# Enhancing State Estimator for Autonomous Racing : Leveraging Multi-modal
System and Managing Computing Resources ††thanks: 1 School of Electrical
Engineering, Korea Advanced Institute of Science and Technologies (KAIST),
Daejeon, Republic of Korea {lee.dk, menu107, ryuchanhoe, sw.nah,
seongwoo.moon<EMAIL_ADDRESS>††thanks: * Corresponding Author ††thanks:
This work was supported by the Technology Innovation Program
(RS-2023-00256794, Development of drone-robot cooperative multimodal delivery
technology for cargo with a maximum weight of 40kg in urban areas) funded By
the Ministry of Trade, Industry & Energy(MOTIE, Korea).

Daegyu Lee1, Hyunwoo Nam1, Chanhoe Ryu1, Sungwon Nah1, Seongwoo Moon1 and D.
Hyunchul Shim1*

###### Abstract

This paper introduces an approach that enhances the state estimator for high-
speed autonomous race cars, addressing challenges from unreliable
measurements, localization failures, and computing resource management. The
proposed robust localization system utilizes a Bayesian-based probabilistic
approach to evaluate multimodal measurements, ensuring the use of credible
data for accurate and reliable localization, even in harsh racing conditions.
To tackle potential localization failures, we present a resilient navigation
system which enables the race car to continue track-following by leveraging
direct perception information in planning and execution, ensuring continuous
performance despite localization disruptions. In addition, efficient computing
is critical to avoid overload and system failure. Hence, we optimize computing
resources using an efficient LiDAR-based state estimation method. Leveraging
CUDA programming and GPU acceleration, we perform nearest points search and
covariance computation efficiently, overcoming CPU bottlenecks.

Simulation and real-world tests validate the system’s performance and
resilience. The proposed approach successfully recovers from failures,
effectively preventing accidents and ensuring safety of the car.

###### Index Terms:

Vehicle location and navigation systems, Intelligent and Autonomous Systems,
Intelligent Vehicle, In-vehicle Navigation Systems

##  I Introduction

High-performance robotics racing is emerging as a powerful platform for
driving development and showcasing state-of-the-art capabilities in autonomous
systems. The first Indy Autonomous Challenge (IAC) started in October 2021,
utilizing the Dallara IL-15 IAC race car, equipped with six mono-cameras,
three radars, three LiDARs, and two real-time kinematic (RTK) GPS (Global
Positioning System). Since then, four races have taken place, including two at
the Consumer Electronics Show (CES) in Las Vegas, one in Texas in November
2022, and the most recent one in Monza, Italy. These competitions emphasize
the development of software capable of effectively handling edge cases in
autonomous driving, even in scenarios involving speeds of up to 300 $km/h$.

Parallel to these developments, a new autonomous racing competition, the
Hyundai Autonomous Challenge (HAC), is emerging in South Korea, featuring
Hyundai Motors’ electric car, the IONIQ 5. While each team can customize the
sensors for their vehicles, there is a common trend toward using LiDAR and
GPS/inertial navigation system (INS) in the high-speed navigation algorithm.
The primary goal of this competition, similar to the IAC, is to stimulate the
advancement of autonomous driving technology.

Figure 1:  (a)Dallara AV-21 of Team KAIST at the Las Vegas Motor Speedway
(LVMS) (b)IONIQ 5 of Team KAIST for HAC 2023.

However, the deployment of high-speed race cars revealed a significant
correlation between accidents and issues with navigation systems. The majority
of participating teams faced challenges arising from unreliable sensor
measurements, particularly impacting localization performance. The GPS, a
critical sensor for localization, posed significant challenges during high-
speed driving, as vibrations often led to drifts in position solutions,
resulting in serious accidents.

In addition to these challenges, common autonomous vehicles encounter
constraints in installing additional computing devices, necessitating
efficient management of computing resources. To address these critical issues
and enhance system robustness, a comprehensive approach to localization
becomes imperative. Multimodal sensing, involving the integration of diverse
sensors, offers a viable solution. In this study, GPS/INS and LiDARs serve as
our primary sensors. Leveraging this multimodal sensing system, we present a
resilient and efficient localization framework, encompassing the following key
components:

  * •

Real-time identification of measurement reliability is achieved through a
probabilistic Bayesian approach. This facilitates selective updates by
filtering out unreliable measurements.

  * •

An efficient LiDAR-based state estimation method, incorporating two core
concepts: computing efficient scan matching for correction and LiDAR odometry
for prediction, both accelerated by CUDA.

  * •

A resilient navigation system that utilizes direct LiDAR measurements to
maintain safe operation of the system, even in the event of localization
failures.

Throughout a series of computer simulations and experiments on real
racetracks, it is validated that the proposed robust localization algorithm
and the efficient navigation module are capable of maintaining high-speed
racing even when all GPS sensors are temporarily deteriorated. Moreover, this
research not only addresses the challenges observed in high-speed race car
deployments but also contributes valuable insights to the broader domain of
autonomous vehicle navigation under challenging conditions, emphasizing the
importance of efficient computing resource management.

The remainder of the paper is organized as follows. Section II introduces
related studies. Section III describes our high-speed, resilient state
estimator leveraging GPS/INS and LiDAR. The experimental results, including
simulated and real-world tests, are discussed in Section IV. Finally, Section
V concludes this paper, and potential improvements and future works are
explored in Section VI.

##  II Related works

###  II-A Localization based on multi-sensor fusion

The key challenge of navigation is adapting, learning, and recovering from
failures [1]. To address this issue, multi-sensor fusion has been studied. A
challenge is to integrate multiple sensors and distinguish between accurate
and unreliable ones. Tightly coupled navigation with global navigation
satellite system (GNSS), LiDAR, inertial measurement unit (IMU) is proposed in
[2]. In addition, Soloviev et al. [3] presented tight coupling of LiDAR and
IMU without GPS. However, these applications are deployed on the limited
scenes due to 2D laser scanner. Recently, robust and precise localization
system is proposed [4]. In the study, they evaluated the localization system
in various city scenes by integrating 3-D LiDAR, GPS, and IMU. However, only
one sensor of each type was used.

In most open sky environments, GPS plays an important role in localization by
providing accurate global position, but their performance might be impacted in
environments with occlusions, multi-path or radio interferences. Therefore,
significant studies on LiDAR-based multi-sensor fusion for GPS-denied
environments have been conducted [5], deploying robots in campus-scale
environments [6] or expanding to city-scale urban areas [7]. In this study, we
utilize GPS/INS and LiDAR for high-speed driving systems, and present research
on multimodal measurement utilizing multiple GPS and LiDAR.

###  II-B Perception based navigation system

Perception based navigation systems offer high accuracy and flexibility in
dynamic environments, particularly by enabling continued navigation even in
the absence of GPS signals. Wall-following navigation based on LiDAR sensors
has demonstrated promising results, as observed in tunnel scenarios [8],
narrow small-scaled environments [9], and in scenarios involving columns or
dynamic obstacles [10]. In these studies, a wall following algorithm was
introduced for mobile platforms, primarily using 2D LiDAR sensors. However,
our system employs 3D LiDAR, requiring ground filtering to distinguish between
walls and the ground. Additionally, when applied to high-speed autonomous
vehicles on a race track, wall following approach presents unique challenges.

RGB images can also be utilized for perception-based navigation system [11].
To improve image-based approach, hybrid methods that combine point-cloud and
images have been proposed in recent studies [12, 13]. Recently, deep
perception-based approaches for autonomous race cars have been studied,
utilizing end-to-end systems to learn control policies from on-board
observations [14]. Collaborative efforts with sim-to-real approaches have also
been explored [15], along with studies on handling cross-domain model transfer
[16]. However, there is an ongoing need to ensure that the output is reliable
and free from fatal errors in real-world scenarios.

