# TOAST: Trajectory Optimization and Simultaneous Tracking Using Shared Neural
Network Dynamics

Taekyung Kim, Hojin Lee, Seongil Hong, and Wonsuk Lee Manuscript received:
January 20, 2022; Revised: April 25, 2022; Accepted: June 1, 2022.This paper
was recommended for publication by Editor C. Gosselin upon evaluation of the
Associate Editor and Reviewers’ comments. (Taekyung Kim and Hojin Lee are co-
first authors.) (Corresponding author: Wonsuk Lee.)The authors are with the
Ground Technology Research Institute, Agency for Defense Development, Daejeon
34186, Republic of Korea {ktk1501, hojini1117, hongsi,
<EMAIL_ADDRESS>Object Identifier (DOI): 10.1109/LRA.2022.3184769

###### Abstract

Neural networks have been increasingly employed in Model Predictive Controller
(MPC) to control nonlinear dynamic systems. However, MPC still poses a problem
that an achievable update rate is insufficient to cope with model uncertainty
and external disturbances. In this paper, we present a novel control scheme
that can design an optimal tracking controller using the neural network
dynamics of the MPC, making it possible to be applied as a plug-and-play
extension for any existing model-based feedforward controller. We also
describe how our method handles a neural network containing history
information, which does not follow a general form of dynamics. The proposed
method is evaluated by its performance in classical control benchmarks with
external disturbances. We also extend our control framework to be applied in
an aggressive autonomous driving task with unknown friction. In all
experiments, our method outperformed the compared methods by a large margin.
Our controller also showed low control chattering levels, demonstrating that
our feedback controller does not interfere with the optimal command of MPC.
111Our video can be found at: https://youtu.be/YQG0yHE5jWw

###### Index Terms:

Model learning for control, robust/adaptive control, autonomous vehicle
navigation, optimization and optimal control, field robots.

##  I INTRODUCTION

Online trajectory optimization, also known as Model Predictive Control (MPC),
has provided promising results in numerous robotic applications. The most
popular current methods for controlling high-dimensional nonlinear systems
involve gradient-based MPC, which was established in the context of
differential dynamic programming [1, 2, 3]. However, this approach poses the
inherent limitation that the algorithm relies on a quadratic approximation for
cost functions and requires smooth dynamics. Furthermore, it is extremely
difficult to include state constraints in the optimization process of
gradient-based MPC.

As a remedy, sampling-based approaches that do not require linear and
quadratic approximations of dynamics and cost functions have been developed.
The key advantage of sampling-based MPC is that it is capable of solving non-
convex optimization problems, allowing engineers to easily encode high-level
robot behaviors using cost function clipping [4, 5]. With the benefit that the
algorithm can directly use the dynamics model without linearization, neural
networks that have been proven effective for approximating the nonlinear
transition models are increasingly employed in sampling-based MPC [6, 7].

Both of these MPC algorithms (called feedforward controllers) optimize an
open-loop control sequence and execute this sequence until the next
optimization update. However, an achievable update rate of the MPC is
insufficient to cope properly with model uncertainty and external
disturbances. This is because the optimization process must incorporate a
number of trajectory rollouts along the future time horizon to reach the
required performance level. In practice, a simply designed local feedback
compensator is often used to correct for trajectory errors on the fastest time
scales [8]. The fundamental problem is that such compensators decouple the
feedback and feedforward control domains, resulting in decreased efficiency
due to the lack of consideration for the feedforward optimization process. As
an alternative, the iterative Linear Quadratic Regulator (iLQR) [9], which is
also known as a Sequential Linear Quadratic (SLQ) solver [10], was proposed to
be used in an online setting for deriving the optimal feedforward trajectory
and the feedback gains simultaneously by solving a single optimal control
problem [2, 11]. However, such schemes have only been considered in gradient-
based settings and they are not scalable to sampling-based algorithms. In
contrast to gradient-based MPC, the optimal feedback controller for sampling-
based MPC has not been fully addressed by prior work.

One particular strategy to deal with this issue is to apply iLQR on top of the
sampling-based MPC for designing an ancillary feedback policy [12, 7].
However, the feedback gain term in iLQR is produced to push the control
sequence to the locally optimal trajectory during the line search optimization
procedure. The feedback gain becomes negligible when the sequence converges
after recursive backward and forward passes. Nonetheless, it is possible to
obtain a considerable feedback gain by intentionally reducing the iterations,
but it is misleading to refer to this as an ideal optimal control gain.
Finally, all the feedback strategies mentioned above have not yet considered a
proper extension to neural network dynamics.

Figure 1: Overview diagram of our TOAST control framework when applied to an
aggressive driving task of an Unmanned Ground Vehicle (UGV). The components
related to the feedforward controller and the feedback controller are depicted
in blue and red, respectively.

In this paper, we propose a novel approach that combines trajectory
optimization and optimal tracking control via sharing neural network dynamics,
which is applicable to both gradient-based and sampling-based MPC. Since our
method is allowed to use the neural network transition models of the system,
it can be applied as a plug-and-play extension to previously completed model-
based control tasks. We also describe how our method handles a neural network
containing history information, in which such a strategy is often employed for
capturing time-varying or higher-order effects. We evaluated our algorithm
using classical control benchmarks in which crosswinds are modeled as external
disturbances. In addition, we demonstrated the feasibility of our idea by
performing a challenging autonomous driving task under varying road friction
conditions. The proposed method called Trajectory Optimization and
Simultaneous Tracking (TOAST) is illustrated in Fig. 1.

##  II METHODOLOGY

In this paper, we consider the problem of designing a generic control
framework that includes MPC and a corresponding optimal tracking controller
that both employ the same learned dynamics. While the idea is general and
applicable to any MPC algorithm, such as iLQG [2] and Differential Dynamic
Programming (DDP) [3], we use Smooth Model Predictive Path Integral (SMPPI)
control [13] as the feedforward controller throughout this paper.

###  II-A Smooth Model Predictive Path Integral Controller

This section provides a brief overview of the SMPPI approach. Consider a
standard discrete time dynamic system in which we denote the state at time $t$
as ${\bf x}_{t}\in\mathbb{R}^{n}$ and the action as ${\bf
u}_{t}\in\mathbb{R}^{m}$. A Model Predictive Path Integral (MPPI) controller,
which is one of the most successful sampling-based approaches in recent years,
draws parallel control sequence samples $U=\\{{\bf u}_{0},{\bf u}_{1},\dots\bf
u_{T-1}\\}$ with a fixed time horizon $T$ [5]. The optimal control trajectory
$U^{*}$ is then obtained using the information theoretic interpretation and
importance sampling. The parallel samples are evaluated with the state cost
function $c({\bf x})$ using a Graphics Processing Unit (GPU). However,
chattering innately occurs in the optimized control sequence due to the nature
of sampling-based algorithms. SMPPI is proposed to attenuate the chattering
without extrinsic smoothing filters. The main idea is to decouple the control
space and action space. Noisy sampling is performed on the control sequences
and then the integral operation is applied on them to obtain action sequences
while smoothing chattering.

###  II-B Designing a Simultaneous Optimal Tracking Controller

Online trajectory optimization methods, as well as SMPPI, have shown effective
performance in various dynamic systems. However, there remains a problem: the
computational demand is too high for a real-world system to process in the
required time frame. In an MPC-controlled system, the same feedforward command
should be maintained for a certain amount of processing time. Control errors
induced by external disturbances or model uncertainty might be difficult to
respond to in such a system.

We propose an optimal feedback controller that minimizes the control errors
that occur while the MPC calculates the next feedforward command. This
ancillary feedback controller does not interfere with the optimal command of
MPC, but rather takes the feedforward command as a reference and performs
tracking control that reduces the error at a faster rate. Our method does not
need to form a new error function to track the feedforward action, nor does it
need to learn separate neural network dynamics. It calculates the optimal
feedback gain adaptively by utilizing the same learned dynamics from MPC.
Here, we emphasize that the feedback controller discussed in this paper is
distinct from the low-level plant controller, which manages actuators based on
the control input and is often referred to as a black-box.

