# A Neck Orthosis with Multi-Directional Variable Stiffness for Persons with
Dropped Head Syndrome

Santiago Price Torrendell1, Hideki Kadone2, Modar Hassan3, Yang Chen4,
Kousei Miura5, and Kenji Suzuki3 Manuscript received: February, 7, 2024;
Revised March, 19, 2024; Accepted May, 5, 2024.This paper was recommended for
publication by Editor Yong-Lae Park upon evaluation of the Associate Editor
and Reviewers’ comments. This work was supported by the Japan Society for the
Promotion of Science KAKENHI [grant number 23H00485], and the Ministry of
Education, Culture, Sports, Science and Technology (MEXT) of Japan.1Santiago
Price Torrendell is with the Graduate School of Systems & Information
Engineering, University of Tsukuba, Tsukuba, Japan
<EMAIL_ADDRESS>Kadone is with the Center for Innovative
Medicine and Engineering, University of Tsukuba Hospital, Tsukuba, Japan
<EMAIL_ADDRESS>Hassan and Kenji Suzuki are with the Institute
of Systems and Information Engineering, University of Tsukuba, Tsukuba, Japan
<EMAIL_ADDRESS><EMAIL_ADDRESS>Chen is with the Artificial
Intelligence Laboratory, University of Tsukuba, Tsukuba, Japan
<EMAIL_ADDRESS>Miura is with the Department of
Orthopaedic Surgery, Faculty of Medicine, University of Tsukuba, Tsukuba,
Japan<EMAIL_ADDRESS>Object Identifier (DOI): see top of this
page.

###### Abstract

Dropped Head Syndrome (DHS) causes a passively correctable neck deformation.
Currently, there is no wearable orthopedic neck brace to fulfill the needs of
persons suffering from DHS. Related works have made progress in this area by
creating mobile neck braces that provide head support to mitigate deformation
while permitting neck mobility, which enhances user-perceived comfort and
quality of life. Specifically, passive designs show great potential for fully
functional devices in the short term due to their inherent simplicity and
compactness, although achieving suitable support presents some challenges.
This work introduces a novel compliant mechanism that provides non-restrictive
adjustable support for the neck’s anterior and posterior flexion movements
while enabling its unconstrained free rotation. The results from the
experiments on non-affected persons suggest that the device provides the
proposed adjustable support that unloads the muscle groups involved in
supporting the head without overloading the antagonist muscle groups.
Simultaneously, it was verified that the free rotation is achieved regardless
of the stiffness configuration of the device.

###### Index Terms:

Prosthetics and Exoskeletons, Soft Sensors and Actuators, Compliant Joints and
Mechanisms.

††publicationid: pubid:  2377-3766 © 2024 IEEE. Personal use of this material
is permitted. Permission from IEEE must be obtained for all other uses, in any
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of this work in other works.

##  I INTRODUCTION

###  I-A Background

Dropped Head Syndrome (DHS) occurs in a set of neurological, muscular, or
neuro-muscular conditions affecting the paraspinal muscles. This syndrome
alters the normal neck posture where the head is shifted forward, technically
known as cervical kyphosis, which leads to chin-on-chest deformity in extreme
cases.

This uncommon inclination of the head has multiple negative effects on the
affected persons’ daily life by causing horizontal gaze disorders, gait
imbalance, and even dysphagia and dyspnea for the more severe cases [1].
Traditionally, the syndrome has been initially addressed by providing a
passive correction using a neck brace [2], which excessively constrains the
cervical range of motion. This restriction creates discomfort for the user
eventually rejecting the device [3].

Surgical intervention is a viable alternative when the non-invasive approach
is ineffective. This practice shows general long-term improvements but it also
often involves post-surgery complications [2]. In this context, there has been
an increasing interest in assistive head orthosis that provides mobile support
which could increase the efficacy of the non-invasive approach [4]. Moreover,
these new generation braces brought a paradigm change for solutions that
simultaneously mitigate neck deformation and restore lost mobility related to
neck muscle weakness.

Figure 1: Illustration of neck mobility with the proposed elastic mechanism
and components of the exoskeleton device. The elastic actuator provides a
nonrestrictive force that partially compensates the head weight during
sagittal flexion relieving the extensor muscles. The assistive force is
regulated by adjusting the stiffness of the elastic mechanism.

###  I-B Related Works

In recent years, several promising assistive braces have been proposed with
passively actuated designs that are simpler and more robust solutions than
active systems [4].

One of the most prominent works in the field is the Sheffield Snood brace, the
only flexible brace certified as a medical device class 1 [5]. The design
consists of a snood made of flexible fabric with removable beam-like
structures which provide a customized elastic support that permits flexion and
rotational motions of the head [5]. A cross-sectional study on 139 neck brace
users showed a clear preference for the flexible snood over the brace used
before [6]. However, the results from this questionnaire showed that the
brace’s effects on breathing and non-swallowing are non-negligible which has
been related to the contact of the anterior side of the neck with the brace by
other studies [3].

Two other prominent works proposed an alternative approach from the Sheffield
brace by compensating the head weight in multiple positions using a rigid link
mechanism attached to the user’s head through a headband that does not require
chin support.

Mohammed et al. developed a passive wheelchair device for DHS patients that
provides support on the sagittal plane for a wide range of motion while
enabling free neck rotation [7]. Electromyographic studies on healthy
participants showed that wearing the device caused a significant reduction and
increment in the average activity of the extensor and flexor muscles
respectively for holding the head in different postures.

Haohan et al. introduced a lightweight three-DoF parallel mechanism to assist
the head motion of patients suffering from Amyotrophic Lateral Sclerosis. The
transmission design optimizes the device’s DoF to the head workspace resulting
in achieving 70% of the range of head rotations [8]. The original mechanism
was further developed as an active and passive device. The active version
employs three motors controlled through a joystick that achieved muscle effort
reduction in healthy participants in preclinical trials [9] and also enhances
the postural control of end users [10]. Additionally, the passive version
utilizes torsional springs that provide elastic support free from control
interfaces but its effects on muscle effort reduction were not statistically
significant [11].

###  I-C Contribution

The presented works show promising results in this new field of research and
highlight the feasibility of a functional passive assistive brace for DHS
patients. However, no prior studies have presented strong evidence that the
proposed device provides suitable support that unloads the weak muscles
without overloading the antagonist group which may exacerbate muscle unbalance
of the neck and aggravate the neck deformity. The optimal mechanisms to
support the neck muscles, and the amount of neck mobility that can be
preserved while supporting DHS are still unknown. In this work, we propose and
investigate a neck exoskeleton that provides support through a compliant
structure based on flexible rods to address the presented research gaps.

