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

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  • Statistical Inference 统计推断
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  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
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  • 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|Applications of SLAM

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

The problem of Simultaneous Localization and Mapping, or SLAM, has attracted immense attention in the robotics literature. SLAM addresses the problem of a mobile robot moving through an environment of which no map is available a priori. The robot makes relative observations of its ego-motion and of features in its environment, both corrupted by noise. The goal of SLAM is to reconstruct a map of the world and the path taken by the robot. SLAM is considered by many to be a key prerequisite to truly autonomous robots [85].
If the true map of the environment were available, estimating the path of the robot would be a straightforward localization problem [16]. Similarly, if the true path of the robot were known, building a map would be a relatively simple task $[63,86]$. However, when both the path of the robot and the map are unknown, localization and mapping must be considered concurrently-hence the name Simultaneous Localization and Mapping.

SLAM is an essential capability for mobile robots traveling in unknown environments where globally accurate position data (e.g. GPS) is not available. In particular, mobile robots have shown significant promise for remote exploration, going places that are too distant [34], too dangerous [88], or simply too costly to allow human access. If robots are to operate autonomously in extreme environments undersea, underground, and on the surfaces of other planets, they must be capable of building maps and navigating reliably according to these maps. Even in benign environments, such as the interiors of buildings, accurate, prior maps are often difficult to acquire. The capability to map an unknown environment allows a robot to be deployed with minimal infrastructure. This is especially important if the environment changes over time.

The maps produced by SLAM algorithms typically serve as the basis for motion planning and exploration. However, the maps often have value in their own right. In July of 2002 , nine miners in the Quecreek Mine in Sommerset, Pennsylvania were trapped underground for three and a half days after accidentally drilling into a nearby abandoned mine. A subsequent investigation attributed the cause of the accident to inaccurate maps [32]. Since the accident, mobile robots and SLAM have been investigated as possible technologies for acquiring accurate maps of abandoned mines. One such robot, shown in Figure $1.1(\mathrm{~b})$, is capable of building $3 \mathrm{D}$ reconstructions of the interior of abandoned mines using SLAM technology [88].

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

The chicken-or-egg relationship between localization and mapping is a consequence of how errors in the robot’s sensor readings are corrupted by error in the robot’s motion. As the robot moves, its pose estimate is corrupted by motion noise. The perceived locations of objects in the world are, in turn, corrupted by both measurement noise and the error in the estimated pose of the robot. Unlike measurement noise, however, error in the robot’s pose will have a systematic effect on the error in the map. In general, this effect can be stated more plainly; error in the robot’s path correlates errors in the map. As a result, the true map cannot be estimated without also estimating the true path of the robot. The relationship between localization and mapping was first identified by Smith and Cheeseman $[82]$ in their seminal paper on SLAM in $1986 .$

Figure $1.2$ shows a set of laser range scans collected by a mobile robot moving through a typical indoor environment. The robot generates estimates of its position using odometers attached to each of its wheels. In Figure 1.2(a), the laser scans are plotted with respect to the estimated position of the robot. Clearly, as error accumulates in the robot’s odometry, the map becomes increasingly inaccurate. Figure $1.2(\mathrm{~b})$ shows the laser readings plotted according to the path of the robot reconstructed by a SLAM algorithm.

Although the relationship between robot path error and map error does make the SLAM problem harder to solve in principle, one can exploit this relationship to factor the SLAM problem into a set of much smaller problems. Each of these smaller problems can be solved efficiently.

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

Two types of information are available to the robot over time: controls and observations. Controls are noisy predictions of the robot’s motion, while observations are noisy measurements of features in the robot’s environment. Each control or observation, coupled with an appropriate noise model, can be thought of as a probabilistic constraint. Each control probabilistically constrains two successive poses of the robot. Observations, on the other hand, constrain the relative positions of the robot and objects in the map. When previously observed map features are revisited, the resulting constraints can be used to update not only the current map feature and robot pose, but also correct map features that were observed in the past. An example of a network of constraints imposed by controls and observations is shown in Figure 1.3.
Initially, the constraints imposed by controls and observations may be relatively weak. However, as map features are repeatedly observed, the constraints will become increasingly rigid. In the limit of an infinite number of observations and controls, the positions of all map features will become fully correlated [19]. The primary goal of SLAM is to estimate this true map and the true pose of the robot, given the currently available set of observations and controls.

