# Nonlinear Estimation for Position-Aided Inertial Navigation Systems

Soulaimane Berkane and Abdelhamid Tayebi This work was supported by the
National Sciences and Engineering Research Council of Canada (NSERC), under
the grants NSERC-DG RGPIN-2020-04759 and NSERC-DG RGPIN-2020-0627. S. Berkane
<EMAIL_ADDRESS>is with the Department of Computer Science and
Engineering, University of Quebec in Outaouais, Gatineau, QC, Canada. A.
Tayebi<EMAIL_ADDRESS>is with Department of Electrical Engineering,
Lakehead University, Thunder Bay, ON, Canada.

###### Abstract

In this work we solve the position-aided 3D navigation problem using a
nonlinear estimation scheme. More precisely, we propose a nonlinear observer
to estimate the full state of the vehicle (position, velocity, orientation and
gyro bias) from IMU and position measurements. The proposed observer does not
introduce additional auxiliary states and is shown to guarantee semi-global
exponential stability without any assumption on the acceleration of the
vehicle. The performance of the observer is shown, through simulation, to
overcome the state-of-the-art approach that assumes negligible accelerations.

##  I Introduction

Inertial navigation systems (INS) are essential devices that allow the
localization and control of autonomous vehicles and robot platforms [1].
Classical INS fuse measurements from on-board accelerometers and gyroscopes
(typically included in an Inertial Measurement Unit (IMU)), to continuously
compute the position, velocity and orientation of a vehicle without any
external reference measurement. However, this classical approach (known also
as dead reckoning) suffers from drift due to measurement errors and unknown
initial conditions [2]. As a result, inertial navigation systems are often
assisted by position sensors such as the Global Positioning System (GPS),
which allow for the correction of position estimates over time, resulting in
small and bounded estimation errors, see, e.g., [3, 4, 5]. Other type of
sensors that can provide range (distance) measurements to known source points
can also be used to provide position information such as GPS pseudo-ranges [6,
7, 8] or Ultra Wide-Band (UWB) radio technology [9, 10]. Kalman-type filters,
such as [11, 4, 12, 13], are considered industry-standard solutions for
inertial navigation systems. However, these filters are often based on
linearization assumptions and may fail when the initial estimation errors are
large. On the other hand, nonlinear observers have been developed for
autonomous navigation applications; see for instance [7, 14, 8, 15, 16, 17,
18]. The advantage of the nonlinear observers is their theoretically proven
stability guarantees, as well as their computational simplicity compared to
the stochastic filters.

In this work, we propose a nonlinear observer for the simultaneous estimation
of the position, linear velocity, attitude, and gyro bias of a rigid body
system. The estimator relies on IMU and inertial position measurements which
are used in typical navigation scenarios. The attitude estimates are directly
obtained on the Special Orthogonal group of rotations $\mathbb{SO}(3)$, thus
avoiding any singularities or ambiguities related to the use of other attitude
parameterizations. The proposed observer, evolving on
$\mathbb{SO}(3)\times\mathbb{R}^{9}$ is shown to guarantee semi-global
exponential stability, which is the strongest stability result that can be
achieved on this state space using smooth observers. In contrast to [5, 7, 15,
18], the proposed observer does not require the introduction of an auxiliary
$3$-dimensional state in the estimation scheme, thus reducing the
computational burden associated with the real-time implementation of the
estimator. The paper is structured as follows. Preliminaries are provided in
Section II while the problem at hand is formulated in Section III. The
observer design and the corresponding stability result are provided in Section
IV (main result). Simulation results using an accelerated trajectory are given
in Section V to show the performance of the proposed observer and to compare
it with a practical adhoc estimation scheme. We conclude the paper with some
remarks in Section VI.

##  II Background

We denote by $\mathbb{R}$ the set of reals and by $\mathbb{N}$ the set of
natural numbers. We denote by $\mathbb{R}^{n}$ the $n$-dimensional Euclidean
space, by $\mathbb{S}^{n}$ the unit $n$-sphere embedded in $\mathbb{R}^{n+1}$
and by $\mathbb{B}_{\epsilon}=\\{x\in\mathbb{R}^{3}:\|x\|\leq\epsilon\\}$ the
closed ball in $\mathbb{R}^{3}$ with radius $\epsilon$. We use $\|x\|$ to
denote the Euclidean norm of a vector $x\in\mathbb{R}^{n}$ and $\|A\|_{F}$ to
denote the Frobenius norm of a matrix $A\in\mathbb{R}^{n\times n}$. Let
$I_{n}$ be the $n$-by-$n$ identity matrix and let $e_{i}$ denote the $i-$th
column of $I_{n}$. The Special Orthogonal group of order three is denoted by
$\mathbb{SO}(3):=\\{A\in\mathbb{R}^{3\times
3}:\;\mathbf{det}(A)=1,\;AA^{\top}=A^{\top}A=I_{3}\\}$. The set
$\mathfrak{so}(3):=\left\\{\Omega\in\mathbb{R}^{3\times
3}\mid\;\Omega^{\top}=-\Omega\right\\}$ denotes the Lie algebra of
$\mathbb{SO}(3)$. For $x,~{}y\in\mathbb{R}^{3}$, the map
$[\cdot]_{\times}:\mathbb{R}^{3}\to\mathfrak{so}(3)$ is defined such that
$[x]_{\times}y=x\times y$ where $\times$ is the vector cross-product on
$\mathbb{R}^{3}$. The inverse isomorphism of the map $[\cdot]_{\times}$ is
defined by $\mathrm{vex}:\mathfrak{so}(3)\to\mathbb{R}^{3}$, such that
$\mathrm{vex}([\omega]_{\times})=\omega$, for all $\omega\in\mathbb{R}^{3}$
and $[\mathrm{vex}(\Omega)]_{\times}=\Omega,$ for all
$\Omega\in\mathfrak{so}(3)$. The composition map
$\psi:=\mathrm{vex}\circ\mathbf{P}_{\mathfrak{so}(3)}$ extends the definition
of $\mathrm{vex}$ to $\mathbb{R}^{3\times 3}$, where
$\mathbf{P}_{\mathfrak{so}(3)}:\mathbb{R}^{3\times 3}\to\mathfrak{so}(3)$ is
the projection map on the Lie algebra $\mathfrak{so}(3)$ such that
$\mathbf{P}_{\mathfrak{so}(3)}(A):=(A-A^{\top})/2$. Accordingly, for a given
$3$-by-$3$ matrix $A=[a_{ij}]_{i,j=1,2,3}$, one has
${\textstyle\psi(A)=\frac{1}{2}[a_{32}-a_{23},a_{13}-a_{31},a_{21}-a_{12}]}$.
We define
${\textstyle|R|:=\frac{1}{4}\mathbf{tr}(I_{3}-R)=\frac{1}{8}\|I_{3}-R\|_{F}^{2}\in[0,1]}$
as the normalized Euclidean distance on $\mathbb{SO}(3)$. Given a scalar
$c>0$, we define the saturation function
$\mathbf{sat}_{c}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that
$\mathbf{sat}_{c}(x):=\min(1,c/\|x\|)x.$ Given two scalars $c,\epsilon>0$, we
also define the smooth projection function
$\mathbf{P}_{c}^{\epsilon}:\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R}^{3}$,
found for instance in [19], as follows:

$\mathbf{P}_{c}^{\epsilon}(\hat{\phi},\mu):=\begin{cases}\mu,&\textrm{if}\;\|\hat{\phi}\|<c\;\textrm{or}\;\hat{\phi}^{\top}\mu\leq
0,\\\
{\scriptstyle\left(I-\theta(\hat{\phi})\frac{\hat{\phi}\hat{\phi}^{\top}}{\|\hat{\phi}\|^{2}}\right)}\mu,&\textrm{otherwise},\end{cases}$
(1)

where $\theta(\hat{\phi}):=\min(1,(\|\hat{\phi}\|-c)/\epsilon)$. The
projection operator $\mathbf{P}_{c}^{\epsilon}(\hat{\phi},\mu)$ is locally
Lipschitz in its arguments. Moreover, provided that $\|\phi\|\leq c$, the
projection map $\mathbf{P}_{c}^{\epsilon}(\hat{\phi},\mu)$ satisfies, along
the trajectories of
$\dot{\hat{\phi}}=\mathbf{P}_{c}^{\epsilon}(\hat{\phi},\mu),\|\hat{\phi}(t)\|\leq
c+\epsilon$, $\forall t\geq 0$.

##  III Problem Formulation

Consider the following dynamics of a rigid-body vehicle:

$\displaystyle\dot{R}$ $\displaystyle=R[\omega]_{\times},$ (2)
$\displaystyle\dot{p}$ $\displaystyle=v,$ (3) $\displaystyle\dot{v}$
$\displaystyle=ge_{3}+Ra_{B},$ (4)

where $p\in\mathbb{R}^{3}$ is the inertial position of the vehicle’s center of
gravity, $v\in\mathbb{R}^{3}$ represents the inertial linear velocity,
$R\in\mathbb{SO}(3)$ is the rotation matrix describing the orientation of the
body-attached frame with respect to the inertial frame, $\omega$ is the
angular velocity of the body-attached frame with respect to the inertial frame
expressed in the body-attached frame, $g$ is the norm of the acceleration due
to gravity, $e_{3}=[0,0,1]^{\top}$ and $a_{B}=R^{\top}a_{I}$ is the “apparent
acceleration”, capturing all non-gravitational forces applied to the vehicle,
expressed in the body-attached frame.

We assume available an inertial measurement unit (IMU) that provides
measurements in the body frame of the angular velocity, the apparent
acceleration and the earth’s magnetic field. These sensors are modelled as
follows:

$\displaystyle\omega^{y}$ $\displaystyle=\omega+b_{\omega},$ (5)
$\displaystyle a_{B}$ $\displaystyle=R^{\top}a_{I},$ (6) $\displaystyle m_{B}$
$\displaystyle=R^{\top}m_{I},$ (7)

where $b_{\omega}$ is a constant unknown gyro bias, $m_{I}$ is the constant
and know earth’s magnetic field and $a_{I}(t)$ is a time-varying unknown
apparent acceleration. For the rotational dynamics, the following is a general
observability assumption commonly used in attitude estimation.

######  Assumption 1.

There exists a constant $c_{0}>0$ such that $\|m_{I}\times a_{I}(t)\|\geq
c_{0}$ for all $t\geq 0$.

Assumption 1 is guaranteed if the time-varying apparent acceleration
$a_{I}(t)$ is non-vanishing and is always non-collinear to the constant
magnetic field vector $m_{I}$. Note that $a_{I}(t)=0$ corresponds to the rigid
body in a free-fall ($\dot{v}=ge_{3}$) which is not likely under normal flight
conditions.

We also assume that we have measurements of the following position output
vector:

$\displaystyle y=C_{p}p,$ (8)

where $C_{p}\in\mathbb{R}^{m\times 3},m\in\mathbb{N},$ is a given output
matrix that satisfies $\mathrm{rank}(C_{p})=3.$ Roughly speaking, this rank
assumption means that the measurement $y$ is sufficient for the construction
of a converging translational observer (assuming perfect knowledge of the
attitude) for (3)-(4). The measurement $y$ can be obtained from different
possible sensors, depending on the application at hand, that provide some
information about the position. For full position measurements obtained from a
GPS for instance, $C_{p}=I_{3}$. For range measurements, obtained from Ultra-
Wide Brand (UWB) sensors for instance, the output $y$ and the matrix $C_{p}$
can be obtained from the range measurements as done in [18, Section 4.3.2];
see also the simulation work in Section V.

The objective of this work is to design a full navigation system that
processes the measurements (7)-(8) and outputs reliable estimates for the
position $p\in\mathbb{R}^{3}$, the linear velocity $v\in\mathbb{R}^{3}$, the
orientation $R\in\mathbb{SO}(3)$, and the gyro bias
$b_{\omega}\in\mathbb{R}^{3}$. More specifically, our goal is to design an
exponentially convergent nonlinear observer that estimates the whole state of
the vehicle using an observer state that evolves on the same manifold as the
system state ($\mathbb{SO}(3)\times\mathbb{R}^{9}$). We further consider the
following mild (realistic) constraints on the trajectory of the vehicle which
are needed to prove the main result:

######  Assumption 2.

There exist constants $c_{1},c_{2},c_{3}>0$ such that
$c_{1}\leq\|a_{I}(t)\|\leq c_{2}$ and $\|\dot{a}_{I}(t)\|\leq c_{3}$ for all
$t\geq 0$.