###  II-C Autonomous racing

In the past few years, scale RC cars have been used to investigate algorithms
for autonomous racing experimentally [17, 18]. A comprehensive overview of the
current autonomous racing platforms, emphasizing the software-hardware co-
evolution to the current stage, is presented in [19]. After a series of IAC
races, each team presented its full stack of autonomous systems and approaches
in [20, 21, 22]. Furthermore, the TUM Autonomous Motorsport team has
introduced planning and control modules, employing strategies such as
minimizing curvature trajectory [23] and optimizing time-optimal trajectories
[24]. They have also explored model-based approaches [25, 26], incorporating
vehicle-model estimation [27] and minimizing lap times considering road
conditions [28]. In addition, when autonomous racing becomes analogous to
human-driven racing competition, behavior planning is crucial for performing
overtaking maneuvers or decision-making, and a reinforcement learning-based
approach has been recognized as significant in this context [29, 30]. Hence,
we studied the game-based predictor to predict future trajectories of racing
competitors for overtaking or decision-making in autonomous racing [31].

##  III Methods

To integrate multimodal measurement system, our full state estimatior is
designed as shown in Fig. 2. We integrate a Kalman filter-based GPS/INS method
with a LiDAR-aided state estimation algorithm.

Figure 2: Proposed state estimator for high-speed race car

###  III-A Multimodal measurement fusion Kalman filter

For multimodal sensor fusion, we use an extended Kalman filter algorithm.
Although Kalman filter enables us to combine information from multiple GPS
units and LiDAR sensors, its performance deteriorates rapidly when unreliable
measurements are fed to the filter. In order to prevent such situations, we
identify measurement reliability in real-time through a probabilistic Bayesian
approach, and selectively update the multimodal measurements.

We propose our multimodal measurement fusion Kalman filter algorithm to deal
with GPS measurement noise due to strong vibrations. To obtain global
positioning using LiDAR, we create a pre-built map of a race track and apply a
scan-matching algorithm.

####  III-A1 Vehicle state estimation formulation

The Kalman filtering process of time-varying stochastic control system with
$k$ multiple measurements is given by

$\displaystyle\mathbf{x}_{t}=\mathbf{F}_{t-1}\mathbf{x}_{t-1}+\mathbf{B}_{t-1}\mathbf{u}_{t-1}+\mathbf{q}_{t},$
(1)
$\displaystyle\mathbf{y}_{t}^{k}=\mathbf{H}_{t-1}^{k}\mathbf{x}_{t-1}+\mathbf{W}_{t},$

where $\mathbf{x}_{t}\in\mathbb{R}^{n}$ is the state,
$\mathbf{y}_{t}^{k}\in\mathbb{R}^{m}$ is the $k$-th measurement, and
$\mathbf{u}_{t}\in\mathbb{R}^{l}$ is the control input.
$\mathbf{q}_{t}\sim\mathcal{N}(0,\mathbf{Q})$ and
$\mathbf{W}_{t}\sim\mathcal{N}(0,\mathbf{R})$ are an additive white Gaussian
noise (AWGN). Specifically, to achieve time-efficient state estimation with
the available computation hardware, we used extended Kalman filter (EKF) with
constant turn rate and velocity (CTRV) model [32]. Therefore, we denote the
vehicle state as

$\displaystyle\mathbf{x}_{t}=[x_{t},y_{t},\theta_{t},b_{t}]^{T},$ (2)
$\displaystyle\mathbf{u}_{t}=[v_{x,t},w_{z,t}]^{T},$
$\displaystyle\mathbf{y}^{k}_{t}=[\mathcal{Z}_{x,t}^{k},\mathcal{Z}_{y,t}^{k},\mathcal{Z}_{\theta,t}^{k}]^{T},$

where $\\{x_{t},y_{t},\theta_{t}\\}\in SE(2)$ are vehicle’s coordinate,
heading, and $b_{t}$ is yaw bias. Control input $\\{v_{x,t},w_{z,t}\\}$ are
longitudinal velocity and yaw velocity, respectively, and they are obtained
from the vehicle electronics and IMU. We assume the evolution of each element
in Eq. 2 is described by and vehicle’s nonlinear kinematics model as

$\displaystyle x_{t}$
$\displaystyle=x_{t-1}+v_{x,t-1}cos(\theta_{t-1}+b_{t-1})dt,$ (3)
$\displaystyle=x_{t-1}+\psi_{t-1}dt$ $\displaystyle y_{t}$
$\displaystyle=y_{t-1}+v_{x,t-1}sin(\theta_{t-1}+b_{t-1})dt,$
$\displaystyle=y_{t-1}+\phi_{t-1}dt$ $\displaystyle\theta_{t}$
$\displaystyle=\theta_{t-1}+w_{z,t-1}dt,$ $\displaystyle b_{t}$
$\displaystyle=b_{t-1},v_{x,t}=v_{x,t-1},w_{z,t}=w_{z,t-1}.$

To extend linear model with Eq. 1, we obtain the discrete-time evolution model
with time-dependent linearized matrix $\mathbf{A}_{t}$ as

$\displaystyle\mathbf{x}_{t}=\mathbf{A}_{t}\cdot\mathbf{x}_{t-1}+\mathbf{q}_{t},$
(4)

where $\mathbf{A}_{t}$ can be represented as

$\displaystyle\mathbf{A}_{t}$ $\displaystyle=$ (5)
$\displaystyle\begin{bmatrix}1&0&\dot{\psi}_{t-1}\cdot
dt&\dot{\psi}_{t-1}\cdot dt&{\psi}_{t-1}/v_{x,t-1}\cdot dt&0\\\
0&1&\dot{\phi}_{t-1}\cdot dt&\dot{\phi}_{t-1}\cdot
dt&{\phi}_{t-1}/v_{x,t-1}\cdot dt&0\\\ 0&0&1&0&0&dt\\\ 0&0&0&1&0&0\\\
0&0&0&0&1&dt\\\ 0&0&0&0&0&1\\\ \end{bmatrix}.$

####  III-A2 Problem definition

Our goal is to estimate vehicle state with $k$ observation model to obtain the
$k$ observations vector as

$\displaystyle\mathbf{y}^{k}_{t}=\mathbf{h}^{k}(\mathbf{x}_{t})+\mathbf{W}_{t}.$
(6)

We can address our localization problem as the posterior probability in terms
of the prior likelihood [33]. Our problem definition starts with optimizing
the overall state of $\hat{\mathbf{x}}_{t}$ with multiple observations
$\mathbf{y}_{t}^{k}$ :

$\displaystyle\\{\hat{\mathbf{x}}_{t}\\}_{t\in[1,t]}=\arg\max[\prod_{i=0}^{t}p(\mathbf{x}_{i}|\mathbf{x}_{i-1})p(\mathbf{y}_{i}^{k}|\mathbf{x}_{i})p(\mathbf{y}_{0}^{k}|\mathbf{x}_{0})p(\mathbf{x}_{0})].$
(7)

We define our localization problem as finding an optimal solution at the
current time $t$; a marginalization can be represented as follows:

$\displaystyle\hat{\mathbf{x}}_{t}=$
$\displaystyle\arg\max_{\mathbf{x}_{t}}[\int
p(\mathbf{y}_{i}^{k}|\mathbf{x}_{t})p(\mathbf{x}_{t}|\mathbf{x}_{t-1})$ (8)
$\displaystyle\prod_{i=0}^{t}p(\mathbf{x}_{i}|\mathbf{x}_{i-1})p(\mathbf{y}_{i}^{k}|\mathbf{x}_{i})p(\mathbf{y}_{0}^{k}|\mathbf{x}_{0})p(\mathbf{x}_{0})d\\{\mathbf{x}_{l}\\}_{l\in[1,t-1]}],$

where we assume each variable as independent and identically distributed. We
further simplify Eq. 8 as

$\displaystyle\hat{\mathbf{x}}_{t}\propto$
$\displaystyle\arg\max_{\mathbf{x}_{i}}[\int
p(\mathbf{y}_{i}^{k}|\mathbf{x}_{i})p(\mathbf{x}_{t}|\mathbf{x}_{t-1})d\\{\mathbf{x}_{l}\\}_{l\in[1,t-1]}].$
(9)

