# Rethinking the modeling of the instrumental response of telescopes with a
differentiable optical model

Tobias Liaudat, Jean-Luc Starck, Martin Kilbinger, Pierre-Antoine Frugier
AIM, CEA, CNRS
Université Paris-Saclay
Université de Paris
F-91191 Gif-sur-Yvette, France
{tobias.liaudat<EMAIL_ADDRESS>
{martin.kilbinger<EMAIL_ADDRESS>https://tobias-
liaudat.github.io

###### Abstract

We propose a paradigm shift in the data-driven modeling of the instrumental
response field of telescopes. By adding a differentiable optical forward model
into the modeling framework, we change the data-driven modeling space from the
pixels to the wavefront. This allows to transfer a great deal of complexity
from the instrumental response into the forward model while being able to
adapt to the observations, remaining data-driven. Our framework allows a way
forward to building powerful models that are physically motivated,
interpretable, and that do not require special calibration data. We show that
for a simplified setting of a space telescope, this framework represents a
real performance breakthrough compared to existing data-driven approaches with
reconstruction errors decreasing $5$ fold at observation resolution and more
than $10$ fold for a $3$x super-resolution. We successfully model chromatic
variations of the instrument’s response only using noisy broad-band in-focus
observations.

##  1 Introduction

In astrophysics and cosmology, as in many other fields of physics, the next
decade will bring about a new generation of extremely powerful instruments,
such as Euclid [1] or the Nancy Grace Roman Space Telescope [2]. A proper
modeling of the instrumental response is an absolute prerequisite to the
ambitious science goals of these missions. Current data-driven modeling
approaches are not able to cope with the stringent error requirements imposed
by the new generation of instruments. They can even be orders of magnitude
away from the prescribed error budget [3], and they are not able to model
chromatic variations (i.e. with wavelength).

In this paper we illustrate how some new technologies brought about by the
Deep Learning (DL) revolution can be leveraged to rethink the way we model the
instrument response or point spread function (PSF). To this end, we follow the
approach in [4], and propose a paradigm shift in the form of a novel framework
that includes a differentiable optical forward model. This allows to shift the
usual data-driven modeling space from the pixels to the wavefront and
translate much of the modeling complexity into the forward model. The use of
DL methods for the modeling of instrumental response fields is limited due to
the complexity of the modeling problem and the fact that current efforts are
blind to the physics of the problem [5]. The present work lays the groundwork
for the introduction of physically-motivated and interpretable DL methods into
the modeling of instrumental response fields.

##  2 Point spread function modeling for space mission telescopes

The modeling of the PSF in the field of view (FOV) can be seen as an inverse
problem. Telescope observations include point-like stars and extended objects
of interest at different positions in the FOV. We consider stars as samples of
the PSF field in the FOV and use them to constrain our model, which we then
use to infer the PSF at target positions. PSF field modeling encompasses
several challenges:

  * 1.

The PSF varies spatially in the FOV. The model needs to capture the spatial
variations of the PSF shape from the stars in order to infer the PSF at target
positions.

  * 2.

The observations are in general under-sampled. The model needs to super-
resolve the output PSFs. This differs from the usual super-resolution (SR)
task, as we do not have several observations of the same object we need to
super-resolve. In this case, we have several samples of the under-sampled PSF
field at different positions in the FOV.

  * 3.

The PSF varies as a function of wavelength. Also known as chromatic
variations, they need to be included in the PSF model for most science goals.
When the instrument has a broad passband, each star observation is integrated
with respect to its wavelength throughout the passband.

###  2.1 Related work

#### Data-driven PSF models

Classical data-driven PSF models only rely on the stars to build the model in
pixel space and are blind to the physics of the inverse problem. They mostly
differ in the way they handle the spatial variations and the super-resolution
[6, 7, 3, 8, 9]. They have difficulties in modeling complex PSF shapes such as
those from space missions which are close to the diffraction limit. There is
no published method capable of successfully modeling the chromatic variations.

#### Parametric PSF models

This class of models, on the contrary, builds a parametric model of the entire
optical system that should be as close as possible to the actual telescope.
Then, a few number of model parameters are fit to the observations, in some
cases wavefronts. Errors will arise when there is a mismatch between the
parametric model and the ground truth. Furthermore, even if, ideally, there
were no mismatch, the optimization of these models in wavefront space is a
degenerate problem. It requires potentially expensive calibration information,
usually in the form of out-of-focus observations, in order to break
degeneracies. Nevertheless, a model of this type has been used for the Hubble
Space Telescope (HST) [10, 11].

#### Phase retrieval with automatic differentiation

Estimating the wavefront of an in-focus observation falls in the category of
phase retrieval problems. Recent works [12, 13], based on a framework
established by [14], are tackling this problem [15] using automatic
differentiation. Their objective is solely to estimate the wavefront of a
single in-focus image. Such methods are unusable for the current PSF modeling
problem as we are interested in the pixel representation at positions where we
do not have observations of the PSF. In the proposed model, the wavefront is
an intermediate product for our goal rather than as an objective in itself.

##  3 Data-driven wavefront PSF model

The stars observed in our FOV are samples of the PSF field. The following
observational model,

$\bar{I}(x_{i},y_{i})=\mathcal{F}\left\\{\int_{\text{passband}}\text{SED}(x_{i},y_{i};\lambda)\;I(x_{i},y_{i};\lambda)\;\text{d}\lambda\right\\}+\mathbf{n}_{i}\;,$
(1)

relates the star observation $\bar{I}(x_{i},y_{i})\in\mathbb{R}^{N\times N}$
in the FOV position $(x_{i},y_{i})\in\mathbb{R}^{2}$ with the objective
modeling quantity $I(x_{i},y_{i};\lambda)\in\mathbb{R}^{M\times M}$, which is
at a higher resolution with respect to the observations ($M>N$). The desired
instrumental response $I$ is integrated in the instrument’s passband weighted
by the star’s normalized spectral energy distribution
$\text{SED}(x_{i},y_{i};\lambda)\in\mathbb{R}_{+}$, which we consider to be
known. Then, it is degraded with the operator $\mathcal{F}:\mathbb{R}^{M\times
M}\to\mathbb{R}^{N\times N}$, which accounts for down-sampling and could
include other types of pixel-level degradation (e.g. sub-pixel shifts, pixel
response). The observational noise is modeled by $\mathbf{n}_{i}$, which we
assume to be Gaussian for simplicity, i.e.
$\mathbf{n}_{i}\sim\mathcal{N}(0,\sigma_{i}^{2}\mathbf{I}_{N})$. The PSF
modeling problem consists in estimating
$\\{I(x_{j},y_{j};\lambda)\\}_{j=1,\ldots,n_{\text{target}}}$ for all the
target positions and wavelengths having as input a set of observations
$\\{\bar{I}(x_{i},y_{i})\\}_{i=1,\ldots,n_{\text{obs}}}$.

Up to now, data-driven models were built using some type of dimensionality
reduction method applied in pixel space, where our observations belong, being
blind to the optical system. The modeling problem in pixel space is too
complex for the quality and amount of observed data, and they cannot cope with
the new challenges of upcoming instruments. We propose a complete paradigm
shift with respect to the way data-driven PSF models have been constructed so
far. We, instead, model the PSF field in the wavefront space without any
special calibration information. More specifically, we model the optical
system’s aberrations as the difference in phase with respect to a wavefront
from a flawless optical system. This crucial change is allowed by the
inclusion of a physics-based forward model that can go from the wavefront to
the pixel representation. The forward model is end-to-end differentiable
thanks to modern automatic differentiation frameworks [16]. It allows
propagating gradients from the reconstruction error to the wavefront
parameters of our PSF model through the optical system and the degradations.
An illustration of the proposed framework can be seen in Figure 1.

Figure 1: A schematic of the proposed framework for data-driven wavefront PSF
modeling.

