# LADFN: Learning Actions for Drift-Free Navigation in Highly Dynamic Scenes

Mohd Omama1 Sundar Sripada V. S.1 Sandeep Chinchali2 K. Madhava Krishna1
1Robotics Research Center, IIIT-Hyderabad, India2ECE Department, The
University of Texas at AustinThe authors thank MathWorks India (Hyderabad) for
their generous financial support.

###### Abstract

We embark on a hitherto unreported problem of an autonomous robot (self-
driving car) navigating in dynamic scenes in a manner that reduces its
localization error and eventual cumulative drift or Absolute Trajectory Error,
which is pronounced in such dynamic scenes. With the hugely popular
Velodyne-16 3D LIDAR as the main sensing modality, and the accurate LIDAR-
based Localization and Mapping algorithm, LOAM, as the state estimation
framework, we show that in the absence of a navigation policy, drift rapidly
accumulates in the presence of moving objects. To overcome this, we learn
actions that lead to drift-minimized navigation through a suitable set of
reward and penalty functions. We use Proximal Policy Optimization, a class of
Deep Reinforcement Learning methods, to learn the actions that result in
drift-minimized trajectories. We show by extensive comparisons on a variety of
synthetic, yet photo-realistic scenes made available through the CARLA
Simulator the superior performance of the proposed framework vis-à-vis methods
that do not adopt such policies.

##  I Introduction

With rapidly increasing popularity of self-driving vehicles, there is a
growing need for their adroit, drift-aware maneuvering in highly dynamic
scenes, as drift particularly takes a hit in the presence of several moving
objects. Highly dynamic scenes pose specific challenges as the signal is
buried in noise for any kind of sensor based-localization/state estimation
methods to prove effective. The autonomous vehicle needs to actively modify
its behavior and navigation actions, often searching for spaces to move into
from where it can reduce its drift and enhance its state estimation accuracy.
To the best of our knowledge, there seems no literature that tackles this
problem head on.

Figure 1: Active navigation significantly reduces autonomous vehicle drift.
Left column represents baseline (Stanley Control) and right column represents
our approach (LADFN). Top Panel: CARLA [1] environments showing ego vehicles
planning their trajectories. Bottom Panel: LOAM [2] map visualizations of the
same scenes. The green lines delineate ground-truth trajectories, and the
purple lines delineate LOAM’s generated odometry. While drift from the green
ground-truth is pronounced on the left, with LADFN there is no such drift or
deviation on the right. For a 100m run, a Stanley-controlled ego drifts in the
$xy$-plane by 14.05m, eventually colliding with nearby traffic, whereas an ego
controlled by our approach drifts by 0.62m, due to traffic collision-avoidance
and active navigation towards nearby features. This drift also affects LOAM’s
[2] mapping potential: the map on the left is distorted due to the ego’s
haphazard odometry, whereas the map on the right remains relatively
unaffected. Figure 2: Pipeline of our approach. First, a map of the input
CARLA [1] scene is built using LOAM [2]. Filtering is applied to remove
dynamic traffic objects, if present. The state obtained from this map is
passed into a deep RL neural network that performs Proximal Policy
Optimization [3], to help choose actions $a\in\mathcal{A}$, our action space.
This model ensures that the ego drives towards feature-rich lanes and avoids
traffic in its way. The action inferences from the model are then are used to
generate a set of discrete waypoints for the ego to traverse. A cubic spline
is fit on these waypoints to generate a smooth trajectory, which is then
tracked using a Stanley Controller [4].

Therefore, in this paper, we present a novel active localization framework
that seeks and learns actions that lead to and result in drift-minimized
navigation in the presence of dynamic obstacles, as shown in Fig. 1. By
suitably-crafted reward functions and PPO [3] as its Deep Reinforcement
Learning architecture, action policies are learned that achieve this cause. We
show by extensive comparisons in a variety of dynamic scenes the efficacy and
superiority of this architecture in achieving drift-minimized navigation vis-
à-vis methods that do not actively navigate to minimize state estimation
errors. Henceforth, we call our system LADFN \- Learning Actions for Drift-
Free Navigation. And for notational convenience, we refer to our autonomous
vehicle as ego vehicle or ego interchangeably.

###  I-A Related Work

State estimation or localization that involves computing the pose of the robot
comes in various flavors with different sensing modalities such as those that
estimate both the agent’s motion and the map [5, 6], or those that only
estimate the agent’s motion or odometry [2, 7]. Prominent amongst these
include the highly popular LOAM [2] and its variants, with LIDAR as the
primary sensing modality or vision-based frameworks with or without direct
availability of depth information [8, 9, 7, 10]. A modest portion of this
literature is devoted towards handling dynamic actors. Herein, the popular
theme is about filtering dynamic obstacles to enhance estimation accuracy
irrespective of the sensing modality [11, 12, 13, 14, 15, 16]. For example,
authors in [17] proposed to detect and eliminate the dynamic regions using a
sparse motion removal (SMR) based on similarity and difference between frames
that is fed to a Bayesian framework for motion detection and filtering. While
[12] used background segmentation methods to filter dynamic agents, [13] used
ego-motion compensation to detect dynamic regions in the current frame with
subsequent vector quantization to segment moving regions in the scene. Other
methods such as [18, 19] jointly estimate stationary and moving agents in a
multi-body framework. Amongst the popular LIDAR based frameworks, [20] use
long term mapping techniques to segment moving parts of the scene, while the
stationary parts of the scene are used for localization and the moving agents
are maintained in the map in the form of their velocity representations. In
[21], a distance-filtering approach is utilized to segment the dynamic
contents of the scene while maintaining stationary parts as raw pointclouds.

The field of active localization, or actively navigating the robot or the ego
to minimize localization errors, has received some attention in the
literature. Active localization was popularized due to [22, 23] in their own
ways. While [22] focused on actively navigating towards less-aliased regions
in the scene so as to converge towards a unimodal estimate of the state from a
multi-modal belief due to perceptual aliasing, the main theme in [23] was
actively navigating to reduce the uncertainty of the unimodal state
distribution. The complexity of active localization was analyzed in [24] and
was shown to be NP-Hard, whereas a randomized algorithm for the same was
presented in [25]. The first and possibly only such formulation for active
localization in a multi-agent setting was presented in [26]. However, none of
the active frameworks talk of navigating an agent to reduce drift in the
presence of moving agents.

This work distinguishes itself from all the prior art in several ways. For
one, it is the first such method to tackle the problem of active navigation in
dynamic on-road scenes. In contrast to the state estimation approaches, the
present formulation showcases the need to go beyond filtering frameworks by
learning actions to maneuver around dynamic actors in a manner that reduces
drift. Unlike previous methods for active navigation that only handled
stationary scenes, the current formulation is specifically tailored towards
highly dynamic on-road scenes.

We use deep RL to learn drift-minimizing behaviours even though alternate
meta-heuristic approaches like GA, BAS have been used in robot control [27].
These approaches suffer from convergence issues with increased search space in
complex problems. A gradient-based RL system generally outperforms gradient-
less meta-heuristic methods.

###  I-B Contributions

  1. 1.

Formulating Active Navigation as an RL Problem: We formalize the problem of
active navigation in the LIDAR setting using RL.

  2. 2.

Drift-Minimizing Reward Function: Our novel reward function facilitates
learning of complex actions that simultaneously optimize for drift and control
costs. We discuss this in detail in III-D.

  3. 3.

Integrating Learned Actions with a Classical Controller: We combine the
learned actions from RL with a Stanley Controller [4] for path tracking.

##  II Pipeline Flow

The complete end-to-end architecture of LADFN is shown in Fig. 2. Our
implementation has the following modules:

  1. 1.

Perception and Dynamic Obstacle Filtering: We use an out-of-the-box
implementation of LOAM [2]. Then, we filter LIDAR points originating from
traffic vehicles.

  2. 2.

Waypoint Generation: We use an RL model to choose actions that provide
discrete waypoints for the ego to traverse. The model is unrolled for $n$
steps to get a trajectory.

  3. 3.

Trajectory Interpolation: We fit a cubic spline on the trajectory to smoothen
it.

  4. 4.

Low-level Control: We use a Stanley Controller [4] to perform path tracking on
the trajectory.

###  II-A Perception and Dynamic Obstacle Filtering

LIDAR Odometry and Mapping (LOAM) [2] is a SLAM technique [28] developed for
3D LIDAR sensors. LOAM splits the complex problem of optimizing several
thousand variables simultaneously by splitting the mapping and localizing
procedures into two: laser odometry and laser mapping. Laser odometry runs at
a high frequency and seeks to localize the LIDAR by calculating its velocity
in every frame. Laser mapping runs at a lower frequency and finds edge-edge
correspondences and plane-plane correspondences. These two processes are
followed by iterative scan matching to get the translation $T$ and rotation
$R$. LOAM is a state-of-the-art odometry solution on several benchmark
datasets, especially KITTI [29]. In our method, LOAM runs in the background
interacting with the LIDAR points originating from CARLA. We use CARLA’s [1]
provided LIDAR segmentation to filter out points coming from dynamic vehicles
and test our approach both in the presence and absence of this filtering. The
odometry output from LOAM is piped into the waypoint generation module,
discussed in detail in Section II-B.

