## robotics代写|寻路算法代写Path Planning Algorithms|Visibility-based Optimal Motion Planning

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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|Observation Of Three-Dimensional Objects

Now, we turn our attention to optimal path planning problems for observing threedimensional objects. As we mentioned in Chap. 2 (Proposition $2.2$ and Remark 2.2) that at least two point-observers are required for total visibility of a solid body in $\mathbb{R}^{3}$. The following lemma gives a sufficient condition for total visibility of a solid body in $\mathbb{R}^{3}$.

Lemma 4.2 Suppose that the object under observation is a compact simply-connected solid body $\partial \mathcal{O} \subset \mathbb{R}^{3}$ with a smooth boundary $\partial \mathcal{O}$ and no outwardnormal intersection points. Then, total visibility of $\mathcal{O}$ can be attained by a finite set of observation points in the observation platform $\mathcal{P}_{h}$ at a constant finite height $h$ along the outward normal above the body surface $\partial \mathcal{O}$.

Proof $4.3$ Since $\mathcal{O}$ is a compact simply-connected solid body with a smooth boundary $\partial \mathcal{O}$ and no outward-normal intersection points, then $\mathcal{P}{h}=\left{z \in \mathbb{R}^{3}: z=\right.$ $x+h n(x), x \in \partial \mathcal{O}}$ is well-defined. For any observation point $z \in \mathcal{P}{h}$, there exists a point $x=z-h n(x) \in \partial \mathcal{O}$ such that $x$ is visible from $z$. Moreover, $\mathcal{V}(z)$, the visible set of $z$, is a compact subset of $\partial \mathcal{O}$ containing $x$ with finite measure $\mu_{2}(\partial \mathcal{V}(z))$. Since $\mu_{2}(\mathcal{O})$ is finite, there exists a finite number $N$ of observation points $z^{(i)} \in \mathcal{P}{h}, i=$ $1, \ldots, N$ such that $\partial \mathcal{O} \subseteq \bigcup{i=1}^{N} \mathcal{V}\left(z^{(i)}\right)$ as guaranteed by the Heine-Borel Covering Theorem.

First, we extend Problem 4.1, the shortest path total visibility problem, to the observation of a 3-D object $\mathcal{O}$ by a single mobile-observer.

Problem 4.1A Single Mobile-Observer Shortest Path Problem. Given an observation platform $\mathcal{P}$ enclosing the observed 3-D object $\mathcal{O}$, and two distinct points $z_{o}, z_{f} \in \mathcal{P}$, find the shortest admissible path $\Gamma \mathcal{P}$ starting at $z_{o}$ and ending at $z_{f}$ such that $\bigcup_{z \in \Gamma_{\mathcal{P}}} \mathcal{V}(z)=\partial \mathcal{O}$.

We observe that for $\mathcal{P}=\mathcal{P}{h}$, once a solution to Problem $3.5$ is obtained, any admissible path starting at $z{o}$, passing through all the points in the observation-point set $\mathcal{P}^{(N)}$, and ending at $z_{f}$, is a candidate to the solution of Problem 4.1A.

To illustrate the nature of the solution to Problem 4.1A, we consider a specific example.

Example $4.5$ Here, the object under observation is a closed spherical ball $\overline{\mathcal{B}}\left(0 ; r_{o}\right)$. The observation platform $\mathcal{P}{h}$ is a sphere $\mathcal{S}\left(0 ; r{o}+h\right)$ with finite observation height $h>0$. From the Principle of Optimality in Dynamic Programming, an optimal path is a concatenation of geodesics starting and ending at distinct specified points $z_{o}, z_{f} \in$ $\mathcal{P}{h}$. Figure $4.13$ shows an optimal path corresponding to $z{o}=\left(0,-\left(r_{o}+h\right), 0\right)$ and $z_{f}=\left(0, r_{o}+h, 0\right)$, with observation height $h$ satisfying $0<\alpha \leq \pi / 4$, where $\alpha=\sin ^{-1}\left(r_{o} /\left(r_{o}+h\right)\right)$ is the half-angle of the observation cone. We observe that the optimal path has length $l_{\min }=3 \pi\left(r_{o}+h\right)$, and has a great-circle loop along the path. For $\alpha>\pi / 4$, there are more than one great-circle loops along the path.

## robotics代写|寻路算法代写Path Planning Algorithms|Particular Case

Let $B=\left{e_{1}, \ldots, e_{n}\right}$ be a specified orthonormal bases for the $n$-dimensional real Euclidean space $\mathbb{R}^{n}$, where $e_{i}$ corresponds to the $i$ th unit basis vector. The representation of a point $x \in \mathbb{R}^{n}$ with respect to $B$ is specified by the column vector $\left[x_{1}, \ldots, x_{n}\right]^{T}$. The usual Euclidean norm of $x$ is denoted by $|x|$. Let $f=f(x)$ be a specified real-valued $C_{2}$-function defined on $\Omega$, a specified simply connected, compact subset of $\mathbb{R}^{2}$ with a smooth boundary $\partial \Omega$. As in Chap. 2 , the graph and vation is the spatial terrain surface described by $G_{f}$ in the world space $\mathcal{W}=\mathbb{R}^{3}$. The observation platform $\mathcal{P}$ on which the observers are attached corresponds to the elevated surface of $G_{f}$ given by $G_{f_{h_{v}}}$, where $f_{h_{v}} \stackrel{\text { def }}{=} f+h_{v}$ with $h_{v}$ being a specified positive number. This implies that for any $x \in \Omega$, the observers are at fixed vertical-height $h_{v}$ above the surface $G_{f}$.

From the extension of Proposition $3.1$ to the case where $\operatorname{dim}(\Omega)=2$, there exists a critical vertical-height $h_{v c}(x)$ for each $x \in \Omega$ such that total visibility is attainable. Consider the nontrivial case where $h_{v}<h_{v c}(x)$ for all $x \in \Omega$ so that the mobile observer must move to achieve total visibility. Let $I_{l f}=\left[0, t_{f}\right]$ denote the observation time interval, where $t_{f}$ may be a finite fixed or variable terminal time. For simplicity, the mobile observer is represented by a point mass $M$. Its position in the world space $\mathcal{W}=\mathbb{R}^{3}$ at any time $t$ is specified by $p(t)$ whose representation with respect to a given orthonormal basis $B$ is denoted by $\left[x_{1}(t), x_{2}(t), x_{3}(t)\right]^{T}$, where $x_{3}(t)$ corresponds to the observer position along the $x_{3}$-axis at time $t$. The motion of

the mobile observer can be described by Newton’s law:
$$\begin{gathered} M \ddot{x}(t)+\nu_{x} \dot{x}(t)=u(t) \ M \ddot{x}{3}(t)+\nu{3}\left(x(t), \dot{x}(t), x_{3}(t), \dot{x}{3}(t)\right)=\xi(t)-M g, \end{gathered}$$ where $x(t)=\left[x{1}(t), x_{2}(t)\right]^{T} ;(u, \xi)$ is the external force with $u=\left[u_{1}, u_{2}\right]^{T}$ being the control; $-M g$ is the gravitational force in the downward direction along the $x_{3}$-axis. $\nu_{x}$ is a given nonnegative friction coefficient; $\nu_{3}$ is a specified real-valued function of its arguments describing the $x_{3}$-component of the friction force. The variables with single and double overdots denote respectively their first and second derivatives with respect to time $t$. Assuming that the mobile observer is constrained to move on $G_{f}$ at all times without slipping, the mobile observer motion satisfies a holonomic constraint:
$$x_{3}(t)=f(x(t)) \text { for all } t \in I_{l_{f}},$$
and a state variable (position) constraint:
$$x(t) \in \Omega \text { for all } t \in I_{I_{f}} .$$
Since $f$ is a $C_{2}$-function on $\Omega$, we may differentiate (5.3) twice with respect to $t$ to obtain
\begin{aligned} &\dot{x}{3}(t)=\nabla{x} f(x(t))^{T} \dot{x}(t) \ &\ddot{x}{3}(t)=\nabla{x} f(x(t))^{T} \ddot{x}(t)+\dot{x}(t)^{T} H_{f}(x(t)) \dot{x}(t), \end{aligned}
where $\nabla_{x}$ denotes the gradient operator with respect to $x$, and $H_{f}(x(t))$ the Hessian matrix of $f$ with respect to $x$ evaluated at $x(t)$. Substituting (5.5) into (5.2) gives the required vertical component $\xi(t)$ of the external force for keeping the mobile observer on the surface $G_{f}$ at all times:
\begin{aligned} \xi(t)=& M\left(\nabla_{x} f(x(t))^{T} \ddot{x}(t)+\dot{x}(t)^{T} H_{f}(x(t)) \dot{x}(t)\right) \ &\left.+\nu_{3}\left(x(t), \dot{x}(t), x_{3}(t), \nabla_{x} f(x(t))^{T} \dot{x}(t)\right) / M+g\right) . \end{aligned}
Assuming that the mobile observer lies on $G_{f}$ at the starting time $t=0$, then
$$x_{3}(0)=f(x(0)), \quad \dot{x}{3}(0)=\nabla{x} f(x(0))^{T} \dot{x}(0) .$$
In the foregoing dynamic model of the mobile observer, we have assumed that the $x$-component of the friction force depend only on $\dot{x}$. In general, they may depend on both $(x, \dot{x})$ and $\left(x_{3}, \dot{x}_{3}\right)$. Also, to simplify the subsequent development, we have not considered the surface contact forces in the foregoing model.

