament. Let the vector ${\boldsymbol{r}}(\tau,\xi)$ satisfies equation
(1). Then, the function ${\boldsymbol{j}}(\tau,\eta)$ satisfies the equation
for continuous Heisenberg spin chain:

$\partial_{\tau}{\boldsymbol{j}}(\tau,\xi)~{}=~{}\alpha{\boldsymbol{j}}(\tau,\xi)\times\partial_{\xi}^{\,2}{\boldsymbol{j}}(\tau,\xi)\,.$
(5)

It is well known that this equation is gauge equivalent to the non-linear
Schrödinger equation [15]. The connection between the LIE and the nonlinear
Schrodinger equation was firstly established in the work [17]. This
equivalence makes it possible to construct new dynamical systems in space
$E_{3}$ that generalize dynamical systems (1) and (2). Despite the formal
definition, such systems, according to the author, open up a new point of view
on a simulation of vortex filaments behavior in a vortex tangle.

First, we will consider the Hamiltonian description of systems (1) and (2) in
terms of non-standard variables. Such a description was proposed in the
authors’ work [18]. The main points of the suggested approach are following.

  * •

We include the circulation $\Gamma$ as an additional independent variable in
our theory. Because this postulate, we take into account the movement of the
surrounding fluid. Indeed, the equations (1) and (2) describe some formal
dynamics of the curve ${\boldsymbol{r}}(\tau,\xi)$ only. Therefore, we
describe the dynamics of a filament by the following set ${\cal A}$ of the
independent variables:

${\cal A}~{}=~{}\\{\,\Gamma\,,\boldsymbol{q}\,,\boldsymbol{j}(\xi)\,\\}\,,$

  * •

We replace the set ${\cal A}$ by the set ${\cal A}^{\prime}$, where

${\cal
A}^{\,\prime}~{}=~{}\bigl{\\{}\,{\boldsymbol{p}}\,,{\boldsymbol{q}}\,;{\boldsymbol{j}}(\xi)\,\bigr{\\}}\,.$
(6)

The momentum ${\boldsymbol{p}}$ is connected with variables $\Gamma$ and
$\boldsymbol{j}(\xi)$ by means of standard hydrodynamical formula

${\boldsymbol{p}}~{}=~{}\frac{\varrho_{0}}{2}\,\int\,\boldsymbol{r}\times\boldsymbol{w}(\boldsymbol{r})dV\,,$
(7)

where $\varrho_{0}$ is a fluid density and vector-function
$\boldsymbol{w}(\boldsymbol{r})$ means the vorticity of the vortex filament.
This function is calculated as

${\boldsymbol{w}}(\boldsymbol{r})~{}=~{}\Gamma\int\limits_{0}^{2\pi}\,\hat{\delta}(\boldsymbol{r}-\boldsymbol{r}(\xi))\partial_{\xi}{\boldsymbol{r}}(\xi)d\xi\,.$
(8)

  * •

The appearance of the momentum components $p_{i}$ as independent variables
makes it possible to define the Galilei boosts in a natural way. As a
consequence, we assume that the central extended Galilei group222We use one
parameter central extension of this group. The mass constant $m_{0}$ (central
charge) is the parameter of this extension.  $\tilde{\mathcal{G}}_{3}$ is the
space-time symmetry group in our model. Therefore, we apply the group-
theoretical approach to the definition of the energy of thin vortex filament.

  * •

The structure of Hamiltonian is motivated by the expressions for Cazimir
functions of Lie algebra of the group $\tilde{\mathcal{G}}_{3}$:

${H}_{0}(p_{1},p_{2},p_{3}\,;{\boldsymbol{j}})~{}=~{}\frac{1}{2m_{0}}\sum_{i=1}^{3}p_{i}^{\,2}+H_{int}[\,{\boldsymbol{j}};\alpha_{1},\dots,\alpha_{n}\,]\,.$
(9)

The term $H_{int}[\,{\boldsymbol{j}};\alpha_{1},\dots,\alpha_{n}\,]$ provides
dynamics of the variables ${\boldsymbol{j}}(\xi)$ that depends on some
parameters $\alpha_{1},\dots,\alpha_{n}$.

The formulas (7) and (8) are written in a compact form:

${\boldsymbol{p}}~{}=~{}\varrho_{0}{R}^{2}\Gamma{\boldsymbol{f}}\,,\qquad{\boldsymbol{f}}~{}=~{}\frac{1}{2}\iint\limits_{0}^{2\pi}\left[\,{\xi-\eta}\,\right]\,{\boldsymbol{j}}(\eta)\times{\boldsymbol{j}}(\xi)d\xi
d\eta\,.$ (10)

Please note that the replacement $\Gamma\to{\boldsymbol{p}}$ leads to the
certain constraints on the set ${\cal A}^{\,\prime}$ in general. We will not
discuss this issue here [26].

The theory has three natural dimensional constants that are relevant to the
physical system being described. These constants are: the fluid’s density
$\varrho_{0}$, the speed of sound in this fluid $v_{0}$ and the mass constant
$m_{0}$. In order to simplify some formulas, we will use the auxiliary
constants333Note that there may be other natural dimensional constants in
theory: for example, the radius $R_{1}$ of the pipe in which the vortex tangle
in question moves. $R_{0}=\sqrt[3\,]{m_{0}/\varrho_{0}}$,
$~{}t_{0}=R_{0}/v_{0}$ and ${\cal E}_{0}=m_{0}v_{0}^{2}$ along with constants
$\varrho_{0}$, $v_{0}$ and $m_{0}$.

