[1]

[1]This work is partially funded under the project "Advanced driver assistance
system (ADAS) for precision maneuvers with single-body and articulated urban
buses", co-financed from the European Union from the European Regional
Development Fund within the Smart Growth Operational Programme 2014-2020
(contract No. POIR.04.01.02-00-0081/17-01).

[type=editor, auid=000,bioid=1, orcid=0000-0003-0097-6612] [1]

Concept, Software Development, Experiments, Writing

[orcid=0000-0003-0975-6077]

Data generation, Control algorithm, Fruitful discussions

Data acquisition and preprocessing, Real-world maps generation

Writing, Methodology, Fruitful discussions

[type=editor, auid=000,bioid=4, orcid=0000-0002-9843-2404]

Writing, Concept of the experiments, Methodology, Fruitful discussions

[cor1]Corresponding author

# Learning from Experience for Rapid Generation of Local Car Maneuvers

Piotr Kicki<EMAIL_ADDRESS>Institute of Robotics and Machine
Intelligence, Poznań University of Technology, ul. Piotrowo 3A, 60-965,
Poznań, Poland  Tomasz Gawron<EMAIL_ADDRESS>Krzysztof Ćwian
<EMAIL_ADDRESS>Mete Ozay<EMAIL_ADDRESS>Piotr Skrzypczyński
<EMAIL_ADDRESS>
###### Abstract

Being able to rapidly respond to the changing scenes and traffic situations by
generating feasible local paths is of pivotal importance for car autonomy. We
propose to train a deep neural network (DNN) to plan feasible and nearly-
optimal paths for kinematically constrained vehicles in small constant time.
Our DNN model is trained using a novel weakly supervised approach and a
gradient-based policy search. On real and simulated scenes, and a large set of
local planning problems, we demonstrate that our approach outperforms the
existing planners with respect to the number of successfully completed tasks.
While the path generation time is about 40 ms, the generated paths are smooth
and comparable to those obtained from conventional path planners.

###### keywords:

motion planning neural networks robotics autonomous driving reinforcement
learning autonomous vehicle navigation

##  1 Introduction

Self-driving cars utilized in city scenarios often require to quickly plan
local maneuvers taking into consideration the constraints imposed by the
kinematics of cars and geometry of scene perceived by vehicle’s sensors.
Although state-of-the-art path planning methods are considered sufficient for
autonomous vehicles [20], their ability to handle highly constrained planning
cases comes at a computational cost that is too high for rapid maneuver
planning. In contrast, parking or overtaking maneuvers are relatively easy for
experienced human drivers, who intuitively generate nearly optimal paths of
their vehicles in a short time horizon, exploiting the prior experience on how
to perform these maneuvers [21]. Unfortunately, conventional path planners
[27] do not integrate the prior experience that comes from learning on similar
cases solved before.

Figure 1: A conceptual scheme of the rapid path generation system. We use
gradient-based policy search for training the proposed neural network to plan
feasible paths.

Therefore, in this paper, we investigate how to learn a path planning policy,
which is then used to rapidly generate feasible and nearly-optimal local paths
for typical maneuvers of a car-like vehicle, being subject to severe kinematic
constraints. For this purpose, we propose a deep neural network, which
generates a feasible path by approximating the oracle planning function
considering the representation of the task (Figure 1). Our approach is not
intended to be a standalone planner, as it is not complete. It should be
invoked by a higher-level planner to speed up operation in constrained
scenarios similar to those already seen. The proposed neural network model is
trained using a gradient-based policy search approach [11] in a reinforcement
learning framework with a differentiable loss function. Although we use
reference paths in the evaluation of the cost function, these paths are
generated automatically by a complete path planning algorithm and are used
only to retreat from the collision areas. Hence they do not constitute a
performance upper bound for the proposed method, which can find paths that are
shorter and/or smoother than the reference ones. In principle, any planning
algorithm able to handle the constraints present in our problem can be applied
to generate the reference paths. In the experiments demonstrated in this
article, we have chosen a variation of the State Lattices method [31] due to
its practical advantages, such as reliability and fast generation of feasible
plans. The main contribution of this paper is twofold:

  1. 1.

An approach for rapid path generation under differential constraints by
approximating the oracle planning function using a novel neural network.

  2. 2.

A novel differentiable loss function which penalizes infeasible paths, since
they violate constraints imposed either by the vehicle kinematics or
environment maps.

Moreover, we propose a dataset of urban environment local maps (based on real
sensory data) and motion planning scenarios that can be used for training and
evaluation of local planners for self-driving cars111Both code and the dataset
are available at https://github.com/pkicki/neural_path_planning. Using this
dataset, we demonstrate the generalization abilities of the learned model of
our network, illustrate its advantages over selected state-of-the-art planning
algorithms, and quantitatively analyze the accuracy of the proposed solution,
planning times, and properties of the generated paths.

##  2 Related Work

Early works on self-driving cars used search-based algorithms, such as the A∗,
in a quantized space [12]. As path planning for car-like vehicles involves
high dimensional search spaces, sampling-based algorithms are considered, due
to their computational efficiency [24]. The basic Rapidly exploring Random
Trees (RRT) algorithm provides no guarantee of the path quality, but RRT∗ [24]
extends this algorithm to an asymptotically optimal version, while Informed
RRT∗ [15] incorporates a heuristic into the RRT∗ algorithm to improve the
convergence rate. The more recent Stable Sparse Random Trees [28] uses a much
more sparse tree, still being asymptotically optimal. Batch Informed Trees
(BIT∗) [17] is an asymptotically optimal anytime algorithm, which quickly
finds feasible solutions, and if the computational time permits, then
converges towards the optimal one. Its latest modification, Advanced BIT∗
[35], applies advanced search techniques to balance between exploration and
exploitation of the sampled state space. An alternative solution is the
Adaptively Informed Trees (AIT∗) algorithm [34]. AIT∗ uses asymmetric
bidirectional search to exploit a problem-specific heuristic, which is updated
during the search. A number of the recent sampling-based algorithms are
implemented in the Open Motion Planning Library (OMPL) [36]. Unfortunately, if
nonholonomic constraints and steering angle limits are present in the vehicle
kinematics, then the performance of sampling-based methods degrades [25]. In
contrast, the State Lattices algorithm [31] is relatively fast while planning
paths for vehicles with nonholonomic constraints. However, if Dubins curves
are applied as motion primitives, then the vehicles have to stop to make
turns, whereas using smoother primitives results in a much higher
computational cost.

Planning frameworks that use optimization and control methods have been
proposed to cope with moving objects and motion prediction. A planner for
aggressive car maneuvers based on a version of RRT∗ and the non-linear Model
Predictive Control method was presented in [1], while a Constrained Iterative
Linear Quadratic Regulator was used in [8] to solve a planning problem with
moving obstacles for on-road driving. Unfortunately, these methods are
computationally intensive and expensive for development and maintenance,
because they require tedious manual parameter tuning for different scenarios.

The time-consuming manual tuning process implemented in the search and
optimization-based path or trajectory planning methods is a consequence of the
fact that none of these methods can exploit the similarity between the task at
hand and the tasks solved before. However, there are works that leverage
previous experience in motion planning for manipulation. An Experience Graph
which represents the underlying connectivity of the space is constructed from
the previous planning episodes in [30] to accelerate motion planning whenever
possible. Demonstrated trajectories are used in [23] to quickly predict
feasible trajectories for novel scenarios in manipulation. This approach
applies a situation descriptor to find similar tasks and descriptions of the
trajectories in the task space. Similarly, entire paths are stored, and then
are re-used or repaired the existing ones to reduce the planning time in the
manipulation in [4].

The success of deep learning in robotics [37] increased the interest in path
and motion planning methods entirely learned from data. Driving data collected
from demonstrations of a human expert can be used to learn a driving policy in
urban scenarios with deep learning, as shown in [7]. A convolutional neural
network model is trained to predict ego vehicle’s future trajectory points
taking as input a bird-view image obtained from the CARLA simulator. In real-
world scenarios, a perception module that transforms raw sensory data into the
symbolic bird-view image has to be deployed on the input of this system, thus
the final performance depends on this module to much extent. A jointly
learnable behavior and trajectory planner for self-driving vehicles was
proposed in [33]. This planner learns a shared cost function used to select
behaviors that handle traffic rules and generate vehicle trajectories
employing a continuous optimization solver. Nevertheless, it learns from a
database of human driving recordings rather than from its own experience.

The deep learning paradigm has been used in the context of learning from
previous experience in manipulation tasks. The recent paper [6] tackles the
problem of learning from previous experience to boost path planning in high-
dimensional continuous state and action spaces. It extends a tree-based
sampling algorithm by introducing learnable neural components that bias the
sampling towards more promising regions. Recurrent neural networks were
employed in the OracleNet [3] to generate new paths using paths obtained from
demonstrations. While this approach was an inspiration for our work, it
differs significantly from the planner presented in this paper, because it
does not use any environment model, relying on the demonstrated paths. Such an
approach is rather infeasible for vehicles in urban scenarios, where the state
space may differ considerably for particular maneuvers and local scenes.
Therefore, we use the reinforcement learning schema instead of direct
demonstrations. The OracleNet concept was then extended to the Motion Planning
Networks (MPNet) framework [32], which generates collision-free paths for the
given start and goal states (configurations) directly taking point clouds as
input. MPNet generalizes to unseen environments, but it still learns through
imitation. The StrOLL algorithm [5] that tackles the problem of lazy search in
graphs for path planning also uses the notion of “oracle”, but it learns
planning policies by imitating an oracle that has full knowledge about the
environment map at training time, and can compute optimal decisions. This
algorithm does not consider non-holonomic constraints or path smoothness, thus
it is not directly applicable to our problem.

##  3 Problem Definition

In this paper, we focus on path planning for a kinematic car with body modeled
as a rectangle (Figure 2) with typical kinematic constraints; no lateral and
longitudinal slip, and limited steering angle $\beta$. Thus, the robot state
can be defined by
$q=\begin{bmatrix}\beta&\theta&x&y\end{bmatrix}^{T}\in\mathcal{Q}$, where
$\theta$, $x$ and $y$ define vehicle orientation and position in the global
coordinate frame. We consider a problem of planning a feasible monotonic,
curvature-continuous path $\mathcal{P}$ from an initial state
$q_{0}\in\mathcal{F}$ to a subset $\mathcal{Q}_{k}\subset\mathcal{F}$ of
desired configurations (which are visualized by a green rectangle in Figure
2), where $\mathcal{F}\subset\mathcal{Q}$ is a free subspace of the
configuration space $\mathcal{Q}$. The vehicle is controlled using an input
vector $u=\begin{bmatrix}\zeta&v\end{bmatrix}^{T}$, where $\zeta$ is angular
velocity of the virtual steering angle, and $v$ is linear velocity of the
guiding point $\Pi_{0}$. The vehicle’s kinematics can be written by

$\dot{q}=\begin{bmatrix}\dot{\beta}\\\ \dot{\theta}\\\ \dot{x}\\\ \dot{y}\\\
\end{bmatrix}=\begin{bmatrix}1&0\\\ 0&\frac{\tan\beta}{L}\\\ 0&\cos\theta\\\
0&\sin\theta\\\ \end{bmatrix}\begin{bmatrix}\zeta\\\ v\end{bmatrix}.$ (1)

We consider a path $\mathcal{P}$ to be feasible when it is collision-free
concerning the whole vehicle body (which is depicted by a blue rectangle in
Figure 2), and it has limited curvature, so that it can be followed by a car
with a limited steering angle, and ends in the subset of desirable
configurations $\mathcal{Q}_{k}$. In our setup, the environment $E$ is modeled
with a $128\times 128$ occupancy grid with resolution of
$0.2\text{\,}\mathrm{m}$, thus we consider a
$$25.6\text{\,}\mathrm{m}$\times$25.6\text{\,}\mathrm{m}$$ local map.

Figure 2: A scheme of the proposed problem. Planning a monotonic and feasible
path from an initial state
${q_{0}=\begin{bmatrix}\beta_{L0}&\theta_{L0}&x_{L0}&y_{L0}\end{bmatrix}^{T}}$
to a subset $\mathcal{Q}_{k}$ of desired configurations (the green rectangle).
Depicted exemplary solution is a spline built with the 5-th degree polynomials
defined in the local coordinate systems associated with the segment’s
endpoints.

##  4 Proposed Solution

The general scheme of the proposed solution to the problem described in
Section 3 is presented in Algorithm 1. Crucial parts of this algorithm, such
as representation of the path $\mathcal{P}$, construction of the policy
$\pi_{\phi}$ as well as the loss function $\mathcal{L}$ and the data used to
train the proposed system are described in the following subsections.

1 Given: Map $E$, initial configuration $q_{0}$, desired configuration
$q_{d}$, initial policy parameters $\phi$;

2 for  _$i\leftarrow 1$ to $N$_ do

3 Evaluate policy $\pi_{\phi}(E,q_{0},q_{d})$ to obtain parameters $S_{i}$ of
the $i$-th segment endpoint;

4 Perform virtual nominal move along the path segment and update $q_{0}$ to
the new vehicle configuration;

5

6 end for

7Parameters $S$ determined by the policy $\pi_{\phi}$ define a path
$\mathcal{P}$;

8 Calculate the loss $\mathcal{L}$ of the path $\mathcal{P}$;

9 if  _training_ then

10 Calculate the gradient of $\mathcal{L}$ with respect to $\phi$;

11 Update policy parameters $\phi$;

12

13 end if

14

Algorithm 1 Our proposed path planning algorithm.