Using Bayes’ theorem, we propose a hyper-parameter $\Theta$ for localization
to conditionally update the measurement as follows:

$\displaystyle\hat{\mathbf{x}}_{t}\propto$
$\displaystyle\arg\max_{\Theta}[\int
p(\mathbf{y}_{i}^{k}|\Theta^{k})p(\Theta^{k}|\mathbf{x}_{i})p(\mathbf{x}_{t}|\mathbf{x}_{t-1})d\\{\mathbf{x}_{l}\\}_{l\in[1,t-1]}].$
(10)

Intuitively, it is the likelihood of the $k$-th measurement
$\mathbf{y}_{t}^{k}$ over all possible parameter values, weighted by the prior
$p(\Theta^{k}|\mathbf{x}_{i})$. If all localization hyper-parameter
$p(\Theta^{k})$ assign high probability to the $k$-th measurement data, then
this is a reasonable criterion for multiple measurements. We define our
multimodal measurement fusion algorithm for high-speed localization as finding
$p(\mathbf{y}_{i}^{k}|\Theta^{k})$ and $p(\Theta^{k}|\mathbf{x}_{i})$.
Therefore, we determine a poor signal quality computing hyper-parameter
$\Theta$ to prevent using weak measurements to correct the Kalman filter.

Figure 3: Diagram of Kalman filter for multimodal measurement and degradation
identification

Our proposed Kalman filter for multimodal measurement and degradation
identification is described in Fig. 3.

####  III-A3 Bayesian-based multimodal measurement update

At a correction step of the Kalman filter, we propose a degradation
identification method with a novel hyper-parameter $p(\Theta^{k})$ derived
from Bayesian decision theory. We assume $p(\Theta^{k}|\mathbf{x}_{t})$ using
the Mahalanobis distance $\Delta_{k}$ [34] as follows:

$\displaystyle p(\Theta^{k}|\mathbf{x}_{t})\triangleq
1-1/(1+exp(-\Delta_{k})),$ (11)

where $\Delta_{k}$ pass through the sigmoid function. Specifically,
$\Delta_{k}$ can be obtained as

$\displaystyle\Delta_{k}=(\mathbf{x}_{t-1}-\mathbf{y}^{k})^{T}\Sigma^{-1}_{k}(\mathbf{x}_{t-1}-\mathbf{y}^{k}),$
(12)

where $\mathbf{y}^{k}$ is the $k$-th measurement and
$\Sigma_{k}^{-1}=\Lambda_{k}$ is the $k$-th measurement precision matrix.
$\Sigma_{k}$ is updated with AWGN $\mathbf{n}_{t}$ and covariance value from
measurement. More precisely, We interpret $\Delta_{k}$ as a Euclidean distance
in an orthogonally transformed new coordinate frame to identify whether
$\mathbf{y}^{k}$ is a reasonable input or not.

We propose a novel method for updating measurements, which is based on
Bayesian decision theory. Let
$p(\mathbf{y}_{i}^{k}|\Theta^{k},\mathbf{x}_{i})$ defined conditional to
$\Delta_{k}$ as follows:

$\displaystyle
p(\mathbf{y}_{i}^{k}|\Theta^{k},\mathbf{x}_{i})=\begin{cases}&\mathbf{y}^{0}\hfill\text{
if }\forall\Delta_{k}\leq\epsilon\\\
&\lambda_{i}\cdot\mathbf{y}^{i}\hfill\text{ if
}\forall\Delta_{k}>\epsilon\text{ and }\forall\Delta_{k}\leq\delta\\\
&\mathbf{y}^{k}\hfill\text{ if }\Delta_{k}\leq\delta\text{ and
}\forall\Delta_{\sim k}>\delta\\\ &\lambda_{reject}\hfill\text{ if
}\forall\Delta_{k}>\delta,\\\ \end{cases}$ (13)

where $\lambda_{k}=1-\Delta_{k}/\sum(\Delta_{i})$, $\delta$, and $\epsilon$
are the measurement update weight, hyper-parameter to determine qualified
measurement, and error($\epsilon<<\delta$), respectively. If all $\Delta_{k}$
is less than $\epsilon$, we use one of the measurements because it is already
qualified. If all $\Delta_{k}$ is less than $\delta$, we implement the
weighted sum of measurement, assuming that all the measurement is in
reasonable condition. In contrast, if $\Delta_{k}$ is less than $\delta$, but
$\Delta_{\sim k}$ is greater than $\delta$, we use the only one feasible
measurement, $\mathbf{y}^{k}$. Here, $\sim k$ in $\Delta_{\sim k}$ denotes
complement set of $\Delta_{k}$. Lastly, if all the distance is unsuitable for
measurement updates, a warning alarm is raised to be ready for the resilient
localization system to be discussed in the next section III-D. If the new
measurement status is rejected, we update our state only depending on control
input $\mathbf{u}_{t}$ without utilizing $\mathbf{y}^{k}$ at all.

###  III-B Unified frame map generation

In a multimodal system, establishing accurate transformations between multiple
global frames is crucial, as depicted in fig. 4. Georeferencing plays a key
role in this process [35]. Seamlessly to utilize a LiDAR scan-matched pose
$\mathbf{y}^{k}_{scan}\in\mathbf{y}^{k}$ as a multimodal measurement,
generating a unified map based on GPS frames is more intuitive, given that
$\mathbf{y}^{k}_{scan}$ can be directly integrated.

To facilitate this process, we employ pose-graph optimization, crafted to
construct a unified 3-D map denoted as $\mathbf{M}^{W}$. This is accomplished
by minimizing the spatial separation between corresponding poses within the
map. Notably, this section introduces a unique pose-graph optimization-based
frame unification method tailored specifically for the racetrack environment.
The outlined approach is depicted in fig. 4.

Figure 4:  The corresponding pose graph is visualized, showcasing the
relationship between the LiDAR point cloud and GPS/INS frames of National
Automobile Racetrack of Monza, Italy.

Using GTSAM framework [36], we update factor graphs using the GPS/INS key pose
$\mathbf{x}^{G}_{i}\in\mathbf{x}_{i}$ and LiDAR key pose
$\mathbf{x}^{L}_{i}\in\mathbf{x}_{i}$. Therefore, the factor graphs can be
represented as

$\displaystyle
f(\mathbf{x}^{G}_{0},\mathbf{x}^{G}_{1},...,\mathbf{x}^{G}_{n-1})=\prod
f_{i}(\mathcal{X}_{i}^{G}),$ (14) $\displaystyle
f(\mathbf{x}^{L}_{0},\mathbf{x}^{L}_{1},...,\mathbf{x}^{L}_{n-1})=\prod
f_{i}(\mathcal{X}_{i}^{L}),$

where $f_{i}$ indicates the connectivity of a factor graph’s each factor.

To update the GPS/INS-based pose graph $f_{i}(\mathcal{X}_{i}^{G})$ and LiDAR-
based pose graph $f_{i}(\mathcal{X}_{i}^{L})$, we process each frame
individually. The update equations for GPS/INS-based odometry measurements
are:

$\displaystyle f_{0}(\mathcal{X}_{0}^{G})=f_{0}(\mathbf{x}_{1}^{G}),\quad$
$\displaystyle\text{for}\ i=0,$ (15) $\displaystyle
f_{i}(\mathcal{X}_{i}^{G})=f_{i}(\mathbf{x}_{i}^{G},\mathbf{x}_{i+1}^{G};\mathbf{o}_{i}^{G}),\quad$
$\displaystyle\text{for}\ i>0.$

For LiDAR odometry measurements:

$\displaystyle f_{0}(\mathcal{X}_{0}^{L})=f_{0}(\mathbf{x}_{1}^{L}),\quad$
$\displaystyle\text{for}\ i=0,$ (16) $\displaystyle
f_{i}(\mathcal{X}_{i}^{L})=f_{i}(\mathbf{x}_{i}^{L},\mathbf{x}_{i+1}^{L};\mathbf{o}_{i}^{L}),\quad$
$\displaystyle\text{for}\ i>0.$

Here, $\mathbf{o}_{i}^{G}$ represents GPS/INS-based odometry measurements, and
$\mathbf{o}_{i}^{L}$ represents LiDAR odometry measurements. For $i=0$, the
initial pose is updated using unary factors $f_{0}(\mathbf{x}_{1}^{G})$ and
$f_{0}(\mathbf{x}_{1}^{L})$ for GPS/INS and LiDAR respectively. For subsequent
poses $(i>0)$, the updates depend on factors
$f_{i}(\mathbf{x}i^{G},\mathbf{x}{i+1}^{G};\mathbf{o}_{i}^{G})$ and
$f_{i}(\mathbf{x}i^{L},\mathbf{x}{i+1}^{L};\mathbf{o}_{i}^{L})$ for GPS/INS
and LiDAR respectively.