## robotics代写|寻路算法代写Path Planning Algorithms|Statement of Problems

Now, a few physically meaningful visibility-based optimal motion planning problems can be stated as follows:

Problem 5.1 Minimum-Time Total Visibility Problem. Let $\mathcal{U}{\mathrm{ad}}=\bigcup{t f \geq 0} \mathcal{U}{\mathrm{ad}}\left(I{t f}\right)$ be the set of all admissible controls. Given $s_{x}(0)=(x(0), \dot{x}(0))$ or the initial state of the mobile observer with initial position $p(0)=(x(0), f(x(0))) \in G_{f}$ and initial velocity $v(0)=\left(\dot{x}(0), \nabla_{x} f(x(0))^{T} \dot{x}(0)\right)$, find the smallest time $t_{f}^{} \geq 0$ and an admissible control $u^{}=u^{}(t)$ defined on $I_{l_{f}^{}}$ such that its corresponding motion or time-dependent path $\Gamma_{i_{f}^{}}=\left{\left(x_{u^{}}(t), f\left(x_{u^{}}(t)\right)\right) \in \mathbb{R}^{3}: t \in I_{t_{f}^{}}\right}$ on the surface $G_{f}$ satisfies the total visibility condition at $t_{f}^{}$ : $$\bigcup_{t \in I_{t_{f}^{}}} \mathcal{V}\left(\left(x_{u^{}}(t), f_{h_{v}}\left(x_{u^{}}(t)\right)\right)=G_{f}\right.$$
or alternatively,
$$\mu_{2}\left{\bigcup_{t \in I_{t}^{}} \Pi_{\Omega} \mathcal{V}\left(\left(x_{u^{}}(t), f_{h_{v}}\left(x_{u^{}}(t)\right)\right)\right}=\mu_{2}{\Omega},\right.$$ where $\mu_{2}{\sigma}$ denotes the Lebesgue measure of the set $\sigma \subset \mathbb{R}^{2}$. In the foregoing problem statement, condition (5.10) only involves the position $x_{u^{}}(t)$, not the velocity $\dot{x}{u^{}}(t)$. In certain physical situations, it is required to move the mobile observer from one rest position to another, i.e. $\dot{x}{u^{}}(0)=0$ and $\dot{x}{u^{}}\left(t{f}^{}\right)=0$. Also, in planetary surface exploration, it is important to avoid paths with steep slopes. This suggests the inclusion of the following gradient constraint in Problem 5.1:
$$\left|\nabla_{x} f\left(x_{u^{}}(t)\right)\right| \leq f_{\max }^{\prime} \text { for all } t \in I_{t_{f}^{}},$$
where $f_{\max }^{\prime}$ is a specified positive number. Now, if the set $\Omega^{\text {def }}=\left{x \in \Omega: | \nabla_{x} f\left(x_{u^{*}}(t)\right)\right.$ $\left.| \leq f_{\max }^{\prime}\right}$ is a simply connected compact subset of $\mathbb{R}^{2}$, then we may replace $\Omega$ in Problem $5.1$ by $\Omega^{\prime}$ to take care of constraint (5.12).

Problem 5.2 Maximum Visibility Problem with Fixed Observation TimeInterval. Given a finite observation time interval $I_{I_{f}}$ and $s_{x}(0)=(x(0), \dot{x}(0))$, or the initial state of the mobile observer with initial position $p(0)=(x(0), f(x(0))) \in G_{f}$ and initial velocity $v(0)=\left(\dot{x}(0), \nabla_{x} f(x(0))^{T} \dot{x}(0)\right)$ at $t=0$, find an admissible control $u^{}=u^{}(t)$ and its corresponding motion or time-dependent path $\Gamma_{t_{f}}^{}=\left{\left(x_{u t^{}}(t), f\left(x_{u}^{*}(t)\right)\right) \in \mathbb{R}^{3}: t \in I_{t f}\right} \subset G_{f}$ such that the visibility functional given by

$$J_{1}(u)=\int_{0}^{t f} \mu_{2}\left{\Pi_{\Omega} \mathcal{V}\left(\left(x_{u}(t), f_{h_{v}}\left(x_{u}(t)\right)\right)\right)\right} d t$$
is defined, and satisfies $J_{1}\left(u^{}\right) \geq J_{1}(u)$ for all $u\left({ }^{}\right) \in \mathcal{U}{\mathrm{ad}}\left(I{t_{f}}\right)$.
Another meaningful visibility functional is given by
$$J_{2}(u)=\mu_{2}\left{\bigcup_{t \in I_{t}} \Pi_{\Omega} \mathcal{V}\left(\left(x_{u}(t), f_{h_{v}}\left(x_{u t}(t)\right)\right)\right)\right} .$$
The foregoing problem with $J_{1}$ replaced by $J_{2}$ corresponds to selecting an admissible control $u^{}(\cdot)$ such that the area of the union of the projected visibility sets on $\Omega$ for all the points along the corresponding time-dependent path $\Gamma_{t_{f}}^{}$ is maximized.

## robotics代写|寻路算法代写Path Planning Algorithms|Particular Case

X3(吨)=F(X(吨)) 对全部 吨∈一世lF,

X(吨)∈Ω 对全部 吨∈一世一世F.

X˙3(吨)=∇XF(X(吨))吨X˙(吨) X¨3(吨)=∇XF(X(吨))吨X¨(吨)+X˙(吨)吨HF(X(吨))X˙(吨),

X(吨)=米(∇XF(X(吨))吨X¨(吨)+X˙(吨)吨HF(X(吨))X˙(吨)) +ν3(X(吨),X˙(吨),X3(吨),∇XF(X(吨))吨X˙(吨))/米+G).

X3(0)=F(X(0)),X˙3(0)=∇XF(X(0))吨X˙(0).

## robotics代写|寻路算法代写Path Planning Algorithms|Statement of Problems

\mu_{2}\left{\bigcup_{t \in I_{t}^{}} \Pi_{\Omega} \mathcal{V}\left(\left(x_{u^{}}(t), f_{h_{v}}\left(x_{u^{}}(t)\right)\right)\right}=\mu_{2}{\Omega},\right。\mu_{2}\left{\bigcup_{t \in I_{t}^{}} \Pi_{\Omega} \mathcal{V}\left(\left(x_{u^{}}(t), f_{h_{v}}\left(x_{u^{}}(t)\right)\right)\right}=\mu_{2}{\Omega},\right。在哪里μ2σ表示集合的 Lebesgue 测度σ⊂R2. 在上述问题陈述中，条件（5.10）只涉及位置X在(吨)，而不是速度X˙在(吨). 在某些物理情况下，需要将移动观察者从一个静止位置移动到另一个静止位置，即X˙在(0)=0和X˙在(吨F)=0. 此外，在行星表面探测中，重要的是要避开陡坡的路径。这表明在问题 5.1 中包含以下梯度约束：
|∇XF(X在(吨))|≤F最大限度′ 对全部 吨∈一世吨F,

J_{2}(u)=\mu_{2}\left{\bigcup_{t \in I_{t}} \Pi_{\Omega} \mathcal{V}\left(\left(x_{u}(t ), f_{h_{v}}\left(x_{ut}(t)\right)\right)\right)\right} 。J_{2}(u)=\mu_{2}\left{\bigcup_{t \in I_{t}} \Pi_{\Omega} \mathcal{V}\left(\left(x_{u}(t ), f_{h_{v}}\left(x_{ut}(t)\right)\right)\right)\right} 。

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## robotics代写|寻路算法代写Path Planning Algorithms|Visibility-Based Optimal Path Planning

statistics-lab™ 为您的留学生涯保驾护航 在代写寻路算法Path Planning Algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写寻路算法Path Planning Algorithms代写方面经验极为丰富，各种代写寻路算法Path Planning Algorithms相关的作业也就用不着说。

• 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|Existence of Solutions

First, we consider Problem 4.1. The following result ensures that the set of all admissible paths $\Gamma$ satisfying condition (1) is nonempty.

