The aim of this paper is to compute all the Frenet apparatus of non-transversal intersection curves (hyper-curves) of three implicit hypersurfaces in Euclidean 4-space. The tangential direction at a ...transversal intersection point can be computed easily, but at a non-transversal intersection point, it is difficult to calculate even the tangent vector. If three normal vectors are parallel at a point, the intersection is “tangential intersection”; and if three normal vectors are not parallel but are linearly dependent at a point, we have “almost tangential” intersection at the intersection point. We give algorithms for each case to find the Frenet vectors (t,n,b1,b2) and the curvatures (k1,k2,k3) of the non-transversal intersection curve.
Understanding the intentions of drivers at intersections is a critical component for autonomous vehicles. Urban intersections that do not have traffic signals are a common epicenter of highly ...variable vehicle movement and interactions. We present a method for predicting driver intent at urban intersections through multi-modal trajectory prediction with uncertainty. Our method is based on recurrent neural networks combined with a mixture density network output layer. To consolidate the multi-modal nature of the output probability distribution, we introduce a clustering algorithm that extracts the set of possible paths that exist in the prediction output and ranks them according to probability. To verify the method's performance and generalizability, we present a real-world dataset that consists of over 23 000 vehicles traversing five different intersections, collected using a vehicle-mounted lidar-based tracking system. An array of metrics is used to demonstrate the performance of the model against several baselines.
We study the dynamical billiards on a symmetric lemon table Q(b), where Q(b) is the intersection of two unit disks with center distance b. We show that there exists δ0>0 such that for all ...b∈(1.5,1.5+δ0) (except possibly a discrete subset), the billiard map Fb on the lemon table Q(b) admits crossing homoclinic and heteroclinic intersections. In particular, such lemon billiards have positive topological entropy.
Given the recent advancements in autonomous driving functions, one of the main challenges is safe and efficient operation in complex traffic situations such as road junctions. There is a need for ...comprehensive testing, either in virtual simulation environments or on real-world test tracks. This paper presents a novel data analysis method including the preparation, analysis and visualization of car crash data, to identify the critical pre-crash scenarios at T- and four-legged junctions as a basis for testing the safety of automated driving systems. The presented method employs k-medoids to cluster historical junction crash data into distinct partitions and then applies the association rules algorithm to each cluster to specify the driving scenarios in more detail. The dataset used consists of 1056 junction crashes in the UK, which were exported from the in-depth "On-the-Spot" database. The study resulted in thirteen crash clusters for T-junctions, and six crash clusters for crossroads. Association rules revealed common crash characteristics, which were the basis for the scenario descriptions. The results support existing findings on road junction accidents and provide benchmark situations for safety performance tests in order to reduce the possible number parameter combinations.
We consider the vertex proper coloring problem for highly restricted instances of geometric intersection graphs of line segments embedded in the plane. Provided a graph in the class UNIT-PURE-
k
...-DIR, corresponding to intersection graphs of unit length segments lying in at most
k
directions with all parallel segments disjoint, and provided explicit coordinates for segments whose intersections induce the graph, we show for
k
=
4
that it is
NP
-complete to decide if a proper 3-coloring exists, and moreover,
#
P
-complete under many-one counting reductions to determine the number of such colorings. In addition, under the more relaxed constraint that segments have at most two distinct lengths, we show these same hardness results hold for finding and counting proper
k
-
1
-colorings for every
k
≥
5
. More generally, we establish that the problem of proper 3-coloring an arbitrary graph with
m
edges can be reduced in
O
m
2
time to the problem of proper 3-coloring a UNIT-PURE-4-DIR graph. This can then be shown to imply that no
2
o
n
time algorithm can exist for proper 3-coloring PURE-4-DIR graphs under the Exponential Time Hypothesis (ETH), and by a slightly more elaborate construction, that no
2
o
n
time algorithm can exist for counting the such colorings under the Counting Exponential Time Hypothesis (#ETH). Finally, we prove an
NP
-hardness result for the optimization problem of finding a maximum order proper 3-colorable induced subgraph of a UNIT-PURE-4-DIR graph.
•Propose an autonomous intersection management algorithm with the max-pressure control considering pedestrians.•Prove the proposed algorithm can achieve the optimal throughput for the combined ...vehicle-pedestrian flow.•Show the efficiency of pedestrians and vehicles are negatively correlated at the intersection in simulations.