######  Assumption 3.

There exists constants $c_{4},c_{5}>0$ such that $\|\omega(t)\|\leq c_{4}$ and
$\|b_{\omega}\|\leq c_{5}$ for all $t\geq 0$.

##  IV Observer Design and Stability Analysis

In this section, we design a navigation observer to estimate the full state
$(p,v,R,b_{\omega})$ consisting of the position, velocity, orientation and
gyro bias. We define the combined translational state
$x:=[p^{\top},v^{\top}]^{\top}\in\mathbb{R}^{6}$. Then, in view of (3)-(4) and
(8), the dynamics of $x$ are written as:

$\displaystyle\dot{x}$ $\displaystyle=Ax+B(ge_{3}+Ra_{B}),$ (9) $\displaystyle
y$ $\displaystyle=Cx,$ (10)

where the matrices $A,B$ and $C$ are defined as follows:

$\displaystyle A=\begin{bmatrix}0_{3\times 3}&I_{3}\\\ 0_{3\times
3}&0_{3\times 3}\end{bmatrix},B:=\begin{bmatrix}0_{3\times 3}\\\
I_{3}\end{bmatrix},C=\begin{bmatrix}C_{p}^{\top}\\\ 0_{3\times
m}\end{bmatrix}^{\top}.$ (11)

The translational dynamics (9)-(10) are those of a linear time-invariant
system with unknown input $a_{I}=Ra_{B}$ representing the apparent
acceleration which is known only in the body-frame. A common approach, in
practice, consists in assuming that the acceleration of the vehicle is
negligible, i.e., $\dot{v}\approx 0$, and therefore $a_{I}\approx-ge_{3}$.
Under this small linear acceleration assumption, the attitude and gyro bias
estimation can be done separately using the following explicit complementary
filter proposed in [20, 5]:

$\displaystyle\dot{\hat{R}}$
$\displaystyle=\hat{R}[\omega_{y}-\hat{b}_{\omega}+k_{R}\sigma_{R}]_{\times},$
(12) $\displaystyle\dot{\hat{b}}_{\omega}$
$\displaystyle=\mathbf{P}_{c_{5}}^{\epsilon_{b}}(\hat{b}_{\omega},-k_{b}\sigma_{R}),$
(13)

with the innovation term

$\displaystyle\sigma_{R}$
$\displaystyle=\rho_{1}(m_{B}\times\hat{R}^{\top}m_{I})+\rho_{2}(a_{B}\times\hat{R}^{\top}(-ge_{3})).$
(14)

Once the attitude is estimated, the translational motion state $x$ can be
estimated using a Luenberger-like observer as follows:

$\displaystyle\dot{\hat{x}}$
$\displaystyle=A\hat{x}+B(ge_{3}+\hat{R}a_{B})+K(y-C\hat{x}),$ (15)

where $K$ is a gain matrix guaranteeing that $(A-KC)$ is Hurwitz, which can be
either constant or tuned via a Riccati equation such as in the Kalman filter.
The cascaded adhoc estimation scheme (12)-(15) will be used as a state-of-the-
art algorithm in our comparison results in Section V.

In this work, however, we design our estimation algorithm without the small
acceleration assumption which compromises the performance of the adhoc scheme
when the vehicle is subject to large linear accelerations. In particular, we
propose the following nonlinear navigation observer on
$\mathbb{SO}(3)\times\mathbb{R}^{9}$:

$\displaystyle\dot{\hat{R}}$
$\displaystyle=\hat{R}[\omega_{y}-\hat{b}_{\omega}+k_{R}\sigma_{R}]_{\times},$
(16) $\displaystyle\dot{\hat{b}}_{\omega}$
$\displaystyle=\mathbf{P}_{c_{5}}^{\epsilon_{b}}(\hat{b}_{\omega},-k_{b}\sigma_{R}),$
(17) $\displaystyle\dot{\hat{x}}$
$\displaystyle=A\hat{x}+B(ge_{3}+\hat{R}a_{B})+K(y-C\hat{x})+\sigma_{x},$ (18)

with initial conditions
$\hat{x}(0)\in\mathbb{R}^{6},\hat{R}(0)\in\mathbb{SO}(3)$ and
$\hat{b}_{\omega}(0)\in\mathbb{B}(c_{5}+\epsilon_{b})$. The innovation terms
$\sigma_{x}:=[\sigma_{p}^{\top},\sigma_{v}^{\top}]^{\top}\in\mathbb{R}^{6}$
and $\sigma_{R}\in\mathbb{R}^{3}$ are defined as follows:

$\displaystyle\sigma_{v}$ $\displaystyle=K_{p}C_{p}\sigma_{p},$ (19)
$\displaystyle\sigma_{p}$
$\displaystyle=k_{R}(K_{v}C_{p})^{-1}[\hat{R}\sigma_{R}]_{\times}\hat{R}a_{B},$
(20) $\displaystyle\sigma_{R}$
$\displaystyle=\rho_{1}(m_{B}\times\hat{R}^{\top}m_{I})+\rho_{2}(a_{B}\times\hat{R}^{\top}\mathbf{sat}_{\hat{c}_{2}}(K_{v}(y-C\hat{x}))).$
(21)

The constant scalars $k_{R},k_{b},\rho_{1},\rho_{2},\epsilon_{b},\hat{c}_{2}$
are positive with $\hat{c}_{2}>\sqrt{8}c_{2}$, the gain matrix
$K=[K_{p}^{\top}\;K_{v}^{\top}]^{\top}\in\mathbb{R}^{6\times m}$ is chosen as
$K=L_{\gamma}K_{0}$ such that $A-K_{0}C$ is Hurwitz,
$L_{\gamma}=\textrm{blockdiag}(\gamma I_{3},\gamma^{2}I_{3})$ and $\gamma\geq
1$. Note that the inverse of the matrix $(K_{v}C_{p})$ in (20) exists since
$(A-K_{0}C)$ is a $2\times 2$ invertible block matrix and the Schur complement
of $I_{3}$ is the matrix $-K_{v}C_{p}$; see [21, Theorem 2.2].

Figure 1: Overall structure of the proposed nonlinear observer in (16)-(18).
The observer uses an IMU and position information to estimate the whole state
of the vehicle. Compared to the adhoc estimation scheme (12)-(15), the
proposed observer introduces coupling (in bold red) between the translational
estimator and the rotational estimator through their innovation terms. This
additional coupling is important to guarantee the stability of the observer
without the low acceleration assumption.

The overall structure of the proposed estimation algorithm is depicted in Fig.
1. Compared to the adhoc estimation scheme (12)-(15), the proposed observer
introduces coupling through $\sigma_{x}$ and $\sigma_{R}$ between the
translational estimator and the rotational estimator. This additional coupling
is important to guarantee the stability of the observer without the low
acceleration assumption. We define the following estimation errors:

$\displaystyle\tilde{x}$ $\displaystyle:=x-\hat{x},$ (22)
$\displaystyle\tilde{R}$ $\displaystyle:=R\hat{R}^{\top},$ (23)
$\displaystyle\tilde{b}_{\omega}$
$\displaystyle:=b_{\omega}-\hat{b}_{\omega}.$ (24)

In view of (2)-(3) (17)-(18) and (22)-(24) one can derive the following
dynamics for the estimation errors:

$\displaystyle\dot{\tilde{x}}$
$\displaystyle=(A-KC)\tilde{x}+B(I-\tilde{R})^{\top}a_{I}-\sigma_{x},$ (25)
$\displaystyle\dot{\tilde{R}}$
$\displaystyle=\tilde{R}[-\hat{R}(\tilde{b}_{\omega}+k_{R}\sigma_{R})]_{\times},$
(26) $\displaystyle\dot{\tilde{b}}_{\omega}$
$\displaystyle=\mathbf{P}_{c_{5}}^{\epsilon_{b}}(\hat{b}_{\omega},k_{b}\sigma_{R}).$
(27)

To prove the convergence of the estimation errors, we introduce the following
auxiliary error variable

$\displaystyle\zeta:=L_{\gamma}^{-1}\left[(A-KC)\tilde{x}+B(I-\tilde{R})^{\top}a_{I}\right].$
(28)

The term between brackets in (28) is an estimation error term that contains a
suitable coupling between the translational estimation error $\tilde{x}$ and
the rotational estimation error $\tilde{R}$. This coupling in the estimation
errors, obtained from (25), allows to design the innovation term $\sigma_{x}$
by inspecting the dynamics of this coupled estimation error. This coupling is
instrumental in showing (exponential) stability of the overall closed-loop
system without requiring an additional observer state as done in our previous
work [18]. Moreover, the matrix $L_{\gamma}^{-1}$ is introduced to assign a
certain time-scaling structure between the different estimation errors in the
same spirit as in [18]; see also [22, 23] for a motivation of this use in the
context of high-gain observers. The high-gain $\gamma$ allows to deal
efficiently with the interconnection between the rotational and translational
dynamics. Now we are ready to state our main result.

######  Theorem 1.

Consider the interconnection of the dynamics (3)-(2) with the observer
(18)-(21) where Assumptions (1)-(3) are satisfied. For each
$\epsilon\in(\frac{1}{2},1)$ and for all initial conditions such that
$\zeta(0)\in\mathbb{R}^{6}$,
$\tilde{R}(0)\in\mho(\epsilon)=\\{\tilde{R}:|\tilde{R}(0)|\leq\epsilon\\}$ and
$\hat{b}_{\omega}(0)\in\mathbb{B}(c_{5}+\epsilon_{b})$, there exist
$k_{R}^{*}>0$ and $\gamma^{*}\geq 1$ such that, for all $k_{R}\geq k_{R}^{*}$
and $\gamma\geq\gamma^{*}$, the estimation errors
$(\zeta,\tilde{R},\tilde{b}_{\omega})$ are globally uniformly bounded and
converge exponentially to zero.

###### Proof.

See Appendix A. ∎

Theorem 1 shows that the proposed navigation observer (16)-(21) guarantees
exponential convergence of the estimation errors to zero starting from any
initial condition inside the set $\mho(\epsilon)$. As $\epsilon$ tends to $1$,
the set $\mho(\epsilon)$ covers the set of all attitude errors with rotation
angle less than $180^{\circ}$. The derived bounds on the gains $k_{R}$ and
$\gamma$ (see the proof) are expected to be very conservative as demonstrated
in the simulation results.

##  V Simulation

In this section, we simulate the proposed nonlinear estimator presented in
Section IV in the case of position range measurements. The real position
trajectory of the vehicle is given by:

$\displaystyle p(t)=\begin{bmatrix}\cos(2\pi t^{2}/100)\\\ \sin(2\pi
t^{2}/100)\\\ 1\end{bmatrix}.$ (29)

This corresponds to a circular trajectory with an angular frequency that
increases over time (given by $t/100$). The attitude trajectory is generated
using the following angular velocity:

$\displaystyle\omega(t)=\begin{bmatrix}\sin(0.2t)\\\ \cos(0.1t)\\\
\sin(0.3t+\pi/6)\end{bmatrix},$ (30)

and an initial attitude $R(0)=\exp([\pi e_{1}/2]_{\times})$. The gyro
measurements are corrupted by a constant bias of $3\;\mathrm{(deg/sec)}$ in
each axis. The inertial earth’s magnetic field is taken as
$m_{I}=[0.033\;0.1\;0.49]^{\top}$ and the earth’s gravity is
$g=9.81\;(\mathrm{m/sec}^{2})$. We also assume available four non-coplanar
source points for range sensing located at:

$\displaystyle a_{1}$ $\displaystyle=[1\;1\;2]^{\top},$ (31) $\displaystyle
a_{2}$ $\displaystyle=[1\;3\;\;0]^{\top},$ (32) $\displaystyle a_{3}$
$\displaystyle=[0\;1\;1]^{\top},$ (33) $\displaystyle a_{4}$
$\displaystyle=[6\;5\;5]^{\top}.$ (34)

The corresponding range measurements are given by

$\displaystyle d_{i}=\|p-a_{i}\|,\quad i=1,\cdots,4.$ (35)

To obtain an output equation of type (8) we proceed as follows. Define the
following measurable scalars

$\displaystyle
y_{i}=\frac{1}{2}\left(d_{i}^{2}-d_{1}^{2}-\|a_{i}\|^{2}+\|a_{1}\|^{2}\right),\quad
i=2,\cdots,4.$ (36)

Then, one can show that $y_{i}=(a_{1}-a_{i})^{\top}p,i=2,\cdots,4$. Define
$y=[y_{2},\cdots,y_{4}]^{\top}$ one has

$\displaystyle y=\begin{bmatrix}(a_{1}-a_{2})^{\top}\\\
(a_{1}-a_{3})^{\top}\\\ (a_{1}-a_{4})^{\top}\end{bmatrix}p:=C_{p}p.$ (37)

It can be verified that $C_{p}$ has a rank of $3$.