By using the GPS/INS-based pose as our main reference, our concept involves
incorporating racing line waypoints, given that race tracks typically have a
loop-shaped configuration. Therefore, we propose a hash-based racetrack ID
matching algorithm to associate a racetrack ID with GPS pose keys and
corresponding LiDAR pose keys. For each GPS/INS pose key $\mathbf{x}_{i}^{G}$
and corresponding LiDAR pose key $\mathbf{x}_{i}^{L}$:

$\text{racetrack\\_dict}[\mathbf{x}_{i}^{G},\mathbf{x}_{i}^{L}]\rightarrow\textit{racetrack\\_geometric\\_id}.$

After successfully finding a loop closing pose from corresponding
racetrack_geometric_id, we update the GPS/INS-based and LiDAR-based pose
factors as follows:

$\displaystyle
f_{i}(\mathcal{X}_{i}^{G})=f_{i}(\mathbf{x}_{i}^{G},\mathbf{x}_{\text{loop}}^{G};\mathbf{o}_{\text{loop}}^{G}),$
(17) $\displaystyle
f_{i}(\mathcal{X}_{i}^{L})=f_{i}(\mathbf{x}_{i}^{L},\mathbf{x}_{\text{loop}}^{L};\mathbf{o}_{\text{loop}}^{L}).$

After conducting several iterative drives on the race course, we perform non-
linear graph optimization using incremental smoothing and mapping(iSAM)-based
optimization [37]. Subsequently, we run an iterative closest point (ICP)
algorithm between the graphs to find a transformation matrix that aligns them.
The transformation matrix is then used to generate a unified map, integrating
the information from both GPS/INS-based and LiDAR-based pose graphs.

###  III-C Efficient LiDAR-based state estimator

####  III-C1 Efficient registration method

We employ the generalized iterative closest point (GICP) variant algorithm
[38, 39], to employ the scan match algorithm by modeling the consequtive scans
$\mathbf{P}=\\{\mathbf{p}_{0},\mathbf{p}_{1},...,\mathbf{p}_{n}\\}$,
$\mathbf{Q}=\\{\mathbf{q}_{0},\mathbf{q}_{1},...,\mathbf{q}_{m}\\}$ as a
Gaussian distribution
$\mathbf{p}_{i}\sim\mathcal{N}(\hat{\mathbf{p}_{i}},C^{\mathbf{p}}_{i})$,
$\mathbf{q}_{i}\sim\mathcal{N}(\hat{\mathbf{q}_{i}},C^{\mathbf{q}}_{i})$.
Subsequently, the transformation error $d_{i}$ can be defined as

$\displaystyle{d_{i}}=\hat{\mathbf{q}_{i}}-\mathbf{T}\hat{\mathbf{p}_{i}}.$
(18)

Thus, the $d_{i}$ distribution can be expressed as

$\displaystyle{d_{i}}$
$\displaystyle\sim\mathcal{N}(\hat{\mathbf{q}_{i}}-\mathbf{T}\hat{\mathbf{p}_{i}},C^{\mathbf{q}}_{i}-\mathbf{T}^{T}C^{\mathbf{p}}_{i}\mathbf{T})$
(19)
$\displaystyle=\mathcal{N}(0,C^{\mathbf{q}}_{i}-\mathbf{T}^{T}C^{\mathbf{p}}_{i}\mathbf{T}).$

Therefore, the registration problem can be represented as minimizing the error
between $\mathbf{P}$ and $\mathbf{Q}$ as

$\displaystyle\mathbf{T}=\arg\min_{\mathbf{T}}(\sum_{i}d_{i}^{T}(C^{\mathbf{q}}_{i}-\mathbf{T}^{T}C^{\mathbf{p}}_{i}\mathbf{T})d_{i}).$
(20)

Here we found that CPU-based registration may encounter bottlenecks due to its
sequential nature. More precisely, finding corresponding points and computing
covariance when estimate the $C^{\mathbf{p}}_{i}$ and $C^{\mathbf{q}}_{i}$ for
relevant points can be computationally expensive, leading to slower
performance, particularly with large and complex point clouds as depicted in
Fig. 5.

By adopting GPU-based methods [39], we can overcome these bottlenecks and
achieve faster and more scalable point cloud registration.

Figure 5: Comparison of Covariance Estimation Methods: CPU-Based vs. GPU-Based
Approaches

By implementing a GPU-based nearest points search and covariance computation
using the kernel descriptors, we can leverage parallel processing capabilities
to significantly accelerate the registration algorithm. GPU computation is
ideal for handling large-scale point cloud data, enabling efficient
computations on multiple points simultaneously.

Here, we consider several kernel descriptors, each with its corresponding
equation, to measure similarity between data points. The Radial basis function
(RBF) kernel ($K_{\text{rbf}}$) [39], The Gaussian kernel
($K_{\text{gauss}}$), the Polynomial kernel ($K_{\text{poly}}$), the Histogram
Intersection kernel (HI) ($K_{\text{hist}}$), and the Laplacian kernel
($K_{\text{laplacian}}$) are presented in Table I. In this study, the
Laplacian method demonstrates the best performance among the considered kernel
descriptors in GPU-based registration methods. Our implementation extends and
optimizes the Voxelized-GICP algorithms, resulting in improved results.

Table I: Kernel Descriptor for GPU-based covariance estimation Kernel Descriptor | Equation
---|---
RBF | $K_{\text{rbf}}(x,y)=\exp\left(-||x-y||^{2}*\sigma\right)$
Gaussian | $K_{\text{gauss}}(x,y)=\exp\left(-\frac{||x-y||^{2}}{2\sigma^{2}}\right)$
Polynomial | $K_{\text{poly}}(x,y)=(\alpha\langle x,y\rangle+c)^{d}$
HI | $K_{\text{hist}}(x,y)=\frac{\sum\min(x[i],y[i])}{\sum x[i]}$
Laplacian | $K_{\text{laplacian}}(x,y)=\exp\left(-\frac{||x-y||}{\sigma}\right)$

####  III-C2 LiDAR-inertial-vehicle model(LIV) odometry for prediction

We propose an efficient LIV odometry algorithm using LiDAR, IMU, and vehicle
state data, including velocity $\hat{\mathbf{v}}_{t}$ and wheel steering angle
$\hat{\delta}_{t}$ to address high-speed challenges. This fusion of inertial
information and vehicle state data ensures precise, real-time pose
predictions, even in the middle of prevalent slips and vibrations in high-
speed driving. By utilizing LIV odometry predictions for the initial
transformation, we aim to enhance the convergence and accuracy of the GICP-
variant algorithm during scan matching providing a more informed starting
point for the iterative scan matching process.

