# Reinforcement learning adaptive fuzzy controller for lighting systems:
application to aircraft cabin

Kritika Vashishtha  Department of Aerospace Engineering, Toronto Metropolitan
University, 350 Victoria St., M5B2K3, Toronto, Ontario, Canada Anas Saad
Department of Aerospace Engineering, Toronto Metropolitan University, 350
Victoria St., M5B2K3, Toronto, Ontario, Canada Reza Faieghi  Department of
Aerospace Engineering, Toronto Metropolitan University, 350 Victoria St.,
M5B2K3, Toronto, Ontario, Canada Corresponding author:
<EMAIL_ADDRESS>Fengfeng Xi  Department of Aerospace Engineering,
Toronto Metropolitan University, 350 Victoria St., M5B2K3, Toronto, Ontario,
Canada

###### Abstract

The lighting requirements are subjective and one light setting cannot work for
all. However, there is little work on developing smart lighting algorithms
that can adapt to user preferences. To address this gap, this paper uses fuzzy
logic and reinforcement learning to develop an adaptive lighting algorithm. In
particular, we develop a baseline fuzzy inference system (FIS) using the
domain knowledge. We use the existing literature to create a FIS that
generates lighting setting recommendations based on environmental conditions
i.e. daily glare index, and user information including age, activity, and
chronotype. Through a feedback mechanism, the user interacts with the
algorithm, correcting the algorithm output to their preferences. We interpret
these corrections as rewards to a Q-learning agent, which tunes the FIS
parameters online to match the user preferences. We implement the algorithm in
an aircraft cabin mockup and conduct an extensive user study to evaluate the
effectiveness of the algorithm and understand its learning behavior. Our
implementation results demonstrate that the developed algorithm possesses the
capability to learn user preferences while successfully adapting to a wide
range of environmental conditions and user characteristics. and can deal with
a diverse spectrum of environmental conditions and user characteristics. This
underscores its viability as a potent solution for intelligent light
management, featuring advanced learning capabilities.

Keywords: fuzzy logic, reinforcement learning, Q-learning, adaptive algorithm

##  1 Introduction

Interior lighting plays a crucial role in enhancing the comfort and well-being
of aircraft passengers. Diverse lighting scenarios create a variety of
atmospheres and experiences for passengers, and can significantly impact
passengers’ mood, perception, alertness, and circadian rhythm. This can range
from creating a sense of relaxation to providing stimulation or entertainment.
Additionally, lighting can assist passengers in adapting to different time
zones and minimizing the effects of jet lag [1, 2].

With the advancement of artificial intelligence and control systems,
intelligent lighting systems are becoming increasingly popular. Previous
research in this area includes the development of a smart lighting system to
elevate visual comfort while minimizing energy consumption [3]. The system
maintains a desired light level where needed while minimizing it where not
required. This concept is further explored in [4], using the Internet of
Things (IoT) to create an autonomous and more efficient lighting management
system in smart cities. Another notable work is [5], which details the design,
implementation, and deployment of a smart emergency light system for
buildings. The system’s key advantage is its integration with existing
building facilities (i.e., emergency lights), and it has been successfully
implemented in smart buildings.

While developing similar systems for aircraft interiors holds great promise
for elevating the flight experience, progress in this area has been limited.
The papers that we were able to find have primarily focused on the
architecture of intelligent lighting systems with little or no emphasis on
developing intelligent light management algorithms [6, 7, 8]. Our objective in
this paper is to present an intelligent algorithm for light management in
aircraft cabins. In particular, we aim to develop an algorithm that can
automatically adjust the intensity of aircraft interior lights based on the
perception of the environment, e.g., outdoor lighting and passengers’
activities. Further, the algorithm should be able to learn the passengers’
preferences and adapt light settings to their liking.

A particular challenge in developing such an algorithm is that lighting
preferences are highly individualistic and influenced by a multitude of
factors. As pointed out in [9], factors such as age, gender, and even the type
of activity being undertaken can significantly impact an individual’s lighting
needs. For instance, the light intensity required for reading a book can
greatly differ from the light intensity preferred while watching a movie or
enjoying a meal. Further, visual comfort and glare sensations are subjective
feelings that cannot be accurately captured by a one-size-fits-all setting.

In this intricate landscape where a comprehensive model is difficult to find,
we propose to use a rule-based algorithm, namely a fuzzy inference system
(FIS). The strength of a FIS lies in its ability to build a controller using
intuitive natural rules. This becomes particularly advantageous in controlling
processes that are difficult to model, such as the nuanced interplay of
environmental conditions and personal preferences in defining optimal lighting
conditions. In fact, FIS has been successfully implemented in related contexts
such as IoT-based traffic light control [10], and also energy-saving
strategies in smart LED lighting systems that account for lighting comfort and
daylight [11]. In addition, our team has had success in using FIS to
automatically adjust the transparency of electrochromic windows and control
glare in aircraft cabins to maintain a comfortable visual environment amidst
varying glare conditions [12].

As we intend to develop an adaptive algorithm that can learn and respond to
the user’s preferences, we need to enhance FIS with an adaptive mechanism that
can interact with the user, interpret the interaction, and take action
accordingly. A standard method of choice to achieve this is reinforcement
learning (RL) [13]. The combination of RL and FIS is an active area of
research with a wide range of applications; see for example [14, 15, 16, 17,
18]. The choice of the RL algorithm depends on the inherent characteristics of
the problem domain. As will be clarified in our developments, for the
intelligent lighting management system, the problem domain is relatively
small, discrete, and model-free. One RL algorithm that is proven effective for
such problems is Q-learning [13]. Therefore, a Q-learning FIS; hereafter
referred to as QFIS, will be our method of choice to develop the intended
adaptive light management algorithm. QFIS has been explored in numerous
studies and has been successfully implemented for various applications; see
for example [19, 20, 21, 22, 23, 24, 25]. The idea behind QFIS is to build a
self-learning FIS whose parameters can be tuned online using Q-learning.

In this study, we first develop a FIS for aircraft cabin light management. We
develop the FIS using domain knowledge and consider inputs such as age,
chronotype, and user activity type. Next, we make the FIS adaptive by
designing a Q-learning algorithm that adjusts the FIS parameters online during
interactions with the user. We present an extensive user study in an aircraft
cabin to evaluate the effectiveness of the algorithm. In our developments, we
adopt the standard formulation of QFIS given in [20]; however, this requires
careful attention in determining the inputs, membership functions (MFs), FIS
rules, the representation of problem domain using Q-tables, the choice of a
reward function, and the learning rate, all of which are detailed in
subsequent sections.

The major contribution of this work is to develop an intelligent lighting
management algorithm for aircraft cabins. To our knowledge, this algorithm is
the first to automatically change lighting settings based on user activities
and preferences. While our focus is on aircraft interiors, the developments
here can be extended to different lighting where an intelligent light
management system is desired, e.g., smart buildings.

##  2 Algorithm Overview

This section provides an overview of the proposed QFIS algorithm for
intelligent lighting management. Figure 1 illustrates the algorithm
architecture. The inputs are categorized into three groups: (1) environment,
(2) passenger, and (3) feedback for the RL agent. For the environmental
information, we use photopic sensors to measure the current ambient lighting
quantified in terms of the daily glare index (DGI). For the passenger
information, we use age, chronotype, and the passenger’s current activity. For
any RL task, a mechanism for interaction with the environment is required. For
this purpose, we use a control knob by which the passenger can correct the
lighting setting at any time. The interactions with the control knob will
constitute the reward feedback for the RL agent. In this approach, if the
algorithm changes lighting conditions and the passenger manually overrides it
via the control knob, the adjustment applied by the passenger constitutes a
negative reward that will be used for learning and adaptation.

Figure 1: Architecture of the proposed QFIS algorithm for intelligent lighting
management

Note that the selection of the above inputs was based on recommendations from
our aircraft manufacturer partner. In practice, the passenger age and
chronotype can be recorded in a user-specific account within the aircraft
infotainment system. Subsequently, the passenger interactions with the control
knob can be recorded in this account for continuous learning and adaptation
over multiple flights. Also, passenger activity can be obtained using a camera
and various algorithms available in the vast literature of vision-based
activity recognition [26].

To generate an appropriate light setting, the environmental and passenger
information enters the FIS module. The role of FIS is to control lighting
based on fuzzy MFs and rule-base that are developed based on expert knowledge.
Once FIS generates the light setting, the RL agent will monitor the passenger
interactions with the control knob. The passenger adjustments will be used to
adapt FIS parameters according to the passenger preferences. As will be
described shortly, the adaptable parameters will modify the shape of MFs and
the weights of rules in the fuzzy inference step.

The next two sections provide the details of the FIS and RL modules.

###  2.1 Fuzzy inference system

Here, we explain the details of the proposed FIS in our algorithm. In the
design of the FIS, we use the Gaussian MFs and the Takagi-Sugeno (TS)
inference system. These choices ensure the smoothness and stability of the
algorithm as it is detailed in [27]. The inputs include age, activity, DGI,
and chronotype. The output is the intensity of interior lights. The structure
of the fuzzy system is comprised of these components: fuzzification, rule-
base, inference engine, and defuzzification, as detailed below.