The forward model is built under Fraunhofer’s diffraction, which is valid for
these astronomical telescopes. This allows to relate the wavefront aberrations
to the pixel PSF by means of the squared absolute value of the fast Fourier
transform (FFT) of the complex pupil function [17, 18]. The sampling can be
controlled by a variable zero-padding of the pupil function. The reconstructed
PSF at any specific wavelength and FOV position can be written as

$\hat{\mathcal{H}}(x_{i},y_{i};\lambda)\propto\underset{p\times p\to M\times
M}{\text{crop}}\underbrace{\Bigg{|}\texttt{FFT}\bigg{\\{}\overbrace{P\odot\exp\Big{[}\frac{2\pi\text{i}}{\lambda}\underbrace{\Phi(x_{i},y_{i})}_{\text{Wavefront
PSF model }}\Big{]}}^{\text{Pupil
function}}\bigg{\\}}\Bigg{|}^{2}}_{\text{Pixel representation}}\;,$ (2)

where $P\in\mathbb{R}^{p\times p}$ represents the telescope’s obscurations,
and $\Phi:\mathbb{R}^{2}\to\mathbb{R}^{p\times p}$ is the wavefront PSF model
that maps a position in the FOV with its corresponding wavefront. Once the
reconstructed PSF $\hat{\mathcal{H}}$ from Equation 2 goes through Equation 1,
it can be matched with the observations (see Figure 1).

With the proposed framework, a big part of the problem’s complexity is shifted
from the PSF model into the forward model that is now encoding all the
diffraction phenomena. The chromatic variation due to diffraction are
described by the $\lambda$ dependence in Equation 2. This allows to simplify
the building of the PSF model block seen in Figure 1. We now need to focus on
a good generalization capability of the PSF to target positions, and the model
adaptation to the observed data. We propose to build the wavefront PSF model,
$\Phi$, using a weighted sum of wavefront features. The weights will vary as a
function of the FOV position $(x,y)$, which writes
$\Phi(x,y)=\sum_{l}f^{\text{Z}}_{l}(x,y)S^{\text{Z}}_{l}+\sum_{k}f^{\text{DD}}_{k}(x,y)S^{\text{DD}}_{k}$
with $f^{\text{Z}}_{l},f^{\text{DD}}_{k}:\mathbb{R}^{2}\to\mathbb{R}$ and
$S^{\text{Z}}_{l},S^{\text{DD}}_{k}\in\mathbb{R}^{p\times p}$. The features
$S^{\text{Z}}$ are not learned, but based on Zernike polynomials [19]. These
provide a good basis for modeling circular apertures as they are orthogonal on
the unit disk. The data-driven (DD) features $S^{\text{DD}}$ are non-
parametric and entirely learned from the data. These DD features adapt to
capture variations that are not well modeled by the Zernike features (or
modes) and also account for mismatches between our forward model and the
ground truth. For $f^{\text{Z}}_{l},f^{\text{DD}}_{k}$ we emphasize on the
generalization, and we use for each FOV position polynomials up to orders
$d^{Z}$ and $d^{DD}$, respectively. For example, if $d^{\text{Z}}=1$ we have
$f^{\text{Z}}_{1}(x,y)=a_{1}+b_{1}x+c_{1}y$.

In a nutshell, we optimize over the polynomial coefficients of
$f^{\text{Z}}_{l},f^{\text{DD}}_{k}$, and the DD features $S^{\text{DD}}$. The
loss function used is the mean squared error weighted by the inverse of the
estimated noise level, e.g.
$L\propto\sum_{i}(1/\hat{\sigma}_{i})\|\bar{I}_{i}-\bar{\mathcal{H}}_{i}\|^{2}_{F}$.
The optimizer used is the Rectified ADAM [20] algorithm. If prior information
is available, it can be added as regularizations at the wavefront level or the
pixel level.

##  4 Experiment

To demonstrate our novel framework, we simulate a simplified FOV with $2000$
star observations for training and $400$ noiseless target stars for testing.
All the positions are randomly distributed in the FOV, and the observations
have a variable signal-to-noise-ratio (SNR) uniformly distributed in the range
$[10,110]$. We use parameters for an optical model close to the Euclid’s VIS
instrument model [1, 21], but do not consider further detector effect. For the
SEDs, we randomly choose for each star one of the $13$ templates from [22] as
done in [23, §5.3]. For the ground truth (GT) model we use $45$ Zernike modes
with a $d^{\text{Z}}$ of $2$ for each mode. The polynomial coefficients are
randomly chosen taking care that the total aberration at any position is
within certain limits. The dimensions used are, $p=256$, $M=64$ and $N=32$.

We compare: i) a wavefront PSF model with $15$ Zernike modes and without DD
part using $d^{\text{Z}}=2$, ii) a model equivalent to i) but increasing the
Zernike modes to $40$, iii) a wavefront PSF model with $15$ Zernike modes,
$d^{\text{Z}}=2$, and $21$ DD features ($d^{\text{DD}}=5$), iv) a widely-used
model PSFEx [7], and v) the current state-of-the-art data-driven model RCA [3]
which has been specially designed for the Euclid mission. All the models are
compared in the reconstruction of the target stars to evaluate their
generalization power in addition to their modeling capabilities. The metrics
analyzed are the root mean squared error (RMSE) of the reconstruction of
target stars, at $1$ and $3$ times the observation resolution.

#### Results

Table 1: Target star reconstruction RMSE at the observation resolution and at SR. | RMSE [$\times 10^{-5}$] (relative)
---|---
PSF model | Resolution x1 | Resolution x3
i) Zernike 15 |  $72.3$ ($10.0\%$) |  $18.3$ ($12.4\%$)
ii) Zernike 40 |  $22.2$ ($3.0\%$) |  $5.75$ ($3.9\%$)
iii) Zernike 15 + DD |  $\mathbf{8.34}$ ($\mathbf{1.1\%}$) |  $\mathbf{4.47}$ ($\mathbf{3.0\%}$)
iv) PSFEx |  $69.2$ ($9.5\%$) |  $66.3$ ($43.0\%$)
v) RCA |  $39.6$ ($5.4\%$) |  $85.3$ ($55.5\%$)

Table 1 summarizes the main results. First, one can notice the breakthrough in
performance by observing the gap between model (iii) and the two models
without the forward model (RCA, PSFEx), almost $5$ times and more than $10$
times lower RMSE for resolutions x$1$ and x$3$, respectively. One can see that
models (i) and (iii) use a reduced number of Zernike modes with respect to the
GT model that uses $45$, three times more. The model (i) is under-performing
considerably, showing the lack of representation of the reduced number of
modes. However, even if we keep a reduced number of Zernike modes we see the
effectiveness of the data-driven features in model (iii) and how they can
adapt to the observations and still generalize to target positions. Despite
the increase in the number of Zernike modes in model (ii), being close to the
GT, the wavefront DD (iii) model still outperforms considerably. A remarkable
fact is the performance gap in the super resolution task between the first
three models that use the forward model and the others. It underlines the
importance of adding prior physical information into the inverse problem to
solve a hard task as SR. We include the RMSE error as a function of wavelength
for model (iii) in Figure 4, achieving an impressive mean relative RMSE of
$3\%$ showing that the model is capturing the chromatic variations of the PSF.
We stress that the other data-driven models, (iv) and (v), are not capable of
modeling the PSF as a function of wavelength. Appendix A provides additional
figures of this experiment.

##  5 Conclusion

We have presented a novel framework for the data-driven modeling of an
instrumental response field which represents a paradigm shift with respect to
current state-of-the-art data-driven methods. We propose to include a
differentiable physics-based forward model that allows to change the modeling
space from pixels to the wavefront. Thus, transferring most of the modeling
complexity into the forward model and simplifying the building of the
instrumental response model. We have shown the importance of a data-driven
term in the model. This term can be successfully learned from noisy under-
sampled observations using the differentiable optical model. When applied to
simplified space telescope simulations, the proposed approach has proven to be
a breakthrough in terms of performance compared to current state-of-the-art
data-driven models. Furthermore, it is now possible to effectively model the
wavelength dependence of the instrumental response from broad-band
observations.