Firstly, we incorporate inertial data, specifically the rotational velocity
$\hat{\mathbf{w}}_{t}=\mathbf{w}_{t}+\mathbf{b}_{t}^{w}+\mathbf{n}_{t}^{w}$
where $\mathbf{b}_{t}^{w}$ and $\mathbf{n}_{t}^{w}$ are bias and noise,
respectively [40]. For rotational pose prediction, we use the quaternion
$\Delta\mathbf{q}_{ij}$ obtained from the IMU pre-integrator. In addition, we
integrate the wheel steering angle
$\hat{\delta}_{t}=\delta_{t}+b_{t}^{\delta}+n_{t}^{\delta}$ and velocity
$\hat{\mathbf{v}}_{t}=\mathbf{v}_{t}+\mathbf{b}_{t}^{v}+\mathbf{n}_{t}^{v}$ to
apply . We predict the rotation matrix for the next time step as:

$\displaystyle\Delta\mathbf{q}_{ij}$
$\displaystyle=\mathbf{q}_{i}^{-}1\otimes\mathbf{q}_{j}+\mathbf{v}_{j}\frac{tan(\delta_{j})}{L}$
(21)
$\displaystyle=\prod_{k=i}^{j-1}\left[\frac{1}{2}(\hat{\mathbf{w}}_{t}-\hat{\mathbf{w}}_{k}^{w})+\mathbf{v}_{k}\frac{tan(\hat{\delta}_{k}-n_{j}^{\delta})}{L}\right]\Delta
t,$

where rotational pre-integration is computed by quaternion multiplication
$\otimes$. In addition, $\Delta t$ represents the precise time difference
between IMU and vehicle information, with a interval of 0.01 seconds. This
approach allows us to estimate the rotational component of the pose even in
the absence of direct inertial measurements.

To address the noise inherent in the inertial sensor’s acceleration
measurements, we directly integrate $\hat{\mathbf{v}}_{t}$ for linear pose
pre-integration to obtain $\Delta\mathbf{p}_{ij}$ as

$\displaystyle\Delta\mathbf{p}_{ij}$
$\displaystyle=\prod_{k=i}^{j-1}\left[\mathbf{v}_{k}+\frac{1}{2}\mathbf{R}\Delta\mathbf{q}_{k}(\hat{\mathbf{v}}_{k}-\mathbf{b}_{k}^{v})\right]\Delta
t.$ (22)

We consider LiDAR odometry’s next scan match initial guess
$\mathbf{x}^{\dagger L}_{j}$ as :

$\mathbf{x}^{\dagger
L}_{j}=\mathbf{x}^{L}_{i}*\begin{bmatrix}\Delta\mathbf{q}_{ij}&\Delta\mathbf{p}_{ij}\\\
0&1\end{bmatrix}.$ (23)

####  III-C3 Scan match for correction

Utilizing a unified frame map, we can leverage the scan match pose
$\mathbf{y}^{k}_{scan}\in\mathbf{y}^{k}$ as a multimodal measurement. Our key
idea is an efficient target map query process of the point cloud map
$\mathbf{M}^{W}$ using a GPU-hash data structure. This process is vital,
especially for large race tracks. By leveraging the GPU-hash data structure,
we associate each pose on the race line with its corresponding point cloud
map. In addition, we employ a sliding-window approach by utilizing points from
$\mathbf{M}_{i}^{\psi}\subset\mathbf{M}^{W}$ [41, 7] instead of processing the
entire unified map $\mathbf{M}^{W}$. Furthermore, the ICP-variant algorithm’s
performance improves through predictive transformations of scan query data
with accurate initial guessing predictions, $\mathbf{x}^{\dagger L}_{j}$.

Finally, a reliability checker enables self-recovery for the LiDAR-based state
estimator in case of scan matching failure or when reliability exceeds safety
thresholds. In such situations, we use the GPS/INS pose $\mathbf{x}_{t}^{G}$
for recovery, leveraging its georeferencing capabilities as the initial pose
for scan matching. A visual representation of our proposed pipeline is
illustrated in Fig. 6.

Figure 6: Proposed pipeline of GPU-accelerated scan match algorithm

###  III-D Wall detection for resilient navigation

Since even the proposed robust localization system temporarily fails due to
harsh conditions on the racetrack, we propose additional wall-following
navigation module which does not depend on the main localization system. It
offers enhanced resilience to the navigation system by providing safe lateral
control in cases of complete localization system malfunction caused by
multiple GPS failures. To detect the walls of the racetrack at extreme driving
speeds, we propose an efficient ground filtering algorithm that only uses
direct LiDAR measurements.

When our proposed multimodal measurement fusion Kalman filter keeps computing
$p(\mathbf{y}_{i}^{k}|\Theta^{k},\mathbf{x}_{i})$ as $\lambda_{reject}$, we
consider this status as positioning degraded situation. To deal with these
critical situations, we have designed our race car such that it follows along
the race track wall, thereby avoiding collision with the wall. To extract a
wall in the racing track, we propose a wall detection algorithm comprising of
fast ground removal and wall clustering algorithms. The algorithm helps
navigate the wall resiliently when a deterioration is detected in the entire
measurement.

####  III-D1 Vertical feature extraction for fast ground filtering

Generally, the race track is mostly banked road, which angles towards the race
track center to help vehicles speed up at curved corners. Thus, the ground
filtering algorithm for LiDAR points has to consider the road gradient.
We propose a novel vertical feature extractor using a hashing algorithm to
consider the sparsity of LiDAR point-cloud data. Let $B\subset\mathbb{R}^{3}$
define the vehicle body coordinates. Here, $B$ annotated values indicating the
information obtained from the vehicle body’s origin—i.e., the center point of
the rear axle. We also define the voxel-filtered LiDAR points
$\mathbf{p}^{B}_{t}=\\{p_{1}^{B},\dots,p_{k}^{B}\\}$ at time $t$, where
$p_{i}^{B}$ is a voxelized point from the incoming LiDAR points. After
voxelization, we project $\mathbf{p}^{B}_{t}$ to the 2-D grid to vote the
points corresponding to the grid-cell, as shown in Fig. 7. Moreover, we
propose a hashing algorithm during the voting to account for the sparsity of
the point cloud.

$\displaystyle H(p_{i}^{B})=index,$ (24) $\displaystyle
H^{-1}(index)=p_{i}^{B},$

where hash function $H(x)$ maps the value $x$ at the table. We iterate
voxelized points $\mathbf{p}^{B}_{t}$ to vote on the corresponding grid-cell
using $f(p_{i}^{B})=g(u_{i},v_{i},\mathbf{n}_{index})$ where
$\mathbf{n}_{index}$ contains the number of points voted and its hashing-
index. Thus, by comparing $\mathbf{n}_{index}$ size with hyper-parameter, we
can efficiently filter out ground points $\mathbf{p}^{ground}_{t}$ from
$\mathbf{p}^{B}_{t}$ in real-time without matrix computation for plane
extraction.

Figure 7:  Description of process for ground points filtering, clustering, and
wall detection. (a) Ground filtering based on a z-axis voting algorithm. The
number of voting to grid-cell is compared to a hyper-parameter to extract the
vertical features. In the illustration, points corresponding to the banked
road have a few counts. (b) Ground-filtered points are illustrated. In
addition, the Euclidean-distance clustering algorithm is implemented. (c) The
length of clusters is used, assuming that the longest right-side cluster is
the wall. Points of the wall are computed to estimate the wall curvature and
distance to the wall.

