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

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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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