##  6 Broader impact

The proposed framework can change the way data-driven instrumental response
fields are currently being modeled. This approach can be of particular
interest for the Euclid space telescope [1], the Vera C. Rubin Observatory
[24] or the Nancy Grace Roman space telescope [2]. Weak gravitational
lensing[25] plays a fundamental role in their science goals, which places
stringent requirements on the point spread function model. Nevertheless, the
framework is sufficiently adaptable for other purposes where a good
characterization of an instrument response is needed. If the Fraunhofer
approximation is not valid anymore, the optical forward model should be
refactored. We believe this work does not entail any negative consequences or
ethical issues. A journal paper based on this work is in preparation. Code to
reproduce the experiences will be released soon.

## Acknowledgments and Disclosure of Funding

The authors would like to thank François Lanusse and Benjamin Remy for useful
comments on the manuscript. This work was granted access to the HPC resources
of IDRIS under the allocation 2021-AD011011554 and 2021-AD011012983 made by
GENCI. This work has made use of the CANDIDE Cluster at the Institut
d’Astrophysique de Paris and made possible by grants from the PNCG and the
DIM-ACAV.

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##  Appendix A Additional figures

In this appendix, we provide additional figures from the numerical experience
of section 4. Figure 2 illustrates examples of the observed stars used as
training and their corresponding wavefront information.

Figure 4 presents the target star reconstruction RMSE as a function of
wavelength at three times the Euclid resolution. We remark that PSF
reconstruction at specific wavelengths is not possible for current data-driven
PSF models. One can notice that the reconstruction error is kept low inside
all of Euclid’s broad passband (i.e. $550$nm to $900$nm). This indicates that
model (iii), in the proposed framework, is not degenerating with respect to
the different wavelengths. This could have been the case as the optimization
is done only using broad-band observations, as seen in Equation 1 and Figure
1, without having access to observations at specific wavelengths.

Figure 3 shows examples of learned data-driven features from model (iii). It
is interesting to see that the learned features are not completely degenerate
and some structure has been learned from the stars with the help of the
automatic differentiation. Figure 5 presents PSF reconstructions at a target
position, the corresponding ground truth PSF, and its residuals. We remark the
very low residuals obtained by model (iii) from the proposed framework. The
model is capturing a good super-resolved representation of the PSFs as well as
a reliable monochromatic, or single wavelength, representation. It is also
proving to generalize well to target positions that have not been used during
the training of the model. One can see the increase of PSF shape complexity
when we super-resolve the images as well as when we reconstruct the PSF at
specific wavelengths. These images represent some of the burden in the PSF
modeling task and the difficulties we face when trying to estimate the PSF of
the third row of Figure 5 from observations like the ones from Figure 2. It is
crucial to change the space of representation of the PSF model from the pixel
to the wavefront in order to include the physics of the problem, as for
example, the diffraction phenomena.

Figure 2: The top row presents examples of star observations at Euclid
resolution from the training dataset. These are examples from different
positions in the FOV and have different signal-to-noise ratios. The bottom row
shows the corresponding ground truth wavefront maps of each observed star.
Figure 3: Examples of learned data-driven features $S^{DD}$. Figure 4: Target
star reconstruction RMSE as a function of wavelength at three times the Euclid
resolution. The model (iii) uses the proposed wavefront framework with $15$
Zernike modes and a data-driven component. The solid line (circular markers)
corresponds to the RMSE on the left axis, and the dashed line (triangular
markers) corresponds to the relative RMSE on the right axis. The error bars
represent the standard deviation of the RMSE over the star reconstructions at
the different target positions. Figure 5: Different type of reconstructions
for a single target position. The first column represents the ground truth
PSF, the second column, the reconstructions from the proposed PSF model using
a data-driven component, and, the third column, the absolute value of the
residuals between the ground truth and our model. Top: Reconstruction at the
Euclid resolution for a broad-band, or polychromatic, observation. Middle:
Reconstruction at three times the Euclid resolution (SR task) for a
polychromatic observation. Bottom: Reconstruction at three times the Euclid
resolution for specific wavelength of $750$nm.


# Impact of Communication Loss on MPC based Cooperative Adaptive Cruise
Control and Platooning

Mahdi Razzaghpour∗, Shahriar Shahram∗, Rodolfo Valiente∗, Yaser P. Fallah∗
∗Connected and Autonomous Vehicle REsearch Lab (CAVREL),
University of Central Florida, Orlando, FL
<EMAIL_ADDRESS>
###### Abstract

Cooperative driving, enabled by communication between automated vehicle
systems, is expected to significantly contribute to transportation safety and
efficiency. Cooperative Adaptive Cruise Control (CACC) and platooning are two
of the main cooperative driving applications that are currently under study.
These applications offer significant improvements over current advanced driver
assistant systems such as adaptive cruise control (ACC). The primary
motivation of CACC and Platooning is to reduce traffic congestion and improve
traffic flow, traffic throughput, and highway capacity. These applications
need an efficient controller to consider the computational cost and ensure
driving comfort and high responsiveness. The advantage of Model Predictive
Control is that we can realize high control performance since all constrain
for these applications can be explicitly dealt with through solving an
optimization problem. These applications highly depend on information update
and Communication reliability for their safety and stability purposes. In this
paper, we propose a Model Predictive Control (MPC) based approach for CACC and
platooning, and examine the impact of communication loss on the performance
and robustness of the control scheme. The results show an improvement in
response time and string stability, demonstrating the potential of cooperation
to attenuate disturbances and improve traffic flow.

###### Index Terms:

Adaptive Cruise Control, Cooperative Adaptive Cruise Control, Platooning,
Model Predictive Control, Cooperative driving, Non-ideal Communication,
Connected Vehicles

##  I Introduction

Automated vehicles (AVs) depend only on the information collected by ego
vehicle’s sensors, i.e. camera, lidar, and radar. On the other hand
cooperative systems use information augmentation and sensor fusion algorithms
to combine ego vehicle’s sensors data with information from Vehicle to
everything (V2X) communication.

Adaptive Cruise Control (ACC) is a radar-based system, which is designed to
enhance driving comfort and convenience by relieving the driver of the need to
adjust the vehicle’s speed to match the speed of the preceding vehicle. The
information received from radar measures the distance to the preceding car in
the same lane. Another capability of this sensor is to measure the relative
velocity of the preceding vehicle in the same lane. ACC does not depend on
inter-vehicular communication or any form of cooperation between vehicles.
Therefore, ACC is not able to provide stable and robust performance with close
spacing between vehicles. In recent years the research community works on two
extensions of ACC called Cooperative Adaptive Cruise Control (CACC) and
Platooning, which relies both on sensor measurements and wireless
communication with other vehicles in the group. The difference between CACC
and Platooning is the different gap regulation strategies. CACC uses a
constant-time-gap-following policy whereas Platooning uses tightly coupled
communication and control for constant clearance or constant distance gap.
Typically, all members of a group broadcast their data like speed and
acceleration to following vehicles in the group, allowing them to react faster
to any changes in traffic conditions when compared to relying on on-board
sensors only.

The majority of current works consider a string of 5 to 10 vehicles at most,
while in this paper we increased the length of string to 25 vehicles and
guaranty string stability. In addition, we use an All-predecessor-leader
following (APLF) topology different from other Information Follow
Topologies[1].

Figure 1: A comparison between connectivity in ACC (top) and CACC or
Platooning (bottom). In ACC every vehicle has access to information of the
preceding one via radar. In CACC and Platooning cases, in addition to radar
information, every vehicle has access to all preceding vehicles’ information
via V2V.

##  II Related Work

Partners for Advanced Transit and Highways (PATH) project demonstrated a
platoon of eight automated vehicles in 1997, following each other in close
formation, with one of them changing lanes and shifting its position in the
platoon formation [2]. Joel et al. shows that ACC can have only a small impact
on highway capacity [3]. On the other hand, Steven et al. expresses that CACC
has the potential to substantially increase highway capacity when it reaches a
moderate to high market penetration [4]. Bart et al. concluded that CACC can
potentially double the capacity of a highway lane at high-CACC market
penetration [5].

