### 机器人代写|SLAM代写机器人导航代考|Proof of the FastSLAM Factorization

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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|Proof of the FastSLAM Factorization

The FastSLAM factorization can be derived directly from the SLAM path posterior (3.2). Using the definition of conditional probability, the SLAM posterior can be rewritten as:
$$p\left(s^{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)=p\left(s^{t} \mid z^{t}, u^{t}, n^{t}\right) p\left(\Theta \mid s^{t}, z^{t}, u^{t}, n^{t}\right)$$
Thus, to derive the factored posterior (3.3), it suffices to show the following for all non-negative values of $t$ :
$$p\left(\Theta \mid s^{t}, z^{t}, u^{t}, n^{t}\right)=\prod_{n=1}^{N} p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)$$
Proof of this statement can be demonstrated through induction. Two intermediate results must be derived in order to achieve this result. The first quantity to be derived is the probability of the observed landmark $\theta_{n_{t}}$ conditioned on the data. This quantity can be rewritten using Bayes Rule.
$$p\left(\theta_{n_{t}} \mid s^{t}, z^{t}, u^{t}, n^{t}\right) \stackrel{\text { Bayes }}{=} \frac{p\left(z_{t} \mid \theta_{n_{t}}, s^{t}, z^{t-1}, u^{t}, n^{t}\right)}{p\left(z_{t} \mid s^{t}, z^{t-1}, u^{t}, n^{t}\right)} p\left(\theta_{n_{t}} \mid s^{t}, z^{t-1}, u^{t}, n^{t}\right)$$
Note that the current observation $z_{t}$ depends solely on the current state of the robot and the landmark being observed. In the rightmost term of (3.6), we similarly notice that the current pose $s_{t}$, the current action $u_{t}$, and the current data association $n_{t}$ have no effect on $\theta_{n_{t}}$ without the current observation $z_{t}$. Thus, all of these variables can be dropped.
$$p\left(\theta_{n_{t}} \mid s^{t}, z^{t}, u^{t}, n^{t}\right) \stackrel{M a r k o v}{=} \frac{p\left(z_{t} \mid \theta_{n_{t}}, s_{t}, n_{t}\right)}{p\left(z_{t} \mid s^{t}, z^{t-1}, u^{t}, n^{t}\right)} p\left(\theta_{n_{t}} \mid s^{t-1}, z^{t-1}, u^{t-1}, n^{t-1}\right)$$
Next, we solve for the rightmost term of (3.7) to get:
$$p\left(\theta_{n_{t}} \mid s^{t-1}, z^{t-1}, u^{t-1}, n^{t-1}\right)=\frac{p\left(z_{t} \mid s^{t}, z^{t-1}, u^{t}, n^{t}\right)}{p\left(z_{t} \mid \theta_{n_{t}}, s_{t}, n_{t}\right)} p\left(\theta_{n_{t}} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)$$

## 机器人代写|SLAM代写机器人导航代考|The FastSLAM 1.0 Algorithm

The factorization of the posterior (3.3) highlights important structure in the SLAM problem that is ignored by SLAM algorithms that estimate an unstructured posterior. This structure suggests that under the appropriate conditioning, no cross-correlations between landmarks have to be maintained explicitly. FastSLAM exploits the factored representation by maintaining $N+1$ filters, one for each term in (3.3). By doing so, all $N+1$ filters are low-dimensional.
FastSLAM estimates the first term in (3.3), the robot path posterior, using a particle filter. The remaining $N$ conditional landmark posteriors $p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)$ are estimated using EKFs. Each EKF tracks a single landmark position, and therefore is low-dimensional and fixed in size. The landmark EKFs are all conditioned on robot paths, with each particle in the particle filter possessing its own set of EKFs. In total, there are $N \cdot M$ EKFs, where $M$ is the total number of particles in the particle filter. The particle filter is depicted graphically in Figure 3.3. Each FastSLAM particle is of the form:
$$S_{t}^{[m]}=\left\langle s^{t,[m]}, \mu_{1, t}^{[m]}, \Sigma_{1, t}^{[m]}, \ldots, \mu_{N, t}^{[m]}, \Sigma_{N, t}^{[m]}\right\rangle$$
The bracketed notation $[m]$ indicates the index of the particle; $s^{t,[m]}$ is the $m$-th particle’s path estimate, and $\mu_{n, t}^{[m]}$ and $\Sigma_{n, t}^{[m]}$ are the mean and covariance of the Gaussian representing the $n$-th feature location conditioned on the path $s^{t,[m]}$. Together all of these quantities form the $m$-th particle $S_{t}^{[m]}$, of which there are a total of $M$ in the FastSLAM posterior. Filtering, that is, calculating the posterior at time $t$ from the one at time $t-1$, involves generating a new particle set $S_{t}$ from $S_{t-1}$, the particle set one time step earlier. The new particle set incorporates the latest control $u_{t}$ and measurement $z_{t}$ (with corresponding data association $n_{t}$ ). This update is performed in four steps.
First, a new robot pose is drawn for each particle that incorporates the latest control. Each pose is added to the appropriate robot path estimate $s^{t-1,[m]}$. Next, the landmark EKFs corresponding to the observed landmark are updated with the new observation. Since the robot path particles are not drawn from the true path posterior, each particle is given an importance weight to reflect this difference. A new set of particles $S_{t}$ is drawn from the weighted particle set using importance resampling. This importance resampling step is necessary to insure that the particles are distributed according to the true posterior (in the limit of infinite particles). The four basic steps of the FastSLAM algorithm $[59]$, shown in Table 3.1, will be explained in detail in the following four sections.

