### robotics代写|寻路算法代写Path Planning Algorithms|Mathematical Preliminaries

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Notions of Visibility

First, we consider objects $\mathcal{O}$ with time-invariant shapes as described in Cases (i) and (ii). We introduce the notion of visibility of a point $y$ on $\partial \mathcal{O}$ from an observation point $z$, assuming no other objects or obstacles are in the world space $\mathcal{W}$. Unless stated otherwise, the object under observation is assumed to be opaque, and the observation point $z \in \overline{\mathcal{O}^{c}}$. The visibility of the point under observation is based on line-of-sight from the observation point $z$.

Definition 2.1 A point $y \in \partial \mathcal{O}$ is visible from an observation point $z \in \overline{\mathcal{O}}$, if (i) $\mathrm{L}(y, z) \subset \mathcal{W}$, and (ii) $\mathrm{L}(y, z) \cap \operatorname{int}(\mathcal{O})=\phi$ (the empty set), where $\mathrm{L}(y, z)$ is the line segment joining points $y$ and $z$ described by $\left{x \in \mathbb{R}^{3}: x=\lambda y+(1-\lambda) z, 0<\right.$ $\lambda<1}$, and int $(\mathcal{O})$ denotes the interior of $\mathcal{O}$. The set $\mathcal{V}(z)$ of all points $y \in \partial \mathcal{O}$ that are visible from a point $z \in \overline{\mathcal{O}^{c}}$ is called the visible set of $z$. The complement of $\mathcal{V}(z)$ relative to $\partial \mathcal{O}$ is called the invisible set of $z$.

Condition (ii) implies that $\mathrm{L}(y, z)$ does not penetrate into the interior of $\mathcal{O}$, and $\mathrm{L}(y, z) \cap \partial \mathcal{O}$ may be nonempty. In the case where diffraction phenomenon occurs, we may replace $\mathrm{L}(y, z)$ in Definition $2.1$ by a smooth arc (in a given class) connecting the points $y$ and $z$. Figure $2.1$ shows the visible and invisible sets of a point $z$ for observing a plane curve and a compact connected set in $\mathbb{R}^{2}$ indicated by solid and dashed curves respectively. Note that the boundary of the object in Fig. 2.1b has a flat portion. Thus, for the indicated observation point $z$ and observed point $y, \mathrm{~L}(y, z) \cap \partial \mathcal{O}$ is nonempty. The visible and invisible sets of an observation point for an opaque solid object in $\mathbb{R}^{3}$ are illustrated by Fig. 2.4.

Remark 2.1 When a point-observer $z$ has finite viewing aperture such as a camera, we may define the visible set of the point-observer $z$ as
$$\mathcal{V}(z)=\left{y \in\left(\mathcal{C}{z} \cap \partial O\right): \lambda z+(1-\lambda) y \notin O \text { for all } \lambda \in\right] 0,1[},$$ where $\mathcal{C}{z}$ is the cone of visibility associated with the point-observer $z$ at its vertex, and has a finite viewing-aperture angle as illustrated in Fig. 2.5. In the case of a camera, its visible set may be enlarged by rotating the camera about the vertex of its viewingaperture cone. This approach may be useful when the object under observation does not change its shape significantly during the rotation period. Throughout this book,Remark $2.2$ In Definition $2.1$, visibility is defined in terms of line-of-sight observations. In practical situations, observations may be accomplished using more complex devices. For example, the observation of an object $\mathcal{O}$ may be made using a reflective surface $R_{f}$ as illustrated in Fig. 2.6. Here, a point $y \in \partial \mathcal{O}$ is visible from a point-observer $z \in \overline{\mathcal{O}^{c}}$, if there exists a ray composed of incident and reflected rays connecting $z$ and $y$. The direction of the reflected ray is determined by the usual Law of Reflection in optics (i.e. angle of incidence equals the angle of reflection). In this work, we only consider line-of-sight observations.