This research aims to develop a compliant mechanism that provides the critical
support needed for DHS while still recruiting the weak muscle groups, and
preserving neck mobility as much as possible. This work presents the
mechanical design of the system, the physical model to estimate the required
moment to assist the head, a simple formulation of the mechanical behavior of
the passive mechanism along with the experimental validation, and non-impaired
participants study where the effects of the actuator stiffness on the EMG
activity of several neck muscles were studied.

##  II METHODS

###  II-A Mechanical design

The device requirements were defined through interviews with a physical
therapist experienced in DHS, resulting in a brace with two distinct
behaviors. On the Sagittal plane, the device offers elastic support to correct
neck deformation, allowing users to maintain a straight head position. This
adjustable support should accommodate each patient’s condition without
exerting a compressive force on the spine, which could hinder recovery.
Horizontally, the mechanism allows free neck rotation for user comfort, which
is safe for the person.

At this research stage, we prioritized a robust support structure over a
lightweight and compact design to ensure accurate actuator assessment,
avoiding potential issues like elastic deformation and loose fittings that
could compromise device performance. Another core design aspect was using
adjustable unions that minimize the fitting time of the device.

Figure 2: Schematic view of the passive actuator and working principle in
three sectioned views

The developed device is presented in Figure 1 which illustrates the behavior
when the user moves the head. On the Sagittal plane, the elastic mechanism
bends during the neck flexion providing an assistive force that supports the
motion while partially compensating the head weight. This assistance would
relieve the extensor muscles that hold the head without overloading the flexor
muscles that perform the motion. Due to the eccentricity between the mechanism
and the cervical vertebrae, a vertical rail connects the mechanism to the head
attachment to minimize both restricting the head motion and compressing the
spine. This element is linked to the horizontal rail of the head attachment
and decouples the head from the system during the neck rotations. Between both
sliders, a universal joint prevents the mechanism from transmitting undesired
torques to the head.

Regarding the elastic mechanism actuation, the system has two well-
differentiated elastic behaviors where the flexible stiffness of the bar array
is controlled by restricting the ascendant movement of the central bar’s lower
end as shown in the diagram from Figure 2. The vertical and bending motions
are tightly coupled [12]. The sliding mechanism transmits normal forces to the
bars deflecting them like a cantilever beam. For a low bending angle
$\theta_{1}$, the stiffness is relatively low as the central bar can move
freely, but when the component touches the cylinder at the angle $\theta_{2}$
the vertical motion is restricted leading to a sudden stiffness increase. The
bending angle where the mode transition occurs will depend on the initial gap
between the cylinder and the plate which is $\delta$. The smaller the value of
$\delta$ is, the earlier the elastic transition will occur.

###  II-B Modeling

####  II-B1 Estimation of the assistive moment

As the main purpose of the elastic mechanism is to assist the head motion on
the sagittal plane, the required assistive moment should ideally balance the
head weight during such a motion. The moment‘s magnitude depends on the head
inclination about the body frame and is estimated through the free body
diagrams (FBDs) of Figure 3. The Actuator and Head FBD shows that the
assistive moment should counterbalance the torques from the reaction force at
the base of the actuator $F_{B}$ and the head weight $F_{H}$, which is
quantified in a moment balance about the C7-T1 level of the cervical spine,
based on previous work [13], on the direction of the Y axis as shown in the
expression 1:

$\displaystyle\begin{split}\sum
My_{C7}=0\rightarrow\text{MM}&=\vec{F_{H}}\times\vec{P_{H}}-AM\\\
AM&=BM-\vec{F_{B}}\times\vec{P_{B}}\end{split}$ (1)

This balance shows the muscle moment MM counteracts the torque due to the head
weight, mitigated by the assistive moment AM. AM results from the difference
between the base moment BM and the eccentric base force $F_{B}$ to the
vertebral joint. Under this formulation, the device’s purpose is to minimize
MM to relieve the effort of the extensor muscles. A device‘s stiffness
superior to the ideal value would overload the flexor muscles to exert a
negative MM to keep the head tilted. The direction and magnitude of $F_{B}$
are deduced from the Actuator diagram. As the elastic structure of the bars
keeps the vertical guide aligned to the head vertical axis Z’, $F_{B}$
counter-balance the horizontal component of the head weight $F_{Hx^{\prime}}$
which is transmitted to the actuator basement. On the perpendicular axis Z’,
the spine compensates the component $F_{Hz^{\prime}}$ while the slider does
not transmit compression effort. As a result, the system is in equilibrium as
the forces on the Z’ and X’ direction is zero as the expression 2 shows.

$\displaystyle\begin{split}\sum
F_{X^{\prime}}&=0\rightarrow\vec{F_{B}}+\vec{F_{H_{X^{\prime}}}}=0\\\ \sum
F_{Z^{\prime}}&=0\rightarrow\vec{F_{S}}+\vec{F_{H_{Z^{\prime}}}}=0\end{split}$
(2)

Based on this model, the ideal moment to fully compensate for the head weight
was estimated from head motion measurements as explained in the experiment
section.

Figure 3: Free Body Diagram showing the forces involved in the interaction
between the device and the user. Figure 4: Cross section of the 7 actuator‘s
bars in a hexagonal arrangement. The 3 red bars are coupled during the High-
Stiffness mode.

####  II-B2 Mathematical model of the actuator

The model from the original work was adapted to the new design to relate the
system stiffness with the material properties and geometric parameters. Under
the hypothesis that the separation of the bars remains constant during bending
due to the spacers, the Belendez solution [14] for a single bar deflection was
used to predict the device response with expression 3. This approach omits the
bars length variation (1-4 mm) during flexion.

$\displaystyle\begin{split}M_{B}(\theta)&=\Gamma(\theta)\cdot E\cdot
I_{eq}\cdot\sqrt{\sin(\theta)}/L\\\
\Gamma(\theta)&=\int_{0}^{\theta}\dfrac{d\gamma}{\sin(\theta)-\sin(\gamma)}\end{split},$
(3)

where $\theta$ and L are the bending angle and length of the bars,
respectively. $I_{eq}$ is the equivalent moment of inertia that depends on the
device‘s stiffness condition. $I_{eq}$ is defined for two stiffness states
used during the device experimental validation: the Loaded and Base Stiffness.

During the Base Stiffness Mode (BS), the vertical motion of the central bar is
unrestricted regardless of the bending angle $\theta$, all the bars work in
parallel presenting a similar curvature. Each bar contribution to the
assistive torque is equal and is equivalent to the response of an ideal bar
$M_{B}^{BS}$ whose second moment is the sum of each real bar contribution as
the expression 4 shows.

$\displaystyle\begin{split}M_{B}^{BS}(\theta)\rightarrow
I_{eq}=\sum^{n_{\text{b}}}_{i=1}I_{i}\\\ \end{split},$ (4)

where $I_{i}$ is the moment of inertia of each beam’s cross-section about its
neutral axis [14] and $n_{b}$ is the total amount of bar which is 7.