The initial conditions for the observer states are
$\hat{p}(0)=\hat{v}(0)=\hat{b}_{\omega}(0)=0$ and $\hat{R}(0)=I_{3}$. The
parameters of the observer are selected as
$k_{R}=2,k_{b}=1,\rho_{1}=\rho_{2}=1,\epsilon_{b}=0.001,\hat{c}_{2}=9\sqrt{8}$
and $\gamma=2$. The gain $K_{0}$ is tuned such that the matrix $(A-K_{0}C)$
has eigenvalues at $\\{-3,-3,-3,-4,-4,-4\\}$. The adhoc estimator (12)-(15) is
implemented with the same parameters and initial conditions. The simulation
results given in Figures 2-6 show that the proposed observer is able to
estimate the position, velocity, acceleration, attitude and gyro bias using
IMU and range measurements. On the other hand, as the acceleration of the
vehicle increases, the adhoc estimator drifts away from the true trajectory
while the proposed estimator is stable against high accelerations.

Figure 2: True and estimated trajectory of the vehicle for the proposed
observer which is compared against the adhoc estimator that assumes negligible
accelerations. The video of the simulation can be found at
https://youtu.be/zbkSDZgh3vU. Figure 3: Position estimation error. Figure 4:
Velocity estimation error. Figure 5: Euler angles (Roll, Pitch, Yaw)
corresponding to the attitude estimation error $\tilde{R}=R\hat{R}^{\top}$.
Figure 6: Gyro bias estimation error.

##  VI Conclusion

In this paper we solved the full state (position, velocity, orientation, gyro
bias) estimation problem for vehicles navigating in $3$-dimensional spaces.
The proposed solution is based on a nonlinear observer evolving on the
configuration space $\mathbb{SO}(3)\times\mathbb{R}^{9}$ that uses IMU and
position measurements as inputs and guarantees semi-global exponential
stability. The proposed solution is particularly suitable in applications with
important accelerations where the traditional cascaded approach (attitude
estimation + position estimation) fails due to the small-accelerations
assumption. The key feature of this observer is the fact that it does not add
computational complexity compared to the traditional approach whereas our
previous solution in [18] requires the introduction of auxiliary states to
achieve the same stability result.

##  Appendix A Proof of Theorem 1

The proof of this result is mainly inspired from our proof in [18] with few
modifications related to the particular closed-loop dynamics at hand. First,
we provide the following easy-to-check facts:

$\displaystyle L_{\gamma}^{-1}AL_{\gamma}=\gamma A,\quad L_{\gamma}^{-1}B$
$\displaystyle=\gamma^{-2}B,\quad CL_{\gamma}=\gamma C.$ (38)

Moreover, in view of the expression of $\sigma_{x}$ in (19)-(20), it is not
difficult to show that

$\displaystyle(A-KC)\sigma_{x}=-k_{R}B[\hat{R}\sigma_{R}]_{\times}\hat{R}a_{B}.$
(39)

Now using the above facts, and in view of (25)-(27), the time-scaled dynamics
of $\zeta$ are given by

$\displaystyle\frac{1}{\gamma}\dot{\zeta}$
$\displaystyle=(A-K_{0}C)\zeta+\frac{1}{\gamma^{3}}Bg(t,\tilde{R},\tilde{b}_{\omega}),$
(40)

where we have defined
$g(t,\tilde{R},\tilde{b}_{\omega}):=(I-\tilde{R})^{\top}\dot{a}_{I}+\tilde{R}^{\top}[a_{I}(t)]_{\times}R(t)\tilde{b}_{\omega}$.
The term $g(t,\tilde{R},\tilde{b}_{\omega})$ is a priori bounded by some
constant $c_{g}>0$ in view of Assumptions 2-3 and the property of the
projection mechanism that guarantees boundedness of the bias estimation error
$\tilde{b}_{\omega}$. Consider the following real-valued positive function:

$\displaystyle\mathbf{V}(\zeta)=\frac{1}{\gamma}\zeta^{\top}P\zeta,$ (41)

where $P$ is the unique solution for the Lyapunov equation
$P(A-K_{0}C)+(A-K_{0}C)^{\top}P=-I$ which is positive definite thanks to the
fact that $(A-K_{0}C)$ is Hurwitz. Denote by $\beta_{1}$ and $\beta_{2}$ the
smallest and largest eigenvalues of $P$, respectively. The time derivative of
$\mathbf{V}(\zeta)$ along the trajectories of (40) satisfies

$\displaystyle\dot{\mathbf{V}}(\zeta)$
$\displaystyle=-\|\zeta\|^{2}+\frac{2}{\gamma^{3}}\zeta^{\top}PBg(t,\tilde{R},\tilde{b}_{\omega}),$
(42)
$\displaystyle\leq-\frac{1}{2}\|\zeta\|^{2}+\|\zeta\|\left(\frac{2\beta_{2}c_{g}}{\gamma^{3}}-\frac{1}{2}\|\zeta\|\right),$
(43)
$\displaystyle\leq-\frac{1}{2}\|\zeta\|^{2},\quad\forall\|\zeta\|\geq\frac{4\beta_{2}c_{g}}{\gamma^{3}},$
(44)
$\displaystyle\leq-\frac{\gamma}{2\beta_{2}}\mathbf{V}(\zeta),\quad\forall\|\zeta\|\geq\frac{4\beta_{2}c_{g}}{\gamma^{3}}.$
(45)

Consider the set
$\Omega(c_{\zeta}):=\\{\zeta:\mathbf{V}(\zeta)\leq\gamma^{-5}c_{\zeta}^{2}\beta_{1}\\}$
for some $c_{\zeta}>0$. Let us pick $\gamma\geq
4\beta_{1}^{-\frac{1}{2}}\beta_{2}^{\frac{3}{2}}c_{g}c_{\zeta}^{-1}$. Then, it
can be shown that when $\zeta$ is outside the set $\Omega(c_{\zeta})$ one has
$\|\zeta\|>4\beta_{2}c_{g}\gamma^{-3}$ and therefore, by (45), the function
$V(\zeta)$ is exponentially decreasing outside $\Omega(c_{\zeta})$. In this
case, $\zeta$ must enter the set $\Omega(c_{\zeta})$ before the following
time:

$\displaystyle\bar{T}^{*}=\frac{2\beta_{2}}{\gamma}\ln\left(\frac{\gamma\mathbf{V}(\zeta(0))}{\beta_{1}c_{\zeta}^{2}}\right),$
(46)

which can be tuned arbitrary small by increasing the value of $\gamma$. On the
other hand, when $\zeta\in\Omega$ one has
$\|\zeta\|^{2}\leq\gamma\beta_{1}^{-1}\mathbf{V}(\zeta)\leq(c_{\zeta}\gamma^{-2})^{2}$.
The following result immediately follows:

$\forall c_{\zeta},T>0,\forall\zeta(0),\exists\gamma_{1}\geq
1\;\textrm{s.th.}\;\gamma\geq\gamma_{1}\Rightarrow\\\
\|\zeta(t)\|\leq\gamma^{-2}c_{\zeta},\;\forall t\geq T.$ (47)

This result shows that the gain $\gamma$ can be tuned to guarantee that the
error variable $\zeta$ converges arbitrary fast to an arbitrary small ball.
Now, we show that the gains can be tuned to guarantee forward invariance of
the set $\mho(\epsilon)$. Let $\epsilon\in(\frac{1}{2},1)$ and let the initial
attitude estimation error be such that $\tilde{R}(0)\in\mho(\epsilon)$. The
time derivative of $|\tilde{R}|^{2}=\mathrm{tr}(I-\tilde{R})/4$, in view of
(26) and making use of [24, Lemma 2], satisfies

$\displaystyle\frac{d}{dt}|\tilde{R}|^{2}$
$\displaystyle=-\frac{1}{4}\mathrm{tr}(\dot{R})$
$\displaystyle=-\frac{1}{4}\mathrm{tr}(\mathbf{P}_{\mathfrak{so}(3)}(\tilde{R})[-\hat{R}(\tilde{b}_{\omega}+k_{R}\sigma_{R})]_{\times})$
(48)
$\displaystyle=-\frac{1}{2}\psi(\tilde{R})^{\top}\hat{R}(\tilde{b}_{\omega}+k_{R}\sigma_{R})$
$\displaystyle\leq\|\tilde{b}_{\omega}\|+k_{R}\|\sigma_{R}\|$ (49)
$\displaystyle\leq
c_{b}+k_{R}(\rho_{1}\|m_{I}\|^{2}+\rho_{2}c_{2}\hat{c}_{2}):=c_{R}.$ (50)

In view of the above inequality on the velocity of $|\tilde{R}|^{2}$, it can
be deduced that the minimum time necessary for the attitude estimation error
$\tilde{R}$ to leave the set $\mho(\epsilon)$ is greater than

$t_{R}:=\frac{\epsilon^{2}-|\tilde{R}(0)|^{2}}{c_{R}}.$

Since we have knowledge about the minimum time necessary for the attitude
estimation error to leave the set $\mho(\epsilon)$, it is possible to prevent
such a scenario by imposing some (high gain) conditions on the gains $\gamma$
and $k_{R}$ as shown next. Pick $0<T\leq t_{R}$ and
$c_{\zeta}=\min(\bar{c}_{\zeta}/k_{R},\hat{c}_{2}-\sqrt{8}c_{2})$ for some
arbitrary $\bar{c}_{\zeta}>0$. By (47) there exists $\gamma^{*}\geq 1$ such
that if one chooses $\gamma\geq\gamma^{*}$ then
$\|\zeta(t)\|\leq\gamma^{-2}c_{\zeta}$ for all $t\geq T$. On the other hand,
in view of (28), we have

$\displaystyle
K_{v}(y-C\hat{x})=-B^{\top}L_{\gamma}\zeta+(I-\tilde{R})^{\top}a_{I}.$ (51)

Therefore, for all $t\geq T$, one has

$\displaystyle\|K_{v}(y-C\hat{x})\|$
$\displaystyle\leq\gamma^{2}\|\zeta\|+\sqrt{8}c_{2}\leq
c_{\zeta}+\sqrt{8}c_{2}\leq\hat{c}_{2}.$ (52)

Consequently, for all $t\geq T$, the saturation function
$\mathbf{sat}_{\hat{c}_{2}}(\cdot)$ in (21) can be removed. Therefore, using
[25, Proposition 3], the innovation term $\sigma_{R}$ in (21) can be written
as follows:

$\displaystyle\sigma_{R}$
$\displaystyle=\rho_{1}(m_{B}\times\hat{R}^{\top}m_{I})+\rho_{2}(a_{B}\times\hat{R}^{\top}(a_{I}-B^{\top}L_{\gamma}\zeta))$
(53)
$\displaystyle=2\hat{R}^{\top}\psi(M\tilde{R})-\rho_{2}(a_{B}\times\hat{R}^{\top}B^{\top}L_{\gamma}\zeta),$
(54)

with $M:=\rho_{1}m_{I}m_{I}^{\top}+\rho_{2}a_{I}a_{I}^{\top}$ which is
positive semidefinite with an rank of $2$ (by Assumption 1). Using the above
expression of $\sigma_{R}$, the time derivative of $|\tilde{R}|^{2}$ satisfies

