### 机器人代写|SLAM代写机器人导航代考|Updating the Landmark Estimates

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• 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|Updating the Landmark Estimates

FastSLAM represents the conditional landmark estimates $p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)$ in (3.3) using low-dimensional EKFs. For now, I will assume that the data associations $n^{t}$ are known. In section 3.4, this restriction will be removed.
Since the landmark estimates are conditioned on the robot’s path, $N$ EKFs are attached to each particle in $S_{t}$. The posterior over the $n$-th landmark position $\theta_{n}$ is easily obtained. Its computation depends on whether $n=n_{t}$, that is, whether or not landmark $\theta_{n}$ was observed at time $t$. For the observed landmark $\theta_{n_{t}}$, we follow the usual procedure of expanding the posterior using Bayes Rule.
$$p\left(\theta_{n_{t}} \mid s^{t}, z^{t}, u^{t}, n^{t}\right) \stackrel{\text { Bayes }}{=} \eta p\left(z_{t} \mid \theta_{n_{t}}, s^{t}, z^{t-1}, u^{t}, n^{t}\right) p\left(\theta_{n_{t}} \mid s^{t}, z^{t-1}, u^{t}, n^{t}\right)$$
Next, the Markov property is used to simplify both terms of the equation. The observation $z_{t}$ only depends on $\theta_{n_{t}}, s_{t}$, and $n_{t}$. Similarly, $\theta_{n_{t}}$ is not affected by $s_{t}, u_{t}$, or $n_{t}$ without the observation $z_{t}$.
$$p\left(\theta_{n_{t}} \mid s^{t}, z^{t}, u^{t}, n^{t}\right) \stackrel{\text { Markov }}{=} \eta p\left(z_{t} \mid \theta_{n_{t}}, s_{t}, n_{t}\right) p\left(\theta_{n_{t}} \mid s^{t-1}, z^{t-1}, u^{t-1}, n^{t-1}\right)$$
For $n \neq n_{t}$, we leave the landmark posterior unchanged.
$$p\left(\theta_{n \neq n_{t}} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)=p\left(\theta_{n \neq n_{t}} \mid s^{t-1}, z^{t-1}, u^{t-1}, n^{t-1}\right)$$
FastSLAM implements the update equation (3.22) using an EKF. As in EKF solutions to SLAM, this filter uses a linear Gaussian approximation for the perceptual model. We note that, with an actual linear Gaussian observation model, the resulting distribution $p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)$ is exactly Gaussian, even if the motion model is non-linear. This is a consequence of sampling over the robot’s pose.

The non-linear measurement model $g\left(s_{t}, \theta_{n_{t}}\right)$ will be approximated using a first-order Taylor expansion. The landmark estimator is conditioned on a fixed robot path, so this expansion is only over $\theta_{n_{t}}$. We will assume that measurement noise is Gaussian with covariance $R_{t}$.
\begin{aligned} \hat{z}{t} &=g\left(s{t}^{[m]}, \mu_{n_{t}, t-1}\right) \ G_{\theta_{n_{t}}} &=\left.\nabla_{\theta_{n_{t}}} g\left(s_{t}, \theta_{n_{t}}\right)\right|{s{t}=s_{t}^{[m]} ; \theta_{n_{t}}=\mu_{n_{t}, t-1}^{[m]}} \ g\left(s_{t}, \theta_{n_{t}}\right) & \approx \hat{z}{t}+G{\theta}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right) \end{aligned}
Under this approximation, the first term of the product (3.22) is distributed as follows:
$$p\left(z_{t} \mid \theta_{i}, s_{t}, n_{t}\right) \sim \mathcal{N}\left(z_{t} ; \hat{z}{t}+G{\theta}\left(\theta_{n_{t}}-\mu_{n_{t}, t-1}^{[m]}\right), R_{t}\right)$$

## 机器人代写|SLAM代写机器人导航代考|Calculating Importance Weights

Samples from the proposal distribution are distributed according to $p\left(s^{t}\right.$ $\left.z^{t-1}, u^{t}, n^{t-1}\right)$, and therefore do not match the desired posterior $p\left(s^{t}\right.$ | $\left.z^{t}, u^{t}, n^{t}\right)$. This difference is corrected through importance sampling. An example of importance sampling is shown in Figure 3.6. Instead of sampling directly from the target distribution (shown as a solid line), samples are drawn from a simpler proposal distribution, a Gaussian (shown as a dashed line). In regions where the target distribution is larger than the proposal distribution, the samples receive higher weights. As a result, samples in this region will be picked more often. In regions where the target distribution is smaller than the proposal distribution, the samples will be given lower weights. In the limit of infinite samples, this procedure will produce samples distributed according to the target distribution.