###  4.1 Path Representation

In the proposed solution, a path $\mathcal{P}$ is represented by a spline of
the 5-th degree polynomials as they have a sufficient level of smoothness, so
they may be tracked by a car with a continuous steering angle. The number of
spline segments ($N$) used to form a path is a free parameter, which depends
on the length and complexity of resulting plans. This parameter is a trade-off
between the ability to represent complex maneuvers and the path computation
time, as it scales linearly with it.

In order to represent the resulting path as a level-curve, which is required
by the control algorithm [18], we defined subsequent segments by functions of
the form $y=f_{i}(x)=\sum_{j=0}^{5}a_{ij}x^{j}$ for
$i\in\\{0,1,\ldots,N-1\\}$, where $a_{ij}$ are the polynomial coefficients,
and both $x$ and $y$ are expressed in the local coordinate frame $L_{i}$
associated with the $i$-th segment endpoint (see Figure 2). Our formulation
enables us to represent paths which are not the functions of the $x$
coordinate. However, it limits the maximum possible turn range during the
maneuver to $N\frac{\pi}{2}$\text{\,}\mathrm{rad}$$, which is enough for most
of the typical local car maneuvers, for sufficiently large $N$.

Assuming that we know the initial state $q_{0}$, we can define the
aforementioned spline using a $N\times 4$ matrix $S$ of parameters of segment
endpoints. Each row of the matrix defines a 2D position $(x,y)$ of an endpoint
as well as the first $\frac{dy}{dx}\rvert_{(x,y)}$ and the second derivative
$\frac{d^{2}y}{dx^{2}}\rvert_{(x,y)}$ of the path at that point. All
parameters in the $i$-th row of the $S$ are expressed with respect to the
$i-1$-th local coordinate system $L_{i-1}$.

###  4.2 Policy Approximation

To solve the problem described in Section 3, instead of applying a search
algorithm to every single problem instance, we rather reformulate the problem
by a Markov Decision Process (MDP), and we aim to find a policy
$\pi_{\phi}(E,q,q_{k})$, parameterized with parameters $\phi$, which is able
to solve a large group of instances of this problem. In the MDP formalism, the
agent (a vehicle in our case) takes states as input, and generates actions
that influence immediate rewards, and subsequently, future states and rewards.
We define the MDP by specifying;

  * •

the state space ${\cal S}$ that contains all possible states of the
environment $s\in{\cal S}$,

  * •

the action space ${\cal A}$ that contains all possible actions the agent can
execute $a\in{\cal A}$, and

  * •

the immediate reward $r(s,a)$ received by the agent if it takes the action $a$
in the state $s$.

A typical MDP [38] is defined as a tuple $\langle{\cal S},{\cal
A},P,r\rangle$, where $P(s_{t+1}|s_{t},a)$ represents the conditional
probability of transition to the next state $s_{t+1}$ given that a specified
action is taken at the current state $s_{t}$ under the action $a$. In the path
planning problem, we can assume that the transitions are deterministic, as
long as the planned paths are feasible. Therefore, in our formulation, the
states consist of the current $q$ and the desired (goal) $q_{k}$
configurations of a vehicle, and the environment (scene) map $E$. Within a
planning episode (i.e. a single maneuver) the map and the goal state remain
unchanged, while the current state evolves along the planned path. The actions
are defined as path segments (see line 1 of Algorithm 1), transitions are
deterministic moves along the planned path segments (see line 1 of Algorithm
1), while the rewards are dependent on the collisions, path curvature, and
reaching the goal (see line 1 of Algorithm 1). Considering the deterministic
transitions, we limit the scope of our work to employing a family of
deterministic policies. Hence, one can see that these policies are in fact
functions, which we call planning functions.

Let’s consider a special type of planning function, which returns a feasible
path for any input if such a path exists. We will call this special function
an oracle planning function. Knowing and being able to evaluate such a
function enables us to solve all planning problems that are covered by our
definition. Although it is impossible to give a closed formula for such a
function in general, the abilities of human drivers to learn how to steer
effectively during rapid local maneuvers suggest that the oracle planning
function may be approximated in some range of its arguments, namely, for short
paths in a set of environments drawn from similar distributions, using the
prior experience. This approach transfers the problem of planning into the
problem of approximation of the oracle planning function.

Neural network models can approximate nonlinear functions of many variables
when they are trained using statistically sufficient training data [22, 10].
Moreover, their inference time is usually short and stable between calls.
Therefore, we propose to learn a deterministic policy $\pi_{\phi}(E,q,q_{k})$
which is parameterized with a neural network, and to use it as the estimate of
the oracle planning function in the rapid path generation problem. In our
considerations, the action is defined by the parameters of the segment
endpoint $S_{i}$, and the reward is differentiable, as it is the loss function
defined in Section 4.3 taken with the minus. Moreover, we assume that
throughout an episode, both the environment representation $E$ and the desired
configurations remain fixed, and the number of steps within an episode is
equal to $N$.

We propose a neural network with the architecture presented in Figure 3. It
consists of 3 main parts: (i) map processor, (ii) configuration processor, and
(iii) parameter estimator. The map processor consists of a series of
convolutions, max-pooling layers, and fully connected layers. The map
processor takes the map as an input and produces the latent representation of
the environment. Similarly, configuration processor transforms the
representations of both actual ($x,y,\sin\theta\cos\theta,\beta$) and desired
($x_{k},y_{k},\sin\theta_{k},\cos\theta_{k}$) configurations using a sequence
of fully connected layers. Then, latent representations are concatenated and
based on this, the segment endpoint parameters are determined. Each parameter
has its own network head, and they differ only in terms of the last layer,
where for $y$, $\frac{dy}{dx}\rvert_{(x,y)}$, and
$\frac{d^{2}y}{dx^{2}}\rvert_{(x,y)}$ heads, there is no activation. For $x$
head, there is a sigmoid at the end, and its output is scaled $10$ times in
order to set maximal length of the single path segment to
$10\text{\,}\mathrm{m}$ and stabilize gradient propagation. Except for these
layers, the rest of the fully connected layers use $\tanh$ non-linearity,
while convolutional layers utilize ReLU activations. Such a network calculates
the parameters of a single segment endpoint. Thus, in order to generate a
spline made of $N$ polynomial segments, we perform $N$ inferences by changing
the representation of the actual state $q$ each time, according to the
previously determined segment.

Figure 3: Architecture of the proposed neural network which is used to
generate path segments.

###  4.3 Our Proposed Differentiable Loss Function

To train the proposed neural network, we need to define a loss function, which
will drive our model towards some oracle planning function. The probably
simplest solution is to use Imitation Learning to copy the behavior of a
complete planning algorithm on multiple path planning tasks. However, this
approach has several important drawbacks. First, the usage of a planning
algorithm requires a significant amount of time to generate training data.
Moreover, the performance of the chosen planner constitutes an upper bound on
the proposed planning policy, which results in paths far from optimal paths,
or increases the time needed to create the training data. Furthermore, the
notion of error in supervised learning drives the solutions towards the
planned paths without taking care of their actual feasibility, which may be an
issue for models with limited capacity, which usually make some errors. Such a
situation is schematically illustrated in Figure 4.

Figure 4: The path $\mathcal{P}_{2}$ has lower error in terms of imitation
learning (lies closer and is more similar to the reference path
$\mathcal{P}_{ref}$) than the path $\mathcal{P}_{1}$. However, the path
$\mathcal{P}_{2}$ is infeasible, in contrast to $\mathcal{P}_{1}$. Such a
situation can be omitted using Reinforcement Learning.

Due to the aforementioned drawbacks of the imitation learning, we exploited
the Reinforcement Learning paradigm in the form of Gradient-based Policy
Search [14]. Thus, we constructed a differentiable loss function, which
encourages the model to act as an oracle planning function – generate feasible
paths, to learn the desired behavior.

The proposed loss function has four components:

  * •

Collision loss $\mathcal{L}_{coll}$, which is used to force the planner to
produce collision-free paths.

  * •

Curvature loss $\mathcal{L}_{curv}$, which is used to ensure that the produced
paths are possible to follow by the car with a limited steering angle.

  * •

Overshoot loss $\mathcal{L}_{over}$, which is used to ensure that the last
segment endpoint lies within a neighborhood $\mathcal{Q}_{k}$ of the goal
configuration $q_{k}$.

  * •

Total curvature loss $\mathcal{L}_{tcurv}$, which is a regularization term
used to ensure that the feasible paths have reasonably low path total
curvature, as it allows for path following with greater velocity.

All losses are summed together to obtain the overall loss, except the total
curvature loss $\mathcal{L}_{tcurv}$, which is added only if path is feasible,
that is, $\mathcal{L}_{coll}+\mathcal{L}_{curv}+\mathcal{L}_{over}=0$.

All the aforementioned loss components are rather not standard, so, we
introduce their definitions. For both collision and curvature losses, we
consider a path as a sequence of $N$ segments, and we divide each segment with
128 points and with orientation, equally distant from each other along the $x$
axis of the local coordinate system. For each of these points, five
characteristic points $\Pi_{ijk}$ on the vehicle model (four corners of the
rectangular body of the vehicle and the guiding point in the middle of the
rear axle) are calculated [39] and used for calculation of the collision loss.
Index $i$ denotes the index of a segment, $j$ denotes the index of a point in
the segment, whereas $k$ denotes the index of the characteristic point ($0$
denotes the guiding point, and the points numbered from $1$ to $4$ are the
corners – see Figure 2). Then, the collision loss is defined by

$\mathcal{L}_{coll}=\sum_{i=1}^{N}\sum_{j=1}^{127}\sum_{k=1}^{5}\sigma(\Pi_{ij},\mathcal{F})d(\mathcal{P}_{r},\Pi_{ijk})l_{ij},$
(2)

where $\mathcal{P}_{r}$ denotes the reference path, $d(X,Y)$ is the smallest
Euclidean distance between $X$ and $Y$, and $l_{ij}$ is the Euclidean distance
between the $(j-1)$-th and the $j$-th point in the $i$-th segment. However,
those distances are taken into account only if the vehicle periphery
$\Pi_{ij}$ is in collision with the environment, which was denoted in (2) by
$\sigma(\Pi_{ij},\mathcal{F})$. Collision $\sigma(\Pi_{ij},\mathcal{F})$ is
determined by checking whether the points on the circumference of the vehicle
lie inside obstacles. In our experiments, we sampled points from the
circumference such that they lie no further than $0.2\text{\,}\mathrm{m}$ from
each other. This ensures that we are checking collisions at the same
resolution as the resolution of the grid map.

Similar to calculation of collision loss, curvature loss $\mathcal{L}_{curv}$
and total curvature loss $\mathcal{L}_{tcurv}$ are also calculated using the
division of segments by

$\mathcal{L}_{curv}=\sum_{i=1}^{N}\sum_{j=1}^{127}\max(|\kappa_{ij}|-\kappa_{max},0)l_{ij}$
(3)

and

$\mathcal{L}_{tcurv}=\gamma\sum_{i=1}^{N}\sum_{j=1}^{127}|\kappa_{i(j+1)}-\kappa_{ij}|,$
(4)

where $\kappa_{ij}$ is a curvature of the path at the $j$-th point in the
$i$-th segment, $\kappa_{max}$ is maximal admissible path curvature, and
$\gamma$ is a regularization parameter, which we set to $10^{-4}$. In turn,
the overshoot loss $\mathcal{L}_{over}$ is defined as a Manhattan Distance [9]
from the subset of the desired configurations $\mathcal{Q}_{k}$.

All parts of the loss function are differentiable almost everywhere, which
enables us to train it directly using the gradient of the proposed loss
function in a Reinforcement Learning paradigm. Among these loss components,
there is only one, which cannot be calculated using the features of the
generated path, thus cannot be trained using Reinforcement Learning methods
exclusively, and it is the collision loss $\mathcal{L}_{coll}$. To evaluate
(2), we need a reference path $\mathcal{P}_{r}$, which we obtain from some
other planner. However, since the supervision is used only to escape from the
forbidden areas of the state space and it is not a performance upper-bound, we
call it weak supervision. Thus, we called the proposed training procedure
Weakly Supervised Gradient-based Policy Search.

###  4.4 Our Proposed Motion Planning Dataset

In order to train our model in a weakly-supervised manner, we need a dataset
of local planning problems. These problems should be solvable using a complete
path planner employing our representation of the path. Therefore, the purpose
of using an auxiliary planner in our method is twofold: Firstly, once a path
is planned, we know that the particular task and scene combination is
solvable, and it is safe to include it in the training data. Secondly, we can
use the planned reference path to drive the learned solution outside the
collision regions. To generate such a dataset, we harvested maps of some real
urban environments. Using the Velodyne HDL-64E LiDAR measurements from the
KITTI dataset [19] and data we have acquired in a town suburbs with a Sick
MRS-6124 LiDAR mounted on a bus, we generated a set of occupation grid maps.
In all cases, the LiDAR scans were registered using the LiDAR Odometry and
Mapping (LOAM) algorithm [40], and the 2D drivable terrain maps were obtained
processing the registered laser scans with an elevation mapping method that
handles properly sparse laser data [2]. Finally, 2D occupancy grids were
produced from the elevation maps by thresholding the elevation values and
setting the drivable (empty) and non-drivable (occupied) cells. In addition,
we used CARLA simulator to gather, much less noisy maps, with multiple
obstacles such as pedestrians and different vehicles. We built some maps of
the Town05 and Town07 to capture both city and rural areas. All the
aforementioned maps have the same grid resolution of 0.2 m/px.