###  2.2 Fuzzification

We use the Gaussian MF for the fuzzification of all inputs. Let $m$ and
$\sigma$ denote the mean and standard deviation of a Gaussian MF, then its
output is expressed as

$\mu\left(x\right)=\exp{\left(-\frac{1}{2}\left(\frac{x-m}{\sigma}\right)^{2}\right)}$
(1)

Figure 2 illustrates the MFs designed for each input. While DGI and age are
numerical inputs, passenger activity and chronotype are categorical. To
process all inputs in the same framework, we consider MFs with zero standard
deviation for the categorical data and pass them through the fuzzification
step. The details of fuzzification for each input are given below.

(a) DGI

(b) Activity

(c) Age

(d) Chronotype

Figure 2: MFs designed for each input of the FIS module

####  2.2.1 Daily glare index

DGI is a common metric used to assess the potential discomfort caused by glare
from daylight sources in indoor spaces. According to [28], DGI values up to 22
are generally deemed acceptable, and higher values tend to become
uncomfortable and even intolerable. As such, we set the DGI value of 22 as a
point of symmetry for the fuzzification of DGI, and use the deviations from
this value to categorize the level of comfort/discomfort of ambient light
(Fig. 2(a)). Note that we overlap the MFs by 15 percent to ensure smooth
transitions from one MF to another. Table 1 shows the parameter values of DGI
MFs.

Table 1: Parameters of DGI MFs MF | $m$ | $\sigma$
---|---|---
Negligible | 14 | 1.5
Acceptable | 18 | 1
Comfortable | 22 | 1
Uncomfortable | 25 | 1
Very Uncomfortable | 29 | 1.8

####  2.2.2 Activity

We consider three types of activities: meeting/entertainment, eating, and
sleeping. As the activity is categorical data, we use MFs with zero standard
deviations and rank them based on their light requirements from low to high.
As shown in Fig. 2(b), we consider the same level of light intensity for
meeting and entertainment and rank them higher than eating which itself ranks
higher than sleeping. As mentioned above, passenger activity is recognized by
a computer vision algorithm, and the type of activities to be considered in
our algorithm can be easily extended to an arbitrary number of activity
categories.

####  2.2.3 Age

For the fuzzification of age, we rely on the findings of [29] in which the
glare thresholds for two major age groups, 20-40 and 40-60 years old, were
studied. The luminosity thresholds for the 40-60 years old category are higher
than those for the 20-40 years group. This information is helpful both in the
design of the MFs and in setting up the rule base. The MFs for age are
illustrated in Fig. 2(c), and the details of their parameters are presented in
Tab. 2. Note that, aside from the 60+ category, we used identical standard
deviations to ensure all age groups have an equal range of 20 years We also
use a five percent overlap between MFs to handle transitions between the age
brackets.

####  2.2.4 Chronotype

Chronotype refers to the inherent preference that individuals have for certain
times of the day. It is a concept used to describe an individual’s natural
sleep-wake patterns and their corresponding preferences for being active or
resting during particular parts of the day. Therefore, it can be used to
tailor light settings to the passengers’ preferences.

There are three primary chronotypes: morning, evening, and intermediate.
Morning chronotypes tend to feel more alert, awake, and productive during the
early morning hours. Evening chronotypes feel more energetic, alert, and
creative during the evening and nighttime hours. Intermediate chronotypes fall
in between morning and evening chronotypes. The relationship between
chronotypes and light exposure is well-studied in the literature [30, 31, 32].
It is shown that generally morning types prefer morning light exposure while
evening types prefer evening light exposure. This allows us to rank the
morning chronotype $>$ intermediate chronotype $>$ evening chronotype in terms
of the tolerance of glare discomfort, leading to the MFs depicted in Fig.
2(d).

Table 2: Parameters of age MFs MF | $m$ | $\sigma$
---|---|---
0-20 years old | 10 | 5
20-40 years old | 30 | 5
40-60 years old | 50 | 5
60+ years old | 75 | 8

###  2.3 Rule-base

With the above selection of MFs, there are five and four MFs allocated to DGI
and age, respectively, while passenger activity and chronotype each have three
MFs. Therefore, there exist $5\times 4\times 3\times 3=180$ possible
combinations of fuzzy sets that must be taken into account in constructing the
rule base. The rules are created to cover all 180 combinations of inputs of
the FIS with each rule corresponding to one of the fuzzy outputs. Due to the
large number of rules, we present them in a supplementary document.

###  2.4 Inference engine and defuzzification

We use the zero-order TS inference engine for the design of the FIS. As
mentioned earlier, this choice ensures the stability of the QFIS algorithm as
shown in [27]. With this choice, the FIS output will be a weighted average of
the output values from the rules.

To formulate the FIS, let
${{\bf{x}}_{t}}={\left[{{x_{1,t}},{x_{2,t}},{x_{3,t}},{x_{4,t}}}\right]^{T}}$
be the vector of FIS inputs at time $t$. The $j$-th rule denoted by $R_{j}$
takes the following form

${R_{j}}={\rm{IF}}\;\;\;{x_{1,t}}\;{\rm{is}}\;A_{1}^{j}\;\;\;{\rm{AND}}\;\;\;\cdots\;\;\;{\rm{AND}}\;\;\;{x_{4,t}}\;{\rm{is}}\;A_{4}^{j}\;\;\;{\rm{THEN}}\;\;\;{f_{j}}\left({{{\bf{x}}_{t}}}\right)={k_{j}},$
(2)

where $A_{i}^{j}$ is the fuzzy set for $x_{i},t$, and the $k_{j}$ is the
consequent parameter. Assuming a weight of $w_{j}$ for $R_{j}$, the the output
of the FIS at time $t$ will take the form

$f\left({{{\bf{x}}_{t}}}\right)=\sum\limits_{j=1}^{N}{{{\bar{w}}_{j}}{k_{j}}}$
(3)

where
${{\bar{w}}_{j}}={w_{j}}{\left({\sum\limits_{j=1}^{N}{{w_{j}}}}\right)^{-1}}$
is the normalized weight, and $N=180$ is the number of rules.

In the proposed algorithm, we make $k_{j}$s adaptable using Q-learning.
Another set of adaptable parameters are the mean values of MFs to be discussed
later. For the baseline FIS, we will use the $k_{j}$ values presented in Tab.
3. The weights associated with each rule represent the degree of activation or
membership of the input variables in the corresponding fuzzy sets. The weights
will be kept constant during Q-learning adaptation. To provide an insight into
the characteristics of the proposed FIS, we present the fuzzy surface plots
for DGI and chronotype, and DGI and age versus light intensity in Fig. 3.

Table 3: Output values chosen for the zero-order TS FIS FIS output category | $k$
---|---
Lights Off (D5) | 0
Darken 4 (D4) | 12.5
Darken 3 (D3) | 25
Darken 2 (D2) | 37.5
Darken 1 (D1) | 50
Light up 1 (LU1) | 62.5
Light up 2 (LU2) | 75
Light up 3 (LU3) | 87.5
Light up 4 (LU4) | 100

Figure 3: Fuzzy surface diagrams for the proposed FIS module

##  3 Reinforcement learning

In this section, we augment the aforementioned FIS with RL, to create the QFIS
algorithm with learning and adaptation capabilities. As discussed earlier,
given the relatively small, discrete, and model-free characteristics of the
problem domain, we will base our approach on Q-learning. Similar to
conventional Q-learning algorithms, we represent the state space using Q
tables.

Let ${{\bf{Q}}_{t}}$ be the Q-table at time $t$. Given the state
${{\bf{x}}_{t}}={\left[{{x_{1t}},{x_{2t}},{x_{3t}},{x_{4t}}}\right]^{T}}\in{{\bf{Q}}_{t}}$,
the FIS module takes the action $f\left({{{\bf{x}}_{t}}}\right)$, leading to a
reward ${r_{t}}$ that is defined using the following proposed reward function

$r_{t}=-\frac{2}{\pi}\arctan\left(\epsilon(f\left({{{\bf{x}}_{t}}}\right)-a)\right),$
(4)

where $\epsilon$ is a bias value that suggests how harsh or forgiving the
reward function is, and $a$ represents the desired light intensity change as
determined by user preferences.

Using $r_{t}$, we will update the Q-table as follows

${{\bf{Q}}_{t+1}}\left({{{\bf{x}}_{t}},\;f\left({{{\bf{x}}_{t}}}\right)}\right)=\;{{\bf{Q}}_{t}}\left({{{\bf{x}}_{t}},\;f\left({{{\bf{x}}_{t}}}\right)}\right)+\;\eta{\Delta_{t}}$
(5)

where $\eta>0$ is the learning rate, and $\Delta_{t}$ is the temporal
difference error defined as follows

${\Delta_{t}}={r_{t}}\left({{{\bf{x}}_{t}},\;f\left({{{\bf{x}}_{t}}}\right)}\right)+\;\gamma{V_{t}}({{\bf{x}}_{t}})-\;{{\bf{Q}}_{t}}\left({{{\bf{x}}_{t}},\;f\left({{{\bf{x}}_{t}}}\right)}\right),$
(6)

where $\gamma>0$ is known as the discount factor and
${V_{t}}({{\bf{x}}_{t}})=\mathop{\max}\limits_{f\left({{{\bf{x}}_{t}}}\right)}\;{{\bf{Q}}_{t}}\left({{{\bf{x}}_{t}},\;f\left({{{\bf{x}}_{t}}}\right)}\right)$.

