robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|The SLAM Problem

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  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|SLAM定位算法代写Simultaneous Localization and Mapping|The SLAM Problem

robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|SLAM Posterior

The pose of the robot at time $t$ will be denoted $s_{t}$. For a robot operating in a planar environment, this pose consists of the robot’s $x-y$ position in the plane and its heading direction. All experimental results presented in this book were generated in planar environments, however the algorithms apply equally well to three-dimensional worlds. The complete trajectory of the robot, consisting of the robot’s pose at every time step, will be written as $s^{t}$.
s^{t}=\left{s_{1}, s_{2}, \ldots, s_{t}\right}
We shall further assume that the robot’s environment can be modeled as a set of $N$ immobile, point landmarks. Point landmarks are commonly used to represent the locations of features extracted from sensor data, such as geometric features in a laser scan or distinctive visual features in a camera image. The set of $N$ landmark locations will be written $\left{\theta_{1}, \ldots, \theta_{N}\right}$. For notational simplicity, the entire map will be written as $\Theta$.

As the robot moves through the environment, it collects relative information about its own motion. This information can be generated using odometers attached to the wheels of the robot, inertial navigation units, or simply by observing the control commands executed by the robot. Regardless of origin, any measurement of the robot’s motion will be referred to generically as a control. The control at time $t$ will be written $u_{t}$. The set of all controls executed by the robot will be written $u^{t}$.
u^{t}=\left{u_{1}, u_{2}, \ldots, u_{t}\right}
As the robot moves through its environment, it observes nearby landmarks. In the most common formulation of the planar SLAM problem, the robot observes both the range and bearing to nearby obstacles. The observation at time $t$ will be written $z_{t}$. The set of all observations collected by the robot will be written $z^{t}$.
z^{t}=\left{z_{1}, z_{2}, \ldots, z_{t}\right}

It is commonly assumed in the SLAM literature that sensor measurements can be decomposed into information about individual landmarks, such that each landmark observation can be incorporated independently from the other measurements. This is a realistic assumption in virtually all successful SLAM implementations, where landmark features are extracted one-by-one from raw sensor data. Thus, we will assume that each observation provides information about the location of exactly one landmark $\theta_{n}$ relative to the robot’s current pose $s_{t}$. The variable $n$ represents the identity of the landmark being observed. In practice, the identities of landmarks usually can not be observed, as many landmarks may look alike. The identity of the landmark corresponding to the observation $z_{t}$ will be written as $n_{t}$, where $n_{t} \in{1, \ldots, N}$. For example, $n_{8}=3$ means that at time $t=8$ the robot observed the third landmark. Landmark identities are commonly referred to as “data associations” or “correspondences.” The set of all data associations will be written $n^{t}$.
n^{t}=\left{n_{1}, n_{2}, \ldots, n_{t}\right}
Again for simplicity, we will assume that the robot receives exactly one measurement $z_{t}$ and executes exactly one control $u_{t}$ per time step. Multiple observations per time step can be processed sequentially, but this leads to a more cumbersome notation.

Using the notation defined above, the primary goal of SLAM is to recover the best estimate of the robot pose $s_{t}$ and the map $\Theta$, given the set of noisy observations $z^{t}$ and controls $u^{t}$. In probabilistic terms, this is expressed by the following posterior:
p\left(s_{t}, \Theta \mid z^{t}, u^{t}\right)
If the set of data associations $n^{t}$ is also given, the posterior can be rewritten as:
p\left(s_{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)

robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|SLAM as a Markov Chain

The SLAM problem can be described best as a probabilistic Markov chain. A graphical depiction of this Markov chain is shown in Figure 2.2. The current pose of the robot $s_{t}$ can be written as a probabilistic function of the pose at the previous time step $s_{t-1}$ and the control $u_{t}$ executed by the robot. This function is referred to as the motion model because it describes how controls drive the motion of the robot. Additionally, the motion model describes how noise in the controls injects uncertainty into the robot’s pose estimate. The motion model is written as:
p\left(s_{t} \mid s_{t-1}, u_{t}\right)
Sensor observations gathered by the robot are also governed by a probabilistic function, commonly referred to as the measurement model. The observation $z_{t}$

is a function of the observed landmark $\theta_{n_{t}}$ and the pose of the robot $s_{t}$. The measurement model describes the physics and the error model of the robot’s sensor. The measurement model is written as:
p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right)
Using the motion model and the measurement model, the SLAM posterior at time $t$ can be computed recursively as function of the posterior at time $t-1$. This recursive update rule, known as the Bayes filter for SLAM, is the basis for the majority of online SLAM algorithms.

robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Bayes Filter Derivation

The Bayes Filter can be derived from the SLAM posterior as follows. First, the posterior (2.6) is rewritten using Bayes Rule.
p\left(s_{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)=\eta p\left(z_{t} \mid s_{t}, \Theta, z^{t-1}, u^{t}, n^{t}\right) p\left(s_{t}, \Theta \mid z^{t-1}, u^{t}, n^{t}\right)
The denominator from Bayes rule is a normalizing constant and is written as $\eta$. Next, we exploit the fact that $z_{t}$ is solely a function of the pose of the robot $s_{t}$, the map $\Theta$, and the latest data association $n_{t}$, previously described as the measurement model. Hence the posterior becomes:
=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) p\left(s_{t}, \Theta \mid z^{t-1}, u^{t}, n^{t}\right)
Now we use the Theorem of Total Probability to condition the rightmost term of $(2.10)$ on the pose of the robot at time $t-1$.
=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \int p\left(s_{t}, \Theta \mid s_{t-1}, z^{t-1}, u^{t}, n^{t}\right) p\left(s_{t-1} \mid z^{t-1}, u^{t}, n^{t}\right) d s_{t-1}

The leftmost term inside the integral can be expanded using the definition of conditional probability.
&=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \
&\int p\left(s_{t} \mid \Theta, s_{t-1}, z^{t-1}, u^{t}, n^{t}\right) p\left(\Theta \mid s_{t-1}, z^{t-1}, u^{t}, n^{t}\right) p\left(s_{t-1} \mid z^{t-1}, u^{t}, n^{t}\right) d s_{t-1}
The first term inside the integral can now be simplified by noting that $s_{t}$ is only a function of $s_{t-1}$ and $u_{t}$, previously described as the motion model.
&=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \
&\qquad \int p\left(s_{t} \mid s_{t-1}, u_{t}\right) p\left(\Theta \mid s_{t-1}, z^{t-1}, u^{t}, n^{t}\right) p\left(s_{t-1} \mid z^{t-1}, u^{t}, n^{t}\right) d s_{t-1}
At this point, the two rightmost terms in the integral can be combined.
=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \int p\left(s_{t} \mid s_{t-1}, u_{t}\right) p\left(s_{t-1}, \Theta \mid z^{t-1}, u^{t}, n^{t}\right) d s_{t-1}
Since the current pose $u_{t}$ and data association $n_{t}$ provide no new information about $s_{t-1}$ or $\Theta$ without the latest observation $z_{t}$, they can be dropped from the rightmost term of the integral. The result is a recursive formula for computing the SLAM posterior at time $t$ given the SLAM posterior at time $t-1$, the motion model $p\left(s_{t} \mid s_{t-1}, u_{t}\right)$, and the measurement model $p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right)$.
&p\left(s_{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)= \
&\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \int p\left(s_{t} \mid s_{t-1}, u_{t}\right) p\left(s_{t-1}, \Theta \mid z^{t-1}, u^{t-1}, n^{t-1}\right) d s_{t-1}

robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|The SLAM Problem


robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|SLAM Posterior

机器人当时的姿势吨将表示s吨. 对于在平面环境中运行的机器人,该位姿由机器人的X−是在平面上的位置及其航向。本书中介绍的所有实验结果都是在平面环境中生成的,但是算法同样适用于三维世界。机器人的完整轨迹,包括机器人在每个时间步的位姿,将被写为s吨.
s^{t}=\left{s_{1}, s_{2}, \ldots, s_{t}\right}s^{t}=\left{s_{1}, s_{2}, \ldots, s_{t}\right}
我们将进一步假设机器人的环境可以建模为一组ñ不动,点地标。点地标通常用于表示从传感器数据中提取的特征的位置,例如激光扫描中的几何特征或相机图像中的独特视觉特征。该组ñ地标位置将被写入\left{\theta_{1}, \ldots, \theta_{N}\right}\left{\theta_{1}, \ldots, \theta_{N}\right}. 为了符号简单,整个地图将被写为θ.

