Decentralized spatial partitioning for multi-vehicle systems in spatiotemporal flow-field

We consider the problem of characterizing a generalized Voronoi diagram/partition of a convex polygon in a two-dimensional Euclidean space that encodes information about the proximity relations between a team of aerial/marine vehicles and arbitrary points in the partition space. These proximity relations are determined by the time required for each vehicle to reach an arbitrary point (time-to-go) in the partition space when driven by a locally optimal feedback control law in the presence of a spatiotemporal drift field. The main contribution of this work is the presentation of a partitioning algorithm, which is decentralized, in the sense that each vehicle can independently compute its corresponding cell from the generalized Voronoi partition without computing or receiving information about the cells of the other vehicles. Finally, we present numerical simulations using data from real drift fields to illustrate the key features of the decentralized solution to the proposed class of spatial partitioning problems.     

Decentralized spatial partitioning algorithms for multi-vehicle systems based on the minimum control effort metric

We consider the problem of characterizing a spatial partition of the position space of a team of vehicles with linear time-varying kinematics. The generalized metric that determines the proximity relations between the vehicles and an arbitrary target point in the partition space is the minimum control effort required for each vehicle to reach the latter point with zero miss distance and exactly zero velocity at a prescribed final time. We show that the solution to the generalized Voronoi partitioning problem can be associated with a special class of spatial partitions known as affine diagrams. Because the combinatorial complexity of the affine diagrams is comparable to the one of the standard Voronoi diagrams, their computation does not pose a significant challenge in applications of multi-vehicle systems. Subsequently, we propose an algorithm for the computation of the spatial partition, which is decentralized in the sense that each vehicle can compute an approximation of its own cell independently from the other vehicles from the same team without utilizing a common spatial mesh. Numerical simulations that illustrate the theoretical developments are also presented.     

Constructing the generalized local Voronoi diagram from laser rangescanner data

We consider the problem of building local free space representation by a circular-base robot, operating in an unknown bounded planar environment populated by obstacles of arbitrary shape. In the unknown environment, the robot depends on its sensors; in our case, this is the laser range scanner. A new structure is represented, termed the generalized local Voronoi diagram (GLVD), constructed directly from sensory data, obtained from an observation of the local environment, that is from the visible region, GLVD is obtained by deleting some edges from the ordinary Voronoi diagram, which is generated by the measured scanned points from the visible region. Crucial for the acquisition of the GLVD is the clusterization of the scanned points. Clusterization means grouping of the scanned points into distinct clusters, each having a specific property, e.g., for a particular point in the cluster, the nearest neighboring point should not lie further than a prescribed distance away. The relevant part of GLVD, the portion of GLVD which is in accordance with the generalized Voronoi diagram of the environment, is determined. Eventual differences between these two structures are discussed and experimental results are presented. With the superposition of several GLVDs we show that this structure can be used to construct a global map of the environment, for which the position and the orientation of the robot is needed     

基于空间邻近的点目标聚类方法

空间聚类是空间数据挖掘和知识发现领域的主要研究方向之一,但点目标空间分布密度的不均匀、分布形状的多样化,以及"多桥"链接问题的存在,使得基于距离和密度的聚类算法不能高效且有效地识别聚集性高的点目标。提出了基于空间邻近的点目标聚类方法,通过Voronoi建模识别点目标间的空间邻近关系,并以Voronoi势力范围来定义相似度准则,最终构建树结构以实现点目标的聚集模式识别。实验将所提算法与K-means、具有噪声的基于密度的聚类(DBSCAN)算法进行比较分析,结果表明算法能够发现密度不均且任意形状分布的点目标集群,同时准确划分"桥"链接的簇,适用于空间点目标异质分布下的聚集模式识别。     

不同交通网络速度的Voronoi图的结晶生成

针对以欧氏距离为度量的Voronoi图所分割必须是均质空间的局限性,为了体现实际分析中的交通网络所导致的空间不均质性,在现有Voronoi图理论成果的基础上,提出了以交通时间距离为度量的基于交通网络的Voronoi图的概念,运用结晶生成法通过C#软件编程实现了不同交通网络速度的基于交通网络的Voronoi图的生成程序。该方法进一步完善和丰富了Voronoi图理论,拓展了Voronoi图的应用范围,体现了实践应用价值。     

一种基于虚拟力的分布式Voronoi控制策略

针对无线移动传感器网络在目标区域的覆盖问题,提出了一种基于移动距离的局部分布式算法,利用Voronoi多边形的特征对目标区域进行有效的分割,运用力学的矢量概念,根据Voronoi图的边和顶点确定虚拟力的方向和大小即节点的移动方向和距离,提出了基于移动距离的分布式Voronoi控制算法,以确定节点移动状态。仿真实验表明,所提算法不仅使得节点在目标区域实现了高覆盖率,同时在时间上也较早地达到了收敛,优化了网络的覆盖控制。     