To adapt the FIS module to the user preferences, we construct adaptation rules
for the MF mean values $m$ and the output values $k$. Let $\bf{m}$ and
$\bf{k}$ be two vectors consisting of all MFs’ mean and FIS output values to
be tuned. Then, the vector of all adaptable parameters is
$\boldsymbol{\phi}={[{{\bf{m}}^{T}}\;{{\bf{k}}^{T}}]^{T}}$. To derive
adaptation laws for these parameters, let us define the following objective
function

$E=\frac{1}{2}\Delta_{t}^{2}.$ (7)

Using the gradient decent approach, the parameters can be updated as follows

$\phi\left({t+1}\right)=\phi\left(t\right)-\eta\frac{{\partial
E}}{{\partial\boldsymbol{\phi}}}$ (8)

We note that

$\frac{{\partial
E}}{{\partial{\boldsymbol{\phi}}}}={\Delta_{t}}\frac{{\partial{\Delta_{t}}}}{{\partial{\boldsymbol{\phi}}}}.$
(9)

Substituting (6) into (9) yields

$\frac{{\partial
E}}{{\partial{\boldsymbol{\phi}}}}=-{\Delta_{t}}\frac{{\partial{Q_{t}}\left({{{\bf{x}}_{t}},f({{\bf{x}}_{t}})}\right)}}{{\partial{\boldsymbol{\phi}}}}$
(10)

Using (3), we have

$\frac{{\partial{{\bf{Q}}_{t}}\left({{{\bf{x}}_{t}},f({{\bf{x}}_{t}})}\right)}}{{\partial{k_{j}}}}={\bar{w}_{j}},$
(11)

where $j$ corresponds to the $j-$th rule. Moreover, we have

$\frac{{\partial{{\bf{Q}}_{t}}\left({{{\bf{x}}_{t}},f({{\bf{x}}_{t}})}\right)}}{{\partial
m_{i}^{j}}}=\frac{{\partial{{\bf{Q}}_{t}}\left({{{\bf{x}}_{t}},f({{\bf{x}}_{t}})}\right)}}{{\partial{w_{j}}}}\frac{{\partial{w_{j}}}}{{\partial
m_{i}^{j}}}\;=\frac{{{k_{j}}-{{\bf{Q}}_{t}}\left({{{\bf{x}}_{t}},f({{\bf{x}}_{t}})}\right)}}{{\sum\nolimits_{j}{{w_{j}}}}}{w_{j}}\frac{{{x_{i}}-m_{i}^{j}}}{{{{\left({\sigma_{i}^{j}}\right)}^{2}}}},$
(12)

where the superscript refers to the $j$-th rule. Note that in driving (11), we
use (1). With the above developments, the parameters update rule can be
summarized as

$\left[{\begin{array}[]{*{20}{c}}{m_{i,t+1}^{j}}\\\
{{k_{j,t+1}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{m_{i,t}^{j}}\\\
{{k_{j,t}}}\end{array}}\right]+\eta{\Delta_{t}}\left[{\begin{array}[]{*{20}{c}}{\frac{{\left({{k_{j}}-{{\bf{Q}}_{t}}\left({{{\bf{x}}_{t}},f({{\bf{x}}_{t}})}\right)}\right)}}{{\sum\nolimits_{j}{{w_{j}}}}}{w_{j}}\frac{{{x_{i}}-m_{i}^{j}}}{{{{\left({\sigma_{i}^{j}}\right)}^{2}}}}}\\\
{{{\bar{w}}_{j}}}\end{array}}\right].$ (13)

###  3.1 Q-table formation

For our intelligent light management algorithm, three scenarios are possible,
including when the FIS output is: (1) correct i.e. it is the same as the
user’s preference, (2) too bright, and a negative reward is given to lower the
intensity levels, and (c) too dark, and a negative reward is given to raise
the intensity levels. Here, a single Q-table will be ineffective as it can not
differentiate between two negative reward scenarios. Therefore, we use two
Q-tables as follows to account for both scenarios. This way, we will be able
to appropriately reward the algorithm for suggesting lighting intensities that
are either too bright or too dark. The combinations of all four inputs of the
algorithm and their corresponding MFs create 180 different states in the
Q-tables.

##  4 Implementation and results

We conducted an extensive user study in an aircraft cabin mockup to assess the
performance of the proposed algorithm. This section details our experimental
setup followed by the test procedures and their results.

###  4.1 Experiments setup

For this purpose, we established an experimental setup using a real aircraft
fuselage within a research laboratory at Toronto Metropolitan University, as
depicted in Fig. 5. To emulate the effect of external sunlight inside the
cabin, we positioned a 1000W Colortran LQF6 floodlight, featuring an output of
roughly 57000 lm and a color temperature of 5000K, outside the aircraft
fuselage. This setup can be seen in Fig. 5.

Figure 4: Aircraft cabin mockup for the experiments

Figure 5: The setup for simulating exterior lighting

Inside the cabin, illumination was provided by an intelligent LED strip,
producing 1600 lm, which was controlled via a Raspberry Pi. We employed a
photopic sensor (PMA 1130-S-420-150K) procured from Solar Light Company,
Glenside, PA, to measure the Direct Glare Index (DGI). The backbone of our
system’s computation was the NVIDIA Jetson TX2, operating on a Linux system.
To make the testing process more efficient and enhance user engagement, we
developed a mobile application. Using this platform, participants were able to
transmit specific inputs, including activity, age, and chronotype, directly to
our algorithm. The FIS utilized these inputs to determine the light intensity
output, which was subsequently conveyed to the LED strip in the form of a duty
cycle. Participants were then exposed to the resulting lighting conditions.
They were given the flexibility to either adjust the light intensity through
the application or continue with the preset configurations. Such feedback
became instrumental for the Q-learning algorithm, enabling it to optimize the
parameters of the FIS. Figure 6 presents a comprehensive outline of our
experimental procedures.

We engaged 10 participants from the 20-40 age bracket and 8 from the 40-60 age
bracket. These age groups were recommended by our industry partner, as most
passengers, especially for business jets, belong to these two age groups.

Figure 6: Flowchart of experimental procedure

###  4.2 FIS results

Before we test the proposed QFIS algorithm, it is important to evaluate the
FIS module of the algorithm to ensure it generates appropriate light settings
for an average individual in each age group. For this purpose, each
participant was placed in a set scenario and asked to correct their preference
for light intensity, using the aforementioned mobile application. The
experiments encompassed three distinct activities, including entertainment,
eating, and sleeping. Based on the recommendations given in [9], the target
light settings for these activities include 450 lux for entertainment, 150-200
lux for eating, and less than 50 lux for sleeping.

Figure 7 illustrates the light intensity prescribed by the FIS module for each
participant across different activities for the two age group categories.
Following the completion of the trials, the results for each activity were
averaged to obtain an understanding of the preferred average lighting settings
in different activities and age ranges (Tab. 4). This data highlights the
difference between the light intensity comfort levels in different
circumstances. Additionally, this data plays an important role in
understanding passengers whose preferences do not fall under the average
patterns in each category.

Figure 7: The baseline FIS light intensity output for the 20-40 years age group (left), and 40-60 years age group (right) Table 4: The average of light intensity values generated by the baseline FIS module Activity | 20 – 40 years age group | 40 - 60 years age group
---|---|---
Entertainment | 72.4 | 83.125
Eating | 62.4 | 67.375
Sleeping | 8.4 | 8.125

###  4.3 QFIS Results

This section explores the learning capabilities of the proposed algorithm.
Detailed below are three different experimental sets, each encompassing
participants from varying age groups with notably distinct lighting
preferences and engaged in different activities:

  1. 1.

Experiments set 1, featuring a participant aged between 20-40 years who
prefers significantly dimmer light settings relative to the typical
preferences of this age group, while engaged in entertainment.

  2. 2.

Experiments set 2, including a participant aged between 40-60 years who leans
towards much brighter light settings compared to the standard settings of this
age range, while having a meal.

  3. 3.

Experiments set 3, focusing on a participant from the 20-40 years age bracket,
while attempting to sleep.

The success criteria for the algorithm constituted two factors:

  1. 1.

The correctness of the tuning: Does the algorithm correctly tune the output
towards the desired passenger light intensity?

  2. 2.

The effectiveness of the tuning: comparing different learning rates $\eta$ and
the number of trials required to tune different scenarios.

####  4.3.1 Experiments set 1

Table 5 presents the input state, the initial algorithm output, and the
participant’s preference in this set of experiments.