###  II-B Waypoint Generation

The Waypoint Generation problem will be formulated as a Markov Decision
Process (MDP) in Section III. While there are a plethora of deep RL algorithms
to approximately solve this MDP, we chose PPO [3] since it is well-established
and well-suited for discrete action spaces that our setting requires. We also
note that the primary novelty of our paper is the MDP formulation for active
navigation to reduce drift, and other RL algorithms should perform well in
practice. For both the actor and the critic, we have a fully connected network
with 6 hidden layers. The number of neurons in each hidden layer are: 64, 64,
32, 32, 16 and 16 respectively. We use ReLU activation along with a batch-size
of 128 and a discount factor $\gamma$ of 0.9999. Our architecture conforms to
the implementation of PPO in Stable-Baselines 3 [30].

We synthetically create a wide range of initial states. These states span
across multiple combinations of ego positions, traffic positions, traffic
speeds, positions of centroid of edge features, etc., During training, we
assume that the position of centroid of edge features remains constant for
that episode. This hypothesis is valid because our training episodes last for
only 10 steps. We created an environment that takes as input at time $t$ the
state $s_{t}$, and the action $a_{t}$, and outputs at time $t+1$ the next
state $s_{t+1}$, and reward $r_{t+1}$. This environment was used to train an
RL agent without directly interacting with the CARLA [1] Simulator. The
episode is terminated either at the completion of 10 steps, or when the ego
breaches lane boundaries, or when the ego collides with the traffic.

###  II-C Trajectory Interpolation

The waypoints obtained from PPO are then passed into a cubic spline planner
for interpolation in order to obtain a smooth trajectory that our ego can
traverse. During inference, we unroll the waypoint generation module $n$ times
to get an $n$-step trajectory. We then fit a cubic spline on it using an
existing framework provided by Sakai et al. [31].

###  II-D Low-level Control

The final step in our process is to traverse the smooth spline and ensure that
the ego is tracking the path with minimal drift. For this purpose we use a
standard Stanley Controller [4], but it should be noted that our approach is
invariant to the choice of the path tracking algorithm. The implementation of
Stanley used is also provided by [31].

##  III Model-Free RL for Drift Minimization

Reinforcement Learning (RL) has been extensively used to solve a vast variety
of Markov Decision Processes (MDPs). A canonical MDP comprises of a state
space $\mathcal{S}$, an action space $\mathcal{A}$, a transitional
probability, $\mathcal{P}$:
$\mathcal{S}\times\mathcal{A}\times\mathcal{S}\rightarrow[0,1]$, that governs
the transition from state $s_{t}$ to state $s_{t+1}$, a reward function,
$\mathcal{R}$:
$\mathcal{S}\times\mathcal{A}\times\mathcal{S}\rightarrow\mathbb{R}$, which
describes a scalar reward that is associated with the transition from state
$s_{t}$ to $s_{t+1}$ by taking an action $a_{t}$, and a discount factor
$\gamma$ which controls the importance we give to future rewards. When we
solve the MDP, we find a policy $\pi$: $\mathcal{S}\rightarrow\mathcal{A}$,
such that it maximises the value function $V_{\pi}$:

$\small
V_{\pi}(s_{0})=\mathbb{E}\bigg{(}\sum_{t=0}^{\infty}\gamma^{t}\mathrm{R}\Big{(}s_{t+1},s_{t},\pi\big{(}s_{t}\big{)}\Big{)}\bigg{)},$
(1)

Now, we will formulate LADFN’s waypoint generation problem as an MDP and
provide a detailed account of the state space $\mathcal{S}$, the action space
$\mathcal{A}$ and the reward $\mathcal{R}$. The dynamics of our MDP depend on
the motion of the ego-vehicle as well as the uncontrolled motion of other
traffic. Specifically, we cannot analytically model where other traffic will
move, nor what the ego will observe at future time-steps from its LIDAR
sensors. Since we do not have an analytical model for the full dynamics of the
MDP, we resort to model-free deep RL to solve for a control policy. We first
define a frame whose $X$-axis points along the direction of the road, and
whose $Y$-axis is perpendicular to the direction of the road. The ego’s
starting coordinate in this frame will be of the form (0, $y_{e}$). We refer
to this frame as the RL frame (Fig. 3).

Figure 3: Our RL Frame representation in a CARLA scene.

###  III-A State

We define the state of the RL agent as a vector such that it captures all
necessary task-relevant information. The entries of the state space vector
will be in RL frame unless stated otherwise. The state vector is as follows:

  * •

$x_{e}$: x-coordinate of ego vehicle

  * •

$y_{e}$: y-coordinate of ego vehicle

  * •

$y_{c}$: y-coordinate of centroid of the edge features which is clipped and
scaled to $[-1,1]$; edge features are discussed in Section III-C

  * •

$edge_{l}$: left limit of the road

  * •

$edge_{r}$: right limit of the road

  * •

$tr_{p_{i}}$: Boolean representing the presence of $i^{th}$ traffic vehicle

  * •

$x_{tr_{i}}$: x-coordinate of the $i^{th}$ traffic vehicle

  * •

$y_{tr_{i}}$: y-coordinate of the $i^{th}$ traffic vehicle

  * •

$v_{tr_{i}}$: velocity of the $i^{th}$ traffic vehicle.

###  III-B Action Space

The RL agent has a set of discrete actions $\mathcal{A}$ that it can choose
from. Each action $a\in\mathcal{A}$, represents the waypoint at the next time
step for the vehicle in the vehicle’s frame of reference. The action $a$ is
defined as a tuple $(a_{x},a_{y})$ where $a_{x}\in\\{1,2,3,5\\}$ represents
$x$-coordinate of the next waypoint, and $a_{y}\in\\{-2,,-1,0,1,2\\}$
represents $y$-coordinate of the next waypoint in the vehicle’s frame. We
ensured that our action space reflects a spectrum of ways in which the vehicle
could move: from slow-straight motion to fast-turning motion.

###  III-C Motivation For Reward Shaping

LOAM [2] geometrically categorizes the LIDAR points as edge or plane features
which are then used in scan-matching. Edge features represent texture-rich
regions which help in accurate scan-matching. We will refer to such regions as
feature-rich. In order to sustain LOAM’s mapping process, and more
importantly, reduce the absolute pose error (APE) in ego-motion, the first
consideration is the ego’s proximity to feature-rich regions. This can be
argued as follows.

Consider two vehicles $ego_{1}$ and $ego_{2}$ and a feature-rich region
$T_{R}$ as shown in Fig. 4, where $ego_{1}$ is closer to $T_{R}$ than
$ego_{2}$. We draw tangents from the two vehicles to $T_{R}$. Let the angles
that these two sets of tangents subtend with $T_{R}$ be $\theta_{1}$ and
$\theta_{2}$ respectively. Also, consider a planar LIDAR which has a
$360^{\circ}$ field view of a single plane and an angular resolution of $n$.
The number of LIDAR rays per unit angle will then be $\frac{n}{360}$. Now, the
number of LIDAR rays that interact with the feature-rich region in both the
cases will be $\frac{n\times\theta_{1}}{360}$ and
$\frac{n\times\theta_{2}}{360}$ respectively. Since, $ego_{1}$ is closer to
$T_{R}$ than $ego_{2}$, $\theta_{1}$ will be larger than $\theta_{2}$,
indicating that more LIDAR rays will interact with $T_{R}$ in the first case
leading to its more dense reconstruction. The same argument can be extended
for 3D LIDAR which is the primary hardware setting in our proposed solution.
Therefore, when the ego is in the proximity of texture-rich regions, LOAM can
more densely reconstruct them since the incoming pointcloud to LOAM will have
more edge features contained in it.

Figure 4: Benefits of active navigation towards feature-rich regions. More
LIDAR rays from $\mathbf{ego_{1}}$ than $\mathbf{ego_{2}}$ interact with the
texture-rich region $T_{R}$, since $\theta_{1}>\theta_{2}$, resulting in a
more dense reconstruction.

The second aspect that plays an impact on the APE is the distance from traffic
vehicles. LIDAR points from dynamic objects lead to erroneous registrations in
the scan-matching algorithm; in our case, the dynamic objects are said traffic
vehicles. Further, proximity to dynamic vehicles can also occlude visibility
of other texture-rich regions. Follow-up papers [32, 33] on LOAM have argued
the same. Further, Thomas et al. [32], have specifically shown that the
proximity to dynamic vehicles leads to substantial error in the rotational
measurement, $R$. Our experimentation in dynamic scenes also resonates with
the findings of these works but unlike us, none of them address drift-
minimizing behaviors.

