### 机器人代写|SLAM代写机器人导航代考|Comparison of FastSLAM to Existing Techniques

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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

## 机器人代写|SLAM代写机器人导航代考|FastSLAM

In this chapter we will describe the basic FastSLAM algorithm, an alternative approach to SLAM that is based on particle filtering.

The FastSLAM algorithm is based on a structural property of the SLAM problem that the EKF fails to exploit. Each control or observation collected by the robot only constrains a small number of state variables. Controls probabilistically constrain the pose of the robot relative to its previous pose, while observations constrain the positions of landmarks relative to the robot. It is only after a large number of these probabilistic constraints are incorporated that the map becomes fully correlated. The EKF, which makes no assumptions about structure in the state variables, fails to take advantage of this sparsity over time.

FastSLAM exploits conditional independences that are a consequence of the sparse structure of the SLAM problem to factor the posterior into a product of low dimensional estimation problems. The resulting algorithm scales efficiently to large maps and is robust to significant ambiguity in data association.

## 机器人代写|SLAM代写机器人导航代考|Particle Filtering

The Kalman Filter and the EKF represent probability distributions using a parameterized model (a multivariate Gaussian). Particle filters, on the other hand, represent distributions using a finite set of sample states, or “particles” $[20,51,75]$. Regions of high probability contain a high density of particles, whereas regions of low probability contain few or no particles. Given enough samples, this non-parametric representation can approximate arbitrarily complex, multi-modal distributions. In the limit of an infinite number of samples, the true distribution can be reconstructed exactly [21], under some very mild assumptions. Given this representation, the Bayes Filter update equation can be implemented using a simple sampling procedure.

Particle filters have been applied successfully to a variety of real world estimation problems $[21,44,81]$. One of the most common examples of particle filtering in robotics is Monte Carlo Localization, or MCL [89]. In MCL, a set of particles is used to represent the distribution of possible poses of a robot relative to a fixed map. An example is shown in Figure 3.1. In this example, the robot is given no prior information about its pose. This complete uncertainty is represented by scattering particles with uniform probability throughout the map, as shown in Figure 3.1(a). Figure 3.1(b) shows the particle filter after incorporating a number of controls and observations. At this point, the posterior has converged to an approximately unimodal distribution.

The capability to track multi-modal beliefs and include non-linear motion and measurement models makes the performance of particle filters particularly robust. However, the number of particles needed to track a given belief may, in the worst case, scale exponentially with the dimensionality of the state space. As such, standard particle filtering algorithms are restricted to problems of relatively low dimensionality. Particle filters are especially ill-suited to the SLAM problem, which may have millions of dimensions. However, the following sections will show how the SLAM problem can be factored into a set of independent landmark estimation problems conditioned on an estimate of the robot’s path. The robot path posterior is of low dimensionality and can be estimated efficiently using a particle filter. The resulting algorithm, called FastSLAM, is an example of a Rao-Blackwellized particle filter [21, 22, 23].

## 机器人代写|SLAM代写机器人导航代考|Factored Posterior Representation

The majority of SLAM approaches are based on estimating the posterior over maps and robot pose.
$$p\left(s_{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)$$
FastSLAM computes a slightly different quantity, the posterior over maps and robot path.s.
$$p\left(s^{t}, \theta \mid z^{t}, u^{t}, n^{t}\right)$$
This subtle difference will allow us to factor the SLAM posterior into a product of simpler terms. Figure $3.2$ revisits the interpretation of the SLAM problem as a Dynamic Bayes Network (DBN). In the scenario depicted by the DBN, the robot observes landmark $\theta_{1}$ at time $t=1, \theta_{2}$ at time $t=2$, and then re-observes landmark $\theta_{1}$ at time $t=3$. The gray shaded area represents the path of the robot from time $t=1$ to the present time. From this diagram, it is evident that there are important conditional independences in the SLAM problem. In particular, if the true path of the robot is known, the position of landmark $\theta_{1}$ is conditionally independent of landmark $\theta_{2}$. Using the terminology of DBNs, the robot’s path “d-separates” the two landmark nodes $\theta_{1}$ and $\theta_{2}$. For a complete description of d-separation see $[74,80]$.

This conditional independence has an important consequence. Given knowledge of the robot’s path, an observation of one landmark will not provide any information about the position of any other landmark. In other words, if an oracle told us the true path of the robot, we could estimate the position of every landmark as an independent quantity. This means that the SLAM posterior (3.2) can be factored into a product of simpler terms.

$$p\left(s^{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)=\underbrace{p\left(s^{t} \mid z^{t}, u^{t}, n^{t}\right)}{\text {path posterior }} \underbrace{\prod{n=1}^{N} p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)}_{\text {landmark estimators }}$$
This factorization, first developed by Murphy [66], states that the SLAM posterior can be separated into a product of a robot path posterior $p\left(s^{t} \mid\right.$ $\left.z^{t}, u^{t}, n^{t}\right)$, and $N$ landmark posteriors conditioned on the robot’s path. It is important to note that this factorization is exact; it follows directly from the structure of the SLAM problem.

## 机器人代写|SLAM代写机器人导航代考|FastSLAM

FastSLAM 算法基于 EKF 未能利用的 SLAM 问题的结构特性。机器人收集的每个控制或观察只约束少量的状态变量。控制以概率方式约束机器人相对于其先前姿势的姿势，而观察约束相对于机器人的地标的位置。只有在合并了大量这些概率约束后，地图才会完全相关。EKF 对状态变量的结构不做任何假设，随着时间的推移无法利用这种稀疏性。

FastSLAM 利用 SLAM 问题的稀疏结构导致的条件独立性，将后验因素分解为低维估计问题的乘积。由此产生的算法可以有效地扩展到大型地图，并且对数据关联中的显着歧义具有鲁棒性。

## 机器人代写|SLAM代写机器人导航代考|Factored Posterior Representation

p(s吨,θ∣和吨,在吨,n吨)
FastSLAM 计算略有不同的数量，即地图和机器人路径的后验。
p(s吨,θ∣和吨,在吨,n吨)

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