For the first test case, the objective was to quickly tune the $k$ parameters
to align with user preferences. The user used the aforementioned mobile
application to interact with the algorithm and provide feedback. The algorithm
calculated a reward from the feedback at every iteration and adapted the $k$
parameter, leaning toward the user preference. To explore the effects of
varying learning rates, we evaluated learning rates from 0.1 to 0.5.

Table 5: Parameters for the experiments set 1 Variable | Category | Value |  |  |
---|---|---|---|---|---
Age | 20-40 | 22 |  |  |
DGI | Comfortable | 22 |  |  |
Chronotype | Evening | 25 |  |  |
Activity | Entertainment | 5 |  |  |
Baseline FIS output | LU2 | 75 |  |  |
Passenger Preference |  $\sim$LU1 |  $\sim$62 |  |  |

Figure 8 illustrates experimental results. While the algorithm starts with the
initial output of 75, it learns from the user inputs and adapts its parameters
to reach the user preferences over time. It is evident that the value of
$\eta$ dramatically influences the number of trials necessary. For instance,
when $\eta=0.5$, the algorithm tuned with approximately 22 percent less number
of trials compared to the case of $\eta=0.1$. This highlights that by choosing
an appropriate value for $\eta$, one can effectively control the learning
behavior of the algorithm.

Figure 8: The change in the algorithm output with varying learning rates in
experiments set 1

Exploring the variations in the $m$ parameters during these trials revealed
they behaved in three distinct manners:

  1. 1.

When the input directly corresponds to the mean of MF, the mean remains
unchanged because it represents the desired value for the input.

  2. 2.

If the input is positioned closely between two MFs, the specific state
oscillates between these two MFs, thus fine-tuning both $k$ parameters
concurrently.

  3. 3.

When the input predominantly aligns with one MF, the mean of that function
undergoes sole adjustment to expedite the tuning, ensuring the output reflects
the passenger’s preference.

To further explore the role of $m$ parameters, we conducted additional
experiments, observing the $m$ values during prolonged trials. We set
$\eta=0.1$ for tuning all $m$ and $k$ parameters. Table 6 shows the final
values of $m$ for the age MFs and their corresponding $k$ values.

Table 6: The shift in $m$ values with different learning rates in experiments set 1 Number of trials | $\eta$ | $m$
---|---|---
48 | 0.5 | 30 to 27.732
59 | 0.4 | 30 to 27.409
71 | 0.3 | 30 to 27.995
87 | 0.2 | 30 to 27.498
136 | 0.1 | 30 to 27.066

To elaborate on the results, recall that while this particular participant
belongs to the age group 20 - 40 years old, their lighting preference aligns
more with a younger age bracket. Therefore, the algorithm struggles to
identify an optimal MF for the user. As $\eta$ of all $m$ and $k$ parameters
are identical, the algorithm tunes all the parameters simultaneously to
generate the optimal output. However, this leads to large shifts in MFs,
ultimately centering them around either 10 or 34. While this adaptation
finally results in an output aligned with the user preference, a significant
drawback is the extended duration required to reach the desired performance.
This can be alleviated by setting a low learning rate (e.g., 0.002) for the
$m$ parameters while maintaining a relatively larger learning rate for the $k$
parameters (e.g., 0.1). With such choices of learning rates, $k$ parameters
can quickly converge to a neighborhood of desired behavior, while slight
changes in the $m$ values will fine-tune the algorithm behavior.

It is noteworthy that the DGI MFs were not affected in this test because the
DGI in the test environment was 22, falling directly on the mean of the
Comfortable DGI MF.

####  4.3.2 Experiments set 2

Table 7 presents the parameters for this set of experiments. Note that this
set of experiments involves a different level of DGI, and new participants
with a different age category, chronotype, and activity compared to
experiments set 1. Furthermore, this participant’s lighting preferences align
more with an older age bracket; therefore, the new experiments evaluate the
algorithm in scenarios opposite to the ones mentioned in the previous section.

Table 7: Parameters for the experiments set 2 Variable | Category | Value
---|---|---
Age | 40-60 | 50
DGI | Negligible | 14
Chronotype | Morning | 5
Activity | Eating | 3
Baseline FIS output | LU3 | 87.5
Passenger Preference |  $\sim$LU4 |  $\sim$100
Figure 9: The change in the algorithm output with varying learning rates in
experiments set 2

We observed the learning behavior of the algorithm for different $\eta$ values
ranging from 0.1 to 0.5. Figure 9 illustrates the results, showing that the
algorithm can effectively learn the passenger preferences and adapt its
parameters correctly for cases where the user prefers more lighting.
Furthermore, when comparing the varying learning rates, using $\eta=0.5$
allowed for 25 percent faster tuning compared to the case of $\eta=0.1$. These
results, coupled with outcomes from experiments set 1, verify the algorithm’s
effectiveness in adapting to the preferences of users with varied
characteristics.

Further, upon closely examining the algorithm’s performance near the target
behavior, it is evident that there is minimal or no overshoot at all. Table 8
presents the final three trials from experiments set 2 with $\eta=0.2$. The
algorithm exceeds the desired value of 100 by a marginal amount, but this
leads to another negative reward that is accounted for by our dual Q-table
configuration. Consequently, once the algorithm exceeds the desired value, it
promptly readjusts to the target in the subsequent trial. It is important to
highlight that while higher learning rates might lead to more noticeable
overshoots; these can be effectively handled with adaptive learning rates.

Table 8: Algorithm overshoot and its correction Trial number | $\eta$ | $r_{t}$ | $f\left({\bf{x}}_{t}\right)$
---|---|---|---
103 | 0.2 | -0.0254 | 99.99203
104 | 0.2 | -0.0159 | 100.005
105 | 0.2 | 0 | 100

####  4.3.3 Experiments set 3

This set of experiments focuses on sleeping, as it is a unique activity in
which the interior lights need to shut off or function as a nightlight.
Through the FIS tests, it was found that the majority of individuals prefer a
light intensity of around 10. However, to challenge the algorithm, we asked a
participant who prefers brighter lighting for sleeping to conduct this set of
experiments. Table 9 shows the parameters for these trials.

Table 9: Parameters for the experiments set 3 Variable | Category | Value |  |  |
---|---|---|---|---|---
Age | 20-40 | 27 |  |  |
DGI | Comfortable | 22 |  |  |
Chronotype | Night | 25 |  |  |
Activity | Sleeping | 2 |  |  |
Baseline FIS output | D4 | 12.5 |  |  |
Passenger Preference | D2 | 35 |  |  |

Figure 10 displays the test results highlighting the effects of various
learning rates. This test displays a higher range of tuning as the participant
required a higher light intensity when compared to the baseline FIS’s initial
output. Observing the behavior $m$ parameters, it was revealed that the user
age predominantly aligned with one MF, and the mean of that MF was tuned
solely to accelerate tuning to match the user preference. This can be seen in
Fig. 10 by the slight increase in the slope of the curves over Table 10
tabulates the shift of the mean from the first test to the last for different
$\eta$ values.

Figure 10: The change in the algorithm output with varying learning rates in experiments set 3 Table 10: The shift in $m$ values ith different learning rates in experiments set 3 Number of trials | $\eta$ | $m$
---|---|---
48 | 0.5 | 30 to 27.732
59 | 0.4 | 30 to 27.409
71 | 0.3 | 30 to 26.995
87 | 0.2 | 30 to 26.498
136 | 0.1 | 30 to 26.066

###  4.4 Summary of results

Through the various experiments conducted, the algorithm’s effectiveness in
adapting to users with a wide spectrum of preferences and characteristics was
verified. Furthermore, the effect of various learning rates was explored,
showing that the learning behavior of the algorithm can be controlled easily
using the learning rate.

The detailed examination of the algorithm behavior revealed that the learning
rate for the $m$ parameters should be smaller than the ones for the $k$
parameters, crucial for a balanced adaptation of algorithm parameters and
minimizing the number of trials needed to match the user preferences.

The number of trials also depends on the difference between the baseline FIS
output, and the passenger preference. This underscores the importance of using
the domain knowledge to build a fine-tuned baseline FIS.

##  5 Conclusion

This paper developed a QFIS algorithm for intelligent lighting management with
a focus on aircraft interiors. Through a comprehensive user study, we have
demonstrated the algorithm’s efficacy in accommodating a diverse spectrum of
user preferences and characteristics. Furthermore, we have conducted an in-
depth analysis of the algorithm’s behavior and studied the impact of its
design parameters.

While our focus was on aircraft interiors, the developed algorithm can be used
in any environment that can benefit from a smart lighting algorithm. As
revealed in our literature review, the existing work on intelligent lighting
systems has primarily focused on hardware architecture, and the software
developments are limited to on-off light settings with no capability to learn
user preferences. As such, our work presents several elements of novelty. To
our knowledge, it is the first to explore the application of FIS and RL for
light management. It uses the domain knowledge to develop an intelligent fuzzy
controller, and at the same time, provides a mechanism for the user to
interact with the algorithm, and thus create a learning behaviour. We were not
able to find similar work with respect to both FIS and RL in similar
applications.