The Grand Cooperative Driving Challenge (GCDC) is a project where multiple
teams tested their CACC vehicles. GCDC 2016 aims at demonstrating how
cooperative and automated vehicles can perform complicated platooning
operations with close-to-reality traffic scenarios. The focus is the
cooperative aspects, where advanced platoon operations (e.g., the cooperative
merge of two parallel platoons) have been demonstrated. In addition, GCDC 2016
introduces a cooperative intersection scenario to demonstrate how cooperative
vehicles solve complicated intersection scenarios safe and efficiently [6].

The Energy ITS project in Japan aimed at the CO2 emission reduction from
automobiles, which includes two themes: an implementation of automated truck
platooning system and an evaluation method of effects of ITS-related systems
and technologies on the CO2 emission reduction[7].

Jeroen et al. implemented CACC on a test fleet consisting of six-passenger
vehicles adopting a constant time gap spacing policy [8]. Darbha et al. had
shown that the time headway can be minimized by using multiple predecessors’
information in the local controller [9]. Studies in [10] have shown that a car
with a velocity of 80km/h following only one predecessor at 25m achieves a 30%
reduction in aerodynamic drag, and a 40% reduction can be attained by
following two predecessors.

Figure 2: Speed profiles for Fifteen-vehicle test for ACC, CACC, and
Platooning (from top to bottom) Figure 3: Acceleration profiles for Fifteen-
vehicle test for ACC, CACC, and Platooning (from top to bottom)

##  III Background

###  III-A ACC

When there are multiple consecutive AVs in ACC, only after the second vehicle
sensed, processed, and responded to the leader’s motion changes, the third
vehicle can infer what happened to the first vehicle from the behavior of the
second vehicle. In an autonomous ACC, the detection and response delay is
cumulative from the leader to the downstream vehicles as shown in Fig 2. The
accumulated delay prevents string stability in long strings of the automated
vehicles[11]. Recent on-the-road experiments have shown that a stream of
autonomous ACC vehicles is string unstable, resulting in a negative impact on
lane capacity[12]. By using Vehicle to Vehicle (V2V) Communication, every
vehicle get information not only from its preceding, as the ACC case, but also
from the vehicles in front of the preceding one as shown in Figure 1.

Depending on the communication topology, different realizations of CACC and
Platooning are possible. In this paper, we used All-predecessor-leader
following (APLF) topology, and all the simulations are based on this topology.
In APLF, in addition to radar information from the exact preceding vehicle,
every vehicle has access to all preceding vehicles’ information and leader
vehicle via V2V [1].

Each vehicle has a unique ID, i.e., $i=0,1,2,\dots,N-1$, where i = 0 is the ID
of the leader vehicle and the following vehicles have IDs
$i\in\\{1,2,\dots,N-1\\}$. At each time slot t, all N vehicles broadcast their
information to their following vehicles. By using both sensor and
communication information, the vehicle string has a quicker response to any
change in road traffic. The lead vehicle in the string (V0) is usually
representing the surrounding traffic or the reference for the following
vehicle. In our simulations, the lead vehicle (V0) is considered as a human-
driven vehicle and provided with a reference trajectory to follow.

###  III-B Cooperation

Using wireless communication, high-risk situations can be detected earlier via
sharing situation awareness between vehicles and vehicles can follow their
predecessors with higher accuracy, faster response, and shorter gaps to
achieve system-wide benefits. Algorithms can use simple data from V2V
communication e.g. speed, acceleration, location, or extensive forms of data
like model-based communication[13], future intention[14, 15], performance
limitation like maximum speed, acceleration, and braking capability for
heterogeneous strings. For instance, intention sharing in these cases allows
following vehicles to start to respond to any speed change, before the
leader’s speed changes measurably. Also via Infrastructure to Vehicle (I2V)
Communication, vehicles can get long-term data like traffic ahead and speed
recommendation.

CACC and Platooning highly depend on communication reliability and information
updates, i.e. Basic Safety Massage (BSM) updates at 10 Hz rate, to maintain
safety and stability. We used the same 10 Hz rate in our simulations. One
upper limit that could be placed on string length is based on the range of the
wireless V2V communication system, assuming that all CACC following vehicles
will require direct communication from the lead vehicle in the string. High-
performance communication technologies, Dedicated Short-Range Communications
(DSRC)[16] and Cellular V2X (CV2X)[17] can provide at least 300 m of
communication range to enable string lengths up to 15 to 25 vehicles. In our
simulations, we consider strings of 5 to 25 vehicles. Also, we assumed that
all vehicles are equipped with V2V communication.

####  III-B1 CACC

In this system, gap policy is a constant time policy. In other words, the
distance between vehicles is proportional to their speed, the higher the speed
is, the longer the distance. CACC only provides longitudinal control of the
vehicle’s motion and the driver has to control the steering[18].

Here, we consider a constant time headway spacing policy where the desired
spacing is

$D_{\text{desired i }}=D_{\text{safety }}+T_{\text{gap }}\cdot V_{i}$ (1)

where i is the vehicle index, $D_{\text{desired i }}$ is the desired gap
between the ith vehicle and its predecessor, $D_{\text{safety }}$ is the
minimum gap, $T_{\text{gap }}$ is the constant time headway representing the
time that it will take the ith vehicle to arrive at the same position as its
predecessor, and $V_{i}$ is the speed of the ith vehicle. Without loss of
generality, in this paper, we assumed $D_{\text{safety }}$ is zero. Also,
$T_{\text{gap }}$ in all simulations for CACC is $0.8$ second.

####  III-B2 Platooning

Platooning has a very close coupling between vehicles to maximize highway
capacity. In this architecture, the separation between vehicles remains
constant and does not change as the vehicle speed changes. The primary goals
of Platooning include increasing lane throughput and reducing aerodynamic drag
by drafting which results in energy efficiency[19]. Platooning can improve
fuel efficiency for vehicles and especially for trucks[7].

This discipline offers performance advantages, but the constant-clearance
following is more difficult to achieve and requires a more formal platoon
architecture. In addition interruptions in communication are more serious from
a safety standpoint. In this paper, the desired distance for all Platooning
simulations is $15$ m.

Figure 4: Difference of Maximum speed and Minimum speed at each time step for
the fifteen vehicle tests in Figure 2 Figure 5: Mean of speed difference for
different string length Vs. different values of PER for CACC and Platooning
(from top to bottom) Figure 6: 95% error for for different string length Vs.
different values of PER for CACC and Platooning (from top to bottom)

##  IV Problem Statement

The purpose of CACC and Platooning is to follow a desired inter-vehicle gap
and the desired velocity between a group of preceding vehicles and a host
vehicle. To validate these requirements, including minimizing the value of
relative gap error and keeping the desired spacing policy, smoothing the
acceleration and deceleration and providing a comfortable ride for passengers,
keeping the velocity and acceleration at a reasonable interval, and providing
the velocity adaptation mechanism in case of changing the velocity of the lead
vehicle, we will introduce three kinds of control objective in this part.

  * •

Collision avoidance: It is the constraint with the highest priority and should
never be violated under any circumstance. The goal of this constraint is to
prevent the following vehicle from colliding with the preceding car.

$D_{i-1,i}\geq d_{\text{safety }}$ (2)

where $D_{i-1,i}$ is the distance between vehicles $i$ and $i-1$

  * •

Adaptive Speed Control: As the objective of cooperation, all vehicles should
be able to track the desired speed at each time instance.

$|v_{desired}-v_{i}|_{t}\longrightarrow 0$ (3)

where $i$ is the index of vehicles.

  * •

Driver Comfort: To reduce the peak force and sudden jerk, acceleration should
be bounded.

$|a_{i}|_{t}\leq Constant$ (4)

where $i$ is the index of vehicles.