Proposition 4.1 Under the conditions of Theorem 3.3, there exists an admissible path $\Gamma \in \Omega$ that satisfies condition (4.2).

Proof 4.I From Theorem $3.3$, there exists a finite point set $P^{(N)}=\left{x^{(k)}, k=\right.$ $1, \ldots, N} \subset \Omega$ that satisfies the total visibility constraint (1). If the specified end points $x_{o}$ and $x_{f} \in P^{(N)}$, then under the assumption that $\Omega$ is simply connected, it is always possible to construct a path $\Gamma$ in $\Omega$ corresponding to a Jordan arc passing through all the points in $P^{(N)}$. In particular, if the line segment joining any pair of points in $P^{(N)}$ lies in $\Omega$, then a Jordan arc composed of straightline segments joining the successive points in $P^{(N)}$ can always be constructed. A trivial case is where $\Omega$ is a compact convex subset of $\mathbb{R}^{2}$. If $x_{o}$ and/or $x_{f} \notin P^{(N)}$, we augment $P^{(N)}$ by these points, and proceed with the construction of a Jordan arc. Finally, from the constructed $\Gamma$, the corresponding path $\Gamma_{\mathcal{P}}$ in $\mathcal{P}$ can be determined from ${(x, g(x)): x \in \Gamma} .$

Remark 4.3 Proposition $4.1$ implies the existence of a Jordan arc passing through a finite set of observation points $\left(x^{(k)}, f_{h_{v}}\left(x^{(k)}\right)\right) \in G_{f_{b_{E}}}$ such that $G_{f}$ is totally visible. In general, the set of observation points for total visibility is not unique. Moreover, the cardinality of this set depends on $f$.

Remark 4.4 Suppose that the line segment $L$ joining the points $x_{o}$ and $x_{f}$ lies in $\Omega$, and $\bigcup_{x \in L} \mathcal{V}\left(\left(x, f_{h_{E}}(x)\right)\right)=G_{f}$, then $L$ is an optimal path. If $\bigcup_{x \in L} \mathcal{V}\left(\left(x, f_{h_{v}}(x)\right)\right) \subset$ $G_{f}$, then for a certain class of $G_{f}$, the optimal path $\Gamma^{*}$ is close to $L$ in the sense of arc length. Minimal excursions from $L$ can be introduced so that the invisible part $G_{f}-\bigcup_{x \in L} \mathcal{V}\left(\left(x, f_{h_{v}}(x)\right)\right)$ becomes visible.

For Problem 4.1″, we first construct the set $\tilde{\Omega} \stackrel{\text { def }}{=}\left{x \in \Omega:|\nabla f(x)| \leq f_{\max }^{\prime}\right}$, and then find the shortest admissible path $\Gamma^{} \subset \bar{\Omega}$ such that $\bigcup_{x \in \Gamma^{}} \mathcal{V}\left(\left(x, f_{h_{v}}(x)\right)\right)=$ $G_{f}$. In general, it is possible that $\bar{\Omega}$ consists of disjoint subsets of $\Omega$, and there may not exist admissible paths in $\bar{\Omega}$ (e.g. $x_{o}$ and $x_{f}$ lie in two disjoint subsets of $\Omega$ separated by a strip on which $|\nabla f(x)|>f_{\max }^{\prime}$ for all points $x$ on this strip). Consequently, Problem 4.1″ has no solution. Note also that the line segment $L$ joining $x_{o}$ and $x_{f}$ may not lie in $\bar{\Omega}$.

Since $f$ is a $C_{1}$-function, the set $\bar{\Omega}$ is compact. Assuming the existence of an admissible path in $\bar{\Omega}$, the observations given in Remarks $4.3$ and $4.4$ are also applicable to this case.

## robotics代写|寻路算法代写Path Planning Algorithms|Optimality Conditions

Here, we develop optimality conditions for Problem $4.1$ under the assumption that a solution exists. From Theorem $3.3$, there exists a finite point set $P^{(N)}=\left{x^{(k)}, k=\right.$ $1, \ldots, N} \subset \Omega$ with $x^{(1)}=x_{o}$ and $x^{(N)}=x_{f}$ such that
$$\bigcup_{k=1}^{N} \mathcal{V}\left(\left(x^{(k)}, f_{h_{v}}\left(x^{(k)}\right)\right)\right)=G_{f}$$
is satisfied. Let $P$ denote the set of all such $P^{(N)}$ ‘s with $2 \leq N<\infty$. The cardinality of $P$ is generally infinite. Now, for a given $P^{(N)} \subset P$, let $\tilde{\mathcal{A}}{P(N)}$ denote the set of all admissible paths $\Gamma$ formed by line segments joining distinct pairs of points in $P^{(N)}$. The cardinality of $\overline{\mathcal{A}}{P(N)}$ is $\leq(N-2)$ !. Thus, Problem $4.1$ reduces to finding a finite point set $P^{(N)} \subset P$ and an admissible path $\Gamma \in \tilde{\mathcal{A}}{P(N)}$ such that the arc length $$\Lambda(\Gamma)=\sum{k=1}^{N-1}\left|x^{(k+1)}-x^{(k)}\right|$$
is minimized. For convenience, the points $x^{(k)}, k=2, \ldots, N-1$ are indexed consecutively along the path $\Gamma$ initiating from $x_{o}$ and moving towards the end point $x_{f}$.
The following simple necessary condition for optimality can be deduced readily from the definition of arc length:

Proposition $4.2$ Let $\Gamma^{}$ be an optimal admissible path for Problem 4.1, and $P^{\left(N^{}\right)}=$ $\left{x_{}^{(k)} \in \Gamma^{}, k=1, \ldots, N^{}\right}$ be a finite point set satisfying (4.16). Then, for any perturbed admissible path $\Gamma \in \overline{\mathcal{A}}{\bar{P}^{\left(N^{}\right)}}$ with finite point set $\left{x{}^{(1)}, \ldots, x_{}^{(k-1)}, x_{}^{(k)}+\right.$ $\left.\delta x, x_{}^{(k+1)}, \ldots, x_{}^{\left(N^{}\right)}\right}$ in which the point perturbation $\delta x$ about $x_{}^{(k)}$ satisfies $x_{}^{(k)}+$ $\delta x \in \Omega$, and condition (4.16) holds, the following inequality:

\begin{aligned} &\left|x_{}^{(k)}+\delta x-x_{}^{(k-1)}\right|+\left|x_{}^{(k+1)}-x_{}^{(k)}-\delta x\right| \geq\left|x_{}^{(k+1)}-x_{}^{(k)}\right| \ &\quad+\left|x_{}^{(k)}-x_{}^{(k-1)}\right|, \quad k=2, \ldots, N^{}-1, \end{aligned} must be satisfied. For Problem 4.2, a necessary condition for optimality can be derived by considering local path perturbations. Let $I=[0,1]$. First, we parameterize an admissible path $\Gamma$ by the scalar parameter $\lambda \in I$, i.e. $$\Gamma=\left{\left(x_{1}, x_{2}\right) \in \Omega: x_{1}=q_{1}(\lambda), x_{2}=q_{2}(\lambda), \lambda \in I\right}$$ where $q_{1}$ and $q_{2}$ are real-valued $C_{1}$-functions on $I$ satisfying $$\left(q_{1}(0), q_{2}(0)\right)=\left(x_{o 1}, x_{o 2}\right) \text { and }\left(q_{1}(1), q_{2}(1)\right)=\left(x_{f 1}, x_{f 2}\right)$$ Let $\Gamma^{}$ and $\Gamma$ denote an optimal path and an admissible perturbed path specified by $q^{}(\lambda)=\left(q_{1}^{}(\lambda), q_{2}^{}(\lambda)\right)$ and $q^{}(\lambda)+\eta(\lambda)=\left(q_{1}^{}(\lambda)+\eta_{1}(\lambda), q_{2}^{}(\lambda)+\eta_{2}(\lambda)\right), \quad \lambda \in I$,
respectively, where $\eta=\left(\eta_{1}, \eta_{2}\right) \in \Sigma_{\Gamma^{}}$, the set of all admissible path perturbations about $\Gamma^{}$ defined by $\Sigma_{\Gamma^{}}=\left{\eta \in C_{1}\left(I ; \mathbb{R}^{2}\right): \eta(0)=(0,0), \eta(1)=(0,0) ; q^{}(\lambda)+\right.$ $\eta(\lambda) \in \Omega$ for all $\lambda \in I}$, with $C_{1}\left(I ; \mathbb{R}^{2}\right)$ being the normed linear space of all continuous functions defined on $I$ with their values in $\mathbb{R}^{2}$ and having continuous first derivatives on $I$, and normed by: $|\eta|_{m}=\sum_{i=1,2}\left(\max {\lambda \in I}\left|\eta{i}(\lambda)\right|+\max {\lambda \in I}\left|\eta{i}^{\prime}(\lambda)\right|\right)$, where $(\cdot)^{\prime}$ denotes differentiation with respect to $\lambda$.