One approach to the SLAM problem would be to estimate the most likely robot pose and map using a batch estimation algorithm similar to those used in the Structure From Motion literature $[43,91]$. While extremely powerful, these techniques operate on the complete set of observations and controls, which grows without bound over time. As a consequence, these algorithms are not appropriate for online operation. Furthermore, these algorithms generally do not estimate the certainty with which different sections of the map are known, an important consideration for a robot exploring an unknown environment.

The most popular online solutions to the SLAM problem attempt to estimate the posterior probability distribution over all possible maps $\Theta$ and robot poses $s_{t}$ conditioned on the full set of controls $u^{t}$ and observations $z^{t}$ at time $t$. The observation at time $t$ will be written as $z_{t}$, while the set of all observations up to time $t$ will be written $z^{t}$. Similarly, the control at time $t$ will be written $u_{t}$, and the set of all controls up to time $t$ will be written $u^{t}$.
Using this notation, the joint posterior distribution over maps and robot poses can be written as:
p\left(s_{t}, \Theta \mid z^{t}, u^{t}\right)
This distribution is referred to as the SLAM posterior. At first glace, posterior estimation may seem even less feasible than the batch estimation approach. However, by making judicious assumptions about how the state of the world evolves, the SLAM posterior can be computed efficiently. Posterior estimation has several advantages over solutions that consider only the most likely state of the world. First, considering a distribution of possible solutions leads to more robust algorithms in noisy environments. Second, uncertainty can be used to compare the information conveyed by different components of the solution. One section of the map may be very uncertain, while other parts of the map are well known.

Any parameterized model can be chosen for the map $\Theta$, however it is typically represented as a set of point features, or “landmarks” [19]. In a real implementation, landmarks may correspond to the locations of features extracted from sensors, such as cameras, sonars, or laser range-finder s. Throughout most of this book we assume the point landmark model, though other representations can be used. Higher order geometric features, such as line segments [70], have also been used to represent maps in SLAM.

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


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

同时定位和映射(SLAM)问题在机器人文献中引起了极大的关注。SLAM 解决了移动机器人在没有先验地图可用的环境中移动的问题。机器人对其自我运动和环境中的特征进行相对观察,两者都被噪声破坏。SLAM 的目标是重建世界地图和机器人所走的路径。许多人认为 SLAM 是真正自主机器人的关键先决条件 [85]。
如果环境的真实地图可用,估计机器人的路径将是一个简单的定位问题[16]。同样,如果机器人的真实路径已知,则构建地图将是一项相对简单的任务[63,86]. 然而,当机器人的路径和地图都未知时,必须同时考虑定位和建图——因此得名同时定位和建图。

SLAM 是移动机器人在无法获得全球准确位置数据(例如 GPS)的未知环境中行驶的基本能力。特别是,移动机器人在远程探索、去太远的地方 [34]、太危险的地方 [88] 或太昂贵而无法让人类进入的地方已经显示出巨大的希望。如果机器人要在海底、地下和其他行星表面的极端环境中自主运行,它们必须能够构建地图并根据这些地图可靠地导航。即使在良性环境中,例如建筑物内部,通常也很难获得准确的先验地图。映射未知环境的能力允许以最少的基础设施部署机器人。如果环境随时间发生变化,这一点尤其重要。

SLAM 算法生成的地图通常用作运动规划和探索的基础。然而,这些地图往往本身就具有价值。2002 年 7 月,宾夕法尼亚州萨默塞特的 Quecreek 矿区的 9 名矿工在意外钻入附近废弃矿井后被困在地下三天半。随后的调查将事故原因归咎于地图不准确[32]。自事故发生以来,移动机器人和 SLAM 已被研究作为获取废弃矿山准确地图的可能技术。一种这样的机器人,如图所示1.1( b), 能够建造3D使用 SLAM 技术重建废弃矿井内部 [88]。