Consider a discrete time nonlinear system dynamics model:

${\bf x}_{t+1}={\bf F}({\bf x}_{t},{\bf u}_{t}).$ (1)

An optimal action trajectory found by MPC is given as $U^{*}=\\{{\bf
u}^{*}_{0},{\bf u}^{*}_{1},\dots\bf u^{*}_{T-1}\\}$ and corresponding state
trajectory is given as $X^{*}=\\{{\bf x}^{*}_{0},{\bf x}^{*}_{1},\dots\bf
x^{*}_{T-1}\\}$. Linearize (1) by using first-order Taylor series expansion:

$\displaystyle{\bf x}_{t+1}$ $\displaystyle\approx{\bf F}({\bf x}_{t}^{*},{\bf
u}_{t}^{*})\,+$ (2) $\displaystyle\underbrace{\left.\frac{\partial{\bf
F}}{\partial{\bf x}_{t}}\right|_{{\bf x}_{t}={\bf x}_{t}^{*}}}_{{\bf
A}_{t}}\left({\bf x}_{t}-{\bf
x}_{t}^{*}\right)+\underbrace{\left.\frac{\partial{\bf F}}{\partial{\bf
u}_{t}}\right|_{{\bf u}_{t}={\bf u}_{t}^{*}}}_{{\bf B}_{t}}\left({\bf
u}_{t}-{\bf u}_{t}^{*}\right).$

As the definition of (1), (2) is equivalent to:

${\bf x}_{t+1}-{\bf x}_{t+1}^{*}\approx{\bf A}_{t}\left({\bf x}_{t}-{\bf
x}_{t}^{*}\right)+{\bf B}_{t}\left({\bf u}_{t}-{\bf u}_{t}^{*}\right).$ (3)

Here, we define new variables ${\bf z}_{t}={\bf x}_{t}-{\bf x}_{t}^{*}$ and
$\bm{\nu}_{t}={\bf u}_{t}-{\bf u}_{t}^{*}$. Then the system dynamics can be
expressed as:

${\bf z}_{t+1}={\bf A}_{t}\,{{\bf z}_{t}}+{\bf B}_{t}\,{\bm{\nu}_{t}}.$ (4)

Assume a quadratic cost function:

$J=\sum_{t=0}^{\infty}{\bf z}_{t}^{\top}\,{{\bf Q}}\,{\bf
z}_{t}+\bm{\nu}_{t}^{\top}\,{{\bf R}}\,\bm{\nu}_{t},$ (5)

where ${\bf Q}\geq 0,{\bf R}>0$. Then, it seems to be formulated as a standard
LQR problem [14]. Here, we use the infinite-time objective function for
computational efficiency. The resulting errors from this simplification would
be sufficiently compensated by the online optimization strategy of MPC. It is
worth mentioning that the dynamics near the optimal trajectory can be
interpreted as being locally linear because the discretized time frame of MPC
would be short enough. As a result, a linear feedback controller such as LQR
can be adopted for this problem.

One should note that the tracking controller will run faster than the MPC. The
discretized time period in (4) is denoted by $\Delta{t}_{1}$, and the one in
(5) is denoted by $\Delta{t}_{2}$. Converting the system dynamics (4) to have
faster sampling period $\Delta{t}_{2}$ results in:

$\displaystyle\tilde{{\bf z}}_{t+1}$ $\displaystyle=\left(1-\frac{\Delta
t_{2}}{\Delta t_{1}}\right)\tilde{{\bf z}}_{t}+\frac{\Delta t_{2}}{\Delta
t_{1}}\left({\bf A}_{t}\tilde{{\bf z}}_{t}+{\bf
B}_{t}\tilde{\bm{\nu}}_{t}\right)$ (6)
$\displaystyle=\underbrace{\left[\frac{\Delta t_{2}}{\Delta t_{1}}{\bf
A}_{t}+\left(1-\frac{\Delta t_{2}}{\Delta
t_{1}}\right)I_{n}\right]}_{\tilde{{\bf A}}_{t}}\tilde{{\bf
z}}_{t}+\underbrace{\frac{\Delta t_{2}}{\Delta t_{1}}{\bf B}_{t}}_{\tilde{{\bf
B}}_{t}}\,\tilde{\bm{\nu}}_{t}$ $\displaystyle={\tilde{{\bf
A}}_{t}}\,\tilde{{\bf z}}_{t}+{\tilde{{\bf B}}_{t}}\,\tilde{\bm{\nu}}_{t},$

where we use notation $\tilde{\left(\cdot\right)}$ to denote variables that
are discretized into a shorter sampling period. Notice that the formulation of
the converted dynamics in (6) is that of a common linear interpolation.

In the discrete time linear quadratic control problem of (5) and (6), we can
solve for $\tilde{\bm{\nu}}_{t}$ to yield the optimal linear feedback policy
using the discrete Riccati equation [15]:

$\tilde{\bm{\nu}}_{t}=-{{\bf K}_{t}}\,\tilde{{{\bf z}}}_{t},$ (7)

where ${\bf K}_{t}$ is feedback gain:

$\displaystyle{\bf K}_{t}$ $\displaystyle=({{\bf R}}+\tilde{{\bf
B}}_{t}^{\top}{{\bf P}_{t}}\tilde{{\bf B}}_{t})^{-1}\tilde{{\bf
B}}_{t}^{\top}{{\bf P}_{t}}\tilde{{\bf A}}_{t},$ (8) $\displaystyle{\bf
P}_{t}$ $\displaystyle=\tilde{{\bf A}}_{t}^{\top}{\bf P}_{t}\tilde{{\bf
A}}_{t}-$ $\displaystyle\quad\tilde{{\bf A}}_{t}^{\top}{\bf P}_{t}\tilde{{\bf
B}}_{t}({\bf R}+\tilde{{\bf B}}_{t}^{\top}{\bf P}_{t}\tilde{{\bf
B}}_{t})^{-1}\tilde{{\bf B}}_{t}^{\top}{\bf P}_{t}\tilde{{\bf A}}_{t}+{\bf
Q}.$

###  II-C Trajectory Optimization and Simultaneous Tracking

Given : ${\bf F}_{\bm{\theta}}$: Neural network dynamics model

${\bf Q},{\bf R}$: Feedback control parameters

while  _task not completed_ do

${\bf x}_{0}^{*}\leftarrow\text{StateEstimation}();$

${\bf u}_{0}^{*}\leftarrow\text{SMPPI}\left({\bf F}_{\bm{\theta}},\,{\bf
x}_{0}^{*}\right);$

${\bf x}_{1}^{*}\leftarrow{\bf F}_{\bm{\theta}}\left({\bf x}_{0}^{*},\,{\bf
u}_{0}^{*}\right);$

${\bf A}_{t},{\bf B}_{t}\leftarrow\text{AutoGrad}\left({\bf
x}_{1}^{*},\,\left[{\bf x}_{0}^{*},\,{\bf u}_{0}^{*}\right]\right);$

$\tilde{{\bf A}}_{t},\,\tilde{{\bf
B}}_{t}\leftarrow\text{ConvertDynamics}\left({\bf A}_{t},\,{\bf
B}_{t}\right);$ $\triangleright$ Eq. (6)

${\bf K}_{t}\leftarrow\text{SolveLQR}\left(\tilde{{\bf A}}_{t},\,\tilde{{\bf
B}}_{t},\,{\bf Q},\,{\bf R}\right);$ $\triangleright$ Eq. (8)

while  _processing time of SMPPI_ do

$\tilde{{\bf x}}_{t}\leftarrow\text{StateEstimation}();$

$\tilde{{\bf x}}_{t}^{*}\leftarrow\text{Interpolation}\left({\bf
x}_{0}^{*},\,{\bf x}_{1}^{*}\right);$ $\triangleright$ Eq. (11)

$\tilde{{\bf u}}_{t}\leftarrow{\bf u}_{0}^{*}-{\bf K}_{t}\left(\tilde{{\bf
x}}_{t}-\tilde{{\bf x}}_{t}^{*}\right);$

$\text{SendToController}(\tilde{{\bf u}}_{t});$

Algorithm 1 TOAST

In the previous section, we demonstrated that the system dynamics for
trajectory optimization can also be utilized by a trajectory following
feedback controller. We now describe how the two controllers in our proposed
control framework, called Trajectory Optimization and Simultaneous Tracking
(TOAST), operate harmoniously. In addition, we will address practical issues
that may arise during implementation.

Expanding (7) we obtain:

$\tilde{{\bf u}}_{t}-\tilde{{\bf u}}_{t}^{*}=-{{\bf K}_{t}}\,\left(\tilde{{\bf
x}}_{t}-\tilde{{\bf x}}_{t}^{*}\right).$ (9)

Note that $\tilde{{\bf u}}_{t}^{*}$ is the optimal feedforward command from
MPC. Here, $\tilde{{\bf u}}_{t}^{*}$ can be replaced with ${\bf u}_{0}^{*}$,
as it will maintain the same command during the processing time
$\Delta{t}_{1}$ of MPC. Then, we have:

$\tilde{{\bf u}}_{t}={\bf u}_{0}^{*}-{{\bf K}_{t}}\,\left(\tilde{{\bf
x}}_{t}-\tilde{{\bf x}}_{t}^{*}\right).$ (10)

Since $\tilde{{\bf x}}_{t}$ can be obtained by measuring directly from the
sensor or only through state estimation algorithms, such as an extended Kalman
filter, it can be updated at a much faster refresh rate than the feedforward
commands.

Let $t_{0}$ be the instant at which the feedforward command ${\bf u}_{0}^{*}$
is received from MPC. And let $t$ be the instant at which the feedback
controller receives the state $\tilde{{\bf x}}_{t}$. Then, $\tilde{{\bf
x}}_{t}^{*}$ in (10) is synchronously updated via linear interpolation:

$\tilde{{\bf x}}_{t}^{*}=\left(1-\frac{t-t_{0}}{\Delta t_{1}}\right){\bf
x}_{0}^{*}+\left(\frac{t-t_{0}}{\Delta t_{1}}\right){\bf x}_{1}^{*}.$ (11)

From the aforementioned derivations, we can derive a generic control scheme
that includes an optimal feedforward controller and a locally linear optimal
feedback policy that adapts to the former. If there are no external
disturbances and sensor noises, $\left(\tilde{{\bf x}}_{t}-\tilde{{\bf
x}}_{t}^{*}\right)$ will remain nearly zero, and the feedforward command ${\bf
u}_{0}^{*}$ will take precedence during $\Delta t_{1}$. Conversely, if the
state of the system deviates from the expected trajectory [as given in (11)],
the feedback command will start to operate to complement the feedforward
command and keep the system on the desired trajectory.