In the Loaded Stiffness Mode (LS), the separation $\delta$ is 0 so the central
bar bottom is completely constrained to move, setting a stretching effort
along the bar. In reaction to the pulling force, the bars on the concave side,
left side in Figure 2, will experience a compressive effort to balance the
inner reaction on the system. Assuming that the coplanar cross sections of the
three coupled bars remain coplanar during flexion, they can be treated as part
of a single elastic beam whose cross sections are the three circular bar
sections arranged in a triangular pattern as shown in Figure 4.

The second moment of the triad bars is determined by the Steiner theorem which
combined with the other three decoupled bars defines the $I_{eq}$ when
exerting the torque $M_{B}^{LS}$ as shown in the expression 5.

$\displaystyle\begin{split}M_{B}^{LS}(\theta)\rightarrow
I_{eq}&=\sum^{n_{b}-n_{c}}_{i=1}I_{i}+I_{3bs}\\\ I_{3bs}&=D_{b}^{2}\cdot
S_{b}\cdot 2/3+\sum^{n_{c}=3}_{i=1}I_{j}\end{split},$ (5)

where $D_{b}$ is the separation of the bars on the ZX plane shown in Figure 3,
$S_{b}$ is the area of every single bar, and $n_{c}$ is the number of coupled
bars.

The estimations for $M_{B}^{BS}$ and $M_{B}^{LS}$ are contrasted to
experimental data in Figure 10.

####  II-B3 Device‘s prototype

The actuator prototype features 1.5mm diameter carbon fiber bars (Unxell) with
an 80mm free length. Other components were 3D-printed in polycarbonate (Fortus
350, Stratasys) and assembled using epoxy adhesive ( J-B KWIKWELD). The
device’s performance was evaluated in a bending test, as described below.

The device used in the user‘s experiments consists of a commercial rigid halo
neck brace (Brand: Hengshui Jingkang, model N211, Instrument classification:
Class I) that incorporates the actuator and transmission mechanism as shown in
Figure 5. The mounting and adjustment can done by a caregiver in under 4
minutes. The body attachments accommodate users with a xiphoid circumference
of 70-100 cm and a head circumference of 70-80 cm, reflecting the average body
dimensions of the Asian elderly population [15], who are potential users of
this device.

Figure 5: Neck brace components. (a)Body attachment with the passive actuator
and slider b) Isolated actuator constituted by the elastic bars inserted in
the body attachment. The middle bar connects to the mobile plate at the
bottom. c) Head attachment and horizontal rail connected to the vertical
slider.

##  III EXPERIMENTS

###  III-A Estimation of the actuator’s ideal moment:

Based on the expression 1, the ideal moment at the base of the actuator BM to
balance the head weight without muscle effort (MM=0) was estimated from a
single-participant experiment. The protocol involves measuring the head motion
using a motion capture system while performing continuing flexion-extension
cycles for one minute. The markers were placed on the head and body based on
previous research [8].

Under this setup, the head inclination angle $\theta$ is measured, which
defines the X’ component of the head weight $F_{H_{X^{\prime}}}$ that is the
opposite of the $F_{B}$ according to 2.

After finishing the trial, the relative position of the markers of the
actuator base and the center of mass of the head were elucidated from a side-
view picture of the participant‘s head while wearing the markers. Therefore,
$P_{B}$ and $P_{H}$ are computed from the markers’ data.

The relative location of the actuator base was estimated at the intersection
of a horizontal axis passing through the c7 vertebra and a vertical one
tangent to the scapula contour. On the sagittal plane, the head center of mass
was referred from the Frankfort and Beuviex Planes as outlined in [16]. By
estimating the head gravitational force $F_{H}$ to 50N, the ideal moment
$BM_{id}$ is computed using the expression 1.

$\displaystyle\begin{split}BM_{id}&=\vec{F_{H}}\times\vec{P_{H}}-\vec{F_{B}}\times\vec{P_{B}}\end{split},$
(6)

This prediction was contrasted with the mechanical characterization results in
Figure 10.

Figure 6: Prototype tested during the mechanical characterization. The test
machine applies a controlled load on the head to bend the bars. Under this
action, the gap cavity reduces until the mobile plate contacts the upper body,
which stiffens the actuator. The bending angle where the stiffening happens is
defined by $\delta$, which is adjusted by rotating the plate.

###  III-B Mechanical characterization

The prototype’s mechanical rigidity was characterized by conducting a bending
test across various increments of the distance parameter ($\delta$). In this
assessment, controlled loads were applied perpendicular to the upper
attachment of the elastic mechanism shown in Figure 6, mimicking the standard
functionality of the entire system. These loads were provided by a
conventional motorized test stand by IMADA, equipped with a force gauge and a
displacement sensor (IMADA MX2-2500N-FA, ZTA-50N). The evaluation involved
determining the moment around the actuator base. Notably, the base was also
used as a reference to estimate the assistive moment in the static analysis.
Lever calculations were derived from the displacement data recorded by the
test stand and the bending angle of the bars, the latter being measured using
an IMU (LP-research LPMS-B2).

(a)  Experimental setup

(b)  EMG sensors placement

Figure 7: User experiment. The participants do a series of randomized head
motions guided by the user interface. The IMUs drive the interface and The
MOCAP markers and the EMG provides synchronized data of the head motion and
the EMG activity of the muscles.

###  III-C Human Experiment

Under the recommendation of a doctor who specialized in DHS, we did a
Biomechanical evaluation of the device as a prior step to a clinical study
that involves persons with DHS. The experiment, which involved 8 volunteers
with no history of neck disorders (7 males and 1 female, age 25.4$\pm$3.2
years, 173.6$\pm$7.3 cm, 68.8$\pm$12.5 Kg), was conducted to evaluate the
brace performance on reducing muscular effort in the neck while performing
head motions that were guided by a Graphical User Interface (GUI). This
experiment was approved by the Research Ethics Board of the University of
Tsukuba. (2022R680)

This experiment compares two stiffness conditions to solely analyze the
adjustable support provided by the elastic mechanism. This approach minimizes
the effects of the mechanism imperfection in the comparison given that this
factor similarly affects both conditions. For the baseline, the mechanism was
set to the Base Stiffness mode which is the least rigid state of the device.
Secondly, the device was set to the Loaded Stiffness mode.

The purpose of the evaluation was to determine the device’s efficacy in
assisting the head during flexion on the sagittal plane while permitting the
free rotation of the neck. This behavior was studied based on three
hypotheses: (a) During Sagittal flexion, healthy participants experience an
EMG reduction on the muscles supporting the head when increasing the brace
stiffness. (b) During axial rotation, the users do not experience significant
changes in muscle activity regardless of the device condition. (c) The
device‘s stiffness does not significantly affect neck mobility.