With the development of vehicle-to-infrastructure and vehicle-to-vehicle technologies, vehicles will be able to communicate with the controller at the intersection. Autonomous driving technology enables vehicles to follow the instructions sent from the controller precisely. Autonomous intersection management considers each vehicle as an agent and coordinates vehicle trajectories to resolve vehicle conflicts inside an intersection. This study proposes an autonomous intersection management algorithm called AIM-ped considering both vehicles and pedestrians which is able to produce the total optimal throughput when combined with max pressure control. This study analyzes the stability properties of the algorithm based on a simpler version of AIM-ped, which is a conflict region model of the autonomous intersection management. To implement the proposed algorithm in simulation, this study combines the max-pressure control with an existing trajectory optimization algorithm to calculate optimal vehicle trajectories. Simulations are conducted to test the effects of pedestrian demand on intersection efficiency. The simulation results show that delays of pedestrians and vehicles are negatively correlated and the proposed algorithm can adapt to the change in the pedestrian demand and activate vehicle movements with conflicting trajectories.
•Signal control algorithm using connected vehicle technology to improve operation.•Considers different development stages of connected vehicle technology.•Modification of automated vehicle ...trajectories to reduce delay and stops.•Proposes a fast branch and bound algorithm to improve the computational efficiency.•Tests show reduction in delay and stops and robustness to measurement errors.
Connected vehicle technology can be beneficial for traffic operations at intersections. The information provided by cars equipped with this technology can be used to design a more efficient signal control strategy. Moreover, it can be possible to control the trajectory of automated vehicles with a centralized controller. This paper builds on a previous signal control algorithm developed for connected vehicles in a simple, single intersection. It improves the previous work by (1) integrating three different stages of technology development; (2) developing a heuristics to switch the signal controls depending on the stage of technology; (3) increasing the computational efficiency with a branch and bound solution method; (4) incorporating trajectory design for automated vehicles; (5) using a Kalman filter to reduce the impact of measurement errors on the final solution. Three categories of vehicles are considered in this paper to represent different stages of this technology: conventional vehicles, connected but non-automated vehicles (connected vehicles), and automated vehicles. The proposed algorithm finds the optimal departure sequence to minimize the total delay based on position information. Within each departure sequence, the algorithm finds the optimal trajectory of automated vehicles that reduces total delay. The optimal departure sequence and trajectories are obtained by a branch and bound method, which shows the potential of generalizing this algorithm to a complex intersection.
Simulations are conducted for different total flows, demand ratios and penetration rates of each technology stage (i.e. proportion of each category of vehicles). This algorithm is compared to an actuated signal control algorithm to evaluate its performance. The simulation results show an evident decrease in the total number of stops and delay when using the connected vehicle algorithm for the tested scenarios with information level of as low as 50%. Robustness of this algorithm to different input parameters and measurement noises are also evaluated. Results show that the algorithm is more sensitive to the arrival pattern in high flow scenarios. Results also show that the algorithm works well with the measurement noises. Finally, the results are used to develop a heuristic to switch between the different control algorithms, according to the total demand and penetration rate of each technology.
Understanding unsaturated flow behaviors in fractured rocks is essential for various applications. A fundamental process in this regard is flow splitting at fracture intersections. However, the ...impact of geometrical properties of fracture intersections on flow splitting is still unclear. This work investigates the combined influence of geometry (intersection angle, fracture apertures, and inclination angle), liquid droplet length, inertia, and dynamic wetting properties on liquid splitting dynamics at fracture intersections. A theoretical model of liquid splitting is developed, considering the factors mentioned above, and numerically solved to predict the flow splitting behavior. The model is validated against carefully-controlled visualized experiments. Our results reveal two distinct splitting behaviors, separated by a critical droplet length. These behaviors shift from a monotonic to a non-monotonic trend with decreasing inclination angle. A comprehensive analysis further clarifies the impacts of the key factors on the splitting ratio, which is defined as the percentage of liquid volume entering the branch fracture. The splitting ratio decreases with increasing inclination angle, indicating a decrease in the gravitational effect on the branch fracture, which is directly proportional to the intersection angle. A non-monotonic relationship exists between the splitting ratio and the aperture ratio of the branch fracture to the main fracture. The results show that as the intersection angle decreases, the splitting ratio increases. Additionally, the influence of dynamic contact angles decreases with increasing intersection angle. These findings enhance our understanding of the impact of geometry on flow dynamics at fracture intersections. The proposed model provides a foundation for simulating and predicting unsaturated flow in complex fractured networks.
A sunflower with
r
petals is a collection of
r
sets over a ground set
X
such that every element in
X
is in no set, every set, or exactly one set. Erdős and Rado
5
showed that a family of sets of ...size
n
contains a sunflower if there are more than
n
!
(
r
-
1
)
n
sets in the family. Alweiss et al.
1
and subsequently, Rao
7
and Bell et al.
2
improved this bound to
(
O
(
r
log
n
)
)
n
. We study the case where the pairwise intersections of the set family are restricted. In particular, we improve the best known bound for set families when the size of the pairwise intersections of any two sets is in a set
L
. We also present a new bound for the special case when the set
L
is the nonnegative integers less than or equal to
d
using the techniques of Alweiss et al.
1
.
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