The Poisson bracket structure was postulated as following:

$\displaystyle\\{p_{i}\,,q_{j}\\}$ $\displaystyle=$
$\displaystyle\,\delta_{ij}\,,\qquad i,j=1,2,3\,,$
$\displaystyle\\{j_{a}(\xi),j_{b}(\eta)\\}$ $\displaystyle=$
$\displaystyle-(2/R_{0}m_{0}v_{0})\,\epsilon_{abc}j_{c}(\xi)\hat{\delta}(\xi-\eta)\,,\qquad\epsilon_{123}=1\,.$
(11)

All other brackets vanish. In accordance with the definition (11), the
function ${\boldsymbol{j}}^{\,2}(\eta)$ annulates the brackets of the
fundamental variables. Thus, the constant ${\boldsymbol{j}}^{\,2}(\eta)$
numbers the symplectic sheets in the phase space of our dynamical system.
Next, we consider a single symplectic sheet with ${\boldsymbol{j}}^{\,2}=1$.

To develop our approach it let us note that the equation (5) lead to the
infinitely series of the integrals of the motion
$I_{n}=I_{n}[\,{\boldsymbol{j}}\,]$, $n=0,1,2,\dots$ [15]. These integrals
generate Hamiltonian flows that lead to the hierarhy of the non-linear
integrable differential equations, which is known as hierarhy AKNS [16]. So,
we use the Hamiltonian

$H_{int}={\mathcal{E}}_{0}\int_{0}^{2\pi}(\partial_{\xi}{\boldsymbol{j}}(\xi))^{2}d\xi=4{\mathcal{E}}_{0}I_{1}$
(12)

to generate the dynamics in accordance with equation (5). As for the full
Hamiltonian (9), it provides the dynamics of the filament according to the
equation (1).

Let’s now add to expression (12) the term that is proportional to the integral
of motion $I_{2}$. Then the resulting full Hamiltonian

$\displaystyle~{}~{}{H}_{0}(p_{1},p_{2},p_{3}\,;{\boldsymbol{j}})~{}=~{}\frac{1}{2m_{0}}\sum_{i=1}^{3}p_{i}^{\,2}~{}+$
$\displaystyle+$
$\displaystyle{{\mathcal{E}}_{0}}\,\left(\alpha_{1}\int\limits_{0}^{2\pi}\bigl{(}\partial_{\xi}{\boldsymbol{j}}(\xi)\bigr{)}^{2}d\xi+\alpha_{2}\int\limits_{0}^{2\pi}\partial_{\xi}{\boldsymbol{j}}(\xi)\bigl{(}{\boldsymbol{j}}(\xi)\,\times\,\partial^{\,2}_{\xi}{\boldsymbol{j}}(\xi)\bigr{)}d\xi\right)$
(13)

will provide the dynamics according to the equation (2).

Considered Hamiltonians $H_{int}$ have a natural geometrical interpretation.
Indeed, taking into account Frenet-Serret equations for the curve (3), we can
find that following simple formula holds:

$H_{int}[{\boldsymbol{j}};\alpha_{1},\alpha_{2}]={{\mathcal{E}}_{0}}\,\left(\alpha_{1}\int\limits_{0}^{2\pi}\bigl{(}k(\xi)\bigr{)}^{2}d\xi+\alpha_{2}\int\limits_{0}^{2\pi}\bigl{(}k(\xi)\bigr{)}^{2}\varkappa(\xi)d\xi\right)\,.$

The function $k(\xi)$ is the curvature of the curve
${\boldsymbol{r}}(\cdot,\xi)$ here and the function $\varkappa(\xi)$ is the
torsion of this curve.

The structure of the Hamiltonian (2) makes it natural to use AKNS hierarchy to
generalize the dynamic system (2). Let the number ${\sf N}>0$ is the natural
number. To describe the hierarchy of the vortex-like filaments, we consider
the Hamiltonian (9), where

$H_{int}[\,{\boldsymbol{j}};\alpha_{1},\alpha_{2}\,]~{}\to~{}H^{\varkappa}_{int}[\,{\boldsymbol{j}};\,\alpha_{1},\dots,\alpha_{\sf
N}\,]\,,$

where

$H^{\varkappa}_{int}[\,{\boldsymbol{j}};\alpha_{1},\dots,\alpha_{\sf
N}\,]~{}=~{}4{\mathcal{E}}_{0}\sum_{k=1}^{\sf
N}\alpha_{k}I_{k}[\,{\boldsymbol{j}}\,]\,,\qquad\alpha_{j}\in{\sf R}\,.$ (14)

The certain number of constants $\alpha_{k}$ may be zero. The fact that
$\alpha_{k_{1}}=0$ for some index $k_{1}$ means that the corresponding
integral of motion $I_{k_{1}}[\,{\boldsymbol{j}}\,]$ does not affect the
dynamics of the filament. To account for this effect, we define the real
number $\varkappa\in[0,1]$ that is written in binary form as

$\varkappa~{}=~{}0,b_{{\sf
N}-1}b_{N-2}\dotsb_{1}b_{0}\,0\,0\,0\,0\,0\,0\,0\dots\,.$ (15)

The numbers $b_{k-1}$ are mapped to constants $\alpha_{k}$ according to the
following rule: $b_{k-1}=0$ for the case $\alpha_{k}=0$ and $b_{k-1}=1$
otherwise . The number $\varkappa$ can take $2^{\sf N}$ values. This number
determines the type of vortex-like filament evolution of concrete vortex loop
in a vortex tangle.