## robotics代写|寻路算法代写Path Planning Algorithms|Observation of Complex Objects

So far, we have considered only the observation of an object consisting of a single connected compact set in the world space $\mathcal{W}$. In more general situations, the object $\mathcal{O}$ under observation may correspond to a collection of disjoint compact subsets $\mathcal{O}{i}, i \in \mathcal{I}$ of $\mathcal{W}$, where $\mathcal{I}$ is a finite or countably infinite index set. The observations of $\mathcal{O}$ are made from points $z \in \mathcal{P} \subset \mathcal{W}$ such that $\mathcal{P} \cap \mathcal{O}$ is empty. In the trivial case where the observation of any object $\mathcal{O}{i}$ can be made from a given observation point $z \in \mathcal{P}$ without considering the remaining subsets $\mathcal{O}{j}, j \in \mathcal{I}-{i}$, then the visible set $\mathcal{V}(z)$ is simply $\bigcup{i \in \mathcal{I}} \mathcal{V}{i}(z)$, where $\mathcal{V}{i}(z) \subset \partial \mathcal{O}{i}$ denotes the visible set of $z$ with respect to $\mathcal{O}{i}$. To illustrate various possible situations involving objects with multiple disjoint subsets, consider a simple example where the object $\mathcal{O} \subset \mathbb{R}^{2}$ consists of two circular disks $D_{i}, i=1,2$ with different radii $r_{1}$ and $r_{2}$ as shown in Fig. 2.23. First, consider the observation point $z^{(1)}$ located between the two disks. Evidently, both $D_{1}$ and $D_{2}$ are partially visible from $z^{(1)}$. From the observation point $z^{(2)}$ (resp. $z^{(3)}$ ), only $D_{1}$ (resp. $D_{2}$ ) is partially visible (as in solar eclipse, where the observation point $z$ is identified with the sun). From the observation $z^{(4)}$, both $D_{1}$ and $D_{2}$ are partially visible. However, only one point in $D_{1}$ is visible from $z^{(4)}$. Finally, both $D_{1}$ and $D_{2}$ are partially visible from the observation point $z^{(5)}$. Moreover, $\mathcal{V}\left(z^{(5)}\right)$, the visible set of $z^{(5)}$ is simply $\mathcal{V}{1}\left(z^{(5)}\right) \cup \mathcal{V}{2}\left(z^{(5)}\right)$, where $\mathcal{V}{i}\left(z^{(5)}\right)$ denotes the visible set of $z^{(5)}$ with respect to $D{i}$, which can be determined independently. This simple example shows

that the structure of the visible sets for objects with multiple disjoint subsets may be very complex, and their determination may be a computationally intensive task.
The following is an example of an object in $\mathcal{W}=\mathbb{R}^{3}$ : composed of an countably infinite number of disjoint compact subsets:

Example $2.4$ Let the object under observation be $\mathcal{O}=\cup_{i \in \mathcal{I}} \mathcal{O}{i}$, where $\mathcal{I}=$ ${1,2, \ldots}$, and $\mathcal{O}{i}$ is a circle with radius $r_{i}$, centered at $\left(x_{1}, x_{2}, x_{3}\right)=\left(0,0, x_{3 i}\right)$ :
$$\mathcal{O}{i}=\left{x=\left(x{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{1}^{2}+x_{2}^{2}=r_{i}^{2}, x_{3}=x_{3 i}=1 / i^{2}\right}, \quad i=1,2, \ldots$$
such that $\mathcal{O}{i} \cap \mathcal{O}{j}$ is empty for all $i, j \in \mathcal{I}, i \neq j$. The observation platform $\mathcal{P}$ may correspond to a nonempty compact subset of $\mathcal{O}^{c}$, the complement of $\mathcal{O}$ relative to $\mathcal{W}$. The visible set of an observation point $z \in \mathcal{P}$ is given by
$$\mathcal{V}(z)=\bigcup_{i \in \mathcal{I}} \mathcal{V}{i}(z)$$ where $\mathcal{V}{i}(z)$ is the set of all boundary points of $\mathcal{O}{i}$ that is visible from $z$. The object $\mathcal{O}$ is said to be totally visible from $z$ if $\mathcal{V}(z)=\bigcup{i \in \mathcal{I}} \partial \mathcal{O}{i}$, i.e. the boundary points of every $\mathcal{O}{i}$ are visible from $z$. Evidently, there does not exist an observation point $z \in \mathcal{O}^{c}$ from which $\mathcal{O}$ is totally visible.