Since the feedback controller utilizes the same dynamics ${\bf F}$ as the
feedforward controller, our control scheme can be applied very effectively to
existing control tasks where the system dynamics models are highly complex and
non-linear. In recent studies of MPC [1, 5] and model-based reinforcement
learning [6], for instance, the dynamics models are approximated by neural
network. Our feedback controller can use the learned dynamics (${\bf
F}_{\bm{\theta}}$) directly to solve optimal tracking problems without
additional resources.

Note that the reference state at the next time step ${\bf x}_{1}^{*}$ in (11)
can not be acquired instantly during feedforward trajectory optimization,
since sampling-based MPC indirectly optimizes the action trajectory $U^{*}$
rather than the state trajectory $X^{*}$. Meanwhile, we implement the
linearization to obtain the Jacobians of neural network dynamics in the
Pytorch autograd automatic differentiation package [16]. For automatically
computing the gradients of ${\bf F}_{\bm{\theta}}$ with respect to ${\bf
x}_{t}$ and ${\bf u}_{t}$, one step forward propagation with the model (1)
must be done first in order to obtain ${\bf x}_{t+1}$. We can take advantage
of this result to interpolate the current reference state to circumvent
additional computation. The overall algorithm of our TOAST framework is shown
in Algorithm 1.

###  II-D Extension to an Aggressive Autonomous Driving Task

(a)  (b)  (c)

Figure 2: We evaluate our algorithm using classical control benchmarks and
aggressive autonomous driving. Crosswinds are modeled in (a) Pendulum, and (b)
Cartpole as external disturbances. (c) A high-fidelity vehicle simulator is
used for the autonomous driving experiments. The vehicle trajectory is also
visualized using our control framework for driving 10 laps around the track in
a counter-clockwise direction.

Our research goal is to control an UGV to drive aggressively under challenging
conditions, such as sharp corners and unknown surfaces with high and low
friction. The vehicle will slide on low-friction corners, causing tire lateral
forces to enter the highly nonlinear zone. Despite longitudinal and lateral
accelerations being robust features for estimating future states of sliding
vehicles, such measurements from real-world sensors are highly noisy. One
effective solution for this problem is to provide a history of the vehicle’s
velocity states and control inputs to the neural network [17, 18]. Following
our earlier study [13] where the results were also promising, we determine the
history length as $4$. Because the history length is relatively short, we
employ a fully-connected neural network for computational efficiency. Note
that other types of model architectures, such as Recurrent Neural Networks
(RNN), can also be utilized if the Jacobians of the model can be obtained.
However, ablation studies on model architecture selection are beyond the scope
of this paper.

Then, the approximated model can be expressed in the following form:

${\bf x}_{t+1}={\bf F}_{\bm{\theta}}\left({\bf x}_{t},\,{\bf u}_{t},\,{\bf
x}_{t-1},\,{\bf u}_{t-1},\,\dots\bf x_{t-H+1},\,{\bf u}_{t-H+1}\right),$ (12)

where $H$ is denoted as the length of the history. Since (12) does not follow
a general form of dynamics, direct linearization is not applicable.
Substituting this dynamics model into the Taylor linearization (2) yields:

$\displaystyle{\bf x}_{t+1}-{\bf x}_{t+1}^{*}\approx{\bf A}_{t}\left({\bf
x}_{t}^{*}+\delta{{\bf x}_{t}}-{\bf x}_{t}^{*}\right)+{\bf B}_{t}\left({\bf
u}_{t}^{*}+\delta{{\bf u}_{t}}-{\bf u}_{t}^{*}\right)$ (13)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\dots$
$\displaystyle+\left.\frac{\partial{\bf F}_{\bm{\theta}}}{\partial{\bf
x}_{t-H+1}}\right|_{{\bf x}_{t-H+1}^{*}}\cancelto{0}{\left({\bf
x}_{t-H+1}^{*}+\delta{{\bf x}_{t-H+1}}-{\bf x}_{t-H+1}^{*}\right)}$
$\displaystyle+\left.\frac{\partial{\bf F}_{\bm{\theta}}}{\partial{\bf
u}_{t-H+1}}\right|_{{\bf u}_{t-H+1}^{*}}\cancelto{0}{\left({\bf
u}_{t-H+1}^{*}+\delta{{\bf u}_{t-H+1}}-{\bf u}_{t-H+1}^{*}\right)},$

where we can replace $\left({\bf x}\right)^{*}+\delta\left({\bf x}\right)$
with $\left({\bf x}\right)$, and replace $\left({\bf
u}\right)^{*}+\delta\left({\bf u}\right)$ as $\left({\bf u}\right)$. The state
and control deviations $\delta\left({\bf x}\right)$ and $\delta\left({\bf
u}\right)$ before the current time step $t$ were presented in the past and
they should be zero at time $t$. As a result of causality, linearization of
the dynamics with historical encoding can be addressed as a current-state-
dependent system.

In a normal vehicle state estimation setting, the state ${\bf x}$ in
approximated dynamics ${\bf F}_{\bm{\theta}}$ consists of the longitudinal
velocity $v_{x}$, the lateral velocity $v_{y}$, and the yaw rate $r$. However,
the original purpose of the tracking controller was not to follow the command
and state at velocity level, but to follow them at position level. We,
therefore, augment the state space with dynamic state variables ${\bf
x}_{d}=\left(v_{x},v_{y},r\right)^{\top}$ and kinematic state variables ${\bf
x}_{k}$ such that:

$\mathbf{x}=\left(\begin{array}[]{c}\mathbf{x}_{d}\\\
\mathbf{x}_{k}\end{array}\right),$ (14)

where ${\bf x}_{k}$ consists of x-position $p_{x}$, y-position $p_{y}$, and
yaw angle $\theta$ on the global coordinate frame. The next kinematic state
can be updated by the current dynamic state:

$\mathbf{x}_{k}(t+1)=\mathbf{x}_{k}(t)+\left(\begin{array}[]{c}\cos(\theta)v_{x}-\sin(\theta)v_{y}\\\
\sin(\theta)v_{x}+\cos(\theta)v_{y}\\\ r\end{array}\right)\Delta t_{1}.$ (15)

The augmented full state is then updated via the neural network ${\bf
F}_{\bm{\theta}}$ and the explicit kinematic function (15). Linearization can
still be done using the automatic differentiation method. The overall control
architecture, as it was applied to an autonomous driving task, is depicted in
Fig. 1.

##  III EXPERIMENTS

Figure 3: Experimental results in four control tasks. The derivative actions
are quantified using the L2 norm.

For performance comparison, we implement four feedback controllers based on
the same feedforward controller, SMPPI. These are: 1) no feedback controller
(abbreviated as “No Feedback”), 2) a feedback controller with a low hand-tuned
gain matrix (abbreviated as “Low Gain”), 3) a feedback controller with a high
hand-tuned gain matrix (abbreviated as “High Gain”), and 4) our TOAST
controller. Note that the TOAST’s minimum and maximum gains observed
throughout the experiments are set to “Low Gain” and “High Gain”,
respectively. We perform a grid search to find the best control parameters
showing the most successful performance for each task. The discretized time
step for feedforward $\Delta{t}_{1}$ is set to $0.1$ s, while the feedback
$\Delta{t}_{2}$ is set to $0.01$ s. In the experiments, we use a neural
network to approximate the system dynamics.

###  III-A Pendulum

We first evaluate the performances of the compared controllers using classical
control benchmarks: Pendulum and Cartpole. To emulate external disturbances,
one can consider adding perturbations in the transition dynamics [19] or
adding a noise in action [20]. We use a more intuitive method by modeling a
continuously varying crosswind. The crosswind applied to the pole’s surface is
converted to imposed torque. This environment is visualized in Fig. 2a.