Therefore, we have the hierarchy of the dynamical systems in framework of
developed theory. These systems generalize the vortex systems (1) and (2)
naturally. In general, the dynamics of the vortex-like filament (3) is
described by ${\sf N}+1$-order non-linear differential equation:

$\partial_{\tau}{\boldsymbol{r}}~{}=~{}\\{H^{\varkappa}_{int}[\,{\boldsymbol{j}};\alpha_{1},\dots,\alpha_{\sf
N}\,]\,,{\boldsymbol{r}}\\}\,.$ (16)

In conclusion of this section, we will say a few words about the geometric
interpretation of such dynamical systems. First, the functions $P_{n}(\xi)$ in
the integrals $I_{n}=\int_{0}^{2\pi}P_{n}(\xi)d\xi$ are polynomials of the
solution $\psi(\xi)$ of the non-linear Schrödinger equation and corresponding
derivatives $\partial^{n}\psi(\xi)$. Second, the function $\psi(\xi)$ can be
reconstructed on the curvature $k(\xi)$ and the torsion $\varkappa(\xi)$ in
accordance with Hasimoto results [17]. Therefore, the Hamiltonians
$H^{\varkappa}_{int}[\,{\boldsymbol{j}};\alpha_{1},\dots,\alpha_{\sf N}\,]$
will be functions of quantities $k(\xi)$ and $\varkappa(\xi)$.

##  3 Quantization

The problem of quantum description of vortices has a very long history. The
author will not take the responsibility to give a full review of the
literature on this issue here. In most of the works that have been cited
above, this problem was also discussed in some way. In addition, it is
necessary to mention the book [19], which also has a detailed review of the
literature on the problem. The quantum Hasimoto transformation LIE $\to$ non-
linear Schrödinger equation was studied in the work [20] recently.

We emphasize that the general dynamics of the vortex filaments is described by
nonlinear equations, even for simplified case of the local induction
approximation. The quantization of the corresponding non-linear integrable
models is complex problem (see, for example, [21, 22]). According to our
initial assumptions, the discussed vortex-like dynamical systems form some
”entangled structure” that models the turbulent flow. The lifetime of vortices
in a turbulent flow is small; this value restricted by the period $T=2\pi/W$,
where symbol $W$ means the vortisity [23]. Thus, it seems appropriate to
consider the quantization of small perturbations of vortex loops. Such
perturbations are described by the linear equations at corresponding
approximation. Therefore, the quantization of these perturbations can be
performed using standard methods. The main question here is follows: what
fundamental variables must parametrize the phase space of the dynamical
systems under consideration? It is well known that quantization of variables
that are canonically equivalent in a classical theory, leads to different
results at the quantum level444 For example, the quantization of such a simple
canonical system as harmonic oscillator in terms of action-angle variables
demonstrates many unexpected and interesting results [24].. Despite more than
a century of development of quantum theory, the following Dirac’s words are
still relevant: ”…methods of quantization are all of the nature of practical
rules, whose application depends on consideration of simplicity” [25].

The quantization of small oscillations of the vortex ring was suggested in the
authors’ article [26] in terms of non-standard Hamiltonian variables (6). As
special result, the set of acceptable values of quantized circulation
$\Gamma_{[n]}$ was calculated within the framework of this approach [27]. As
it turns out, this set is wider than the standard one $\Gamma_{n}\propto n$,
where $n$ is natural number. The developed theory allows us to interpret a
quantized vortex as a certain particle with an internal structure.
Consequently, we can apply the methods of the theory of many bodies to
describe the processes of the interaction, creation and annihilation of the
vortices [28].

Let us make the overview of our quantization scheme. First, we consider the
single closed vortex-like filament that has some fixed shape. This filamet is
homotopically equivalent to a circle and is characterized by a certain tangent
vector ${\boldsymbol{j}}_{0}(\eta)={\boldsymbol{j}}_{0}(\tau,\eta)$:

${\boldsymbol{j}}_{0}(\eta+2\pi)={\boldsymbol{j}}_{0}(\eta)\,,\qquad{\boldsymbol{j}}_{0}^{2}(\eta)=1\,.$

We consider the small perturbation of this tangent vector:

$\boldsymbol{j}_{0}(\tau,\xi)~{}\to~{}{\boldsymbol{j}_{0}}(\xi)+\varepsilon{\boldsymbol{j}}_{p}(\tau,\xi)\,,\qquad\varepsilon<<1\,.$
(17)

Additionally, we suppose that these excitations be transverse:

${\boldsymbol{j}}_{p}(\tau,\xi){\boldsymbol{j}_{0}}(\xi)~{}\equiv~{}0\,$ (18)

The small perturbation of the circular vortex filament without the restriction
(18) has been studied in the work [29]. Various aspects of the theory related
to vortex ring oscillations have also been studied in the works [30, 31].
Besides that, the small perturbations of straight vortex filaments were
studied in the work [32].

The condition (18) leads to decomposition

${\boldsymbol{j}}_{p}(\tau,\xi)~{}=~{}j_{\rho}(\tau,\xi){\boldsymbol{e}}_{\rho}+j_{z}(\tau,\xi){\boldsymbol{e}_{z}}\,,$
(19)

where the vectors
$\\{{\boldsymbol{e}_{\rho}}\,,{\boldsymbol{j}_{0}}\,,{\boldsymbol{e}_{z}}\,\\}$
denote the local orthonormal basis in the corresponding point of the loop. For
convenience, we define the complex-valued function

${\mathfrak{J}}(\tau,\xi)~{}=~{}j_{\rho}(\tau,\xi)+{\rm i}j_{z}(\tau,\xi)\,.$

This function allows us define the complex amplitudes
${\mathfrak{j}}_{\,n}(\tau)$:

${\mathfrak{J}}(\tau,\xi)~{}=~{}\sum_{n}{\mathfrak{j}}_{\,n}(\tau)\,e^{\,i\,\,n\xi}\,.$
(20)

For example, in the simplest case (1) amplitude ${\mathfrak{J}}(\tau,\xi)$
satisfies the linear equation ($A=1$) [26]

$\partial_{\tau}{\mathfrak{J}}=-i\partial_{\xi}^{\,2}{\mathfrak{J}}-\frac{i}{2}\Bigl{(}{\mathfrak{J}}-\overline{\,{\mathfrak{J}}\,}\,\Bigr{)}\,.$
(21)