From all those maps, we sampled vehicle centered local maps (128$\times$128
px). We tied the global coordinate system in the middle column and the 120-th
row of the map image, oriented upwards, and situated the vehicle there. The
obtained local map could be augmented with up to 15 randomly placed
rectangular obstacles. For each local map, we generated multiple plans with
random final configurations. The plans have been obtained using a modified
State Lattices (SL) planner utilizing 73 polynomial path primitives. To speed
up the generation process, we guided the state lattice search using a Dubins
distance as a heuristic. Such a lattice-based approach makes it easy to ensure
that the time and memory needed to generate a plan are bounded. Furthermore,
one can easily tune the search resolution (it corresponds to the number of
primitives) and can guarantee the desired accuracy of reaching the final
configuration (here 0.2 m in terms of Dubins distance). Sample problems of the
proposed dataset (grid maps and auxiliary reference paths) are visualized in
Figure 5.

Figure 5: Visualization of sample maps of the proposed dataset. Maps together
with exemplary paths planned with the State Lattices algorithm, which connects
the actual robot state with some randomly drawn configurations. Scenarios in
the first row are obtained from CARLA Town05 and Town07, whereas in the second
row, we show maps which are obtained from real LiDAR data.

In this manner, we generated 23517 local maps and 134509 scenarios. Firstly,
we excluded the test set, which was generated entirely from a single map of
our own Sick LiDAR dataset. It contains 1014 local maps and 8128 scenarios.
Secondly, we collected maps using; (i) another map from our dataset, (ii) maps
built from the KITTI dataset, and (iii) maps generated in CARLA. Then, we
randomly selected 1996 maps to construct a validation set, and the rest of the
maps was used as a training set. Finally, we generated paths for those maps,
and obtained 11008 and 115319 scenarios in the validation and training set,
respectively. We examine the generalization ability of the models to
previously unseen environments by using data collected from entirely different
environments in the training and test sets. We would like to share our dataset
with the community as a benchmark for kinematically constrained motion
planning222Dataset is available at
https://drive.google.com/file/d/1TACIbwO6L4qTL3lzfvcPc3HDiVmxtd77/view?usp=sharing.

##  5 Experimental Results

In our experiments, we modeled a Kia Rio III, whose physical dimensions are
the following: distance from rear axle to rear bumper is
$L_{B}=$0.67\text{\,}\mathrm{m}$$, distance from rear axle to front bumper is
$L_{F}=$3.375\text{\,}\mathrm{m}$$, distance between axles is
$L=$2.8\text{\,}\mathrm{m}$$ and vehicle width is
$W=$1.72\text{\,}\mathrm{m}$$. Moreover, for the length $L$ and maximal
steering angle $\beta_{max}=$0.57\text{\,}\mathrm{rad}$$, the maximal
admissible path curvature is $\kappa_{max}=$0.227\text{\,}{\mathrm{m}}^{-1}$$.
The subset of the desired states $\mathcal{Q}_{k}$ is defined around a given
desired state $q_{k}$, such that its elements should not lie further than
$0.2\text{\,}\mathrm{m}$ from the $q_{k}$ both in $x$ and $y$ axes, and the
orientation $\theta$ should satisfy
$|\theta-\theta_{k}|<$0.05\text{\,}\mathrm{rad}$$. We trained our planner
using Adam [26] optimizer with learning rate equal to $10^{-4}$, and batch
size of 128. The models were trained for 400 epochs. It took 33 hours to train
the models using NVIDIA GTX1080Ti GPU.

###  5.1 Impact of the number of segments of the generated path

At the first stage of the experiments, we evaluated how the number of path
segments $N$ affects the learning process, runtime, and accuracy of the
proposed machine learning based path planner. Figure 6 presents the learning
curves of the models (i.e., change of their accuracy on both training and
validation sets) trained with a parameter $N\in\\{2,4,6\\}$. One can see that
the accuracy of all models grows rapidly from the very first epoch, and
achieves 80% in less than 40 epochs. The fastest learning rate was obtained
for $N=4$, however, the learning curves approach a similar level of
performance for both $N=4$ and $N=6$ at the end. Notice that the accuracy of
models obtained for the validation sets is only slightly worse than for the
training sets, and it does not decrease over time. This observation may
suggest that the models have learned some transformation from the task
definition into the solution path, which is applicable beyond the training
examples to unseen desired poses and environment maps.

Figure 6: Change of accuracy of the models (on training and validation sets)
for $N\in\\{2,4,6\\}$ through the learning process.

The number of the predicted segments $N$ affects also the inference time. In
Figure 7, we present the mean and standard deviations of model running times,
for different $N$ values, together with its accuracy on the test set. One can
see that models trained with larger $N$ are slower, but achieve superior
performance. Therefore, we observe a tradeoff between the performance on the
test set and the inference time. The accuracy increases with larger $N$, since
larger $N$ allows performing much more complicated maneuvers, which may be
necessary for highly confined spaces. Since all the obtained times are
relatively low (all networks guarantee that the path computation time is below
$50\text{\,}\mathrm{ms}$), we decided to choose the neural network model
trained for $N=6$ in the other experiments, as it achieved the best accuracy
in the analyses, and it allows planners to perform more complex maneuvers
compared to the other values.

Figure 7: Mean inference time and their standard deviation, as well as the
test set accuracy of the proposed neural networks for $N\in\\{2,4,6\\}$.

###  5.2 Validation on the test set

Exemplary paths generated by the proposed planner are presented both in Figure
8 and Figure 9, together with the sets of reachable configurations. One can
see that our planner learned to plan for wide areas of the accessible state
space in the previously unseen environments. Obviously, our planner is not
complete. Thus, some regions of the state space are not reachable using plans
generated by our network. However, it still can perform complex maneuvers in
narrow passages. Moreover, it anticipates the environment using information
from the occupancy map, and it is able to detect cars, pedestrians, and other
obstacles, taking their presence into account while planning (Figure 8).
However, it ignores distant obstacles by paying attention to the objects that
possibly collide with the path. We also show that our planner can use both
high-quality maps obtained from the CARLA simulator, as well as lower quality
maps, which are obtained from real LiDAR measurements (Figure 9).

Figure 8: Visualizations of the sets of the final states (denoted by heatmaps)
for which the planner generated feasible paths (so-called reachable sets).
Darker dots represent wider ranges of possible end orientations of the
vehicle. Colored lines depict paths drawn by the four corners of the vehicle
and the guiding point while moving along the path provided by the network. The
planner takes into account the obstacles located on the path to the goal (2nd
image), whereas ignores the distant obstacles (3rd image). Moreover, it reacts
to the obstacle size (4th image) and plans accordingly. Figure 9:
Visualizations of the reachable sets and exemplary paths. Maps in the first
row are obtained from the test set made with real LiDAR data, whereas in the
second row, we show maps which are obtained from CARLA Town04 (not used in the
training).

In Table 1, we compare the accuracy and quality of our planner to selected
planning algorithms. We use the OMPL [36] implementations of those, equipped
with the Dubins curves as primitives, which makes the planning problem way
easier because in this case, the graph of possible solutions is significantly
smaller. Notice that our solution ensures the continuous virtual steering
wheel angle throughout the path, while the rest of the planners allows for
instant changes of this angle. In Dubins paths, curvatures change instantly
from 0 to a constant value between line segments and circle arcs. Hence,
Dubins paths can be followed by a car-like vehicle defined in Section 3 only
if we apply an infinite steering input or stop the vehicle to change the
steering angle. Therefore, in our case, we cannot utilize a convenient
steering function, which can connect two arbitrary vehicle configurations with
Dubins curves. When such a steering function is not present, the performance
of sampling-based planners decays rapidly [25].

For accuracy, we compute a ratio of valid plans to all scenarios from the test
set. We consider that the plan is invalid, if it violates kinematics or
environmental constraints, or if the planner exceeds the maximal allowed
planning time, which is set to $50\text{\,}\mathrm{ms}$. The proposed method
obtained the best accuracy among other planners. However, it produces paths,
which are on average a bit longer and require more turns. It is known that
Dubins curves optimize the path length. However, its convergence is not so
clear in terms of accumulated turns. We conjecture that our solution results
in more turns, because of insufficient regularization of the total curvature,
which enables planning slightly wavy paths, even if there is no need for these
micro-turns.

Table 1: Accuracy, accumulated turn and length of the paths generated by different planning methods for the $50\text{\,}\mathrm{ms}$ planning time limit. Both accumulated turns and lengths are reported only for paths valid for all planners (except SST and RRT* due to their low accuracy). Planner | SST[28] | RRT*[24] | AIT*[34] | BIT*[16] | ABIT*[35] | ours
---|---|---|---|---|---|---
Accuracy [%] | 3.95 | 15.97 | 84.12 | 84.19 | 84.15 | 92.24
Accumulated turn [$\text{\,}\mathrm{rad}$] | - | - | 0.63$\pm$0.43 | 0.63$\pm$0.43 | 0.63$\pm$0.43 | 0.78$\pm$0.55
Length [$\text{\,}\mathrm{m}$] | - | - | 11.6$\pm$6.1 | 11.6$\pm$6.1 | 11.6$\pm$6.1 | 11.8$\pm$6.2

A more detailed comparison of the accuracy of the planners is presented in
Figure 10. This chart gives an interpretation of how the accuracy of the given
planner changes with respect to the maximal allowed planning time. The
proposed planner outperforms all other tested planners for the planning
horizon up to $1\text{\,}\mathrm{s}$. But, even for the
$10\text{\,}\mathrm{s}$ limit, it is more accurate than BIT*, ABIT*, and AIT*.
This shows that our method solves correctly a large part of the scenarios
within $50\text{\,}\mathrm{ms}$, and its planning time is almost constant and
equal to $41\pm$1\text{\,}\mathrm{ms}$$. However, our method is not able to
improve after that point (as it is not meant to be able to). One can consider
coupling our method together with RRT* or SST, as if there is more than
$1\text{\,}\mathrm{s}$ available, then they may improve the generated
solution. For all these measurements, we used Intel Core i7-9750H based PC.

Figure 10: Accuracy [%] for all planners on the test set with respect to the
maximal allowed planning time [$\text{\,}\mathrm{s}$]. Our planner outperforms
all other planners when the planning time horizon is below
$1\text{\,}\mathrm{s}$.

###  5.3 Ablation study

The total curvature loss $\mathcal{L}_{tcurv}$, defined in (4) is the only
term in the loss function, which is not necessary to obtain a feasible path.
Hence, we include a brief ablation study to analyze the impact of this part of
our loss function (4.3) on the planner accuracy and the quality of generated
paths. For this study, we trained and tested a new model in the same
conditions as the basic one (for $N=6$) but without $\mathcal{L}_{tcurv}$ in
the loss function. It obtained slightly better performance in terms of the
accuracy on the test set achieving 92.53%. However, the main idea behind this
loss term is visible in the comparison of the mean accumulated turn, which is
equal to $0.86\text{\,}\mathrm{rad}$ for the model trained without
$\mathcal{L}_{tcurv}$ term, and $0.78\text{\,}\mathrm{rad}$ for the model
trained with $\mathcal{L}_{tcurv}$. A more intuitive view on the importance of
the total curvature loss is presented in Figure 11, where we compare paths
generated by the aforementioned models in several different scenarios. Even
though in all cases the maximal admissible curvature is not exceeded, paths in
the first row (generated by the model with full loss function) are even
visually much smoother than those in the second row (generated by the model
trained without $\mathcal{L}_{tcurv}$ term). Such behavior is desirable as it
limits the sudden and frequent turns which are usually unnecessary to perform
the maneuver, may limit maximal allowed velocity, and cause problems in
control. Moreover, maneuvers performed smoothly, without redundant turns, are
far closer to the expectations of the human drivers and give a feeling of safe
driving.

Figure 11: Exemplary paths generated by the proposed neural network based
motion planner trained with the standard loss function (defined in Section
4.3) – first row, and with modified loss function without
$\mathcal{L}_{tcurv}$ term (defined in (4)) – second row. Presence of the
total curvature loss term results in much smoother looking and easier to
follow paths.

###  5.4 Experiments in CARLA

Since the planning time of our method is so short, it can be used even to
handle dynamic environments. We validated that claim using the CARLA simulator
[13]. In the performed simulations, we use CARLA API to obtain the maps of the
ego-vehicle local environment and the actual configuration of the ego-vehicle.
Then, using that information, and a predefined list of control points, which
represent the information obtained from a higher-level motion planner333In the
CARLA experiments, the routes were defined manually, but in principle, any
complete path planner, e.g. based on Probabilistic Road Maps can be applied
here., the proposed maneuver generation algorithm predicts the path leading to
the next control point. Finally, this path is followed using the VFO kinematic
controller defined in [29]. In such a scenario, the planner runs every
$50\text{\,}\mathrm{ms}$ and produces new paths to follow, which enables it to
avoid collisions and fix possible path infeasibility resulting both from the
controller errors, and inaccuracy of the network.

