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|Visibility-based Optimal Motion Planning

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:
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
&\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),
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:
\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) .
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|Visibility-based Optimal Motion Planning


robotics代写|寻路算法代写Path Planning Algorithms|Observation Of Three-Dimensional Objects

现在,我们将注意力转向观察三维物体的最优路径规划问题。正如我们在第一章中提到的。2(命题2.2和备注 2.2) 至少需要两个点观察者才能在R3. 以下引理给出了实体在R3.

引理 4.2 假设被观察的物体是一个紧致的单连通固体∂这⊂R3具有平滑的边界∂这并且没有外法相交点。然后,总能见度这可以通过观察平台中的一组有限的观察点来获得磷H在恒定的有限高度H沿着体表上方的外法线∂这.

证明4.3自从这是具有光滑边界的紧致简连实体∂这并且没有向外法线交点,那么\mathcal{P}{h}=\left{z \in \mathbb{R}^{3}: z=\right.$ $x+h n(x), x \in \partial \mathcal{O}}\mathcal{P}{h}=\left{z \in \mathbb{R}^{3}: z=\right.$ $x+h n(x), x \in \partial \mathcal{O}}是明确的。对于任何观察点和∈磷H, 存在一个点X=和−Hn(X)∈∂这这样X从可见和. 而且,在(和),可见集和, 是的紧子集∂这包含X有限度μ2(∂在(和)). 自从μ2(这)是有限的,存在一个有限的数ñ观察点和(一世)∈磷H,一世= 1,…,ñ这样∂这⊆⋃一世=1ñ在(和(一世))由 Heine-Borel 覆盖定理保证。

首先,我们将问题 4.1(最短路径总能见度问题)扩展到对 3-D 对象的观察这由单个移动观察者。

问题 4.1 单个移动观察者最短路径问题。给定一个观察平台磷封闭观察到的 3-D 对象这, 和两个不同的点和这,和F∈磷, 找到最短允许路径Γ磷开始于和这并结束于和F这样⋃和∈Γ磷在(和)=∂这.

我们观察到对于磷=磷H, 一旦解决问题3.5获得,任何可接受的路径开始于和这,通过观察点集中的所有点磷(ñ), 并结束于和F, 是问题 4.1A 解决方案的候选项。

为了说明问题 4.1A 解决方案的性质,我们考虑一个具体的例子。

例子4.5这里,被观察的物体是一个封闭的球体乙¯(0;r这). 观景台磷H是一个球体小号(0;r这+H)观察高度有限H>0. 根据动态规划中的最优性原理,最优路径是在不同指定点开始和结束的测地线的串联和这,和F∈ 磷H. 数字4.13显示对应于的最佳路径和这=(0,−(r这+H),0)和和F=(0,r这+H,0), 有观察高度H令人满意的0<一种≤圆周率/4, 在哪里一种=罪−1⁡(r这/(r这+H))是观察锥的半角。我们观察到最优路径有长度l分钟=3圆周率(r这+H),并且沿着路径有一个大圆环。为了一种>圆周率/4,沿路径有不止一个大圆环。

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

让B=\left{e_{1}, \ldots, e_{n}\right}B=\left{e_{1}, \ldots, e_{n}\right}是指定的正交基n维实欧几里得空间Rn, 在哪里和一世对应于一世th 单位基向量。一个点的表示X∈Rn关于乙由列向量指定[X1,…,Xn]吨. 通常的欧几里得范数X表示为|X|. 让F=F(X)是一个指定的实值C2-函数定义在Ω,一个指定的简单连接,紧凑的子集R2具有平滑的边界∂Ω. 就像在第一章中一样。2 , graph 和 vation 是描述的空间地形表面GF在世界空间在=R3. 观景台磷观察者所在的位置对应于高架表面GF由GFH在, 在哪里FH在= 定义 F+H在和H在是一个指定的正数。这意味着对于任何X∈Ω,观察者处于固定的垂直高度H在表面之上GF.