$\displaystyle\frac{d}{dt}|\tilde{R}|^{2}$
$\displaystyle=-k_{R}\psi(\tilde{R})^{\top}\psi(M\tilde{R})-\frac{1}{2}\psi(\tilde{R})^{\top}(\hat{R}\tilde{b}_{\omega}-$
(55) $\displaystyle\qquad k_{R}\rho_{2}(\hat{R}a_{B})\times
B^{\top}L_{\gamma}\zeta)$ (56)
$\displaystyle\leq-4k_{R}\lambda_{\min}^{\mathbf{E}(M)}|\tilde{R}|^{2}(1-|\tilde{R}|^{2})+|\tilde{R}|\|\tilde{b}_{\omega}\|+$
(57) $\displaystyle\qquad\gamma k_{R}\rho_{2}c_{2}|\tilde{R}|\|\zeta\|$ (58)
$\displaystyle\leq-4k_{R}\lambda_{\min}^{\mathbf{E}(M)}|\tilde{R}(t)|^{2}(1-|\tilde{R}(t)|^{2})+c_{b}+\rho_{2}c_{2}\bar{c}_{\zeta}$
(59)

for all $t\geq T$, where inequalities from [24, Lemma 2] have been used with
$\mathbf{E}(M):=\frac{1}{2}(\mathbf{tr}(M)-M^{\top})$ for any $M$. Note that
the matrix $\mathbf{E}(M)$ is positive definite in view of Assumption 1. Now
assume that $|\tilde{R}(t)|=\epsilon$ and
$k_{R}>(c_{b}+\rho_{2}c_{2}\bar{c}_{\zeta})/(4\lambda_{\min}^{\mathbf{E}(M)}\epsilon^{2}(1-\epsilon^{2}))$
then one has

$\displaystyle\frac{d}{dt}|\tilde{R}(t)|^{2}$
$\displaystyle\leq-4k_{R}\lambda_{\min}^{\bar{A}}\epsilon^{2}(1-\epsilon^{2})+c_{b}+\rho_{2}c_{2}\bar{c}_{\zeta}<0,$

for all $t\geq T$. This implies that $|\tilde{R}(t)|$ is strictly decreasing
whenever $|\tilde{R}(t)|=\epsilon$. It follows from the continuity of the
solution that $\tilde{R}(t)$ will never leave the ball $\mho(\epsilon)$ for
all $t\geq T$. Recall also that $|\tilde{R}(t)|\leq\epsilon$ for all $t\leq T$
(since $T\leq t_{R}$). This implies that the set $\mho(\epsilon)$ is forward
invariant. Now, let us show the exponential convergence after $t\geq T$.
Consider the following Lyapunov function candidate:

$\mathbf{W}(\zeta,\tilde{R},\hat{R},\tilde{b}_{\omega})=|\tilde{R}|^{2}+\frac{\mu
k_{R}}{2k_{b}}\tilde{b}_{\omega}^{\top}\tilde{b}_{\omega}+\\\
\mu\tilde{b}_{\omega}^{\top}\hat{R}^{\top}\psi(\tilde{R})+\gamma^{5}\mathbf{V}(\zeta),$
(60)

where $\mu$ is some positive constant scalar and $\mathbf{V}(\zeta)$ is
defined in (41). Let
$\varsigma:=[|\tilde{R}|,\|\tilde{b}_{\omega}\|,\|\zeta\|]^{\top}$ and
following a similar procedure to [18, (A.11)-(A.14)], we can show that there
exist positive definite matrices $P_{1},P_{2},P_{3}$ such that the
$\varsigma^{\top}P_{1}\varsigma\leq\mathbf{W}\leq\varsigma^{\top}P_{2}\varsigma$
and $\dot{\mathbf{W}}\leq-\varsigma^{\top}P_{3}\varsigma$ provided that the
gains satisfy

$\displaystyle\mu$
$\displaystyle<\lambda_{\min}^{\mathbf{E}(M)}(1-\epsilon^{2})/\alpha_{2},$
$\displaystyle k_{R}$ $\displaystyle>\max\left\\{2\mu
k_{b},\frac{2\alpha_{1}\mu^{2}+(1+2c_{\omega}\mu)^{2}}{2\mu\lambda_{\min}^{\mathbf{E}(M)}(1-\epsilon^{2})}\right\\},$
$\displaystyle\gamma$
$\displaystyle>\max\left\\{\frac{4(\beta_{2})^{2}c_{2}^{2}}{\mu},\frac{k_{R}\rho_{2}c_{2}+4\sqrt{2}\beta_{2}c_{3}+\mu(\alpha_{3}+k_{R}\alpha_{4})^{2}}{4k_{R}\lambda_{\min}^{\mathbf{E}(M)}(1-\epsilon^{2})}\right\\},$

where $\alpha_{1}=8c_{b}^{2}+4k_{b}\lambda_{\max}^{\mathbf{E}(M)}$,
$\alpha_{2}=8\lambda_{\max}^{\mathbf{E}(M)}c_{b}(\sqrt{2}+4))$,
$\alpha_{3}=2k_{b}\rho_{2}c_{2}$ and
$\alpha_{4}=2\rho_{2}c_{b}c_{2}(\sqrt{2}+4)$. Consequently, the error variable
$\varsigma$ converges exponentially to zero after $t\geq T$.

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# Identification and classification of TESS variable stars

L. A. Balona
South African Astronomical Observatory, P.O. Box 9, Observatory 7935, Cape
Town, South Africa E-mail<EMAIL_ADDRESS>
(Accepted …. Received …)

###### Abstract

Visual classification of the variability classes of over 120 000 Kepler, K2
and TESS stars is presented. The sample is mainly based on stars with known
spectral types. Since variability classification often requires the location
of the star in the H–R diagram, a catalog of effective temperatures was
compiled. Luminosities were estimated from Gaia DR3 parallaxes. The different
classes of variable found in this survey are discussed. Examples of light
curves and periodograms for common variability classes are shown. A catalogue
of projected rotational velocities is also included.

###### keywords:

catalogues; stars:general; stars:variable; stars:fundamental parameters

††pagerange: Identification and classification of TESS variable
stars–References††pubyear: 2011

##  1 Introduction

The first step in any scientific investigation of a group of objects with
unknown properties is to look for similarities among them and develop a
classification system. One may then begin to investigate the physical
circumstances which could lead to the observed properties. In astrophysics,
the classification of stars into different variability classes has been one of
the most important steps in understanding stellar structure and evolution.

Until a decade ago, practically all available information on stellar
variability was obtained through ground-based observations and classification
by visual inspection of the light curves. These results were collected in the
General Catalogue of Variable Stars (GCVS, Samus et al. 2017). Over the years,
the definition of the various variability classes has evolved. Some classes
have fallen away or merged, while new ones have been discovered. The increase
in data resulting from space photometry has, however, overwhelmed the capacity
of the GCVS.

Given that there are now many millions of stars observed from space with light
curves of unprecedented accuracy, variability classification has become a
daunting task. There is a pressing need for automation, which requires a
precise definition of each variability class. The danger is that unexpected
classes of variability might escape attention. There is also a need for a
large sample of stars classified by visual inspection to act as a learning
sample for an artificial intelligence code. Automated variable star
classification using ground-based photometry was performed by Pojmanski et al.
(2005), and in the Kepler field by Blomme et al. (2010, 2011) and also by
Audenaert et al. (2021) for TESS data.

It is evident that a certain amount of familiarity with the appearance of the
light curves and periodograms for different variable star classes is an
important requirement for visual classification. The author has been active in
the variable star community for several decades, and has visually classified
over 20 000 stars observed by Kepler and K2. The lessons learnt in this
exercise were very valuable.

Before embarking on this project, it was necessary to place limitations on the
number of objects to be classified. The initial goal was to include stars with
known spectral types, since knowledge of the spectral type is often crucial in
allocating a variability class. A further restriction was made by limiting the
stars in the sample to 12.5 mag or brighter. Since the author’s main interest
lies in the hot main sequence, the sample was further restricted to stars
earlier than G. Using the SIMBAD website (Wenger et al., 2000), a master
catalogue was constructed for stars in the Kepler field. With the advent of
TESS, this was expanded to an all-sky catalogue of 718 000 stars, which
includes many fainter stars and stars of interest with no spectral type.

The master catalogue includes the RA, Dec, star names, variability class,
$UBV$ magnitudes, Strömgren $uvby\beta$, effective temperature, luminosity,
extinction, projected rotation velocity, rotation period, peculiarity code and
spectral type. In this paper the master catalogue is not presented, because
only a subset of these stars, i.e. stars observed by Kepler, K2 and TESS, have
been classified by the author.

While the master catalogue contains 718 000 stars, the number of stars
actually selected for classification is much smaller: about 20 000 stars
observed by Kepler and K2 and about 100 000 stars observed by TESS. The main
result of this paper is a presentation of variability classification of these
stars together with the best estimate of their luminosities and effective
temperatures. The luminosity of each star was derived from GAIA DR3 parallaxes
(Gaia Collaboration et al., 2016, 2022).

Since the variability class depends on the location of the star in the H–R
diagram, approximate effective temperatures and luminosities are essential.
For this purpose, and also because the effective temperature and luminosity
are fundamental parameters in any study, the literature was searched for
sources of effective temperature. The resulting catalogue contains over 900
000 entries for nearly 635 000 stars and is not limited to the Kepler, K2 or
TESS stars.

Another important parameter is the projected rotational velocity, $v\sin i$.
The last update of this catalogue (Glebocki & Gnacinski, 2005) is quite old
and an updated version with over 75 000 entries for about 52 000 stars is
presented. Like the effective temperature catalogue, the rotational velocity
catalog is not restricted to Kepler, K2 or TESS stars.

##  2 Effective temperatures, luminosities and rotation

There are several ways in which $T_{\rm eff}$ may be estimated and not all
should be treated with equal weight. The most reliable method of obtaining
$T_{\rm eff}$ is by fitting spectral line profiles using suitable atmospheric
models. Spectroscopic modelling may be restricted to the Balmer lines, which
is less expensive in resources, but may also include other lines. It is clear
values of $T_{\rm eff}$ based on spectroscopic fitting should be used in
preference to any other method.

Narrow-band photometry, such as the Strömgren and Geneva systems, in
combination with an appropriate calibration, is a well-known technique for
deriving precise $T_{\rm eff}$ estimates. Interferometry, when available, can
also be a very useful source of reliable effective temperatures. If a
spectroscopic estimate is not available, these estimates of $T_{\rm eff}$, if
available, are to be preferred.

Another method of obtaining $T_{\rm eff}$ is by fitting several photometric
measurements at different wavelengths to a model (the spectral energy
distribution, SED). While this is a good method, the $U$ band needs to be
included if it is to be used for B stars. Johnson $UBV$ photometry can be used
to derive the reddening for B stars using the Q method. The de-reddened
colours then provide a good estimate of $T_{\rm eff}$.

Values of $T_{\rm eff}$ in the Kepler Input Catalogue (KIC, Brown et al. 2011)
are derived from 2MASS photometry as well as Sloan filters, such as the griz
filters. The effective temperatures in the TESS Input Catalogue (TIC, Stassun
et al. 2018) are derived from the GAIA DR2 catalog as a base together with a
merger of a large number of other photometric catalogs, including 2MASS,
UCAC4, APASS, SDSS, WISE, etc. Care are must be taken in using the KIC and TIC
values of $T_{\rm eff}$ for B stars because $U$-band measurements are
generally absent.

Finally, if the spectral type and class is known, a crude value of $T_{\rm
eff}$ may be inferred using a table of effective temperatures as a function of
spectral type and spectral class. This is very often sufficient for breaking
an ambiguity in the variability class.

For each entry in the effective temperature catalogue, the method used to
derive $T_{\rm eff}$ is recorded by a code. For example, spectroscopic values
of $T_{\rm eff}$ are given the code SPC, estimates from Strömgren photometry
the code STR, estimates based on the SED have code SED, those from the TESS
Input Catalogue have code TIC, etc. Each code can be assigned a priority
level. The software calculates the “best” value of $T_{\rm eff}$ using the
priority level, in accordance with the guidelines mentioned above. For
example, if several entries of $T_{\rm eff}$ are present for the same star,
some with the SPC code and others with the SED and TIC codes, it is clear that
only those measurements with the SPC code should be used. The others should be
ignored and the “best” value of $T_{\rm eff}$ will be the average of all SPC
values.