Figure 12 presents a view from the (simulated) camera, as well as the map
given by the environment. On the map, we presented one of the possible paths
to follow, together with the visualization of the accessible set.

Figure 12: Visualization of the online path planning in the CARLA simulator.

In Figure 13, we present a sequence of snapshots from the CARLA simulator
taken in a scenario of crossing the intersection. While our car drives
straight through the junction, a police car tries to block its way. The
vehicle re-plans every $50\text{\,}\mathrm{ms}$, thus reacts immediately (note
the limited sensing range – about $25\text{\,}\mathrm{m}$) and starts to
follow a new path to avoid a collision. In Figure 14, we present a similar
sequence, but for the scenario of navigation through a traffic jam. This
scenario emphasizes the ability of the proposed planning algorithm to navigate
in narrow passages. Although the initial path prediction is slightly
inaccurate, after a small movement along the planned path, the online re-
planning process allows generating a valid maneuver.

Figure 13: Visualization of the online replanning which enables the car to
avoid collisions with dynamic obstacles. In the bottom left corner, there are
maps, which present the planned path. This sequence is also available as a
supplementary video clip (https://youtu.be/dnfIp5A-k6Q). Figure 14:
Visualization of the online replanning process, which enables the car to avoid
collisions with static obstacles. Even though at the very first moment, the
planner produces a path that collides slightly with the obstacle, after a
small movement, it replans, and the produced path is feasible. In the upper
left corner, there are maps, which present the planned path. This sequence is
also available as a supplementary video clip (https://youtu.be/pj02xUgyCrU).

##  6 Conclusion

We present a novel learning-based approach to local maneuver generation for
self-driving cars. Our method contests the typical approach to motion planning
based on state-space search algorithms and proposes to approximate an oracle
planning function instead. Our proposed approach lets one omit the time-
consuming execution of the algorithm, by learning how to plan off-line
(optimizing the approximator of the oracle planning function), and then only
reusing the gathered knowledge on-line (using a fast approximator). Although
this approach gives no guarantees about the completeness, it is fast and
flexible, as its quality depends on the data used in the training process, and
the expressiveness and generalization abilities of the model. To achieve rapid
path generation, an oracle planning function is approximated by a deep neural
network trained using a weakly supervised gradient-based policy search. In our
approach, we do not model the oracle planning function with the use of human-
generated data, but instead, we define it implicitly by the employment of a
novel loss function.

Our neural planner can handle typical local maneuvers, such as overtaking,
avoiding obstacles, or parking in almost constant time, rapidly generating
feasible paths that are nearly optimal with respect to their length, and
ensure continuous steering angle. The proposed approach outperforms several
conventional path planners (including those very recent) concerning the number
of successfully completed tasks (achieving the accuracy of 92%), being also by
far the fastest one. Extensive tests on local urban environments maps obtained
from real LiDAR data and generated from the CARLA simulator demonstrate good
generalization abilities of the proposed network model, and the efficiency of
its weakly supervised training procedure. Experiments performed in the CARLA
simulator show not only the ability of the network to produce plans, which are
possible to follow with the kinematic controller but also the advantage gained
because of the fast path replanning. Ablation study presents the impact of
using total curvature loss term (4) on the generated path smoothness. We also
provide an open-source code of our system and a public dataset of motion
planning problems with solutions that can be used by others to test the
proposed planner or to compare against it. Planned future work concerns the
possibility to use our path generator combined with higher-level planners, as
it is fast and accurate but not complete.

## References

  * Arab et al. [2016] Arab, A., Yu, K., Yi, J., Song, D., 2016\.  Motion planning for aggressive autonomous vehicle maneuvers, in: IEEE International Conference on Automation Science and Engineering, pp. 221–226.
  * Belter et al. [2012] Belter, D., Labȩcki, P., Skrzypczyński, P., 2012.  Estimating terrain elevation maps from sparse and uncertain multi-sensor data, in: IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 715–722.
  * Bency et al. [2019] Bency, M.J., Qureshi, A.H., Yip, M.C., 2019.  Neural path planning: Fixed time, near-optimal path generation via oracle imitation.
  * Berenson et al. [2012] Berenson, D., Abbeel, P., Goldberg, K., 2012.  A robot path planning framework that learns from experience, in: IEEE International Conference on Robotics and Automation, Saint Paul. pp. 3671–3678.
  * Bhardwaj et al. [2019] Bhardwaj, M., Choudhury, S., Boots, B., Srinivasa, S.S., 2019\.  Leveraging experience in lazy search, in: Robotics: Science and Systems, Freiburg im Breisgau.
  * Chen et al. [2020] Chen, B., Dai, B., Lin, Q., Ye, G., Liu, H., Song, L., 2020.  Learning to plan in high dimensions via neural exploration-exploitation trees, in: 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020.
  * Chen et al. [2019a] Chen, J., Yuan, B., Tomizuka, M., 2019a.  Deep imitation learning for autonomous driving in generic urban scenarios with enhanced safety, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2884–2890.
  * Chen et al. [2019b] Chen, J., Zhan, W., Tomizuka, M., 2019b.  Autonomous driving motion planning with constrained iterative lqr.  IEEE Transactions on Intelligent Vehicles 4, 244–254.
  * Craw [2017] Craw, S., 2017.  Manhattan distance, in: Sammut, C., Webb, G.I. (Eds.), Encyclopedia of Machine Learning and Data Mining, Springer, Boston. pp. 790–791.
  * Cybenko [1989] Cybenko, G., 1989.  Approximation by superpositions of a sigmoidal function.  Math. Control. Signals Syst. 2, 303–314.
  * Deisenroth et al. [2013] Deisenroth, M.P., Neumann, G., Peters, J., 2013.  A survey on policy search for robotics.  Foundations and Trends in Robotics 2, 1–141.
  * Dolgov et al. [2009] Dolgov, D., Thrun, S., Montemerlo, M., Diebel, J., 2009\.  Path planning for autonomous driving in unknown environments, in: Experimental Robotics, Springer. pp. 55–64.
  * Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V., 2017.  CARLA: An open urban driving simulator, in: Proceedings of the 1st Annual Conference on Robot Learning, pp. 1–16.
  * Duan et al. [2016] Duan, Y., Chen, X., Houthooft, R., Schulman, J., Abbeel, P., 2016.  Benchmarking deep reinforcement learning for continuous control, in: Balcan, M.F., Weinberger, K.Q. (Eds.), Proceedings of The 33rd International Conference on Machine Learning, PMLR, New York, USA. pp. 1329–1338.
  * Gammell et al. [2018] Gammell, J.D., Barfoot, T.D., Srinivasa, S.S., 2018.  Informed sampling for asymptotically optimal path planning.  IEEE Transactions on Robotics 34, 966–984.
  * Gammell et al. [2020] Gammell, J.D., Barfoot, T.D., Srinivasa, S.S., 2020.  Batch informed trees (bit*): Informed asymptotically optimal anytime search.  The International Journal of Robotics Research 39, 543–567.
  * Gammell et al. [2015] Gammell, J.D., Srinivasa, S.S., Barfoot, T.D., 2015.  Batch informed trees (BIT∗): Sampling-based optimal planning via the heuristically guided search of implicit random geometric graphs, in: IEEE International Conference on Robotics and Automation, pp. 3067–3074.
  * Gawron and Michałek [2018] Gawron, T., Michałek, M.M., 2018\.  A G3-continuous extend procedure for path planning of mobile robots with limited motion curvature and state constraints.  Applied Sciences 8.
  * Geiger et al. [2013] Geiger, A., Lenz, P., Stiller, C., Urtasun, R., 2013\.  Vision meets robotics: The KITTI dataset.  The International Journal of Robotics Research .
  * González et al. [2016] González, D., Pírez, J., Milanís, V., Nashashibi, F., 2016\.  A review of motion planning techniques for automated vehicles.  IEEE Transactions on Intelligent Transportation Systems 17, 1135–1145.
  * Groeger [2000] Groeger, J.A., 2000.  Understanding Driving: Applying Cognitive Psychology to a Complex Everyday Task.  Psychology Press, East Sussex.
  * Hornik [1991] Hornik, K., 1991.  Approximation capabilities of multilayer feedforward networks.  Neural Networks 4, 251 – 257.
  * Jetchev and Toussaint [2013] Jetchev, N., Toussaint, M., 2013\.  Fast motion planning from experience: trajectory prediction for speeding up movement generation.  Autonomous Robots 34, 1573–7527.
  * Karaman and Frazzoli [2011] Karaman, S., Frazzoli, E., 2011\.  Sampling-based algorithms for optimal motion planning.  The International Journal of Robotics Research 30, 846–894.
  * Karaman and Frazzoli [2013] Karaman, S., Frazzoli, E., 2013\.  Sampling-based optimal motion planning for non-holonomic dynamical systems, in: IEEE International Conference on Robotics and Automation, Karlsruhe. pp. 5041–5047.
  * Kingma and Ba [2015] Kingma, D.P., Ba, J., 2015.  Adam: A method for stochastic optimization, in: Bengio, Y., LeCun, Y. (Eds.), 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings.
  * LaValle [2006] LaValle, S.M., 2006.  Planning Algorithms.  Cambridge University Press.
  * Li et al. [2016] Li, Y., Littlefield, Z., Bekris, K.E., 2016.  Asymptotically optimal sampling-based kinodynamic planning.  The International Journal of Robotics Research 35, 528–564.
  * Michałek and Gawron [2018] Michałek, M.M., Gawron, T., 2018\.  Vfo path following control with guarantees of positionally constrained transients for unicycle-like robots with constrained control input.  Journal of Intelligent & Robotic Systems 89, 191–210.
  * Phillips et al. [2012] Phillips, M., Cohen, B., Chitta, S., Likhachev, M., 2012\.  E-Graphs: Bootstrapping planning with experience graphs, in: Robotics, Science and Systems Conference.
  * Pivtoraiko et al. [2009] Pivtoraiko, M., Knepper, R.A., Kelly, A., 2009.  Differentially constrained mobile robot motion planning in state lattices.  Journal of Field Robot. 26, 308–333.
  * Qureshi et al. [2019] Qureshi, A.H., Simeonov, A., Bency, M.J., Yip, M.C., 2019\.  Motion planning networks, in: IEEE International Conference on Robotics and Automation, pp. 2118–2124.
  * Sadat et al. [2019] Sadat, A., Ren, M., Pokrovsky, A., Lin, Y., Yumer, E., Urtasun, R., 2019\.  Jointly learnable behavior and trajectory planning for self-driving vehicles, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3949–3956.
  * Strub and Gammell [2020a] Strub, M.P., Gammell, J.D., 2020a.  Adaptively informed trees (AIT∗): Fast asymptotically optimal path planning through adaptive heuristics.
  * Strub and Gammell [2020b] Strub, M.P., Gammell, J.D., 2020b.  Advanced BIT∗ (ABIT∗): Sampling-based planning with advanced graph-search techniques, in: IEEE International Conference on Robotics and Automation, Paris. pp. 3671–3678.
  * Sucan et al. [2012] Sucan, I.A., Moll, M., Kavraki, L.E., 2012.  The open motion planning library.  IEEE Robotics & Automation Magazine 19, 72–82.
  * Sünderhauf et al. [2018] Sünderhauf, N., Brock, O., Scheirer, W., Hadsell, R., Fox, D., Leitner, J., Upcroft, B., Abbeel, P., Burgard, W., Milford, M., Corke, P., 2018.  The limits and potentials of deep learning for robotics.  The International Journal of Robotics Research 37, 405–420.
  * Sutton and Barto [1998] Sutton, R.S., Barto, A.G., 1998\.  Introduction to Reinforcement Learning.  MIT Press, Cambridge.
  * Yoon et al. [2017] Yoon, S., Lee, D., Jung, J., Shim, D., 2017.  Spline-based rrt* using piecewise continuous collision-checking algorithm for car-like vehicles.  Journal of Intelligent & Robotic Systems 90, 1–13.
  * Zhang and Singh [2017] Zhang, J., Singh, S., 2017.  Low-drift and real-time lidar odometry and mapping.  Autonomous Robots 41, 401–416.


$K$ there are $2^{20}$ choices for $S$.

The BOSS bundles are degree $12$ threefolds $Z\subset{\bf P}^{5}$ that are
conic bundles over quartic surfaces in ${\bf P}^{3}$. These conic bundles were
discovered in [BOSS] and further studied in [IR2], which we follow (see also
[Al], [Ku2], [DMS], [MS+]). We will explicitly construct an embedding
$Z\hookrightarrow S^{[2]}$ for a degree $16$ K3 surface $S$. We show in
Proposition 5.3 that this implies that the BOSS ${\bf P}^{1}$-fibrations are
isomorphic to exceptional divisors in hyper-Kähler fourfolds.