It is worth noting that the results presented in this paper can be extended in
several directions. For example, an adaptive learning rate can be utilized to
accelerate the learning process of the algorithm when the baseline FIS and
user preferences are significantly different. Mimicking human decisions is
very complex and there are numerous factors that stimulate visual comfort.
While the baseline FIS developed here was based on thorough research in the
literature and discussions with an industry partner active in developing
aircraft lighting systems, a deeper study of human lighting preferences (e.g.,
light color, human eye conditions) can help design a more effective baseline
FIS, subsequently reducing the gap between user preferences and the algorithm
output, and reducing the need for prolonged learning. Moreover, our user study
included 18 participants. While this relatively limited number of participants
does not compromise the rigor of our developments, a larger user study can
help improve our knowledge about humans’ lighting preferences and potentially
inform new design directions that can help accelerate the adoption of such
algorithms in real practice.

In the future, we will work on integrating the developed algorithm with smart
window systems and will conduct a larger user study.

## Acknolwedgement

This research was partially supported by the Natural Sciences and Engineering
Research Council of Canada Discovery Grants program.

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## Supplementary Material

The fuzzy rules for the rule-base are as follows.

Age | Activity | DGI | Chronotype | Output
---|---|---|---|---
0-20 | Sleeping | 14 | Morning | D1
0-20 | Sleeping | 14 | Intermediate | D1
0-20 | Sleeping | 14 | Evening | D1
0-20 | Sleeping | 18 | Morning | D1
0-20 | Sleeping | 18 | Intermediate | D1
0-20 | Sleeping | 18 | Evening | D1
0-20 | Sleeping | 22 | Morning | D1
0-20 | Sleeping | 22 | Intermediate | D1
0-20 | Sleeping | 22 | Evening | D1
0-20 | Sleeping | 25 | Morning | D5
0-20 | Sleeping | 25 | Intermediate | D5
0-20 | Sleeping | 25 | Evening | D5
0-20 | Sleeping | 29 | Morning | D5
0-20 | Sleeping | 29 | Intermediate | D5
0-20 | Sleeping | 29 | Evening | D5
0-20 | Eating | 14 | Morning | LU2
0-20 | Eating | 14 | Intermediate | LU2
0-20 | Eating | 14 | Evening | LU1
0-20 | Eating | 18 | Morning | LU1
0-20 | Eating | 18 | Intermediate | LU1
0-20 | Eating | 18 | Evening | D1
0-20 | Eating | 22 | Morning | LU1
0-20 | Eating | 22 | Intermediate | D1
0-20 | Eating | 22 | Evening | D1
0-20 | Eating | 25 | Morning | D1
0-20 | Eating | 25 | Intermediate | D2
0-20 | Eating | 25 | Evening | D2
0-20 | Eating | 29 | Morning | D2
0-20 | Eating | 29 | Intermediate | D3
0-20 | Eating | 29 | Evening | D3
0-20 | Meeting/Entertainment | 14 | Morning | LU3
0-20 | Meeting/Entertainment | 14 | Intermediate | LU3
0-20 | Meeting/Entertainment | 14 | Evening | LU2
0-20 | Meeting/Entertainment | 18 | Morning | LU2
0-20 | Meeting/Entertainment | 18 | Intermediate | LU2
0-20 | Meeting/Entertainment | 18 | Evening | LU1
0-20 | Meeting/Entertainment | 22 | Morning | LU2
0-20 | Meeting/Entertainment | 22 | Intermediate | LU1
0-20 | Meeting/Entertainment | 22 | Evening | LU1
0-20 | Meeting/Entertainment | 25 | Morning | LU1
0-20 | Meeting/Entertainment | 25 | Intermediate | D1
0-20 | Meeting/Entertainment | 25 | Evening | D1
0-20 | Meeting/Entertainment | 29 | Morning | D2
0-20 | Meeting/Entertainment | 29 | Intermediate | D3
0-20 | Meeting/Entertainment | 29 | Evening | D3
20-40 | Sleeping | 14 | Morning | D1
20-40 | Sleeping | 14 | Intermediate | D1
20-40 | Sleeping | 14 | Evening | D1
20-40 | Sleeping | 18 | Morning | D1
20-40 | Sleeping | 18 | Intermediate | D1
20-40 | Sleeping | 18 | Evening | D1
20-40 | Sleeping | 22 | Morning | D1
20-40 | Sleeping | 22 | Intermediate | D1
20-40 | Sleeping | 22 | Evening | D1
20-40 | Sleeping | 25 | Morning | D5
20-40 | Sleeping | 25 | Intermediate | D5
20-40 | Sleeping | 25 | Evening | D5
20-40 | Sleeping | 29 | Morning | D5
20-40 | Sleeping | 29 | Intermediate | D5
20-40 | Sleeping | 29 | Evening | D5
20-40 | Eating | 14 | Morning | LU3
20-40 | Eating | 14 | Intermediate | LU2
20-40 | Eating | 14 | Evening | LU2
20-40 | Eating | 18 | Morning | LU2
20-40 | Eating | 18 | Intermediate | LU2
20-40 | Eating | 18 | Evening | LU1
20-40 | Eating | 22 | Morning | LU1
20-40 | Eating | 22 | Intermediate | LU1
20-40 | Eating | 22 | Evening | LU1
20-40 | Eating | 25 | Morning | LU1
20-40 | Eating | 25 | Intermediate | D1
20-40 | Eating | 25 | Evening | D1
20-40 | Eating | 29 | Morning | D2
20-40 | Eating | 29 | Intermediate | D2
20-40 | Eating | 29 | Evening | D3
20-40 | Meeting/Entertainment | 14 | Morning | LU4
20-40 | Meeting/Entertainment | 14 | Intermediate | LU3
20-40 | Meeting/Entertainment | 14 | Evening | LU3
20-40 | Meeting/Entertainment | 18 | Morning | LU3
20-40 | Meeting/Entertainment | 18 | Intermediate | LU3
20-40 | Meeting/Entertainment | 18 | Evening | LU2
20-40 | Meeting/Entertainment | 22 | Morning | LU2
20-40 | Meeting/Entertainment | 22 | Intermediate | LU2
20-40 | Meeting/Entertainment | 22 | Evening | LU1
20-40 | Meeting/Entertainment | 25 | Morning | LU1
20-40 | Meeting/Entertainment | 25 | Intermediate | D1
20-40 | Meeting/Entertainment | 25 | Evening | D1
20-40 | Meeting/Entertainment | 29 | Morning | D2
20-40 | Meeting/Entertainment | 29 | Intermediate | D2
20-40 | Meeting/Entertainment | 29 | Evening | D3
40-60 | Sleeping | 14 | Morning | D1
40-60 | Sleeping | 14 | Intermediate | D1
40-60 | Sleeping | 14 | Evening | D1
40-60 | Sleeping | 18 | Morning | D1
40-60 | Sleeping | 18 | Intermediate | D1
40-60 | Sleeping | 18 | Evening | D1
40-60 | Sleeping | 22 | Morning | D1
40-60 | Sleeping | 22 | Intermediate | D1
40-60 | Sleeping | 22 | Evening | D1
40-60 | Sleeping | 25 | Morning | D5
40-60 | Sleeping | 25 | Intermediate | D5
40-60 | Sleeping | 25 | Evening | D5
40-60 | Sleeping | 29 | Morning | D5
40-60 | Sleeping | 29 | Intermediate | D5
40-60 | Sleeping | 29 | Evening | D5
40-60 | Eating | 14 | Morning | LU3
40-60 | Eating | 14 | Intermediate | LU2
40-60 | Eating | 14 | Evening | LU2
40-60 | Eating | 18 | Morning | LU3
40-60 | Eating | 18 | Intermediate | LU3
40-60 | Eating | 18 | Evening | LU2
40-60 | Eating | 22 | Morning | LU2
40-60 | Eating | 22 | Intermediate | LU1
40-60 | Eating | 22 | Evening | LU1
40-60 | Eating | 25 | Morning | LU1
40-60 | Eating | 25 | Intermediate | D1
40-60 | Eating | 25 | Evening | D1
40-60 | Eating | 29 | Morning | D1
40-60 | Eating | 29 | Intermediate | D2
40-60 | Eating | 29 | Evening | D3
40-60 | Meeting/Entertainment | 14 | Morning | LU3
40-60 | Meeting/Entertainment | 14 | Intermediate | LU2
40-60 | Meeting/Entertainment | 14 | Evening | LU2
40-60 | Meeting/Entertainment | 18 | Morning | LU3
40-60 | Meeting/Entertainment | 18 | Intermediate | LU3
40-60 | Meeting/Entertainment | 18 | Evening | LU2
40-60 | Meeting/Entertainment | 22 | Morning | LU3
40-60 | Meeting/Entertainment | 22 | Intermediate | LU3
40-60 | Meeting/Entertainment | 22 | Evening | LU2
40-60 | Meeting/Entertainment | 25 | Morning | LU1
40-60 | Meeting/Entertainment | 25 | Intermediate | D1
40-60 | Meeting/Entertainment | 25 | Evening | D1
40-60 | Meeting/Entertainment | 29 | Morning | D1
40-60 | Meeting/Entertainment | 29 | Intermediate | D2
40-60 | Meeting/Entertainment | 29 | Evening | D3
60+ | Sleeping | 14 | Morning | D1
60+ | Sleeping | 14 | Intermediate | D1
60+ | Sleeping | 14 | Evening | D1
60+ | Sleeping | 18 | Morning | D1
60+ | Sleeping | 18 | Intermediate | D1
60+ | Sleeping | 18 | Evening | D1
60+ | Sleeping | 22 | Morning | D1
60+ | Sleeping | 22 | Intermediate | D1
60+ | Sleeping | 22 | Evening | D1
60+ | Sleeping | 25 | Morning | D5
60+ | Sleeping | 25 | Intermediate | D5
60+ | Sleeping | 25 | Evening | D5
60+ | Sleeping | 29 | Morning | D5
60+ | Sleeping | 29 | Intermediate | D5
60+ | Sleeping | 29 | Evening | D5
60+ | Eating | 14 | Morning | LU2
60+ | Eating | 14 | Intermediate | LU2
60+ | Eating | 14 | Evening | LU1
60+ | Eating | 18 | Morning | LU2
60+ | Eating | 18 | Intermediate | LU1
60+ | Eating | 18 | Evening | LU1
60+ | Eating | 22 | Morning | LU2
60+ | Eating | 22 | Intermediate | LU2
60+ | Eating | 22 | Evening | LU1
60+ | Eating | 25 | Morning | LU1
60+ | Eating | 25 | Intermediate | LU1
60+ | Eating | 25 | Evening | D1
60+ | Eating | 29 | Morning | D1
60+ | Eating | 29 | Intermediate | D2
60+ | Eating | 29 | Evening | D3
60+ | Meeting/Entertainment | 14 | Morning | LU3
60+ | Meeting/Entertainment | 14 | Intermediate | LU2
60+ | Meeting/Entertainment | 14 | Evening | LU2
60+ | Meeting/Entertainment | 18 | Morning | LU3
60+ | Meeting/Entertainment | 18 | Intermediate | LU2
60+ | Meeting/Entertainment | 18 | Evening | LU2
60+ | Meeting/Entertainment | 22 | Morning | LU3
60+ | Meeting/Entertainment | 22 | Intermediate | LU3
60+ | Meeting/Entertainment | 22 | Evening | LU2
60+ | Meeting/Entertainment | 25 | Morning | LU1
60+ | Meeting/Entertainment | 25 | Intermediate | LU1
60+ | Meeting/Entertainment | 25 | Evening | D1
60+ | Meeting/Entertainment | 29 | Morning | D1
60+ | Meeting/Entertainment | 29 | Intermediate | D2
60+ | Meeting/Entertainment | 29 | Evening | D3