For this purpose, each code (SPC, STR, SED…) is allocated a priority. The
software recognises the code, evaluates the priority, and calculates an
average of the $T_{\rm eff}$ values with highest priority, ignoring all other
values. The allocation of priority to codes is flexible and will be different
for B stars. In fact, for B stars only values of $T_{\rm eff}$ from
spectroscopic, narrow-band photometry and the Q method should be used.
However, a more pragmatic approach was taken: if $T_{\rm eff}$ differs by less
than 20 percent from the value derived from the spectral type, it is accepted.
If none of these methods are available, then the spectral type serves as an
estimate of $T_{\rm eff}$.

From time to time, as new sources of $T_{\rm eff}$ become available, the
master catalogue is updated with the new “best” values of $T_{\rm eff}$. The
priorities used are listed in Table 1. The highest priority is priority 1, the
lowest is priority 10.

Table 1: Priority values for effective temperature estimation used in this work are given in the first two columns, the second column applicable to B stars only. The method is listed in the third column while the last column shows the codes used in the effective temperature catalogue. p | p(B) | Method | Code
---|---|---|---
1 | 1 | Spectroscopy, interferometry | SPC,INF
2 | 2 | Narrow-band photometry | BCD,STR,GEN
3 | 3 | KIC photometry | KIC
4 | 4 | TIC and EPIC photometry | TIC,EPI
5 | 5 | Low-dispersion spectroscopy | LMT
6 | 6 | (Anders et al., 2022) | PAT
7 | 7 | SED | SED
8 | - | Wide-band photometry | PHT,IFM
- | 3 | Q-method | BV0
10 | 10 | Spectral type |

Table 2 shows an extract from the effective temperature catalogue.. The
catalogue contains the RA and Dec (J2000), the star name the effective
temperature and its error (or zero if this is unknown), the bibliographic code
and the code used in assigning priorities.

In compiling this catalogue, the PASTEL catalogue (Soubiran et al., 2016)
provided a very important starting point. Subsequent entries were made by
searching the SIMBAD database. About 6 percent of the effective temperatures
from Anders et al. (2022), which are based on fits to multicolour photometry
(coded as PAT in the table), are inconsistent with the spectral type and were
rejected.

Table 2: An extract from the effective temperature catalogue. The full
catalogue is available electronically.

RA | Dec | Name | $T_{\rm eff}$ | e$T_{\rm eff}$ | Bibcode | Code
---|---|---|---|---|---|---
0.3480992 | 19.0760842 | TYC 1181-837-1 | 6066 | 0 | 2022yCat.1354….0A | PAT
0.3481996 | 30.8280173 | TIC 83957915 | 6138 | 0 | 2018AJ….156..102S | TIC
0.3481996 | 30.8280173 | TIC 83957915 | 6233 | 0 | 2022yCat.1354….0A | PAT
0.3486018 | 39.6107104 | HD 224873 | 5147 | 90 | 2009A&A…504..829G | SPC
0.3486018 | 39.6107104 | HD 224873 | 5386 | 80 | 2011A&A…530A.138C | SPC
0.3498366 | 34.2818478 | StKM 2-1808 | 4263 | 0 | 2022yCat.1354….0A | PAT
0.3510299 | 0.2515648 | HD 224877 | 6125 | 0 | 2022yCat.1354….0A | PAT
0.3510299 | 0.2515648 | HD 224877 | 6156 | 134 | 2020AJ….160..120J | SPC
0.3519965 | -30.6495347 | TIC 70741976 | 6612 | 0 | 1994MNRAS.268..119B | LAB
0.3519965 | -30.6495347 | TIC 70741976 | 6727 | 0 | 2018AJ….156..102S | TIC
0.3519965 | -30.6495347 | TIC 70741976 | 6775 | 0 | 1995A&A…293…75E | SPC
0.3519965 | -30.6495347 | TIC 70741976 | 7013 | 0 | 2022yCat.1354….0A | PAT

The projected rotational velocity, $v\sin i$, is essential in studies of
stellar rotation. For this purpose, the compilation by Glebocki & Gnacinski
(2005) is very important, but rather old. For this purpose, a literature
search for values of $v\sin i$ published after this date was initiated. The
updated catalogue contains over 75 000 entries for nearly 52 000 stars. Unlike
the effective temperature catalogue, the adopted value of $v\sin i$ is simply
taken as the mean value of all measurements for the particular star. An
extract of this catalog is shown in Table 3. An entry of zero for the error in
$v\sin i$ means that no error is listed in the reference.

Table 3: An extract from the projected rotational velocity catalogue. The full
catalogue is available electronically.

RA | Dec | Name | $v\sin i$ | e$v\sin i$ | Bibcode
---|---|---|---|---|---
280.2494688 | 43.9151077 | KIC 8073705 | 3.6 | 0.0 | 2018AJ….156..254W
280.2494688 | 43.9151077 | KIC 8073705 | 3.6 | 1.0 | 2017AJ….154..107P
280.2510393 | 54.9261528 | TIC 359632996 | 8.0 | 0.0 | 2004A&A…418..989N
280.2765993 | -26.7405823 | HD 172407 | 9.0 | 0.0 | 2004A&A…418..989N
280.2906696 | 43.9269491 | KIC 8073767 | 48.4 | 17.0 | 2022A&A…662A..66X
280.2913298 | 27.9122106 | HD 336659 | 29.2 | 0.0 | 2011ApJ…732…39C
280.2915445 | 43.9257190 | KIC 8073771 | 55.5 | 15.2 | 2022A&A…662A..66X
280.3092843 | -41.4399169 | HD 172283 | 9.0 | 0.0 | 2004A&A…418..989N
280.3192664 | -63.5343778 | HD 171825 | 2.0 | 0.0 | 2004A&A…418..989N
280.3348041 | 4.5581620 | HD 172675 | 6.0 | 0.0 | 2004A&A…418..989N
280.3405802 | 7.9036620 | HD 172718 | 5.0 | 0.0 | 2004A&A…418..989N
280.3408131 | 19.5564334 | HD 349207 | 108.2 | 19.7 | 2022A&A…662A..66X

##  3 Effective temperature and luminosity from the spectral type

Luminosities for all stars in the master catalogue were derived from GAIA DR3
parallaxes (Gaia Collaboration et al., 2016, 2022) using the Gaia $G$
magnitude and corresponding bolometric correction from Chen et al. (2019). The
interstellar extinction in $G$ is taken from Anders et al. (2022) which is
based on a 3D extinction map by Green et al. (2019). For the few stars that
are not listed in the catalogue by Anders et al. (2022), the 3D extinction map
by Gontcharov (2017) was used. The typical error in $\log L/L_{\odot}$ is 0.07
dex which includes the error in the parallax, an error of 0.02 mag in $G$, and
an error of 0.05 mag in both the extinction, $A_{G}$, and the bolometric
correction.

Spectral types are mostly from the catalogue of Skiff (2014) supplemented by
later publications when required. If more than one spectral type is available,
the one which provides the most information is selected for inclusion in the
master catalogue.

Given the effective temperature calculated in the manner described above and
the luminosities from the Gaia parallax, it is possible to derive an
approximate calibration for $T_{\rm eff}$ and $\log L/L_{\odot}$ as a function
of spectral type and class. This calibration is shown in Table 4.

The relative standard deviation in $T_{\rm eff}$ derived from the spectral
type for luminosity class V is approximately 5 percent of $T_{\rm eff}$ for
stars later than B and 10–15 percent for B stars. For giants (class III), this
is about 10–15 percent for stars later than B, and about 15–20 percent for B
stars. The relative standard deviation in luminosity, $\log L/L_{\odot}$, is
about the same as for $T_{\rm eff}$.

Table 4: Calibration of effective temperature and luminosity as a function of
spectral type and class.