Consequently,

${\mathfrak{J}}(\tau,\xi)~{}=~{}\sum_{n}{\mathfrak{j}}_{\,n}\,e^{\,i\,[\,n\xi+n\sqrt{n^{2}-1}\,\tau\,]}\,,$
(22)

where the amplitudes ${\mathfrak{j}}_{\,n}$ satisfy the condition
$\overline{\,{\mathfrak{j}}\,}_{\,-n}~{}=~{}2\left[n\sqrt{n^{2}-1}-n^{2}+\frac{1}{2}\right]{\mathfrak{j}}_{\,n}\,.$

Dynamics

${\mathfrak{j}}_{\,-n}~{}\to~{}{\mathfrak{j}}_{\,-n}(\tau)~{}=~{}{\mathfrak{j}}_{\,-n}e^{\,i\,n\sqrt{n^{2}-1}\,\tau\,}$

is generated (for the conditional time $t=t_{0}\tau$) by Hamiltonian

${H}_{0}(p_{1},p_{2},p_{3}\,;{\sf
j})~{}=~{}\frac{{\boldsymbol{p}}^{\,2}}{2m_{0}}~{}+~{}{\mathcal{E}}_{0}\sum_{n>1}|\,{\sf
j}_{\,-n}|^{2}n\sqrt{n^{2}-1}\,$ (23)

and Poisson brackets

$\displaystyle\\{p_{i}\,,q_{j}\\}$ $\displaystyle=$
$\displaystyle\delta_{ij}\,,\qquad i,j=x,y,z\,,$ $\displaystyle\\{{\sf
j}_{\,m},\overline{\,\sf j\,}_{\,n}\\}$ $\displaystyle=$
$\displaystyle(i/m_{0}R_{0}v_{0})\,\delta_{mn}\,,\qquad m,n=-1,-2,\dots$ (24)

In the general case (14), we have more complex but also linear equations for
the amplitudes ${\mathfrak{J}}(\tau,\xi)$.

We assume that the non-exited filament $\boldsymbol{r}_{0}(\tau,\xi)$ (see Eq.
(3), where ${\boldsymbol{j}}(\xi)\equiv{\boldsymbol{j}_{0}}(\xi)$), is
described by the finite numbers of additional555to the variables
${\boldsymbol{j}}_{p}(\tau,\xi)$ dynamical variables, except for ”external”
variables ${\boldsymbol{p}}$ and ${\boldsymbol{q}}$. To simplify our
subsequent studies, we consider the non-exited filament as a ring of radius
$R$. In case of the Eq. (2), the following formula for the filament
$\boldsymbol{r}_{0}(\tau,\xi)$ is fulfilled:

$\boldsymbol{r}_{0}(\tau,\xi)~{}=~{}\boldsymbol{q}+R\bigl{(}\,\cos(\xi+\phi_{0}+\beta\tau)\boldsymbol{e}_{x}\,,~{}\sin(\xi+\phi_{0}+\beta\tau)\boldsymbol{e}_{y}\,,~{}A\tau\boldsymbol{e}_{z}\,\bigr{)}\,,$
(25)

where the angle $\phi_{0}\in[0,2\pi)$ and $\beta=B/R$. Therefore, the
variables

$\Delta R~{}=~{}\sqrt{R^{2}-R_{0}^{2}}\,,\qquad\phi(\tau)=\phi_{0}+\beta\tau$

can be considered as the additional variables. Because the inequality $R\geq
R_{0}$ is fulfilled, the constant $R_{0}$ can characterize the average size of
a molecular cluster in a flow.

To simplify our consideration, we consider here the quantization of the
variables ${\boldsymbol{p}}$, ${\boldsymbol{q}}$ and
${\boldsymbol{j}}_{p}(\tau,\xi)$ only. As regards of the quantization of the
additional variables that define the filament $\boldsymbol{r}_{0}(\tau,\xi)$,
this procedure was fulfilled in the articles [28, 33] in simplest case (25).

The constructed set of the classical dynamical variables defines the
quantization scheme uniquely. The quantum states of a single vortex loop are
the vectors of the Hilbert space

$\boldsymbol{H}_{1}~{}=~{}\boldsymbol{H}_{pq}\otimes\boldsymbol{H}_{j}\,,$
(26)

where the symbol $\boldsymbol{H}_{pq}$ denotes the Hilbert space of a free
structureless particle in $3D$ space ${\sf R}_{3}$. As usually, we suppose
that $\boldsymbol{H}_{pq}=L^{2}({\sf R}_{3})$. The symbol $\boldsymbol{H}_{j}$
denotes the space for the number of the harmonic oscillators (infinite in
general) which correspond to the variables ${\mathfrak{j}}_{\,-n}$. In order
to simplify the following formulas, we assume that a finite number $K$ of
modes ${\mathfrak{j}}_{-k}$ can be excited only for every vortex filament.
From the author’s point of view, the proposed quantization scheme is a mirror
image of Lord Kelvin’s old idea that particles can be considered as some kind
of vortex-like structures [34]. In our approach, we describe the closed
vortex-like filaments as structured particles.

As it is mentioned above, our quantization scheme make it possible to describe
the interaction of vortex-like structures in terms of the many-body theory.
The details can be found in the articles [28, 33] for certain special case.
Here we will go the brief overview only.