To handle the problems described above, the ego should be aware of the
feature-rich regions and plan its trajectory accordingly. It should also avoid
proximity with traffic wherever possible. Our proposed solution comprises of
an RL agent that generates waypoints which are then traversed with the aid of
a low-level controller. The input to the RL agent is a state that includes the
centroid of edge features, the ego’s pose, and the traffic vehicles’ poses. We
calculate the centroid using queried edge features obtained through LOAM. The
pose of the ego is directly queried from LOAM. We also assume that we have the
speed and pose information of the traffic readily available due to recent
advancements in LIDAR-based semantic segmentation and tracking [34].

###  III-D Reward

The reward was designed with two primary objectives: navigation and
perception. The navigation objectives comprise of the follows.

  1. 1.

Forward motion: We incentivize the ego to move forward towards the goal.

  2. 2.

Road boundary constraint: $edge_{l}\leq y_{e}\leq edge_{r}$, i.e., the ego
should move within the road boundaries.

  3. 3.

Collision constraint: $\lVert(x_{tr_{i}},y_{tr_{i}})-(x_{e},y_{e})\rVert\geq
3$ meters, i.e., the ego must not collide with the traffic.

The perception objectives comprise of the follows.

  1. 1.

Feature proximity: minimize $|y_{e}-y_{c}|$, i.e., the ego should try to
maintain proximity with the feature-rich regions.

  2. 2.

Traffic distance: $\lVert(x_{tr_{i}},y_{tr_{i}})-(x_{e},y_{e})\rVert\leq 10m$,
where the ego should try to avoid proximity with traffic.

Our reward $\mathcal{R}$ models both the objectives described above and is
defined as:

$\small\mathcal{R}=\mathcal{P}\times\mathcal{T}+(1-\mathcal{P})\times\mathcal{F}+\mathcal{G},$
(2)

where $\mathcal{P}=\lor{p}_{i}$, which is the logical OR over all the traffic-
proximity Boolean $p_{i}$. $p_{i}$ will be described in detail in the Traffic
Reward section. We now explain the other three terms in equation (2).

####  III-D1 The Goal Reaching Reward, $\mathcal{G}$

$\mathcal{G}$ returns a positive reward for forward motion along the $X$-axis
towards the goal. Although a simple term, this ensures that the ego
prioritizes moving towards the finish line besides pursuing feature-rich
regions and avoiding traffic. The other navigation constraints like traffic
and road limits are also modeled here. The ego is penalized if it collides
with the traffic or moves out of the road, and the episode terminates.

####  III-D2 The Feature Reward, $\mathcal{F}$

The feature reward needs to be shaped in a way that the ego is rewarded if it
maintains proximity with feature-rich regions. To achieve this, we need two
behaviors that are captured in two terms in (3) as follows:

##### The first term

The ego should move towards the left if the left side of the road is feature-
rich and vice-versa. The lateral position of the feature-rich region is
described by $y_{c}$, which is clipped and scaled to a normalized range of
$[-1,1]$. A value of $-1$ means the centroid of features is towards the left
end of the road, and a value of $1$ means the centroid is towards the right.
To formulate the feature reward, we also need $y_{e}$ in the range $[-1,1]$.
The term
$(2((y_{e}-{y_{e}}_{\mathrm{min}})/({y_{e}}_{\mathrm{max}}-{y_{e}}_{\mathrm{min}}))-1)$
handles that. We first raise $y_{c}$ to an odd power, $O$, and then multiply
it with normalized $y_{e}$. This means that the reward will be positive if
both $y_{c}$ and normalized $y_{e}$ have the same sign, implying they are on
the same side of the road; and the reward will be negative if they are on the
opposite sides of the road. Further, $y_{c}^{O}$ forces the term following it
to quickly drop to 0 when $y_{c}$ is close to 0, ensuring that this first term
has no impact when there are feature-rich regions on both sides of the ego.

##### The second term

If there are features on both sides of the road, the ego must prevent
unnecessary lateral motion. To achieve this, we multiply the absolute of our
lateral motion $|a_{y}|$ with $(1-|y_{c}^{O}|)$. In this case, $|y_{c}^{O}|$
will be close to zero since features are on both sides and $(1-|y_{c}^{O}|)$
will tend to $1$. Further, $(1-|y_{c}^{O}|)$ drops to zero when $y_{c}$ is
close to 1, ensuring that this second term has no impact when there are
features on only one side of the road (left or right).

The complete mathematical formulation of the feature reward is given by:

$\small\begin{split}\mathcal{F}&=K_{1}\times\bigg{(}y_{c}^{O}\times\Big{(}\frac{2\times(y_{e}-{y_{e}}_{\mathrm{min}})}{{y_{e}}_{\mathrm{max}}-{y_{e}}_{\mathrm{min}}}\Big{)}-1)\bigg{)}\\\
&-K_{2}\times(1-|y_{c}^{O}|)\times|a_{y}|,\end{split}$ (3)

where $K_{1}$ and $K_{2}$ are constants.

####  III-D3 The Traffic Reward, $\mathcal{T}$

We define a traffic-proximity region
$\lVert(x_{tr_{i}},y_{tr_{i}})-(x_{e},y_{e})\rVert\leq 10m$ where the effect
of traffic is large on localization. The traffic reward should be formulated
such that the ego tries to move out of this region. We design a reward which
incentivizes the ego if the relative difference in forward direction between
the ego and the traffic increases. It also returns a positive reward if the
ego increases its lateral distance with the traffic car, and a negative reward
otherwise. The traffic reward can be mathematically written as:

$\small\mathcal{T}=\sum_{i=1}^{n}p_{i}\Big{[}\big{(}K_{3}\times|a_{x}-v_{tr_{i}}|\big{)}+K_{4}\times\big{(}|y_{tr_{i}}^{t}-y_{e}^{t}|-|y_{tr_{i}}^{t-1}-y_{e}^{t-1}|\big{)}\Big{]},$
(4)

where $p_{i}$ is a Boolean which is 1 only when the ego is in a traffic-
proximity region, otherwise this traffic-vehicle has no impact on the reward,
and $K_{3}$ and $K_{4}$ are constants. The summation term outside accumulates
the rewards/penalties from each traffic vehicle $i$.

The waypoint generation MDP is solved using deep RL using an architecture
described in Section II-B. We now evaluate our RL-based active navigation
framework, LADFN, in challenging autonomous driving scenarios.

##  IV Experiments

###  IV-A Experimental Setup

The performance of LIDAR odometry algorithms can depend a lot on the physics
of the environment as well as that of the ego. Hence, we use CARLA simulator
[1] for our experimentation as it emulates real-world physics accurately. It
is also a state-of-the-art high-fidelity simulation environment for autonomous
driving and so it is ideal for our application. We simulate a 3D LIDAR in
CARLA with 16 channels and a vertical field of view of $30^{\circ}$
($-15^{\circ}$ to $15^{\circ}$). The LIDAR emits 300,000 points per second
which are evenly distributed over all channels. While running our experiments,
we use three different LIDAR ranges: 45m, 50m, and 55m. These ranges
correspond to the the maximum distance that can be measured by the LIDAR. The
variety of LIDAR ranges during the experiments helps us analyze the
performance of our method for various sensor ranges.

Figure 5: LADFN significantly reduces drift with a minor change in the path
taken to reach the goal. The LOAM-generated trajectories are shown in red, the
ground-truth trajectories are shown in black, and the green structures
correspond to feature-rich regions. The left column shows Vanilla Stanley + F
while the right column shows LADFN + F. Each row corresponds to a different
scene. These scenes were run with a LIDAR range of $45$ meters and dynamic
obstacle filtering enabled.

###  IV-B Scene Description

We have created a test bed of 5 different scenes in CARLA [1]. These scenes
have a run length ranging from 100 to 500 meters, and include static as well
as dynamic scenarios. We have created the scenes such that they have varying
distributions of feature-rich regions and dynamic vehicles. Scenes 1 and 4 are
static and don’t have any dynamic traffic actors whereas scenes 2, 3 and 5
contain moving traffic cars or vans around the ego. The results shown have up
to two dynamic vehicles, but the method is easily scalable to an arbitrary
traffic count.

Figure 6: LADFN significantly reduces drift than competing benchmarks on a
diversity of CARLA scenes. Firstly, we want to show that LADFN leads to lower
average drift than the benchmark. This is clearly shown in the first column
where LADFN reduces Stanley’s drift by at least $1.48\times$. Secondly, we
provide a framework for analyzing the performance of goal-reaching tasks, seen
in the final translation offset plots in the second column, where once again,
LADFN consistently outperforms the Vanilla Stanley controller by at least
$1.63\times$. Thirdly, we show that rotational errors are significantly
reduced by LADFN in the third column. In dynamic scenes (2,3,5), dynamic
obstacle filtering was enabled. In all columns, F in the plot legends denotes
that filtering of dynamic obstacles was carried out. Diagonal hatches ’/’
inside bars represent collision occurrence of the ego with its surroundings in
the experiment.