We designed the neural network dynamics with two fully-connected hidden
layers, each of which has $32$ neurons. The model predicts the residual
difference between the current state and the next state:

$\begin{bmatrix}\theta_{t+1}\\\
\dot{\theta}_{t+1}\end{bmatrix}=\begin{bmatrix}\theta_{t}\\\
\dot{\theta}_{t}\end{bmatrix}+{\bf
F}_{\bm{\theta}}\left(\begin{bmatrix}\sin(\theta_{t})\\\ \cos(\theta_{t})\\\
\dot{\theta}_{t}\\\ {\bf u}_{t}\end{bmatrix}\right).$ (16)

We designed two distinct tasks: 1) swing-up the pole just as in other control
studies, and 2) swing-down the pole to maintain a static state. The swing-down
task is designed to assess the chattering level of the controller when the
system is in a stable condition. The running cost function of the swing-up
task is formulated as:

$c({\bf
x}\equiv[\theta,\dot{\theta}]^{\top})=\theta^{2}+0.1\,{\dot{\theta}^{2}}.$
(17)

Likewise, the cost function of the swing-down task is formulated to keep the
pole downward against external disturbances:

$c({\bf
x}\equiv[\theta,\dot{\theta}]^{\top})=(\theta+\pi)^{2}+0.1\,{\dot{\theta}^{2}}.$
(18)

While only MPC was activated, a dataset was collected online with $1000$
random bootstrap data and the model was trained with the dataset every $50$
time-steps until the pole was balanced successfully in an upright position.
After training, the feedback controller was activated for the experiments. For
each task, the dynamics were trained with $10$ different fixed random seeds,
and the controllers were tested at $50$ random locations for each of the pre-
trained dynamics. In the first half of the $500$ tests, random winds were
applied, whereas sinusoidal winds were applied in the second half. Each
controller was tested under exactly the same conditions except for the
feedback controller. We analyzed the state costs for the swing-up task, and
the derivative actions for the swing-down task to quantify the degree of
chattering (shown in Fig. 3). The tracking errors of “No Feedback” and TOAST
during a Pendulum swing-up test are depicted in Fig. 4.

Figure 4: Visualization of tracking errors during a Pendulum swing-up task.
The feedforward controller optimizes a new trajectory every $0.1$ s. At that
time, the tracking error is updated to zero (red squares). TOAST is
effectively regulating the trajectory deviations caused by external
disturbances.

For SMPPI, $1000$ trajectories with $15$ time-steps were sampled for
trajectory optimization. For TOAST, the control parameters were set as
follows: ${\bf Q}=\text{Diag}(1,\,0.1)$, ${\bf R}=\text{Diag}(0.001)$.

###  III-B Cartpole

The majority of Cartpole’s experimental settings follow the Pendulum. We
modify the common Cartpole environment to provide a continuous action space
with a range of $(-10,10)$ N, in which the force is applied to the cart
horizontally (shown in Fig. 2b). We use neural network dynamics with the same
architecture as Pendulum:

$\begin{bmatrix}\dot{x}_{t+1}\\\
\dot{\theta}_{t+1}\end{bmatrix}=\begin{bmatrix}\dot{x}_{t}\\\
\dot{\theta}_{t}\end{bmatrix}+{\bf
F}_{\bm{\theta}}\left(\begin{bmatrix}\sin(\theta_{t})\\\ \cos(\theta_{t})\\\
\dot{\theta}_{t}\\\ {\bf u}_{t}\end{bmatrix}\right),$ (19)

where the velocity of the cart $\dot{x}$ can be derived using only the model
inputs. Then, the kinematic variables can be obtained using Euler integration
as described in Section II-D:

$\begin{bmatrix}x_{t+1}\\\ \theta_{t+1}\end{bmatrix}=\begin{bmatrix}x_{t}\\\
\theta_{t}\end{bmatrix}+\begin{bmatrix}\dot{x}_{t}\\\
\dot{\theta}_{t}\end{bmatrix}\Delta t.$ (20)

The running cost function of the swing-up task is formulated as:

$c({\bf
x}\equiv[x,\dot{x},\theta,\dot{\theta}]^{\top})=50\,x^{2}+10\,\dot{x}^{2}+10\,\theta^{2}+{\dot{\theta}^{2}}.$
(21)

The cost function of the swing-down task follows the same method as for
Pendulum.

For TOAST, the control parameters were set as follows: ${\bf
Q}=\text{Diag}(1,\,1,\,10,\,10)$, ${\bf R}=\text{Diag}(0.001)$. The signs of
the hand-tuned gains for “Low Gain” and “High Gain” were switched depending on
whether the pole was placed in the upper half-plane or the lower half-plane.
This is because the action in Cartpole, unlike in Pendulum, is applied to the
cart horizontally.

The experimental results of four tasks are shown in Fig. 3. The results show
that both “High Gain” and TOAST could keep the pole upward against the
continuously varying external disturbances. MPC without a feedback controller
failed to control the systems in most cases. “Low Gain” performed better than
“No Feedback”, but still failed to achieve the desired performance level. On
the other hand, “Low Gain” and “No Feedback” showed low action chattering
levels in the swing-down tasks, as expected. TOAST also demonstrated a much
lower chattering level than “High Gain”, suggesting that our proposed feedback
controller does not interfere with MPC when the tracking errors are assumed to
be trivial.

###  III-C Aggressive Autonomous Driving

We also evaluated the performances of different controllers in a high-fidelity
vehicle simulator, the IPG CarMaker. It has been widely used to validate
precisely nonlinear vehicle dynamics [21, 22, 23]. We built a race track with
a length of $1016$ m, two moderate curves, and four sharp curves (see Fig.
2c). A Volvo XC90 is used as the control vehicle. Unlike the electric vehicles
used in recent learning-based aggressive autonomous driving studies [5, 24],
our vehicle used an automatic transmission for gear-shifting. Therefore, we
employ desired speed ($v_{des}$), instead of throttle, as a high-level
controller’s input and let a low-level plant controller manage the throttle
and brake. Thus, the control input ${\bf u}$ consists of steering angle
$\delta$ and desired speed $v_{des}$.

####  III-C1 Training Neural Network Vehicle Dynamics

Our fully-connected neural network approximates the nonlinear vehicle
dynamics. Note that only the dynamic state variables are considered as input
to the neural network. We denote $\left|{\bf x}_{d}\right|$ and $\left|{\bf
u}\right|$ as the size of the controller’s state and action. The history of
states and actions, which has a size of $(\left|{\bf x}_{d}\right|+\left|{\bf
u}\right|)\cdot H$, is forward propagated to the neural network. Then, the
output of the network predicts the residual difference between the current
state and the next state, which has the size of $\left|{\bf x}_{d}\right|$.
The network was implemented as an MLP with four hidden layers. We applied a
hyperbolic tangent (tanh) activation function, and the network was trained
with the mean squared error loss function and Adam optimizer.

We collected a human-controlled driving dataset, with a data rate of $10$ Hz,
to train our network by expanding the methods used in our previous work [23].
We found that the dataset should comprise three types of distinct maneuvers in
order for the neural network to accurately represent the vehicle dynamics for
various friction conditions:

  1. i)

Zig-zag driving at low speeds ($20-25$ km/h) on the race track, in both
clockwise and counter-clockwise directions. Each driving in a different
direction was treated as a separate maneuver.

  2. ii)

High speed driving on the race track in both directions, trying to maintain
$40$ km/h as much as possible.

  3. iii)

Sliding maneuvers at the friction limits on flat ground, in combinations of
acceleration and deceleration with various steering angles.

The above five maneuvers were done with seven different friction coefficients:
$[0.4,0.5,0.6,0.7,0.8,0.9,1.0]$. These $35$ maneuvers were logged for two
minutes each, obtaining a total of $70$ minutes of driving data. We divided
the data into $70\,\%$ for training and $30\,\%$ for testing after randomizing
the data to break temporal correlations. The trained model was saved when it
showed the best result for test error. Then, it was evaluated with the
validation dataset. We collected the validation data on the race track in both
directions. The default friction coefficient was set to $0.8$ and the friction
coefficients of the six curves were set to $[0.95,0.85,0.75,0.65,0.55,0.45]$,
respectively, which were previously not included in the training data.

The test and validation errors of our neural network are shown in Table I.
Root Mean Square Error (RMSE) is denoted as $\mathbf{E}_{RMS}$ and the max
error is denoted as $\mathbf{E}_{max}$. The results show that our neural
network can make accurate one-step predictions under a variety of driving
conditions.

TABLE I: Test and validation errors of the neural network vehicle dynamics. |  $v_{x}$ [m/s] |  $v_{y}$ [m/s] |  $r$ [rad/s]
---|---|---|---
| $\mathbf{E}_{RMS}$ | $\mathbf{E}_{max}$ | $\mathbf{E}_{RMS}$ | $\mathbf{E}_{max}$ | $\mathbf{E}_{RMS}$ | $\mathbf{E}_{max}$
Test | 0.0273 | 0.3940 | 0.0178 | 0.3180 | 0.0147 | 0.3455
Val. | 0.0220 | 0.3753 | 0.0114 | 0.1127 | 0.0095 | 0.1413

####  III-C2 Experimental Setup

We designed the state-dependent cost function $c({\bf x})$ in MPC to have the
following form:

$\displaystyle c({\bf x})$ $\displaystyle=\alpha_{1}{\text{Track}({\bf
x})}+\alpha_{2}{\text{Speed}({\bf x})}+\alpha_{3}{\text{Slip}({\bf x})},$ (22)
$\displaystyle\text{Track}({\bf x})$
$\displaystyle=(0.9)^{t}\,{10000\,\textbf{M}(p_{x},p_{y})},$
$\displaystyle\text{Speed}({\bf x})$ $\displaystyle=(v_{x}-v_{ref})^{2},$
$\displaystyle\text{Slip}({\bf x})$
$\displaystyle=\sigma^{2}+10000I\,(\\{\left|\sigma\right|>0.2\\}),$

where $I$ is an indicator function. $\textbf{M}(p_{x},p_{y})$ in track cost is
the two-dimensional cost map value at the global position $(p_{x},p_{y})$.
Thanks to the sampling-based derivation of SMPPI, we can provide an impulse-
like penalty in the cost function. The speed cost is a simple quadratic cost
to achieve the reference vehicle speed $v_{ref}$. The slip cost imposes both
soft and hard costs to discourage slip angle in the trajectory
$(\sigma=-\text{arctan}(\frac{v_{y}}{\lVert v_{x}\rVert}))$. The trajectory
expected to have a slip angle greater than $0.2$ rad (approximately
$11.46\,^{\circ}$) will be penalized since it has the potential to make the
vehicle unstable.