| V | IV | III | II | I
---|---|---|---|---|---
| $T_{\rm eff}$ | $\log\tfrac{L}{L_{\odot}}$ | $T_{\rm eff}$ | $\log\tfrac{L}{L_{\odot}}$ | $T_{\rm eff}$ | $\log\tfrac{L}{L_{\odot}}$ | $T_{\rm eff}$ | $\log\tfrac{L}{L_{\odot}}$ | $T_{\rm eff}$ | $\log\tfrac{L}{L_{\odot}}$
O3 | 37500 | 5.50 | 37500 | 5.50 | 37500 | 5.50 | 36200 | 5.50 | 35000 | 5.50
O4 | 36600 | 5.34 | 36600 | 5.44 | 36600 | 5.44 | 34600 | 5.44 | 33400 | 5.50
O5 | 35000 | 5.24 | 35000 | 5.37 | 35000 | 5.50 | 33000 | 5.50 | 31800 | 5.50
O6 | 34100 | 5.10 | 34100 | 5.25 | 34100 | 5.40 | 31400 | 5.45 | 30300 | 5.50
O7 | 32800 | 4.92 | 32800 | 5.05 | 32800 | 5.28 | 29800 | 5.30 | 28700 | 5.30
O8 | 30700 | 4.71 | 30700 | 4.85 | 30700 | 5.01 | 28200 | 5.02 | 27100 | 5.02
O9 | 28700 | 4.51 | 28700 | 4.76 | 28700 | 4.90 | 26600 | 4.95 | 25600 | 5.00
B0 | 25900 | 4.24 | 25900 | 4.47 | 25900 | 4.59 | 25000 | 4.62 | 24000 | 4.65
B1 | 23200 | 3.97 | 23200 | 4.20 | 23200 | 4.46 | 23400 | 4.52 | 22400 | 4.60
B2 | 22500 | 3.68 | 22500 | 3.88 | 22500 | 4.07 | 21800 | 4.25 | 20800 | 4.40
B3 | 20800 | 3.48 | 20000 | 3.62 | 19000 | 3.83 | 20200 | 3.91 | 19200 | 4.09
B4 | 18600 | 3.20 | 18200 | 3.41 | 17900 | 3.63 | 18600 | 3.90 | 17700 | 4.10
B5 | 17500 | 3.05 | 17000 | 3.18 | 16200 | 3.30 | 17000 | 3.50 | 16100 | 3.90
B6 | 15400 | 2.74 | 15200 | 2.87 | 15000 | 3.02 | 15400 | 3.40 | 14600 | 3.90
B7 | 14200 | 2.62 | 14000 | 2.74 | 13700 | 2.83 | 13800 | 3.30 | 13000 | 3.90
B8 | 12600 | 2.31 | 12600 | 2.49 | 12600 | 2.66 | 12200 | 3.20 | 11400 | 3.88
B9 | 11300 | 1.99 | 11300 | 2.20 | 11300 | 2.41 | 10600 | 3.00 | 9830 | 3.88
A0 | 9170 | 1.63 | 9170 | 1.76 | 9170 | 1.92 | 9000 | 2.60 | 8860 | 3.87
A1 | 8740 | 1.44 | 8740 | 1.50 | 8740 | 1.62 | 8420 | 2.40 | 8100 | 3.86
A2 | 8500 | 1.42 | 8500 | 1.49 | 8500 | 1.56 | 8290 | 2.30 | 8000 | 3.85
A3 | 8260 | 1.35 | 8260 | 1.42 | 8260 | 1.49 | 8150 | 2.30 | 7900 | 3.84
A4 | 8210 | 1.32 | 8210 | 1.39 | 8210 | 1.46 | 8020 | 2.30 | 7800 | 3.83
A5 | 7950 | 1.25 | 7950 | 1.32 | 7950 | 1.40 | 7890 | 2.30 | 7700 | 3.82
A6 | 7930 | 1.25 | 7930 | 1.34 | 7930 | 1.44 | 7750 | 2.30 | 7600 | 3.82
A7 | 7790 | 1.21 | 7790 | 1.28 | 7790 | 1.35 | 7620 | 2.30 | 7500 | 3.81
A8 | 7630 | 1.19 | 7630 | 1.28 | 7630 | 1.37 | 7490 | 2.30 | 7400 | 3.81
A9 | 7610 | 1.12 | 7610 | 1.22 | 7610 | 1.31 | 7360 | 2.30 | 7300 | 3.80
F0 | 7410 | 1.03 | 7410 | 1.16 | 7410 | 1.29 | 7200 | 2.30 | 7200 | 3.80
F1 | 7210 | 1.03 | 7210 | 1.14 | 7210 | 1.25 | 7090 | 2.30 | 7100 | 3.70
F2 | 6900 | 0.94 | 6900 | 1.08 | 7030 | 1.23 | 6960 | 2.30 | 6900 | 3.65
F3 | 6710 | 0.86 | 6710 | 1.03 | 6910 | 1.21 | 6820 | 2.30 | 6800 | 3.60
F4 | 6830 | 0.87 | 6830 | 1.04 | 6620 | 1.20 | 6690 | 2.32 | 6700 | 3.55
F5 | 6570 | 0.76 | 6570 | 0.98 | 6510 | 1.21 | 6560 | 2.35 | 6600 | 3.50
F6 | 6510 | 0.72 | 6510 | 0.92 | 6450 | 1.23 | 6420 | 2.35 | 6500 | 3.45
F7 | 6370 | 0.61 | 6370 | 0.91 | 6380 | 1.25 | 6290 | 2.36 | 6200 | 3.40
F8 | 6270 | 0.54 | 6270 | 0.85 | 6280 | 1.27 | 6160 | 2.36 | 6000 | 3.35
F9 | 6190 | 0.47 | 6190 | 0.79 | 6180 | 1.28 | 6020 | 2.35 | 5900 | 3.35
G0 | 6040 | 0.43 | 6040 | 0.68 | 6040 | 1.35 | 5890 | 2.37 | 5800 | 3.32
G1 | 5890 | 0.40 | 5890 | 0.65 | 5890 | 1.49 | 5760 | 2.41 | 5700 | 3.33
G2 | 5740 | 0.38 | 5740 | 0.65 | 5740 | 1.70 | 5620 | 2.50 | 5600 | 3.30
G3 | 5600 | 0.34 | 5600 | 0.59 | 5600 | 1.83 | 5490 | 2.55 | 5500 | 3.30
G4 | 5450 | 0.29 | 5450 | 0.60 | 5450 | 1.95 | 5360 | 2.60 | 5400 | 3.25
G5 | 5300 | 0.27 | 5300 | 0.68 | 5300 | 2.09 | 5230 | 2.70 | 5300 | 3.25
G6 | 5150 | 0.24 | 5150 | 0.76 | 5150 | 2.31 | 5090 | 2.75 | 5150 | 3.20
G7 | 5000 | 0.20 | 5000 | 0.89 | 5000 | 2.51 | 4960 | 2.85 | 5000 | 3.20
G8 | 4850 | 0.17 | 4850 | 1.08 | 4850 | 2.60 | 4800 | 2.87 | 4850 | 3.15
G9 | 4700 | 0.14 | 4700 | 1.30 | 4700 | 2.66 | 4690 | 2.90 | 4700 | 3.10
K0 | 4900 | 0.10 | 4900 | 1.37 | 4114 | 2.72 | 4114 | 2.93 | 4114 | 3.15
K1 | 4760 | -0.35 | 4760 | 0.89 | 4057 | 2.21 | 4057 | 3.36 | 4057 | 5.00
K2 | 4620 | -0.40 | 4620 | 0.92 | 4000 | 2.31 | 4000 | 3.39 | 4000 | 5.10
K3 | 4480 | -0.48 | 4480 | 0.97 | 3900 | 2.40 | 3900 | 3.44 | 3900 | 5.40
K4 | 4340 | -0.51 | 4340 | 1.01 | 3800 | 2.54 | 3800 | 3.50 | 3800 | 5.50
K5 | 4200 | -0.58 | 4200 | 1.03 | 3700 | 2.64 | 3700 | 3.56 | 3700 | 5.48
K6 | 4060 | -0.67 | 4060 | 1.09 | 3717 | 2.67 | 3717 | 3.55 | 3717 | 5.47
K7 | 3920 | -0.73 | 3920 | 1.15 | 3733 | 2.70 | 3733 | 3.54 | 3733 | 5.42
K8 | 3780 | -0.81 | 3780 | 1.23 | 3750 | 2.73 | 3750 | 3.57 | 3750 | 4.65
K9 | 3640 | -0.84 | 3640 | 1.32 | 3725 | 2.78 | 3725 | 3.58 | 3725 | 5.42
M0 | 3500 | -0.90 | 3500 | 1.42 | 3700 | 2.97 | 3700 | 3.77 | 3700 | 5.50
M1 | 3333 | -1.21 | 3333 | 1.55 | 3510 | 2.97 | 3510 | 3.77 | 3510 | 5.53
M2 | 3167 | -1.49 | 3167 | 1.71 | 3320 | 3.08 | 3320 | 3.92 | 3320 | 5.68
M3 | 3000 | -1.75 | 3000 | 1.89 | 3130 | 3.27 | 3130 | 4.15 | 3130 | 5.87
M4 | 2833 | -1.98 | 2833 | 2.10 | 2940 | 3.44 | 2940 | 4.36 | 2940 | 6.04
M5 | 2667 | -2.21 | 2667 | 2.32 | 2750 | 3.71 | 2750 | 4.67 | 2750 | 6.31
M6 | 2500 | -2.35 | 2500 | 2.61 | 2560 | 3.98 | 2560 | 4.98 | 2560 | 6.62
M7 | 2333 | -2.43 | 2333 | 2.97 | 2370 | 4.32 | 2370 | 5.40 | 2370 | 6.96
M8 | 2167 | -2.45 | 2167 | 3.39 | 2180 | 4.71 | 2180 | 5.87 | 2180 | 7.43
M9 | 2000 | -2.38 | 2000 | 3.90 | 1990 | 5.25 | 1990 | 6.45 | 1990 | 8.01

Figure 1: Examples of light curves and periodograms for different classes of
variable stars observed by TESS. The brightness scale is parts per thousand.

##  4 The light curve and periodogram

The MAST website provides a bulk download script for all light curves in a
particular TESS sector. The star number in this script is compared with the
TIC number in the master catalogue. If there is a match, the light curve is
downloaded as a FITS file. Typically, about 1000–4000 light curve files are
downloaded per sector.

The next step is to extract the light curve from the FITS file to a text file.
It is important to be able to generate the full path name of the light curve
from the TIC number. The light curve of TIC 010510382, for example, is stored
in file T010/010510382/010510382.dat. The text file contains three columns:
the time (BJD), the simple aperture (SAP) light curve and the pre-search data
conditioning (PDC) light curve in parts per thousand. If the star has
previously been observed, the latest light curve is simply appended to the
existing light curve. This procedure has been automated using suitable
software.

The next step is calculation of the periodogram and extraction of significant
frequencies, which is usually done by successive prewhitening. While this
technique is perfectly adequate for ground-based light curves, problems arise
when the signal-to-noise ratio, S/N, is very high, as it is in space
photometry (Balona, 2014a). In addition, successive prewhitening is very slow
and impractical for the task at hand.

This problem was solved by mapping the observed time series to a time series
which is spaced by exactly 2 min, the spacing in the TESS observations. Where
gaps exist in the time series, the data value is set to zero. This allows the
periodogram to be calculated using the fast Fourier transform (FFT). For this
purpose the number of data points needs to be a power of two. Normally, one
would take the power of two just greater than the number of points in the time
series and pad the excess with zeros. However, for greater resolution of the
periodogram peaks, the time series was extended further by as much as a factor
of 4 by zero padding. In this way one can obtained well-resolved periodograms
of 1000–4000 stars in just a few hours. The periodogram is stored in the same
folder as the light curve.

It is well known that the FFT is just another way of representing the data. It
neither adds nor detracts from the information content. All frequencies which
can be extracted are already visible in the FFT. These can be detected simply
by noting the peaks in the FFT. This process is no different from trying to
extract spectral line wavelengths in a crowded stellar spectrum. One picks the
strongest lines, fits suitable curves to each line profile and removes them.
Repeating the process one more time is sufficient to extract most of the
remaining significant spectral wavelengths. The code which applies the FFT
technique to the light curve also extracts the frequencies and amplitudes in
the same way, the window function being used to model each peak.

As each frequency is extracted, the S/N ratio is calculated by measuring the
amplitude relative to a smooth curve representing the periodogram background.
The extracted information, which is stored in the same folder as the light
curve and periodogram, contains the frequency, amplitude and their respective
errors as well as the S/N. Peaks are deemed significant if S/N $>4.7$. After
the periodogram for each star has been calculated, variability classification
may begin. In this process, the periodogram plays a dominant role. In fact, in
most cases it is not really possible to perform visual classification without
the periodogram.

Before describing the next stage in the classification process, it is first
necessary to discuss all the variability classes that were encountered.

##  5 Variability classes

Variability classification is a subjective process; the distinction between
variability classes is often not clear unless strict guidelines are provided.
The informal manner in which these classes were described for ground-based
observations is no longer adequate in the age of space photometry.

For example, the $\gamma$ Dor class was originally defined as “F stars with
multiple low frequencies”. This was perfectly adequate because the only F
stars with multiple frequencies known at that time were the $\delta$ Sct
stars. Since only high frequencies were observed in $\delta$ Sct stars (in
agreement with the models), this definition of the $\gamma$ Dor stars
presented no problem. The definition of “low frequencies” was typically taken
as frequencies below about 5 d-1 (approximately the lowest frequency observed
in ground-based observations of $\delta$ Sct stars), but the exact value was
not part of the definition. In order to drive low frequencies in $\gamma$ Dor
stars, a different mechanism from that in $\delta$ Sct stars was required
(Guzik et al., 2000), cementing the distinction between the two classes.

However, the distinction soon became blurred with the discovery of $\delta$
Sct+$\gamma$ Dor hybrids, i.e. $\delta$ Sct stars with both high and low
frequencies (Handler et al., 2002). The few hybrids then known suggested that
these were stars on the boundary between the two classes and could still be
accommodated by theory. Later, when Kepler observations became available, it
was found that practically all $\delta$ Sct stars contain low frequencies
(Grigahcène et al., 2010).

The situation became even less clear when Gaia parallaxes became available. It
was found that $\delta$ Sct stars occupy the same region of the H–R diagram as
$\gamma$ Dor stars (Balona, 2018). It seems that $\gamma$ Dor stars are not a
separate class after all. Under the circumstances, it is difficult to know
whether one should perhaps rename the $\gamma$ Dor class in a way which unites
it with the $\delta$ Sct class. Until a suitable model can be found to explain
the low frequencies in $\delta$ Sct stars and how high frequencies are damped
in $\gamma$ Dor stars, the problem will remain unresolved.

The same situation has recently been found in regard to the relationship
between $\delta$ Sct and roAp stars. It turns out that many ostensibly non-
peculiar stars pulsate with the high frequencies seen in roAp stars (above 60
d-1). Furthermore, the fraction of $\delta$ Sct stars among Ap stars is no
different from that in non-peculiar stars. Moreover, these $\delta$ Sct Ap
stars have frequency peaks covering the whole frequency range. In other words,
the roAp stars are just an arbitrary selection of Ap $\delta$ Sct stars with
no particular significance (Balona, 2022a).

The significance perhaps lies in a subset of roAp stars in which equally-
spaced multiplets, with spacing equal to the rotation frequency, are generated
due to the effect of the strong magnetic field. This occurs at high
frequencies when the pulsational wavelength is comparable to the atmospheric
scale height. Perhaps it is more appropriate to restrict the name “roAp” to
these stars. In any case, the class is probably too entrenched and we continue
to use it for now to avoid confusion. The definition adopted here for roAp is
any Ap star with frequencies in excess of 60 d-1.

These high frequencies are also present in non-peculiar $\delta$ Sct stars
where they were temporarily labeled “roA” stars by Balona (2022a), but this
name is now dropped. Historically, the upper frequency limit for unstable
modes in purely radiative models of $\delta$ Sct stars is about 50 d-1. Here
we label all $\delta$ Sct stars with independent frequencies higher than 50
d-1 as DSCTH (i.e. $\delta$ Sct high frequency) and those with independent
frequencies higher than 60 d-1 DSCTU ($\delta$ Sct ultra-high frequency). An
independent frequency is one which is not a linear combination of two lower
frequencies or multiples of lower frequencies. Whether or not the subclasses
DSCTH and DSCTU prove useful will have to be decided once we have a theory
that can explain these pulsations.