## robotics代写|寻路算法代写Path Planning Algorithms|Numerical Algorithms

To facilitate the development of numerical algorithms for the optimal path planning problems, a mesh on $\Omega$ using standard methods such as Delaunay triangulation is established. Then $G_{f}$ is approximated by a polyhedral surface $\hat{G}{f}$, in particular, a surface formed by triangular patches. In practical situations, the function $f=f(x)$ is usually given in the form of numerical data. An approximation of $G{f}$ can be obtained by interpolation of the given numerical data. Here, algorithms are developed for the approximate Problems $4.1$ and $4.2$ that make use of the numerical data directly.
For the Shortest Path Problem 4.1, consider the simplest case where the line segment $L$ joining the end points $x_{o}$ and $x_{f}$ lies in $\Omega$. Remark $4.4$ suggests that a possible approach to obtaining a solution to the approximate Problem $4.1$ is to seek first a finite number of points along $L$ having maximal visibility, and then introduce additional points close to $L$ to achieve total visibility. Finally, a Jordan-arc with minimal length passing through all the observation points is constructed.

Suppose that on $\Omega$, a mesh consisting of points $x^{(k)}, k=1, \ldots, M$ along with the approximate surface $\hat{G}{f}$ formed by triangular patches have been established. For convenience, let $x^{(1)}=x{o}$ and $x^{(M)}=x_{f}$. Let $\hat{\mathcal{G}}$ denote the set of all Jordan arcs connecting $x^{(1)}$ and $x^{(M)}$ formed by line segments corresponding to the edges of the triangles. The basic steps in our algorithm for determining an optimal path for the approximate Problem $4.1$ without assuming that the line segment $L$ joining $x_{o}$ and $x_{f}$ lies in $\Omega$ are as follows:

Step 3 Determine $\hat{\mathcal{G}}^{}=\left{\Gamma_{j}^{}, j=1, \ldots, K\right}$, the set of all Jordan arcs in $\hat{\mathcal{G}}$ having the shortest path length.
Step 4 Select a path in $\hat{\mathcal{G}}^{}$, say $\Gamma_{i}^{}$.
Step 5 Compute the visible set $\mathcal{V}\left(\left(x^{(k)}, f_{h_{v}}\left(x^{(k)}\right)\right)\right)$ corresponding to each point $x^{(k)}$ along the path $\Gamma_{i}^{}$ and its projection $\Pi_{\Omega} \mathcal{V}\left(\left(x^{(k)}, f_{h_{v}}\left(x^{(k)}\right)\right)\right)$. Step 6 Determine whether there exists a combination of points $x^{(k)}$ along $\Gamma_{i}^{}$ such that the union of their visible sets $\mathcal{V}\left(\left(x^{(k)}, f_{h_{v}}\left(x^{(k)}\right)\right)\right)$ is equal to $G_{f}$.
If YES, then $\Gamma_{i}^{}$ is an optimal path for the approximate Problem 4.1, STOP; if NO, select a neighboring path of $\Gamma_{i}^{}$ in the sense of arc length, and GO TO Step $5 .$

Remark $4.5$ In Steps 2 and 3, instead of considering triangles in $\Omega$, we may consider triangular patches associated with the polyhedral approximation of the surface $G_{f}$. Efficient algorithms for computing the shortest arc length such as those due to Sharir and Shorr [3], O’Rourke et al. [4], and Lawler [5] may be used.

Remark $4.6$ Step 5 involves the computation of visible sets $\mathcal{V}\left(\left(x^{(k)}, f_{h_{x}}\left(x^{(k)}\right)\right)\right)$ associated with points $x^{(k)}$ along the path $\Gamma_{i}^{*}$, an NP-hard problem in computational geometry. This task can be accomplished using a suitable algorithm developed recently by Balmes and Wang [1]. The complexity of that algorithm is $O\left(n p^{2}\right)$, where $n$ and $p$ are the number of observation points and the number of triangular patches respectively. This algorithm can be easily modified to take into account the limited aperture of cameras or sensors.

## robotics代写|寻路算法代写Path Planning Algorithms|Optimality Conditions

⋃ķ=1ñ在((X(ķ),FH在(X(ķ))))=GF

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。

## robotics代写|寻路算法代写Path Planning Algorithms|Computational Algorithms

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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|Computational Algorithms

In practical situations, the object under observation $\mathcal{O}$ and the observation platform $\mathcal{P}$ are usually given in the form of numerical data. Approximations of $\mathcal{O}$ and $\mathcal{P}$ can be obtained by interpolation of the given numerical data. In what follows, we shall develop numerical algorithms for the approximate solution of Problems $3.1-3.3$ using the numerical data directly.

Consider the simplest case discussed in Sect. 3.1, Case (i), where the observed object $\mathcal{O}$ and the observation platform $\mathcal{P}$ correspond respectively the graphs of specified real-valued $C_{1}$-functions $f=f(x)$ and $g=g(x)$ defined on $\Omega=[a, b]$, a compact interval of $\mathbb{R}$ satisfying $g(x)>f(x)$ for all $x \in \Omega$. Let the given numerical data be composed of the values of $f$ at uniformly spaced mesh points $x^{(i)}=a+(i-1) \Delta x, i=1, \ldots, N ; \Delta x=(b-a) /(N-1)$. Then, the first derivative of $f$ at $x^{(i)}$ can be approximated by the usual forward difference $D f\left(x^{(i)}\right)=\left(f\left(x^{(i+1)}\right)-f\left(x^{(i)}\right)\right) / \Delta x$. For Problem 3.1, the critical height profile $h_{c}=h_{c}\left(x^{(i)}\right)$ can be computed via the steps outlined in the proof of Lemma $3.1$.
Next, we consider the approximate numerical solution of Problem 3.2. An essential first step is to compute the visible set of any point $z^{(i)}=\left(x^{(i)}, g\left(x^{(i)}\right)\right), i=$ $1, \ldots, N$. This task can be accomplished by considering the points along the line segments $\mathrm{L}\left(z^{(i)}, y^{(j)}\right)$ joining $z^{(i)}$ and points $y^{(j)}=\left(x^{(j)}, f\left(x^{(j)}\right)\right), j=1, \ldots, N$.
For $1 \leq j<i$, the points along the line segment $L\left(z^{(i)}, y^{(j)}\right)$ are given by $\left{\left(x^{(k)}, w\left(x^{(k)}\right)\right), j \leq k \leq i\right}$, where
$$w\left(x^{(k)}\right)=g\left(x^{(i)}\right)+\left(f\left(x^{(j)}\right)-g\left(x^{(i)}\right)\right)((k-i) /(j-i)), \quad j \leq k \leq i$$

If $w\left(x^{(k)}\right) \geq f\left(x^{(k)}\right)$ for $j \leq k \leq i$, then the point $\left(x^{(j)}, f\left(x^{(j)}\right)\right)$ belongs to $\mathcal{V}\left(\left(x^{(i)}, g\left(x^{(i)}\right)\right)\right)$, the visible set of $\left(x^{(i)}, g\left(x^{(i)}\right)\right)$ (see Fig.3.18a).

Similarly, for $i<j \leq N$, the points along the line segment $\mathrm{L}\left(z^{(i)}, y^{(j)}\right)$ joining $z^{(i)}$ and points $y^{(j)}=\left(x^{(j)}, f\left(x^{(j)}\right)\right)$ are given by $\left{\left(x^{(k)}, w\left(x^{(k)}\right)\right), j \leq k \leq N\right}$, where
$$w\left(x^{(k)}\right)=g\left(x^{(i)}\right)+\left(f\left(x^{(i)}\right)-g\left(x^{(j)}\right)\right)((k-i) /(i-j)), \quad i \leq k \leq j$$
If $w\left(x^{(k)}\right) \geq f\left(x^{(k)}\right)$ for $i \leq k \leq j$, then the point $\left(x^{(j)}, f\left(x^{(j)}\right)\right)$ belongs to $\mathcal{V}\left(\left(x^{(i)}, g\left(x^{(i)}\right)\right)\right)$ (see Fig.3.18b).

Having computed the visible set of each point in $\mathcal{P}=G_{g}$, its corresponding measure as a function of $x$ can be readily determined. Thus, an approximate numerical solution to Problem $3.2$ can be found simply by finding those points $x^{(i)}$ that correspond to the maximum value for the measure of the visible sets.