# Macro to micro phase separation in a collection of chiral active swimmers

Vivek Semwal<EMAIL_ADDRESS>Indian Institute of Technology
(BHU) Varanasi, India 221005  Jayam Joshi<EMAIL_ADDRESS>Indian Institute of Technology (BHU) Varanasi, India 221005  Shradha Mishra
<EMAIL_ADDRESS>Indian Institute of Technology (BHU) Varanasi, India
221005

###### Abstract

We studied a collection of chiral active particles (CAP) on a two dimensional
substrate using extensive numerical study. Particles interact through soft
repulsive interaction. The activity and chirality of particles is tuned by
varying their self-propulsion speed and angular velocity respectively.
Kinetics and steady state properties of particles are studied for different
chirality and activity. The phase diagram of system on the plane of activity
and chirality shows three distinct phases. For small chirality when activity
is dominant, particles show enhanced dynamics and macroscopic phase separation
of ordered clusters is observed. For moderate chirality, micro clustered phase
is observed in which small clusters with moderate ordering are formed. For
large chirality, when chirality dominates, no clustering is found because
particle motion is mainly confined to its location. Our study gives a detail
insight into the effect of chirality on the properties of collection of CAP,
which can be useful to understand the dynamics and steady state of many
natural micro swimmers.

††preprint:

##  I Introduction

Active Brownian particles (ABPs) are prominent example of active matter Toner
_et al._ (2005); Kumar  _et al._ (2014); Ghosh and Fischer (2009); Liao and
Klapp (2018); Levis and Liebchen (2018). ABPs combine Brownian motion with
self-propulsion. Motile microorganisms are frequently characterised as ABPs,
in addition to artificial self-propelled microparticles Bechinger  _et al._
(2016); Liao and Klapp (2018); Levis and Liebchen (2018). Even bacteria that
conduct a run-and-tumble action Peruani and Bär (2013); Bechinger  _et al._
(2016); Shaebani  _et al._ (2020); Mino  _et al._ (2011), such as Escherichia
coli, have been effectively classified as ABPs. One of the remarkable property
of ABP’s are motility induced phase separation (MIPS) without any cohesive
interaction among the particles Gonnella  _et al._ (2015); Kümmel  _et al._
(2013); Bechinger  _et al._ (2016); Kumar  _et al._ (2021); Jepson  _et al._
(2013); Leptos  _et al._ (2009). Most of theoretical and simulation study of
ABP is focused on systems without chirality Gonnella  _et al._ (2015). But
chirality is an inherent property in many natural active particles Bechinger
_et al._ (2016); Elgeti  _et al._ (2010); Menzel (2015); Kraft  _et al._
(2013); Ghosh and Fischer (2009); Liao and Klapp (2018). Hence effect of
chrality on the properties of ABPs is an important question to be asked.

In Keaveny and Shelley (2009); Keaveny  _et al._ (2013), motion of
microswimmers is studied in the presence of chirality. Chirality leads to the
deviation of particle trajectory from the straight line motion. In a recent
review Löwen (2016), it is shown that an individual or collection of chiral or
circle swimmer can show interesting properties. When present in bulk they can
also show active turbulence Keaveny  _et al._ (2013); Winkler (2016); Ginelli
(2016); Levis and Liebchen (2018). Our study is motivated with recent study of
Ma and Ni (2022) dynamic clustering of chiral active particles. Chirality of
particles suppress the motility induced phase separation present for nonchiral
active Brownian particles Cates and Tailleur (2015). Hence effect of chirality
on the properties of individual and collective behaviour of active particles
can give good understanding of another class of nonequilibrium system called
chiral active particles (CAPs). Most of the recent study of CAPs have focused
on the effect of chirality on the kinetics or steady state properties of
active particles on the variation of activity or packing densities of the
particles Ghosh and Fischer (2009); Liao and Klapp (2018); Levis and Liebchen
(2018). The study of properties of active particles, on the variation of
chirality is very scarce. But such study can provide a good understanding of
effect of chirality on the properties of active particles. The two extremes:
small or large chirality is trivial: for the first case we expect the results
of ABPs in the collection and for the later case we expect mainly the confined
circular motion. But what happen when we slowly tune the chirality from small
to large values still unexplored. In the present work we focus on this effect
of variation of chirality on the properties of active particles.

Here we show the kinetics and steady state properties of CAPs on the variation
of chirality and activity. The system is found in three distinct phases: (i)
for small chirality, when activity dominates, system shows the enhanced
diffusionMa and Ni (2022); Omar  _et al._ (2021); Klamser  _et al._ (2018) and
macroscopic clustering as found in MIPS. In the second phase where both
activity and chirality are in competition, we find some clustering, but no
macro phase separation. For larger chirality, the chirality dominates over
activity and dynamics of particles is mostly confined to its location and no
clustering is observed.

Our article divided in the following sections: In section II, we give the
detailed description of our model. In section III discuss about the results of
numerical simulation of the system. In section V we conclude our result and
discussion about the future directions of our study.

##  II Model

Our system consists of $N$ chiral active particles (CAP) of radius $a_{0}$ on
a two-dimensional substrate. On the substrate each $i^{th}$ particle is
represented by its position vector ${\bf r}_{i}(t)$ and orientation
$\theta_{i}(t)$, at time $t$. The dynamics of the particle is governed by the
overdamped Langevin equation Semwal  _et al._ (2021a); Ramaswamy (2017); Fodor
and Marchetti (2018); Uhlenbeck and Ornstein (1930); Semwal  _et al._ (2021b)

$\partial_{t}{\bf{r}}_{i}=v{\bf\hat{{n}}_{i}}+\mu_{1}\sum_{j\neq
i}{\bf{F}}_{ij}$ (1)
$\partial_{t}\theta_{i}=\omega+\sqrt{2D_{r}}\bf{\eta_{i}}$ (2)

Figure 1: (color online) Trajectory of the particle for (a) small
$\omega_{0}=0.001$, (b) intermediate $\omega_{0}=0.2$ and (c) large
$\omega_{0}=100$. The blue and red arrows show the starting and end of the
trajectory for a fixed time interval. The numbers on the axes show the
coordinates of the particle at different times. Notice the change in the
extend of the trajectory for three different $\omega_{0}$’s.