Figure 8: Motion guidance provided by the interface. The user controls the
position of the grey marker and the green guides the movements. During a
cycle, the reference marker moves, and the user has to match both avatars
(Approaching phase). When the target is reached, the user keeps the head
position for 10 seconds (Holding phase). Finally, the reference marker returns
to the neutral position and the user follows (Recovery phase)

####  III-C1 Instrumentation

The muscle EMG activity was measured using Trigno Mini surface electrodes
(20–500 Hz, Common Mode Rejection Ratio of 80 dB, input impedance exceeding
1000 $\Omega$, Delsys Inc., Boston, MA). After properly cleaning the skin in
contact with the electrodes, the sensors were mounted on both
Stercleidomastoid (SCM) and Splenius Capitis (SPL), which are the main
superficial muscles involved in head mobility [17]. As Figure 7b shows, the
acquisition electrodes for each SCM muscle were placed along a line drawn from
the sternal notch to the mastoid process, at 1/3 the length of the line from
the mastoid process [18]. The electrodes that measure the SPLs were placed on
the palpable portion between the upper trapezius and SCM located between 6 and
8 cm lateral of the median line at the level of C4 [18].

The head kinematic was measured by adding motion capture markers (MX System,
Vicon Motion Systems, Ltd., U.K.) on the body and head attachment as shown in
Figure 7.

Simultaneously, the GUI received current head orientation about the body in
real-time by one pair of IMU sensors (LPMS-B2 Life Performance Research,
Tokyo, Japan) placed on both attachments, as Figure 7a shows.

The visual interface is based on the package RVIZ from ROS (Noetic Nynjemy
distribution). The IMU data acquisition was computed using the library
OpenZen111https://bitbucket.org/lpresearch/openzenros/src/master/.

####  III-C2 Protocol

The experimental procedure is mainly based on the work of M.N. Mahmood et al,
[7] and comprises two sessions of 5 minutes that evaluate the device
performance when using the device in two different configurations. The device
was set to the Base Stiffness mode in the first session and the Loaded
Stiffness in the second Session. A break time of 5 minutes between sessions
was set to prevent the participants’ muscular fatigue.

During each trial, the participant moves the head following the instructions
of the interface on the monitor in front of them. As a first step, there is a
10-minute practice session to familiarize the participant with the system
while wearing the device. The interface presents the frontal and side view of
two markers with the shape of a human head. The gray marker mimics the user‘s
head orientation measured by the IMUs while the green guides the prescribed
motion. The participant’s task is to follow this reference and make both
markers match. Figure 8 illustrates the actions for completing one motion
cycle from the sequence, which can be divided into three phases: approaching,
holding, and recovery. Initially, the user keeps the head straight and both
markers’ positions stay neutral. After 3 seconds, the approaching phase begins
when the green marker orientation changes and the user moves the head
intending to match both markers again. The marker matching initiates the
holding phase where the user‘s marker is kept near the target for 10 seconds.
After the elapsed holding time, the recovery phase starts where the reference
returns to the neutral position, and the user follows.
The reference marker repeats the described cycle in a randomly ordered
sequence involving twelve postures during the holding phase, each repeated
twice. The included head inclinations were $\pm$15 and $\pm$40 degrees on the
Sagital plane that corresponds to frontal and back flexion, $\pm$15 and
$\pm$30∘ on the Transverse plane for right and left rotation,(15 and 40∘) on
the Coronal plane.
The user performed these movements guided by the GUI as indicated in Figure 8
to keep the head at each position for 10 seconds.

####  III-C3 Postprocessing

After the experiment is concluded, the EMG data is manually segmented using
the head orientation calculated from synchronized MOCAP data. The markers
determine the head orientation about a common frame from which the neck
inclines. The head inclination is represented by roll-pitch-yaw angles to
segment the movement phases. For example, Figure 9 shows the head orientation
and EMG data during one trial after the segmentation.

Figure 9: Segmented head orientation and EMG activity during Flexion (F) and
Rotation (R). The three phases are delimited by vertical dashed lines in each
colored motion cycle.

After the segmentation, the raw EMG was filtered with the following steps
based on prior work [19]: the DC component removal, bandpass filter (15-400
Hz), rectification, and local integration by a moving window of 250 ms width.

##  IV RESULTS

###  IV-A Mechanical characterization

The experimental bending stiffness of the elastic mechanism for different
$\delta$ values is contrasted with the theoretical predictions and required
assistive moment in Figure 10. As discussed in the previous prototype [12],
the device presents a hysteresis during the mechanical cycle linked to
friction between the sliding parts. The experimental curves show both elastic
modes are differentiated where the low-stiffness linear response occurs first
followed by a sudden stiffness increase indicated by the magenta dots. This
transition occurs earlier as the distance $\delta$ is smaller. By convention,
the condition $\delta=\inf$ is denominated Base Stiffness, and when the gap is
nearly non-existent, $\delta=0.3mm$ is the Loaded Stiffness, Regarding the
theoretical model, the experimental results for the Loaded and Base Stiffness
are within the numerical prediction showing agreement between both approaches.
Beyond the Loaded Stiffness condition, higher assistive forces can be achieved
by compressing the adjustable plate against the rigid restriction
($\delta=-1.7mm$) which preloads the mechanism. In conclusion, these results
show that the prototype can compensate up to 55$\%$ of the head weight during
sagittal flexion.

Figure 10: Bending performance. The experiment results present a clear
stiffness transition that depends on the parameter $\delta$. The predicted
moment of the actuator model (3) for the Loaded and Base Stiffness conditions
($M_{B}^{LS}$ 5 and $M_{B}^{BS}$ 4) show close similarities with experimental
results. The device demonstrated to provide up to 55$\%$ of the required
Moment $BM_{id}$ deduced from the mocap experiment (6).

###  IV-B Human Experiment

The average muscle activity for the SCM and SPL group during the Holding Phase
is compared between the Loaded and Base Stiffness conditions in Figure 11 for
each neck motion. Each subfigure presents the results for movements on the
coronal and transverse plane on a bar plot for each muscle group. Each graph
presents the mean value and deviation of the average EMG value from the
studied group during the Loaded Stiffness and Base Stiffness conditions, which
were compared using the two-sided Wilcoxon Signed Rank Test. Before computing
the average EMG, each participant’s mean activity during the target motion was
calculated from the holding phase defined in the protocol and normalized by
the maximum recorded value during the execution of rotational motions, the
associated movement to the greater activity of the studied muscles [17].
Additionally, the head inclination was computed to segment the EMG activity
and assess the participant‘s performance following the GUI instructions. Table
I shows the mean pitch and yaw during the holding phase for flexion and
rotation movements respectively:

TABLE I:  Mean angular position during each motion Motion | Flexion | Rotation
---|---|---
| LS | HS | LS | HS
40 |  $36$ p m 3 |  $39$ p m 1 | $30$ |  $30$ p m 3
15 |  $14$ p m 2 |  $14$ p m 1 | $15$ |  $11$ p m 1
-15 |  $-12$ p m 3 |  $-9$ p m 4 | $-15$ |  $-15$ p m 6
|  |  | $-30$ |  $-31$ p m 3

The backward flexion at 40 degrees was omitted because the vertical slider
restricted mobility for some cases leading to dispairing results. For similar
reasons, lateral flexion results show dissimilar results to the presence of
friction in some trials which induced an additional resistance that restricted
the mobility on the coronal plane.