###  IV-C Metrics and Benchmarks

We benchmark our approach against what we call the Vanilla Stanley Controller.
With Vanilla Stanley, the ego will not actively change its direction or speed
to account for drift minimization. The ego will drive with a speed that is
similar to the traffic around it. Hence, it represents perception-unaware
control, a realistic scenario where the ego is tracking a trajectory without
having any idea of the perception goals. Since there is no previous work in
this area, Vanilla Stanley is an ideal benchmark to compare against.

In dynamic scenes, a general approach to decrease trajectory error is to
filter out dynamic vehicles from the LIDAR scans. Hence we have another
benchmark which we refer to as Vanilla Stanley + F which is just a Vanilla
Stanley Controller running with filtering enabled. We test and compare our
proposed approach both in the presence and absence of dynamic obstacle
filtering. This helps us understand that even when filtering is available,
active navigation is essential to further reduce drift in the trajectory.

We use three different metrics for evaluating our approach:

  1. 1.

Average Drift: The average translation drift in $xy$-plane for the entire
trajectory.

  2. 2.

Final Drift: Final translation drift in $xy$-plane at the end of the
trajectory.

  3. 3.

Rotational Offset: Yaw offset with respect to the ground-truth yaw at the goal
location.

Metric 1 gives a general idea of the quality of localization achieved for the
entire trajectory. Metric 2 gives an idea of how the methods will perform on
goal-reaching tasks. Metric 3 serves two purposes: like Metric 2, this is also
a representation for goal-reaching tasks, and further, it gives an idea of the
quality of the future trajectory, since rotational errors in odometry have a
compounding effect on the future trajectory.

###  IV-D Qualitative Results

Fig. 5 shows a qualitative view of the trajectory and drift in all the five
scenes. In the right column, as the estimated LOAM trajectory in red overlaps
the ground-truth trajectory in black, unlike the baseline shown in the left
column where the red deviates from the black, it can be clearly seen that with
our approach, the drift is substantially reduced in every case. We can see
that the ego tries to maintain proximity with the feature-rich regions shown
in green. Rows 2, 3, and 5 correspond to dynamic scenes with filtering enabled
and so traffic vehicles are not visible in the LIDAR scans shown in the Fig.
5. We observe that even in the presence of filtering, our approach outperforms
the baseline, highlighting the need for active navigation in drift
minimization.

We achieve this due to the complex behaviours learned by our model, the
examples of which can be seen in the following link:
https://mohdomama.github.io/LADFN/.

###  IV-E Quantitative Results

We depict the quantitative comparisons of our approach with the baselines in
the five rows of Fig. 6 corresponding to the five scenes. Rows 1 and 4 do not
have dynamic actors, and the comparison is between LADFN and Vanilla Stanley,
shown with orange and violet bars. Rows 2, 3 and 5 correspond to the scenes
populated with dynamic actors. Here, we include the filtering module in both
the Stanley Controller and LADFN, denoted as LADFN + F and Vanilla Stanley +
F, and shown with green and blue bars. In some scenes, the LOAM estimates have
significantly deviated resulting in collision for the baseline Vanilla
Stanley, these are shown with hatched violet bars.

Our first step is to show that LADFN leads to a lower average drift than
benchmark competitors. This is clearly shown in column 1. In static scenes, we
get up to $3.7\times$ improvement in average drift using LADFN. In dynamic
scenes, with filtering enabled, we get up to $4.8\times$ improvement in
average drift using LADFN, while with filtering disabled, the drift in Vanilla
Stanley is often so high that it leads to collision. This is not the case with
LADFN which remains relatively unaffected, even with filtering disabled. It
should also be noted that the impact of dynamic vehicles and feature-proximity
on benchmark competitors gets pronounced as the LIDAR range decreases, while
LADFN is resilient to these changes in LIDAR range.

The second column and the third column show the final drift and the final yaw
offset at the end of the run respectively. They give an idea of the goal-
reaching capabilities of all the approaches along with an intuition of how bad
the future trajectory would be affected if the run was continued. We see that
LADFN outperforms the benchmarks here with similar patterns as in column 1.

##  V Conclusion

Future autonomous vehicles must actively navigate to reduce their localization
and perception errors as well as optimize for conventional metrics such as
obstacle avoidance and time-efficient navigation. This paper presents a
principled method, based on deep RL, to synthesize navigation behaviors that
significantly reduce localization error for autonomous vehicles using state-
of-the-art LIDAR SLAM methods. Our key technical insight was to adopt a
hierarchical RL approach, which focuses on selecting drift-minimizing
waypoints that can be efficiently tracked by a low-level Stanley Controller
[4].

In future work, we plan to implement our perception-aware RL navigation
algorithm on a small mobile robot that operates in crowded, dynamic scenes.
Moreover, we plan to work towards theoretical guarantees based on a model
predictive control (MPC) framework where a robot must jointly optimize for
safe, time-optimal waypoint control as well as minimizing localization errors.
We intend to explore CEM-MPC based batch optimization over GPU [35] to
manifest complex behaviors in real-time.

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# Two-dimensional anisotropic non-Hermitian Lieb lattice

L. C. Xie  School of Physics, Nankai University, Tianjin 300071, China  H. C.
Wu  School of Physics, Nankai University, Tianjin 300071, China  X. Z. Zhang
<EMAIL_ADDRESS>College of Physics and Materials Science, Tianjin Normal
University, Tianjin 300387, China  L. Jin<EMAIL_ADDRESS>School of
Physics, Nankai University, Tianjin 300071, China  Z. Song  School of Physics,
Nankai University, Tianjin 300071, China

###### Abstract

We study an anisotropic two-dimensional non-Hermitian Lieb lattice, where the
staggered gain and loss present in the horizontal and vertical directions,
respectively. The intra-cell nonreciprocal coupling generates magnetic flux
enclosed in the unit cell of the Lieb lattice and creates nontrivial topology.
The active and dissipative topological edge states are along the horizontal
and vertical directions, respectively. The two-dimensional non-Hermitian Lieb
lattice also supports passive topological corner state. At appropriate
magnetic flux, the non-Hermiticity can alter the corner state from one corner
to the opposite corner as the non-Hermiticity increases. The gapless phase of
the Lieb lattice is characterized by different configurations of exceptional
points in the Brillouin zone. The topology of the anisotropic non-Hermitian
Lieb lattices can be verified in many experimental platforms including the
optical waveguide lattices, photonic crystals, and electronic circuits.

##  I Introduction

Over the last two decades, the ubiquitous effect of dissipation has proven to
induce astonishing non-Hermitian features, rather than just being an
inescapable nuisance. Within this young field, the treasure hunt is sprouting
into fascinating new directions ranging from the complex optical media Gupta
et al. (2021), non-equilibrium open systems with gain and/or loss Hu et al.
(2014); Kaiser et al. (2014); Mitrano et al. (2016); Cantaluppi et al. (2018);
Sato et al. (2019); McIver et al. (2020); Booker et al. (2020); Tindall et al.
(2020) to strongly correlated systems as a result of finite-lifetime
quasiparticles Zhang et al. (2017); Shen and Fu (2018); Yoshida et al. (2018);
Zhang et al. (2020a); Zhang and Song (2020); Nakagawa et al. (2020, 2021).
Most recently, topological characterization and dynamic control of non-
Hermitian models are hot areas of research Longhi (2019a); Kawabata et al.
(2019); Zhang and Song (2019); Luo and Zhang (2019); Yuce and Ramezani (2019);
Du et al. (2019); Ge et al. (2019); Lieu (2019); Jiang et al. (2020); Xu et
al. (2020); Wu et al. (2020); Liu et al. (2020); Xu and Chen (2020); Wojcik et
al. (2020); Kawabata et al. (2020a); Cui et al. (2020); Li et al. (2020a);
Zeng et al. (2020); Wu and An (2020); Guo et al. (2020); Zhang et al. (2020b,
c); Zhang and Gong (2020); Zhou et al (2020); Yokomizo and Murakami (2020);
Wang et al. (2021a); Lang et al. (2021); Xu et al. (2021); Rivero and Ge
(2021); Hou et al. (2021); Longhi (2021); Schiffer et al. (2021); Zhang et al.
(2021). Among the most relevant features observed in non-Hermitian systems,
the appearance of an interface significantly alters the entire spectrum,
leading to the exponential localization of all eigenmodes at the interface,
which goes beyond the expectations for Hermitian systems. This unique non-
Hermitian effect is dubbed as non-Hermitian skin effect Yao and Wang (2018).
As a consequence, the conventional bulk-boundary correspondence breaks down. A
correct description requires one to extend Bloch band theory into the
generalized Brillouin zone (BZ) Yao and Wang (2018); Zhang et al. (2020d);
Yang et al. (2020). Inspired by these exciting advances, different generalized
versions of bulk-boundary correspondence based on redefining the bulk
topological indices to incorporate the impact of non-Hermitian skin effect
have been proposed Imura and Takane (2019); Song et al. (2019a, b); Yokomizo
and Murakami (2019); Jiang et al. (2019); Yi and Yang (2020); Longhi (2019b);
Wang et al. (2019); Borgnia et al. (2020); Longhi (2020); Okuma et al. (2020);
Zhu et al. (2020a); Zeng and Xu (2020); Kawabata et al. (2020b); Silberstein
et al. (2020); Wang et al. (2020); Koch and Budich (2020). Furthermore, non-
Hermitian skin effect itself is also a topological effect manifested by the
spectral winding on the complex energy plane with a reference energy.
Importantly, the non-Hermitian topological phenomena have been observed
experimentally in various experimental platforms including the optical
waveguide lattices, photonic crystals, and electronic circuits Pocock et al.
(2019); Ezawa (2019); Zhu et al. (2020b); Li et al. (2020b); Yin and Peng
(2020); Qin et al. (2020); Sone et al. (2020); Xiao et al. (2020); Ota et al.
(2020); Sun and Zhou (2020); Zhu et al. (2020c); Zhang and Franz (2020); Wang
et al. (2021b); Parto et al. (2021).

