### robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|The Extended Kalman Filter

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|The Extended Kalman Filter

The dominant approach to the SLAM problem was introduced in a seminal paper by Smith and Cheeseman $[82]$ in 1986 , and first developed into an implemented system by Moutarlier and Chatila $[64,65]$. This approach uses the Extended Kalman Filter (EKF) to estimate the posterior over robot pose and maps. The EKF approximates the SLAM posterior as a high-dimensional Gaussian over all features in the map and the robot pose. The off-diagonal elements of the covariance matrix of this multivariate Gaussian encode the correlations between pairs of state variables. By estimating the covariance between all pairs of state variables, the EKF is expressive enough to represent the correlated errors that characterize the SLAM problem. An example of the EKF run on simulated data is shown in Figure 1.4(a). The corresponding covariance matrix, drawn as a correlation matrix, is shown in Figure 1.4(b). The darker the matrix element, the higher the correlation between the state variables corresponding to the element’s row and column. While the EKF has become the dominant approach to SLAM, it suffers from two problems that complicate its application in large, real-world environments: quadratic complexity and sensitivity to failures in data association.

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Quadratic Complexity

The first drawback of the EKF as a solution to the SLAM problem is computational complexity. Both the computation time and memory required by the

EKF scale quadratically with the number of landmarks in the map [70], limiting its application to relatively small maps. Quadratic complexity is a consequence of the Gaussian representation employed by the EKF. The uncertainty of the SLAM posterior is represented as a covariance matrix encoding the correlations between all possible pairs of state variables. In a two-dimensional world, the covariance matrix contains $2 N+3$ by $2 N+3$ entries, where $N$ is the total number of landmarks in the map. Thus, it is easy to see how the memory required to store this covariance matrix grows with $N^{2}$.

Becanse the correlations between all pairs of state variables are maintained, any sensor observation incorporated into the EKF will necessarily affect all of the other state variables. To incorporate a sensor observation, the EKF algorithm must perform an operation on every element in the covariance matrix, which requires quadratic time.

In practice, the full EKF is rarely applied to the SLAM problem. The sensor update step can be made computationally tractable by using any of a variety of approximate EKF methods. These approximations will be discussed further in Section 1.5.

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Single-Hypothesis Data Association

The second problem with EKF-based SLAM approaches is related to data association, the mapping between observations and landmarks. The SLAM problem is most commonly formulated given known data association, as in

(1.2). In the real world, the associations between observations and landmarks are hidden variables that must be determined in order to estimate the robot pose and the landmar.

The standard approach to data association in EKFs is to assign every observation to a landmark using a maximum likelihood rule; i.e. every observation is assigned to the landmark most likely to have generated it. If the probability of an observation belonging to an existing landmark is too low, it is considered for inclusion as a new landmark. Since the EKF has no mechanism for representing uncertainty over data associations, the effect of incorporating an observation given the wrong data association can never be undone. If a large number of readings are incorporated incorrectly into the EKF, the filter will diverge. Sensitivity to incorrect data association is a well known failure mode of the EKF [18].

The accuracy of data association in the EKF can be improved substantially by considering the associations of multiple observations simultaneously, at some computational cost $[1,68]$. However, this does not address the underlying data association problem with the EKF; namely that it chooses a single data association hypothesis at every time step. The correct association for a given observation is not always the most probable choice when it is first considered. In fact, the true association for an observation may initially appear to be quite improbable. Future observations may be required to provide enough information to clearly identify the association as correct. Any EKF algorithm that maintains a single data association per time step, will inevitably pick wrong associations. If these associations can never be revised, repeated mistakes can cause the filter to diverge.

Multiple data association hypotheses can always be considered by maintaining multiple copies of the EKF, one for each probable data association hypothesis $[77]$. However, the computational and memory requirements of the EKF make this approach infeasible for the SLAM problem.

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|The Extended Kalman Filter

Smith 和 Cheeseman 在一篇开创性论文中介绍了 SLAM 问题的主要方法[82]1986 年，由 Moutarlier 和 Chatila 首次开发成一个已实现的系统[64,65]. 这种方法使用扩展卡尔曼滤波器 (EKF) 来估计机器人姿态和地图的后验。EKF 将 SLAM 后验近似为地图中所有特征和机器人姿态的高维高斯。该多元高斯的协方差矩阵的非对角元素对状态变量对之间的相关性进行编码。通过估计所有状态变量对之间的协方差，EKF 的表达能力足以表示表征 SLAM 问题的相关误差。图 1.4(a) 显示了在模拟数据上运行 EKF 的示例。绘制为相关矩阵的相应协方差矩阵如图 1.4(b) 所示。矩阵元素越黑，表示该元素所在行列对应的状态变量之间的相关性越高。

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Quadratic Complexity

EKF 作为 SLAM 问题的解决方案的第一个缺点是计算复杂性。所需的计算时间和内存

EKF 与地图中地标的数量成二次方[70]，将其应用限制在相对较小的地图上。二次复杂度是 EKF 采用的高斯表示的结果。SLAM 后验的不确定性表示为一个协方差矩阵，该矩阵编码所有可能的状态变量对之间的相关性。在二维世界中，协方差矩阵包含2ñ+3经过2ñ+3条目，其中ñ是地图中地标的总数。因此，很容易看出存储这个协方差矩阵所需的内存是如何增长的ñ2.

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Single-Hypothesis Data Association

(1.2)。在现实世界中，观察和地标之间的关联是隐藏变量，必须确定这些变量才能估计机器人姿势和地标。

EKF 中数据关联的标准方法是使用最大似然规则将每个观察值分配给一个界标；即，每个观察都分配给最有可能产生它的地标。如果观测属于现有地标的概率太低，则考虑将其包含为新地标。由于 EKF 没有表示数据关联不确定性的机制，因此在给定错误数据关联的情况下合并观察的效果永远无法消除。如果大量读数被错误地合并到 EKF 中，过滤器就会发散。对不正确数据关联的敏感性是 EKF [18] 众所周知的故障模式。

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## MATLAB代写

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