### 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∗

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