Power图的离散生成

Power图是一种特殊的加权Voronoi图，该图中每个生成元点pi都带有权值wi．给出了一种直接构造Power图的算法．以每个生成元点Pi为圆心，Power距离,√wi为半径画圆；然后将这些圆以不同颜色填充，并以相同速率向外扩展这些圆的边界，直到屏幕上所有像素点都涂上颜色为止，环绕Pi的新边界构成Power图．该算法改进了在Voronoi图基础上构造Power图的传统方法，具有较高的效率．     

Relay pursuit of a maneuvering target using dynamic Voronoi diagrams

This paper addresses the problem of the pursuit of a maneuvering target by a group of pursuers distributed in the plane. This pursuit problem is solved by associating it with a Voronoi-like partitioning problem that characterizes the set of initial positions from which the target can be intercepted by a given pursuer faster than any other pursuer from the same group. In the formulation of this partitioning problem, the target does not necessarily travel along prescribed trajectories, as it is typically assumed in the literature, but, instead, it can apply an “evading” strategy in an effort to delay or, if possible, escape capture. We characterize an approximate solution to this problem by associating it with a standard Voronoi partitioning problem. Subsequently, we propose a relay pursuit strategy, that is, a special group pursuit scheme such that, at each instant of time, only one pursuer is assigned the task of capturing the maneuvering target. During the course of the relay pursuit, the pursuer–target assignment changes dynamically with time based on the (time varying) proximity relations between the pursuers and the target. This proximity information is encoded in the solution of the Voronoi-like partitioning problem. Simulation results are presented to highlight the theoretical developments.     

Region-expansion for the Voronoi diagram of 3D spheres

Given a set of spheres in 3D, constructing its Voronoi diagram in Euclidean distance metric is not easy at all even though many mathematical properties of its structure are known. This Voronoi diagram has been known for many important applications from science and engineering. In this paper, we characterize the Voronoi diagram of spheres in three-dimensional Euclidean space, which is also known as an additively weighted Voronoi diagram, and propose an algorithm to construct the diagram. Starting with the ordinary Voronoi diagram of the centers of the spheres, the proposed region-expansion algorithm constructs the desired diagram by expanding the Voronoi region of each sphere, one after another. We also show that the whole Voronoi diagram of n spheres can be constructed in O(n³) time in the worst case.     

Approximating the Generalized Voronoi Diagram of Closely Spaced Objects

We present an algorithm to compute an approximation of the generalized Voronoi diagram (GVD) on arbitrary collections of 2D or 3D geometric objects. In particular, we focus on datasets with closely spaced objects; GVD approximation is expensive and sometimes intractable on these datasets using previous algorithms. With our approach, the GVD can be computed using commodity hardware even on datasets with many, extremely tightly packed objects. Our approach is to subdivide the space with an octree that is represented with an adjacency structure. We then use a novel adaptive distance transform to compute the distance function on octree vertices. The computed distance field is sampled more densely in areas of close object spacing, enabling robust and parallelizable GVD surface generation. We demonstrate our method on a variety of data and show example applications of the GVD in 2D and 3D.     

Analytical Solution of the Minimum Time Slew Maneuver Problem for an Axially Symmetric Spacecraft in the Class of Conical Motions

The conventional problem of the time-optimal slew of a spacecraft considered as a solid body with a single symmetry axis subject to arbitrary boundary conditions for the attitude and angular velocity is considered in the quaternion statement. By making certain changes of variables, the original dynamic Euler equations are simplified, and the problem turns into the optimal slew problem for a solid body with a spherical distribution of mass containing one additional scalar differential equation. For this problem, a new analytical solution in the class of conical motions is found; in this solution, the initial and terminal attitudes of the space vehicle belong to the same cone realized under a bounded control. A modification of the optimal slew problem in the class of generalized conical motions is made that makes it possible to obtain its analytical solution under arbitrary boundary conditions for the attitude and angular velocity of the spacecraft. A numerical example of a spacecraft’s conical motion and examples demonstrating the proximity of the solutions of the conventional and modified optimal slew problems of an axially symmetric spacecraft are discussed.     

A deterministic annealing approach for parsimonious design ofpiecewise regression models

A new learning algorithm is proposed for piecewise regression modeling. It employs the technique of deterministic annealing to design space partition regression functions. While the performance of traditional space partition regression functions such as CART and MARS is limited by a simple tree-structured partition and by a hierarchical approach for design, the deterministic annealing algorithm enables the joint optimization of a more powerful piecewise structure based on a Voronoi partition. The new method is demonstrated to achieve consistent performance improvements over regular CART as well as over its extension to allow arbitrary hyperplane boundaries. Comparison tests, on several benchmark data sets from the regression literature, are provided     

满足本地化差分隐私的众包位置数据采集

针对位置数据众包采集中个人位置隐私泄露的问题,提出了一种满足本地化差分隐私的位置数据众包采集方法。首先,使用逐点插入法构造维诺图,对路网空间进行分割;然后,采用满足本地化差分隐私的随机扰动的方式对每个维诺格中的位置数据进行扰动;再次,设计了一种在扰动数据集上进行空间范围查询的方法,获得对真实结果的无偏估计;最后,在空间范围查询下进行了实验验证,并与保护隐私的轨迹数据采集（PTDC）算法进行了对比,算法查询误差率最坏不超过40%,最好情况在20%以下,运行时间在8 s以内,在隐私保护度高于PTDC算法的前提下,上述参数优于PTDC算法。     

