robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Structure and Sparsity in SLAM

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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|Structure and Sparsity in SLAM

robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Structure and Sparsity in SLAM

At any given time, the observations and controls accumulated by the robot constrain only a small subset of the state variables. This sparsity in the dependencies between the data and the state variables can be exploited to compute the SLAM posterior in a more efficient manner. For example, two landmarks separated by a large distance are often weakly correlated. Moreover, nearby pairs of distantly separated landmarks will have very similar correlations. A number of approximate EKF SLAM algorithms exploit these properties by breaking the complete map into a set of smaller submaps. Thus, the large EKF can be decomposed into a number of loosely coupled, smaller EKFs. This approach has resulted in a number of efficient, approximate EKF algorithms that require linear time [36], or even constant time $[1,5,7,49]$ to incorporate sensor observations (given known data association).

While spatially factoring the SLAM problem does lead to efficient EKFbased algorithms, the new algorithms face the same difficulties with data association as the original EKF algorithm. This book presents an alternative solution to the SLAM problem which exploits sparsity in the dependencies between state variables over time. In addition to enabling efficient computation of the SLAM posterior, this approach can maintain multiple data association hypotheses. The result is a SLAM algorithm that can be employed in large environments with significant data association ambiguity.

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

As shown in Section 1.2, correlations between elements of the map only arise through robot pose uncertainty. Thus, if the robot’s true path were known, the landmark positions could be estimated independently. Stated probabilistically, knowledge of the robot’s true path renders estimates of landmark positions to be conditionally independent.

Proof of this statement can be seen by drawing the SLAM problem as a Dynamic Bayes Network, as shown in Figure 1.5. The robot’s pose at time $t$ is denoted $s_{t}$. This pose is a probabilistic function of the previous pose of the robot $s_{t-1}$ and the control $u_{t}$ executed by the robot. The observation at time $t$, written $z_{t}$, is likewise determined by the pose $s_{t}$ and the landmark being observed $\theta_{n_{t}}$. In the scenario depicted in Figure $1.5$, the robot observes landmark 1 at $t=1$ and $t=3$, and observes landmark 2 at $t=2$. The gray region highlights the complete path of the robot $s_{1} \ldots s_{t}$. It is apparent from this network that this path “d-separates” [80] the nodes representing the two landmarks. In other words, if the true path of the robot is known, no information about the location of the first landmark can tell us anything about the location of the second landmark.

As a result of this relationship, the SLAM posterior given known data association (1.2) can be rewritten as the following product:
p\left(s^{t}, \theta \mid z^{t}, u^{t}, n^{t}\right)=\underbrace{p\left(s^{t} \mid z^{t}, u^{t}, n^{t}\right)}{\text {path posterior }} \underbrace{\prod{n=1}^{N} p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)}_{\text {landmark estimators }}
This factorization states that the full SLAM posterior can be decomposed into a product of $N+1$ recursive estimators: one estimator over robot paths, and $N$ independent estimators over landmark positions, each conditioned on the path estimate. This factorization was first presented by Murphy [66]. It is important to note that this factorization is exact, not approximate. It is a result of fundamental structure in the SLAM problem. A complete proof of this factorization will be given in Chapter 3 .

The factored posterior (1.4) can be approximated efficiently using a particle filter $[20,51,75]$, with each particle representing a sample path of the robot. Attached to each particle are $N$ independent landmark estimators (implemented as EKFs), one for each landmark in the map. Since the landmark filters estimate the positions of individual landmarks, each filter is low dimensional. In total there are $N, M$ Kalman filters. The resulting algorithm for updating this particle filter will be called FastSLAM. Readers familiar with the statistical literature should note that FastSLAM is an instance of the Rao-Blackwellized Particle Filter [23], by virtue of the fact that it combines a sampled representation with closed form calculations of certain marginals.
There are four steps to recursively updating the particle filter given a new control and observation, as shown in Table 1.1. The first step is to propose a new robot pose for each particle that is consistent with the previous pose and the new control. Next, the landmark filter in each particle that corresponds with the latest observation is updated using to the standard EKF update equations. Each particle is given an importance weight, and a new set of samples is drawn according to these weights. This importance resampling step corrects for the fact that the proposal distribution and the posterior distribution are not the same. This update procedure converges asymptotically to the true posterior distribution as the number of samples goes to infinity. In practice, FastSLAM generates a good reconstruction of the posterior with a relatively small number of particles (i.e. $M=100$ ).

Initially, factoring the SLAM posterior using the robot’s path may seem like a poor choice because the length of the path grows over time. Thus, one might expect the dimensionality of a filter estimating the posterior over robot paths to also grow over time. However, this is not the case for FastSLAM. As will be shown in Chapter 3, the landmark update equations and the importance weights only depend on the latest pose of the robot $s_{t}$, allowing us to silently forget the rest of the robot’s path. As a result, each FastSLAM particle only needs to maintain an estimate of the current pose of the robot. Thus the dimensionality of the particle filter stays fixed over time.