Information from the radar is used to reduce gap error between the desired gap
and the actual gap in ACC. In addition, for CACC and Platooning the local
vehicle control objective can now be defined as regulating the error to be
zero subject to the above-mentioned constraints for all preceding vehicles.

###  IV-A Vehicle Model

We define the state vector as:

$Z=[x\hskip 5.69054pty\hskip 5.69054pt\varphi\hskip 5.69054ptv]$ (5)

Where $x$ is the x-position, $y$ is the y-position, $\varphi$ is the yaw-angle
and $v$ is the velocity. The input vector is defined as:

$u=[a\hskip 5.69054pt\delta]$ (6)

Where $a$ is the acceleration and $\delta$ is the steering angle.

Finally our vehicle model is:

$\dot{Z}(t)=A^{\prime}Z(t)+B^{\prime}u(t)$ (7)

$Z(t+1)=AZ(t)+Bu(t)+C$ (8)

Where $Z(t+1)$ is the output of control or new state and

$A=\begin{pmatrix}1&0&cos(\bar{\phi})dt&-\bar{v}sin(\bar{\phi})dt\\\
0&1&sin(\bar{\phi})dt&\bar{v}cos(\bar{\phi})dt\\\ 0&0&1&0\\\
0&0&\frac{tan(\bar{\delta})}{L}dt&1\end{pmatrix}$ (9)

$B=\begin{pmatrix}0&0\\\ 0&0\\\ dt&0\\\
0&\frac{\bar{v}}{L{cos\textsuperscript{2}(\bar{\delta})}}dt\end{pmatrix}$ (10)

$C=\begin{pmatrix}\bar{v}sin(\bar{\phi})\bar{\phi}dt\\\
-\bar{v}cos(\bar{\phi})\bar{\phi}dt\\\ 0\\\
\frac{\bar{v}\bar{\delta}}{L{cos\textsuperscript{2}(\bar{\delta})}}dt\end{pmatrix}$
(11)

$\dot{\phi}=\frac{v.tan(\delta)}{L}$ (12)

L is the length of vehicle (4.5 m) and the bar sign stands for the mean value
of the variable between each simulation time.

###  IV-B Model Predictive Control

The advantage of Model Predictive Control (MPC) for our design is that it
allows to incorporate operating constraints, i.e. regulates the velocity of
the vehicle while satisfying the desired gap.

The control inputs are calculated by solving the General constrained
optimization problem below during each simulation time step.

$\displaystyle\begin{array}[]{c}\min_{u}J=\sum_{k=1}^{T}(Z_{i,t+k}^{\top}Q_{i,i}Z_{i,t+k}+\Delta
u_{i,t+k}^{\top}R_{\Delta u}\Delta u_{i,t+k}\\\
+u_{i,t+k}^{\top}R_{u}u_{i,t+k})+\left(Z_{i,i-1}^{\top}Q_{i,i-1}Z_{i,i-1}\right)\\\
+\sum_{j=0}^{i-2}\left(Z_{i,j}^{\top}Q_{i,j}Z_{i,j}\right)\\\\[10.0pt] \text{
subject to }\\\\[10.0pt] Z_{t}^{min}\leqslant Z_{t+k}\leqslant
Z_{t}^{max}\\\\[10.0pt] u_{t}^{\operatorname{min}}\leqslant u_{t}\leqslant
u_{t}^{max}\\\\[10.0pt] \Delta u_{t}^{min}\leqslant\Delta u_{t}\leqslant\Delta
u_{t}^{\operatorname{max}}\end{array}$ (13)

where $t$ is the current time, $T$ is the prediction horizon, $\Delta u_{t}$
is the control input change, and the suffix “min” and “max” denotes lower
bound and upper bound. $Q_{i,i}$, $R_{\Delta u}$, $R_{u}$ are the positive-
semi definite weight matrices for the ego vehicle trajectory error, the change
rate, and the magnitude of the control input, respectively. The second
summation term is the place where MPC Consider ACC, and the third term is
where MPC Consider CACC or Platooning based on the application. $Z_{i,j}$ is
the relative measurement between vehicle $i$ and $j$ where relative gap and
relative speed are considered. $Q_{i,j}$ is a matrix with weights for relative
gap and relative speed. In this paper $Q_{i,j}$ is identical for different $i$
and $j$. The linear MPC optimization problem above results in a Quadratic
Programming (QP) problem.

MPC needs to solve an optimization problem considering future prediction in
every simulation period. So MPC requires a plant model to predict the future
behavior of the plant along a finite time horizon. At each simulation instant,
optimal control inputs are calculated for 10 prediction horizons. The control
signal for the current instant is applied to the system and future states of
the system are predicted by using the aforementioned vehicle model.

###  IV-C Communication Loss

In CAVs, vehicles frequently communicate their state information (e.g.,
position, speed, and heading) to other vehicles (e.g., every 100ms) over a
broadcast wireless link. Also, each vehicle should keep track of the movements
of neighboring vehicles based on the received information. These two sub-
components of communication/networking and estimation together provide real-
time situational awareness.

In this paper, we assume that the packet delivery succeeds with the same
constant probability on all communication links. Packet loss is an independent
and identically distributed random process[20]. At any given instance, the
data is either received or lost. Each packet is assumed to be received with
probability $P$ and missed with probability $1-P$ which is the same as PER
[21].

For each vehicle $i$, if the packet from its neighbor $j$ is transmitted
successfully, then $i$ will use the newly received information for local
control. Otherwise, vehicle $i$ will update the latest received information
based on Constant speed for the local control. In this scheme, we considered
that the vehicle’s speed will remain unchanged during each inter-packet gap.
The algorithm uses the speed received from the last received information until
it receives the next packet.

###  IV-D String Stability

The Goal of String Stability is to guarantee the reduction of disturbances
propagated from the leading vehicle to the rest of vehicles in a string for a
specific design, avoiding that leading vehicle speed changes cause
amplification in the rest of the vehicles. It can be defined as

$|SS(s)|=\left|\frac{X_{i}(s)}{X_{i-1}(s)}\right|\leq 1,\quad i\geq 2$ (14)

where $i$ indicates the index of the vehicle in the string.

Sinan et al. obtains bounds on Maximum Allowable Transmission Intervals (MATI)
and Maximum Allowable Delays (MAD) while string stability is still guaranteed
[22]. For various initial conditions quantitative numbers for $T_{MATI}$ and
$T_{MAD}$ can be obtained with the help of a numerical search algorithm. These
times are around 1.5 seconds. In our Simulations, we use 10 Hz communication
rate. As it is proved by results, our architecture is stable. Note that if the
communication rate is sufficiently high, then the probability of receiving a
packet will be large even if the PER is high.

The analysis shows that the ACC string is stable if there are relatively large
headway-time values (larger than 0.5 sec), whereas the CACC system is string
stable for all headway-time values that were considered. For additional
analysis refer to [1].

##  V Results

Figure 7: 95% error comparison for different string length for 3 scenarios :
ACC , Platooning best case scenario (PER=0) and Platooning worse case scenario
(PER=0.6) Figure 8: Mean of speed difference comparison for different string
length for 5 scenarios : ACC , CACC best case scenario (PER=0) and CACC worse
case scenario (PER=0.6), Platooning best case scenario (PER=0) and Platooning
worse case scenario (PER=0.6)

The speed, and acceleration response of a 15 vehicle string for ACC, CACC, and
Platooning are shown in Figure 2 and Figure 3. In all plots, the leader was
supposed to reduce its speed from $20\ m/s$ to $15\ m/s$ and accelerate to
reach to $25\ m/s$. As Figure 2 shows, in the case of ACC, any change in
leader’s speed propagated in the string with a delay as discussed in the
Background. Each vehicle is controlled to follow its predecessor while
maintaining a desired distance. These simulations involve 15 vehicles, with a
maximum acceleration of $1\ m/s^{2}$. It takes around 15 seconds for
decelerating and around 20 seconds for accelerating that the whole string
adapts to the new speed. CACC works based on a constant time gap, and
Platooning works based on a constant distance gap. Using the same setup for
CACC and Platooning, in both cases, it takes less than 10 seconds for
decelerating and around 10 seconds for accelerating that the whole string
adapts to the new speed. Also, it was seen that as we increase the index for
vehicles (as we go away from the leader), the error value decreases. It is
simply because those vehicles start to respond to changes very quickly.

