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

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## 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}

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