robotics代写|寻路算法代写Path Planning Algorithms|Optical Imaging of Global Air Circulation

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  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
  • Statistical Machine Learning 统计机器学习
  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础
robotics代写|寻路算法代写Path Planning Algorithms|Optical Imaging of Global Air Circulation

robotics代写|寻路算法代写Path Planning Algorithms|Optical Imaging of Global Air Circulation

Although complex mathematical/computational models for the global air circulation in the Earth’s atmosphere have been developed for weather prediction, the input data for these models are often derived from real-time optical imaging of the Earth’s atmosphere exhibited by cloud formation patterns. This task can be accomplished by implementing an observer system composed of cameras attached to a set of geosynchronous and low-orbit satellites. Since the cameras have finite viewing apertures, a basic problem is to determine the minimum number of cameras and their attitudes for complete coverage of Earth’s atmosphere. The satellites should be capable of communicating with each other to form a real-time global-circulation monitoring network.

An asteroid is usually a small irregularly-shaped solid body with nearly uniform mass density. When such an asteroid enters the spatial region where the Earth’s gravitational force becomes significant, it is of interest to predict its motion or path in the world or observation space, and to determine its surface and material properties. The second task can be accomplished by using one or more spacecraft equipped with cameras, radars and sensors moving in the vicinity of the asteroid. An important problem is to select the spacecraft motion so that the asteroid’s surface can be mapped out completely. In the case where the observation time duration is specified, the main task is to map out the asteroid’s surface as much as possible within the given observation time duration. For a single spacecraft, it is required to select a path/motion satisfying certain constraints such that maximum or complete visual coverage of the asteroid surface is attained (see Fig. 1.1). Here, the object under observation is a moving solid body in a three-dimensional world space. In this problem, a basic difficulty is that complete information about the asteroid surface is usually unavailable before launching of the spacecraft. Thus, one cannot preplan its observation path. The asteroid surface information must be acquired progressively as the spacecraft moves in the vicinity of the asteroid. We shall discuss practical methods for overcoming this difficulty later.

robotics代写|寻路算法代写Path Planning Algorithms|Path Planning on Structured Network

By a structured network defined on a given terrain, we mean a set of specified fixed nodes interconnected by a set of well-defined bidirectional or unidirectional paths on the terrain. Given a pair of starting and terminal nodes, it is required to find a path connecting these nodes such that an observer (e.g. a mobile robot or rover equipped with camera) attains maximum visual coverage of the terrain. This is an optimal path planning problem. For example, a tourist bus guide wishes to find the shortest sightseeing route in a city such that maximum visual coverage of the attractions can be attained. When the observation time interval is specified, the tourist bus guide may wish to plan the bus motion such that similar objective is achieved over the given time interval. This is a visibility-based optimal motion planning problem.

robotics代写|寻路算法代写Path Planning Algorithms|Objects Under Observation

Let the world space $\mathcal{W}$ be a specified subset of the $n$-dimensional Euclidean space $\mathbb{R}^{n}, n \in{2,3}$ in which a vector $x$ is an ordered n-tuple $\left(x_{1}, \ldots, x_{n}\right)$ of real numbers $x_{i}, i=1, \ldots, n$. The scaler product between two vectors $x$ and $x^{\prime}$ in $\mathbb{R}^{n}$ is denoted by $\left(x, x^{\prime}\right)$. The representation of a vector $x$ with respect to a given bases $B=\left{e_{1}, \ldots, e_{n}\right}$ for $\mathbb{R}^{n}$ is denoted by the column vector $[x]$. When ambiguity does not occur, the bracket notation [ -] is dropped for brevity. We assume that an object under observation $\mathcal{O}$ is a compact (closed and bounded) connected subset of $\mathcal{W}$ with boundary $\partial \mathcal{O}$. The observation of the object $\mathcal{O}$ is made from point-observers or observation points on an observation platform $\mathcal{P}$ corresponding to a given subset of $\mathcal{W}$. In what follows, we shall focus our attention mainly on the practically important case where $\mathcal{P} \subset \overline{\mathcal{O}^{c}}$ (the closure of the complement of $\mathcal{O}$ relative to $\mathcal{W}$ ). We assume that $\mathcal{P}$ is transparent in the sense that it can be penetrated by the line segment or an arc connecting an observation point in $\mathcal{P}$ and an observed point in $\mathcal{O}$. For opaque or non-transparent objects, only the boundary $\partial \mathcal{O}$ can be observed by point-observers in $\mathcal{P}$. The difficulty associated with the visibility problems to be considered generally depends on the dimensions and geometric properties of $\mathcal{O}$ and $\mathcal{P}$. A few cases of special interest are given in the sequel.
Case (i) $\mathcal{W}=\mathbb{R}^{2}$
(a) $\mathcal{O}$ is a plane curve which can be represented by $G_{f}=\left{(x, f(x)) \in \mathbb{R}^{2}: x \in \Omega\right}$, the graph of a real-valued $C_{m}$-function $f=f(x)$ defined on a compact interval $\Omega \subset \mathbb{R}$ (i.e. $f(\cdot) \in C_{m}(\mathbb{R} ; \Omega), m \geq 1$, the space of all real-valued functions having continuous derivatives on $\Omega$ up to the $m$ th order). The observations of the