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

FastSLAM has two main advantages over the EKF. First, by factoring the estimation of the map into in separate landmark estimators conditioned on the robot path posterior, FastSLAM is able to compute the full SLAM posterior in an efficient manner. The motion update, the landmark updates, and the computation of the importance weights can all be accomplished in constant time per particle. The resampling step, if implemented naively, can be performed in linear time. However, this step can be implemented in logarithmic time by organizing each particle as a binary tree of landmark estimators, instead of an array. The $\log (N)$ FastSLAM algorithm can be used to build a map with over a million landmarks using a standard desktop computer.

robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Structure and Sparsity in SLAM


robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Structure and Sparsity in SLAM

在任何给定时间,机器人积累的观察和控制只约束状态变量的一小部分。可以利用数据和状态变量之间依赖关系的稀疏性以更有效的方式计算 SLAM 后验。例如,相隔很远的两个地标通常是弱相关的。此外,附近的远距离地标对将具有非常相似的相关性。许多近似 EKF SLAM 算法通过将完整地图分解为一组较小的子地图来利用这些属性。因此,大型 EKF 可以分解为许多松散耦合的较小 EKF。这种方法产生了许多需要线性时间 [36] 甚至恒定时间的高效、近似 EKF 算法[1,5,7,49]结合传感器观察(给定已知的数据关联)。

虽然对 SLAM 问题进行空间分解确实会产生高效的基于 EKF 的算法,但新算法在数据关联方面面临与原始 EKF 算法相同的困难。本书介绍了 SLAM 问题的另一种解决方案,该解决方案利用了状态变量之间随时间的依赖关系的稀疏性。除了能够高效计算 SLAM 后验之外,这种方法还可以维护多个数据关联假设。结果是一种 SLAM 算法,可以在具有显着数据关联模糊性的大型环境中使用。

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

如第 1.2 节所示,地图元素之间的相关性仅通过机器人姿势不确定性产生。因此,如果机器人的真实路径已知,则可以独立估计地标位置。从概率上讲,对机器人真实路径的了解使得对地标位置的估计是有条件独立的。

可以通过将 SLAM 问题绘制为动态贝叶斯网络来证明这一说法,如图 1.5 所示。机器人当时的姿势吨表示s吨. 这个位姿是机器人前一个位姿的概率函数s吨−1和控制在吨由机器人执行。当时的观察吨, 写和吨, 同样由姿势决定s吨和正在观察的地标θn吨. 在图所示的场景中1.5,机器人在吨=1和吨=3,并在吨=2. 灰色区域突出显示机器人的完整路径s1…s吨. 从这个网络中可以明显看出,这条路径“d-分离”[80]代表两个地标的节点。换句话说,如果机器人的真实路径已知,则没有关于第一个地标位置的信息可以告诉我们关于第二个地标位置的任何信息。

由于这种关系,给定已知数据关联 (1.2) 的 SLAM 后验可以重写为以下乘积:
p\left(s^{t}, \theta \mid z^{t}, u^{ t}, n^{t}\right)=\underbrace{p\left(s^{t} \mid z^{t}, u^{t}, n^{t}\right)} {\text {路径后 }} \underbrace{\prod {n=1}^{N} p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^ {t}\right)}_{\text {landmark estimators }}
这个因式分解表明完整的 SLAM 后验可以分解为ñ+1递归估计器:机器人路径上的一个估计器,以及ñ地标位置上的独立估计器,每个估计器都以路径估计为条件。这种因式分解首先由 Murphy [66] 提出。需要注意的是,这种分解是精确的,而不是近似的。这是 SLAM 问题中基本结构的结果。这种因式分解的完整证明将在第 3 章中给出。

因子后验 (1.4) 可以使用粒子滤波器有效地逼近[20,51,75],每个粒子代表机器人的一个样本路径。附着在每个粒子上的是ñ独立的地标估计器(实现为 EKF),用于地图中的每个地标。由于地标过滤器估计单个地标的位置,因此每个过滤器都是低维的。总共有ñ,米卡尔曼滤波器。用于更新此粒子过滤器的最终算法将称为 FastSLAM。熟悉统计文献的读者应该注意到,FastSLAM 是 Rao-Blackwellized 粒子滤波器 [23] 的一个实例,因为它结合了采样表示和某些边缘的封闭形式计算。
给定新的控制和观察,递归更新粒子滤波器有四个步骤,如表 1.1 所示。第一步是为每个粒子提出一个新的机器人姿态,该姿态与之前的姿态和新的控制一致。接下来,使用标准 EKF 更新方程更新每个粒子中与最新观察相对应的地标滤波器。每个粒子都被赋予一个重要性权重,并根据这些权重抽取一组新的样本。这个重要性重采样步骤纠正了提议分布和后验分布不同的事实。随着样本数量趋于无穷大,这个更新过程渐近收敛到真实的后验分布。在实践中,米=100 ).

最初,使用机器人的路径来分解 SLAM 后验似乎是一个糟糕的选择,因为路径的长度会随着时间的推移而增长。因此,人们可能期望估计机器人路径后验的过滤器的维度也会随着时间的推移而增长。然而,FastSLAM 并非如此。如第 3 章所示,地标更新方程和重要性权重仅取决于机器人的最新位姿s吨,让我们默默地忘记机器人的其余路径。因此,每个 FastSLAM 粒子只需要保持对机器人当前姿态的估计即可。因此,粒子滤波器的维数随时间保持固定。

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

与 EKF 相比,FastSLAM 有两个主要优势。首先,通过将地图的估计分解为以机器人后路为条件的独立地标估计器,FastSLAM 能够以有效的方式计算完整的 SLAM 后验。运动更新、地标更新和重要性权重的计算都可以在每个粒子的恒定时间内完成。重采样步骤,如果简单地实施,可以在线性时间内执行。然而,这个步骤可以通过将每个粒子组织为地标估计器的二叉树而不是数组来在对数时间内实现。这日志⁡(ñ)FastSLAM 算法可用于使用标准台式计算机构建具有超过一百万个地标的地图。

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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。





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