####  5.3.1. An incidence correspondence

Let $V_{6}$ be a six dimensional complex vector space with a symplectic form.
There is natural decomposition $\wedge^{3}V_{14}=V_{14}\oplus V_{6}$ as
representations of the symplectic group of $V_{6}$. Let $\Sigma=LG(3,V_{6})$
be the Grassmannian of Lagrangian (i.e. maximally isotropic) subspaces in
$V_{6}$. There is a (Plücker) embedding $\Sigma\hookrightarrow{\bf P}V_{14}$.
For $p\in{\bf P}V_{6}$ we denote by $Q_{p}\subset\Sigma$ the subvariety of
Lagrangian subspaces containing $p$. Then $Q_{p}$ is isomorphic to a smooth
3-dimensional quadric and it is cut out in $\Sigma$ by a four dimensional
linear subspace ${\bf P}^{4}_{p}\subset{\bf P}V_{14}$, so $Q_{p}={\bf
P}^{4}_{p}\cap\Sigma$ ([IR2, §2.4]).

We consider the incidence variety (see [IR2, p.394])

$J\,:=\,\\{(p,\omega)\,\in\,{\bf P}V_{6}\times{\bf P}V_{14}^{*}\,:\;{\bf
P}^{4}_{p}\subset{\bf P}^{12}_{\omega}\\}~{},$

here ${\bf P}^{12}_{\omega}\subset{\bf P}V_{14}$ is the hyperplane defined by
$\omega\in V_{14}^{*}$. The image of $J$ under the projection to ${\bf
P}V_{14}^{*}$ is a quartic hypersurface $F=\pi(J)\subset{\bf P}V_{14}^{*}$.
The singular locus $\Omega$ of $F$ is $9$-dimensional variety and the fiber of
$\pi$ over a point in $F-\Omega$ is a smooth conic ([IR2, Prop.2.5.5]).

####  5.3.2. The K3 surfaces $S$ and $K$

Let $S$ be a K3 surface of degree $16$ with Picard rank one. Mukai proved that
there is, up to projective equivalence, a unique ${\bf P}^{9}={\bf
P}^{9}_{S}\subset{\bf P}V_{14}$ such that $S={\bf P}^{9}_{S}\cap\Sigma$. The
Mukai dual of $S$ is the quartic K3 surface $K={\bf P}^{3}_{K}\cap F$, where
${\bf P}^{3}_{K}\subset{\bf P}V_{14}^{*}$ is the dual of ${\bf P}^{9}_{S}$.

The BOSS conic bundle over $K$ defined by $S$ is the restriction of $\pi$ to
$Z:=\pi^{-1}(K)\rightarrow K$.

$\begin{array}[]{rrcl}\pi\colon&J&\longrightarrow&\pi(J)\,=\,F\quad\subset\;{\bf
P}V_{14}^{*}\\\ &\cup&&\cup\\\ &Z&\longrightarrow&K\,=\,{\bf P}^{3}_{K}\cap
F~{}.\end{array}$

For $(p,\omega)\in Z\subset J$, we have $\omega\in K\subset{\bf P}^{3}_{K}$
and ${\bf P}^{4}_{p}\subset{\bf P}^{12}_{\omega}$. Dualizing
$\\{\omega\\}\subset{\bf P}^{3}_{K}$ we get ${\bf P}^{9}_{S}\subset{\bf
P}^{12}_{\omega}$ and thus ${\bf P}^{4}_{p}\cap{\bf P}^{9}_{S}\subset{\bf
P}^{12}_{\omega}$. In particular $\dim{\bf P}^{4}_{p}\cap{\bf P}^{9}_{S}\geq
1$.

There is an embedding $Z\subset{\bf P}V_{6}$ given by $Z=\\{p\in{\bf
P}V_{6}:\,\dim({\bf P}^{4}_{p}\cap{\bf P}^{9}_{S})\geq 1\,\\}$ ([Al, Remark
3]) and the ideal of the image, a threefold of degree $12$, is generated by
ten quintics. This is the classical description of the BOSS bundle ([BOSS]).

####  5.3.3. The BOSS bundle as divisor on a Hilbert square.

Let $(p,\omega)\in Z\subset J$, so $\dim{\bf P}^{4}_{p}\cap{\bf P}^{9}_{S}\geq
1$. As $S=\Sigma\cap{\bf P}^{9}_{S}$ and $\Sigma\cap{\bf P}^{4}_{p}=Q_{p}$ we
see that

$S\,\cap\,({\bf P}^{4}_{p}\,\cap\,{\bf P}^{9}_{S})\,=\,Q_{p}\,\cap\,({\bf
P}^{4}_{p}\cap{\bf P}^{9}_{S})~{}.$

If $\dim{\bf P}^{4}_{p}\cap{\bf P}^{9}_{S}>1$ then, since $Q_{p}$ is a quadric
threefold in ${\bf P}^{4}_{p}$, $S$ will contain a conic which contradicts our
assumption that $Pic(S)$ has rank one. Hence ${\bf P}^{4}_{p}\cap{\bf
P}^{9}_{S}$ is a line. This line is not contained in $S$ since $Pic(S)$ has
rank one. Thus the line ${\bf P}^{4}_{p}\cap{\bf P}^{9}_{S}$ intersects
$Q_{p}$ and hence $S$ in a zero cycle of degree two. This gives a map

$\psi:\,Z\,\longrightarrow\,S^{[2]},\qquad(p,\omega)\,\longmapsto S\cap{\bf
P}^{4}_{p}\cap{\bf P}^{9}_{S}~{}.$

This map is an embedding and there is a BN contraction on $S^{[2]}$ which
induces $\pi:Z\rightarrow K$.

######  Proposition 5.3.

A general point in the Heegner divisor ${\mathcal{D}}^{(1)}_{16,16,\beta}$ is
given by $(S^{[2]},K_{16})$ where $(S,h)$ is a general K3 surface of degree
$16$, $K_{16}=\langle 3H-8\xi,\,H-3\xi\rangle$ with $H\in Pic(S^{[2]})$ the
divisor class defined by $h$ and $2\xi$ the class of the divisor parametrizing
non-reduced subschemes.

The map $\psi:Z\rightarrow S^{[2]}$ is an embedding and its image is the
exceptional divisor $E$ of the corresponding BN contraction. The ${\bf
P}^{1}$-fibration $Z\rightarrow K$ is over the Mukai dual quartic K3 surface
$(K,h)$ of $S$. The Brauer class of the BOSS bundle has (unique) invariant
$Bh=1/2$.

Proof. As in [DM, Example 5.3] we compute the nef cone of $S^{[2]}$. The
Picard group of $S^{[2]}$ is generated by $H,\,\xi$, and $H^{2}=16$, $H\xi=0$
and $\xi^{2}=-2$. Since the Pell-type equation $a^{2}-32b^{2}=5$ has no
solutions mod $5$, hence has no solutions at all, the nef cone and the movable
cone coincide and thus $S^{[2]}$ has a unique birational hyper-Kähler model
(cf. [DM, §5]) and in particular it does not admit any flops.

The movable cone has extremal rays $H$ and $H-8(b/a)\xi$ where $(a,b)$ is the
solution of $a^{2}-8b^{2}=1$, with $a,b>0$ and $a$ minimal. This minimal
solution is $(a,b)=(3,1)$, so the nef cone has extremal rays $H$ and
$3H-8\xi$. The $(-2)$-classes perpendicular to these rays are $\xi$ and
$H-3\xi$. Notice that indeed $(H-3\xi)(3H-8\xi)=3h^{2}+24\xi^{2}=3\cdot
16-24\cdot 2=0$ and $(H-3\xi)^{2}=16-9\cdot 2=-2$.

In particular, the divisor with class $H-3\xi$ is contracted by the big and
nef class $H_{16}=3H-8\xi$. Notice that $H_{16}^{2}=9\cdot 16-64\cdot
2=16(9-8)=16$, as we know should be the case for $(S^{[2]},H_{16})$ to be in
${\mathcal{D}}^{(1)}_{16,16,\beta}$.

If $\psi$ is not injective, there are distinct $p,q\in{\bf P}V_{6}$ such that
the lines $l_{p}:={\bf P}^{4}_{p}\cap{\bf P}^{9}_{S}$ and $l_{q}:={\bf
P}^{4}_{q}\cap{\bf P}^{9}_{S}$ are the same. Hence $Q_{p}\cap Q_{q}$ is not
empty. As the quadrics $Q_{p},Q_{q}\subset\Sigma$ are either disjoint or
intersect along a line, they must then intersect along a line $l$. As $l$
contains the degree two 0-cycle cut out by $l_{p}$ on $S$, we have $l=l_{p}$.
But $l\subset Q_{p}\cap Q_{q}\subset\Sigma$ hence $l\subset S$, which again
contradicts that the rank of $Pic(S)$ is one. Finally, $\psi$ is an embedding
because the map ${\bf P}V_{6}\ni p\to{\bf P}^{4}_{p}\in G(5,14)$ is an
embedding.

To prove that the ${\bf P}^{1}$-bundle $Z\subset S^{[2]}$ can be contracted we
use [O3, Prop. 3.24], which asserts that $h^{0}(S^{[2]},Z)=1$ and $Z^{2}<0$
for the BB-form. From Theorem 3.1 we then deduce that $Z$ can be contracted
after a series of flops. But we already observed that $S^{[2]}$ does not admit
flops and hence $Z$ can be contracted, so its class is on an extremal ray.

The extremal ray defined by $\xi$ defines the HC contraction ${\bf
P}{\mathcal{T}}_{S}\rightarrow S$ and since $Z$ is contracted to $K$, a K3 of
degree four, it must define the other extremal ray. From [BOSS] we know that
$K_{Z}^{3}=12$. Hence, as in the proof of Proposition 4.1, $Z^{2}=-2$, so $Z$
defines a BN contraction and $Z$ has class $H-3\xi\in Pic(S^{[2]})$. From
Table 1 it follows that the period point lies in
${\mathcal{D}}_{16,16,\beta}^{(1)}$ and the B-field representative for
$\beta\in Br(T)_{2}$ given there is also the representative for $\alpha\in
Br(K)_{2}$ by Theorem 4.2.  $\Box$

###  5.4. The Heegner divisor ${\mathcal{D}}_{6,6}^{(2)}$ and nodal cubic
fourfolds

For $2d=6$, a general point of the Heegner divisor
${\mathcal{D}}^{(2)}_{2d,2d}$ is isomorphic to $S^{[2]}$ where $(S,h)$ is a
general K3 surface of degree $6$. These K3 surfaces and their Hilbert squares
are related to nodal cubic fourfolds and their Fano varieties cf. [Ha, 4.2,
Lemma 6.3.1], [DM, Example 6.4], [GO, 4.3].

Such a Hilbert square admits two divisorial contractions, one is HC and the
other is BN. From Corollary 4.5 the BN divisor is the projectivisation of the
rank two, stable, Mukai bundle ${\mathcal{V}}$ with
$v({\mathcal{V}})=s=(2,h,2)$ and $s^{2}=-2$ (see Table 1). In particular, the
associated Brauer class is trivial.

The K3 surface $S\subset{\bf P}^{4}$ is the complete intersection of a smooth
quadric and a cubic hypersurface. Taking this quadric to be a hyperplane
section of the Grassmannian $Gr(2,4)\subset{\bf P}^{5}$, in [GO, Lemma 4.5] it
is shown that the Mukai bundle ${\mathcal{V}}$ can be chosen to be the
restriction of the dual of the universal bundle on $Gr(2,4)$ to $S$.

###  5.5. The Heegner divisor ${\mathcal{D}}_{22,22}^{(2)}$ and Debarre-Voisin
fourfolds

Let $(K,h)$ be a K3 surface of degree $22$ and denote by $h$, $2\xi\in
Pic(K^{[2]})$ the classes corresponding to $h$ and the class of the divisor
parametrizing non-reduced subschemes respectively. In [DV], Debarre and Voisin
show that the general hyper-Kähler fourfold with period in
${\mathcal{P}}^{(2)}_{22}$ is the zero locus of a section of
$\wedge^{3}{\mathcal{E}}_{6}$, where ${\mathcal{E}}_{6}$ is the tautological
rank $6$ vector bundle on the Grassmannian $Gr(6,10)$. Moreover, these
varieties are shown to be deformations of $(K^{[2]},H)$, where

$H=10h\,-\,33\xi,\qquad\mbox{so}\quad H^{2}\,=\,10^{2}\cdot
22+33^{2}\cdot(-2)\,=\,22(100-99)=22~{}.$

Note that $HD\equiv 0\mod 2$ for all $D\in H^{2}(K^{[2]},{\bf Z})$, hence the
divisibility $\gamma$ of $H$ is indeed $2$.

In [DV, Lemma 3.6] it is shown that $\phi_{H}$ is birationally a contraction
of a divisor $D^{\prime}$ to a K3 surface $Y^{3}_{\sigma}$ of degree $22$ (cf.
[DV, Prop. 3.2]). In view of the divisibility of $H$, Table 1 shows that this
contraction gives a point in ${\mathcal{D}}^{(2)}_{22,22,\alpha}$.

In [DV, Claim, p.79] it is shown that $D^{\prime}$ is not reduced, but has
multiplicity two and combining with [DV, Lemma 3.5] one finds that
$D^{\prime}=2E$ and the class of $E\in H^{2}(S^{[2]},{\bf Z})$ is

$E\,:=\,3h-10\delta,\qquad\mbox{so}\quad E^{2}\,=\,-2,\qquad HE\,=\,0~{}.$

By Corollary 4.5 the BN divisor $E$ is the projectivisation of the rank two,
stable, Mukai bundle ${\mathcal{V}}$ with $v({\mathcal{V}})=s=(2,h,6)$ and
$s^{2}=-2$.