For the rest of the simulations, we used 5 to 25 vehicles in the string and
run multiple experiments for different PER values and the leader’s trajectory
has a time-varying speed with a maximum of $20\ m/s$ and a minimum of $10\
m/s$ to makes it look like a realistic highway driving scenario. To capture a
statistical sense of worst-case behavior, and due to characteristics of
wireless networks, we choose 95 percentile of the error’s absolute value. As
in [11] we defined the error for CACC as the absolute value of the difference
between actual time gap and desired time gap $(0.8s)$ in seconds, while in
platooning the error is defined as the absolute value of the difference
between actual distance gap and desired distance gap $(15m)$ in meters.

The difference between Maximum speed and Minimum speed considering all string
members at each time step is a good measure for traffic flow, we defined this
metric as speed difference and used it to measure the performance. In Figure 4
we plot the speed difference for ACC, CACC, and Platooning for the results of
the simulations presented in Figure 2.

Figure 5 shows the mean of speed difference for CACC and Platooning for
different string length and in variable PER scenarios. Platooning is more
sensitive to packet loss compared to CACC. By increasing the length of the
string, the average speed difference increase in both cases as expected. For
25 vehicles and PER of $0.6$ in platooning, it reaches to $0.67\ m/s$ while in
CACC it reaches to $0.26\ m/s$. In Figure 6 we showed the 95% error for CACC
and Platooning for various string length and different values of PER.

Figure 7 shows the 95% error for Platooning and ACC. It shows that Platooning
highly reduces the 95% error even at the highest PER (worse case scenario)
compared to ACC. CACC error is measured in seconds, so we did not put it in
this comparison. Finally Figure 8 shows that Platooning and CACC reduce the
average speed oscillations and allow the traffic flow to be maintained at peak
throughput even at the highest PER compared to ACC.

##  VI Conclusion

In this paper, we explored the performance of the CACC and Platooning with
random loss. In contrast with ACC, CACC and Platooning showed improvements in
response time and string stability, indicating the potential for a cooperative
system to attenuate disturbances and improve traffic flow stability. We also
provide statistics for errors in gap policy and speed variation of the string
members for different channel situations (PER from 0 to 0.6) and different
lengths of string (from 5 vehicles to 25 vehicles). Compared to the constant
distance policy (Platooning), the constant time headway spacing policy (CACC)
improves scalability and string stability. Even though a larger time headway
can improve the string stability in case of random loss, but it also
negatively affects some of CACC benefits such as increasing road throughput.
The study shows that communication rate and loss ratio have a significant
influence on the string stability and performance of the proposed
architecture. Reducing speed oscillations will reduce vehicle crashes and
improve traffic flow and comfort. In the future, we will model a realistic
link model for highway scenarios, and use that model as a source of
uncertainty and randomness. The effectiveness of the proposed control
algorithm is carried out only numerically and additional experimental study is
needed in the future to support the numerical results.

##  VII Acknowledgement

This research was supported by the National Science Foundation under grant
number CNS-1932037.

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\put(260.0,18.0){\vector(1,0){35.0}}\put(260.0,0.0){\vector(1,0){35.0}}\put(260.0,-18.0){\vector(1,0){35.0}}
\par\put(173.0,6.0){$\Downarrow$}\put(223.0,6.0){$\Downarrow$}\put(273.0,6.0){$\Downarrow$}
\put(173.0,-12.0){$\Downarrow$}\put(223.0,-12.0){$\Downarrow$}\put(273.0,-12.0){$\Downarrow$}
\par\end{picture}

$\bullet$$\bullet$$\bullet$

;

$\Downarrow$$\Downarrow$$\Downarrow$

$\Downarrow$$\Downarrow$$\Downarrow$

$\bullet$$\bullet$$\bullet$$\bullet$

.

$\Downarrow$$\Downarrow$$\Downarrow$$\Downarrow$$\Downarrow$$\Downarrow$

Then, for each of the above diagrams, built the n-category $h\mathbf{4}$ as
follows:

the structure for the k-cells with $0\leq k<n$ is the same as in
$n\mathbb{B}$;

the n-cells constitute the smallest preorder on (n-1)-cells with the same
initial and terminal (n-2)-cell such that

(1) the one diagram corresponding to the vertically/horizontally composable
triples of horizontally/vertically composable pairs in question is in
$h\mathbf{4}$ and,

(2) with such preorder, $h\mathbf{4}$ is in fact a n-category;

notice that there is a trivial preorder satisfying (1) and (2), the one which
relates every ordered pair of (n-1)-cells with the same initial and terminal
(n-2)-cell, and so the smallest preorder can be obtained intersecting all such
preorders satisfying (1) and (2).

The n-category

$n\mathbb{E}=\coprod_{0\leq i<j<n}\coprod_{h\in H_{ij}}h\mathbf{4},$

such that $H_{ij}$ is the set of all vertically/horizontally composable
triples of horizontally/vertically composable pairs of n-cells in
$n\mathbb{B}$, with j-cells as arrows and i-cells as objects in $n\mathbb{B}$.

Then, there is an e.d.m. $np:n\mathbb{E}\rightarrow n\mathbb{B}$ which
projects the corresponding copy of $h\mathbf{4}$, for every $h\in H_{ij}$,
$0\leq i<j<n$.171717Remark that $h\mathbf{4}$ is really a n-preorder, as
defined just below at the beginning of the following section 11.

##  11\. The reflection of n-categories into n-preorders has stable units and
a monotone-light factorization

Let $nPreord$ be the full subcategory of $nCat$ determined by the objects
$C:n\mathbb{P}\rightarrow Set$ such that $Cd^{n(n-1)}$ and $Cc^{n(n-1)}$ are
jointly monic (cf. diagram $(9.1)$), that is,

$C(P^{n(n-1)})=$$C(P_{n(n-1)})$$Cr^{n(n-1)}$$Cm^{n(n-1)}$$Cq^{n(n-1)}$$C(P_{n})$$Cc^{n(n-1)}$$Ce^{n(n-1)}$$Cd^{n(n-1)}$$C(P_{n-1})$$(11.1)$

is a preordered set.

There is a reflection

$H\vdash
I:nCat$$nPreord$,$(11.2)$$a$$b$$a$$b$,$f$$g$$\mapsto$$f$$g$$\Downarrow$$\theta$$\Downarrow$$\leq$

which identifies all n-cells which have the same domain and codomain with
respect to $P^{n(n-1)}$.181818Notice that, in this picture $(11.2)$, $f$ and
$g$ stand for generic (n-1)-cells and $a$ and $b$ for (n-2)-cells. That is,
the reflector $I$ takes the middle vertical category $C(P^{n(n-1)})$ (see
diagram $(9.1)$) to its image by the well known reflection $Cat\rightarrow
Preord$ from categories into preordered sets (see [9]).

###  11.1. Stable units

The reflection $Cat\rightarrow Preord$ from categories into preorders is known
to be a base monoidal reflection in the sense of Definition 4.1, considering
$Cat$ equipped with its cartesian symmetric monoidal structure. It has also
stable units in the sense of [2] (cf. section 8).

Therefore one can iterate it n times, obtaining exactly the reflection
$(11.2)$, also with stable units according to Proposition 8.1:

######  Theorem 11.1.

The reflection $H\vdash I:nCat\rightarrow nPreord$ has stable units, for every
$n\geq 1$.

###  11.2. Monotone-light factorization for n-categories via n-preorders

######  Theorem 11.2.

The reflection $H\vdash I:nCat\rightarrow nPreord$ does have a monotone-light
factorization, for every $n\geq 1$.