To obtain an approximate numerical solution to Problem 3.3, we first compute the characteristic function of $\Pi_{\Omega} \mathcal{V}\left(\left(x^{(i)}, g\left(x^{(i)}\right)\right)\right)$, denoted by $\Phi\left(\Pi_{\Omega} \mathcal{V}\left(\left(x^{(i)}, g\left(x^{(i)}\right)\right)\right)\right.$. Now, for each point $x^{(i)} \in \Omega, \Phi\left(\Pi_{\Omega} \mathcal{V}\left(\left(x^{(i)}, g\left(x^{(i)}\right)\right)\right)\right.$ can be represented by a binary string $\mathbf{s}^{(i)}$ of length $N$. A unit string $\mathbf{s}^{(i)}$ consisting of all 1 ‘s implies that $\Omega$ is totally visible from $\left(x^{(i)}, g\left(x^{(i)}\right)\right)$. If there are no such strings, we proceed by seeking pairs of distinct strings $\mathbf{s}^{(i)}$ and $\mathbf{s}^{(j)}$ such that $\mathbf{s}^{(i)} \vee \mathbf{s}^{(j)}$ is equal to the unit string, where $\vee$ denotes the logic “OR” operation between the corresponding components of $s^{(i)}$ and $\mathbf{s}^{(j)}$. If there are no such string pairs, we seek triplets of distinct strings $\mathbf{s}^{(i)}, \mathbf{s}^{(j)}, \mathbf{s}^{(k)}$ such that $\mathbf{s}^{(i)} \vee \mathbf{s}^{(j)} \vee \mathbf{s}^{(k)}$ is equal to a unit string. This process is continued until a finite set of strings $\mathbf{s}^{(i)}, i=1, \ldots, P$ such that a unit string $\bigvee_{i=1}^{P} \mathbf{s}^{(i)}$ is found. Clearly, the smallest set of such strings is an approximate numerical solution to Problem 3.3.
For the case where the given numerical data are composed of the values of $f$ and $g$ at specified mesh points $x^{(i)}$ in a two-dimensional domain $\Omega$, approximate numerical solution to Problem $3.3$ can be obtained via the following steps:

(i) Approximate $G_{f}$ and $G_{g}$ by polyhedral surfaces $\hat{G}{f}$ and $\hat{G}{g}$ respectively.
(ii) Compute the visible sets corresponding to the vertex points in $\hat{G}_{g}$.
(iii) Compute the projections of the visible sets on $\Omega$, and their measures.
(iv) Determine a minimal mesh-point set such that the union of the corresponding visible sets is equal to $\Omega$.

Step (i) can be accomplished by using Delaunay triangulation to obtain approximate surfaces in the form of triangular patches (see Appendix C). Step (ii) corresponds to a problem in computational geometry involving the intersection of a flat cone (with its vertex at an observation point on $G_{g}$ ) and a triangular patch on $\hat{G}{f}$ in $\mathbb{R}^{3}$ [11]. Step (iii) involves straightforward computation. The final step (iv) corresponds to a “Set Covering Problem” (see Appendix A) which can be reformulated as an integer programming problem. It has been shown by Cole and Sharir [12] that this problem (with $\hat{G}{g}$ coinciding with the approximate observed surface $\hat{G}{f}$ ) is NP-hard. An algorithm integrating the foregoing steps has been developed by Balmes and Wang [13]. The general idea behind this algorithm is to hop over all the observation points $z^{(i)} \in \hat{G}{g}$ and try to determine whether or not a specific triangle is visible.

## robotics代写|寻路算法代写Path Planning Algorithms|Numerical Examples

In what follows, we shall present a few numerical examples to illustrate the application of some of the algorithms discussed in Sect. 3.3.

Example $3.7$ Optimal Sensor Placement in Micromachined Structures. Consider the optimal sensor placement problem for a model of a one-dimensional micromachined solid structure whose spatial profile $G_{f}$ and observation platform $G_{g}$ are shown in Fig. $3.23$, where $g=f_{h_{v}}$. The critical vertical-height profile $G_{h_{v c}}$ for $f$ computed by the steps outlined in the proof of Lemma $3.1$ is also shown in Fig. 3.23. Since $G_{h_{v c}} \cap G_{g}$ is empty, $G_{f}$ is not totally visible from any point in $G_{g}$. The projection of the visible set $\mathcal{V}\left(\left(x^{(i)}, g\left(x^{(i)}\right)\right)\right)$ on the spatial domain $\Omega=[0,20] \mu \mathrm{m}$ as a function of the normalized $x^{(i)}$ (graph of the set-valued mapping $x^{(i)} \rightarrow \Pi_{\Omega} \mathcal{V}\left(\left(x^{(i)}, g\left(x^{(i)}\right)\right)\right.$ )

on $\Omega$ into $\Omega$ ) is shown in Fig. $3.24$, where $x^{(i)}$ is the $i$ th mesh point. The corresponding measure as a function of $x^{(i)}$ is shown in Fig.3.25. We note from Fig. $3.24$ that every point in the diagonal line lies in its visible set, or every $x \in \Omega$ is a fixed point of $\Pi_{I} \mathcal{V}((\cdot, g(\cdot)))$ as expected. Moreover, at least three sensors are needed to cover the entire $G_{f}$, but their locations are non-unique. From Fig. 3.25, it is evident that the solution to the approximate optimal sensor placement problem is given by $x^{}=13.5$ $\mu \mathrm{m}$ and $\mu_{1}\left{\Pi_{\Omega} \mathcal{V}\left(\left(x^{}, g\left(x^{*}\right)\right)\right)\right}=16.75 \mu \mathrm{m}$.

Next, we consider a more complex structure formed by micromachined components having various shapes embedded in a flat bottom plane. It is required to determine the minimum number and locations of optical sensors attached to a platform above the observed surface for health monitoring and inter-structure communication. Figures $3.26$ and $3.27$ show the observed surface $G_{f}$ formed by the bottom plane and micromachined components with various geometric shapes. Here, two different observation platforms are considered. The first one corresponds to setting the sensors at a fixed distance $(10 \mu)$ above the observed surface. This case is motivated by the fact that micromachined structures are usually fabricated in layers by etching. The second observation platform corresponds to the case where the sensors lie in a plane at 5 microns above the observed surface. These two cases represent the most important ones for monitoring a micromachined structure. Evidently, the minimum number and locations of the sensors for total visibility are not obvious intuitively. Figure $3.28 \mathrm{a}$, b show respectively the surfaces of visible-set measures for the first and second cases respectively. In the first case, the salient features of the observed surface are also reproduced in the surface of visible-set measures, but their order and position are shifted. For the second case, there is not much to say except that the visible sets of the observation points above the higher part of the observed surface have the smallest measure.

## robotics代写|寻路算法代写Path Planning Algorithms|Observation of Two-Dimensional Objects

First, consider the case where the object under observation $\mathcal{O}$ and the observation platform $\mathcal{P}$ are described respectively by $G_{f}$ and $G_{g}$ (the graphs of given real-valued $C_{1}$-functions $f=f(x)$ and $g=g(x)$ defined on a given compact set $\Omega \subset \mathbb{R}^{2}$

satisfying $g(x)>f(x)$ for all $x \in \Omega$ ). For this case, along any admissible path $\Gamma_{\mathcal{P}} \subset \mathcal{P}$, the corresponding arc $\mathcal{C}\left(\Gamma_{\mathcal{P}}\right)={(x, f(x)):(x, g(x)) \in \Gamma \mathcal{P}} \subset G_{f}$ is a Jordan arc, and so is $\Pi_{\Omega} \mathcal{C}\left(\Gamma_{\mathcal{P}}\right)$, the projection of $\mathcal{C}\left(\Gamma_{\mathcal{P}}\right)$ on $\Omega$.

Problem 4.1 Single Mobile Point-Observer Shortest Path Problem. Given an observation platform $\mathcal{P}=G_{g}$ and two distinct points $z_{o}=\left(x_{o}, g\left(x_{o}\right)\right), z_{f}=$ $\left(x_{f}, g\left(x_{f}\right)\right) \in \mathcal{P}$, find the shortest admissible path $\Gamma_{\mathcal{P}}^{} \in \mathcal{A}{\mathcal{P}}$ starting at $z{o}$ and ending at $z_{f}$ such that
$$\bigcup_{z \in \Gamma_{P}^{}} \mathcal{V}(z)=G_{f}$$
In many situations, instead of considering admissible paths $\Gamma p$ in $\mathcal{A}{\mathcal{P}}$, it is more convenient to consider admissible paths $\Gamma \in \mathcal{A}{\Omega}$ (the set of all admissible paths $\Gamma$ in $\Omega$ ). Thus, we have the following modified version of Problem 4.1.