The Hilbert square $S^{[2]}$ does admit flops, in fact it has the $-10$ class
$2h-7\delta$, and the moving cone is divided in two chambers. Let
$\overline{S^{[2]}}$ be the flop of $S^{[2]}$ along this $-10$ class. It
follows that $H$ induces a BN contraction on $\overline{S^{[2]}}$. See also
[DHOV] for polarized Hilbert squares that are Debarre-Voisin fourfolds.

###  5.6. The Heegner divisor ${\mathcal{D}}_{38,38}^{(2)}$ and Iliev-Ranestad
fourfolds

The general hyper-Kähler fourfold with period in ${\mathcal{P}}^{(2)}_{38}$ is
a variety of sums of powers $VSP(F,10)$ where $F$ is a cubic threefold in
${\bf P}^{5}$ and points of $VSP(F,10)$ correspond to the ways of writing the
equation of $F$ as a sum of ten cubes of linear forms ([IR1] and [IR3]).

In [RV, §5] it is shown that for a general $F\in V_{V-ap}$, where $V_{V-ap}$
is the divisor in the moduli space of cubic fourfolds consisting of cubics
apolar to a Veronese surface, the variety $VSP(F,10)$ is actually singular
along a K3 surface $S$ and its desingularization

$X=\widetilde{VSP(F,10)}\,\longrightarrow\,Y=VSP(F,10)$

is a divisorial contraction and the exceptional divisor $E$ is a ${\bf
P}^{1}$-fibration over $S$. From our results we deduce the following which
implies that a general point in ${\mathcal{D}}^{(2)}_{38,38,\alpha}$
corresponds, birationally, to the contraction above.

######  Proposition 5.4.

For a general $F\in V_{V-ap}$, the desingularization $X\rightarrow Y$ is a BN
contraction and $X=\widetilde{VSP(F,10)}$ is birational to $S^{[2]}$ where
$S=Sing(Y)$ is a K3 surface of degree $38$. The Brauer class of the conic
bundle $E\rightarrow S$ is trivial and $E$ is isomorphic to the
projectivisation of the Mukai bundle on $S$ with Mukai vector $s=(2,h,10)$
where $h^{2}=38$.

The Picard lattice of $S^{[2]}$, which is isomorphic to ${\bf Z}h\oplus{\bf
Z}\xi=<38>\oplus<-2>$, contains the $-10$ class $2h-9\delta$ of divisibility
$2$ that defines a Mukai flop from $S^{[2]}$ to $\overline{S^{[2]}}$. On
$\overline{S^{[2]}}$ the BN class $E=39h-170\xi$ is contracted by the morphism
associated to the divisor $H=170h-741\delta$ of degree $38$.

Proof. Since the contraction is given by a movable divisor $H$ with
divisibility $\gamma=2$ we conclude from Table 1, where all the possible
divisorial contractions are listed, that we are in the case of
${\mathcal{D}}_{8k+6,8k+6}^{(2)}$ and $k=4$. In $Pic(S^{[2]})$ we must have
$H=ah+b\xi$, with $h$ the class induced by the polarization of degree $38$ on
$S$, and $38=H^{2}=38a^{2}-2b^{2}$ and $a$ should be even (since $\gamma=2$)
whereas the exceptional divisor $E$ should be perpendicular to $H$, with
$E^{2}=-2$ and moreover the divisibility of $E$ is one, so $E=ch_{38}+d\delta$
and $c$ should be odd. Thus $38c^{2}-2d^{2}=-2$ and $(c,d)$ is a solution of
the Pell equation $d^{2}-19c^{2}=1$ whose solutions are generated by
$(c,d)=(39,170)$. Then $HE=0$ (with $H\cdot h_{38}>0$) gives $(a,b)=(170,\pm
19\cdot 39)$, so up to isometry, these are the classes of $H$ and $E$.  $\Box$

##  6\. The Heegner divisor ${\mathcal{D}}^{(1)}_{4,16}$, EPW sextics and a
special EPW sextic

###  6.1. The hyper-Kähler fourfolds in ${\mathcal{D}}^{(1)}_{4,16,\alpha}$

Let $(X,H)$ be a general hyper-Kähler fourfold of K3(2)-type with an ample
divisor class $H$ such that $q(H)=4$. Then

$\varphi_{H}\colon X\longrightarrow Y\subset{\bf P}^{9}$

is a birational morphism as can be checked in the case where $X=S_{4}^{[2]}$,
where $(S,h)$ is a degree $4$ polarized K3 surface, and $H$ is defined by $h$.
Indeed, the map given by $H$ factorises through the HC contraction $S^{[2]}\to
Sym^{2}(S)$ and an embedding $Sym^{2}(S)\subset Sym^{2}({\bf P})\subset{\bf
P}^{9}$, so it is birational.

Now we assume that $X$ has Picard rank two, that $H$ is only big and nef and
moreover that it induces a BN contraction. According to Table 1, the period
point of $(X,H)$ then lies in ${\mathcal{D}}^{(1)}_{4,16,\alpha}$. Let the
contraction be given by $X\supset E\to K\subset Y\subset{\bf P}^{9}$, then $K$
is a K3 surface of degree $4$.

The Brauer class of $E$ in $Br(K)_{2}$ is non-trivial for the general $(X,H)$
(see Proposition 3.6).

We show in §6.2 that these hyper-Kähler fourfolds are also double EPW sextics.
The natural involution on the fourfold $X$ permutes two conic bundles, one of
which is $E$, the other will be denoted by $E^{\prime}$. Both conic bundles
turn out to be birational to the complete intersection of the EPW sextic (in a
${\bf P}^{5}$) with a quadric, but this intersection is not a normal variety.

Since the general case remains somewhat mysterious, we study in §6.3 a
codimension one subvariety of the divisor ${\mathcal{D}}^{(1)}_{4,16,\alpha}$
which parametrizes Hilbert squares of degree four K3 surfaces with a degree
four rational curve. In that case $E$ is a trivial conic bundle but certain
aspects of the geometry are still interesting. Finally, in §6.4 we consider
the concrete case of a certain rational quartic curve on the quartic Fermat
surface in ${\bf P}^{3}$. It turns out that then the singular surface of the
EPW sextic has 60 isolated singular points, which makes the EPW sextic rather
special.

###  6.2. The divisor ${\mathcal{D}}^{(1)}_{4,16,\alpha}$ and EPW sextics.

In this section we provide some information on the $-2$-divisor $E\subset X$
which is a conic bundle over a quartic surface $K$. Since the period point of
$X$ lies in ${\mathcal{D}}^{(1)}_{4,16,\alpha}$, the Picard group is, with now
$\Lambda=H^{2}(X,{\bf Z})$:

$Pic(X)\,={\bf Z}H\,\oplus\,{\bf Z}E\,=\,\,<4>\,\oplus\,<-2>~{},\quad\
\gamma:=\mbox{div}_{\Lambda}(H)\,=\,1,\quad\det(Pic(X)^{\perp_{\Lambda}})=16~{}.$

The nef cone of $X$ has extremal rays spanned by $H$ and $H^{\prime}:=3H-4E$:

$\overline{Nef(X)}\,=\,{\bf R}_{\geq 0}H\,+\,{\bf R}_{\geq 0}H^{\prime},\qquad
H^{\prime}:=3H-4E,\quad H^{2}\,=\,(H^{\prime})^{2}\,=4~{}.$

see [BM1, §13]

The perpendiculars of the extremal rays are

$H^{\perp}={\bf Z}E,\quad(H^{\prime})^{\perp}={\bf Z}E^{\prime},\qquad
E^{\prime}\,:=\,2H-3E~{},\qquad E^{2}\,=\,(E^{\prime})^{2}\,=\,-2~{}.$

Hence $\phi_{H}$, $\phi_{H^{\prime}}$ contracts $E$, $E^{\prime}$
respectively.

Notice that $X$ has the ample class $L:=H-E$ of square two and thus
$\phi_{L}:X\rightarrow{\bf P}^{5}$ exhibits $X$ as a double EPW sextic. The
covering involution $\iota$ induces an involution $\iota^{*}$ on $Pic(X)$
which is minus the reflection in the orthogonal complement of $L$.

$\phi_{L}:\,X\,\longrightarrow\,Y\,:=\,X/\iota\,\subset\,{\bf P}^{5},\qquad
L\,:=\,H-E~{}.$

The involution $\iota^{*}$ interchanges the two extremal rays, and also $E$,
$E^{\prime}$ are exchanged. Thus these two divisors, which are conic bundles
over quartic surfaces, are mapped to the same threefold
$F=\phi_{L}(E)=\phi_{L}(2H-3E)\subset{\bf P}^{5}$, of degree $12$ by
Proposition 4.1.

Notice that the sum of the classes of the two $-2$-divisors is
$E+E^{\prime}=2L$. Since $\phi_{L}(X)=Y$, an EPW sextic, $Y$ is not contained
in any quadric and thus the pull-back map
$\phi_{L}^{*}:H^{0}({\mathcal{O}}_{{\bf P}^{5}}(2))\rightarrow H^{0}(2L)$ is
injective and also surjective for dimension reasons. Thus there is a quadric
$Q\subset{\bf P}^{5}$ which cuts out the image $F$ of the $-2$-divisors in
$Y$. As this intersection is a threefold of degree $2\cdot 6=12$ and also $F$
has degree 12 we conclude that $F=Q\cap Y$. By adjunction we compute the
canonical sheaf $\omega_{F}=\mathcal{O}_{F}(2)$. Now since $\phi_{L}|_{E}$
does not contract curves and the canonical divisor $K_{E}=E|_{E}\neq 2L|_{E}$
we conclude that the image $F$ of $E$ is not normal.

###  6.3. Example: Hilbert squares of certain degree four K3’s

We construct a more explicit example of fourfolds, actually Hilbert squares,
in ${\mathcal{D}}^{(1)}_{4,16,\alpha}$. Let $(S,h)$ be a degree $4$ K3 surface
with a $(-2)$-curve $n$ so that $hn=4$. Notice that the quartic curve
$n\subset{\bf P}^{3}$ lies on a quadric surface and that the residual curve
$n^{\prime}$ is again a rational quartic with class $n^{\prime}:=2h-n$. The
linear system $h_{12}:=h+2n$ has degree

$h_{12}^{2}\,=(h+2n)^{2}=\,h^{2}\,+\,4hn\,+\,4n^{2}\,=\,4+16-8\,=\,12,\qquad
h_{12}n\,=\,4-4=0~{}.$

So in general $\phi_{h_{12}}:S\rightarrow S_{12}\subset{\bf P}^{7}$ contracts
the rational curve $n$ and $S_{12}$ is a nodal degree $12$ K3 surface.

Let $H_{12}$ be the divisor class on the Hilbert square $S^{[2]}$ defined by
$h_{12}$ on $S$ and let

$H\,:=\,H_{12}-2\xi,\qquad H^{2}\,=\,12-8\,=\,4~{}.$

The divisor $n$ on $S$ defines the divisor $E:=n+S$ which is birational to
${\bf P}^{1}\times S$. The class of $E$ is $[E]=n=(n,0)\in H^{2}(S^{[2]},{\bf
Z})=H^{2}(S,{\bf Z})\oplus{\bf Z}\xi$, where $2\xi$ is the divisor on
$S^{(2)}$ parametrizing the non-reduced subschemes. Using the BB-form we have:

$N\,:=\,[E]\,=\,[n+S]~{},\qquad HN\,=\,0,\qquad N^{2}\,=\,-2~{}.$

Consider the subgroup

$K\,:=\,{\bf Z}H\,\oplus\,{\bf Z}N\qquad(\subset H^{2}(S^{[2]},{\bf Z}))~{},$

with the BB-form on $K$ we find the lattice $K\cong<4>\oplus<-2>$.

######  Lemma 6.1.

The pairs $(S^{[2]},K)$ are parametrized by a codimension $1$ subvariety of
the irreducible component ${\mathcal{D}}^{(1)}_{4,16}$ of the Heegner divisor
${\mathcal{D}}^{(1)}_{4,16}$. The line bundle $L$ on $S^{[2]}$ with $L^{2}=2$
from §6.2 is given by

$L\,=\,H\,-\,E\,=\,H_{10}\,-\,2\xi\;\in\,Pic(S^{[2]})~{},$

where $H_{10}$ is induced by $h_{10}:=h+n\in Pic(S)$.

Proof. We may assume that $h,n\in H^{2}(S,{\bf Z})=U^{3}\oplus E_{8}(-1)^{2}$
are in the first two components $U^{2}$ and are given by

$h\,=\,(1,2)_{1},\qquad n\,=\,(0,4)_{1}+(1,-1)_{2}\qquad(\subset U^{3}\oplus
E_{8}(_{1})^{2})~{}.$

Then $h_{12}=h+2n=(1,10)_{1}+(2,-2)_{2}$ and in $\Lambda=H^{2}(S,{\bf
Z})\oplus{\bf Z}\xi$ we have $K=<H,N>$ with

$H\,=\,H_{12}\,-\,2\xi\,=\,(1,10)_{1}\,+\,2(1,-1)_{2}\,-\,2\xi,\qquad
N\,=\,(0,4)_{1}+(1,-1)_{2}~{}.$

Since $H^{2}=4$ and the divisibility of $N$ in $\Lambda$ is one (since
$N\cdot(0,1)_{2}=1$) we see from Table 1 that $(S^{[2]},K)$ lies in
${\mathcal{D}}^{(1)}_{4,16}$.