The exceptional point (EP) uniquely presents in the non-Hermitian Hamiltonian
Berry (2004); Heiss (2012); Lee (2016); Miri and Alù (2019). At the EPs, the
eigenstate coalescence occurs. The order of the EP depends on the geometric
multiplicity of the corresponding eigenvalue. In general, the band touching
induces EPs in one-dimensional system. Intriguingly, the EPs are topologically
stable at generic points in the BZ in the sense that they will not disappear
suddenly, but move, split, and merge in the BZ until merging in pairs.

In this paper, we study an anisotropic two-dimensional (2D) non-Hermitian Lieb
lattice. The gain and loss present in the horizontal and vertical directions,
respectively; and induce the active and dissipative topological edge states.
The Lieb lattice supports passive topological corner state, which is created
at the appropriate cooperation between the non-Hermiticity and the magnetic
flux. The non-Hermitian term affects the corner states in a subtle way: when
the intra-cell nonreciprocal coupling induced magnetic flux is $\pi/2$, the
corner states are always localized at one corner; however, when the intra-cell
nonreciprocal coupling induced magnetic flux is $-\pi/2$, the corner states
slowly evolve into the opposite corner as the non-Hermiticity increases. The
type and configuration of EPs in the BZ distinguish the gapless phases. Our
findings shed light on the influence of non-Hermiticity for the 2D Lieb
lattice.

The remainder of the paper is organized as follows. In Sec. II, we introduce
the anisotropic 2D non-Hermitian Lieb lattice. Section III presents the
topology of non-Hermitian Lieb lattice and the topological edge states in the
gapped phase. The corner states under the influence of non-Hermiticity are
elaborated. Section IV highlights the enriched gapless band structures
characterized by the EPs. Our conclusion is summarized in Sec. V.

##  II 2D non-Hermitian Lieb lattice

We consider an anisotropic 2D non-Hermitian Lieb lattice. The Hamiltonian is
written in the form of

$\displaystyle H$ $\displaystyle=\sum\limits_{l,n}(\kappa
A_{l,n}^{\dagger}B_{l,n}+\kappa
B_{l,n}^{\dagger}C_{l,n}+imC_{l,n}^{\dagger}A_{l,n}$ $\displaystyle+\kappa
A_{l,n}^{\dagger}B_{l+1,n}+\kappa A_{l,n}^{\dagger}B_{l,n+1})+\mathrm{H.c.}$
$\displaystyle+i\gamma(A_{l,n}^{\dagger}A_{l,n}-C_{l,n}^{\dagger}C_{l,n}),$
(1)

where $A^{\dagger}$, $B^{\dagger}$, and $C^{\dagger}$ ($A$, $B$, and $C$)
denote the creation (annihilation) operators for the three sublattices in each
unit cell. The non-Hermitian Lieb lattice is schematically illustrated in Fig.
1. $\kappa$ and $m$ are the coupling strengths. The intra-cell nonreciprocal
coupling $im$ between the sublattices $A$ and $C$ induces effective magnetic
flux $\pi/2$ enclosed in the unit cell. The non-Hermiticity $\gamma$
originates from the balanced gain and loss that presented in the sublattices
$A$ and $C$ in the horizontal and vertical directions, respectively. The
proposed non-Hermitian Lieb lattice can be experimentally implemented in the
platforms of ultracold atomic gas in optical lattices, photonic crystals, and
coupled resonators based on the nowadays technology Y. X. Xiao et al. (2020);
Goldman et al. (2011); Guzmán-Silva et al. (2014); Vicencio et al. (2015);
Taie et al. (2015); Mukherjee et al. (2015); Whittaker et al. (2018).

Figure 1: Schematic illustration of the 2D non-Hermitian Lieb lattice. The
system consists of three sublattices denoted by green, black and red solid
circles, respectively. The non-Hermiticity arises from the gain in red and
loss in green. The black arrow indicates nonreciprocal coupling and each unit
cell is threaded by a magnetic flux. Figure 2: Phase diagram in the
$m$-$\gamma$ parameter space. The color regions show the three gapless phases
distinguished by the EPs in the BZ. In white regions show the gapped phases
distinguished by the Chern numbers.

Applying the Fourier transformation
$a_{l,n}=N^{-1}\sum_{\mathbf{k}}e^{i\mathbf{k\cdot r}}a_{\mathbf{k}}$
$(a=A,B,C)$ for the three sublattices, the Hamiltonian $H$ rewritten in the
Nambu representation reads

$H=\sum_{\mathbf{k}}H\left(\mathbf{k}\right)=\sum_{\mathbf{k}}\psi_{\mathbf{k}}^{\dagger}h\left(\mathbf{k}\right)\psi_{\mathbf{k}},$
(2)

where the basis is
$\psi_{\mathbf{k}}=[A_{\mathbf{k}},B_{\mathbf{k}},C_{\mathbf{k}}]^{T}$ and
$h\left(\mathbf{k}\right)$ is a $3\times 3$ matrix

$h\left(\mathbf{k}\right)=\left(\begin{array}[]{ccc}i\gamma&\kappa(e^{ik_{x}}+1)&-im\\\
\kappa(e^{-ik_{x}}+1)&0&\kappa(e^{ik_{y}}+1)\\\
im&\kappa(e^{-ik_{y}}+1)&-i\gamma\end{array}\right).$ (3)

The Hamiltonians in the momentum subspaces commute with each other
$[H\left(\mathbf{k}\right),H(\mathbf{k}^{\prime})]=0$.

At $m=\gamma=0$, the lattice is a standard Lieb lattice and supports a flat
band. In the absence of the gain and loss $\gamma=0$, the lattice has a flat
band and the band energy is tuned by the coupling strength $m$ Vidal et al.
(1998); Wu et al. (2007); Chalker et al. (2010); Apaja et al. (2010); Bermudez
et al. (2011); Bodyfelt et al. (2014); Julku et al. (2016); Shukla (2018). The
wave transport in the flat band is completely suppressed because of the
momentum-independent dispersion relation, leading to a strong localization of
the eigenstates. This provides an ideal platform to investigate various
interesting strongly correlated phenomena Goda et al. (2006); Maksymenko et
al. (2012); Peotta and Törmä (2015); Kauppila et al. (2016).

The presence of gain and loss drastically alters the spectrum and results in
exotic phenomena comparing to the standard Lieb lattice. We investigate
$h\left(\mathbf{k}\right)$ to show the insights of the non-Hermitian Lieb
lattice. The energy band properties of $h\left(\mathbf{k}\right)$ are
determined from solving the secular equation
$\mathrm{det}[h\left(\mathbf{k}\right)-E\left(\mathbf{k}\right)I]=0$, where
$I$ is the identity matrix. The algebra after the basis transformation shows a
cubic equation

$E^{3}\left(\mathbf{k}\right)+p(h_{x},h_{y})E\left(\mathbf{k}\right)+q(h_{x},h_{y})=0,$
(4)

where

$\displaystyle p(h_{x},h_{y})$ $\displaystyle=$
$\displaystyle\gamma^{2}-h_{x}^{2}-h_{y}^{2}-m^{2},$ (5) $\displaystyle
q(h_{x},h_{y})$ $\displaystyle=$ $\displaystyle
i\gamma(h_{y}^{2}-h_{x}^{2})+2h_{x}h_{y}m\sin[(k_{x}+k_{y})/2]$ (6)

with $h_{\epsilon}^{2}=|\kappa(e^{ik_{\epsilon}}+1)|^{2}$ ($\epsilon=x,$ $y$).
The energy bands and eigenstates are straightforwardly obtained through
solving the cubic equation.