####  III-D2 Wall following navigation

To extract the plane feature from the point-cloud, we implement a random
sample consensus (RANSAC)-variant algorithm [42, 43]. However, a distance
parameter determines the inlier or outlier points, which is unsuitable for
curved wall areas. Our approach uses a common tree-based Euclidean distance
algorithm to find a wall from the ground-filtered points
$\mathbf{p}^{filtered}_{t}$, as shown in Fig. 7. We implement a CUDA-based
Euclidean distance clustering algorithm [44, 45] to find cluster $C_{j}$ [46]:

$\displaystyle
C_{j}=\arg\min_{i}\parallel{p}^{filtered}_{i}-\mu_{j}\parallel_{2},$ (25)
$\displaystyle\mu_{j}=\frac{\sum_{i=1}^{m}1\\{c^{i}=j\\}{p}^{filtered}_{i}}{\sum_{i=1}^{m}1\\{c^{i}=j\\}},$

where Eq. 25 is the subject condition for clustering algorithm. We assume the
longest cluster as wall cluster $\mathbf{w}_{i}(x_{w,i},y_{w,i})$ :

$\displaystyle\mathbf{w}_{i}(x,y)=\mathbf{P}_{z}\cdot\arg\max_{x}(\mathcal{H}(C_{j}(x_{w,i},y_{w,i},z_{w,i}))),$
(26)

where $\mathcal{H}(x)$ extracts the length of clusters, and $\mathbf{P}_{z}$
is the z-directional projection matrix.
We then obtain a coefficient of the polynomial regression model
$\hat{\bm{\beta}}_{w}$ of $\mathbf{w}_{i}(x_{w,i},y_{w,i})$ as follows:

$\displaystyle\hat{\bm{\beta}}_{w}=(\mathbf{X}_{w}^{T}\mathbf{X}_{w})^{-1}\mathbf{X}_{w}^{T}\mathbf{y}_{w},$
(27)

where $\mathbf{X}_{w}=\\{x_{w,i}\in\mathbf{w}_{i}(x_{w,i},y_{w,i})\\}_{i=1:n}$
and $\mathbf{y}_{w}=\\{y_{w,i}\in\mathbf{w}_{i}(x_{w,i},y_{w,i})\\}_{i=1:n}$.
We next estimate the distance to the wall used for collision warning as
follows:

$\displaystyle d_{w}=\mathbf{y}_{w}(x_{w,0}),$ (28)

where $x_{w,0}$ denotes the origin point on body coordinates of the vehicle’s
$x$-directional axis. Thus, we can calculate the desired lateral shift from
the current position, $d_{w}-d_{gap}$, where $d_{gap}$ denotes the desired
distance from wall. Subsequently, we are able to generate a wall following
path $\mathbf{p}_{w}$ that maintains a desired distance from the wall as

$\displaystyle\mathbf{p}_{w}(x_{w,i})=\hat{\beta}_{2}{x_{w,i}^{2}}+\hat{\beta}_{1}{x_{w,i}+(d_{w}-d_{gap})},$
(29)

where $\\{\hat{\beta}_{1},\hat{\beta}_{2}\\}\subset\hat{\bm{\beta}}_{w}$. The
path is generated using the second-order polynomial curve fitting to address
the characteristics of oval tracks. Afterwards, the generated path is being
handled by the existing controller to track the given path accordingly. More
details about the controller can be found in our other study [47].

##  IV Results

###  IV-A Test scenarios

During our evaluation, we conducted tests at speeds of up to 248.8 $km/h$
across four different race events held in different countries. The maximum
speeds achieved for each event are detailed in Table II.

The US tracks, including Lucas Oil Raceway, Indianapolis Motor Speedway, Las
Vegas Motor Speedway, and Texas Motor Speedway, provided a suitable
environment to test high-speed scenarios. On the other hand, the Monza Circuit
in Italy and Everland Speedway in South Korea offered more dynamic scenarios
with challenging corners such as hairpin turns and chicanes. These dynamic
features posed significant challenges for state estimation, especially in
dealing with lateral derivatives and dynamic rotational driving.

Figure 8:  GG-diagram illustrating different test scenarios. Longitudinal and
lateral acceleration data are collected using IMU sensors. (a) GG-diagram of
Everland Speedway, driven by Hyundai IONIQ 5. (b) GG-diagram of Monza Circuit,
driven by Dallara Indy Lights car.

Figure 8 depicts the gg-diagram of the Monza and Everland circuits, offering
insights into the test scenarios by plotting longitudinal acceleration against
lateral acceleration.

Table II: Field tests performance summary.

Track | LOR | IMS | LVMS | TMS | MC | ES
---|---|---|---|---|---|---
| Maximum
---
speed
| 27.74 (m/s)
---
99.9 (km/h)
| 41.08
---
147.9
| 69.11
---
248.8
| 57.55
---
205.4
| 55.72
---
200.6
| 42.04
---
151.34

LOR : Lucas Oil Raceway, US IMS : Indianapolis Motor Speedway, US
LVMS : Las Vegas Motor Speedway, US TMS : Texas Motor Speedway, US
MC : Monza Circuit, Italy ES : Everland Speedway, South Korea

###  IV-B Vehicle platform

####  IV-B1 Indy autonomous challenge

The race car was based on a Dallara Indy Lights IL-15 chassis. The computation
hardware was equipped with an ADLink x64 computer system based on an Intel
Xeon with eight physical CPU cores and an NVIDIA RTX 8000 GPU. The sensor unit
has two independent NovAtel PwrPak7-Ds, with a full multi-frequency
integrating GNSS engine and embedded micro-electro-mechanical system (MEMS)
IMU, three Luminar LIDARs, three Aptiv radar, and six Allied Vision cameras.
In addition, vehicle states, such as wheel speeds and brake pressures, are
available. More detailed information is represented in Table III.

Table III: Computing and networking hardware specifications Device | Specification
---|---
CPU | Intel Xeon E 2278 GE – 3.30 GHz (16T, 8C)
GPU | Nvidia Quadro RTX 8000 x 1 (PCIe slot)
RAM | 64 GB
Ethernet ports | 3 x GigE RJ45 (In-built)
| \+ 2 x 40GbE QSFP+ (PCIe-based)
CAN ports | 4 x In-built ports, 2 x port on PCIe card
Network switch | Cisco IE-5000-12s12p-10G – 12 GigE copper
| 12 GigE fiber, 4x 10G uplink, PTP GM Clock
Wireless | Cisco/Fluidmesh FM4500 - up to 500Mbp
GPS | 2 x NovAtel PwrPak7 + HxGN SmartNet RTK
LiDAR | 3 x Luminar Hydra 120∘
RGB camera | 6 x optical camera
Radar | 3 x Delphi(1 x ESR + 2 x MRR)

####  IV-B2 Hyundai Autonomous Challenge

For the Hyundai competition, the race car was built on an Intel Xeon-based
computation hardware with an NVIDIA GeForce RTX 3080 GPU. The vehicle was
equipped with an array of sensors, including the Ouster OS2-128 LiDAR and two
Velodyne VLP 16 puck LiDAR sensors mounted on the side parts. Additionally,
the car featured four GMSL Seconix cameras for visual perception, while the
GPS system consisted of a Ublox F9P receiver and a Microstrain 3DM-GQ7 INS for
localization. Detailed specifications of the computation and sensor hardware
used in the Hyundai competition are provided in Table IV.

Table IV: Hyundai Competition’s Computing and Sensor Hardware Specifications Device | Specification
---|---
CPU | Intel Xeon E 2278 G – 3.40 GHz (16T, 8C)
GPU | Nvidia GeForce RTX 3080 (PCIe slot)
RAM | 64GB
Ethernet ports | PCIe-PoE550X 2-port 10GbE
CAN ports | PEAK PCAN-PCI Express FD 2Ch
Network switch | Tp-link TL-SX1008-10G
Wireless | KT LTE Egg - up to 500Mbp
GPS | Ublox F9P + Microstrain 3DM-GQ7
RTK | Network RTK + Synerex MRD-1000V2
LiDAR | Ouster OS2-128 + 2 x Velodyne VLP 16
RGB camera | 4 x GMSL Seconix camera

###  IV-C Multimodal measurement fusion Kalman filter

####  IV-C1 Resilience to simulated error and noise

Figure 9:  Large-noise was artificially added to the multimodal measurements
and the single modality system. It is crucial to note that the artificial
noise does not accurately replicate the intricacies of real-world GPS noise;
it serves as an approximation and should not be conflated with actual GPS
noise. (a) A simulated error on the LVMS race track. The degradation of both
GPS is represented in magnified view. (b) Proposed algorithm successfully
estimates the position robustly, compared to single modality system.
Specifically, the accuracy performance was maintained even when one of the two
degradation occurred.