当机器人在环境中移动时,它会收集有关其自身运动的相关信息。这些信息可以使用附在机器人轮子上的里程表、惯性导航单元或简单地通过观察机器人执行的控制命令来生成。无论起源如何,对机器人运动的任何测量都将统称为控制。当时的控制吨会写在吨. 机器人执行的所有控制的集合将被写入在吨.
u^{t}=\left{u_{1}, u_{2}, \ldots, u_{t}\right}u^{t}=\left{u_{1}, u_{2}, \ldots, u_{t}\right}
当机器人在其环境中移动时,它会观察附近的地标。在平面 SLAM 问题的最常见公式中,机器人观察到附近障碍物的范围和方位。当时的观察吨会写和吨. 机器人收集的所有观测值的集合将被写入和吨.
z^{t}=\left{z_{1}, z_{2}, \ldots, z_{t}\right}z^{t}=\left{z_{1}, z_{2}, \ldots, z_{t}\right}

在 SLAM 文献中通常假设传感器测量可以分解为有关单个地标的信息,这样每个地标观察可以独立于其他测量值合并。在几乎所有成功的 SLAM 实现中,这是一个现实的假设,其中地标特征是从原始传感器数据中一一提取的。因此,我们将假设每个观测都提供了关于恰好一个地标位置的信息θn相对于机器人的当前姿势s吨. 变量n表示被观察的地标的身份。在实践中,通常无法观察到地标的身份,因为许多地标可能看起来很相似。与观测对应的地标的身份和吨将被写为n吨, 在哪里n吨∈1,…,ñ. 例如,n8=3意味着当时吨=8机器人观察到了第三个地标。地标身份通常被称为“数据关联”或“通信”。将写入所有数据关联的集合n吨.
n^{t}=\left{n_{1}, n_{2}, \ldots, n_{t}\right}n^{t}=\left{n_{1}, n_{2}, \ldots, n_{t}\right}

使用上面定义的符号,SLAM 的主要目标是恢复机器人姿态的最佳估计s吨和地图θ,给定一组嘈杂的观测值和吨和控制在吨. 在概率方面,这由以下后验表示:

robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|SLAM as a Markov Chain

SLAM 问题可以最好地描述为概率马尔可夫链。该马尔可夫链的图形描述如图 2.2 所示。机器人当前位姿s吨可以写成前一个时间步的位姿的概率函数s吨−1和控制在吨由机器人执行。这个函数被称为运动模型,因为它描述了控制如何驱动机器人的运动。此外,运动模型描述了控制中的噪声如何将不确定性注入到机器人的姿态估计中。运动模型写为:

是观察到的地标的函数θn吨和机器人的姿势s吨. 测量模型描述了机器人传感器的物理特性和误差模型。测量模型写为:
使用运动模型和测量模型,SLAM 后验时间吨可以递归地计算为时间的后验函数吨−1. 这种递归更新规则,称为 SLAM 的贝叶斯过滤器,是大多数在线 SLAM 算法的基础。

robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Bayes Filter Derivation

贝叶斯滤波器可以从 SLAM 后验推导出来,如下所示。首先,使用贝叶斯规则重写后验(2.6)。
贝叶斯规则的分母是归一化常数,写为这. 接下来,我们利用以下事实和吨仅是机器人姿态的函数s吨, 地图θ,以及最新的数据关联n吨,之前描述为测量模型。因此后验变为:

=这p(和吨∣s吨,θ,n吨) ∫p(s吨∣θ,s吨−1,和吨−1,在吨,n吨)p(θ∣s吨−1,和吨−1,在吨,n吨)p(s吨−1∣和吨−1,在吨,n吨)ds吨−1
=这p(和吨∣s吨,θ,n吨) ∫p(s吨∣s吨−1,在吨)p(θ∣s吨−1,和吨−1,在吨,n吨)p(s吨−1∣和吨−1,在吨,n吨)ds吨−1
由于当前姿势在吨和数据关联n吨不提供有关的新信息s吨−1或者θ没有最新的观察和吨,它们可以从积分的最右边项中删除。结果是用于计算 SLAM 后验的递归公式吨给定 SLAM 后验时间吨−1, 运动模型p(s吨∣s吨−1,在吨), 和测量模型p(和吨∣s吨,θ,n吨).
p(s吨,θ∣和吨,在吨,n吨)= 这p(和吨∣s吨,θ,n吨)∫p(s吨∣s吨−1,在吨)p(s吨−1,θ∣和吨−1,在吨−1,n吨−1)ds吨−1

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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。


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