For SMPPI, the number of parallel samples was $10000$ and the number of time
steps was $35$. For TOAST, the control parameters were set as follows:

$\displaystyle{\bf Q}$
$\displaystyle=\text{Diag}\begin{pmatrix}5,&5,&0.01,&0.05,&0.05,&0.025\end{pmatrix},$
(23) $\displaystyle{\bf R}$
$\displaystyle=\text{Diag}\begin{pmatrix}0.4,&0.1\end{pmatrix}.$

Figure 5: Visualization of the time-varying feedback gains with different
controllers while they are driving the same section. (a) “Low Gain”, (b) “High
Gain”, and (c) TOAST. For clarity, only the gains corresponding to the
steering angle are visualized. We applied rotation on the hand-tuned gains of
$p_{x}$ and $p_{y}$ from global to vehicle coordinate frame. TOAST adaptively
changes the gains according to the time-varying dynamic characteristics of the
system. Figure 6: Vehicle trajectories of the compared controllers. The
friction coefficient of Corner #2 (in gray) is 0.60 and Corner #3 (in brown)
is 0.65.

####  III-C3 Experimental Results

We evaluated the control performance of the four controllers on the race
track. The goal of the controllers was to complete whole laps in a clockwise
direction.

The race track was adjusted to be more challenging to drive than it was when
the training dataset was collected. The friction coefficients $\mu$ were
modified to have values of $[0.55,0.60,0.65,0.70,0.75,0.80]$ at the corners,
respectively, and to have a default of $1.0$. During ten laps around the
course, we measured average lap times on the six corners at a reference speed
of $40$ km/h. When the vehicle left the track, it was placed at the starting
point and started a new lap. All of the methods used the same pre-trained
model and no further training was given throughout the experiments. The
results are shown in Table II.

TABLE II: Average lap times on the six corners of different control methods. The minimum speed (in [km/h]) and maximum slip angle (in [∘]) at each corner are also analyzed with our method. The success rates (SR) are also displayed. Lap time [s] | #1 | #2 | #3 | #4 | #5 | #6 | SR
---|---|---|---|---|---|---|---
No Feedback | 7.8 | 9.4 | 12.2 | 8.7 | 7.0 | 11.9 | 1/10
Low Gain | 7.9 | 8.9 | 12.2 | 8.9 | 6.9 | 12.2 | 3/10
High Gain | 8.3 | 8.7 | 12.2 | 10.0 | 7.6 | 13.6 | 10/10
TOAST | 7.7 | 7.8 | 11.8 | 9.5 | 8.3 | 9.4 | 10/10
Min. Speed | 33.2 | 24.4 | 22.6 | 33.5 | 26.7 | 26.7 |
Max. Slip | 3.4 | 7.2 | 6.8 | 2.8 | 7.4 | 4.2 |

“No Feedback” and “Low Gain” barely got through Corner #2 and Corner #3, which
are the most challenging sections of the track. They completed the full course
with only one and three out of ten laps, respectively. The learned dynamics
predicted that the vehicle would slide greatly if it did not slow down due to
the low friction surfaces. Therefore, SMPPI generated trajectories that
applied brakes first, then steered the vehicle and slowly accelerated it to
pass the corners. However, the vehicle failed to follow the planned trajectory
and understeering occurred due to model uncertainty. In contrast, “High Gain”
and “TOAST” completed entire laps in $100\%$ of the trials. However, the
results show that TOAST has a faster lap time than “High Gain” in most cases.
This is because if the feedback controller keeps a fixed high gain all the
time, it frequently interferes with the control sequence of the MPC and thus
loses the optimality of the solution. Our controller computes the optimal
feedback gain for each optimal control sequence based on the contextual
information encoded in the neural network dynamics. As a result, a low gain
can be maintained in situations where feedback compensation is not greatly
required. We show the time-varying feedback gains of the compared controllers
in Fig. 5. For TOAST, the overall average speed during ten laps was $33.4$
km/h and the maximum slip angle was $7.4\,^{\circ}$. We visualize the vehicle
trajectories on Corner #2 and Corner #3 in Fig. 6.

Figure 7: Derivative action for each control variable. They are quantified by
the L2 norm.

We also analyzed the degree of chattering on control values with “High Gain”
and TOAST. It is a well-known fact that rapid changes in the action commands
are a burden to the actuators. In the straight sections of the track with high
friction coefficients, the vehicle is stable and the requirement for a
feedback controller becomes negligible. Therefore, we analyzed the derivative
actions during $10$ laps excluding the corners. The results are shown in Fig.
7. For “High Gain”, the average derivative actions of the steering angle
($\delta$) and the desired speed ($v_{des}$) were $1.31$, and $0.81$,
respectively. On the other hand, those of TOAST were $0.96$ and $0.46$. The
results demonstrate that the adaptive manner of our proposed method can
alleviate chattering.

##  IV CONCLUSION

We presented a novel control scheme that combines online trajectory
optimization and optimal tracking control using the same learned dynamics. We
described how to convert a single dynamics model so that it could be used for
controllers on two different time scales. We also explained how to handle
dynamics models containing history information for our method. The performance
of the proposed method was evaluated using two control benchmarks,
demonstrating that our ancillary feedback controller is capable of regulating
tracking errors without interfering with the optimal feedforward control
sequence. We also extended our work to an autonomous aggressive driving task,
suggesting its scalability. In all experiments, our method outperformed the
compared methods by a large margin. Our controller also showed control
chattering levels that were $24-43\%$ lower than for the controller with high
hand-tuned gains in the driving task.

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2024

1Department of M.Tech Integrated (CSE), Noida Institute of Engineering and
Technology, Greater Noida, 201310, U.P, India
2Department of CS, Noida Institute of Engineering and Technology, Greater
Noida, 201310, U.P, India.

# Exoplanet Detection : A Detailed Analysis

Mahima Kaushik1,* 0000-0003-2650-6758 Aditee Mattoo1 and Dr. Ritesh Rastogi2

###### Abstract

The exoplanet detection is the most exciting and challenging field of
astronomy. The discovery of many exoplanets has revolutionized our
understanding of the formation and evolution of planetary systems and has
showed new ways to search for extra terrestrial life. In recent years, some
primary methods of exoplanet detection like transit, radial velocity,
gravitational microlensing, direct imaging and astrometry have played a
important role for the discovery of exoplanets. In this paper we explored
detection methodologies with all the implications and analytics of comparison
between them. Here we also discussed on different machine learning algorithms
for exoplanet detection and visualization. Finally, concluded with the
significant discoveries made by some missions and their implications on our
understanding for the properties, environmental conditions and importance of
exoplanets in the universe.

###### keywords:

Exoplanet Detection—Transit—Missions—Exoplanets—Gaia.

<EMAIL_ADDRESS>
10.48550/arXiv.2404.09143 #### 000 2024 1-10 10

##  1 Introduction

Since the discovery of exoplanets began, they have continually surprised us
with the hidden secrets of the cosmos. Here on earth, wealth is often
associated with the possession of precious rocks and metals. But what about
worlds made of these rocks and metals? For instance, 55 Cancri e is an
exoplanet made of diamonds and orbits a sun-like star. It was discovered by
McArthur McArthur $et~{}al$. (2004) using the radial velocity method by
observing the spectrum of one of the parent stars in a binary system located
40 light-years from the planet Earth. Exoplanets are poised to become a
significant part of human civilization in various aspects, such as energy
sources, colonization of other planets, transportation, space exploration,
employment, and many others. Since the first exoplanet was discovered,
astronomers have been working tirelessly to discover other extrasolar planets.
Numerous missions have been conducted by several space agencies to study
exoplanets, and the results have been truly fulfilling. Till date,
approximately 5500+ exoplanets have been discovered Akeson $et~{}al$. (2013),
with many of them undergoing detailed studies.

Figure 1: Exoplanet detection methods, shown on a logarithmic mass scale, show
current and upcoming measuring possibilities. Open arrows show previous
evaluations, whereas solid arrows represent discoveries. Data collected from
the NASA Exoplanet Archive (2018) Perryman (2018).