We performed simulated tests to evaluate the capability of the proposed
algorithm. We introduced significant artificial noise into the recorded
multimodal data, which included measurements from two GPS and one LiDAR
sensors. This artificial noise was generated with a standard deviation of
approximately 5m and 10m, as illustrated in Fig. 9. It is important to clarify
that the artificial noise, denoted as $\sigma_{sim}$, is employed to assess
the system’s robustness. However, it is crucial to note that $\sigma_{sim}$
does not accurately replicate the intricacies of real-world GPS noise; it
serves as an approximation and should not be conflated with actual GPS noise.

During the evaluation scenario, the vehicle averaged 225 $km/h$ and reached
244 $km/h$ at maximum speed. We compared the estimated pose error with the
ground truth under conditions where one or two of the multimodal measurements
were degraded, as well as in a single-modality system under identical
conditions. When compared to a system based on a single modality, the proposed
algorithm successfully estimates the position robustly. Specifically, the
accuracy performance was maintained even when one of the two degradation
occurred as depicted in Fig. 9.

####  IV-C2 Real-world test

Figure 10:  Result of GPS log data and its estimated position using the
proposed localization at IMS. Despite the degradation, the proposed
localization algorithm estimated position resiliently throughout the track.
(a)-(d) A magnified representation of the track is shown with partial
degradation of the GPS data and an estimated result. (e) Errors from two GPS
units are shown during 20km driving. Initially, the GPS signal deteriorated
frequently, and corner sections drifted frequently. Ground-truth is manually
generated to compute error.

Regarding the specifics of implementation, we set $\epsilon$ as 0.2 $m$ and
$\delta$ as 5.0 $m$ in Eq. 13. Figures 10-10 present the magnified view of GPS
and estimation data from the first day of the competition in IMS.
Specifically, on the first competition day, we experienced more difficulties
with faultly GPS antennas and heavey vibration, leading to a degradation of
GPS performance. As depicted in Fig. 10, errors from two GPS units are shown
during 20km driving. Despite frequent degradation, we were able to effectively
reject degraded measurements through the methods proposed.

The operation of the tight INS/GNSS integration mode in the presence of heavy
vibrations resulted in significant positioning degradation, particularly in
the earlier IAC competitions. However, with the advancements in technology and
the implementation of mechanical damping systems to mitigate vibrations, this
issue has been gradually addressed and is now less prevalent in recent
competitions.

Figure 10 shows the data from the first IAC competition in Indianapolis,
illustrating the impact of vibrations on positioning accuracy.

###  IV-D Unified frame map generation

We evaluate the integrated robust state estimator, combining GPS/INS and
efficient LiDAR-based state estimators, to assess its performance in high-
speed and dynamic driving scenarios on the race track. Our evaluation aimed to
validate the state estimator’s accuracy and reliability under challenging
conditions.

Additionally, we introduced simulated LiDAR point cloud noise to create
scenarios where the scan matching algorithm might fail during the drive. This
allowed us to assess the robustness of the integrated system and evaluate its
ability to recover the LiDAR-based state estimation using the GPS/INS modality
as a reliable fallback. A evaluation video can be found at the following link:
111https://youtu.be/qnaOq1lsJOo.

###  IV-E Efficient LiDAR-based state estimator

We conducted a comprehensive evaluation using a dataset. Our LiDAR-based state
estimator emphasizes that we can extend the capabilities of our high-speed
racecar beyond reliance solely on GPS-based systems, even in GPS-denied
environments such as overpasses and tunnels.

####  IV-E1 Efficient registration method

To enable dense registration using 128-channel or solid-state LiDAR, we
encountered a significant CPU bottleneck during the covariance estimation
process, particularly while performing the corresponding points search
algorithm. To address this limitation and enhance computational efficiency, we
successfully utilized CUDA programming, leveraging GPU acceleration for the
nearest points search and covariance computation. This approach effectively
overcame the performance constraints associated with CPU-based methods.

Furthermore, we conducted an in-depth investigation into the impact of various
kernel descriptors on the rotational error within our registration method. To
ensure efficient computation, we systematically tested multiple kernel
descriptors. Remarkably, our proposed Laplacian method emerged as the most
effective, yielding the lowest errors across all evaluation metrics. Detailed
results are presented in Table V.

Table V: Comprehensive Comparison of Kernel Descriptors

Kernel Descriptor | rmse | mean | median | std
---|---|---|---|---
Polynomial | 164.93 | 143.97 | 125.01 | 80.47
HI | 15.94 | 14.75 | 17.19 | 6.04
Gaussian | 23.28 | 19.70 | 18.36 | 12.39
RBF | 11.14 | 9.69 | 8.14 | 5.48
Laplacian | 10.67 | 9.24 | 8.17 | 5.32

* Leaf size = 1.0 m

####  IV-E2 Evaluation on challenging dataset

We conducted an evaluation of our efficient LIV odometry on the demanding
Monza circuit to assess its performance in handling high lateral acceleration
and dynamic rotation at speeds of up to 200 $km/h$ on the road course circuit.

When the inertial and vehicle models are not integrated in dynamic scenarios,
our LiDAR-only odometry exhibits significant errors and struggles to cope with
environments where similar features are repeated as shown in Fig 11.

Figure 11:  Comparison of LiDAR odometry with and without IMU and vehicle
information integration. (a) LiDAR odometry with preintegrated IMU and vehicle
information. (b) LiDAR-only odometry without preintegrated IMU and vehicle
information.

To address this limitation, we integrated the LiDAR data with inertial and
vehicle motion models, resulting in improved odometry performance and
robustness in high-speed and dynamic driving scenarios. Our integrated
approach effectively overcomes challenges posed by repeated features and
yields more accurate and reliable state estimation, as demonstrated in the
Monza circuit evaluation.

A evaluation video can be found at the following link:
222https://youtu.be/C-H2i3tJvJ4.

####  IV-E3 Comprehensive evaluation with benchmarks

We conducted a comprehensive evaluation of our efficient LiDAR-based method by
comparing it with publicly available LiDAR odometry algorithms. Our proposed
cuda-accelerated method, which utilizes a pre-built map, demonstrated superior
performance even with significantly lower CPU usage compared to other methods.

While our proposed method may exhibit slightly less accuracy than methods
using CPU-based covariance estimation, this discrepancy can be compensated for
through pose graph optimization. Given that our deployment environment is a
race track, we found that our loop closing method, which utilizes race track
geometry, efficiently identifies re-visited positions and detects loops.

To further illustrate the effectiveness of our approach, we provide benchmark
comparisons and present the results of LiDAR odometry evaluation on the
Everland Speedway in Figure 12 and Table VI, respectively. The data showcases
the robustness and efficiency of our method in real-world scenarios.

Furthermore, we emphasize the importance of managing computing resources. In
situations where autonomous driving algorithms with substantial computing
requirements are already running on a single PC, efficient deployment of the
LiDAR-based approach without overloading the CPU is crucial.

Table VI: Comparison with the state-of-the-arts on ES dataset.

Method | rmse | mean | median | std | cpu(%)
---|---|---|---|---|---
mean | max | min
BLIO [48] | 2.745 | 2.632 | 2.585 | 0.779 | 10.8 | 23.81 | 7.73
DLO [49] | 6.142 | 4.775 | 3.414 | 3.862 | 3.22 | 4.92 | 2.24
Faster-LIO [50] | 3.574 | 3.138 | 3.199 | 1.710 | 2.94 | 3.92 | 2.29
Fast-LIO [51] | 3.801 | 3.319 | 3.430 | 1.853 | 2.45 | 2.97 | 2.13
KISS-ICP [52] | 29.802 | 24.655 | 22.907 | 16.74 | 8.28 | 12.79 | 5.63
LEGO-LOAM [53] | 22.270 | 20.359 | 18.099 | 9.026 | 3.88 | 7.45 | 1.20
Proposed(cpu) | 12.45 | 10.89 | 9.28 | 6.04 | 6.26 | 11.06 | 4.64
Proposed(cuda) | 25.618 | 21.064 | 18.44 | 14.581 | 1.17 | 1.43 | 1.04
Proposed(cuda)_with_map | 1.887 | 1.579 | 1.372 | 1.032 | 1.65 | 1.97 | 1.43
Proposed(cuda)_with_pgo | 6.764 | 5.647 | 4.859 | 3.722 | 3.68 | 6.53 | 1.19

*Due to the high computational demands of DLO, KISS-ICP cannot be executed using a leaf size of 1.0m on dense point cloud data. Therefore, all algorithms were evaluated using data downsampled to 2.0m with identical conditions.