For FastSLAM, the importance weight of each particle $w_{t}^{[i]}$ is equal to the ratio of the SLAM posterior and the proposal distribution described previously.

$$w_{t}^{[m]}=\frac{\text { target distribution }}{\text { proposal distribution }}=\frac{p\left(s^{t,[m]} \mid z^{t}, u^{t}, n^{t}\right)}{p\left(s^{t,[m]} \mid z^{t-1}, u^{t}, n^{t-1}\right)}$$
The numerator of (3.37) can be expanded using Bayes Rule. The normalizing constant in Bayes Rule can be safely ignored because the particle weights will be normalized before resampling.
$$w_{t}^{[m]} \stackrel{\text { Bayes }}{\propto} \frac{p\left(z_{t} \mid s^{t,[m]}, z^{t-1}, u^{t}, n^{t}\right) p\left(s^{t,[m]} \mid z^{t-1}, u^{t}, n^{t}\right)}{p\left(s^{t,[m]} \mid z^{t-1}, u^{t}, n^{t-1}\right)}$$
The second term of the numerator is not conditioned on the latest observation $z_{t}$, so the data association $n_{t}$ cannot provide any information about the robot’s path. Therefore it can be dropped.
$$\begin{gathered} w_{t}^{[m]} \stackrel{\text { Markov }}{=} \frac{p\left(z_{t} \mid s^{t,[m]}, z^{t-1}, u^{t}, n^{t}\right) p\left(s^{t,[m]} \mid z^{t-1}, u^{t}, n^{t-1}\right)}{p\left(s^{t,[m]} \mid z^{t-1}, u^{t}, n^{t-1}\right)} \ =p\left(z_{t} \mid s^{t,[m]}, z^{t-1}, u^{t}, n^{t}\right) \end{gathered}$$
The landmark estimator is an EKF, so this observation likelihood can be computed in closed form. This probability is commonly computed in terms of “innovation,” or the difference between the actual observation $z_{t}$ and the predicted observation $\hat{z}{t}$. The sequence of innovations in the EKF is Gaussian with zero mean and covariance $Z{n_{t}, t}$, where $Z_{n_{t}, t}$ is the innovation covariance matrix defined in (3.31) [3]. The probability of the observation $z_{t}$ is equal to the probability of the innovation $z_{t}-\hat{z}{t}$ being generated by this Gaussian, which can be written as: $$w{t}^{[m]}=\frac{1}{\sqrt{\mid 2 \pi Z_{n_{t}, t}} \mid} \exp \left{-\frac{1}{2}\left(z_{t}-\hat{z}{n{t}, t}\right)^{T}\left[Z_{n_{t}, t}\right]^{-1}\left(z_{t}-\hat{z}{n{t}, t}\right)\right}$$
Calculating the importance weight is a constant-time operation per particle. This calculation depends only on the dimensionality of the observation, which is constant for a given application.

## 机器人代写|SLAM代写机器人导航代考|Importance Resampling

Once the temporary particles have been assigned weights, a new set of samples $S_{t}$ is drawn from this set with replacement, with probabilities in proportion to the weights. A variety of sampling techniques for drawing $S_{t}$ can be found in [9]. In particular, Madow’s systematic sampling algorithm [56] is simple to implement and produces accurate results.

Implemented naively, resampling requires time linear in the number of landmarks $N$. This is due to the fact that each particle must be copied to the new particle set, and the length of each particle is proportional to $N$. In general, only a small fraction of the total landmarks will be observed at any one time, so copying the entire particle can be quite inefficient. In Section 3.7, we will show how a more sophisticated particle representation can eliminate unnecessary copying and reduce the computational requirement of FastSLAM to $O(M \log N)$.

At first glace, factoring the SLAM problem using the path of the robot may seem like a bad idea, because the length of the FastSLAM particles will grow over time. However, none of the the FastSLAM update equations depend on the total path length $t$. In fact, only the most recent pose $s_{t-1}^{[m]}$ is used to update the particle set. Consequently, we can silently “forget” all but the most recent robot pose in the parameterization of each particle. This avoids the obvious computational problem that would result if the dimensionality of the particle filter grows over time.

## 机器人代写|SLAM代写机器人导航代考|Updating the Landmark Estimates

FastSLAM 表示条件地标估计p(θn∣s吨,和吨,在吨,n吨)在 (3.3) 中使用低维 EKF。现在，我将假设数据关联n吨是已知的。在第 3.4 节中，此限制将被删除。

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

p(θn≠n吨∣s吨,和吨,在吨,n吨)=p(θn≠n吨∣s吨−1,和吨−1,在吨−1,n吨−1)
FastSLAM 使用 EKF 实现更新方程（3.22）。与 SLAM 的 EKF 解决方案一样，该滤波器对感知模型使用线性高斯近似。我们注意到，使用实际的线性高斯观测模型，得到的分布p(θn∣s吨,和吨,在吨,n吨)是高斯的，即使运动模型是非线性的。这是对机器人姿势进行采样的结果。

p(和吨∣θ一世,s吨,n吨)∼ñ(和吨;和^吨+Gθ(θn吨−μn吨,吨−1[米]),R吨)

## 机器人代写|SLAM代写机器人导航代考|Calculating Importance Weights

(3.37) 的分子可以用贝叶斯法则展开。可以安全地忽略贝叶斯规则中的归一化常数，因为粒子权重将在重采样之前进行归一化。

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