(a)  Flexion on the Sagittal Plane

(b)  Axial Rotation

Figure 11: Comparison of the averaged EMG activity and neck ROM in the Holding
Phase between the device’s Base and Loaded Stiffness (ST) modes during
sagittal flexion and axial rotation. The studied muscles were
Sternocleidomastoid (SCM) and Splenius Capitis (SPL). The significant
decrement in both SPLs’ activity during flexion evidences an increment of the
assistive support for the higher stiffness without significantly affecting the
effort to perform rotation. The small ROM difference between conditions
indicates that the device‘s stiffness adjustment does not significantly affect
the mobility of the neck

During the backward flexion, the SCM EMG activity significantly decreases
while the mean bending angle between both conditions defers 3 degrees within
the values uncertainties.

During sagittal flexion, the SPL activity significantly decreases in the
Loaded Stiffness condition. Also, there is a slight increase when the bending
angle is 40 degrees. Finally, there are no major changes during axial rotation
regarding the mobility of muscle effort.

##  V DISCUSSION

The results for flexion on the sagittal plane present evidence that the
assistive force of the device can be regulated by adjusting the system
stiffness, given that the SPL muscles supporting the head show an EMG activity
reduction without significantly affecting mobility. In contrast, the SCM group
does not present significant differences in the activity suggesting that the
assistive force does not surpass the required value to compensate for the head
weight, which can be concluded from the bending test results in Figure 10.
This aspect was considered in the mechanical design to prevent over-training
the muscles involved in bending the head. Given these results, it is expected
that DHS users will experience a similar muscular unloading compensating the
muscle weakness and reducing the neck deformity but permitting a wide range of
motion when needed. Adjusting the assistive force would allow the user to keep
using the weak muscle group potentially preventing or lowering their atrophy.

In contrast with related work, this research presents the first result of a
passive device that decreases muscle effort on supporting the head weight
during motion on the sagittal plane without overloading the muscles involved
in the active flexion. Passive Columbia Neck brace [11] actuated by torsional
springs shows no statistically significant differences in the average neck
muscle activity whether the springs are used or not, while the study on the
wheelchair brace developed in Vrije [7] indicates a reduction on the Upper
Trapecious at the expense of a significant increment of the SCM group during
forward flexion. Studies on the Sheffield snood analyzed motion restriction
but no muscle effort [5].

The EMG results for back flexion are analogous to the previous case. The SCM
muscles supporting the head show a muscle activity reduction for the Loaded
Stiffness condition while the SPL group is not significantly affected. In
conclusion, these results emphasize that the user did not experience major
effort for performing this motion, which facilitates executing tasks like
reaching objects above the line of sight.

Thirdly, small and large rotations for both sides were analyzed showing a
symmetric behavior. Each case shows no significant changes in the muscle
activity and mobility between conditions suggesting that the slider guide
permits a free rotation regardless of the device‘s stiffness.

Regarding the EMG analysis, the results for the approaching and recovery
phases were not included due to insignificant changes in muscle activation
between conditions. Inter-participant differences in movement speeds affecting
muscle activation may occlude a possible EMG reduction. The different
velocities were due to the interface‘s limited feedback as the reference
avatar of the interface instantly moved from the origin to the target position
during the motion phases. A larger scale study with a GUI that guides the user
during the three phases could lead to clearer results for the muscle activity
during each phase. A field of study still in an early stage of development
[20].

A limitation of this study concerns the current prototype‘s fitness and
compactness. The entire device weighs 2.5 Kg which is way over the recommended
value achieved by other authors [8]. Given that the elastic module of the
system weighs 100 grams, we estimate that the orthosis weight can be reduced
to 1.2 Kg by using topology-optimized 3D-printed components. Regarding the
fitting, the vest attachment ensures an effective force transfer at the cost
of depending on another person to self-fit the device which is a disadvantage.
Alternative attachments that facilitate the donning and doffing of the device
will be considered in the future. Another limitation of this work is the
absence of a condition where the participant does not wear the device.
Including such a condition would permit quantifying the effects of the
device‘s mechanical imperfections on muscle activity. Despite this additional
information that would expand the research scope, we consider that the
presented results highlight the device‘s capacity to provide adjustable
support, which is central for neck orthosis for DHS patients. Future studies
will consider these conditions for a more comprehensive device analysis.

In summary, the presented results suggest the proposed device provides
adjustable support for the head in the sagittal plane without severely
restricting mobility during sagittal flexion or axial rotation. These features
could provide an ideal balance of support and mobility to DHS patients that
current devices lack. The upcoming experiments will evaluate these aspects.

##  VI CONCLUSION

This work introduces a novel portable passive neck brace for DHS patients that
assists head motions in the sagittal plane and permits the free rotation of
the neck. The device assistance can be regulated by an elastic mechanism of
adjustable stiffness whose actuation principle was theorized and validated
through a mechanical test. The neck brace effects on neck muscle activity were
studied in 8 healthy participants for different stiffness values of the
actuator showing significant decrements in the extensor muscles for motions in
the sagittal plane for the higher stiffer conditions. Additionally, no
significant changes in mobility or EMG activity were measured during neck
rotation suggesting that the device does not restrict such a motion. These
results present evidence that the proposed device reduces neck muscle effort
while enabling a wide range of motion on the sagittal and axial plane, an
essential feature for a life support device for DHS patients. Future
evaluations will test this hypothesis on real patients.

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# Neutrino oscillations in the galactic dark matter halo

Roberto A. Lineros INSPIRE-HEP profile<EMAIL_ADDRESS>
###### Abstract

The observation of PeV neutrinos is an open window to study New Physics
processes. Among all possible neutrino observables, the neutrino flavor
composition can reveal underlying interactions during the neutrino
propagation. We study the effects on neutrino oscillations of dark matter-
neutrino interactions. We estimate the size of the interaction strength to
produce a sizable deviation with respect to the flavor composition from
oscillations in vacuum. We found that the dark matter distribution produces
flavor compositions non reproducible by other New Physics phenomena. Besides,
the dark matter effect predicts flavor compositions which depend on the
neutrino’s arrival direction. This feature might be observed in neutrino
telescopes like IceCube and KM3NET with access to different sky sections. This
effect presents a novel way to test Dark Matter particle models.