Exoplanets come in various types such as earth-size, neptune-size, super-
earth, jupiter-size, sub-earth, super-jupiters, brown dwarfs, and stellar
companions. There are also various techniques for detecting exoplanets Wei
(2018), including radial velocity, transit method, astrometry, direct imaging,
and gravitational microlensing. Each method offers distinct perspectives,
strengths, limitations, and other details, often represented in tabular and
graphical forms. By analyzing the light that passes through the atmosphere of
a planet, scientists can understand its composition. Some exoplanets have been
found to have atmospheres containing elements such as carbon dioxide, methane,
and even water vapor Perryman (2018). If we compare a piece of wood and a
piece of diamond to see which one is more expensive, the correct answer is
wood. This is because, so far, earth is the only place in the universe where
wood is produced, while diamonds are present on different exoplanets like 55
Cancri e. Gliese 667Cc is a super-earth exoplanet that lies 22 light-years
away from earth. It’s a potentially habitable super-earth with a mass of 3.8
times that of earth. Humans are still in the race of hunting for those
habitable exoplanets, on which we can grow more plants and expand human
colonization beyond Earth.

51 Pegasi b, the first exoplanet, orbiting around a star similar to sun, was
found using the radial velocity approach by Mayor & Queloz. Several space
agencies are running missions specifically for exoplanet detection using
different detection methods Shafi (2021). These missions include hubble,
spitzer, corot, kepler/K2, gaia, TESS, CHEOPs, webb, and many more, which
continuously monitor the universe, and astronomers keep detecting exoplanets
with their help. In the future, there will be missions for specific exoplanets
to understand their properties in detail and their environmental conditions.
However, as of now, it’s hard to travel to even nearby exoplanets within a
lifetime. Further more, introduced space missions briefly, including details
about data formats and other details. Astronomers use various machine learning
Kumari $et~{}al$. (2023) and data cleaning algorithms to work on the data that
comes from these missions. Such algorithms include the BLS algorithm, light
curve analyzer, CNN, and other algorithms. all exoplanets are formed by normal
matter but some scientists assumes that some exoplanets could me formed
through dark matter Bai $et~{}al$. (2023). Introduction about exoplanets with
there basic details are going to be dicsussed in the next section.

##  2 The exoplanets

Extrasolar planets, also referred to as exoplanets, are planets that circle
stars outside of our solar system. From the first exoplanet to be discovered
in the early 1990s to the current booming field with thousands of confirmed
planets, the growth of exoplanet investigation has been incredible wikipedia
(2022). Section 2 discusses the crucial roles that various detection methods
have had in this process. Technological developments like adaptive optics,
space-based observatories like kepler and TESS, and high-precision
spectrographs have made these approaches far more useful Woodrum $et~{}al$.
(2023). The discovery of smaller and farther-off planets has been made
feasible by these developments. Exoplanet research is essential because it can
increase our knowledge of planetary systems and the conditions necessary for
habitability and the possibility that life exists elsewhere in the cosmos.

###  2.1 Stars That Host Planets

On average, each star Scalo $et~{}al$. (2007) is believed to host at least one
planet. It’s estimated that around 20% of sun like stars have an ”earth-sized”
planet in their habitable zones. The majority of known exoplanets orbit stars
similar to our Sun, specifically those in the main-sequence spectral
categories F, G, or K Lee $et~{}al$. (2012) Perryman (2018) Meunier $et~{}al$.
(2023). Planets detectable by the radial-velocity method are less common
around lower mass stars, such as the red dwarfs in the M spectral group
Ananyeva $et~{}al$. (2023). Despite this, the transit method, which excels at
detecting smaller planets, has enabled the kepler telescope to identify dozens
of planets orbiting near red dwarfs.

Figure 2: Diagram showing the habitable zones surrounding certain stars. The
green area indicates where water on a planet’s surface can stay liquid at the
ideal distance from the star. The blue area is too cool, and the red area is
too heated.

###  2.2 Detection methods

Various detection methods Rouan & Lagrange (2023) exist for exoplanets, and
astronomers continuously work toward developing new ways of detection. Some
effective methods include radial velocity, which led to the discovery of the
1st exoplanet in the late ’90s; the transit method, which analyzes light
curves and periodograms Bozza $et~{}al$. (2016); direct imaging, capturing
exoplanets through images; gravitational microlensing; and astrometry Perryman
(2018). Table 1 shows the total number of exoplanets discovered by various
methods of detection till beginning of 2024 NASA (2024). data taken from NASA
exoplanet archive Akeson $et~{}al$. (2013).

Table 1: Exoplanets discovered by various methods till date

Method of discovery | Number of planets
---|---
Astrometry | 3
Disk Kinematics | 1
Radial Velocity | 1071
Eclipse timing variations | 17
Orbital brightness modulations | 9
Transit | 4146
Transit timing variations | 28
Pulsar timing variations | 7
Microlensing | 204
Imaging | 69
Pulsation timing variations | 2

###  2.3 Types of exoplanets

As of the conclusion of 2023, the existence of over 5500 planets had been
confirmed. In an attempt to better understand these celestial bodies,
scientists have undertaken numerous initiatives to categorize them based on
various attributes, including their size, mass, and composition Perryman
(2018).Table 2 shows the classification of exoplanets based on mass, size and
chemical compositions of known gases such as $ZnS$, $H_{2}O$, $CO_{2}$ &
$CH_{4}$ Forestano $et~{}al$. (2023).

Figure 3: planet masses are categorized based on solar system objects, as
suggested by Stevens & Gaudi (2013). The solar system’s planet masses are
displayed on the right, with the size of the circles corresponding to the
planets’ radii Perryman (2018) Mathur $et~{}al$. (2021).

Table 2: Types of exoplanets based on various factors

Factor | Planet type | Parameter
---|---|---
Size | Earth size | ($1.25-2R_{\oplus}$)
| Super Earth size | ($2-6R_{\oplus}$)
| Jupiter size | ($6-15R_{\oplus}$)
Mass | Earth | ($0.1M_{\oplus}-2M_{\oplus}$)
| Sub Earth | ($10^{-8}M_{\oplus}-0.1M_{\oplus}$)
| Super Earth | ($2M_{\oplus}-10M_{\oplus}$)
| Neptunes | ($10M_{\oplus}-100M_{\oplus}$)
| Jupiters | ($100M_{\oplus}-10^{3}M_{\oplus}$)
| Super Jupiters | ($10^{3}M_{\oplus}-13M_{J}$)
| Brown Dwarfs | ($13M_{J}-0.07M_{\odot}$)
| Stellar Companions | ($0.07M_{\odot}-1M_{\odot}$)
Composition | Hot super Earths | ($187-1.12$)
(stellar flux) | Warm Super Earths | ($1.12-0.30$)
| Cold super Earth | ($0.30-0.0030$)
| Hot Rocky | ($182-1.0$)
| Warm Rocky | ($1.0-0.28$)
| Cold Rocky | ($0.28-0.0035$)
| Hot Sub Neptunes | ($188-1.15$)
| Warm Sub Neptunes | ($1.15-0.32$)
| Cold Sub Neptunes | ($0.32-0.0.0030$)
| Hot Sub Jovians | ($220-1.65$)
| Warm Sub Jovians | ($1.65-0.45$)
| Cold Sub Jovians | ($0.45-0.0030$)
| Hot Jovians | ($220-1.65$)
| Warm Jovians | ($1.65-0.40$)
| Cold Jovians | ($0.40-0.0025$)

###  2.4 Atmospheric conditions

Finding transiting exoplanets is a significant breakthrough that provides
previously unattainable knowledge about the composition and physical
properties of these far-off worlds Yakovlev $et~{}al$. (2022). Transit and
eclipse spectroscopy allows the investigation of atmospheric dynamics Walker
$et~{}al$. (2018) and heat transport processes, especially for hot Jupiters,
while the resulting densities, derived from detailed measurements of masses
and radii, offer important hints regarding internal structures and
compositions. These discoveries have been made possible in large part by
drawing on knowledge amassed over the previous 50 years from research on the
planets and satellites of the solar system Jakka (2023).

Future developments in photometric and spectroscopic observations, such as
those brought about by the 2020 launch of the James Webb Space Telescope
(JWST), are very promising. Advanced diagnostics may reveal the precise
atmospheric compositions of exoplanets that are located in the ”habitable zone
Jagtap $et~{}al$. (2021), ” which is where liquid water may be present. These
compositions may include indications of either primitive or advanced life
forms. These developments advance not only the hunt for extraterrestrial life
but also help answer the important question of how common or uncommon life is
in the universe Seager & Deming (2010). The dynamic interplay between
theoretical models and observational methods is set to propel future research
and transform our knowledge of exoplanetary systems.

##  3 Exoplanet detection methods

As discussed previously there are various detection methods which played an
important role to present this unique field of study. all methods works on
various different properties such as light, noise Smith $et~{}al$. (2008),
EMwaves etc. some of them are explained here.

###  3.1 Radial Velocity Method

The radial velocity technique for identifying exoplanets involves tracking the
reflexive movement of a star, which is triggered by the gravitational force of
a planet in orbit. This method assesses three visible characteristics of the
star that are altered by the planet’s existence:

Radial Velocity: When a planet orbits a star, both celestial bodies revolve
around a shared center of mass, termed the barycenter. The star’s motion
around this point results in a periodic fluctuation in its radial velocity
Hatzes (2016), which signifies its speed moving towards or away from an
observer located on Earth. It is discovered by spectroscopic studies that this
radial velocity shift exists. The spectral lines of a star change towards the
blue end of the spectrum as it travels towards the observer, and to the red
end when it goes away (blueshift and redshift, respectively) Bozza $et~{}al$.
(2016) Chen (2023).