The first term on the right hand side (RHS) of Eq. 1 is due to the activity of
the particle, and $v$ is its self-propulsion speed. The particle moves along
its unit orientation direction vector
${\bf\hat{n}}_{i}=(\cos{\theta_{i}},\sin{\theta_{i}})$ with speed $v$. The
second term represents the steric force, ${\bf
F}_{ij}=F_{ij}{\bf\hat{r}}_{ij}$, which takes care of the repulsive
interaction, acting on the $i^{th}$ particle due to its neighbouring particles
in contact with it. Hence we consider $F_{ij}=k(2a_{0}-r_{ij})$ if
$r_{ij}\leq{2a_{0}}$ and $F_{ij}=0$ if $r_{ij}\geq{2a_{0}}$, where $r_{ij}$ is
the centre to centre distance between $i^{th}$ and $j^{th}$ particles,
$r_{ij}=|{{\bf r}_{i}-{\bf r}_{j}}|$. $k$ is the strength of the force,
$F_{ij}$ is tuned by the mobility $\mu_{1}$ of the particles. Further, the
orientation of the particle is updated by Eq. (2) where $\omega$ is the
chirality (angular velocity) of the particle and $\eta_{i}$ is Gaussian white
noise term. $D_{r}$ is the rotational diffusivity. The smallest time step
considered is $\delta t=0.005$. We define the dimensionless chirality as
$\omega_{0}=\frac{\omega}{D_{r}}$, and the dimensionless activity of the
particles as $v_{0}=\frac{v}{a_{0}\times D_{r}}$. We start our simulation with
random position and orientation of particles on the substrate and evolve the
system by integrating the Eqs. 2 and 2 using Euler’s integration scheme. The
system is simulated for total simulation time steps of $10^{6}$. One
simulation step is counted after update of all the particles once. All the
physical quantities calculated here are averaged $50$ realizations. The tuning
parameters are chirality $\omega_{0}$ and activity $v_{0}$. We tuned
$\omega_{0}$ from $0$ to $100$ and $v_{0}$ from $1$ to $10$. Simulation is
performed in box of size $L=150\times a_{0}$, with packing fraction
$\frac{N\pi\sigma^{2}}{L^{2}}=0.6$.

##  III Results

We first observe the trajectory of a single particle in the collection for
different chirality. In Fig. 1(a-c) we show the trajectory for three different
values of chirality $\omega_{0}=0.001$, $0.2$ and $100$ respectively and for
fixed self-propulsion speed $v_{0}=10$. For small $\omega_{0}=0.001$, Fig.
1(a) trajectory looks very extended. The blue and black arrows show the
starting and end point of trajectory respectively. For intermediate
$\omega_{0}=0.2$, Fig. 1(b), trajectory looks localised for early time and
then extended towards the end. For large $\omega_{0}=100$, Fig. 1(c)
trajectory is always confined within a small region. For comparison the time
difference from the start and end of the trajectory is kept the same for all
three cases.

To quantify the above observation, we measure the mean square displacement of
the particles for different chirality. We define the particles mean square
displacement as $\Delta(t)=<(r_{i}(t+t_{0})-r_{i}(t_{0}))^{2}>$ where $<..>$
denotes average over all the particles, many reference times $t_{0}$ and over
different realisations. In general the dynamics of active particles shows an
early time ballistic dynamics and then crossover to late time diffusion. In
Fig. 2 we show the behaviour of MSD for fixed activity $v_{0}=10.0$ and
varying the chirality. For zero chirality system shows a very clear crossover
from early time ballistic to late time diffsuion. As we introduce chirality,
$\Delta(t)$ shows oscillations, which is due to the oscillations of particles
trajectory for finite chirality. The periodic oscillations increases and
crossover time decreases on increasing $\omega_{0}$. We extract the typical
crossover time $t_{c}(\omega_{0})$ by fitting the MSD with the persistent
random walk= $\Delta(t)=4D_{eff}t[1-exp(-t/t_{c})]$, for different
$\omega_{0}$ and plot is shown in the inset of Fig. 2(a). The crossover time
remains almost constant for smaller chirality $\omega_{0}\leq 0.1$ and then
show a smooth decay for intermediate $0.1<\omega_{0}<10.0$ and then decays
sharply for larger $\omega_{0}>10.0$. For large chirality the dynamics of
particle is no longer diffusive for late time and hence $t_{c}$ cannot be
calculated.
To further characterise the dynamics of particle we also calculated the late
time effective diffusivity $D_{eff}(\omega_{0})$ for different $\omega_{0}$
for three different $v_{0}=1,10$ and $100$. The late time effective
diffusivity $D_{eff}$ is obtained by
$D_{eff}=\lim_{t\to\infty}\frac{\Delta(t)}{4t}$ The plot of
$D_{eff}(\omega_{0})$ vs. $\omega_{0}$ is shown in Fig. 2(b). For large
activities $v_{0}=10$ and $100$, we find that for small chirality
$\omega_{0}\leq 0.1$, the $D_{eff}(\omega_{0})$ remains flat and then for the
intermediate $\omega_{0}\in(0.1,10)$, shows a shallow region with slow decay
and finally for larger chirality decay sharply to very small values. For
smaller activities $v_{0}=1.0$, the dynamics for larger chirality is mostly
confined hence the $D_{eff}$ cannot be defined.
For comparison we also calculated the MSD for noninteracting single chiral
particle analytically. The calculation is performed by making the interaction
force term zero. In the absence of interaction, the overdamped Langevin
equations 3 and 4 reduces to:

$\partial_{t}{\bf{r}}_{i}=v{\bf\hat{{n}}_{i}}$ (3)
$\partial_{t}\theta_{i}=\omega+\sqrt{2D_{r}}\bf{\eta_{i}}$ (4)

these equations can be solved to obtain the mean square displacement of chiral
active particle without interaction. The first and second moments of
$\theta(t)$ are simply $<\theta(t)>=\theta_{0}+\omega t$ and
$<\theta^{2}(t)>=\omega^{2}t^{2}+2D_{r}t$, where, $\theta_{0}=\theta(t=0)=0$.
Using this, the second moment of $r(t)$ or the mean square displacement is
obtained:

$<r^{2}(t)>=\Delta(t)=\frac{2v^{2}}{(\omega^{2}+D_{r}^{2})^{2}}[\omega^{2}-D_{r}^{2}+D_{r}(D_{r}^{2}+\omega^{2})t\\\
+e^{-D_{r}t}((D_{r}^{2}-\omega^{2})cos(\omega t)-2D_{r}\omega sin(\omega t))]$

The late time diffusivity or $D_{eff}$ is defined to be
$\lim_{t\to\infty}\frac{\Delta(t)}{4t}$. Hence in the absence of interaction
the diffusivity is given by the expression:

$D_{eff}=\frac{v^{2}D_{r}}{2(\omega^{2}+D_{r}^{2})}$ (5)

Writing $D_{eff}$ in terms of dimensionless chirality $\omega_{0}$,

$D_{eff}=\frac{v_{0}^{2}}{2D_{r}(\omega_{0}^{2}+1)}$ (6)

In Fig. 2(b), lines are from the analytical expression of $D_{eff}$ as given
in Eq. 6. For small chirality for all activities the $D_{eff}$ for
noninteracting case is smaller than the interacting full numerical simulation.
Later we are going to show for the same range of chirality particles show the
clustering. Hence in this regime collective dynamic of particles is
responsible for the enhanced dynamics in comparison to the noninteracting
single particle dynamics. As we increase chirality, the noninteracting
$D_{eff}$ decay smoothly to zero values for all activities, whereas for
interacting case, it shows a small plateau for intermediate chirality. Hence
we can say that the second region with intermediate chirality is all due to
the interaction between the particles. Later we are going to explore the
system more for the three different regions using numerical study of
interacting system. Further we classify the three regions for the interacting
system as region $I$, $II$ and $III$ as marked in Fig. 2(b). In next sections
we discuss in detail the how does the clustering of particles is changed due
to the effectively three different dynamics in the three regions.