curve are made from points $z=(x, w) \in \mathcal{P} \subset$ Epi $_{f}{ }{\text {def }}^{=}\left{\left(x^{\prime}, w^{\prime}\right) \in \Omega \times \mathbb{R}: w^{\prime} \geq\right.$ $\left.f\left(x^{r}\right)\right}$, the epigraph of $f$ (See Fig. 2.1a). We may also consider observations from points below the curve $G{f}$, i.e. $z \in\left{\left(x^{\prime}, w^{\prime}\right) \in \Omega \times \mathbb{R}: w^{\prime}<f\left(x^{\prime}\right)\right}$.
(b) $\mathcal{O}$ is an opaque object represented by a compact connected subset of $\mathcal{W}$. It can be described by $\bigcap_{k \in{1, \ldots, K}}\left{x \in \mathcal{W}: g_{k}(x) \leq 0\right}$, and has interior points and a piecewise smooth boundary $\partial \mathcal{O}$, where the $g_{k}$ ‘s are specified functions in $C_{m}(\mathbb{R} ; \mathcal{W}), m \geq 1$. The point-observers or observation points lie in $\mathcal{O}^{c}$ (See Fig. 2.1b).
Case (ii) $\mathcal{W}=\mathbb{R}^{3}$
(a) $\mathcal{O}$ is an opaque smooth surface described by $G_{f}$, the graph of a real-valued $C_{m}{ }^{-}$ function $(m \geq 1) f=f(x), x=\left(x_{1}, x_{2}\right) \in \Omega$, a compact connected subset of $\mathbb{R}^{2}$ (See Fig. 2.2). The observation points $z=(x, w)$ are located in Epi $\mathrm{i}{f} \subset \mathbb{R}^{3}$. (b) $\mathcal{O}$ is an opaque solid body represented by a nonempty compact connected subset of $\mathbb{R}^{3}$ with interior points and a piecewise smooth boundary $\partial \mathcal{O}$ which may be expressed as a level set of a real-valued continuous function $g=g(x)$ defined on $\mathbb{R}^{3}$, i.e. $\mathcal{O}=\left{x \in \mathbb{R}^{3}: g(x)=\alpha\right}$ or $g^{-1}(\alpha)$, where $\alpha \in \mathbb{R}$. As in Case (i)(b), the object $\mathcal{O}$ may also be described by $\bigcap{k \in{1, \ldots . K}}\left{x \in \mathcal{W}: g_{k}(x) \leq 0\right}$, where the $g_{k}$ ‘s are specified functions in $C_{m}(\mathbb{R} ; \mathcal{W})$. The observation points lie in $\mathcal{O}^{\text {}}$.

robotics代写|寻路算法代写Path Planning Algorithms|Optical Imaging of Global Air Circulation


robotics代写|寻路算法代写Path Planning Algorithms|Optical Imaging of Global Air Circulation


小行星通常是具有几乎均匀质量密度的不规则形状的小型固体。当这样的小行星进入地球引力变得显着的空间区域时,预测其在世界或观测空间中的运动或路径,并确定其表面和材料特性是有意义的。第二项任务可以通过使用在小行星附近移动的一个或多个配备有照相机、雷达和传感器的航天器来完成。一个重要的问题是选择航天器的运动,以便可以完整地绘制出小行星的表面。在指定观测时间的情况下,主要任务是在给定的观测时间范围内尽可能地绘制出小行星的表面。对于单个航天器,需要选择满足某些约束的路径/运动,以便获得小行星表面的最大或完全视觉覆盖(见图 1.1)。在这里,被观察的物体是一个在三维世界空间中移动的固体。在这个问题中,一个基本的困难是在航天器发射之前通常无法获得有关小行星表面的完整信息。因此,人们无法预先计划其观察路径。随着航天器在小行星附近移动,必须逐步获取小行星表面信息。稍后我们将讨论克服这一困难的实用方法。被观察的物体是一个在三维世界空间中移动的固体。在这个问题中,一个基本的困难是在航天器发射之前通常无法获得有关小行星表面的完整信息。因此,人们无法预先计划其观察路径。随着航天器在小行星附近移动,必须逐步获取小行星表面信息。稍后我们将讨论克服这一困难的实用方法。被观察的物体是一个在三维世界空间中移动的固体。在这个问题中,一个基本的困难是在航天器发射之前通常无法获得有关小行星表面的完整信息。因此,人们无法预先计划其观察路径。随着航天器在小行星附近移动,必须逐步获取小行星表面信息。稍后我们将讨论克服这一困难的实用方法。