##  III Topological Phases

Figure 3: Phase diagram in the $m$-$\gamma$ parameter space focused on the
gapped regions. Here $\kappa$ is taken as the unit without loss of generality.
In gapped regions, the system can be either in topologically trivial or non-
trivial phase characterized by the Chern number. $C=\pm 1$ indicates the Chen
number of the lowest band of the topologically non-trivial phase. Except for
the white region marked with $C=1$, all other white regions satisfy $C=0$. For
the topologically non-trivial phase, edge states exist for the system under
OBC in the $x$ or $y$ direction. Figure 4: Energy spectrum of the
topologically nontrivial phase under PBC in the $x$ direction, but under OBC
in the $y$ direction. Here $\kappa$ is taken as the unit without loss of
generality, and the total number of the unit cells $N$ is $10$. (a)-(b)
$m=-1,$ $\gamma=0.1$ depicts the topologically nontrivial phase with gapless
edge states. (c)-(d) the probability distribution of $|\psi_{1}\rangle$ and
$|\psi_{2}\rangle$ by supposing $k_{x}=1.4\pi$. Different colour bars indicate
different sublattices, respectively.

In this section, we discuss the topological phase of the system based on the
non-Hermitian topological band theory Shen et al. (2018). For the non-
Hermitian Hamiltonian, the separable energy band means that any two bands of
the system are not degenerate at any point $\mathbf{k}$ in the momentum space
(i.e., $E_{\alpha}(\mathbf{k})\neq E_{\beta}(\mathbf{k})$, $\alpha$, $\beta$
are the band index). The system experiences topological phase transition when
the closing of separable bands occurs.

For separable bands, the topology of the anisotropic 2D non-Hermitian Lieb
lattice can be characterized by the Chern number. Clearly, there is a fixed
magnetic flux in the unit cell because of the nonreciprocal coupling strength
between the next nearest neighbor, which breaks the time-reversal symmetry and
ensures the existence of the Chern number. For nonzero Chern number, the
topological edge states exist for the system under OBC in the $x$ or $y$
direction; for zero Chern number, the system has trivial edge states or no
edge states when imposing OBC in the $x$ or $y$ direction.

Separable bands and Chern number. Figure 3 is the phase diagram. The nonzero
Chern number are marked. The white regions except for the topologically
nontrivial region marked with $C=\pm 1$ are all topologically trivial phases
with the zero Chern number. The non-Hermiticity greatly affects the
topological property of the Lieb lattice. When the non-Hermitian term is
introduced, the gapped regions shrink and the gapless regions appear; a
typical feature is the existence of the EPs. Moreover, the large non-
Hermiticity destroys the nontrivial topology of the Lieb lattice as shown in
the phase diagram Fig. 2.

The Chern number of the separable energy band is well-defined as the integral
of Berry curvature on the entire Brillouin zone.

$C=\frac{1}{2\pi}\iint_{\mathrm{BZ}}\mathrm{d}k_{x}\mathrm{d}k_{y}\Omega,$ (7)

where $\Omega=\nabla\times\mathcal{A}$ with
$\mathcal{A}=-i\left\langle\varphi(\mathbf{k})\right|\nabla\left|\varphi(\mathbf{k})\right\rangle$
and $\left|\varphi(\mathbf{k})\right\rangle$ is the eigenstate of the band.
The Chern number evaluated under PBC predicts the number of gapless edge
states of the system under OBC Shen et al. (2018). In the topologically
nontrivial phase, the Chern number for the lower band is $C=\pm 1$, the Chern
number for the middle band is $C=0$, and the Chern number for the upper band
is $C=\mp 1$. In the topologically trivial phase, the Chern number for the
energy bands are all zeros.

As a consequence, the gapless edge states between the middle band and the
other bands exist for the Lieb lattice under OBC in the $x$ or the $y$
direction. Figure 4(a) depicts the energy bands when the system under PBC in
the $x$ direction, but under OBC in the $y$ direction. Figure 5(a) shows the
energy band when PBC is applied in the $y$ direction but OBC is applied in the
$x$ direction. The non-Hermitian Hamiltonian $h\left(\mathbf{k}\right)$ in the
momentum space has the particle-hole symmetry,
$\mathcal{T}_{-}h^{\ast}\left(\mathbf{k}\right)\mathcal{T}_{-}^{-1}=-h\left(-\mathbf{k}\right)$;
the unitary operator $\mathcal{T}_{-}$ is defined as
$\mathcal{T}_{-}=\mathrm{diag}\left(1,-1,1\right)$. Thus, the band energies
satisfy $E_{\mathbf{k}}=-E_{-\mathbf{k}}^{\ast}$ as reflected from Fig. 4(a).

Figure 5: Energy spectrum of the gapless edge states under PBC in the $y$
direction, but under OBC in the $x$ direction. The (a) real and (b) imaginary
parts of energy band at $m=-1,$ $\gamma=0.1$. The edge states (c)
$|\psi_{1}\rangle$ and (d) $|\psi_{2}\rangle$ at $k_{y}=1.4\pi$. Color sticks
indicate the probabilities for the corresponding sublattices. The total number
of unit cells $N$ is $10$ and $\kappa=1$.

Edge states. In the topologically nontrivial phase of the non-Hermitian Lieb
lattice, we examine the properties of the gapless edge states. The edge states
localized on the boundary in the $x$ and$\ y$ direction exhibit the gain and
loss, respectively. We discuss the $C=-1$ region as an example. The energy
spectra for the lattice under PBC in the $x$ direction and OBC in the $y$
direction are shown in Figs. 4(a)-4(b). Four edge states appear in pairs
within the band gap. Straightforward algebra shows that

$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle\sqrt{m^{2}-\gamma^{2}},$
(8) $\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\sqrt{4\kappa^{2}\cos^{2}(k_{x}/2)-\gamma^{2}/4}+i\gamma/2,$ (9)
$\displaystyle E_{3}$ $\displaystyle=$
$\displaystyle-\sqrt{m^{2}-\gamma^{2}},$ (10) $\displaystyle E_{4}$
$\displaystyle=$
$\displaystyle-\sqrt{4\kappa^{2}\cos^{2}(k_{x}/2)-\gamma^{2}/4}+i\gamma/2.$
(11)

The edge state energies between the middle and lowest bands satisfy
$E_{3}=-E_{1}$ and $E_{4}=-E_{2}^{\ast}$ under the particle-hole symmetry.
$E_{1}$ and $E_{3}$ are opposite in pair and independent of the momentum
$k_{x}$. $E_{2}$ and $E_{4}$ are complex with constant gain rate $i\gamma/2$.
The wavefunction of the edge state for $E_{\mu}$ is denoted as
$|\psi_{\mu}\rangle$ with $\mu=1,2,3,4$. We set the expression of the edge
states as
$|\psi_{\mu}\rangle=\left(\psi_{1A},\psi_{1B},\psi_{1C},\cdots,\psi_{NA},\psi_{NB},\psi_{NC}\right)$
with $N$ being the total number of the unit cells. To analytically obtain the
wavefunction of the edge states, we consider the lattice size at the
limitation of infinity large $N\rightarrow\infty$.

For the edge state $|\psi_{1}\rangle$, the component of $|\psi_{1}\rangle$ at
the sublattice $B$ is $\psi_{nB}=0$ with $n$ being the index of the unit cell
in the $y$ direction. The components of $|\psi_{1}\rangle$ in the first unit
cell at the bottom are
$\left(\psi_{1A},\psi_{1B},\psi_{1C}\right)=\left(1,0,e^{i\phi}\right)$ with
$\phi=$sgn$(m)\arccos(\gamma/m)$. The components of $|\psi_{1}\rangle$ in the
$n$-th unit cell satisfy the recursion relation
$\left(\psi_{nA},\psi_{nB},\psi_{nC}\right)=\rho^{n-1}\left(\psi_{1A},\psi_{1B},\psi_{1C}\right)$
with $\rho=-e^{-i\phi}-e^{-i(\phi+k_{x})}-1$. The probabilities of edge states
for the sublattices $A$ and $C$ in every unit cell are identical and decay
exponentially from bottom to top of the Lieb lattice as shown in Fig. 4(c).
For the edges state $|\psi_{3}\rangle$, we have similar wavefunction
distribution. The components of $|\psi_{3}\rangle$ in the first unit cell at
the bottom are
$\left(\psi_{1A},\psi_{1B},\psi_{1C}\right)=\left(1,0,e^{-i\phi}\right)$ and
decay as the index of the unit cells at the rate $\rho=-e^{i\phi}-e^{i(\phi-
k_{x})}-1$.

Figure 6: Corner states of the Lieb lattice at the energies $E_{1}$ and
$E_{3}$. (a) The dimerized structure of the corner state, being parity-time-
symmetric. (b) The probability distribution of the corner state. The
parameters are $\kappa=1$, $m=-1$, $\gamma=0.1$ and the lattice consists of
8$\times$8 unit cells.