从命题的延伸3.1到的情况暗淡⁡(Ω)=2, 存在临界垂直高度H在C(X)对于每个X∈Ω这样才能获得完全的可见性。考虑不平凡的情况,其中H在<H在C(X)对全部X∈Ω因此移动观察者必须移动以实现完全可见性。让一世lF=[0,吨F]表示观察时间间隔,其中吨F可以是有限的固定或可变终止时间。为简单起见,移动观察者由点质量表示米. 它在世界空间中的位置在=R3随时吨由指定p(吨)其关于给定正交基的表示乙表示为[X1(吨),X2(吨),X3(吨)]吨, 在哪里X3(吨)对应于沿观察者的位置X3-轴在时间吨. 的动议

米X¨(吨)+νXX˙(吨)=在(吨) 米X¨3(吨)+ν3(X(吨),X˙(吨),X3(吨),X˙3(吨))=X(吨)−米G,在哪里X(吨)=[X1(吨),X2(吨)]吨;(在,X)是外力与在=[在1,在2]吨作为控制;−米G是沿向下方向的重力X3-轴。νX是给定的非负摩擦系数;ν3是其参数的指定实值函数,描述X3- 摩擦力的分量。带有单点和双点的变量分别表示它们关于时间的一阶和二阶导数吨. 假设移动观察者被限制继续前进GF在任何时候都没有滑动,移动的观察者运动满足一个完整的约束:
X3(吨)=F(X(吨)) 对全部 吨∈一世lF,
X(吨)∈Ω 对全部 吨∈一世一世F.
自从F是一个C2- 功能开启Ω, 我们可以将 (5.3) 微分两次吨获得
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),
在哪里∇X表示关于的梯度算子X, 和HF(X(吨))的 Hessian 矩阵F关于X评价为X(吨). 将 (5.5) 代入 (5.2) 得到所需的垂直分量X(吨)使移动观察者保持在表面上的外力GF每时每刻:
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).
假设移动观察者位于GF在开始时间吨=0, 然后
在上述移动观察者的动态模型中,我们假设X- 摩擦力的分量仅取决于X˙. 一般来说,它们可能取决于两者(X,X˙)和(X3,X˙3). 另外,为了简化后续开发,我们没有考虑上述模型中的表面接触力。

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


问题 5.1 最小时间总能见度问题。让在一种d=⋃吨F≥0在一种d(一世吨F)是所有允许控制的集合。给定sX(0)=(X(0),X˙(0))或具有初始位置的移动观察者的初始状态p(0)=(X(0),F(X(0)))∈GF和初速度在(0)=(X˙(0),∇XF(X(0))吨X˙(0)), 找到最小的时间吨F≥0和一个可接受的控制在=在(吨)定义于一世lF使得其相应的运动或时间相关的路径\Gamma_{i_{f}^{}}=\left{\left(x_{u^{}}(t), f\left(x_{u^{}}(t)\right)\right) \在 \mathbb{R}^{3}: t \in I_{t_{f}^{}}\right}\Gamma_{i_{f}^{}}=\left{\left(x_{u^{}}(t), f\left(x_{u^{}}(t)\right)\right) \在 \mathbb{R}^{3}: t \in I_{t_{f}^{}}\right}在表面上GF满足总能见度条件吨F :⋃吨∈一世吨F在((X在(吨),FH在(X在(吨)))=GF
\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,
在哪里F最大限度′是一个指定的正数。现在,如果集合\Omega^{\text {def }}=\left{x \in \Omega: | \nabla_{x} f\left(x_{u^{*}}(t)\right)\right.$ $\left.| \leq f_{\max }^{\prime}\right}\Omega^{\text {def }}=\left{x \in \Omega: | \nabla_{x} f\left(x_{u^{*}}(t)\right)\right.$ $\left.| \leq f_{\max }^{\prime}\right}是一个简单连通的紧子集R2,那么我们可以替换Ω在问题5.1经过Ω′照顾约束(5.12)。

问题 5.2 固定观察时间间隔的最大能见度问题。给定一个有限的观察时间间隔一世一世F和sX(0)=(X(0),X˙(0)),或具有初始位置的移动观察者的初始状态p(0)=(X(0),F(X(0)))∈GF和初速度在(0)=(X˙(0),∇XF(X(0))吨X˙(0))在吨=0, 找到一个可接受的控制在=在(吨)及其相应的运动或时间相关路径\Gamma_{t_{f}}^{}=\left{\left(x_{ut^{}}(t), f\left(x_{u}^{*}(t)\right)\right) \in \mathbb{R}^{3}: t \in I_{t f}\right} \subset G_{f}\Gamma_{t_{f}}^{}=\left{\left(x_{ut^{}}(t), f\left(x_{u}^{*}(t)\right)\right) \in \mathbb{R}^{3}: t \in I_{t f}\right} \subset G_{f}使得可见性函数由下式给出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 tJ_{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
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} 。

robotics代写|寻路算法代写Path Planning Algorithms 请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。







术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。



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