###### Proof.

The statement is a consequence of the central result of [1] (cf. Corollary 6.2
in [8]), because $H\vdash I$ has stable units (cf. Theorem 11.1) and for every
$n\mathbb{B}\in nCat$ there is an e.d.m. $np:n\mathbb{E}\rightarrow
n\mathbb{B}$ with $n\mathbb{E}\in nPreord$ (cf. Example 10.1). ∎

In the following section 12, it will be proved that the monotone-light
factorization system is not trivial, for every $n\geq 1$. That is, it does not
coincide with the reflective factorization system associated to the reflection
of $nCat$ into $nPreord$.

##  12\. Vertical and stably-vertical n-functors

In this section, it will be given a characterization of the class of vertical
morphisms $\mathcal{E}_{I}$ in the reflective factorization system
$(\mathcal{E}_{I},\mathcal{M}_{I})$, and of the class of the stably-vertical
morphisms $\mathcal{E}^{\prime}_{I}$
($\subseteq\mathcal{E}_{I}$)191919$\mathcal{E}^{\prime}_{I}$ is the largest
subclass of $\mathcal{E}_{I}$ stable under pullbacks. The terminologies
“vertical morphisms” and “stably-vertical morphisms” were introduced in [5].
in the monotone-light factorization system
$(\mathcal{E}^{\prime}_{I},\mathcal{M}^{*}_{I})$, both associated to the
reflection $I:nCat\rightarrow nPreord$. Then, since $\mathcal{E}^{\prime}_{I}$
is a proper class of $\mathcal{E}_{I}$, one concludes that
$(\mathcal{E}^{\prime}_{I},\mathcal{M}^{*}_{I})$ is a non-trivial monotone-
light factorization system.

Consider a n-functor $f:A\rightarrow B$, which is obviously determined by the
$n+1$ functions $f_{0}:A(P_{0})\rightarrow B(P_{0})$,
$f_{1}:A(P_{1})\rightarrow B(P_{1})$,…, $f_{n}:A(P_{n})\rightarrow B(P_{n})$
(cf. diagram $(9.1)$), so that we may make the identification
$f=(f_{n},...,f_{1},f_{0})$.

######  Proposition 12.1.

A n-functor $f=(f_{n},...,f_{1},f_{0}):A\rightarrow B$ belongs to the class
$\mathcal{E}_{I}$ of vertical n-functors if and only if the following two
conditions hold:

  1. (1)

$f_{0}$, $f_{1}$,…, $f_{n-1}$ are bijections;

  2. (2)

for every two elements $h$ and $h^{\prime}$ in $A(P_{n-1})$,

if $Hom_{B(P^{n(n-1)})}(f_{n-1}h,f_{n-1}h^{\prime})$ is nonempty

then so is $Hom_{A(P^{n(n-1)})}(h,h^{\prime})$.

###### Proof.

The n-functor $f=(f_{n},...,f_{1},f_{0})$ belongs to $\mathcal{E}_{I}$ if and
only if $If$ is an isomorphism (cf. [1, §3.1]), that is, $If_{0}$, $If_{1}$,…,
$If_{n}$ are bijections. Since $If_{0}=f_{0}$,…, $If_{n-1}=f_{n-1}$, the fact
that $f\in\mathcal{E}_{I}$ implies and is implied by (1) and (2) is trivial. ∎

######  Proposition 12.2.

A n-functor $f=(f_{n},...,f_{1},f_{0}):A\rightarrow B$ belongs to the class
$\mathcal{E}^{\prime}_{I}$ of stably-vertical n-functors if and only if the
following two conditions hold:

  1. (1)

$f_{0}$, $f_{1}$,…, $f_{n-1}$ are bijections;

  2. (2)

for every two elements $h$ and $h^{\prime}$ in $A(P_{n-1})$,

$f$ induces a surjection

$Hom_{A(P^{n(n-1)})}(h,h^{\prime})\rightarrow
Hom_{B(P^{n(n-1)})}(f_{n-1}h,f_{n-1}h^{\prime})$

($f$ is a “full functor on n-cells”).

###### Proof.

As every pullback $g^{*}(f)=\pi_{1}:C\times_{B}A\rightarrow C$ in $nCat$ of
$f$ along any n-functor $g:C\rightarrow B$ is calculated pointwise, and being
$(f_{n},f_{n-1}):A(P^{n(n-1)})\rightarrow B(P^{n(n-1)})$ a stably-vertical
functor for the reflection $Cat\rightarrow Preord$, that is, $f_{n-1}$ a
bijection and $(f_{n},f_{n-1})$ a full functor (cf. Propositions 4.4 and 3.2
in [9]), then (1) and (2) imply that $g^{*}(f)$ belongs to $\mathcal{E}_{I}$
(cf. last Proposition 12.1).

Hence, $f\in\mathcal{E}^{\prime}_{I}$ if (1) and (2) hold.

If $f\in\mathcal{E}^{\prime}_{I}$, then $f\in\mathcal{E}_{I}$
($\mathcal{E}^{\prime}_{I}\subseteq\mathcal{E}_{I}$), and therefore (1) holds.

Suppose now that (2) does not hold, so that there is
$\theta:f_{n-1}h\rightarrow f_{n-1}h^{\prime}$ not in the image of $f$, and
consider the n-category $C$ generated202020Generated in the sense of Example
10.1, that is, $C$ has the same structure of $B$, with the exception of
n-cells, whose structure is the least which includes the following diagram. by
the diagram
$f_{n-2}(a)$$f_{n-2}(a^{\prime})$,$f_{1}h$$f_{1}h^{\prime}$$\Downarrow$$\theta$
and let $g$ be the inclusion of $C$ in $B$. Then, $C\times_{B}A$ contains
$f_{n-2}(a)$$f_{n-2}(a^{\prime})$,$f_{1}h$$f_{1}h^{\prime}$

with no non-identity 2-cells, and so $g^{*}(f)$ is not in $\mathcal{E}_{I}$.

Hence, if $f\in\mathcal{E}^{\prime}_{I}$ then (1) and (2) must hold.∎

It is evident that $\mathcal{E}^{\prime}_{I}$ is a proper class of
$\mathcal{E}_{I}$, therefore the monotone-light factorization system
$(\mathcal{E}^{\prime}_{I},\mathcal{M}^{*}_{I})$ is non-trivial
($\neq(\mathcal{E}_{I},\mathcal{M}_{I})$).

##  13\. Trivial coverings for n-categories via n-preorders

A n-functor $f:A\rightarrow B$ belongs to the class $\mathcal{M}_{I}$ of
trivial coverings (with respect to the reflection $H\vdash I:nCat\rightarrow
nPreord$) if and only if the following square

$B$$A$$I(B)$$I(A)$$f$$\eta_{A}$$If$$(13.1)$$\eta_{B}$

is a pullback diagram, where $\eta_{A}$ and $\eta_{B}$ are unit morphisms for
the reflection $H\vdash I:nCat\rightarrow nPreord$ (cf. [2, Theorem 4.1]).

Since the pullback (as any limit) is calculated pointwise in $nCat$ (cf.
Corollary 10.1), then $f\in\mathcal{M}_{I}$ if and only if the following
squares are pullbacks, corresponding to the pointwise components of $\eta_{A}$
and of $\eta_{B}$ (cf. diagram $(9.1)$), $0\leq l\leq n$ and $0\leq i<j<k\leq
n$:

$B(P_{l})$$A(P_{l})$$I(B)(P_{l})$$I(A)(P_{l})$$f_{P_{l}}$$\eta_{A(P_{l})}$$If_{P_{l}}$;($D_{l}$)$\eta_{B(P_{l})}$

$B(P_{ji})$$A(P_{ji})$$I(B)(P_{ji})$$I(A)(P_{ji})$$f_{P_{ji}}$$\eta_{A(P_{ji})}$$If_{P_{ji}}$;($D_{ji}$)$\eta_{B(P_{ji})}$

$B(P_{kj})$$A(P_{kj})$$I(B)(P_{kj})$$I(A)(P_{kj})$$f_{P_{kj}}$$\eta_{A(P_{kj})}$$If_{P_{kj}}$;($D_{kj}$)$\eta_{B(P_{kj})}$

$B(P_{ki})$$A(P_{ki})$$I(B)(P_{ki})$$I(A)(P_{ki})$$f_{P_{ki}}$$\eta_{A(P_{ki})}$$If_{P_{ki}}$;($D_{ki}$)$\eta_{B(P_{ki})}$

$B(P_{kji})$$A(P_{kji})$$I(B)(P_{kji})$.$I(A)(P_{kji})$$f_{P_{kji}}$$\eta_{A(P_{kji})}$$If_{P_{kji}}$($D_{kji}$)$\eta_{B(P_{kji})}$

The squares $(D_{1}),(D_{2}),...,(D_{n-1})$ are pullbacks since
$\eta_{A(P_{l})}$ and $\eta_{B(P_{l})}$ are identity maps for $l<n$ (cf.
diagram $(9.1)$ and the definition of the reflection $H\vdash
I:nCat\rightarrow nPreord$ in $(11.2)$).