##  1 Introduction

The observation and confirmation of neutrino oscillations is one of the
earliest evidences of New Physics (NP) beyond the Standard Model (SM) of
particle physics. Roughly speaking, neutrino flavors ($\nu_{e}$, $\nu_{\mu}$,
and $\nu_{\tau}$) mix while they propagate through space. The misalignment
between flavor and mass eigenstates is the cause of the oscillations.

The overwhelming observational evidence of neutrino oscillations challenges
the SM because it is a clear indication that neutrinos are massive. A
conservative estimation of neutrino masses is $\sum_{i=1}^{3}m_{\nu}\lesssim
0.72\leavevmode\nobreak\ {\rm eV}$ which comes from cosmological observations
and the analysis done by the PLANCK Collaboration Ade:2015xua . Moreover,
oscillation experiments are sensitive to the neutrino square-mass differences
$(\Delta m_{ij}^{2}=m_{j}^{2}-m_{i}^{2})$, the mixing angles between
flavor/mass eigenstates $(\theta_{ij})$, and the CP-phase $\delta$ (for
details check Forero:2014bxa  and references within). Currently, the precision
in the neutrino parameters determination allows us to probe possible NP
effects.

Neutrinos are unique among other SM particles. They are the lightest fermions
and electrically neutral. Besides, neutrinos can interact with SM particles
via the exchange of $Z$ and $W^{\pm}$ bosons. This gives them the property to
cross through anything without almost interactions. Therefore neutrinos can
unveil regions of the Universe which are hidden to other type of observations.
In this case, neutrino telescopes like IceCube Aartsen:2016oji , ANTARES
Michael:2015gjn , and KM3Net Margiotta:2014gza , among others, give
complementary information with respect to other astroparticle messengers (e.g.
gamma- and cosmic-rays) becoming unique tools to understand the Universe. It
is important to highlight the IceCube observatory and its leading role on
revealing the Universe with the observation of PeV neutrinos Aartsen:2015trq .
This opens the door to test New Physics effects in the neutrino physics
sector. Recently, IceCube has presented results on the flavor composition of
astrophysical neutrinos Aartsen:2015zva . The reported best fit point in the
flavor space is far from the expected flavor composition. The expected
composition assumes neutrino oscillations in vacuum and initial neutrino
flavor from astrophysics-motivated sources. In some scenarios, the deviation
could be related to the presence of exotic sources of PeV neutrinos
Bustamante:2015waa . Alternatively to such hypothesis, the deviation could
come from effects from the presence of Dark Matter (DM). It is quite well
established that the Universe is fulfilled with DM and its presence has a
large effect in the cosmological evolution of the Universe. In fact, the DM
relic abundance is quite well estimated using observables extracted from
anisotropies of the Cosmic Microwave Background, structure formation, among
others. The combination of all these observables sets the DM relic abundance
to $\Omega_{\rm DM}h^{2}=0.1198\pm 0.0015$ Ade:2015xua .

The presence of DM is _per se_ an evidence of NP and thus any DM candidate
must come from models beyond-SM. In some cases, the DM candidate is related to
neutrino physics (e.g. Lattanzi:2014mia  and references within). Regardless of
the DM nature, it is tantalizing to consider that DM could affect some
neutrino observables. In this manuscript, we summarized the results presented
in deSalas:2016svi  which studies the neutrino flavor composition of neutrinos
propagating in the galactic DM halo.

##  2 Neutrino oscillations through a DM medium

The Milky Way galaxy is embedded in a DM halo. The precise DM distribution
($\rho_{\rm DM}$) inside the Milky Way is still under study. However, it is
expected that its DM distribution is in the ballpark of a NFW
1996ApJ…462..563N  or an isothermal profile distribution 1980ApJS…44…73B .

Neutrinos produced inside the Milky Way must travel across the galactic DM
halo. In this scenario, the DM effects on the neutrino oscillation can be
modeled using the Mikheyev-Smirnov-Wolfenstein (MSW) effect Wolfenstein:1977ue
; Mikheev:1986gs . In such case the total hamiltonian in the flavor basis is

$\displaystyle\mathcal{H}_{\rm tot}(x)=\mathcal{H}_{\rm
vac}+\mathcal{V}(x)\,,$ (1)

where $\mathcal{H}_{\rm vac}$ is the (global phase subtracted) hamiltonian in
vacuum,

$\displaystyle\mathcal{H}_{\rm
vac}=\frac{1}{2E}U_{0}\left[\begin{array}[]{ccc}0&0&0\\\ 0&\Delta
m^{2}_{21}&0\\\ 0&0&\Delta m^{2}_{31}\end{array}\right]U_{0}^{\dagger}\,,$ (5)

which depends on the neutrino square-mass difference $\Delta
m^{2}_{ij}=m^{2}_{i}-m^{2}_{j}$, the lepton mixing matrix $U_{0}$
Agashe:2014kda , and the neutrino energy $E$. The spatial dependence of the
hamiltonian comes from the effective potential $\mathcal{V}(x)$ in the flavor
basis,

$\displaystyle\mathcal{V}(x)=\mathcal{V_{\oplus}}\,\hat{\rho}_{\rm DM}(x)\,,$
(6)

where $\hat{\rho}_{\rm DM}(x)$ is the DM distribution (Sun’s position
normalized) and $\mathcal{V_{\oplus}}$ is the value of the effective potential
matrix at the Sun’s position. Depending of the interpretation, the effective
potential can be also written mimicking the weak interaction MSW effective
potential:

$\displaystyle\mathcal{V}(x)={\lambda^{\prime}}\,G_{F}^{\prime}N_{\rm
DM}(x)\,,$ (7)

where $\lambda^{\prime}$ is the couplings matrix in the flavor basis,
$G_{F}^{\prime}=G_{F}\,m_{Z}^{2}/m_{Z^{\prime}}^{2}$ is a scaled Fermi
constant for a mediator with mass $m_{Z^{\prime}}$, and $N_{\rm
DM}(x)=\rho_{\rm DM}(x)/m_{\rm DM}$ is the number density of DM particles. The
relation between Eqs. 6 and 7 gives that:

$\displaystyle\mathcal{V}_{\oplus}=\frac{{\lambda^{\prime}}\,G_{F}^{\prime}\,\rho_{\rm
DM}(x_{\oplus})}{m_{\rm DM}}\,,$ (8)

where we use $\rho_{\rm DM}(x_{\oplus})=0.4\,{\rm GeV}/{\rm cm^{3}}$ as a
common value for all the DM distributions. Each description of
$\mathcal{V}(x)$ remarks different aspects of the effect: Eq. 6 remarks the
strength of the effective potential and spatial effects and Eq. 7 remarks a
possible particle model extension. In both cases, the results are the
equivalent and provide interesting insights.