Figure 2: (color online) (a) Plot of MSD, $\Delta$(t) vs. time $t$ for
different value of chirality. Dashed and solid lines are of slope $2$ and $1$
respectively. (inset) crossover time $t_{c}$ vs. $\omega_{0}$. Dotted orange
lines show three different regime I, II and III. (b) $D_{eff}$ vs.
$\omega_{0}$ for three different values of $v_{0}$. Different symbols show
numerical data, straight line shows the plot of Eq. 6

###  III.1 Cluster size distribution

In the previous section we studied the effect of chirality on the effective
dynamics of the system. Now we study how the change in effective dynamics
changes the clustering and phase behaviour of particles in the system. In Fig.
4(a-c) we show the snapshots of particles for three different $\omega_{0}$
$=0.001,1.0$ and $50$ respectively in regions I, II and III for activity
$v_{0}=10.0$. For the I region, Fig. 4(a) we clearly see the macroscopic
clustering in the system, As we to to the region II, Fig. 4(b) we see the
microscopic clustering, Finally in region III, Fig. 4(c) there is no
clustering and system is homogeneously distributed. Further we calculate the
cluster size distribution (CSD) for different chirality for activity
$v_{0}=10$ as shown in Fig. 3(a). The CSD is defined using the particles
connected by a most probable distance $r_{0}$. In Fig. 3(a) we plot the
normalised CSD $P(n)$ vs. $n$ for different chirality $\omega_{0}$,
specifically chosen in the three regions of the $D_{eff}$ plot shown in Fig.
2(b), where $n$ is the size of the cluster. We find that for small
$\omega_{0}=0.2$, $P(n)$ decays as power law $\simeq\frac{1}{n^{\alpha}}$ for
large $n$, with exponent $\alpha=2$ as reported in previous studies Ding and
Aidun (2006). As we increase $\omega_{0}$ and system transits into the second
phase $P(n)$ still decay as power law, but the exponent $\alpha\simeq 3$, On
further increasing the chirality of the particle $P(n)$ decays exponentially
with $n$. Hence the three regions which are defined based on the effective
dynamics of particles in the steady state: also lead to different types of
clustering of particles. We also calculated the average cluster size
$n_{av}(\omega_{0})$ for different $\omega_{0}$ and found that it also shows
three different regions as shown in Fig. 3(b) for $v_{0}=10$.
$n_{av}(\omega_{0})$ is defined as $n_{av}(\omega_{0})=\int nP(n)dn$. In the
first region as shown in Fig.3(b), system shows the formation of macroscopic
clusters with an average cluster size $n_{av}(\omega_{0})$ around 75 in the
(I) phase, $n_{av}(\omega_{0})$ varies from $70$ to $25$ for the $\omega_{0}$
in the (II) phase. As we further increase $\omega_{0}$, system enters in to
the third region and homogeneous state of the particles and $n_{av}$ decay
sharply to very small values. Now we try to understand how the chirality
affects the structural ordering of particles in the system.

|
---|---
(a) | (b)
Figure 3: (a) $P(n)$ vs. $n$ for $v_{0}=50$ for three different values of
$\omega_{0}=0,0.2$ and $50$ chosen in three different regions. Inset: shows
the zoomed plot of $P(n)$ vs. $n$ for large $n$. The solid and dashed lines
are lines of slope $-2$ and $-3$ respectively. The dotted dashed curve is
exponential fitted curve. (b) $n_{av}(\omega_{0})$ vs. $\omega_{0}$. Dotted
orange lines show the different regions I, II and III.

###  III.2 Structural ordering

We further try to understand the effect of chirality on phase separation and
structural ordering in the system. We define local density around a particle
$\rho_{loc}$ as function of chirality for different activities. In general a
nonchiral system shows a motility induced phase separation (MIPS) on tuning
the activity or packing fraction Klamser  _et al._ (2018). Here we are
interested on the effect of chirality keeping packing fraction fixed and for
different activity. We define the local density $\rho_{loc}$ with the help of
the number of particles surrounding a given particle.
$\rho_{loc}=\frac{n_{p}}{6}$, where $n_{p}$ is the number of particles
surrounding the given particle. For a perfectly packed surrounding we expect
number is $6$ for hexagonal close packed (HCP) structure and then we define
that the $\rho_{loc}=1$. Hence in this way for a given snapshot of the system
we have a distribution of $\rho_{loc}$, $P(\rho_{loc})$ where the probability
distribution function $P(\rho_{loc})$ is obtained by looking $\rho_{loc}$ of
each particle. For a perfect clustered phase the $\rho_{loc}$ will be peaked
around $1$ for completely homogeneous phase $\rho_{loc}$ will approach the
mean packing density $\rho_{loc}=0.6$ of the system. In Fig. 5(a) we show the
plot of location of peaks $\rho_{0}$ of $P(\rho_{loc})$ vs. $\omega_{0}$ for
three different values of activities $v_{0}=1,10,30$. We find that for small
$v_{0}=1$, $P(\rho_{loc})$ has only one peak. For small $\omega_{0}$,
$\rho_{0}$ remains flat close to $0.7$ and then decay to mean packing density
$0.6$ for larger chirality. For activity $v_{0}=10$, the $P(\rho_{loc})$ is
bimodal (data not shown) and location of two peaks at smaller and larger
$\rho_{0}$’s is shown in Fig. 5(a). For small chirality or in the I region the
two peaks are widely separated and as we enter the II region the difference
between two peaks diminishes and for IIIrd region or for high chirality
$\rho_{0}$ approaches value close to mean packing density. For larger activity
$v_{0}=30$, in the I region $\rho_{0}$ remains close to 1 and the smoothly
decay to moderate values in the II region and finally approaches to $0.6$ in
the III region. For large activities, we do not find bimodal distribution of
$P(\rho_{loc})$ due to the very strong clustering in the system.

Now to further understand the effect of chirality on the structural ordering
in the system.

|  |
---|---|---
(a) | (b) | (c)
Figure 4: Snapshots in three different regions (a) for $\omega_{0}=0.001$, (b)
for $\omega_{0}=1.0$ (b) and (c) for $\omega_{0}=50$ and all are for
$v_{0}=10$ .

We define the local bond order parameter $\psi_{6}(t)$ Mermin (1968); Lechner
and Dellago (2008)

$\psi_{6}(t)=\frac{1}{N_{p}}\sum_{k=1}^{N_{p}}\sqrt{\frac{1}{N_{k}}\sum_{j=1}^{N_{k}}e^{i6\theta_{kj}}}$
(7)

$\psi_{6}(t)$ measures the amount of hexagonal ordering in the system. For the
perfect hexagonal close packed (HCP) structure, $\psi_{6}(t)$ will be close to
$1$ and for perfect random arrangement it is close to $0$. In Fig. 5(b) we
show the plot of mean value of $\psi_{6}(t)$, $\Psi_{6}(t)=<\psi_{6}(t)>$,
where the mean $<..>$ is average over time in the steady state and over
realisations. For $v_{0}=1$, $\Psi_{6}$ remains close to small values $0.08$
for small chirality and decay on increasing $\omega_{0}$. For large activity
and small chirality (region I) the $\Psi_{6}\simeq 0.65$ and remains flat in
the first region and then smoothly decay in region II and finally approaches
very small values of the $\mathcal{O}(10^{-}2)$ as system approaches III
region. Hence very clearly not only density shows the three types of
clustering in three regions, but structural ordering also distinctly shows
three regions with variation of chirality for high activities.

|
---|---
(a) | (b)
Figure 5: (a) $\rho_{0}$ vs. $\omega_{0}$ for different values of $v_{0}$. .
(b) Plot of $\Psi_{6}$ vs. $\omega_{0}$ for different values of $v_{0}$.
Dotted orange lines have the same meaning as in Fig. 3

##  IV Phase Diagram

Based on the above results of $\Psi_{6}$ we have drawn the phase diagram in
the plane of ($v_{0}$, $\omega_{0}$). The system is found in three distinct
phases: (i) Homogeneous State (HS) defined as small structural ordering
$\Psi_{6}<0.025$, $\rho_{0}$ close to mean packing density $0.6$, exponential
CSD shown by triangles in the figure 6. This phase is found for large
chirality and all activities. (ii) Microscopic cluster (MIC), defined as
moderate structural ordering $0.025<\Psi_{6}<0.25$, $\rho_{0}$ decays to
moderate values and CSD decay algebrically with larger exponent $3$. This
phase is shown as circles in the phase diagram. (iii) Macroscopic cluster
(MAC), with large $\Psi_{6}>0.25$, large $\rho_{0}$ close to $1$ and CSD
decays algebrically with power close to $2$. This phase is shown using squares
in the phase diagram. The color shows the value of $\Psi_{6}$ for different
parameters ($v_{0}$, $\omega_{0}$).

Figure 6: (color online) Phase diagram of the system in ($\omega_{0}$,
$v_{0}$) plane. Squares show the MAC phase, circles show the MIC phase and
rectangle shows the HS phase. Color bar shows the magnitude of $\Psi_{6}$
order parameter.

##  V Discussion

In conclusion we show the effect of varying the chirality on the collection of
circle micro swimmers. The competition between activity and chirality leads to
three distinct phases as we slowly tune the chirality. For small chirality
when linear motion dominates, effective dynamics of particles is enhanced, in
comparison to single chiral particle with the same chirality. It leads to
macroscopic clustering of particles. For intermediate chirality when linear
and circle motion are comparable, the particles show weaker clustering with
small cluster formation. The effective dynamics is suppressed. For strong
circle swimmer of large chirality the motion of particle is mostly confined to
its own location and no clustering is observed. The interaction among the
particle leads to such three distinct phases, whereas for noninteracting
chiral system only two types of dynamics is observed. The presence of three
distinct phases for different chirality gives detail understanding of effect
of chirality on the particles dynamics and steady state properties. This can
be useful to use sorting of particles, based on their chirality. Hence our
study can be useful in pharmaceutical industry as well as in clinical
therapeutics and sorting of drugsAbeylath and Turos (2008); Bechinger  _et
al._ (2016); Nourhani  _et al._ (2015); Mijalkov and Volpe (2013).

##  VI conflict of interest

Here is no conflict of interest.

##  VII Acknowledgement

The authors gratefully acknowledge the DST support for funding this project.
S. Mishra thanks DST, SERB (INDIA), Project No. ECR/2017/000659 for partial
financial support.

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