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

定位和映射之间的先有鸡还是先有蛋的关系是机器人传感器读数中的错误如何被机器人运动中的错误破坏的结果。当机器人移动时,它的姿态估计会被运动噪声破坏。反过来,世界中物体的感知位置会受到测量噪声和机器人估计姿态误差的影响。然而,与测量噪声不同,机器人姿态的误差将对地图中的误差产生系统性影响。一般来说,这种效果可以说得更清楚。机器人路径中的错误与地图中的错误相关。结果,如果不估计机器人的真实路径,就无法估计真实地图。本地化和映射之间的关系首先由 Smith 和 Cheeseman 确定[82]在他们关于 SLAM 的开创性论文中1986.

数字1.2显示了移动机器人在典型室内环境中移动时收集的一组激光范围扫描。机器人使用连接在每个轮子上的里程表来估计其位置。在图 1.2(a) 中,激光扫描是相对于机器人的估计位置绘制的。显然,随着机器人里程计误差的累积,地图变得越来越不准确。数字1.2( b)显示了根据 SLAM 算法重建的机器人路径绘制的激光读数。

尽管机器人路径误差和地图误差之间的关系确实使 SLAM 问题在原则上更难解决,但可以利用这种关系将 SLAM 问题分解为一组小得多的问题。这些小问题中的每一个都可以有效地解决。

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

随着时间的推移,机器人可以获得两种类型的信息:控制和观察。控制是对机器人运动的噪声预测,而观察是对机器人环境中特征的噪声测量。每个控制或观察,加上适当的噪声模型,都可以被认为是一个概率约束。每个控制概率地约束机器人的两个连续姿势。另一方面,观察限制了机器人和地图中物体的相对位置。当重新访问先前观察到的地图特征时,生成的约束不仅可用于更新当前的地图特征和机器人姿态,还可以用于更新过去观察到的地图特征。图 1.3 显示了控制和观察施加的约束网络示例。
最初,控制和观察施加的约束可能相对较弱。然而,随着地图特征的反复观察,约束将变得越来越严格。在无限数量的观察和控制的限制下,所有地图特征的位置将变得完全相关[19]。SLAM 的主要目标是在给定当前可用的一组观察和控制的情况下估计此真实地图和机器人的真实姿势。

解决 SLAM 问题的一种方法是使用类似于“运动结构”文献中使用的批量估计算法来估计最可能的机器人姿势和地图[43,91]. 虽然功能非常强大,但这些技术对完整的观察和控制集进行操作,这些观察和控制会随着时间的推移而不受限制地增长。因此,这些算法不适用于在线操作。此外,这些算法通常不会估计已知地图不同部分的确定性,这是机器人探索未知环境的重要考虑因素。

SLAM 问题最流行的在线解决方案试图估计所有可能地图的后验概率分布θ和机器人姿势s吨以全套控制为条件在吨和观察和吨有时吨. 当时的观察吨将被写为和吨, 而到时间的所有观测值的集合吨会写和吨. 同样,控制时间吨会写在吨, 以及截至时间的所有控件的集合吨会写在吨.
这种分布称为 SLAM 后验。乍一看,后验估计似乎比批量估计方法更不可行。然而,通过对世界状态如何演变做出明智的假设,可以有效地计算 SLAM 后验。与仅考虑世界最可能状态的解决方案相比,后验估计有几个优势。首先,考虑可能解决方案的分布会导致在嘈杂环境中算法更加稳健。其次,不确定性可用于比较解决方案的不同组件所传达的信息。地图的一个部分可能非常不确定,而地图的其他部分是众所周知的。

可以为地图选择任何参数化模型θ,但是它通常表示为一组点特征或“地标”[19]。在实际实现中,地标可能对应于从传感器(例如相机、声纳或激光测距仪)中提取的特征的位置。在本书的大部分内容中,我们都假设点地标模型,尽管可以使用其他表示。高阶几何特征,如线段 [70],也已用于表示 SLAM 中的地图。

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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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