In planetary explorations using mobile robots, one may encounter cavities and tunnel-like structures on the planet surface. Here, the object $\mathcal{O}$ under observation is the surface inside these structures. To describe $\mathcal{O}$ mathematically, consider the simple idealized case in the world space $\mathcal{W}=\mathbb{R}^{2}$ where $\mathcal{O}$ is the union of the graphs of two real-valued $C_{1}$ functions $f_{i}=f_{i}(x), i=1,2$ defined on the interval $[a, b]$ as shown in Fig. 2.24. In Fig. 2.24a, $G_{f_{1}}$ and $G_{f_{2}}$ intersect at $x=b$ where $f_{1}(b)=f_{2}(b)$.

## robotics代写|寻路算法代写Path Planning Algorithms|Concluding Remarks

In this chapter, we have introduced various notions of visibility associated with line-of-sight observation of an object from a point-observer. These notions are also applicable to target interception by replacing the point-observer with a point source, and the object under observation with a target (e.g. a laser source emitting a beam toward the target). We may rename the “visible set of a point observer”as the “impact set of the point source”. Thus, total visibility of an object from a point-observer corresponds to total impact of the target boundary by a point source, i.e. each point of the target boundary can be impacted by at least one straight beam emitted from the source. In practical applications, it may be of interest to expose a particular part of the target surface to the source. This task can be accomplished only when that particular part lies in the impact set of the source.
Exercises
Ex.2.1. Let $\mathcal{W}=\mathbb{R}^{2}$. For each of the objects $\mathcal{O} \subset \mathcal{W}$ whose boundaries are composed of circular arcs and straightline segments (see Fig.2.26), determine the smallest number and locations of point-observers in $\mathcal{O}^{c}$ for total visibility of $\mathcal{O}$.

Ex.2.2. Let $\mathcal{W}=\mathbb{R}^{2}$. The object $\mathcal{O}$ under observation is formed by the union of two circular disks with different radii $r_{1}, r_{2}>0$.
(i) Find the visible sets of various observation points $z \in \mathcal{O}^{c}$ for each of the following cases:
(a) two disks are tangent to each other;
(b) two disks are disjoint with their centers separated by a finite distance $r_{1}+$ $r_{2}+d, d>0$.

## robotics代写|寻路算法代写Path Planning Algorithms|Notions of Visibility

\mathcal{V}(z)=\left{y \in\left(\mathcal{C}{z} \cap \partial O\right): \lambda z+(1-\lambda) y \notin O \text { 对于所有 } \lambda\in\right] 0.1[},\mathcal{V}(z)=\left{y \in\left(\mathcal{C}{z} \cap \partial O\right): \lambda z+(1-\lambda) y \notin O \text { 对于所有 } \lambda\in\right] 0.1[},在哪里C和是与点观察者相关的可见锥和如图 2.5 所示，在其顶点处具有有限的视角。在相机的情况下，可以通过围绕其视孔锥的顶点旋转相机来扩大其可见集。当被观察的物体在旋转期间没有显着改变其形状时，这种方法可能很有用。在本书中，备注2.2在定义中2.1，可见性是根据视线观察定义的。在实际情况下，可以使用更复杂的设备完成观察。例如，观察一个物体这可以使用反射面制成RF如图 2.6 所示。在这里，一个点是∈∂这从点观察者可见和∈这C¯, 如果存在一条由入射光线和反射光线组成的光线和和是. 反射光线的方向由光学中通常的反射定律确定（即入射角等于反射角）。在这项工作中，我们只考虑视线观察。

## robotics代写|寻路算法代写Path Planning Algorithms|Observation of Complex Objects

\mathcal{O}{i}=\left{x=\left(x{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{1} ^{2}+x_{2}^{2}=r_{i}^{2}, x_{3}=x_{3 i}=1 / i^{2}\right}, \quad i=1 ,2, \ldots\mathcal{O}{i}=\left{x=\left(x{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{1} ^{2}+x_{2}^{2}=r_{i}^{2}, x_{3}=x_{3 i}=1 / i^{2}\right}, \quad i=1 ,2, \ldots

## robotics代写|寻路算法代写Path Planning Algorithms|Concluding Remarks

Ex.2.1。让在=R2. 对于每个对象这⊂在其边界由圆弧和直线段组成（见图 2.26），确定点观测器的最小数量和位置这C总能见度这.

(i) 找到各个观察点的可见集和∈这C对于以下每种情况：
(a) 两个圆盘彼此相切；
(b) 两个圆盘不相交，它们的中心相隔有限距离r1+ r2+d,d>0.

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