For the edge state $|\psi_{2}\rangle$, the component of $|\psi_{2}\rangle$ at
the sublattice $C$ vanishes, $\psi_{nC}=0$. The components of
$|\psi_{2}\rangle$ in the first unit cell at the top are
$\left(\psi_{NA},\psi_{NB},\psi_{NC}\right)=\left(e^{i(\phi+k_{x}/2)},1,0\right)$
with $\phi=\arcsin(\gamma/[4\cos(k_{x}/2)])$ for $m>0$, and
$\pi-\arcsin(\gamma/[4\cos(k_{x}/2)])$ for $m<0$. The components of
$|\psi_{2}\rangle$ in the $n$-th unit cell satisfy the recursion relation
$\left(\psi_{nA},\psi_{nB},\psi_{nC}\right)=\rho^{N-n}\left(\psi_{NA},\psi_{NB},\psi_{NC}\right)$
with $\rho=-1-ime^{i(\phi+k_{x}/2)}$. The probabilities of the edge states for
the sublattices $A$ and $B$ in every unit cell are identical and decay
exponentially from top to bottom of the Lieb lattice as shown in Fig. 4(d).
For the edge state $|\psi_{4}\rangle$, the result is similar. The components
of $|\psi_{4}\rangle$ in the first unit cell at the top are
$\left(\psi_{NA},\psi_{NB},\psi_{NC}\right)=\left(-e^{i(k_{x}/2-\phi)},1,0\right)$
and decay as the index of the unit cells at the rate
$\rho=-1+ime^{i(k_{x}/2-\phi)}$ with $\phi=\arcsin(\gamma/[4\cos(k_{x}/2)])$
for $m<0$, and $\pi-\arcsin(\gamma/[4\cos(k_{x}/2)])$ for $m>0$.

Figure 7: The profiles of corner state in the real space for finite-size non-
Hermitian Lieb lattice affected by the non-Hermiticity. The 2D lattice
consists of 8$\times$8 unit cells. (a) and (f) The plots of energy levels
close to the corner states (red line) as functions of $\gamma$. $m=1$ for the
upper panels and $m=-1$ for the lower panels. (b, g) $\gamma=0$, (c, h)
$\gamma=0.4$, (d, i) $\gamma=0.5$, (e, j) $\gamma=0.8$. The other parameter is
$\kappa=1$.

The energy spectra for the lattice under PBC in the $y$ direction and OBC in
the $x$ direction are shown in Figs. 5(a)-5(b). There are still four edge
states in pairs within the band gap, in the form of

$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle\sqrt{m^{2}-\gamma^{2}},$
(12) $\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\sqrt{4\kappa^{2}\cos^{2}(k_{y}/2)-\gamma^{2}/4}-i\gamma/2,$
(13) $\displaystyle E_{3}$ $\displaystyle=$
$\displaystyle-\sqrt{m^{2}-\gamma^{2}},$ (14) $\displaystyle E_{4}$
$\displaystyle=$
$\displaystyle-\sqrt{4\kappa^{2}\cos^{2}(k_{y}/2)-\gamma^{2}/4}-i\gamma/2.$
(15)

$E_{1}$ and $E_{3}$ are independent of the momentum $k_{y}$ with
$E_{3}=-E_{1}$. $E_{2}$ and $E_{4}$ are complex with constant loss rate
$-i\gamma/2$ with $E_{4}=-E_{2}^{\ast}$. We analyze the wave functions of the
four edge states in the same way as above.

For the edge state $|\psi_{1}\rangle$, the component of $|\psi_{1}\rangle$ at
the sublattice $B$ is $\psi_{\upsilon B}=0$ with $\upsilon$ being the index of
the unit cell in the $x$ direction. The components of $|\psi_{1}\rangle$ in
the first unit cell at the right are
$\left(\psi_{NA},\psi_{NB},\psi_{NC}\right)=(e^{-i\phi},0,1)$ with
$\phi=$sgn$(m)\arccos(\gamma/m)$. The components of $|\psi_{1}\rangle$ in the
$\upsilon$-th unit cell satisfy the recursion relation $\left(\psi_{\upsilon
A},\psi_{\upsilon B},\psi_{\upsilon
C}\right)=\rho^{N-\upsilon}(\psi_{NA},\psi_{NB},\psi_{NC})$ with
$\rho=-1-e^{i\phi}-e^{i(k_{y}+\phi)}$. The probabilities of edge states for
the sublattices $A$ and $C$ in every unit cell are identical and decay
exponentially from right to left of the Lieb lattice as shown in Fig. 5(c).
For the edges state $|\psi_{3}\rangle$, we have similar wavefunction
distribution. The components of $|\psi_{3}\rangle$ in the first unit cell at
the right are $\left(\psi_{1A},\psi_{1B},\psi_{1C}\right)=(e^{i\phi},0,1)$ and
decay as the index of the unit cells at the rate
$\rho=-1-e^{-i\phi}-e^{i(k_{y}-\phi)}$.

For the edge state $|\psi_{2}\rangle$, the component of $|\psi_{2}\rangle$ at
the sublattice $A$ vanishes, $\psi_{\upsilon A}=0$. The components of
$|\psi_{2}\rangle$ in the first unit cell at the left are
$\left(\psi_{1A},\psi_{1B},\psi_{1C}\right)=(0,1,e^{-i(\phi+k_{y}/2)})$ with
$\phi=\arcsin(\gamma/[4\cos(k_{y}/2)])$ for $m>0$, and
$\pi-\arcsin(\gamma/[4\cos(k_{y}/2)])$ for $m<0$. The components of
$|\psi_{2}\rangle$ in the $\upsilon$-th unit cell satisfy the recursion
relation $\left(\psi_{\upsilon A},\psi_{\upsilon B},\psi_{\upsilon
C}\right)=\rho^{\upsilon-1}\left(\psi_{1A},\psi_{1B},\psi_{1C}\right)$ with
$\rho=-1+ime^{-i(\phi+k_{y}/2)}$. The probabilities of the edge states for the
sublattices $B$ and $C$ in every unit cell are identical and decay
exponentially from left to right of the Lieb lattice as shown in Fig. 5(d).
For the edge state $|\psi_{4}\rangle$, the result is similar. The components
of $|\psi_{4}\rangle$ in the first unit cell at the right are
$\left(\psi_{NA},\psi_{NB},\psi_{NC}\right)=(0,1,-e^{i(\phi-k_{y}/2)})$ and
decay as the index of the unit cells at the rate $\rho=-1-ime^{i(\phi-
k_{y}/2)}$ with $\phi=\arcsin(\gamma/[4\cos(k_{y}/2)])$ for $m<0$, and
$\pi-\arcsin(\gamma/[4\cos(k_{y}/2)])$ for $m>0$.

Figure 8: Complex energy band structures of the non-Hermitian Bloch
Hamiltonian $h(\mathbf{k})$. The system parameters are determined by the
representative points and traces the variation of EP in different region of
Fig. 2. The EPs projected on the $k_{x}$-$k_{y}$ plane: (a)
$m=1,\gamma=0.388$, (b) $m=1,\gamma=0.5$, (c) $m=0.5,\gamma=0.5$, (d)
$m=1,\gamma=1$, (e) $m=1,\gamma=2$, (f) $m=1,\gamma=3$, (g) $m=1,\gamma=4$,
(h) $m=1,\gamma=4.162$, (i) $m=-\sqrt{2},\gamma=0$ and (j)
$m=\sqrt{2},\gamma=0$. The other system parameter is $\kappa=1$. The solid
black dots, red black dots and black circles represent EP2, EP3 and DP2
respectively. Notably, the variation of $\gamma$ can induce the splitting and
the merging of the different types of EP so that the system can exhibit rich
structure.

From Figs. 4(c) and 5(c), the edge state energies are independent of the
momentum and the edge states locate on the sublattices $A$ and $C$ for the
non-Hermitian Lieb lattice under open boundary in either the $x$ or the $y$
direction. If the boundaries of the non-Hermitian Lieb lattice on both the $x$
and the $y$ directions are open, the corner state appears at the bottom-right
of the non-Hermitian Lieb lattice. The structure of the corner state is
schematically illustrated in Fig. 6(a) and the probability distribution of the
corner state is shown in Fig. 6(b). The destructive interference plays the
crucial role to form the dimerized pattern of the corner state. In the
Hermitian case with zero $\gamma$, numerical simulation for the finite system
shows that there exist two corner states with energy
$E_{c}=\pm\left|m\right|$. The corresponding corner states appear at the
bottom-right of the lattice as shown in Figs. 7(b) and 7(g), where the
sublattice $B$ is unoccupied. For the case with nonzero $\gamma$, the non-
Hermitian term affects the corner states in a subtle way. Numerical results
are plotted in Fig. 7, the influence of the gain and loss is demonstrated.

We find that (i) for $\gamma/m<0$, the magnetic flux enclosed in the unit cell
is $\pi/2$. The corner states always appear at the bottom-right of the
lattice, and become more localized with the increase of non-Hermiticity
$\left|\gamma\right|$. while (ii) for $\gamma/m>0$, the magnetic flux enclosed
in the unit cell is $-\pi/2$. The corner state becomes more extended as
$\left|\gamma\right|$ increases, and the bottom-right corner states slowly
evolve into the top-left corner state. A new corner state appears at the top-
left of the lattice as $\left|\gamma\right|$ across about $0.5$.
Interestingly, the occupation of the $B$ sublattice is dominant in this case.