robotics代写|寻路算法代写Path Planning Algorithms|Path Planning on Structured Network


robotics代写|寻路算法代写Path Planning Algorithms|Objects Under Observation

让世界空间在是指定的子集n维欧几里得空间Rn,n∈2,3其中一个向量X是一个有序的 n 元组(X1,…,Xn)实数X一世,一世=1,…,n. 两个向量之间的标量积X和X′在Rn表示为(X,X′). 向量的表示X关于给定的碱基B=\left{e_{1}, \ldots, e_{n}\right}B=\left{e_{1}, \ldots, e_{n}\right}为了Rn由列向量表示[X]. 当不发生歧义时,为简洁起见,括号符号 [-] 被删除。我们假设一个被观察的物体这是一个紧凑的(封闭且有界的)连通子集在有边界∂这. 对象的观察这由观测平台上的点观测器或观测点组成磷对应于给定的子集在. 在下文中,我们将主要关注实际重要的情况,其中磷⊂这C¯(补的关闭这关系到在)。我们假设磷是透明的,因为它可以被线段或连接观察点的弧穿透磷和一个观察点这. 对于不透明或不透明的物体,只有边界∂这点观察者可以观察到磷. 与要考虑的可见性问题相关的困难通常取决于尺寸和几何特性这和磷. 续集中给出了一些特别感兴趣的案例。
(一种)这是一条平面曲线,可以表示为G_{f}=\left{(x, f(x)) \in \mathbb{R}^{2}: x \in \Omega\right}G_{f}=\left{(x, f(x)) \in \mathbb{R}^{2}: x \in \Omega\right}, 实值图C米-功能F=F(X)在紧区间上定义Ω⊂R(IEF(⋅)∈C米(R;Ω),米≥1, 所有具有连续导数的实值函数的空间Ω至米次订单)。的观察

曲线由点组成和=(X,在)∈磷⊂和_{f}{ }{\text {def }}^{=}\left{\left(x^{\prime}, w^{\prime}\right) \in \Omega \times \mathbb{R} : w^{\prime} \geq\right.$ $\left.f\left(x^{r}\right)\right}_{f}{ }{\text {def }}^{=}\left{\left(x^{\prime}, w^{\prime}\right) \in \Omega \times \mathbb{R} : w^{\prime} \geq\right.$ $\left.f\left(x^{r}\right)\right}, 题词F(见图 2.1a)。我们也可以考虑曲线下方点的观察结果GF, IEz \in\left{\left(x^{\prime}, w^{\prime}\right) \in \Omega \times \mathbb{R}: w^{\prime}<f\left(x^ {\素数}\右)\右}z \in\left{\left(x^{\prime}, w^{\prime}\right) \in \Omega \times \mathbb{R}: w^{\prime}<f\left(x^ {\素数}\右)\右}.
(二)这是一个不透明的对象,由 的紧凑连接子集表示在. 可以描述为\bigcap_{k \in{1, \ldots, K}}\left{x \in \mathcal{W}: g_{k}(x) \leq 0\right}\bigcap_{k \in{1, \ldots, K}}\left{x \in \mathcal{W}: g_{k}(x) \leq 0\right},并且具有内部点和分段平滑边界∂这, 其中Gķ是指定的函数C米(R;在),米≥1. 点观察者或观察点位于这C(见图 2.1b)。
(一种)这是一个不透明的光滑表面,由GF, 实值图C米−功能(米≥1)F=F(X),X=(X1,X2)∈Ω, 的紧连通子集R2(见图 2.2)。观察点和=(X,在)位于 Epi一世F⊂R3. (二)这是一个不透明的实体,由 的非空紧连接子集表示R3具有内部点和分段平滑边界∂这可以表示为实值连续函数的水平集G=G(X)定义于R3, IE\mathcal{O}=\left{x \in \mathbb{R}^{3}: g(x)=\alpha\right}\mathcal{O}=\left{x \in \mathbb{R}^{3}: g(x)=\alpha\right}或者G−1(一种), 在哪里一种∈R. 与情况 (i)(b) 一样,对象这也可以描述为\bigcap{k \in{1, \ldots 。K}}\left{x \in \mathcal{W}: g_{k}(x) \leq 0\right}\bigcap{k \in{1, \ldots 。K}}\left{x \in \mathcal{W}: g_{k}(x) \leq 0\right}, 其中Gķ是指定的函数C米(R;在). 观测点位于这.

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中,其中问题和解决方案以熟悉的数学符号表示。典型用途包括:数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发,包括图形用户界面构建MATLAB 是一个交互式系统,其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题,尤其是那些具有矩阵和向量公式的问题,而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问,这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展,得到了许多用户的投入。在大学环境中,它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域,MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要,工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数(M 文件)的综合集合,可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。



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