The evolution of neutrino states in a medium is controlled by

$\displaystyle i\partial_{t}\Psi=\mathcal{H}_{\rm tot}\Psi\,,$ (9)

where $\Psi$ is the neutrino field. The relation between flavor states and
mass states is given by

$\displaystyle|\nu_{\alpha}\rangle=\sum_{k}U^{*}_{\alpha k}|\nu_{k}\rangle\,,$
(10)

where the greek index $\alpha$ corresponds to a flavor index (e, $\mu$,
$\tau$), the latin index $k$ refers to $\mathcal{H}_{\rm tot}$ eigenstates
with effective mass $m_{k,{\rm eff}}$, and $U$ is the unitary matrix which
diagonalizes $\mathcal{H}_{\rm tot}$:

$\displaystyle\mathcal{H}_{\mathrm{tot}}^{m}=U^{\dagger}\mathcal{H}_{\mathrm{tot}}U=\frac{1}{2E}\left(\begin{array}[]{ccc}0&0&0\\\
0&\Delta m_{21,\mathrm{eff}}^{2}&0\\\ 0&0&\Delta
m_{31,\mathrm{eff}}^{2}\end{array}\right)\,,$ (14)

where $\mathcal{H}_{\mathrm{tot}}^{m}$ is the (global phase subtracted) total
hamiltonian in the mass-state basis.

The typical distance between sources and Earth ranges from parsecs to
kiloparsecs. This distance is extremely large with respect to oscillation
length at the PeV energy making neutrinos to loose coherence. This allow us to
simplify the temporal/spatial evolution of neutrino states because of the
large distance averaging effect of neutrino oscillations. In the case of a
spatially homogeneous potential $\mathcal{V}(x)=\mathcal{V}$, the observed
flavor composition is:

$\displaystyle
f_{\beta}=\sum_{\alpha=e,\mu,\tau}\left(\sum_{i=1,3}\left|U_{\beta i}U_{\alpha
i}^{*}\right|^{2}f^{0}_{\alpha}\right)\,,$ (15)

where $f^{0}_{\alpha}=(f^{0}_{e},f^{0}_{\mu},f^{0}_{\tau})$ is the flavor
composition at the source point, and $U$ is the unitary matrix that
diagonalizes the total hamiltonian $\mathcal{H}_{\mathrm{tot}}$.

A more interesting case occurs when the effective potential varies along the
neutrino path to Earth. In this case, we may need to solve Eq. 9 and follow
the evolution of $f_{\beta}$ in different stages along the neutrino path.
However, the DM profile does not vary fast enough and the evolution of
neutrino flavors satisfies the adiabatic condition (see deSalas:2016svi  for
details). In this case, the flavor composition at Earth is

$\displaystyle
f^{\oplus}_{\beta}=\sum_{\alpha=e,\mu,\tau}\left(\sum_{i=1,3}\left|U^{\oplus}_{\beta
i}U_{\alpha i}^{0*}\right|^{2}f^{0}_{\alpha}\right)\,,$ (16)

where $U^{\oplus}$ diagonalizes $\mathcal{H}_{\rm tot}$ evaluated at Earth’s
position, while $U^{0}$ diagonalizes $\mathcal{H}_{\rm tot}$ evaluated at the
source. The difference between $U$’s arises because the effective potential
scales according to $\rho_{\rm DM}$. Let just remark that the matrix texture
would not to change during the neutrino path.

##  3 Results and Discussion

The DM effect on the neutrino oscillations depends on the size of
$\mathcal{V}_{\oplus}$ and its texture. We know from the standard MSW effect
that if the effective potential is smaller than $\mathcal{H}^{\rm vac}$ the
flavor composition converges to the solution in vacuum. On the other hand, if
it is larger than $\mathcal{H}^{\rm vac}$, the composition is equal to the
composition at the source. Therefore, a rough estimation is
$\mathcal{V}\sim\Delta m_{ij}^{2}/2E$ at some point during the neutrino path.
For neutrinos at PeV range, the effective potential in the range
$\mathcal{V}\sim 10^{-17}$ – $10^{-21}$ eV may affect non-trivially the
observed flavor composition. More details about the exact range and texture of
$\mathcal{V}$ are in deSalas:2016svi .

In Fig. 1, we present 2 ternary plots in the flavor space. In both plots, we
consider neutrinos with $E_{\nu}=1$ PeV homogeneously produced in the galatic
disk. In the left panel, we assume an initial flavor composition of
$f^{0}=(1:2:0)$ which is compatible with neutrinos coming from pion decays.
Besides, we randomly generate the matrix $\mathcal{V}_{\oplus}$ keeping
$\left|\mathcal{V}_{\oplus}\right|<10^{-17}$ eV. The black pentagon indicates
the expected composition assuming oscillations in vacuum: $f^{\oplus,{\rm
vac}}\simeq(1:1:1)$. The red region corresponds to the accessible area if the
DM distribution is constant. We highlight that our region coincides with the
one coming from Lorentz-violation effects Arguelles:2015dca . The blue region
corresponds to same setup but using a NFW profile. In this case, the
accessible region for the NFW profile is larger than the constant profile
case. This indicates that spatial dependence of the effective potential does
largely affect the observed flavor composition. The green star indicates
IceCube’s best fit. Unfortunately, its location is just in the interface among
both areas.

The right panel of same figure shows the accessible region when maximum value
of $\mathcal{V}$ is restricted. This case uses $f^{0}=(1:0:0)$ and a NFW
profile. The black pentagon is the expected flavor composition in vacuum and
it corresponds to $f^{\oplus,{\rm vac}}\simeq(0.56:0.22:0.22)$. Here, we show
that the effect of the DM distribution can cover almost the full flavor
triangle for $\mathcal{V}$ values as large as $10^{-17}$ eV. Smaller values of
the potential shrinks the area, converging to an small area around the vacuum
solution.

Figure 1: Ternary plots for the neutrino flavor composition at Earth. Left
panel shows the accessible flavor regions comparing homogeneous and a NFW DM
profile. Right panel shows how the size of the effective potential affects the
accessible regions assuming a NFW profile. More details are in deSalas:2016svi
.

The $\mathcal{V}$ scale may indicate a beyond-SM solution. To this end, we
consider DM models like asymmetric-DM, scalar DM, and fuzzy-DM Hu:2000ke . All
of them could generate such scale providing interesting classes of models. For
the interpretation in terms of particle models, we need to calculate the
neutrino mean free path,

$\displaystyle l_{\nu}=\left(\sigma\,N_{\rm DM}\right)^{-1}\,,$ (17)

which indicates the distance that neutrinos can cover without (hard)
scattering on DM. For distances $\mathcal{O}\left({\rm kpc}\right)$, i.e.
Milky Way size, we required roughly $\sigma\lesssim\times 10^{-21}\,{\rm
cm}^{2}\times m_{\rm DM}/{\rm GeV}$. This condition ensures that the DM
interaction only affects the neutrino oscillation.