  * •

The space of a quantum states of a ”vortex tangle” is constructed as the Fock
space

${\mathfrak{H}}~{}=~{}\bigoplus_{M=0}^{\infty}\boldsymbol{H}_{M}\,,$

where

$\boldsymbol{H}_{M}~{}=~{}\bigotimes_{i=0}^{M}\boldsymbol{H}_{1}~{}=~{}\underbrace{\,\bigl{[}\boldsymbol{H}_{pq}\otimes\boldsymbol{H}_{j}\bigr{]}\otimes\dots\otimes\bigl{[}\boldsymbol{H}_{pq}\otimes\boldsymbol{H}_{j}\bigr{]}\,}_{M}=\boldsymbol{H}_{pq}^{M}\otimes\boldsymbol{H}_{j}^{M}\,.$
(27)

Any vector $|\Phi^{M}\rangle\in\boldsymbol{H}_{M}$ has a form

$\displaystyle|\Phi^{M}\rangle$ $\displaystyle=$
$\displaystyle\sum_{n_{1},\dots,n_{M}}\int\,\cdots\int
d\boldsymbol{p}_{1}\dots\boldsymbol{p}_{N}\,\times$ $\displaystyle\times$
$\displaystyle\,f^{M}_{{\mathfrak{n}}_{1},\dots,{\mathfrak{n}}_{M}}(\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{M})\,|\boldsymbol{p}_{1}\rangle\dots|\boldsymbol{p}_{M}\rangle|{\mathfrak{n}}_{1}\rangle\dots|{\mathfrak{n}}_{M}\rangle\,,$
(28)

where the functions $f^{M}$ are symmetrical both indexes ${\mathfrak{n}}$ and
arguments $\boldsymbol{p_{\ell}}$. The vector $|\boldsymbol{p}_{\ell}\rangle$
are eigenvector of the operator $\hat{\boldsymbol{p}}_{\ell}$ that corresponds
to the filament with index ${}^{\prime\prime}{\ell}\,^{\prime\prime}$. Here
and further, we use the summation index
${}^{\prime\prime}{\ell}\,^{\prime\prime}$ to number the vortex filaments
only. As regards of the ”numbers” ${\mathfrak{n}}_{\ell}$, these symbols are
reduced form for occupation numbers $n_{k}({\ell})$:

${\mathfrak{n}}_{\ell}~{}=~{}[\,n_{1}(\ell)\,\dots,n_{K}(\ell)\,]\,.$

The occupation number $n_{k}(\ell)$ describes the excitation of the variable
${\mathfrak{j}}_{-k}$ (see Eq. (20)) for the filament with the index
${}^{\prime\prime}\ell\,^{\prime\prime}$.

  * •

Hamiltonian

$\hat{\mathcal{H}}~{}=~{}\hat{\mathcal{H}}_{0}+\hat{U}\,,$

where operator $\hat{\mathcal{H}}_{0}$ means the Hamiltonian for the system of
the non - interacting vortex-like structures.

To write down these operators, let’s first introduce notation for some
operators in the Fock space ${\mathfrak{H}}$:

  1. 1.

The creation and annihilation operators
${\hat{a}}^{+}(\boldsymbol{p},{\mathfrak{n}})$,
${\hat{a}}(\boldsymbol{p},{\mathfrak{n}})$ act in the Fock space
${\mathfrak{H}}$. They are defined in a standard way:

$\Bigl{(}{\hat{a}}(\boldsymbol{p},{\mathfrak{n}})f^{M}_{{\mathfrak{n}}_{1},\dots,{\mathfrak{n}}_{M}}\Bigr{)}_{{\mathfrak{n}}_{1},\dots,{\mathfrak{n}}_{M-1}}(\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{M-1})~{}=~{}\sqrt{M}f^{M}_{{\mathfrak{n}}_{1},\dots,{\mathfrak{n}}_{M-1},{\mathfrak{n}}}(\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{M-1},\boldsymbol{p})\,,$
$\Bigl{(}{\hat{a}}^{+}(\boldsymbol{p},{\mathfrak{n}})f^{M}_{{\mathfrak{n}}_{1},\dots,{\mathfrak{n}}_{M}}\Bigr{)}_{{\mathfrak{n}}_{1},\dots,{\mathfrak{n}}_{M+1}}(\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{M+1})~{}=~{}{\hskip
162.1807pt}$
$~{}=~{}\Bigl{(}{1}/{\sqrt{M+1}}\Bigr{)}\sum_{j=1}^{M+1}\delta_{{\mathfrak{n}},{\mathfrak{n}}_{j}}\delta(\boldsymbol{p}-\boldsymbol{p}_{j})f^{M}_{{\mathfrak{n}}_{1},\dots,\not{\mathfrak{n}}_{j},\dots,{\mathfrak{n}}_{M+1}}(\boldsymbol{p}_{1},\dots,\not\boldsymbol{p}_{j}\dots,\boldsymbol{p}_{M+1})\,.{\hskip
14.22636pt}$

As usual, the crossed-out symbol means its absence.

  2. 2.

Operators ${\hat{a}}^{+}_{m}(\ell)$, ${\hat{a}}_{m}(\ell)$ act in every
Hilbert space $\boldsymbol{H}_{j}^{M}$. These operators create or annihilate
the oscillator energy level for the mode ${\mathfrak{j}}_{\,-m}$ in the
$\ell$-th multiplier
$\bigl{[}\boldsymbol{H}_{pq}\otimes\boldsymbol{H}_{j}\bigr{]}$ of
decomposition (27). For example, the standard quantization rule for the value
$|{\mathfrak{j}}_{\,-m}|^{2}$ looks like this:

$|\,{\mathfrak{j}}_{\,-m}\,|^{2}~{}\to~{}\frac{\hbar}{2}\left[\,{\hat{a}}^{+}_{m}(\ell){\hat{a}}_{m}(\ell)+{\hat{a}}_{m}(\ell){\hat{a}}^{+}_{m}(\ell)\,\right]\,.$

Thus, summarizing of our consideration, we can write down a general formula
for the operator $\hat{\mathcal{H}}_{0}$:

$\hat{\mathcal{H}}_{0}~{}=~{}\bigoplus_{M=0}^{\infty}\hat{\mathcal{H}}_{0}^{M}\,,$

where operators $\hat{\mathcal{H}}_{0}^{M}$ act in the spaces
$\boldsymbol{H}_{M}$. These operators have following form:

$\displaystyle~{}\hat{\mathcal{H}}_{0}^{M}~{}=~{}\frac{\hbar^{2}}{2m_{0}}\sum_{{\mathfrak{n}}}\int\boldsymbol{p}^{2}{\hat{a}}^{+}(\boldsymbol{p},{\mathfrak{n}}){\hat{a}}(\boldsymbol{p},{\mathfrak{n}})~{}+$
(29) $\displaystyle+$
$\displaystyle\frac{\hbar}{2t_{0}}\sum_{\ell=1}^{M}\sum_{m=1}^{K}\Omega_{m\ell}(\varkappa_{\ell}\,;\alpha_{1}(\ell),\dots,\alpha_{\sf
N}(\ell))\,\left[{\hat{a}}^{+}_{m}(\ell){\hat{a}}_{m}(\ell)+{\hat{a}}_{m}(\ell){\hat{a}}^{+}_{m}(\ell)\right]\,.$

The function $\Omega_{m\ell}(\varkappa_{\ell}\,;\alpha_{1},\dots,\alpha_{\sf
N})$ corresponds to the vortex-like structure with number $\ell$ in the
considered vortex tangle. Let us recall that the number ${\sf N}$ determines
the level of our model (see Eq. (15)). The explicit form of the functions
$\Omega_{mn}(\varkappa_{n}\,;\alpha_{1},\dots,\alpha_{\sf N})$ can be deduced
from the detail study of linearized dynamical equations (16). For example, let
the number ${\sf N}$ is equal to one and corresponding number $b_{0}$ is also
equal to one: ${\sf N}=1$, $b_{0}=1$. This simplest case corresponds to a
vortex tangle that contains vortex loops described by the equation (1). The
meaning of the constants $\alpha_{1}(\ell)$, $i=1,\dots,M$ is as follows:
$\alpha_{1}(\ell)\propto|\boldsymbol{v}(\ell)|$, where the vector
$\boldsymbol{v}(\ell)$ is the velosity of the vortex loop that numbered by
index ${}^{\prime\prime}\ell\,^{\prime\prime}$. The operator
$\hat{\mathcal{H}}_{0}^{M}$ has the following form:

$\displaystyle\hat{\mathcal{H}}_{0}^{M}$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{0}M}\sum_{{\mathfrak{n}}}\int\boldsymbol{p}^{2}{\hat{a}}^{+}(\boldsymbol{p},{\mathfrak{n}}){\hat{a}}(\boldsymbol{p},{\mathfrak{n}})~{}+$
$\displaystyle+$
$\displaystyle\frac{\hbar}{2t_{0}}\sum_{\ell=1}^{M}\sum_{k=1}^{K}\alpha_{1}(\ell)k\sqrt{k^{2}-1}\,\left[{\hat{a}}^{+}_{k}(\ell){\hat{a}}_{k}(\ell)+{\hat{a}}_{k}(\ell){\hat{a}}^{+}_{k}(\ell)\right]\,.$
(30)

Let this considered simplest vortex tangle is located in a closed bounded
domain $D\subset{\sf R}_{3}$. Then, the eigenvalue promlem

$\hat{\mathcal{H}}_{0}^{M}|\Phi_{[\sigma]}^{M}\rangle~{}=~{}E_{[\sigma]}|\Phi_{[\sigma]}^{M}\rangle\,,\qquad\quad|\Phi_{[\sigma]}^{M}\rangle\in{\boldsymbol{H}}_{M}\,,$

has following solution. The eigenvectors $|\Phi_{[\sigma]}^{M}\rangle$ have
form (• ‣ 3), where

$\displaystyle~{}f^{M}_{{\mathfrak{n}}_{1},\dots,{\mathfrak{n}}_{M}}(\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{M})~{}=~{}\frac{1}{(M!)^{2}}\times$
$\displaystyle\times$
$\displaystyle~{}\left(\sum_{\lfloor\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{M}\rceil}\prod_{\ell=1}^{M}f_{w(\ell)}(\boldsymbol{p}_{\ell})\delta\bigl{(}\boldsymbol{p}_{\ell}^{2}-\lambda_{w(\ell)}^{2}\bigr{)}\right)\left(\sum_{\lfloor{\mathfrak{n}}_{1},\dots,{\mathfrak{n}}_{M}\rceil}\prod_{\ell=1}^{M}\delta_{{\mathfrak{s}}_{\ell}{\mathfrak{n}}_{\ell}}\right)\,,$

where the symbol $\lfloor c_{1},\dots,c_{M}\rceil$ means some rearrangement of
the corresponding values $c_{1},\dots,c_{M}$ (summations are performed on all
rearrangements). Symbols ${\mathfrak{s}}_{\ell}$ mean the reduced form for
certain set of the occupation numbers $s_{k}({\ell})$:

${\mathfrak{s}}_{\ell}~{}=~{}[\,s_{1}(\ell)\,\dots,s_{K}(\ell)\,]\,.$

The function $f_{i}(\boldsymbol{p})\delta(\boldsymbol{p}^{2}-\lambda_{i}^{2})$
is the momentum representation of the eigenfunction
${\tilde{f}}_{i}(\boldsymbol{q})$ which corresponds to certain eigenvalue
$\lambda_{i}^{2}$ for the Laplace operator $\Delta$ in the domain $D$:

$\Delta{\tilde{f}}_{i}(\boldsymbol{q})=-\lambda_{i}^{2}{\tilde{f}}_{i}(\boldsymbol{q})\,,\qquad\boldsymbol{q}\in
D\,,\quad{\tilde{f}}_{i}(\boldsymbol{q})\Big{|}_{\boldsymbol{q}\in\partial
D}=~{}0\,.$