Yet another example of the inadequacy of the current system was the
realization that what appear to be $\gamma$ Dor stars are to be found at much
higher effective temperatures than predicted by the models (Balona et al.,
2016). These hot $\gamma$ Dor stars are to be found all along the upper main
sequence and merge with what would be generally described as SPB stars (Balona
& Ozuyar, 2020a). In order to avoid this confusion it is necessary to define
an effective temperature which separates the two groups. This was taken to be
$T_{\rm eff}=10\,000$ K in Balona & Ozuyar (2020a), a value that is continued
in this paper.

Note that many stars are assigned more than one variability class. For
example, a $\delta$ Sct star which is also suspected to show rotational peaks
is classified as DSCT+ROT. The criterion used in pulsating stars with
suspected rotational peaks is that the first harmonic must be clearly present
and both peaks must be of reasonably high amplitude.

The notation SPB/BCEP means that the star may be either classified as an SPB
or BCEP because it falls within the defined $T_{\rm eff}$ range for both
stars, but the frequency peaks straddle the defined limits for both classes.
In other words “+” means “and”; “/” means “or”.

A brief discussion of all variability classes used in this work now follows.
Where there is a possibility of confusion with another class, the adopted
limits on $T_{\rm eff}$ and frequency are given. This means that what is
classified as a $\gamma$ Dor star may be re-classified as an SPB star, or
vice-versa, if there is a change in the adopted $T_{\rm eff}$. Pictorial
examples of what the light curves and periodograms might look like are shown
in Fig. 1.

###  5.1 Dash

When a dash is allocated to a variability class it simply means that no
significant variability was detected. It is also sometimes allocated to a star
where there seems to be some indistinct, complex low-frequency variations
which cannot be classified.

###  5.2 Pulsating variables

$\alpha$ Cyg (ACYG) Variables are supergiants with low frequencies which may
be of pulsational origin or perhaps some other cause (Bowman et al., 2019).
The light often shows smooth irregular variations, but may also be variable on
shorter timescales. In the GCVS the designation is limited to B and A
supergiants, but here we extend it to F supergiants as well.

$\beta$ Cep (BCEP) are pulsating O8–B6 main sequence stars with multiple
periods in the range 0.1–0.6 d. The GCVS also defines a group of short-period
$\beta$ Cep stars (BCEPS), but this class is obsolete. The long period end of
this definition is more appropriate to SPB stars and in this work BCEP stars
are limited to multiple frequencies in excess of 3 d-1 and with $T_{\rm
eff}>18000$ K (Balona & Ozuyar, 2020a). The class BCEPH denotes a BCEP with
one or more frequencies higher than 50 d-1.

The Maia variables (MAIA) are main sequence mid- to late B-type stars which
pulsate in multiple high frequencies very similar to $\delta$ Sct stars
(Balona et al., 2015, 2016). MAIAH and MAIAU are Maia variables with
frequencies higher than 50 and 60 d-1 respectively. The Maia stars merge with
the $\beta$ Cep stars at high temperatures and with the $\delta$ Sct stars at
lower temperatures. For this reason, the limit $10000<T_{\rm eff}<18000$ K was
imposed to properly define the MAIA class. At low frequencies, Maia variables
merge with the SPB stars. Therefore it is also necessary to impose a frequency
limit of 5 d-1. It has been suggested that Maia stars are just rapidly
rotating SPB stars (Salmon et al., 2014), but this is not supported by $v\sin
i$ measurements (Balona & Ozuyar, 2020a). The pulsations may be due to high
degree modes (Daszyńska-Daszkiewicz et al., 2017).

In the GCVS the slowly pulsating B stars (SPB stars) are labeled as LPB (long-
period pulsating B stars). In this paper we use the name SPB. These are
multiperiodic mid- and late-B stars with frequencies less than about 5 d-1.
However, TESS has shown that these variables are present among all main-
sequence B stars, including the hottest B stars (Balona & Ozuyar, 2020a). They
can be confused with the $\beta$ Cep variables, many of which have frequencies
as low as 3 d-1. In order to avoid this confusion, the designation SPB is used
for stars with $T_{\rm eff}>18000$ K with frequencies less than 3 d-1. For
cooler B stars ($10000<T_{\rm eff}<18000$ K), the frequency limit is extended
to 5 d-1 to avoid confusion with Maia variables (Balona & Ozuyar, 2020a). The
SPB and $\gamma$ Dor stars seem to form a continuous sequence across all B, A
and F main sequence stars for which no explanation exists at this time. For
this reason, the SPB class is restricted to stars hotter than 10 000 K.

The $\gamma$ Dor stars (GDOR), $\delta$ Sct (DSCT) and roAp stars have already
been discussed. To avoid confusion with $\delta$ Sct and SPB star, the
frequencies in GDOR stars must be lower than 5 d-1 and $T_{\rm eff}<10\,000$
K. The $\delta$ Sct stars must have $T_{\rm eff}<10\,000$ K to avoid confusion
with Maia variables.

SX Phe (SXPHE) stars resemble the $\delta$ Sct variables but belong to
spherical component or old disk galactic population. They are mainly present
in globular clusters.

Solar-type oscillators (SOLR) are generally F or K stars with very low
amplitude pulsations similar to the 5-min solar oscillations. Large numbers of
these stars were detected by Kepler, but are much more difficult to identify
with TESS owing to the larger noise level. The SOLR stars listed in this paper
are all Kepler stars.

Cepheids (CEP) are F and G supergiants with radial pulsations. Later spectral
types have longer periods. The GCVS has three other subclass, CEP(B), DCEP and
DCEPS, but these are not used here.

The RR Lyraes (RR, RRAB, RRC) are radially pulsating Population II giants.
They are present in large numbers in globular clusters. The general
designation is RR, but most of the stars observed by Kepler and TESS are of
the RRAB class with asymmetric light curves with a steep ascending branch. A
few RRC stars, which have symmetrical almost sinusoidal light curves, have
also been observed by TESS.

The W Vir stars (CW) are Population II pulsating variable). The subclasses CWA
and CWB are W Vir stars with periods respectively longer and shorter than 8 d.

The RV Tau (RV) stars are radially pulsating supergiants with F/G spectral
types at maximum and K/M at minimum with alternating primary and secondary
minima. The minima vary in depth so that the primary minimum may become the
secondary minimum and vice-versa. The period between two primary minima is
30–150 d. The subclass RVA does not vary in mean magnitude, while RVB does
vary in mean magnitude.

The Mira (M) variables are long-period variable giants with characteristic
late-type emission spectra (Me, Ce, Se) and semi-regular pulsations with
periods 80–1000 d.

Semiregular variables (SR) are giants or supergiants of intermediate and late
spectral types showing noticeable light periodicity (from 20 to over 2000 d),
but with irregular interruptions. The GCVS divides these into subclasses SRA
(persistent periodicity) and SRB (poorly defined periodicity). The SRD class
are F, G or K supergiants with 30–1100 d periods.

In the GCVS the V361 Hya stars are labeled RPHS (rapidly pulsating hot
subdwarfs), but there is no class for the slowly pulsating V1093 Her stars.
The label SPHS is used here. The separation between the two classes is taken
to be about 300 d-1 in frequency, which means that it is not possible to
detect pure RPHS stars with TESS. Those that are labeled RPHS are taken from
the literature. Originally, there seemed to be a distinction in effective
temperature between RPHS and SPHS stars, the SPHS being cooler. This
distinction seems to have disappeared with further observations (Krzesinski &
Balona, 2022). Most pulsating B subdwarfs appear to be hybrids (SPHS+RPHS).

PVTEL variables are helium supergiant Bp stars with weak hydrogen lines and
enhanced lines of He and C. They pulsate with periods of approximately 0.1–1
d. The BLAP (blue large-amplitude pulsator) are a proposed new class of
degenerate star (Pietrukowicz et al., 2013). At this stage little is known
about them.

ZZ Ceti variables (ZZ) are nonradially pulsating white dwarfs with periods in
the range 0.5–25 min. Flares are sometimes observed which are usually
attributed to a cool companion. The ZZA are of DA spectral type, while ZZB are
DB stars. The ZZO class was introduced in the GCVS for ZZ Cet variables of the
DO spectral type showing HeII and and CIV absorption lines in their spectra.
This also includes the GW Vir class.

###  5.3 Rotational variables

In the GCVS, the BY Dra (BY) variables are emission-line dwarfs of dKe-dMe
spectral type showing quasi-periodic light changes with periods from a
fraction of a day to 120 d. The light variability is caused by spots. Many
such star also flare. The FK Com stars (FKCOM) are rapidly-rotating G or K
giants with non-uniform surface brightness. The light variation is due to
rotation.

In the GCVS the $\alpha^{2}$ CVn (ACV) stars are defined by a strictly
periodic variation in B8p–A7p stars. They exhibit magnetic field and
brightness changes (periods of 0.5–160 days or more). These peculiar main-
sequence stars display strong lines of Si, Sr, Cr and rare earths. The light
modulation is attributed to surface patches with differing abundances. In this
work, the ACV designation is applied to any Ap/Fp star with $T_{\rm
eff}<10000$ K which shows strict periodicity compatible with the expected
rotation frequency.

In the GCVS, SX Ari (SXARI) variables are main sequence peculiar B stars with
variable intensity He I and Si III lines and strong magnetic fields. The light
period is the same as the rotational period. These are the high-temperature
analogs of the ACV variables. In this paper, SX Ari stars are peculiar B stars
with $T_{\rm eff}>10000$ K. We distinguish between the SX Ari stars and the
He-weak and He-rich variables in that SXARI stars must have anomalous
abundances of Si or Hg/Mn. This is a subject which requires a deeper study.
The He-variable stars are usually classified as ROT if they have a strict
light period consistent with rotation.

The label MRP describes stars which emit radio pulses, two per rotation period
(Das et al., 2022). They appear to be ACV or SXARI stars.

Even a brief examination of Kepler or TESS photometry will show that a
significant fraction of A and B stars have an isolated low frequency, low
amplitude peak at less than 4 d-1, consistent with what one might expect for
the rotational frequency. The harmonic of the peak is often present. Many
studies have shown that these are indeed to be identified as the rotational
frequency (see Balona 2022b and references therein). The GCVS does not have a
designation for hot stars with rotational modulation. For this reason, the ROT
class was introduced. A subclass, the ROTD stars, is reserved for stars which
show a sharp rotational peak at a slightly higher frequency than a broad peak
which is suspected to be due to inertial modes (the “hump and spike” stars
Balona 2014b; Saio et al. 2018). It is often not possible to distinguish
between rotational modulation and low-level eclipses and for this reason the
ROT class is restricted to frequency peaks with amplitudes less than 10 ppt.
However, most of the ROT variables have amplitudes less than 0.5 ppt (Balona,
2022b).

###  5.4 Eclipsing variables

The general GCVS classification for eclipsing stars is E. However, more
specific subclasses have been introduced. Note that not much attention has
been payed to specific subclasses of eclipsing variables in this work.
Generally, any star in which eclipses are suspected are classified as EA or
EB.

The EA class, or Algol ($\beta$ Per)-class eclipsing systems, are binaries
with spherical or slightly ellipsoidal components. This class is used when it
is possible to specify the moments of the beginning and end of the eclipses.
Between eclipses the light remains almost constant or varies insignificantly
because of reflection effects.

The EB ($\beta$ Lyr) eclipsing systems have ellipsoidal components and light
curves for which it is impossible to specify the exact times of onset and end
of eclipses. The depth of secondary minimum is usually considerably smaller
than that of the primary minimum.

AM Her variables (AM) are close binary systems consisting of a dK or dM star
and a compact object with a strong magnetic field.

DM are detached main sequence binary systems in which the inner Roche lobes
are not filled.

The HRTB (heartbeat stars, Welsh et al. 2011) are variable binary star systems
in eccentric orbits with pulsations driven by tidal forces.

The ellipsoidal variables (ELL) are close binary systems with ellipsoidal
components. The combined brightness varies with a period equal to the orbital
period, but there are no eclipses.