Problem 4.1′ Given an observation platform $\mathcal{P}=G_{g}$ and $z_{o}=\left(x_{o}, g\left(x_{o}\right)\right), z_{f}=$ $\left(x_{f}, g\left(x_{f}\right)\right) \in \mathcal{P}$ such that $x_{o} \neq x_{f}$, find the shortest admissible path $\Gamma^{} \in \mathcal{A}{\Omega}$ starting at $x{o}$ and ending at $x_{f}$ such that
$$\bigcup_{x \in \Gamma^{+}} \Pi_{\Omega} \mathcal{V}((x, g(x)))=\Omega .$$
Remark $4.1$ Evidently, the shortest path $\Gamma^{}$ in $\Omega$ generally does not imply that the corresponding path $\Gamma_{\mathcal{P}}$ in $\mathcal{P}$ has the shortest length, hence Problems $4.1$ and 4.1′ are generally not equivalent. Nevertheless, it is still useful to consider Problem $4.1^{\prime}$ ‘, since its solution provides insight into the solution of corresponding Problem 4.1. Next, we observe that for $\mathcal{P}=G_{f+h_{v}}$ (the constant vertical-height observation platform), once a solution to Problem 3.5 (Minimal Observation-Point Set Problem) is obtained, any admissible path starting at $z_{o}$, passing through all the points in

the observation-point set $\mathcal{P}^{(N)}$, and ending at $z_{f}$, is a candidate to the solution of Problem 4.1.

In planetary surface exploration, it is important to avoid paths with steep slopes. This requirement can be satisfied by including the following gradient constraint in Problem 4.1:
$$|\nabla f(x)| \leq f_{\max }^{\prime} \text { for all } x \in \Gamma^{*}$$
where $f_{\max }^{\prime}$ is a specified positive number. This modified problem will be referred to hereafter as Problem 4.1″.

## robotics代写|寻路算法代写Path Planning Algorithms|Computational Algorithms

(i) 近似值GF和GG由多面体曲面 $\hat{G} {f}一种nd\hat{G} {g}r和sp和C吨一世在和l是.(一世一世)C这米p在吨和吨H和在一世s一世bl和s和吨sC这rr和sp这nd一世nG吨这吨H和在和r吨和Xp这一世n吨s一世n\hat{G}_{g}.(一世一世一世)C这米p在吨和吨H和pr这j和C吨一世这ns这F吨H和在一世s一世bl和s和吨s这n\欧米茄,一种nd吨H和一世r米和一种s在r和s.(一世在)D和吨和r米一世n和一种米一世n一世米一种l米和sH−p这一世n吨s和吨s在CH吨H一种吨吨H和在n一世这n这F吨H和C这rr和sp这nd一世nG在一世s一世bl和s和吨s一世s和q在一种l吨这\欧米茄$。

## robotics代写|寻路算法代写Path Planning Algorithms|Observation of Two-Dimensional Objects

⋃和∈Γ磷在(和)=GF

⋃X∈Γ+圆周率Ω在((X,G(X)))=Ω.

|∇F(X)|≤F最大限度′ 对全部 X∈Γ∗

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。

## robotics代写|寻路算法代写Path Planning Algorithms|Static Optimal Visibility Problems

statistics-lab™ 为您的留学生涯保驾护航 在代写寻路算法Path Planning Algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写寻路算法Path Planning Algorithms代写方面经验极为丰富，各种代写寻路算法Path Planning Algorithms相关的作业也就用不着说。

• 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|Single Point-Observer Static Optimal Visibility Problems

Consider the simplest case where the observed object $\mathcal{O}$ and the observation platform $\mathcal{P}$ are respectively the graphs of specified real-valued $C_{1}$-functions $f=f(x)$ and $g=g(x)$ defined on $\Omega$, a simply connected, compact subset of $\mathbb{R}^{n}, n \in{1,2}$ such that
$$g(x)>f(x) \text { for all } x \in \Omega$$
As mentioned in Remark 2.3, a special observation platform having practical importance is the constant vertical-height platform corresponding to the elevated profile of $f$ defined by the graph of $f_{h_{v}} \stackrel{\text { def }}{=} f+h_{v}$, where $h_{v}$ is a given positive number specifying the vertical-height of the point-observer above $\mathcal{O}=G_{f} \stackrel{\text { def }}{=}\left{(x, f(x)) \in \mathbb{R}^{n+1}\right.$ : $x \in \Omega}$. Since $f$ is a $C_{1}$-function defined on a compact set $\Omega, G_{f}$ is also compact. Moreover, for any point-observer at $(x, g(x)) \in G_{g}$, its visible set $\mathcal{V}((x, g(x)))$ and its projection on $\Omega$ (denoted by $\Pi_{\Omega} \mathcal{V}((x, g(x)))$ ) are compact. Thus, we may regard $(x, g(x)) \rightarrow \mathcal{V}((x, g(x)))$ (resp. $\left.\Pi_{\Omega} \mathcal{V}((x, g(x)))\right)$ as a set-valued mapping on $G_{g}$ into $2^{G_{f}}$ (resp. $\left.2^{\Omega}\right)$. In general, $\mathcal{V}((x, g(x)))$ and $\Pi_{\Omega}(\mathcal{V}((x, g(x))))$ may be the union of disjoint compact subsets of $G_{f}$ and $\Omega$ respectively. This situation is illustrated by the example shown in Fig. $3.1$ with the point-observer at $\left(x_{o}, g\left(x_{o}\right)\right) \in G_{g}$ and

$\Omega=[0,1]$. It can be seen that $\Pi_{\Omega} \mathcal{V}\left(\left(x_{o}, g\left(x_{o}\right)\right)\right)=\left[0, \hat{x}{1}\right] \cup\left{\hat{x}{2}\right} \cup\left[\hat{x}{3}, \hat{x}{4}\right] \cup\left[\hat{x}{5}, \hat{x}{6}\right]$ As in Example 2.1, this example also shows that the visible set of a point-observer may contain isolated points.

Now, we consider two optimal visibility problems associated with observation of the object $\mathcal{O}=G_{f}$ from point-observers located in Epi ${ }_{f}$, the epigraph of $f$.

Problem 3.1 Minimum Vertical-height Total Visibility Problem. Given $f=$ $f(x)$ defined on $\Omega$, find the minimum vertical-height $h_{v}^{} \geq 0$ and a point $x^{} \in \Omega$ such that $G_{f}$ is totally visible from the point-observer at $\left(x^{}, f_{h_{v}^{}}\left(x^{*}\right)\right)$.

Problem 3.2 Maximum Visibility Problem. Given real-valued $C_{1}$-functions $f$ and $g$ defined on $\Omega$ satisfying condition (3.1), find a point $x^{} \in \Omega$ such that $J_{g}\left(x^{}\right) \geq$ $J_{g}(x)$ for all $x \in \Omega$, where $J_{g}(x) \stackrel{\text { def }}{=} \mu_{1}\left{\Pi_{\Omega} \mathcal{V}((x, g(x)))\right}$, the Lebesgue measure of $\Pi_{\Omega} \mathcal{V}((x, g(x)))$.

If we set $g$ in Problem $3.2$ to $f_{h_{v}}$ for a given $h_{v}>0$, then we have the practically important Constant Vertical-height Maximum Visibility Problem.

Remark 3.1 In Problem 3.2, we may choose to maximize the total measure of $\mathcal{V}((x, g(x)))$ instead of the total measure of $\Pi_{\Omega} \mathcal{V}((x, g(x)))$ at the expense of increased computational complexity.

To fix ideas, we first consider the foregoing problems for the case with an onedimensional domain $\Omega$ and present some results which are relevant to the solution of more general optimal visibility problems. Then, similar problems for the case of a 2 -dimensional $\Omega$ will be discussed.