Formula (3) allows us to find eigenvalues:

$\displaystyle E_{[\sigma]}\bigl{(}\alpha_{1}(1),\dots,\alpha_{1}(M)\bigr{)}$
$\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m_{0}}\sum_{\ell=1}^{M}\lambda_{w(\ell)}^{2}~{}+~{}$
$\displaystyle+$
$\displaystyle\frac{\hbar}{t_{0}}\sum_{\ell=1}^{M}\sum_{k=1}^{K}\alpha_{1}(\ell)\,k\sqrt{k^{2}-1}\left(s_{k}(\ell)+\frac{1}{2}\right)\,.$
(31)

To simplify the writing of these formulas, we have introduced a notation for
the multi-index $[\sigma]$:

$[\sigma]~{}=~{}\bigl{[}\,w(1),\dots,w(M);s_{1}(1),\dots,s_{1}(M);\dots;s_{K}(1),\dots,s_{K}(M)\,\bigr{]}\,.$

Let’s analyze the structure of formula (3). The terms, which are proportional
to the value of $\hbar$, give the main contribution to the energy. These terms
represent the energy of oscillatory modes of the vortex rings. Terms that are
proportional to the value of $\hbar^{2}$ can be interpreted as certain ”fine
structure” of oscillatory levels. This structure is completely determined by
the geometry of the domain in which the considered vortex system is moving.

The interaction Hamiltonian $\hat{U}$ is written as follows:

$\hat{U}~{}=~{}\sum_{m,n}\varepsilon_{mn}\hat{U}_{m\leftrightarrow n}\,.$ (32)

The constants $\varepsilon_{mn}$ are the coupling constants which define the
intensity of the vortex interaction. Operators $\hat{U}_{m\leftrightarrow n}$
define the reconnection of the considered vortex filaments in the flow. They
are constructed in terms of the creation and annihilation operators
${\hat{a}}^{+}(\boldsymbol{p},{\mathfrak{n}})$ and
${\hat{a}}(\boldsymbol{p},{\mathfrak{n}})$. Operator
$\hat{U}_{m\leftrightarrow n}$ describes the transformation of $m$ vortex
rings into $n$ rings and vice versa, $n\to m$. For example, the theory with
$U=\varepsilon\,\hat{U}_{1\to 2}$ was investigated in the paper [28], where
the model of the turbulent flow in it’s initial stage was considered. We must
emphasize that the variables ${\boldsymbol{p}}$ and ${\boldsymbol{q}}$
describe the virtual particles corresponding to the external degrees of
freedom of the vortex-like loops. Therefore, the Hamiltonian $\hat{U}$
describes, generally speaking, a non-local interaction. These questions, as
well as corresponding examples of the operator $\hat{U}_{2\leftrightarrow 2}$,
were discussed in the article [33]. In this study, we will not dwell on this
in detail.

##  4 Analysis and outlooks

Despite the presence of a number of uncertain parameters, the proposed theory
represents certain new viewpoint to the description of turbulent flow. Some of
these parameters appear to be random numbers. Apparently, this is unavoidable
when describing a turbulence. Nevertheless, the approach developed in this
study offers a new look at the relationship between ”random” and ”integrable”
dynamics in the description of a fluids motion. According to the author, the
proposed model can stimulate a vortex tangle even in the case of $U=0$,
especially for the large number of ${\sf N}$. Of course, the case where
$U\not=0$ is more interesting and realistic.

The proposed theory makes it possible to calculate the partition function
${\mathcal{Z}}$ for any concrete domain $D$. In the simplest case considered
above, we can write the following expression:

${\mathcal{Z}}=\prod_{\ell=1}^{M}\sum_{n}\exp\left(-\frac{\langle
E_{n}(\ell)\rangle}{{\sf k}_{B}{\sf T}}\right)\,,$

where the numbers $E_{n}(\ell)$ are valid energy values for the vortex
filament with number ${}^{\prime\prime}\ell\,^{\prime\prime}$, symbol
$\langle\dots\rangle$ means some averaging over the parameters
$(\,\alpha_{1}(\ell),\dots,\alpha_{\sf N}(\ell)\,)$, the value ${\sf T}$ is
the temperature and the constant ${\sf k}_{B}$ is Boltzmann constant. This
possibility is an important feature of the proposed approach from a practical
viewpoint. Indeed, the function ${\mathcal{Z}}$ allows us to calculate
explicitly the thermodynamic characteristics of the simulated turbulent flow.

It is interesting to consider the limits $K\to\infty$ and ${\sf N}\to\infty$.
It is clear that the proposed model, within such limits, will require the
application of certain quantum field theory methods. For example, expression
(3) for the energy levels would require a renormalization procedure. The limit
${\sf N}\to\infty$ leads, in general, to the non-local equations for the
single vortex-like structures. As a cosequense, we have an additional
possibility to describe the non-local effects. In this case, the number
$\varkappa$ (see Eq.(15)) takes any values in the segment $[0,1]$. Note that
non-local terms in LIE equation appear when describing the self-stretching
[32]. Moreover there is another option to introduce non-local effects here: we
can consider the additional non-local integrals $I_{-n}$, $n=1,2,3,\dots$
which can be introduced [15] for the non-linear Schrödinger equation. All
these issues, as well as the physical interpretation of the proposed dynamical
systems, require separate studies.

##  5 Novelty, limitations and possible applications

Firstly, the proposed quantization scheme leads to the richer circulation
spectrum than the standard one. This issue is studied in the works of the
author [26, 27, 33] in detail, so we do not discuss the relevant formulas
here. From the author’s point of view, this extention of a circulation
spectrum provides more opportunities for practical modeling of a quantum
turbulent flow.

Secondly, it is known that the standard approach to the energy definition of a
thin ($a\to 0$) vortex filament leads to problems due to the divergence of the
integrals. In this study, we applied a group-theoretical approach to the
definition of energy.