Recall that $L=H-E=H_{12}-N-2\xi$ in $Pic(S^{[2]})$. Since $H_{12}$ is induced
by $h+2n\in Pic(S)$, $H_{12}-N$ is induced by $h+n$.  $\Box$

###  6.4. The Fermat quartic and a special EPW sextic

We consider the following smooth rational quartic curve $n$ in ${\bf P}^{3}$:

$n\;:=\;\\{\,(s^{4}:\,st^{3}:\,s^{3}t:\,t^{4})\,\in\,{\bf
P}^{3}:\;(s:t)\,\in\,{\bf P}^{1}\,\\}~{}.$

This curve lies in the smooth quartic surface

$S:\qquad
f\,:=\,z_{1}^{4}\,+\,z_{2}^{4}\,-\,z_{0}z_{3}(z_{0}^{2}+z_{3}^{2})\,=\,0~{}.$

Notice that $S$ is (projectively) isomorphic to the quartic Fermat surface
(substitute $z_{0}:=z_{0}+z_{3},\,z_{3}:=z_{0}-z_{3}$).

The intersection of $S$ with the quadric

$Q:\qquad q\,:=\,z_{0}z_{3}\,-\,z_{1}z_{2}\,=\,0$

consists of $n$ and another smooth rational degree four curve $n^{\prime}$
obtained by permuting the second and third coordinates ($z_{1}\leftrightarrow
z_{2}$) on $n$:

$S_{4}\cdot Q\,=\,n\,+\,n^{\prime}~{},\qquad
n^{\prime}\,:=\,\\{\,(s^{4}:\,s^{3}t:\,st^{3}:\,t^{4})\,\in\,{\bf
P}^{3}\,\\}~{}.$

Thus $2h_{4}=n+n^{\prime}$ and $h_{10}=h+n=3h-n^{\prime}$. So we see that
$h_{10}$ is given by the restriction to $S$ of the cubics on ${\bf P}^{3}$
containing $n^{\prime}$. Notice that these cubics are independent of $S$ and
depend only on the choice of $n^{\prime}$.

The ideal of $n^{\prime}$ is generated by the quadratic polynomial $q$ and
three cubics:

$f_{1}\,:=\,-z_{0}^{2}z_{2}\,+\,z_{1}^{3},\qquad
f_{2}\,:=\,z_{0}z_{2}^{2}\,-\,z_{1}^{2}z_{3},\qquad
f_{3}\,:=\,-z_{1}z_{3}^{2}\,+\,z_{2}^{3}~{}.$

Thus the map $\phi_{h_{10}}$ is defined as follows:

$\phi_{h_{10}}\,:\,S\,\longrightarrow\,S_{10}\,\subset{\bf
P}^{6},\qquad\phi_{h_{10}}\,=\,(z_{0}q:\,z_{1}q:\,z_{2}q:\,z_{3}q:\,f_{1}:\,f_{2}:\,f_{3})~{}.$

Notice that $\phi_{h_{10}}(n)$ is the conic $C_{n}\subset S_{10}$ defined by
$y_{0}=\ldots=y_{3}=y_{4}y_{6}+y_{5}^{2}=0$.

####  6.4.1. The image $Z$ of ${\bf P}^{3}$ in ${\bf P}^{6}$

The coordinate functions of the map $\phi_{h_{10}}$ do not depend on the
quartic surface $S_{4}$ but only on the quartic curve $n^{\prime}$. The image
of ${\bf P}^{3}$ under $\phi_{h_{10}}$ is (according to Magma) a smooth
threefold $Z$ of degree $5$ in ${\bf P}^{6}$ which is the intersection of five
quadrics defined by the polynomials:

$F_{0}\,:=\,-y_{0}y_{5}\,+\,y_{1}^{2}\,-\,y_{2}y_{4},\quad
F_{1}\,:=\,y_{0}y_{2}\,-\,y_{1}y_{5}\,-\,y_{3}y_{4},\quad
F_{2}\,:=\,-y_{1}y_{2}\,+\,y_{4}y_{6}\,+\,y_{5}^{2},$
$F_{3}\,:=\,-y_{0}y_{6}\,+\,y_{1}y_{3}\,+\,y_{2}y_{5},\quad
F_{4}\,:=\,-y_{1}y_{6}\,+\,y_{2}^{2}\,+\,y_{3}y_{5}~{}.$

The birational inverse of $\phi_{h_{10}}:{\bf P}^{3}\rightarrow Z$ is the
projection of $Z$ given by

$(y_{0}:\ldots:y_{6})\mapsto(y_{0}:\ldots:y_{3}),$

its base locus in $Z$ is the conic $C_{N}$.

####  6.4.2. The Del Pezzo quintic threefold.

We recall the definition from [MU] (cf. [KPS, Section 5.1]) of a quintic Fano
threefold $Z_{MU}$ of index $2$ in ${\bf P}^{6}$, known as the Del Pezzo
quintic threefold, which is also a linear section of $Gr(2,5)\subset{\bf
P}^{9}$. We show that it is isomorphic to $Z$. (More intrinsically this could
be done by studying the projection of $Z_{MU}$ from a conic to ${\bf P}^{3}$,
the inverse map is then given by the cubics in the ideal of a degree four
rational curve.)

The variety $Z_{MU}$ is defined by the following $5$ quadratic polynomials
([MU, p.505]):

$A_{0}\,:=\,h_{0}h_{4}-4h_{1}h_{3}+3h_{2}^{2},\quad
A_{1}:=h_{0}h_{5}-3h_{1}h_{4}+2h_{2}h_{3},\quad
A_{2}:=h_{0}h_{6}-9h_{2}h_{4}+8h_{3}^{2},$
$A_{3}:=h_{1}h_{6}-3h_{2}h_{5}+2h_{3}h_{4},\quad
A_{4}:=h_{2}h_{6}-4h_{3}h_{5}+3h_{4}^{2}~{}.$

To verify that $Z\cong Z_{MU}$ it suffices to verify that for $j=0,\ldots,4$
one has:

$F_{j}(18h_{1},-3r^{3}h_{2},-3r^{4}h_{4},-18r^{2}h_{5},r^{2}h_{0},12rh_{3},18h_{6})\,=\,324A_{j}(h_{0},\ldots,h_{6})~{},\qquad(r^{5}\,=\,18)~{}.$

####  6.4.3. The image $S_{10}$ of $S$

The K3 surface $S_{10}=\phi_{h_{10}}(S)\subset{\bf P}^{6}$ is the intersection
of $Z=\phi_{h_{10}}({\bf P}^{3})$ with a sixth quadric. In the case we
consider here, one can take the quadric defined by

$F_{5}\,:=\,y_{0}^{2}+y_{1}y_{4}+y_{2}y_{6}+y_{3}^{2}~{}.$

####  6.4.4. The map from $S^{[2]}$ to ${\bf P}^{5}$

From now on we identify the K3 surfaces $S,S_{10}$. Recall the rational map

$\phi_{L}:\,S^{[2]}\,\longrightarrow\,Y\quad(\subset\,{\bf P}^{5})$

given by the divisor class $L=H_{10}-2\xi$, where $H_{10}$ corresponds to
$h_{10}\in Pic(S)$, so $H_{10}^{2}=10$, and $2\xi$ is the divisor
parametrizing non-reduced subschemes. It is defined by $H-2\xi$ is given by
associating to $p+q\in S^{[2]}$ the hyperplane in the ${\bf P}^{5}$ of
quadrics defining $S$ which vanish on the line spanned by $p$ and $q$. This
map can be given explicitly as in the proof of Proposition 2.2 of [DG+]: if
$p,q\in S$ and $F$ is a quadratic polynomial vanishing on $S$, then $F(\lambda
p+\mu q)=2\lambda\mu B(p,q)$ for a bilinear form $B$. The bilinear forms
defined by the six defining quadrics are the coordinate functions of a
rational map $S\times S\rightarrow{\bf P}^{5}$ which factors over $S^{[2]}$ to
give the map defined by $L$.

####  6.4.5. The EPW sextic $Y$ in ${\bf P}^{5}$

The map $\phi_{L}$ still defines a (rational) 2:1 map onto an EPW sextic in
${\bf P}^{5}$. We checked this explicitly by using the bilinear forms $B_{j}$
associated to the following six quadratic forms $F_{j}$ (notice the non-
standard ordering):

$F_{5},\;F_{0},\;F_{1},\;F_{2},\;F_{4},\;F_{3}~{}.$

We used Magma for the computations below. The image of $S^{[2]}$ under
$\phi_{L}$ is indeed a sextic fourfold $Y$ whose singular locus has dimension
$2$ and degree $40$, as expected for an EPW sextic. The explicit equation has
$53$ terms (the coordinates on ${\bf P}^{5}$ are $(x_{0}:\ldots:x_{5})$):

$\begin{array}[]{l}-4x_{0}^{3}x_{1}x_{3}x_{4}\,+\,x_{0}^{3}x_{1}x_{5}^{2}\,+\,x_{0}^{3}x_{2}^{2}x_{4}\,+\,x_{0}^{3}x_{2}x_{3}x_{5}\,+\,x_{0}^{3}x_{3}^{3}\,+\,4x_{0}^{2}x_{1}^{3}x_{4}\,-\,3x_{0}^{2}x_{1}^{2}x_{2}x_{5}\,-\,3x_{0}^{2}x_{1}^{2}x_{3}^{2}\\\
\,+\,3x_{0}^{2}x_{1}x_{2}^{2}x_{3}\,+\,4x_{0}^{2}x_{1}x_{4}^{3}\,-\,x_{0}^{2}x_{2}^{4}\,-\,3x_{0}^{2}x_{2}x_{4}^{2}x_{5}\,-\,3x_{0}^{2}x_{3}^{2}x_{4}^{2}\,+\,3x_{0}^{2}x_{3}x_{4}x_{5}^{2}\,-\,x_{0}^{2}x_{5}^{4}\,+\,3x_{0}x_{1}^{4}x_{3}\\\
\,-\,2x_{0}x_{1}^{3}x_{2}^{2}\,+\,6x_{0}x_{1}^{2}x_{3}x_{4}^{2}\,-\,6x_{0}x_{1}^{2}x_{4}x_{5}^{2}\,-\,6x_{0}x_{1}x_{2}^{2}x_{4}^{2}\,-\,12x_{0}x_{1}x_{2}x_{3}x_{4}x_{5}\,+\,8x_{0}x_{1}x_{2}x_{5}^{3}\\\
\,-\,4x_{0}x_{1}x_{3}^{3}x_{4}\,+\,3x_{0}x_{1}x_{3}^{2}x_{5}^{2}\,+\,8x_{0}x_{2}^{3}x_{4}x_{5}\,+\,3x_{0}x_{2}^{2}x_{3}^{2}x_{4}\,+\,x_{0}x_{2}x_{3}^{3}x_{5}\,+\,3x_{0}x_{3}x_{4}^{4}\,-\,2x_{0}x_{4}^{3}x_{5}^{2}\\\
\,-\,x_{1}^{6}\,-\,3x_{1}^{4}x_{4}^{2}\,+\,4x_{1}^{3}x_{2}x_{4}x_{5}\,+\,4x_{1}^{3}x_{3}^{2}x_{4}\,-\,2x_{1}^{3}x_{3}x_{5}^{2}\,-\,6x_{1}^{2}x_{2}^{2}x_{3}x_{4}\,-\,3x_{1}^{2}x_{2}x_{3}^{2}x_{5}\,-\,3x_{1}^{2}x_{4}^{4}\\\
\,+\,4x_{1}x_{2}^{4}x_{4}\,+\,8x_{1}x_{2}^{3}x_{3}x_{5}\,+\,x_{1}x_{2}^{2}x_{3}^{3}\,+\,4x_{1}x_{2}x_{4}^{3}x_{5}\,+\,4x_{1}x_{3}^{2}x_{4}^{3}\,-\,6x_{1}x_{3}x_{4}^{2}x_{5}^{2}\,+\,4x_{1}x_{4}x_{5}^{4}\\\
\,-\,4x_{2}^{5}x_{5}\,-\,x_{2}^{4}x_{3}^{2}\,-\,2x_{2}^{2}x_{3}x_{4}^{3}\,-\,3x_{2}x_{3}^{2}x_{4}^{2}x_{5}\,+\,8x_{2}x_{3}x_{4}x_{5}^{3}\,-\,4x_{2}x_{5}^{5}\,+\,x_{3}^{3}x_{4}x_{5}^{2}\,-\,x_{3}^{2}x_{5}^{4}\,-\,x_{4}^{6}~{}.\end{array}$

####  6.4.6. The trivial conic bundle

The divisor $E=n+S\subset S^{[2]}$ is the limit of a conic bundle in the
general $X\in{\mathcal{D}}^{(1)}_{4,16,\alpha}$. The image $F=\phi_{L}(E)$ is
the image of $C_{n}\times S_{10}\subset({\bf P}^{6})^{2}$, notice that
$C_{n}\times S_{10}\subset C_{n}\times Z$. We first computed the image of
$C_{n}\times Z$ under the map given by the bilinear forms $B_{j}(p,q)$, it is
the quadric in ${\bf P}^{5}$ defined by:

$Q_{n}\,:\quad q_{N}\,=\,0,\qquad\mbox{with}\quad
q_{n}\,:=\,x_{1}x_{4}\,-\,x_{2}x_{5}~{}.$

Thus we have an interpretation of the quadric that cuts out $F=\phi_{L}(E)$ in
terms of the geometry of $S_{10}$. The trivial conic bundle $n\times S\subset
S^{2}$ maps first of all to a subvariety of $S^{[2]}$ that is birationally
isomorphic to the trivial bundle ${\bf P}^{1}$-bundle over $S$, and the image
of this subvariety under $\phi_{L}$ is $F=Q_{n}\cap Y$.