## robotics代写|寻路算法代写Path Planning Algorithms|Multiple Point-Observer Static Optimal Visibility Problems

So far, we have considered various optimal visibility problems involving a single stationary point-observer. When total visibility of the observed object cannot be

achieved by a single stationary point-observer, it is natural to ask whether total visibility can be attained by a finite (preferably smallest) number of stationary pointobservers. Before answering this question, we first establish a few properties of visible sets which are useful in the subsequent development. To simplify our discussion, we consider the case where the object $\mathcal{O}$ under observation is a surface in $\mathbb{R}^{3}$ described by $G_{f}$, the graph of a real-valued continuous function $f=f(x)$ defined on $\Omega$, a compact subset of $\mathbb{R}^{2}$. The observation points are restricted to a constant vertical-height observation platform $\mathcal{P}{h{v}}=G_{f}+h_{v}$ *

Lemma $3.4$ Every point $x^{\prime} \in \Omega$ is a fixed-point of the set-valued mapping $x \rightarrow$ $\Pi_{\Omega} \mathcal{V}\left(\left(x, f_{h_{z}}(x)\right)\right)$ on $\Omega$ into $2^{\Omega}$. Moreover, at a point $x^{\prime} \in \Omega$ where the mapping $\Pi_{\Omega} \mathcal{\nu}\left(\cdot, f_{h_{v}}(\cdot)\right)$ is continuous with respect to the Euclidean metric $\rho_{E}$ on $\Omega$, and Hausdorff metric $\rho_{H}$ on $2^{\Omega}$, there exists an open ball $\mathcal{B}\left(x^{\prime} ; \delta\right)=\left{x \in \mathbb{R}^{2}:\left|x-x^{\prime}\right|<\right.$ $\delta}$ about $x^{\prime}$ with radius $\delta>0$ such that $\left(\mathcal{B}\left(x^{\prime} ; \delta\right) \cap \Omega\right) \subset \Pi_{\Omega} \mathcal{V}\left(\left(x^{\prime}, f_{h_{v}}\left(x^{\prime}\right)\right)\right)$.

Proof Let $x^{\prime}$ be any point in $\Omega$. Then, the point $\left(x^{\prime}, f\left(x^{\prime}\right)\right)$ is always visible from the point $\left(x^{\prime}, f_{h_{v}}\left(x^{\prime}\right)\right) \in$ Epi $_{f}$. Hence, $\left(x^{\prime}, f\left(x^{\prime}\right)\right) \in \mathcal{V}\left(\left(x^{\prime}, f_{h_{v}}\left(x^{\prime}\right)\right)\right)$, and $x^{\prime} \in$ $\Pi_{\Omega} \mathcal{V}\left(\left(x^{\prime}, f_{h_{v}}\left(x^{\prime}\right)\right)\right)$, or $x^{\prime}$ is a fixed point of $\Pi_{\Omega} \mathcal{V}\left(\left(\cdot, f_{h_{v}}(\cdot)\right)\right)$. At a point $x^{\prime} \in \Omega$ where the mapping $\Pi_{\Omega} \mathcal{V}\left(\cdot, f_{h_{v}}(\cdot)\right)$ is continuous with respect to the metrics $\rho_{E}$ and $\rho_{H}$, there exists an open ball $\mathcal{B}\left(x^{\prime} ; \delta\right)$ with radius $\delta>0$ such that for every $x \in \mathcal{B}\left(x^{\prime} ; \delta\right) \cap \Omega$, the point $(x, f(x))$ is visible from $\left(x, f_{h_{v}}(x)\right)$. Thus, the desired result follows.

Theorem $3.3$ Assume that the spatial domain $\Omega$ has a $C_{1}$-boundary $\partial \Omega$, and the mapping $x \rightarrow \Pi_{\Omega} \mathcal{V}\left(\left(x, f_{h_{v}}(x)\right)\right)$ from $\partial \Omega$ into $2^{\Omega}$ is continuous with respect to metrics $\rho_{E}$ and $\rho_{H}$. Then there exists an integer $N \geq 1$, and a finite point set $P(N)=\left{x^{(k)}, k=1, \ldots, N\right} \subset \Omega$ such that $\Omega=\bigcup_{k=1}^{N} \Pi_{\Omega} \mathcal{V}\left(\left(x^{(k)}, f_{h_{v}}\left(x^{(k)}\right)\right)\right)$, or equivalently, $G_{f}$ is totally visible from the finite point set $\left{\left(x^{(k)}, f_{h_{v}}\left(x^{(k)}\right)\right): x^{(k)} \in\right.$ $\left.P^{(N)}\right} .$

Proof Since $\partial \Omega$ is $C_{1}$ and compact; and $f$ restricted to $\partial \Omega$ is a $C_{1}$-function, it follows from Lemma $2.1$ that we can find a finite point set $P_{1}=\left{x^{(k)} \in \partial \Omega, k=1, \ldots, M\right}$ such that $\bigcup_{k=1}^{M} \mathcal{B}\left(x^{(k)} ; \delta_{\min }\right)$ forms a boundary layer $L_{B}$ about $\partial \Omega$, where $\delta_{\min }$ is the minimum radius of the open balls $\mathcal{B}(x ; \delta)$ (having properties specified in Lemma 3.4) over all $x \in \partial \Omega$, and
$$L_{B}=\bigcup_{k=1}^{M}\left(\mathcal{B}\left(x^{(k)} ; \delta_{\min }\right) \cap \Omega\right)$$

## robotics代写|寻路算法代写Path Planning Algorithms|Non-simply Connected Objects

For a 3D non-simply connected object, the determination of minimum number of point-observers for total visibility is generally a difficult problem. We shall examine a few cases where explicit solutions to Problem $3.4$ are obtainable.
(i) Toroidal Objects: First, consider the case where the solid object $\mathcal{O}$ whose surface $\partial \mathcal{O}$ under observation is a $3-\mathrm{D}$ torus described by:
$$\partial \mathcal{O}=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{3}^{2}=r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2}\right}$$
where $r$ is the radius of the circular torus tube, and $R$ is the distance from the torus center to the center of the tube satisfying $R>r$. The observation points are restricted to the exterior of $\mathcal{O}$ and its boundary surface $\partial \mathcal{O}$. Moreover, we require that the distances of the observation points $z^{(i)}$ from the torus center are $>R+r$ and $\leq \Delta$, a specified distance $>R+r$ (See Fig. 3.10). This case is relevant to the problem of sensor placement for observing a toroidal plasma such as that in the tokamak machine. The solution to Problem $3.4$ can be constructed by making use of the geometric symmetry of the torus with respect to the $x_{3}$-axis. To obtain the largest visible sets for each observation point, two of the point observers $z^{(1)}$ and $z^{(2)}$ should be at the maximum allowable distance $\Delta$ from the torus center on the $x_{3}$-axis. The visible sets of these observation points are given by

$$\mathcal{V}\left(z^{(1)}\right)=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{3}=\sqrt{r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right.}\right)^{2}$$
if $\left(R-r \cos \left(\theta_{4}\right)\right)^{2}<x_{1}^{2}+x_{2}^{2} \leq\left(R+r \cos \left(\theta_{3}\right)\right)^{2}$;
$$\left.x_{3}^{2}=r^{2}-\left(R^{2}-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2}, \text { if } r^{2} \leq x_{1}^{2}+x_{2}^{2} \leq\left(R-r \cos \left(\theta_{4}\right)\right)^{2}\right},$$
$$\mathcal{V}\left(z^{(2)}\right)=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{3}=-\sqrt{r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2}}\right.$$
if $\left(R-r \cos \left(\theta_{4}\right)\right)^{2}<x_{1}^{2}+x_{2}^{2} \leq\left(R+r \cos \left(\theta_{3}\right)\right)^{2}$;
$$\left.x_{3}^{2}=r^{2}-\left(R^{2}-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2}, \text { if } r^{2} \leq x_{1}^{2}+x_{2}^{2} \leq\left(R-r \cos \left(\theta_{4}\right)\right)^{2}\right},$$
where
$$\theta_{1}=\tan ^{-1}(\Delta / R), \quad \theta_{2}=\cos ^{-1}\left(r / \sqrt{\Delta^{2}+R^{2}}\right), \quad \theta_{3}=\pi-\theta_{1}-\theta_{2}, \quad \theta_{4}=\theta_{2}-\theta_{1} .$$
The invisible set of these points corresponds to $\partial \mathcal{O}-\left(\mathcal{V}\left(z^{(1)}\right) \cup \mathcal{V}\left(z^{(2)}\right)\right)$ which is a circular band given by
$$\begin{array}{r} B_{d}=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{3}^{2}=r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2},\right. \ \text { if } \left.\left(R-r \cos \left(\theta_{3}\right)\right)^{2} \leq x_{1}^{2}+x_{2}^{2} \leq(R+r)^{2}\right} \end{array}$$
The remaining problem is determine the minimum number of point-observers at distance $\Delta$ from the torus center to attain total visibility of $B_{d}$. This problem corresponds to finding a $N$-polygon whose vertices lie on the circle $\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}\right.$ : $\left.x_{1}^{2}+x_{2}^{2}=\Delta^{2}, x_{3}=0\right}$ with the smallest $N$, that circumscribes the circle with radius $R+r$. The minimum number of point-observers for total visibility of $\partial \mathcal{O}$ is $2+N$. Figure $3.10$ shows the location of the point-observers for total visibility of $\partial \mathcal{O}$ for a special case. In this case, the circle $\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{1}^{2}+x_{2}^{2}=(R+r)^{2}, x_{3}=0\right}$ can be circumscribed by a square whose corners correspond to the point-observers at a distance $\Delta$ from the torus center. Thus, the minimum number of point-observers for total visibility of $\partial \mathcal{O}$ is six.