Figure 12: Comparison of different localization methods (a) V-FAST-GICP, (b)
KISS-ICP, (c) GICP, (d) Proposed method. Ground truth is obtained by
leveraging GPS signals during the map construction process (e) BLIO, (f) FAST-
LIO, (g) Faster-LIO, (h) DLO, (i) KISS-ICP

###  IV-F Resilient navigation during GPS degradation

This study proposed a resilient navigation system that enables the race car to
continue to follow the race track even in the event of a localization failure.
We computed wall following path $\mathbf{p}_{w}$ using Eq. 29 utilizing direct
perception information until the completion of localization recover. As shown
in Fig. 13,13, our resilient navigation system came up with critical
degradation on all GPS units. In situations of localization failure caused by
completely degraded measurement, our resilient navigation algorithm allowed
the race car to maintain stability and continue driving away from wall. When
analyzing the lateral steering command that follows the racing line based on
localization, we found the negative directional steering command causing the
vehicle to crash onto the wall, as illustrated in 13. More details can be seen
online video 333 https://youtu.be/fiSqdMDmjGo.

Figure 13:  Proposed resilient navigation system which prevents crashes caused
by critical GPS degradation. (a) The proposed algorithm keeps the racing car
at a certain distance from the wall for approximately 2 s to prevent crashing.
(b) After receiving reasonable measurement data, the final steering command
uses the localization-based command.

###  IV-G Resilient navigation under incorrect desired path

During a test run, an incorrect desired path was accidentally given to the
system. This happened due to the bias originated from the satellite imagery of
the racetrack when the desired path was computed. As shown in Fig. 14, the
incorrect desired path, also referred to as an abnormal path, could lead
dangerously narrow gap to the race wall. When the distance between the wheel
and wall became smaller than the emergency threshold as depicted in Fig. 14,
the resilient navigation system was triggered and provided counter-directional
control. Thanks to the resilient navigation system, the wheel to wall distance
was successfully maintained bigger than the emergency threshold even though
the vehicle kept trying to track the incorrect desired path and approaching to
the race wall.

Figure 14:  Proposed resilient navigation system prevents abnormal cases from
collisions. (a) A magnified representation of the red-shaded region of entire
path. An error parameter was present for the track boundary of the abnormal
path, causing the wheel to wall distance abnormally small, that can
potentially lead to collision with the wall. (b) The subplot below refers to
the final merged command. The resilience system is triggered when the distance
between the wheel and the wall falls below the established emergency
threshold. Positive steering values indicate counter-directional control
against the wall.

##  V Conclusion

Our study introduces a comprehensive approach to enhance the state estimator
for high-speed autonomous race cars, addressing challenges such as unreliable
measurements, localization failures, and computing resource management. The
proposed robust localization system utilizes Bayesian-based probabilistic
evaluation of multimodal measurements, ensuring precise and reliable
localization under harsh racing conditions. Additionally, a georeferenced
LiDAR-based state estimator is seamlessly employed as a multimodal pose
measurement, leveraging CUDA programming and GPU acceleration to overcome CPU
bottlenecks. To handle potential localization failures, our resilient
navigation system enables continuous track-following using direct perception
information. Validation through real-world and simulation tests demonstrates
robust performance, preventing accidents and ensuring race car safety.

##  VI Discussion & Future works

In this section, we delve into the limitations and concerns to offer detailed
insights into real-world applications. Our study employs a multimodal
measurement fusion method, emphasizing the prioritization of GPS/INS
measurements.

Furthermore, in the generation of a unified frame map, we integrate RTK-GPS
for centimeter-level accuracy. While our robust unified map generation
framework, utilizing pose graph optimization, is generally resilient against
typical noise, there is a concern regarding potential degradation in map
quality when faced with significant noise.

We designed a real-world test to operate the vehicle solely using the LiDAR-
based state estimator without GPS/INS. However, further exploration is needed
to assess the reliability of the LiDAR-only method, especially in high-speed
scenarios. Therefore, in this study, we conducted a comprehensive evaluation
using a dataset.

As part of future work, we plan to explore advanced vision-based methods for
drivable area perception, taking advantage of the evolving capabilities of
edge computing. This exploration is motivated by a limitation in our current
wall-following method, particularly when deploying on road-course tracks like
Monza and Everland, where continuous walls are absent. In attempting to
overcome this limitation, we experimented with a vision-based approach;
however, a reliable and robust solution is yet to be established.

## Acknowledgment

We thank Dr Hoam Chung at Monash University for constructive feedback about
the organization of the manuscript.

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|  Daegyu Lee received his B.S. degree in Automotive Engineering from Kookmin
University, Seoul, South Korea, in 2018, and earned his M.S. and Ph.D. degrees
in the Division of Future Vehicle and Electrical Engineering from the Korea
Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea, in
2020 and 2024, respectively. He is currently a researcher at the Electronics
and Telecommunications Research Institute (ETRI), Daejeon, South Korea.
Daegyu’s research interests lie in autonomous systems and localization,
focusing on robust and learning-based approaches.
---|---
|  Hyunwoo Nam received his B.S. degree in Automobile and IT convergence,
Kookmin University, Seoul, South Korea, in 2022, and his M.S. degree in
Robotics Program from the Korea Advanced Institute of Science and Technology
(KAIST), Daejeon, South Korea, in 2024. He is currently pursuing a Ph.D.
degree in Electrical Engineering at KAIST. His research interests include
autonomous systems and localization based on unmanned ground vehicles
---|---
|  Chanhoe Ryu received his B.S. degree in Electrical Engineering, University
of California at Los Angeles (UCLA), Los Angeles, CA, USA, in 2019, and he is
currently pursuing a M.S. degree in Electrical Engineering from the Korea
Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea.
His research interests include autonomous systems, perception and motion
planning on unmanned ground vehicles.
---|---
|  Sungwon Nah received his B.S. degree department of Mechanical Engineering
from Konkuk University, Seoul, SouthKorea, in 2022, and his M.S. degree in
Robotics Program from the Korea Advanced Institute of Science and Technology
(KAIST), Daejeon, South Korea, in 2024. He is currently pursuing a Ph.D.
degree in Electrical Engineering at KAIST. His research interests include
autonomous racing, 3D object detection, and trajectory prediction.
---|---
|  Seongwoo Moon received his B.S. degree in Electronic Engineering from
Kyungpook National University, Daegu, South Korea, in 2022, and his M.S.
degree in Electrical Engineering from the Korea Advanced Institute of Science
and Technology (KAIST), Daejeon, South Korea, in 2024. He is currently
pursuing a Ph.D. degree in Electrical Engineering at KAIST. His research
interests include robotics, motion planning, and control based on unmanned
ground vehicles.
---|---
|  D.Hyunchul Shim received his B.S. and M.S. degrees in Mechanical Design and
Production Engineering from Seoul National University, Seoul, South Korea, in
1991 and 1993, respectively, and his Ph.D. degree in Mechanical Engineering
from the University of California at Berkeley, Berkeley, CA, USA, in 2000. He
worked with the Hyundai Motor Company and Maxtor Corporation from 1993 to 1994
and from 2001 to 2005, respectively. In 2007, he joined the Department of
Aerospace Engineering, KAIST, Daejeon, South Korea, and is currently a tenured
Professor with the Department of Electrical Engineering, and Adjunct
Professor, Graduate School of AI, KAIST. His research interests include
control systems, autonomous vehicles, and robotics. He is also the Director of
the Korea Civil RPAS Research Center.
---|---