####  6.4.7. The singular surface of $Y$

The singular surface of $Y$ is rather remarkable in that its singular locus
consists of $60$ points, which we call the very singular points of $Y$. These
points are rational over the splitting field $L$ of the polynomial $x^{4}-2$.
Notice that the roots of this polynomial are $\alpha_{k}:=i^{k}\sqrt[4]{2}$,
$k=0,1,2,3$, and thus $i=\alpha_{1}/\alpha_{0}\in L$, with $i^{2}=-1$, and
also $\zeta_{8}:=(1+i)/\sqrt{2}\in L$, with $\zeta_{8}^{2}=i$. Thus $L$
contains the field ${\bf Q}(\zeta_{8})$ of eight-roots of unity. Only two of
the singular points are defined over ${\bf Q}$, there are $11$ other Galois
orbits, one of length $2$, six of length $4$ and four of length $8$.

####  6.4.8. The very singular points

The lines on the Del Pezzo quintic threefold $Z$ in ${\bf P}^{6}$ are
parametrized by ${\bf P}^{2}$. The general line can be found as the image of a
secant line to the rational curve $n^{\prime}\subset{\bf P}^{3}$ under the map
$\phi$. Since $Z$ is defined by the last 5 quadrics of the six in the list
given in Section 6.4.5 that define the K3 surface $S\subset{\bf P}^{6}$, the
first quadric will intersect a line on $Z$, not contained in $S_{10}$, in two
points. This gives a rational map ${\bf P}^{2}\rightarrow S^{[2]}$. Under the
map $S^{[2]}\rightarrow Y$ the image of this ${\bf P}^{2}$ is contracted to
$(1:0:\ldots:0)\in Y$, which is in fact one of the sixty very singular points
of $Y$. See [O1, section 3.2] and [Be] for the map $S^{[2]}\rightarrow Y$.

Let $C\subset S_{10}$ be a smooth conic and let ${\bf P}_{C}\subset{\bf
P}^{6}$ be the plane it spans. Any quadric containing $S_{10}$ either contains
${\bf P}_{C}$ or intersects ${\bf P}_{C}$ in $C$. If the quadric $F=0$
contains ${\bf P}_{C}$, then for any two points $p,q\in C$ one has $B(p,q)=0$
where $B$ is the associated bilinear form. Thus $C^{[2]}\cong{\bf
P}^{2}\subset S^{[2]}$ is contracted under $\phi_{L}:S^{[2]}\rightarrow Y$ and
the image is a very singular point of $Y$ (cf. [O2, Section 4] but notice that
$S_{10}$ does contain lines, in fact there are $20$ lines on $S_{10}$, for
example $(is:it:\zeta_{8}t:s:\zeta_{8}^{3}t:0:t)$ parametrizes a line on
$S_{10}$). Similarly, in case the conic $C=l\cup m$ is reducible, the image of
$l\times m\subset S^{2}$ in $S^{[2]}$ is contracted by $\phi_{L}$.

In particular the plane $y_{0}=\ldots=y_{3}=0$ spanned by the conic
$C_{n}\subset S_{10}$ lies in the quadric $F_{i}=0$ except if $i=2$ and thus
$S^{2}C_{n}$ is contracted to the very singular point $(0:0:0:1:0:0)$. The
other conics that are contracted are not defined over the rationals (there are
$59$ conics on $S_{10}$, $34$ of them are reducible).

## References

  * [Al] A. Alzati, A concrete example of symplectic duality among K-3 surface, Beiträge Algebra Geom. 51 (2010) 391–415.
  * [AJ] M. Artin, A.J. de Jong, Stable orders over surfaces, unpublished manuscript (2004), see: http://www.math.lsa.umich.edu/courses/711/ordersms-num.pdf.
  * [BM1] A. Bayer, E. Macrì, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math., 198 (2014) 505–590.
  * [B] A. Beauville, Determinantal Hypersurfaces, Michigan Math. J. 48 (2000) 39–64.
  * [Be] P. Beri, EPW sextics associated to K3 surfaces, arXiv:2011.12223.
  * [BKK] G. Bini, G. Kapustka, M. Kapustka, Symmetric locally free resolutions and rationality problems, arXiv:2005.02092.
  * [BOSS] R. Braun, G. Ottaviani, M. Schneider, F.-O. Schreyer, Classification of conic bundles in $P^{5}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996) 69–97.
  * [C] A. Calderaru, Derived categories of twisted sheaves on Calabi–Yau manifolds, Ph.D. Thesis, Cornell University (2000).
  * [CI] D. Chan, C. Ingalls, Conic bundles and Clifford algebras, New trends in noncommutative algebra, 53–75, Contemp. Math., 562, AMS, 2012.
  * [DHOV] O. Debarre, F. Han, K. O’Grady, C. Voisin, Hilbert squares of K3 surfaces and Debarre-Voisin varieties, J. Éc. polytech. Math. 7 (2020) 653–710.
  * [DK] O. Debarre, A. Kuznetsov, _Gushel-Mukai varieties: Moduli_ , Internat. J. Math. 31 (2020) 2050013 (59 pages).
  * [DM] O. Debarre, E. Macrì, On the period map for polarized hyperkähler fourfolds, Int. Math. Res. Not. IMRN 2019 22 (2019) 6887–6923.
  * [Dr] S. Druel, Quelques remarques sur la decomposition de Zariski divisorielle sur les varietes dont la première classe de Chern est nulle. Math. Z. 267 (2011) 413–423.
  * [DV] O. Debarre, C. Voisin, Hyper-Kähler fourfolds and Grassmann geometry, J. Reine Angew. Math. 649 (2010) 63–87.
  * [DMS] W. Decker, N. Manolache, and F.-O. Schreyer, Geometry of the Horrocks bundle on $P^{5}$, Complex projective geometry, London Math. Soc. Lecture Note Ser., 179, pp. 128–148, Cambridge Univ. Press 1992.
  * [DG+] M. Donten-Bury, B. van Geemen, G. Kapustka, M. Kapustka, J. Wisniewski, _A very special EPW sextic and two hyper-Kähler fourfolds._ Geom. Topol. 21 (2017) 1179–1230.
  * [Fr] R. Friedman, Algebraic surfaces and holomorphic vector bundles. Springer 1998.
  * [vG] B. van Geemen, _Some remarks on Brauer groups of K3 surfaces_ , Adv. Math. 197 (2005) 222–247.
  * [GO] F. Gounelas, J.C. Ottem, _Remarks on the positivity of the cotangent bundle of a K3 surface_ , Épijournal de Géométrie Algébrique 4 (2020).
  * [HAG] R. Hartshorne, Algebraic Geometry. GTM 52, Springer.
  * [HM] J. Harris, I. Morrison, Moduli of Curves. GTM 187, Springer.
  * [Ha] B. Hassett, Special cubic fourfolds, Compositio Math. 120 (2000) 1–23.
  * [HT1] B. Hassett, Y. Tschinkel, Rational curves on holomorphic symplectic fourfolds, Geom. Funct. Anal. 11 (2001) 1201–1228.
  * [HT2] B. Hasset, Y. Tschinkel, _Extremal rays and automorphisms of holomorphic symplectic varieties_ , in: “K3 surfaces and their Moduli”, Progr. Math, Vol. 315, pp. 73–95, Birkhhäuser 2016.
  * [Hat] A. Hatcher, Algebraic topology. Cambridge University Press 2002.
  * [HoSt] N. Hoffmann, U. Stuhler, Moduli schemes of generically simple Azumaya modules, Doc. Math. 10 (2005), 369–389.
  * [Hu1] D. Huybrechts, Compact hyperkähler manifolds: Basic results, Invent. Math. 135 (1999) 63–113.
  * [Hu2] D. Huybrechts, Generalized Calabi-Yau structures, K3 surfaces, and B-fields, Internat. J. Math. 19 (2005) 13–36.
  * [Hu3] D. Huybrechts, Lectures on K3 surfaces. Cambridge University Press 2016.
  * [HS1] D. Huybrechts, P. Stellari, Equivalences of twisted K3 surfaces, Math. Ann. 332 (2005) 901–936.
  * [HS2] D. Huybrechts, P. Stellari, Proof of Caldararu’s conjecture, in: “Moduli spaces and arithmetic of algebraic varieties”. Adv. Stud. Pure Math. 45 (2007), 31–42.
  * [IR1] A. Iliev and K. Ranestad, K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds, Trans. Amer. Math. Soc. 353 (2001) 1455–1468.
  * [IR2] A. Iliev and K. Ranestad, Geometry of the Lagrangian Grassmannian LG(3,6) with applications to Brill-Noether Loci, Michigan Math. J. 53 (2005) 383–417.
  * [IR3] A. Iliev and K. Ranestad, Addendum to “K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds”, C. R. Acad. Bulgare Sci. 60 (2007) 1265–1270.
  * [IOOV] C. Ingalls, A. Obus, E. Ozman, B. Viray, Unramified brauer classes on cyclic covers of the projective plane, Brauer groups and obstruction problems: moduli spaces and arithmetic, Progr. Math. Vol. 320, pp. 115–153, Birkhäuser 2013.
  * [IK] C. Ingalls and M. Khalid, An explicit derived equivalence of Azumaya algebras on K3 surfaces via Koszul duality, J. of Algebra 432 (2015) 300–327.
  * [KV] G. Kapustka, A. Verra, On Morin configurations of higher length, Int. Math. Res. Not., https://doi.org/10.1093/imrn/rnaa170
  * [KLM] A. Knutsen, M. Lelli-Chiesa, G. Mongardi, Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds, Trans. Amer. Math. Soc. 371 (2019) 1403–1438.
  * [Ko] J. Kollár, Severi-Brauer varieties; a geometric treatment, arXiv:1606.04368.
  * [Ku1] A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008) 1340–1369.
  * [Ku2] A. Kuznetsov, Derived categories of cubic fourfolds, Cohomological and Geometric Approaches to Rationality Problems. New Perspectives, Progr. Math, Vol. 282, pp. 219–243, Birkhäuser 2010.
  * [KPS] A. Kuznetsov, Y. Prokhorov, C. Shramov, Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Jpn. J. Math. 13 (2018) 109–185.
  * [MS] E. Macrì, P. Stellari, Fano varieties of cubic fourfolds containing a plane, Math. Ann. 354 (2012) 1147–1176.
  * [Ma] E. Markman, Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections, Kyoto J. Math. 53 (2013) 345–403.
  * [MS+] K. McKinnie, J. Sawon, S. Tanimoto, A. Várilly-Alvarado, Brauer groups on K3 surfaces and arithmetic applications, Brauer groups and obstruction problems, Progr. Math., 320, 177–218, Birkhäuser 2017.
  * [Mu] S. Mukai, Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. USA 86 (1989) 3000–3002.
  * [MU] S. Mukai, H. Umemura, Minimal rational threefolds, in: Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, 1983, pp. 490–518.
  * [Nk] T. Nakashima, Stable vector bundles with $c_{1}=0$ on K3 surfaces, Arch. Math. 74 (2000) 148–153.
  * [Nm] Y. Namikawa, Deformation theory of singular symplectic n-folds, Math. Ann. 319 (2001) 597–623.
  * [Ni] V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izv. 14 (1980) 103–167.
  * [O1] K. G. O’Grady, Dual double EPW-sextics and their periods, Pure Appl. Math. Q. 4 (2008) 427–468.
  * [O2] K. G. O’Grady, Double covers of EPW-sextics, Michigan Math. J. 62 (2013) 143–184.
  * [O3] K. G. O’Grady, Periods of double EPW-sextics, Math. Z. 280 (2015) 485–524.
  * [RV] K. Ranestad, C. Voisin, _Variety of power sums and divisors in the moduli space of cubic fourfolds_ , Doc. Math. 22 (2017) 455–504.
  * [Sar] V. G. Sarkisov, On conic bundle structures, Math. USSR Izvestiya 20 (1982) 355–390.
  * [S] C. Shramov, Fiberwise bimeromorphic maps of conic bundles, Internat. J. Math. 30 (2019) 1950059, 12 pp.
  * [Vo] C. Voisin, Théorème de Torelli pour les cubiques de $P^{5}$, Invent. Math. 86 (1986) 577–601.
  * [Y1] K. Yoshioka, _Moduli spaces of twisted sheaves on a projective variety_ , Moduli Spaces and Arithmetic Geometry (Kyoto, 2004), 1–30, Adv. Stud. Pure Math. 45, Tokyo, 2006.