Next, we consider a variation of the foregoing case in which the object under observation is a toroidal cavity whose wall is described by (3.20). It is desirable to observe the cavity wall by means of point-observers located on the wall and in the interior of the cavity. We may classify this case as an “Interior Observation-Point Set Problem”, and the former case as an “Exterior Observation-Point Set Problem”.

## robotics代写|寻路算法代写Path Planning Algorithms|Single Point-Observer Static Optimal Visibility Problems

G(X)>F(X) 对全部 X∈Ω

Ω=[0,1]. 可以看出\Pi_{\Omega} \mathcal{V}\left(\left(x_{o}, g\left(x_{o}\right)\right)\right)=\left[0, \hat{x} {1}\right] \cup\left{\hat{x}{2}\right} \cup\left[\hat{x}{3}, \hat{x}{4}\right] \cup\左[\hat{x}{5}, \hat{x}{6}\right]\Pi_{\Omega} \mathcal{V}\left(\left(x_{o}, g\left(x_{o}\right)\right)\right)=\left[0, \hat{x} {1}\right] \cup\left{\hat{x}{2}\right} \cup\left[\hat{x}{3}, \hat{x}{4}\right] \cup\左[\hat{x}{5}, \hat{x}{6}\right]与示例 2.1 一样，该示例还表明点观察器的可见集可能包含孤立点。

## robotics代写|寻路算法代写Path Planning Algorithms|Non-simply Connected Objects

(i) 环形物体：首先，考虑固体物体的情况这谁的表面∂这在观察中是一个3−D圆环描述为：
\partial \mathcal{O}=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{3}^{2 }=r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2}\right}\partial \mathcal{O}=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{3}^{2 }=r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2}\right}

\mathcal{V}\left(z^{(2)}\right)=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^ {3}：x_{3}=-\sqrt{r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{ 2}}\对。$$if \left(Rr \cos \left(\theta_{4}\right)\right)^{2}<x_{1}^{2}+x_{2}^{2} \leq\left (R+r \cos \left(\theta_{3}\right)\right)^{2};$$ \left.x_{3}^{2}=r^{2}-\left(R^{2}-\sqrt{x_{1}^{2}+x_{2}^{2}} \right)^{2}, \text { if } r^{2} \leq x_{1}^{2}+x_{2}^{2} \leq\left(Rr \cos \left(\theta_ {4}\right)\right)^{2}\right},\mathcal{V}\left(z^{(2)}\right)=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^ {3}：x_{3}=-\sqrt{r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{ 2}}\对。$$if \left(Rr \cos \left(\theta_{4}\right)\right)^{2}<x_{1}^{2}+x_{2}^{2} \leq\left (R+r \cos \left(\theta_{3}\right)\right)^{2};$$ \left.x_{3}^{2}=r^{2}-\left(R^{2}-\sqrt{x_{1}^{2}+x_{2}^{2}} \right)^{2}, \text { if } r^{2} \leq x_{1}^{2}+x_{2}^{2} \leq\left(Rr \cos \left(\theta_ {4}\right)\right)^{2}\right},

θ1=棕褐色−1⁡(Δ/R),θ2=因−1⁡(r/Δ2+R2),θ3=圆周率−θ1−θ2,θ4=θ2−θ1.

\begin{array}{r} B_{d}=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{ 3}^{2}=r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2},\right. \ \text { if } \left.\left(Rr \cos \left(\theta_{3}\right)\right)^{2} \leq x_{1}^{2}+x_{2}^{ 2} \leq(R+r)^{2}\right} \end{数组}\begin{array}{r} B_{d}=\left{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{ 3}^{2}=r^{2}-\left(R-\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2},\right. \ \text { if } \left.\left(Rr \cos \left(\theta_{3}\right)\right)^{2} \leq x_{1}^{2}+x_{2}^{ 2} \leq(R+r)^{2}\right} \end{数组}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写寻路算法Path Planning Algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写寻路算法Path Planning Algorithms代写方面经验极为丰富，各种代写寻路算法Path Planning Algorithms相关的作业也就用不着说。

• 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.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写寻路算法Path Planning Algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写寻路算法Path Planning Algorithms代写方面经验极为丰富，各种代写寻路算法Path Planning Algorithms相关的作业也就用不着说。

• 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|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|Objects Under Observation

（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

(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.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。

## robotics代写|寻路算法代写Path Planning Algorithms|Mapping of Planetary Surface

statistics-lab™ 为您的留学生涯保驾护航 在代写寻路算法Path Planning Algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写寻路算法Path Planning Algorithms代写方面经验极为丰富，各种代写寻路算法Path Planning Algorithms相关的作业也就用不着说。

• 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|Mapping of Planetary Surface

In planetary exploration, one or more spacecraft or artificial satellites with onboard cameras, sensors and radar altimeters moving in the vicinity of a planet may be used to map out the planet surface and its physical properties. It is desirable to choose appropriate trajectories for the spacecraft or artificial satellite such that a specified part of the planet surface can be mapped out completely. Here, the object under observation is a 2-dimensional surface embedded in a 3-dimensional world space. The observers correspond to moving cameras and sensors with finite viewing apertures. In the case of multiple spacecraft or artificial satellites, the observation may be made in a cooperative manner so that complete surface mapping can be accomplished by using a minimal amount of non-redundant observation data. One may develop cooperative strategies based on the chosen spacecraft trajectories, or in conjunction with the motion planning task.

The placement of fixed cameras for observing a $3 \mathrm{D}$-object in the world space for analysis and action is a basic task in surveillance and monitoring systems. The cameras generally have finite viewing apertures, and they are mounted on fixed observation platforms. For complete visual coverage of the object, more than one camera are needed. A basic problem is to determine the minimum number of cameras and their locations for complete visual coverage of the object under observation.

## robotics代写|寻路算法代写Path Planning Algorithms|Radio Repeater Allocation

Modern cellular telephone and wireless communication networks make use of multiple radio or optical repeaters to cover a given service area. These repeaters receive radio or electromagnetic-wave signals from the users via line-of-sight transmission, and relay the signals to other users in the network. In the planning and design of the repeater network, it is desirable to use a minimum number of stationary repeaters to achieve complete coverage of a given service area. A basic problem is to determine the minimum number of repeaters and their locations in a specified spatial domain such that complete coverage of the service area is attained. The service area and the allowable area for repeater installation are generally not identical.

The identification of cancer or abnormal cells by means of computer-aided analysis of microscopic observation of a sample collection of living cells is of great interest in biomedical applications. To keep the cells alive during the observation period, they are usually immersed in a liquid medium. To obtain 3D images of the cells, more than one cameras placed on a platform outside or immersed inside the liquid medium are required. Thus, a basic problem is to determine the minimum number of cameras and their locations for a given observation platform. Recently, studies involving the interaction of living cells call for the manipulation of living cells using microscopic images. The image information may be used for the feedback control of cell movements. In this application, it is necessary to ensure that the cell properties such as geometric shapes are unaffected by the observation and actuation processes. For example, when active electromagnetic sensors such as laser-based sensors and manipulators are used for observation and actuation, the electromagnetic pressure exerted on the cell-surface produced by the sensors and actuators may affect the cell shape and structural properties.

## robotics代写|寻路算法代写Path Planning Algorithms|Health-Monitoring and Control of Micro-distributed

In the health monitoring and control of micro-distributed systems such as microopto-electromechanical systems composed of micro-machined solid structures, it is required to observe the structural surface by means of a finite number of discrete optical sensors. An optimum design problem is to determine the minimum number of these sensors and their locations to observe the entire structural surface. This problem is akin to the well-known “Art Gallery Problem” first posed by Klee [12], i.e. determine the minimum number and locations of point guards inside an n-wall polygonal art gallery room such that every wall can be seen by at least one-guard. In the Art Gallery Problem, the observation points (locations of the guards) are in the interior or on the boundary of a polygonal spatial domain. Here, the object under observation is a surface or a 2 -dimensional manifold in the 3 -dimensional Euclidean space, and the observation points are restricted to another surface which does not intersect the observed one.

In the surveillance of a specified terrestrial domain and exploration of a planetary surface, single or multiple Unmanned Aerial Vehicles (UAV’s) and robotic rovers equipped with cameras may be used. It is desirable to find their motions such that complete visual coverage of the terrestrial domain or maximum amount of sensor data can be obtained along their corresponding paths in the spatial domain. These paths may be determined before launching the UAV’s or robotic rovers based on known terrestrial data. The mobile-observer motions may also be determined in real-time based on the observed terrestrial and/or sensor data accumulated along the past path up to the present time.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。