Angular (Astrometric) Position in the Sky: The star moves in a modest circular
or elliptical path around the shared center of mass due to the gravitational
attraction of an orbiting planet Pham $et~{}al$. (2024) Bozza $et~{}al$.
(2016). The star’s angular location in the sky changes periodically,
indicating its motion. Astrometric measures entail monitoring the star’s
apparent location with accuracy over time. Measurements of radial velocity
have been used more frequently than this approach, which is more difficult
Perryman (2018) Chen (2023).

Time of Arrival of Periodic Reference Signal: Variations in the arrival times
of periodic signals, such pulsar pulses or eclipse timings in binary star
systems, can be caused by the gravitational interaction between a star and its
circling planet Bozza $et~{}al$. (2016). Timing variations in these signals of
reference may indicate the existence of an unseen planet and offer additional
evidence of its impact Fischer $et~{}al$. (2015).

Radial velocity Dumusque $et~{}al$. (2017) measurements revealed the first
credible discovery of an exoplanet in 1995 Perryman (2018). Since then, a
number of exoplanets have been found using this method because to
technological and observational breakthroughs. Via radial velocity metrics,
1075 exoplanets had been found as of 02/05/2024; some of these planets have
been observed to transit their host stars, allowing the star’s light to
periodically fade as the planet passes in front of it.

###  3.2 Transit method

Astronomy achieved a major milestone in 1999 with the ground-breaking
observation of the first transit of an extrasolar planet Perryman (2018) Chen
(2023). The ’hot Jupiter’ system HD 209458 was observed photometrically
Perryman (2018), which allowed for this observation. A gas giant planet known
as a ”hot Jupiter” has a close orbit around its host star, which increases the
planet’s temperature somewhat. other star systems found by radial velocity
surveys came into focus after this finding Fischer $et~{}al$. (2015). After a
planet passes in front of its host star, causing a periodic dimming of the
star’s brightness, researchers kept an eye on these systems in the hopes of
finding other transits Yang (2023). Identifying exoplanets became increasingly
dependent on this methodology, particularly for those that were not initially
found via the radial velocity method Bozza $et~{}al$. (2016). Dedicated
surveys were rapidly launched to perform ”blind hunts,” initially from space
then from ground-based observatories Perryman (2018). The goal of these
surveys was to find new planets only by using the periodic transit
fingerprints observed in star light curves. With this method, scientists were
able to find a wide variety of exoplanets without any prior awareness of their
presence Chen (2023).

Figure 4: Exoplanets detection methods. fig 4.1 represents Number of
exoplanets detected by Radial velocity method in previous years, fig 4.2
represents Number of exoplanets detected by Transit method in previous years,
fig 4.3 represents Number of exoplanets detected by Gravitational Microlensing
method in previous years, fig 4.4 represents Number of exoplanets detected by
Direct Imaging method in previous years, fig 4.5 represents Number of
exoplanets detected by Timing method in previous years, fig 4.6 represents
Number of exoplanets detected by Astrometry method in previous years .

In the field of exoplanet research, transiting planets are especially
significant Afanasev (2018). The light curves acquired during transits yield
important details on planet sizes since the radii of the planets may be
estimated due to the star’s periodic dimming of light Perryman (2018) Fischer
$et~{}al$. (2015). Through techniques like radial velocity measurements,
astronomers can establish the transiting planet’s mass as well as its density,
providing preliminary information about the planet’s composition Bozza
$et~{}al$. (2016) Chen (2023).

Figure 5: Representing the periodogram of exoplanet KIC 6922244

Around the end of the year of 2023, exoplanet research had undergone a major
transformation. There were now close to 4153 transiting planets known, with
the Kepler space observatory detecting nearly 2778 of them as reported by
NASA. Kepler began operations in 2009 and has been a key player in the area of
transiting exoplanet research, providing an abundance of data that has made
significant effects Panahi $et~{}al$. (2022) Chen (2023). A wide range of
exoplanetary systems have been found as a result of the joint efforts of
ground-based surveys Perryman (2018), such HAT and WASP, which revealed bright
and easily detectable transiting systems, and Kepler’s numerous discoveries,
particularly those of close-orbiting multi-planet systems Afanasev (2018).
These systems contribute to our understanding of the diversity of planetary
systems outside of our solar system by displaying a wide range of physical
attributes and circumstances.

###  3.3 Direct imaging method

”Imaging” in the context of modern exoplanet research is the process of
recognizing a planet as an individual source of light Perryman (2018) Chen
(2023). This may happen as a result of the planet’s own thermal emission or
light reflected from the host star. It is important that this imaging idea is
not the same as reaching the planetary surface’s spatial resolution Bozza
$et~{}al$. (2016). Additionally, it is distinct from techniques that detect
fluctuations in the total brightness Manfredi $et~{}al$. (2020) of the planet
and star system as a result of things like secondary eclipses, reflected
light, or changes in the stellar spectrum Birkby (2018) throughout a transit.
In 2005, the first imaging detections were reported, namely the identification
of huge, youthful, self-luminous planets in broad orbits Bozza $et~{}al$.
(2016) Perryman (2018). The NASA exoplanet archive listed 44 of these
discoveries as at the end of 2017 Perryman (2018). Prominent instances
comprise planets discovered in well researched debris disk systems such as
$\beta$ Pic and fomalhaut, in along with multi-planet systems like HR 8799
(which includes four planets) and LkCa 15 (containing two planets) Perryman
(2018). There are certain deep learning models are used to inhance the
capabilities of detection such as deep PACO Flasseur $et~{}al$. (2024)
Researchers have been able to determine the orbital velocity of planets in
systems such as $\beta$ Pic, HR 8799, and fomalhaut thanks to multi-epoch
research. Over the past ten years, developments in imaging instrument
technology Daglayan $et~{}al$. (2023) Hebrard $et~{}al$. (2018) have led to an
emergence of cutting-edge devices specifically intended for exoplanet imaging
Beuzit $et~{}al$. (2006). VLT-SPHERE, Gemini-GPI Galicher $et~{}al$. (2014),
and Subaru-HiCIAO are a few examples Fischer $et~{}al$. (2015). It is
important to remember that these devices can only directly picture massive
planets that are young, warm, and self-luminous at this time Perryman (2018).
It is expected that with the next 30–40-meter telescopes, direct detection of
point source pictures resulting from reflected starlight would become possible
Bozza $et~{}al$. (2016) Perryman (2018). On the other hand, picturing planets
similar to earth is still a distant objective Chen (2023). It is crucial to
take into account imaging Koechlin $et~{}al$. (2005) at radio and X-ray
wavelengths while studying star-planet systems. These other wavelength ranges
may provide light on many facets of the interactions that take place between
stars and their planets Flasseur $et~{}al$. (2023). There is talk about the
potential to expand imaging capabilities to X-ray and radio wavelengths, which
implies a more thorough investigation of observational methods in exoplanet
studies Rosa & do Nascimento Dias (2023) Daglayan $et~{}al$. (2023).

###  3.4 Gravitational Microlensing

The general relativity-based observation method known as gravitational
microlensing Perryman (2018). Using gravitational microlensing, scientists
found 210 exoplanets by the end of 2023. This technique uses the gravitational
potential of a foreground object (the lens) to bend light beams from a distant
background object (the source) Bozza $et~{}al$. (2016) Perryman (2018).
Depending on how the source, lens, and observer are aligned, the outcome is a
distorted image of the source, sometimes in many images Chen (2023) Beaulieu
(2023). The effects of gravitational lensing can be observed, leading to the
classification into various domains Jones (2008). Robust lensing, observable
at the level of individual objects, is further classified into macro- and
microlensing. Whereas microlensing deals with discrete, unresolved multiple
visuals, macrolensing generates multiple resolved images, or ”arcs.” Weak
lensing is a different regime that is statistically noticed without a clear
resolution of individual objects. Specific circumstances within gravitational
lensing are described by the introduction of specialized words. Mesolensing is
the term used to characterize the astrometric displacement that accompanied a
high-proper motion foreground star, whereas nanolensing is the lensing caused
by planetary-mass objects Gaudi (2012). Relative motion between the source,
lens, and observer causes time-varying amplification of pictures .
Depending on the characteristics of the radiation source and the lens, this
amplification takes place across an assortment of intervals, from hours to
years. A more complex light curve may be seen from the background source if
the foreground lensing object is gravitationally complex, such a star with
planets or a cluster of galaxies Wright & Gaudi (2012). This complexity
results from the changing alignment geometry’s time-varying magnification.
Significant discoveries have been made using gravitational microlensing
Tamanini & Danielski (2019) in exoplanet exploration. In 2004, a 4MJ planet
was first clearly detected at approximately 4 astronomical units (au) Perryman
(2018). A 5M $\oplus$ planet was then found in 2006. An observation of a two-
planet system in 2008 made it possible to quantify orbital motion during the
lensing event. In 2015, the lens mass and microlens parallax were measured. In
early 2018, evidence for planets flying freely was proposed. These results
validate gravitational microlensing as an independent and potent probe of
exoplanets, especially over an important mass and orbital radius region Bozza
$et~{}al$. (2016).

