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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

Adding a new landmark to FastSLAM can be difficult decision to make, just as with EKF-based algorithms. This is especially true when an individual measurement is insufficient to constrain the new landmark in all dimensions [13]. If the measurement function $g\left(\theta_{n_{t}}, s_{t}\right)$ is invertible, however, a single measurement is sufficient to initialize a new landmark. Each observation defines a Gaussian:
$$\mathcal{N}\left(z_{t} ; \hat{z}{t}+G{\theta_{n_{t}}}\left(\theta_{n_{t}}-\mu_{n_{t}, t}^{[m]}\right), R_{t}\right)$$
This Gaussian can be written explicitly as:
\begin{aligned} \frac{1}{\sqrt{\left|2 \pi R_{t}\right|}} \exp {&-\frac{1}{2}\left(z_{t}-\hat{z}{t}-G{\theta_{n_{t}}}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right)\right)^{T} \ &\left.R_{t}^{-1}\left(z_{t}-\hat{z}{t}-G{\theta_{n_{t}}}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right)\right)\right} \end{aligned}
We define a function J to be equal to the negative of the exponent of this Gaussian:
$$J=\frac{1}{2}\left(z_{t}-\hat{z}{t}-G{\theta_{n_{t}}}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right)\right)^{T} R_{t}^{-1}\left(z_{t}-\hat{z}{t}-G{\theta_{n_{t}}}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right)\right)$$

The second derivative of $J$ with respect to $\theta_{n_{t}}$ will be the inverse of the covariance matrix of the Gaussian in landmark coordinates.
\begin{aligned} \frac{\partial J}{\partial \theta_{n_{t}}} &=-\left(z_{t}-\hat{z}{t}-G{\theta_{n_{t}}}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right)\right)^{T} R_{t}^{-1} G_{\theta_{n_{t}}} \ \frac{\partial^{2} J}{\partial \theta_{n_{t}}^{2}} &=G_{\theta_{n_{t}}}^{T} R_{t}^{-1} G_{\theta_{n_{t}}} \end{aligned}
Consequently, an invertible observation can be used to create a new landmark as follows.
\begin{aligned} \mu_{n_{t}, t}^{[m]} &=g^{-1}\left(s_{t}^{[m]}, z_{t}\right) \ \Sigma_{n_{t}, t}^{[m]} &=\left(G_{\theta_{n_{t}}, t}^{T} R^{-1} G_{\theta_{n_{t}}, t}\right)^{-1} \ w_{t}^{[m]} &=p_{0} \end{aligned}
In practice, a simpler initialization procedure also works well. Instead of computing the correct initial covariance, the covariance can be computed by setting the variance of each landmark parameter to a large initial value $K$ and incorporating the first observation. Higher values of $K$ lead to closer approximations of the true covariance, but can also lead to numerical instability.
\begin{aligned} \mu_{n_{t}, t}^{[m]} &=g^{-1}\left(s_{t}^{[m]}, z_{t}\right) \ \Sigma_{n_{t}, t}^{[m]} &=K \cdot I \end{aligned}
Initialization techniques for situations in which $g$ is not invertible (e.g. bearings-only SLAM) are discussed in $[12,13]$. These situations require the accumulation of multiple observations in order to estimate the location of a landmark. FastSLAM is currently being applied to the problem of bearingsonly SLAM [83].

机器人代写|SLAM代写机器人导航代考|Summary of the FastSLAM Algorithm

Table $3.2$ summarizes the FastSLAM algorithm with unknown data association. Particles in the complete FastSLAM algorithm have the form:
$$S_{t}^{[m]}=\left\langle s_{t}^{[m]} \mid N_{t}^{[m]}, \mu_{1, t}^{[m]}, \Sigma_{1, t}^{[m]}, \ldots, \mu_{N_{t}^{[m]}, t}^{[m]}, \Sigma_{N_{t}^{[m]}, t}^{[m]}\right\rangle$$
In addition to the latest robot pose $s_{t}^{[m]}$ and the feature estimates $\mu_{n, t}^{[m]}$ and $\Sigma_{n, t}^{[m]}$, each particle maintains the number of features $N_{t}^{[m]}$ in its local map. It is interesting to note that each particle may have a different number of landmarks. This is an expressive representation, but it can lead to difficulties in determining the most probable map.