Thirdly, as already noted above, constructed quantum systems are ”inverse
realization” of Lord Kelvin’s old idea that any particles can be considered as
some kind of vortex-like structures [34]. In our approach, we describe the
closed vortex-like filament as some structured particle. In a sense, the fluid
in which the vortex in question moves plays the role of ”aether”. Note that
such ”aether” is quite real here, and not hypothetical.

The proposed quantization scheme for a system of vortex-like structures takes
into account the movement of the surrounding fluid in a minimal way [26].
Apparently, this fact imposes some limitations on the applicability of the
constructed model. Undoubtedly, this issue requires further research.

##  6 Concluding remarks

The next stage in the development of the proposed model should be numerical
simulation of some observable characteristics of the flow for both various
types of interaction (32) and various types of domain $D\subset{\sf R}_{3}$.
All these issues, as well as the physical interpretation and subsequent
theoretical investigation of the constructed dynamical systems, require
separate studies.

## References

  * [1] R. P. Feynman. Progress in Low Temperature Physics. ed. by C. J. Gorter. V. 1. P. 17. Amsterdam: North-Holland. (1955).
  * [2] R.J. Donnely. Quantum Vortices in Helium II Cambrige Univ. Press. (1991).
  * [3] U. Frisch. Turbulence. The Legacy of A.N. Kolmogorov.  Cambrige Univ. Press. (1996).
  * [4] J.I. Polanco, N.P. Müller, G. Krstulovic. Nature Communication. V. 12. Pp. 1 -11 (2021) article number 7090.
  * [5] W. F. Vinen. Journal of Low Temperature Physics. V. 145, pp. 7–24 (2006)
  * [6] R. Aarts, A numerical study of quantized vortices in He II. Technische Universiteit Eindhoven. (1993)
  * [7] M. Tsubota, K. Fujimoto and S. Yui. J. Low. Temp. Phys. 188, 119 (2017).
  * [8] Koji Fukagata. Nature Computational Science. V. 2. pp. 68–69 (2022).
  * [9] N.P. Möller, J.I. Polanco, G. Krstulovic. Phys. Rev. X 11, 011053 (2021).
  * [10] N. Gourianov, M. Lubasch, …D. Jaksch. Nature Computational Science V. 2. pp. 30 – 37 (2022).
  * [11] Z. Zhang, I. Danaila, E. Lévêque, L. Danaila. Journ. of Fluid Mechanics, 962, A22 (2023).
  * [12] Carlo F. Barenghi. Tangled vortex lines: dynamics, geometry and topology of quantum turbulence. in: Knotted Fields.  ed. by R.L. Ricca and X. Liu, Springer (2022).
  * [13] S.K. Nemirovskii. Phys. Rev. B 97, 134511 (2018).
  * [14] S.V. Alekseenko, P.A. Kuibin, V.L. Okulov, Theory of concentrated vortices. Springer-Verlag, Berlin Heidelberg (2007).
  * [15] L.A. Takhtajan, L.D. Faddeev, Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin. 1987.
  * [16] M.J Ablowitz, D.J. Kaup, A.C. Newell, H. Segur. The inverse scattering transform – Fourier analysis for non-linear problems. Stud. Appl. Math., V.53. pp. 249 – 315 (1974)
  * [17] H. Hasimoto. A soliton on a vortex filament // J. Fluid Mech. 23. p. 61 - 97. 1972.
  * [18] S.V. Talalov. // Mod. Phys. Lett. B. V. B32. No 33. 1850410. 2018. arXiv:1807.08922 [math-ph].
  * [19] Sonin E.B. Dynamics of Quantized Vortex in Superfluids. Cambridge University Press (2016).
  * [20] R. A. Van Gorder. Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation. Phys. Rev. E 91, 053201. (2015)
  * [21] L. A. Takhtadzhan, L. D. Faddeev. The Quantum Method of the Inverse Problem and the Heizenberg XYZ Model // Russ. Math. Surv. V.34. 11. 1979\. DOI 10.1070/RM1979v034n05ABEH003909
  * [22] E.K. Sklyanin, St. Petersburg Mathematical Journal (Formely Leningrad Mathematical Journal). 1. No 2. (1989). 189 – 206.
  * [23] H. Tennekes. in book: Handbook of Turbulence. ed. by W. Frost and T.H. Moulden. Plenum press, N.Y. (1977)
  * [24] Kastrup H. A. Annalen der Physik. 519. Issues 7-8. 439 - 528 (2007).
  * [25] Dirac P.A.M. Generalized hamiltonian dynamics. Canadian Journal of Mathematics. V.2. No 2. pp. 129 - 148 (1950).
  * [26] S.V. Talalov. Eur. Journ. Mech B/Fluids. 92. pp. 100 - 106. (2022). arXiv: math-ph/2112.04859v1.
  * [27] S.V. Talalov. Physics of Fluids. 34. 041702 (2022)
  * [28] S.V. Talalov. Physics of Fluids. 35. 045132 (2023)
  * [29] Ricca R.L. Chaos. 3. Issue 1. 83–91 (1993)
  * [30] V.F. Kop’ev, S.A. Chernyshev. Vortex ring oscillations, the development of turbulence in vortex rings and generation of sound. Phys. Usp. 43 663–690 (2000).
  * [31] L. Kiknadze, Yu. Mamaladze. Journ. Low Temperature Physics. V. 126. Nos 1/2. Jan. (2002).
  * [32] A.J. Majda and A.L. Bertozzi. Vorticity and Incompressible Flow.  Cambridge Univ. Press. (2002).
  * [33] S.V. Talalov. Phys. Rev. Fluids. 8. 034607 (2023)
  * [34] W. Thomson. On vortex atoms. Proc. R. Soc. Edinburgh 6, 94 (1869).