### 机器人代写|SLAM代写机器人导航代考|FastSLAM with Unknown Data Association

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• 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 with Unknown Data Association

The biggest limitation of the FastSLAM algorithm described thus far is the assumption that the data associations $n^{t}$ are known. In practice, this is rarely the case. This section extends the FastSLAM algorithm to domains in which the mapping between observations and landmarks is not known [57]. The classical solution to the data association problem in SLAM is to chose $n_{t}$ such that it maximizes the likelihood of the sensor measurement $z_{t}$ given all available data [18].
$$\hat{n}{t}=\underset{n{t}}{\operatorname{argmax}} p\left(z_{t} \mid n_{t}, \hat{n}^{t-1}, s^{t}, z^{t-1}, u^{t}\right)$$
The term $p\left(z_{t} \mid n_{t}, \hat{n}^{t-1}, s^{t}, z^{t-1}, u^{t}\right)$ is referred to as a likelihood, and this approach is an example of a maximum likelihood (ML) estimator. ML data association is also called “nearest neighbor” data association, interpreting the negative log likelihood as a distance function. For Gaussians, the negative log likelihood is Mahalanobis distance, and the estimator selects data associations by minimizing this Mahalanobis distance.

In the EKF-based SLAM approaches described in Chapter 2, a single data association is chosen for the entire filter. As a result, these algorithms tend to be brittle to failures in data association. A single data association error can induce significant errors in the map, which in turn cause new data association errors, often with fatal consequences. A better understanding of how uncertainty in the SLAM posterior generates data association ambiguity will demonstrate how simple data association heuristics often fail.

## 机器人代写|SLAM代写机器人导航代考|Data Association Uncertainty

Two factors contribute to uncertainty in the SLAM posterior: measurement noise and motion noise. As measurement noise increases, the distributions of possible observations of every landmark become more uncertain. If measurement noise is sufficiently high, the distributions of observations from nearby landmarks will begin to overlap substantially. This overlap leads to ambiguity

in the identity of the landmarks. We will refer to data association ambiguity caused by measurement noise as measurement ambiguity. An example of measurement ambiguity is shown in Figure 3.7. The two ellipses depict the range of probable observations from two different landmarks. The observation, shown as an black circle, plausibly could have come from either landmark.
Attributing an observation to the wrong landmark due to measurement ambiguity will increase the error of the map and robot pose, but its impact will be relatively minor. Since the observation could have been generated by either landmark with high probability, the effect of the observation on the landmark positions and the robot pose will be small. The covariance of one landmark will be slightly overestimated, while the covariance of the second will be slightly underestimated. If multiple observations are incorporated per control, a data association mistake due to measurement ambiguity of one observation will have relatively little impact on the data association decisions for the other observations.

Ambiguity in data association caused by motion noise can have much more severe consequences on estimation accuracy. Higher motion noise will lead to higher pose uncertainty after incorporating a control. If this pose uncertainty is high enough, assuming different robot poses in this distribution will imply drastically different ML data association hypotheses for the subsequent observations. This motion ambiguity, shown in Figure $3.8$, is easily induced if there is significant rotational error in the robot’s motion. Moreover, if multiple observations are incorporated per control, the pose of the robot will correlate the data association decisions of all of the observations. If the SLAM algorithm chooses the wrong data association for a single observation due to motion ambiguity, the rest of the data associations also will be wrong with high probability.

## 机器人代写|SLAM代写机器人导航代考|Per-Particle Data Association

Unlike most EKF-based SLAM algorithms, FastSLAM takes a multi-hypothesis approach to the data association problem. Each particle represents a different hypothesized path of the robot, so data association decisions can be made on a per-particle basis. Particles that pick the correct data association will receive high weights because they explain the observations well. Particles that pick wrong associations will receive low weights and be removed in a future resampling step.

Per-particle data association has several important advantages over standard ML data association. First, it factors robot pose uncertainty out of the data association problem. Since motion ambiguity is the more severe form of data association ambiguity, conditioning the data association decisions on hypothesized robot paths seems like a logical choice. Given the scenario in Figure 3.8, some of the particles would draw new robot poses consistent with data association hypothesis on the left, while others would draw poses consistent with the data association hypothesis on the right.

Doing data association on a per-particle basis also makes the data association problem easier. In the EKF, the uncertainty of a landmark position is due to both uncertainty in the pose of the robot and measurement error. In FastSLAM, uncertainty of the robot pose is represented by the entire particle set. The landmark filters in a single particle are not affected by motion noise because they are conditioned on a specific robot path. This is especially useful if the robot has noisy motion and an accurate sensor.

Another consequence of per-particle data association is implicit, delayeddecision making. At any given time, some fraction of the particles will receive plausible, yet wrong, data associations. In the future, the robot may receive a new observation that clearly refutes these previous assignments. At this point, the particles with wrong data associations will receive low weight and likely be removed from the filter. As a result of this process, the effect of a wrong data association decision made in the past can be removed from the filter. Moreover, no heuristics are needed in order to remove incorrect old associations from the filter. This is done in a statistically valid manner, simply as a consequence of the resampling step.

## 机器人代写|SLAM代写机器人导航代考|FastSLAM with Unknown Data Association

n^吨=最大参数n吨p(和吨∣n吨,n^吨−1,s吨,和吨−1,在吨)

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