机器人代写|SLAM代写机器人导航代考|Greedy Mutual Exclusion

If multiple observations are incorporated simultaneously, the simplest approach to data association is to consider the identity of each observation independently. However, the data associations of each observation are clearly correlated, as was shown in Section 3.4. The data associations are correlated

through error in the robot pose, and they also must all obey a mutual exclusion constraint; more than one observation cannot be associated with the same landmark at the same time. Considering the data associations jointly does address these problems $[1,14,68]$, but these techniques are computationally expensive for large numbers of simultaneous observations.

FastSLAM addresses the first problem, motion ambiguity, by sampling over robot poses and data associations. Each set of data association decisions is conditioned on a particular robot path. Thus, the data associations can be chosen independently without fear that pose error will corrupt all of the decisions. Some of the particles will chose the correct data associations, while others draw inconsistent robot poses, pick incorrect data associations, and receive low weights. Picking associations independently per particle still ignores the issue of mutual exclusion, however. Mutual exclusion is particularly useful for deciding when to add new landmarks in noisy environments. Instead of assigning an observation of an unseen landmark to an existing landmark, mutual exclusion will force the creation of a new landmark if both features are observed.

Proper handling of mutual exclusion requires that the data associations of all observations be considered simultaneously. However, mutual exclusion can also be enforced in a greedy fashion. Each observation is processed sequentially and ignores the landmarks associated with previously assigned observations. With a single data association hypothesis, applying mutual exclusion greedily can lead to failures in noisy environments. It can work well in FastSLAM, though, because the motion ambiguity that commonly causes greedy mutual exclusion failures is largely factored out by sampling over the the robot’s path. Furthermore, errors due to the greedy nature of the algorithm can also be minimized by processing the observations in different orders for each particle.

SLAM代写

ñ(和吨;和^吨+Gθn吨(θn吨−μn吨,吨[米]),R吨)

\begin{aligned} \frac{1}{\sqrt{\left|2 \pi R_{t}\right|}} \exp {&-\frac{1}{2}\left(z_{t}- \hat{z}{t}-G{\theta_{n_{t}}}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]} \right)\right)^{T} \ &\left.R_{t}^{-1}\left(z_{t}-\hat{z}{t}-G{\theta_{n_{t} }}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right)\right)\right} \end{aligned}\begin{aligned} \frac{1}{\sqrt{\left|2 \pi R_{t}\right|}} \exp {&-\frac{1}{2}\left(z_{t}- \hat{z}{t}-G{\theta_{n_{t}}}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]} \right)\right)^{T} \ &\left.R_{t}^{-1}\left(z_{t}-\hat{z}{t}-G{\theta_{n_{t} }}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right)\right)\right} \end{aligned}

Ĵ=12(和吨−和^吨−Gθn吨(θn吨−μn吨,吨−1[米]))吨R吨−1(和吨−和^吨−Gθn吨(θn吨−μn吨,吨−1[米]))

∂Ĵ∂θn吨=−(和吨−和^吨−Gθn吨(θn吨−μn吨,吨−1[米]))吨R吨−1Gθn吨 ∂2Ĵ∂θn吨2=Gθn吨吨R吨−1Gθn吨

μn吨,吨[米]=G−1(s吨[米],和吨) Σn吨,吨[米]=(Gθn吨,吨吨R−1Gθn吨,吨)−1 在吨[米]=p0

μn吨,吨[米]=G−1(s吨[米],和吨) Σn吨,吨[米]=ķ⋅一世

机器人代写|SLAM代写机器人导航代考|Greedy Mutual Exclusion

FastSLAM 通过对机器人姿势和数据关联进行采样来解决第一个问题，即运动模糊性。每组数据关联决策都以特定的机器人路径为条件。因此，可以独立选择数据关联，而不必担心姿势错误会破坏所有决策。一些粒子会选择正确的数据关联，而其他粒子会绘制不一致的机器人姿势，选择不正确的数据关联，并获得低权重。然而，每个粒子独立地选择关联仍然忽略了互斥问题。互斥对于决定何时在嘈杂环境中添加新地标特别有用。如果两个特征都被观察到，互斥将强制创建新的地标，而不是将未见地标的观察分配给现有地标。

有限元方法代写

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MATLAB代写

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