##  IV Band structure of gapless phase characterized by the EPs

In this section, we investigate the band structure of the non-Hermitian Lieb
lattice, where the existence of the EP in the spectrum is featured for the
gapless phase. The EP is unique for the non-Hermitian physics and is
associated with the level coalescence, where not only the eigen energies but
also the eigenstates become the same. Many interest effects without Hermitian
counterparts arise around the EP, ranging from the square root frequency
dependence Peng et al. (2014), the nontrivial topological property resulting
from the Riemann sheet structures Rotter and Bird (2015); Zhou et al. (2018),
to unidirectional reflectionless and coherent perfect absorption Zhang et al.
(2013); Jin et al. (2016); Parto et al. (2021).

To calculate the EP of the non-Hermitian Lieb lattice, we define $\Delta$ as
the discriminant of the cubic equation (4). The real and imaginary parts of
$\Delta$ read

$\displaystyle\mathrm{Im}(\Delta)$ $\displaystyle=$ $\displaystyle m\gamma
h_{x}h_{y}\sin[(k_{x}+k_{y})/2](h_{y}^{2}-h_{x}^{2}),$ (16)
$\displaystyle\mathrm{Re}(\Delta)$ $\displaystyle=$ $\displaystyle
p^{3}/27+\mathrm{Re}(q^{2})/4.$ (17)

$\Delta=0$ with nonzero $\gamma$ signifies the EP in the spectrum, where the
energy bands are contacted. The phase diagram for the band structure
determined from Eqs. (16)-(17) is depicted in Fig. 2. The non-Hermitian system
is gapless when the energy bands are inseparable. In the gapless phase, at
least two of the bands touch and $\mathbf{k}_{\mathrm{EP}}$ is the EP in the
BZ; besides, the EPs might present in different energy bands at the same
momentum $\mathbf{k}_{\mathrm{EP}}$. The gapless phase is divided into three
regions in terms of the types of EPs rather than the exceptional ring Pie et
al. (2016), denoted as yellow, green, and cyan. In the yellow regions, there
are four EP2s (two-state coalescence); six EP2s and two EP3s (three-state
coalescence) present in the green regions; and the cyan regions have four
EP2s. The EP merge or split when the system parameters cross the solid lines
as the boundaries between colored regions.

Figure 9: Energy bands structures of the non-Hermitian Hamiltonian in the
momentum space as the counterpart of Fig. 8. In all panels, the vertical axis
represents the real part of eigenvalues and the color indicates the imaginary
part of eigenvalues. (a) $m=1,\gamma=0.388$, (b) $m=1,\gamma=0.5$, (c)
$m=0.5,\gamma=0.5$, (d) $m=1,\gamma=1$, (e) $m=1,\gamma=2$, (f)
$m=1,\gamma=3$, (g) $m=1,\gamma=4$, (h) $m=1,\gamma=4.162$, (i)
$m=-\sqrt{2},\gamma=0$ and (j) $m=\sqrt{2},\gamma=0$. The other system
parameter is $\kappa=1$.

We take $\kappa=1$ as an illustration and analytically determine the EPs and
the boundaries within the gapless phases from Im$(\Delta)=0$ and
Re$(\Delta)=0$. The EPs may appear at $k_{x}=\pi$, $k_{y}=\pi$, and
$\left|k_{x}\right|=\left|k_{y}\right|$.

(i) The EP2s appear at $k_{x}=k_{y}$, we obtain $h_{x}^{2}=h_{y}^{2}$ and
$\mathrm{Im}(\Delta)=0$. The cyan and green regions in the phase diagram have
EP2s in this case, where the energies at the EP2s are real with
$\mathrm{Im}(E)=0$. From Re$(\Delta)=0$, we obtain

$4(\gamma^{2}-m^{2}-2h_{x}^{2})^{3}=27h_{x}^{6}m^{2}(h_{x}^{2}-4),$ (18)

which determines the green and cyan regions in Fig. 2. The purple boundary and
the boundary $\left|\gamma\right|=\left|m\right|$ in red enclose the cyan
region with four EP2s, but the blue boundary and the boundary
$\left|\gamma\right|=\left|m\right|$ enclose the green region with two EP2s.
The representative points in the two regions are shown in Fig. 8(b) and Fig.
8(e) respectively. We notice that on the dashed line $k_{x}=k_{y}$, the former
has four EP2s, and the latter has only two. At the boundary, the number of
EP2s will be reduced due to merging. In Fig. 8(a) and Fig. 8(d), we notice
that the four EP2s in the cyan region merge into two at the purple or red
boundaries. In Fig. 8(c) and Fig. 8(f), we notice that the two EP2s in the
green region merge into one EP at the blue or black boundaries.

(ii) The EP3s appear at $k_{x}=-k_{y}$, we obtain $\sin(k_{x}+k_{y})/2=0$ and
$\mathrm{Im}(\Delta)=0$. From Re$(\Delta)=0$, we obtain

$\gamma^{2}-m^{2}=2h_{x}^{2},$ (19)

which determines the green region. The boundary $\gamma^{2}-m^{2}=8$ in blue
and the boundary $\left|\gamma\right|=\left|m\right|$ enclose the green region
with two EP3s. This indicates that the green region has EP3s in addition to
EP2s. The EP3s are the red dots in Fig. 8(e), where the three band coalesce at
zero energy $E=0$. Two EP3s merge to one at $(k_{x},k_{y})=(0,0)$ in Fig. 8(f)
at the blue boundary and one at $(k_{x},k_{y})=(\pi,\pi)$ in Figs. 8(c)-(d) at
$\left|\gamma\right|=\left|m\right|$. The non-Hermitian Lieb lattice does not
hold the anti-unitary symmetries and the EP3s are not topological stable
Delplace et al. (2016); Kawabata et al. (2016); Bessho et al. (2016).

(iii) The EP2s appear at $k_{x}=\pi$ or $k_{y}=\pi$, which leads to $h_{x}=0$
or $h_{y}=0$ and $\mathrm{Im}(\Delta)=0$. From Re$(\Delta)=0$, we obtain

$4(\gamma^{2}-m^{2}-h_{x}^{2})^{3}=27\gamma^{2}h_{x}^{4},$ (20)

which determines the green and yellow regions of the phase diagram in Fig. 2.
The boundary $(\gamma^{2}-m^{2}-4)^{3}=108\gamma^{2}$ in green and the
boundary $\left|\gamma\right|=\left|m\right|$ enclose these two regions with
four EP2s. At these four EP2s, the coalesced band energies are imaginary with
Re$(E)=0$. The representative configurations are shown in Figs. 8(d)-8(h),
showing the movement and merging of EPs. Four EP2s move along the two dashed
lines $k_{x}=\pi$ and $k_{y}=\pi$ in Figs. 8(e)-8(g) until they merge into two
EP2s at $(k_{x},k_{y})=(0,\pi),(\pi,0)$ at the green boundary in Fig. 8(h) or
merge as one EP at $(k_{x},k_{y})=(\pi,\pi)$ at
$\left|\gamma\right|=\left|m\right|$ in Fig. 8(d). The energy bands for Fig. 8
are shown in Fig. 9, where the bulk Fermi-arc is observed.

In the phase diagram, the cyan region has four EP2s at $k_{x}=k_{y}$; the
green region has two EP2s at $k_{x}=k_{y}$, two EP3s at $k_{x}=-k_{y}$, and
four EP2s at $k_{x}=\pi$ and $k_{y}=\pi$; the yellow region has four EP2s at
$k_{x}=\pi$ and $k_{y}=\pi$. Under the OBC, the edge states exist in the
gapless region in cyan; they are the remnant of topological features inherited
from the Hermitian Lieb lattice; however, the edge state does not exist in the
gapless regions in green and yellow. At even larger gain and loss, the non-
Hermitian Lieb lattice enters the trivial phase.

##  V Conclusion

To conclude, we have proposed an anisotropic 2D non-Hermitian Lieb lattice,
which has gain and loss along the horizontal and vertical directions,
respectively. The nonreciprocal intra-cell coupling creates non-trivial
topology. The gain and loss result in active and dissipative topological edge
states with net gain and net loss, respectively. The non-Hermitian Lieb
lattice also supports passive topological edge states. Interestingly, the gain
and loss can alter the localization position of the corner state. When the
magnetic flux enclosed in the unit cell is $\pi/2$, the corner states are
always in one corner of the lattice; however, when the magnetic flux enclosed
in the unit cell is $-\pi/2$, the corner states slowly evolve into the
opposite corner as the non-Hermiticity increases. The interplay between the
magnetic flux and non-Hermiticity produces rich band structures featured from
different types of EPs. The topological properties and rich band structures of
the non-Hermitian Lieb lattice benefit our understanding of three-band lattice
and robust non-Hermitian transport.

## Acknowledgement

We acknowledge the support of the National Natural Science Foundation of China
(Grants No. 11975128, No. 11975166, and No. 11874225). X.Z.Z. is also
supported by the Program for Innovative Research in University of Tianjin
(Grant No. TD13-5077).

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