###  3.5 Astrometry

Table 3: Comparative analysis of methods

Detection method |  Principle |  Measurements |  Advantages |  speciality |  Drawbacks |  Equipement
---|---|---|---|---|---|---
Radial velocity |  Gravitational Wobble |  $m_{p},T,a$ |  large amount of planets can be observe at once, precision |  large planets, small cool stars |  its for only planets closer to observer |  Spectrometer
Transit photometry |  Light Dip |  $r_{p},T,a$ |  High senstivity, Large sample |  precise evaluation of properties, short period planets |  False positives,intersecting and sort period planets |  Photometer, TESS, Kepler, CoRoT
Direct Imaging |  Visual Observation |  $m_{p},T,a$ |  characterization , atmospheres |  large size and a, hot, planet close to observer |  needed stability, glare |  cornograph and infrared telescope
Gravitational Microlensing |  Gravitational Lensing |  $m_{p}$ |  Distant planets, galactic scale |  background towards center of galaxy, large a |  doubtful alignment, single trial |  Warsaw telescope, OGLE
Astrometry |  Stellar Wobble |  RA, Dec,.. |  High precision, Long baselines |  precise |  extremely hard |  Gaia, keck

The exact measurement of the motions and positions of different celestial
objects, from bodies of the solar system and stars within our galaxy to,
theoretically, entire galaxies and clusters, is known as astrophysicist.
Determining stellar parallaxes and proper movements has historically been a
major goal of astrometry, as these are essential for comprehending the
characteristics of host stars Perryman (2018). In the present development,
astrometry plays a new role: it seeks to identify the crosswise element of the
displacement of a host star due to an orbiting planet’s gravity. Measures of
radial velocity, which are sensitive to the concomitant movement of the
photocenter along the line-of-sight, are closely associated with this dynamic
manifestation in the sky Saffe $et~{}al$. (2005).

It has been difficult to achieve high-accuracy astrometry; hipparcos and the
fine guidance sensors on the hubble space telescope, for example, achieved
accuracies of about 1 milliarcsecond (mas). Still, this degree of accuracy has
only just touched the tip of the iceberg in terms of how much star position
displacements caused by planets in orbit can be identified. By the end of
2023, astrometry had only found three exoplanets, two of which were 5MJ and
2.26MJ in mass, and one of which was a substellar partner with a mass 28 times
that of jupiter (28MJ). These discoveries were recorded in the NASA exoplanet
archive. Since the gaia Ranalli $et~{}al$. (2018) mission was launched in
2013, the field of astrophysicists has seen a dramatic change Panahi
$et~{}al$. (2022). Over a billion stars with magnitudes up to V = 20–21 are
being actively measured by gaia, with accuracies of about 20–25
microarcseconds (µas) for stars with a magnitude of 15 Perryman (2018). Our
knowledge of planetary systems is predicted to be completely transformed by
this astounding increase in precision Hodgkin $et~{}al$. (2021) Panahi
$et~{}al$. (2022). Several thousand planetary systems with correct planet
masses and absolute orbits free of the sin i ambiguity (the uncertainty
surrounding the inclination angle of the planetary orbit) should be found
thanks to gaia’s large dataset Panahi $et~{}al$. (2022) Perryman (2018).
Moreover, gaia will provide insightful data on planetary systems’ co-
planarity, illuminating the configuration and behaviour of several planets
within a system wikipedia (2022).

##  4 Comparative analysis of detection methods

Count of detected exoplanets by specific method is the proof of its
importance. Every method of extrasolar planet detection works on different
principles and scientific concepts Perryman (2018) Chen (2023). While transit
photometry and radial velocity methods are playing main role for the majority
detection. after discovery there are various parameters and properties to know
for detecting habitable planet, which is possible through other methods
Meunier $et~{}al$. (2012). hence every detection method have its own
advantages and disadvantages Fischer $et~{}al$. (2015). The transit photometry
method coveres wide range of stars but its false positive rate is high
Tembhare $et~{}al$. (2023). transit method works for star-planet systems,
which have orbits aligned in such a way that, they can be visible from earth.
those exoplanets should move in between host star and earth. Radial velocity
method is highly prolific and works for many planets but they must be close to
the spectator McArthur $et~{}al$. (2004). Timing method works for limited
materiality, but provide accurate data of exoplanets residing far from earth.
Direct imaging also works for few stars but straight forward detection of
exoplanets. Large orbital radii planets can be detected by gravitational
microlensing, but here measurements alignments can’t be repeated and
questionable Hebrard $et~{}al$. (2018). The majority of precise searches
employ multiple methodologies to collect multifaceted information on a certain
planet. In particular exoplanet HD 86266 c alternative name TOI-652 c detected
by primary transit method in 2020 and gave radius parameter details. Table 3
shows the comparative analysis of detection methods with important parameters.

##  5 Space missions for exoplanets

There are several missions are working towards exoplanets and there
discoveries. these missions are running by various space agencies like NASA,
ESA, JAXA, SETI etc. some missions ended up working and some are currently
doing good work. here is the details of some space missions for exoplanets.
The landscape of exoplanet exploration has been significantly shaped by a
series of pioneering space missions, each contributing unique insights into
the vast diversity of planetary systems beyond our solar neighborhood Galicher
$et~{}al$. (2014). Among these missions, TESS (Transiting Exoplanet Survey
Satellite) Kane $et~{}al$. (2020) and GAIA Ranalli $et~{}al$. (2018) stand as
ongoing endeavors,

Table 4: Space missions for exoplanet detection

Mission |  Status
---|---
TESS |  Ongoing
GAIA |  Ongoing
Kepler |  completed (2013)
K2 |  Completed (2018)
CHEOPS |  On going
JWST |  On going
ARIEL |  Planned
RST |  Planned
TOLIMAN |  Planned
PLATO |  Planned
CoRot |  Completed(2006)
SPitzer |  Completed (2020)

tirelessly scanning the celestial expanse for transiting exoplanets and
providing precise astrometric data for over a billion stars,
respectively.There are various machine learning techniques Barboza $et~{}al$.
(2020) are used to study and detect exoplanets Jin $et~{}al$. (2022). Table 4
shows the list of missions working on exoplanet wikipedia (2022) .

Kepler, although completed in 2013, left an enduring legacy by unveiling
thousands of exoplanets through the meticulous scrutiny of stellar light
curves. Its successor, K2, continued this mission, expanding our understanding
of exoplanets until its completion in 2018. Ongoing missions such as CHEOPS
(Characterizing Exoplanet Satellite) and JWST (James Webb Space Telescope)
Alderson $et~{}al$. (2023) hold promise for further revelations Agrawal
$et~{}al$. (2015). CHEOPS focuses on characterizing known exoplanets,
scrutinizing their atmospheres and compositions, while JWST, a joint venture
by NASA, ESA, and CSA, is poised to revolutionize infrared astronomy and
deepen our understanding of exoplanet atmospheres Hebrard $et~{}al$. (2018).

Looking ahead, a new wave of planned missions, including ARIEL (Atmospheric
Remote-sensing Infrared Exoplanet Large-survey), RST (Roman Space Telescope),
TOLIMAN, PLATO (PLAnetary Transits and Oscillations of stars), and more,
signal a future of expanded horizons and enhanced capabilities in exoplanet
detection and characterization. Reflecting on the completed missions, CoRot
and Spitzer played pivotal roles, with CoRot contributing to the discovery of
rocky exoplanets and Spitzer providing invaluable infrared observations until
its completion in 2020. Together, these missions have propelled exoplanet
science into a dynamic era, where each undertaking adds layers to the evolving
narrative of the cosmos beyond our solar system wikipedia (2022).

##  6 Conclusion and future scope

A wide range of creative techniques used by astronomers and researchers have
allowed the science of exoplanet detection to advance and diversify quite
significantly. this detailed analysis of various exoplanet detection methods
has illuminated the diverse approaches and technological advancements that
have shaped our understanding of distant planetary systems. Each method, from
the radial velocity and transit techniques to direct imaging, gravitational
microlensing, astrometry, and pulsar timing, contributes uniquely to the
growing catalog of known exoplanets. The precision of radial velocity
measurements, the sensitivity of transits, and the potential for direct
characterization through imaging collectively showcase the robustness of our
exoplanet detection capabilities. Astrometry, with its emphasis on high
precision and long baselines, offers complementary insights, especially for
massive planets in close orbits.

Future work in exoplanet detection should strive towards enhancing the synergy
between these methods. Integrating data from multiple detection techniques
will not only strengthen the validity of individual findings but also provide
a more comprehensive view of exoplanetary systems. Additionally, advancements
in technology, particularly in the development of next-generation space
telescopes and ground-based observatories, hold the promise of further
refining existing methods and unveiling new avenues for exoplanet discovery.
Continued collaboration between astronomers, astrophysicists, and instrument
engineers will be crucial for pushing the boundaries of exoplanet science,
ultimately enriching our understanding of the vast diversity of planetary
systems beyond our solar neighborhood.

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