The mean free path depends on same parameters as Eq. 8. The cross section for
a $t$-channel interaction in the high energy regime ($\sqrt{s}\gg
m_{Z}^{\prime},m_{\rm DM}$) becomes

$\displaystyle\sigma\simeq\lambda^{\prime}\left(\frac{m_{Z^{\prime}}}{\rm
GeV}\right)^{-2}\times 3.75\times 10^{-29}{\rm cm}^{2}\,.$ (18)

Looking inside Eq. 8, we can estimate the numerical value of two parameters
among $\lambda^{\prime}$, $m_{\rm DM}$ and $G_{F}^{\prime}$ (or
$m_{Z^{\prime}}$) by reproducing the required value of $\mathcal{V}$ and
$l_{\nu}$. We found that fuzzy-DM models with $m_{\rm DM}\simeq 10^{-23}$ eV
align better if the interaction strength is similar to the one of Weak
interactions. Models with masses in the GeV–keV range, inspired by WIMPs,
majorons, etc. (e.g. Hirsch:2013ola ; Lineros:2014jba ; Rojas:2017sih ),
requires mediators with masses $m_{Z}^{\prime}\sim 10^{-6}(10^{-2})$ eV which
would generate an observable effect in the neutrino oscillation. We study
other possibilities including making the mean free path larger than the
observable universe (see details in deSalas:2016svi ).

Another interesting aspect of the DM effects is the dependence on the arrival
direction. Galactic neutrinos originated in regions closer to the galactic
center would be affected differently that those originated in the Milky Way’s
outskirts. This is important because neutrino telescopes like IceCube and
KM3NET are sensitive to different patches of the sky. Indeed, this effect
indicates that observed flavor composition in both telescopes may be
different. The case of extragalactic neutrinos are also interesting because it
allows us to explore DM densities in extragalactic halos.

##  4 Conclusions

The observation of PeV neutrinos allows us to study the Universe as never done
before. Similarly, the presence of DM in the Universe is a clear signal of
physics beyond-SM.

Neutrinos traveling to Earth must propagate through the galactic DM halo.
Assuming that DM and neutrinos do interact, we study the effect of such
interactions at the level of neutrino oscillations. We found that the effect
can produce flavor compositions at Earth different than those expected from
neutrino oscillations in vacuum. Besides, the spatial dependence generates
flavor compositions outside the region obtained from other NP effects.

The interpretation in terms of DM models gives interesting constructions. For
DM particles with masses in the keV and GeV range, the neutrino-DM interaction
could be mediated by new particles with masses in the sub-eV range. The
implication for the Weak-like interaction is that DM mass has to be extremely
small $\mathcal{O}(10^{-23}\,{\rm eV})$. This solution is compatible with
fuzzy-DM models.

Better statistics of the neutrino flavor composition might indicate a
dependence with respect to the neutrino arrival direction. This feature gives
a nice prediction of this effect.

I would like to thank the RICAP 2016 organizers for a very interesting
conference. I thank P. F. de Salas and M. Tórtola for their great work done in
the paper referred in this manuscript. I also thank the hospitality of the
Theoretical Physics group at the U. Libre de Bruxelles and the _Interactions
Fondamentales en Physique et en Astrophysique_ (IFPA) group at U. Liège. R. A.
L. acknowledges the support of the Juan de la Cierva contract JCI-2012-12901
(MINECO) and the Spanish MESS via the Servicio Público de Empleo Estatal.

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##  Appendix A Modular Arithmetic Architectures

For implementing modular arithmetic, we use hardware-friendly algorithms and
optimizations so that we can optimize these fundamental arithmetic blocks for
both area and speed. Modular addition and subtraction are the simplest blocks
and they are implemented using adder and subtractor circuits made of
configurable fabric logic, i.e., LUTs.

For implementing modular multiplication, we use the DSP units available in
FPGAs so that the highest throughput can be achieved. We used several layers
of pipeline registers to meet high clock-frequency constraints. One integer
multiplier (60-bit or 54-bit) consumes 10 DSP slices. The most commonly used
methods for performing modular reduction are based on the Barrett or
Montgomery methods. The SEAL library [SEA20] and the hardware architecture of
the HEAX processor [RLPD20] use the Barrett reduction technique for reducing
the results of integer multiplications. Both Barrett and Montgomery methods
are based on multiplications. Another method for implementing the modular
reduction operation is using a table-based modular reduction approach as
proposed in [RJV+18]. In this method, the result of a multiplication is
reduced in multiple steps where each step reduces a part near the most
significant bit using a pre-computed look-up table. Although this method does
not use any multipliers, it increases LUT utilization on FPGA and does not
provide an optimal solution when the modulus is constant.

When the modulus is selected as a sparse prime, the reduction operation can be
performed efficiently using only add and shift operations as proposed in
[ZYC+20]. In this work, we use reduction-friendly pseudo-Mersenne primes and
employ the fast add-shift-based modular reduction method to save both DSPs and
LUTs [ZYC+20]. For example, the first modulus in the RNS-basis ($q_{0}$) is a
60-bit sparse prime $2^{59}+2^{25}+2^{22}-2^{20}+1$ and the result of a
multiplication (120 bit) is reduced using the relation
$2^{59}\equiv-2^{25}-2^{22}+2^{20}-1\pmod{q_{0}}$ recursively. This approach
saves up to 45% LUT units compared to the table-based modular reduction
method.

##  Appendix B NTT Algorithm

Algorithm 10 Decimation-in-time (DIT) Forward NTT

In: $\boldsymbol{a}\in R_{Q,N}$ and $\psi\in\mathbb{Z}_{q}$ ($2N$-th root of
unity)
Out: $\tilde{\boldsymbol{a}}=\mathtt{NTT}(\boldsymbol{a})\in R_{Q,N}$

1:$\tilde{\boldsymbol{a}}\leftarrow\mathtt{BitReverse}(\boldsymbol{a})$
$\triangleright$ Permutation of coefficients

2:for $m=2$ to $N$ by $m=2m$ do

3: $w\leftarrow 1$, $\psi_{m}\leftarrow\psi^{N/m}$

4: for $j=0$ to $m/2-1$ do $\triangleright$ Butterfly loop

5: for $k=0$ to $N-1$ by $m$ do

6:
$u\leftarrow\tilde{\boldsymbol{a}}[k+j],t\leftarrow\tilde{\boldsymbol{a}}[k+j+m/2]$

7: $\tilde{\boldsymbol{a}}[k+j]\leftarrow u+w\cdot t$

8: $\tilde{\boldsymbol{a}}[k+j+m/2]\leftarrow u-w\cdot t$

9: end for

10: $w\leftarrow w\cdot\psi_{m}$

11: end for

12:end for