Voronoi图划分实现位置数据发布隐私保护

针对位置这一特殊数据发布的隐私问题，提出了基于Voronoi图预划分的隐私保护策略。该策略通过信息熵计算处理待发布位置与敏感位置关联关系，并利用关联最低位置作为图心建立Voronoi图。进而利用Voronoi单元格特性将待发布的位置信息替换为图心位置，以此实现敏感信息隐藏的目的。在信息隐藏的基础上，利用广义差分隐私原理，提出了基于位置发布数据的[ε]-敏感位置关联隐私模型，并证明所提出的算法能够满足该模型。最后，通过比较实验进一步证明了所提出的算法在隐私保护能力和发布数据可用性方面的优势，并对实验结果进行了详细的成因分析。     

The Zermelo–Voronoi diagram: A dynamic partition problem

We consider a Voronoi-like partition problem in the plane for a given finite set of generators. Each element in this partition is uniquely associated with a particular generator in the following sense: an agent that resides within a set of the partition at a given time will arrive at the generator associated with this set faster than any other agent that resides anywhere outside this set at the same instant of time. The agent’s motion is affected by the presence of a temporally varying drift, which is induced by local winds/currents. As a result, the minimum time to a destination is not equivalent to the minimum distance traveled. This simple fact has important ramifications over the partitioning problem. It is shown that this problem can be interpreted as a Dynamic Voronoi Diagram problem, where the generators are not fixed, but rather they are moving targets to be reached in minimum time. The problem is solved by first reducing it to a standard Voronoi Diagram by means of a time-varying coordinate transformation. We then extend the approach to solve the dual problem where the generators are the initial locations of a given set of agents distributed over the plane, such that each element in the partition consists of the terminal positions that can be reached by the corresponding agent faster than any other agent.     

基于Voronoi图的定性路径推理

动态空间知识的表示与推理是定性空间推理研究的重要内容.基于Voronoi图及其动态变化,提出运动路径定性表示与推理方法.先根据Voronoi图空间邻近关系定义Voronoi图生成子空间关系,进一步定义定性位置及概念邻域,并应用概念相邻的定性位置序列给出定性路径表示.再由动态Voronoi图的边集变化和给出的概念邻域中定性位置间最短路径的启发式算法,设计并实现具有观察者角度的定性路径推理算法.最后,实验分析并验证该方法的有效性.     

基于Voronoi图的方向区域查询方法

针对空间中方向区域查询效率不高的问题,通过引入Voronoi图,利用其特性对数据空间进行划分,提出了基于Voronoi图的方向区域查询方法.该方法在基于Delaunay三角网生成的Voronoi图索引结构基础上,将首结点与查询对象连线形成有向线段,利用Voronoi图可以通过邻接生成点延展的特点确定查询对象的位置,通过...     

Approximation of Generalized Voronoi Diagrams by Ordinary Voronoi Diagrams

A numerically robust algorithm for the ordinary Voronoi diagrams is applied to the approximation of various types of generalized Voronoi diagrams. The generalized Voronoi diagrams treated here include Voronoi diagrams for figures, additively weighted Voronoi diagrams, Voronoi diagrams in a river, Voronoi diagrams in a Riemannian plane, and Voronoi diagrams with respect to collision-avoiding shortest paths. The construction of these generalized Voronoi diagrams is reduced to the construction of the ordinary Voronoi diagrams. The methods proposed here can save much time which is otherwise necessary for writing a computer program for each type of generalized Voronoi diagram.     

GEOMETRIC TASK DECOMPOSITION IN A MULTI-AGENT ENVIRONMENT

ABSTRACT

Task decomposition in a multi-agent environment is often performed online. This paper proposes a method for sub-task allocation that can be performed before the agents are deployed, reducing the need for communication among agents during their mission. The proposed method uses a Voronoi diagram to partition the task-space among team members and includes two phases: static and dynamic. Static decomposition (performed in simulation before the start of the mission) repeatedly partitions the task-space by generating random diagrams and measuring the efficacy of the corresponding sub-task allocation. If necessary, dynamic decomposition (performed in simulation after the start of a mission) modifies the result of a static decomposition (i.e., in case of resource limitations for some agents). Empirical results are reported for the problem of surveillance of an arbitrary region by a team of agents.     

障碍空间中基于Voronoi图的k最近邻查询

为了提升障碍空间中k最近邻查询的效率,研究了障碍空间中基于Voronoi图的k最近邻查询方法,提出了在障碍空间基于Voronoi图的kNN-Obs算法。该算法采用了两个过程:过滤过程和精炼过程。过滤过程主要是利用Voronoi图的过滤功能,较大程度地减少了被查询点的个数。精炼过程主要根据障碍距离和邻接生成点对候选集内对象进行第二次筛选。进一步给出了处理新增加点的ADDkNN-Obs算法和处理删除点的DENkNN-Obs算法。实验表明该算法在处理障碍空间中的k最近邻问题时具有优势。