## 机器人代写|SLAM代写机器人导航代考|Sampling a New Pose

The particle set $S_{t}$ is calculated incrementally, from the set $S_{t-1}$ at time $t-1$, the observation $z_{t}$, and the control $u_{t}$. Since we cannot draw samples directly from the SLAM posterior at time $t$, we will instead draw samples from a simpler distribution called the proposal distribution, and correct for the difference using a technique called importance sampling.

In general, importance sampling is an algorithm for drawing samples from functions for which no direct sampling procedure exists [55]. Each sample drawn from the proposal distribution is given a weight equal to the ratio of the posterior distribution to the proposal distribution at that point in the sample space. A new set of unweighted samples is drawn from the weighted set with probabilities in proportion to the weights. This process is an instantiation of Rubin’s Sampling Importance Resampling (SIR) algorithm [79].

The proposal distribution of FastSLAM generate guesses of the robot’s pose at time $t$ given each particle $S_{t-1}^{[m]}$. This guess is obtained by sampling from the probabilistic motion model.
$$s_{t}^{[m]} \sim p\left(s_{t} \mid u_{t}, s_{t-1}^{[m]}\right)$$
This estimate is added to a temporary set of particles, along with the path $s^{t-1,[m]}$. Under the assumption that the set of particles $S_{t-1}$ is distributed according to $p\left(s^{t-1} \mid z^{t-1}, u^{t-1}, n^{t-1}\right)$, which is asymptotically correct, the new particles drawn from the proposal distribution are distributed according to:
$$p\left(s^{t} \mid z^{t-1}, u^{t}, n^{t-1}\right)$$
It is important to note that the motion model can be any non-linear function. This is in contrast to the EKF, which requires the motion model to be

linearized. The only practical limitation on the measurement model is that samples can be drawn from it conveniently. Regardless of the proposal distribution, drawing a new pose is a constant-time operation for every particle. It does not depend on the size of the map.

A simple four parameter motion model was used for all of the planar robot experiments in this book. This model assumes that the velocity of the robot is constant over the time interval covered by each control. Each control $u_{t}$ is two-dimensional and can be written as a translational velocity $v_{t}$ and a rotational velocity $\omega_{t}$. The model further assumes that the error in the controls is Gaussian. Note that this does not imply that error in the robot’s motion will also be Gaussian; the robot’s motion is a non-linear function of the controls and the control noise.

The errors in translational and rotational velocity have an additive and a multiplicative component. Throughout this book, the notation $\mathcal{N}(x ; \mu, \Sigma)$ will be used to denote a normal distribution over the variable $x$ with mean $\mu$ and covariance $\Sigma$.
\begin{aligned} v_{t}^{\prime} & \sim \mathcal{N}\left(v_{t}, \alpha_{1} v_{t}+\alpha_{2}\right) \ \omega_{t}^{\prime} & \sim \mathcal{N}\left(\omega_{t}, \alpha_{3} \omega_{t}+\alpha_{4}\right) \end{aligned}
This motion model is able to represent the slip and skid errors errors that occur in typical ground vehicles [8]. The first step to drawing a new robot pose from this model is to draw a new translational and rotational velocity according to the observed control. The new pose $s_{t}$ can be calculated by simulating the new control forward from the previous pose $s_{t-1}^{[m]}$. Figure $3.4$ shows 250 samples drawn from this motion model given a curved trajectory. In this simulated example, the translational error of the robot is low, while the rotational error is high.

## 机器人代写|SLAM代写机器人导航代考|Proof of the FastSLAM Factorization

FastSLAM 分解可以直接从 SLAM 路径后验 (3.2) 推导出来。使用条件概率的定义，SLAM 后验可以重写为：
p(s吨,θ∣和吨,在吨,n吨)=p(s吨∣和吨,在吨,n吨)p(θ∣s吨,和吨,在吨,n吨)

p(θ∣s吨,和吨,在吨,n吨)=∏n=1ñp(θn∣s吨,和吨,在吨,n吨)

p(θn吨∣s吨,和吨,在吨,n吨)= 贝叶斯 p(和吨∣θn吨,s吨,和吨−1,在吨,n吨)p(和吨∣s吨,和吨−1,在吨,n吨)p(θn吨∣s吨,和吨−1,在吨,n吨)

p(θn吨∣s吨,和吨,在吨,n吨)=米一种rķ这在p(和吨∣θn吨,s吨,n吨)p(和吨∣s吨,和吨−1,在吨,n吨)p(θn吨∣s吨−1,和吨−1,在吨−1,n吨−1)

p(θn吨∣s吨−1,和吨−1,在吨−1,n吨−1)=p(和吨∣s吨,和吨−1,在吨,n吨)p(和吨∣θn吨,s吨,n吨)p(θn吨∣s吨,和吨,在吨,n吨)

## 机器人代写|SLAM代写机器人导航代考|The FastSLAM 1.0 Algorithm

FastSLAM 使用粒子滤波器估计 (3.3) 中的第一项，机器人路径后验。其余ñ条件地标后验p(θn∣s吨,和吨,在吨,n吨)使用 EKF 估计。每个 EKF 跟踪单个地标位置，因此是低维且大小固定的。地标 EKF 都以机器人路径为条件，粒子过滤器中的每个粒子都拥有自己的一组 EKF。总共有ñ⋅米EKF，其中米是粒子过滤器中的粒子总数。图 3.3 以图形方式描述了粒子过滤器。每个 FastSLAM 粒子的形式为：

## 机器人代写|SLAM代写机器人导航代考|Sampling a New Pose

FastSLAM 的proposal distribution 对机器人的位姿进行猜测吨给定每个粒子小号吨−1[米]. 这个猜测是通过从概率运动模型中采样获得的。
s吨[米]∼p(s吨∣在吨,s吨−1[米])

p(s吨∣和吨−1,在吨,n吨−1)

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