Notice that if diagram $(9.1)$ is restricted to the precategory diagram
$P^{n(n-1)}$, one obtains from $(13.1)$ the following square in $Cat$, with
unit morphisms of the reflection of all categories into preorders
$Cat\rightarrow Preord$ (cf. [9]),

$B(P^{n(n-1)})$$A(P^{n(n-1)})$$I(B)(P^{n(n-1)})$.$I(A)(P^{n(n-1)})$$f_{P^{n(n-1)}}$$\eta_{A(P^{n(n-1)})}$$If_{P^{n(n-1)}}$$(13.2)$$\eta_{B(P^{n(n-1)})}$

It is known (cf. [9, Proposition 3.1]) that this square is a pullback in $Cat$
if and only if, for every two objects $h$ and $h^{\prime}$ in $A(P_{n-1})$
with $Hom_{A(P^{n(n-1)})}(h,h^{\prime})$ nonempty, the map

$Hom_{A(P^{n(n-1)})}(h,h^{\prime})\rightarrow
Hom_{B(P^{n(n-1)})}(f_{n-1}h,f_{n-1}h^{\prime})$

induced by $f$ is a bijection.

A necessary condition for the n-functor $f$ to be a trivial covering was just
stated; the following Lemma 13.1 will help to show that this necessary
condition is also sufficient in next Proposition 13.1.

######  Lemma 13.1.

Consider the following commutative parallelepiped

$B_{2}$$\eta_{B,2}$$r^{B}$$q^{B}$$B_{1}$$\eta_{B,1}$$d^{B}$$c^{B}$$B_{0}$$\eta_{B,0}$$f_{2}$$f_{1}$$f_{0}$$A_{2}$$\eta_{A,2}$$r^{A}$$q^{A}$$A_{1}$$\eta_{A,1}$$d^{A}$$c^{A}$$A_{0}$$\eta_{A,0}$$I(B)_{2}$$Ir^{B}$$Iq^{B}$$I(B)_{1}$$Id^{B}$$Ic^{B}$$I(B)_{0}$$If_{2}$$If_{1}$$If_{0}$$I(A)_{2}$$Ir^{A}$$Iq^{A}$$I(A)_{1}$$Id^{A}$$Ic^{A}$$I(A)_{0}$$(13.3)$

where the five squares $c^{A}q^{A}=d^{A}r^{A}$, $c^{B}q^{B}=d^{B}r^{B}$,
$Ic^{A}Iq^{A}=Id^{A}Ir^{A}$, $If_{0}\eta_{A,0}=\eta_{B,0}f_{0}$ and
$If_{1}\eta_{A,1}=\eta_{B,1}f_{1}$ are pullbacks.

Then, the square $If_{2}\eta_{A,2}=\eta_{B,2}f_{2}$ is also a
pullback.212121The notation used in diagram $(13.3)$ is arbitrary and was
chosen to make the application of Lemma 13.1 to this section easily
understandable.

###### Proof.

The proof is obtained by an obvious diagram chase. ∎

######  Proposition 13.1.

A n-functor $f:A\rightarrow B$ is a trivial covering for the reflection
$H\vdash I:nCat\rightarrow nPreord$ (in notation, $f\in\mathcal{M}_{I}$) if
and only if, for every two objects $h$ and $h^{\prime}$ in $A(P_{n-1})$ with
$Hom_{A(P^{n(n-1)})}(h,h^{\prime})$ nonempty, the map

$Hom_{A(P^{n(n-1)})}(h,h^{\prime})\rightarrow
Hom_{B(P^{n(n-1)})}(f_{1}h,f_{1}h^{\prime})$

induced by $f$ is a bijection.

###### Proof.

In the considerations just above, it was showed that the statement warrants
that the square $(D_{n})$ is a pullback, adding to the fact that
$(D_{0}),(D_{1}),...,(D_{n-1})$ are all pullbacks.

Then, all the others squares $(D_{ji}),(D_{kj}),(D_{ki})$ and $(D_{kji})$ must
also be pullbacks according to Lemma 13.1, $0\leq i<j<k\leq n$. ∎

##  14\. Coverings for n-categories via n-preorders

A n-functor $f:A\rightarrow B$ belongs to the class $\mathcal{M}^{*}_{I}$ of
coverings (with respect to the reflection $H\vdash I:nCat\rightarrow nPreord$)
if there is some effective descent morphism (also called monadic extension in
categorical Galois theory) $p:C\rightarrow B$ in $nCat$ with codomain $B$ such
that the pullback $p^{*}(f):C\times_{B}A\rightarrow C$ of $f$ along $p$ is a
trivial covering ($p^{*}(f)\in\mathcal{M}_{I}$).

The following Lemma 14.1 can be found in [9, Lemma 4.2], in the context of the
reflection of categories into preorders, but for n-categories via n-preorders
the proof is exactly the same, since the same conditions hold (cf. Theorem
11.1 and Example 10.1). The next Proposition 14.1 characterizes the coverings
for n-categories via n-preorders.

######  Lemma 14.1.

A n-functor $f:A\rightarrow B$ in $nCat$ is a covering (for the reflection
$H\vdash I:nCat\rightarrow nPreord$) if and only if, for every n-functor
$\varphi:X\rightarrow B$ over $B$ from any n-preorder $X$, the pullback
$X\times_{B}A$ of $f$ along $\varphi$ is also a n-preorder.

######  Proposition 14.1.

A n-functor $f:A\rightarrow B$ in $nCat$ is a covering (for the reflection
$H\vdash I:nCat\rightarrow nPreord$) if and only if it is faithful with
respect to n-cells, that is, for every pair $g,g^{\prime}\in A(P_{n-1})$, the
map

$Hom_{A(P^{n(n-1)})}(g,g^{\prime})\rightarrow
Hom_{B(P^{n(n-1)})}(f_{n-1}g,f_{n-1}g^{\prime})$

induced by $f$ is an injection.

###### Proof.

Consider again the n-preorder $T$ generated222222Cf. the footnote in the proof
of Proposition 12.2. by the diagram
$a$$a^{\prime}$.$h$$h^{\prime}$$\Downarrow$$\leq$

If $f$ is not faithful with respect to n-cells (in the sense of the
statement), then, by including $T$ in $B$, one could obtain a pullback
$T\times_{B}A$ that is not a n-preorder.

Therefore, $f$ is not a covering, by the previous Lemma 14.1.

Reciprocally, consider any n-functor $\varphi:X\rightarrow B$ such that $X$ is
a n-preorder.

If $f$ is faithful with respect to n-cells (in the sense of the statement),
then the pullback $X\times_{B}A$ is a n-preorder, given the nature of $X$.
Hence, $f$ is a covering, by the previous Lemma 14.1. ∎

## Acknowledgement

This work was supported by The Center for Research and Development in
Mathematics and Applications (CIDMA) through the Portuguese Foundation for
Science and Technology

(FCT - Fundação para a Ciência e a Tecnologia),

references UIDB/04106/2020 and UIDP/04106/2020.

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