In the GCVS the designation EP is given to stars with eclipses due to planets.
In this work we use EP for any eclipse-like variation with an amplitude less
than about 10 ppt. The periodogram typically contains several low-amplitude
harmonics which could either be due to low-level eclipses or rotation. The
distinction is made mainly on the number of visible harmonics. If only one or
two harmonics are seen, then the star is classed as ROT, otherwise as EP. It
is quite possible that many stars classified as EP are actually rotational
variables or even planetary systems. An analysis of these stars for planetary
systems would be interesting as very few A and B stars are known to host
planets.

W UMa class eclipsing variables (EW) have periods shorter than 1 d and the
primary and secondary minima are equal or almost equal in depth. In the GCVS
this designation is used only for F or later type stars.

GS eclipsing systems are giants or supergiants where one of the components may
be a main sequence star.

The RS CVn stars (RS) are eruptive variables in a binary system showing Ca II
H and K emission. The light period is close to the orbital period and they are
X-ray sources.

Table 5: A list of Kepler, K2 and TESS stars classified by the author. The
full catalogue is available electronically. The column labeled p is the
priority number for the adopted effective temperature. The rotation period,
$P_{\rm rot}$ is given where applicable. The column marked pec is a code for
the stellar peculiarity.

RA | Dec | Name | Var | $T_{\rm eff}$ | $\log\tfrac{L}{L_{\odot}}$ | p | $v\sin i$ | $P_{\rm rot}$ | pec | Sp. Type
---|---|---|---|---|---|---|---|---|---|---
279.8504662 | -47.1368375 | TIC 339907982 | EB | 31000 | 1.62 | 4 |  |  | 9 | sdB,sdB,sdO/BsdB
279.8654723 | 25.1250158 | TIC 316904504 | - | 7970 | 1.20 | 4 |  |  |  | A0
279.8681614 | 45.3642605 | TIC 383751023 | - | 7819 | 1.09 | 1 |  |  | 1 | kA2hA6mF4
279.8695086 | 43.5142894 | KIC 7797592 | ROT | 6905 | 0.83 | 3 |  | 1.326 |  | [F5V]
279.8759807 | -21.9655564 | EPIC 216652481 | ROT/SPB | 12600 | 3.39 | 10 | 174.0 | 1.786 | 6 | B8IIIe
279.8769148 | -48.5085760 | TIC 304451865 | DSCT | 7289 | 1.14 | 4 |  |  |  | A8/9V
279.8783231 | 18.3319300 | TIC 345978356 | - | 5971 | 0.11 | 6 |  |  |  | G0
279.8834254 | -55.3068028 | TIC 120202126 | ROT | 6449 | 0.57 | 4 |  | 4.651 |  | F3V
279.8867664 | -72.9482134 | TIC 344167219 | - | 6559 | 0.46 | 4 |  |  |  | F3V
279.8876674 | 40.9350562 | TIC 157696928 | SPB | 10423 | 2.51 | 1 | 106.0 |  |  | B8:Vn + A0III

###  5.5 Eruptive and irregular variables

The designation FLARE is attached to any object where evidence of a flaring
event is present in the light curve.

The Be stars (BE, BE+ROT) are eruptive B stars in which Balmer line emission
is seen, though the emission might disappear and re-appear at different times.
Recent observations from TESS have shown that about 75 percent of Be stars
vary with short quasi-periods in the range 0.5–2.0 d (Balona & Ozuyar, 2020b).
It is suggested that the eruptions and subsequent rotational modulation is a
result of localized surface activity (flaring) which ejects material. This
material is channelled to diametrically opposed regions where the geometric
and magnetic equators intersect. The light and line-profile variation is due
to obscuration by these trapped clouds which eventually disperse into the
circumstellar disc (Balona & Ozuyar, 2021; Balona, 2022b).

Both the periodogram and the light curve are very distinctive. The periodogram
normally shows a low-frequency group (suspected to be a result of
circumstellar material) as well as two other groups. One group can be
identified at about the expected rotational frequency, while the other group
is at about twice the rotational frequency. This is a consequence of
obscuration by the diametrically located trapped clouds. The amount of trapped
material varies with time, sometimes resulting in the disappearance of the
highest frequency group. What is clear is that the peaks are broad and vary in
phase and amplitude with a timescale of weeks to months (Balona & Ozuyar,
2020b, 2021). The light curve often shows distinctive eclipse-like dips, quite
different from the light curves of SPB and $\gamma$ Dor stars which vary in
the same period range, but with strict coherent frequencies.

In this work, Be stars which show the three groups are labeled as BE+ROT while
those which only show long-period variations are classed as BE. The ROT
designation refers to the possible non-coherent rotational light variation of
the two higher-frequency groups as described. The distinctive periodogram and
light curve is sometimes seen in ostensibly non-Be stars. These may be
incipient Be stars which are designated BEV or BEV+ROT.

The $\gamma$ Cas (GCAS) stars are Be stars with X-ray emission. The
$\sigma^{2}$ Ori variables (SORI) are Be stars in which the light variation is
due to circumstellar matter trapped by a strong tilted dipole magnetic field.
Most Wolf-Rayet (WR) stars show irregular light variations, but periodicity is
present in a few stars (WR+BCEP or WR+ROT).

UV are variables of the UV Ceti class, which are KVe–MVe stars sometimes
displaying flare activity with amplitudes from several tenths of a magnitude
up to several magnitudes.

The S Doradus (SDOR) are eruptive, high luminosity Bp–Fp stars showing
irregular (sometimes cyclic) light changes with amplitudes 1–7 mag in V. As a
rule, these stars are associated with diffuse nebulae and surrounded by
expanding envelopes.

U Geminorum-class variables (UG), quite often called dwarf novae, are close
binary systems consisting of a dwarf or subgiant K-M star that fills the
volume of its inner Roche lobe and a white dwarf surrounded by an accretion
disk. Orbital periods are in the range 0.05-0.5 d. These are divided into
subclasses UGSU (SU UMa systems) and UGSS (SS Cyg systems) and UGZ (Z Cam
systems).

Stars of variability class I are an homogeneous set of poorly studied
irregular variables. This can be divided into class IA (early-type, O–A) and
IB (late type). The IN type are irregular, eruptive variables connected with
bright or dark diffuse nebulae or observed in the regions of these nebulae.
These again are divided into early type (INA) and late type (INB). The INS and
ISA classes show rapid light variations. INT or IT are Orion variables of the
T Tauri type. NL are poorly studied eruptive nova-like variables. PN stars are
the nuclei of planetary nebulae. The LB class describes late-type stars with
slow irregular variations.

The NB, NC and NL classes are novae with fast (NA), slow (NB) and very slow
(NC) light variations. The NL class are poorly studied nova-like variables.

XP denotes an X-ray pulsar source. The subclass XNG describes a nova-like
transient system consisting an early-type giant or supergiant with a compact
companion.

##  6 Classification

Once the TESS stars have been downloaded and the periodograms calculated, the
next step is to display the light curves and periodograms of unclassified
stars for visual classification. It is imperative that these curves be
displayed with the utmost ease to reduce the time required for classification.
For this purpose, software was developed which enabled the display of these
curves by simply clicking on a list of TIC star numbers.

The periodogram is the most important source of information for classifying
purposes. The process requires two passes. In the first pass, the full
frequency range of the periodogram is displayed (0–360 d-1) so that high
frequencies can be detected. In this pass, many $\delta$ Sct stars and
eclipsing binaries are normally seen. In the second pass, the displayed
frequency range is narrowed to 0–10 d-1. This pass proceeds more slowly as
other classes of variable are identified. In every new TESS release there are
around 1000–2000 previously unclassified stars. The process of classification
is usually accomplished in 2–4 days.

Among the stars in the master catalogue observed by Kepler, K2 and TESS, only
about 2 percent appear in the GCVS. The variability class derived by visual
inspection is nearly always in good agreement with the GCVS class, but
corrections were sometimes made.

An extract of the catalog, which is complete up to TESS sector 55, is shown in
Table 5. In this table, stars without spectral type have been assigned
spectral types in accordance with the measured $T_{\rm eff}$ and $\log
L/L_{\odot}$. These spectral types are enclosed in square brackets. The
stellar peculiarity has been assigned a code in accordance with the following
scheme: 1: Am (CP1); 2: Ap (CP2); 3: Hg/Mn (CP3); 4: He-weak (CP4); 5: He-rich
(CP5); 6: Classical Be; 7: Herbig Be; 8: $\lambda$ Boo; 9: sdOB.

There are 20 744 stars observed by Kepler, 5 877 stars observed by K2 and 94
186 TESS stars (total 120 807 stars). The location of several types of
pulsating star in the H-R diagram is shown in Fig. 2.

Figure 2: Location of some pulsating stars in the H–R diagram. The solid line is the zero-age main sequence from solar composition models by Bertelli et al. (2009).  Table 6: Number of stars in the most common classes and sub-classes. Class | N | Class | N
---|---|---|---
TOTAL | 120807 | FLARE (all) | 1146
Non-variable (-) | 45064 | Be (all) | 723
|  | BE+ROT | 589
Rotational (all) | 42983 | SORI | 15
ROT | 1302 | BEV | 49
ROTD | 289 | GCAS | 3
BY | 8 |  |
FKCOM | 2 | BCEP (all) | 700
ACV | 996 |  |
SXARI | 675 | Maia (all) | 518
|  | MAIA | 475
$\delta$ Sct (all) | 13755 | MAIAH | 28
DSCT | 10164 | MAIAU | 15
DSCTH | 1468 |  |
DSCTU | 2001 | ACYG (all) | 426
ROAP | 121 |  |
|  | Tidal (all) | 459
GDOR (all) | 6128 | ELL | 374
|  | HRTB | 85
Eclipsing (all): | 5928 |  |
EA | 2722 | SPHS | 178
EB | 1996 |  |
EP | 970 | CEP | 132
EW | 216 | CW | 16
SOLR (all) | 2415 | RRAB | 198
SPB (all) | 1286 | RRC | 52

##  7 Conclusions

Over the last decade, the author has been visually classifying the light
variations of stars observed by Kepler, K2 and TESS. The classification system
has been kept as close as possible to that of the General Catalogue of
Variable Stars, but a few modifications were inevitable. The results are
presented as a table of over 120 000 stars. These data could be used as a
learning sample for an artificial intelligence code which can then be applied
to all TESS light curves. Table 6 summarizes the number of stars in each class
and subclass for the most common classes.

The informal and rather vague definitions of variability classes in the
General Catalogue of Variable Stars is no longer appropriate to the large
number of stars observed from space photometry. In this work we have attempted
to define the classes according to limits of effective temperature and
frequency, where necessary. There is a pressing need for a more comprehensive
theoretical understanding of stellar pulsation along the main sequence. It is
quite clear that preconceived assumptions regarding the atmospheres in upper
main sequence stars need to be relaxed. Until the models are able to replicate
the locations of the stars in the H–R diagram and their pulsational
frequencies, no progress can be made in understanding the different classes of
pulsational variables that have been observed.

In order to support this project, the author has also been collecting
estimates of effective temperature from the literature. This is presented as a
table of over 900 000 individual measurements for nearly 635 000 stars with
literature references and a code which indicates the method used. A catalogue
of projected rotational velocities containing over 75 000 individual
measurements for nearly 52 000 stars with bibliographic reference code is also
presented.

The assigned variability class represents the author’s opinion and it is
possible that others would assign a different class for many stars. There is
no “correct” variability class. There are, no doubt, errors in these three
catalogues which have escaped the author’s attention and there is no claim of
completeness in any of these catalogues. They are presented in the hope that
they will prove useful to the community.

## Acknowledgments

LAB wishes to thank the National Research Foundation of South Africa for
financial support.

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA Explorer Program. Funding for the TESS
Asteroseismic Science Operations Centre is provided by the Danish National
Research Foundation (Grant agreement no.: DNRF106), ESA PRODEX (PEA
4000119301) and Stellar Astrophysics Centre (SAC) at Aarhus University. We
thank the TESS and TASC/TASOC teams for their support of the present work.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of the SIMBAD database, operated at CDS, Strasbourg, France.

The data presented in this paper were obtained from the Mikulski Archive for
Space Telescopes (MAST). STScI is operated by the Association of Universities
for Research in Astronomy, Inc., under NASA contract NAS5-2655.

This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France.

## Data availability

All data are incorporated into the article and its online supplementary
material. The data are available at:

https://sites.google.com/view/tessvariables/home

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