### 机器人代写|SLAM代写机器人导航代考|Log FastSLAM

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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

## 机器人代写|SLAM代写机器人导航代考|Log FastSLAM

The computational complexity of the FastSLAM algorithm presented up to this point requires time $O(M, N)$ where $M$ is the number of particles, and $N$ is the number of landmarks in the map. The linear complexity in $M$ is unavoidable, given that we have to process $M$ particles for every update. This linear complexity in $N$ is due to the importance resampling step in Section 3.3.4. Since the sampling is done with replacement, a single particle in the weighted particle set may be duplicated several times in $S_{t}$. The simplest way to implement this is to repeatedly copy the entire particle into the new particle set. Since the length of the particles depends linearly on $N$, this copying operation is also linear in the size of the map.

The wholesale copying of particles from the old set into the new set is an overly conservative approach. The majority of the landmark filters remain unchanged at every time step. Indeed, since the sampling is done with replacement, many of the landmark filters will be completely identical.

These observations suggest that with proper bookkeeping, a more efficient particle representation might allow duplicate landmark filters to be shared between particles, resulting in a more efficient implementation of FastSLAM. This can be done by changing the particle representation from an array of landmark filters to a binary tree. An example landmark tree is shown in Figure $3.11$ for a map with eight landmarks. In the figure, the landmarks are organized by an arbitrary landmark number $K$. In situations in which data association is unknown, the tree could be organized spatially as in a k-d tree.
Note that the landmark parameters $\mu_{n}, \Sigma_{n}$ are located at the leaves of the tree. Each non-leaf node in the tree contains pointers to up to two subtrees. Any subtree can be shared between multiple particles’ landmark trees. Sharing subtrees makes the update procedure more complicated to implement, but results in a tremendous savings in both memory and computation. Assuming that the tree is balanced, accessing a leaf requires a binary search, which will take $\log (N)$ time, on average.

The $\log (N)$ FastSLAM algorithm can be illustrated by tracing the effect of a control and an observation on the landmark trees. Each new particle in $S_{t}$ will differ from its generating particle in $S_{t-1}$ in two ways. First, each will posses a different pose estimate from (3.17), and second, the observed feature’s Gaussian will be updated as specified in (3.29)-(3.34). All other Gaussians will be equivalent to the generating particle. Thus, when copying the particle to $S_{t}$, only a single path from the root of the tree to the updated Gaussian needs to be duplicated. The length of this path is logarithmic in $N$, on average.
An example is shown in Figure $3.12$. Here we assume that $n_{t}=3$, that is, only the landmark Gaussian parameters $\mu_{3}^{[m]}, \Sigma_{3}^{[m]}$ are updated. Instead of duplicating the entire tree, a single path is duplicated, from the root to the third Gaussian. This path is an incomplete tree. The tree is completed by copying the missing pointers from the tree of the generating particle. Thus, branches that leave the modified path will point to the unmodified subtrees of the generating particle. Clearly, generating this modified tree takes time logarithmic in $N$. Moreover, accessing a Gaussian also takes time logarithmic in $N$, since the number of steps required to navigate to a leaf of the tree is equivalent to the length of the path. Thus, both generating and accessing a partial tree can be done in time $O(\log N)$. $M$ new particles are generated at every update step, so the resulting FastSLAM algorithm requires time $O(M \log N)$.

## 机器人代写|SLAM代写机器人导航代考|Garbage Collection

Organizing particles as binary trees naturally raises the question of garbage collection. Subtrees are constantly being shared and split between particles. When a subtree is no longer referenced as a part of any particle description,

the memory allocated to this subtree must be freed. Otherwise, the memory required by FastSLAM will grow without bound.

Whenever landmarks are shared between particles, the shared landmarks always form complete subtrees. In other words, if a particular node of the landmark tree is shared between multiple particles, all of the nodes descendants will also be shared. This greatly simplifies garbage collection in FastSLAM, because the landmark trees can be freed recursively.

Garbage collection in FastSLAM can be implemented using reference counts attached to each node in the landmark tree. Each counter counts the number of times the given node is pointed to by other nodes. A newly created node, for example, receives a reference count of 1 . When a new reference is made to a node, the reference count is incremented. When a link is removed, the reference count is decremented. If the reference count reaches zero, the reference counts of the node’s children are decreased, and the node’s memory is freed. This process is then applied recursively to all children of the node with a zero reference count. This process will require $O(M \log N)$ time on average. Furthermore, it is an optimal deallocation algorithm, in that all unneeded memory is freed immediately when it is no longer referenced.

## 机器人代写|SLAM代写机器人导航代考|Victoria Park

The FastSLAM algorithm was tested on a benchmark SLAM data set from the University of Sydney [37]. An instrumented vehicle, shown in Figure 3.13, equipped with a laser range finder was repeatedly driven through Victoria Park, in Sydney, Australia. Victoria Park is an ideal setting for testing SLAM algorithms because the park’s trees are distinctive features in the robot’s laser scans. Encoders measured the vehicle’s velocity and steering angle. Range and bearing measurements to nearby trees were extracted from the laser data using a local minima detector. The vehicle was driven around for approximately 30 minutes, covering a distance of over $4 \mathrm{~km}$. The vehicle is also equipped with GPS in order to capture ground truth data. Due to occlusion by foliage and buildings, ground truth data is only available for part of the overall traverse. While ground truth is available for the robot’s path, no ground truth data is available for the locations of the landmarks.

Since the robot is driving over uneven terrain, the measured controls are fairly noisy. Figure $3.14$ (a) shows the path of the robot obtained by integrating the estimated controls. After 30 minutes of driving, the estimated position of the robot is well over 100 meters away from its true position measured by GPS. The laser data, on the other hand, is a very accurate measure of range and bearing. However, not all objects in the robot’s field of view are trees, or even static objects. As a result, the feature detector produced relatively accurate observations of trees, but also generated frequent outliers.

Data association for this experiment was done using per-particle ML data association. Since the accuracy of the observations is high relative to the average density of landmarks, data association in the Victoria Park data set is a relatively straightforward problem. In a later experiment, more difficult data association problems will be simulated by adding extra control noise.
The output of FastSLAM is shown in Figure $3.14(\mathrm{~b})$ and (c). The GPS path is shown as a dashed line, and the output of FastSLAM is shown as a solid line. The RMS error of the resulting path is just over 4 meters over the $4 \mathrm{~km}$ traverse. This experiment was run with 100 particles.

## 机器人代写|SLAM代写机器人导航代考|Garbage Collection

FastSLAM 中的垃圾收集可以使用附加到地标树中每个节点的引用计数来实现。每个计数器计算给定节点被其他节点指向的次数。例如，一个新创建的节点接收到的引用计数为 1 。当对节点进行新引用时，引用计数会增加。删除链接时，引用计数会减少。如果引用计数达到零，则该节点的子节点的引用计数减少，并且该节点的内存被释放。然后将此过程递归地应用于具有零引用计数的节点的所有子节点。这个过程将需要这(米日志⁡ñ)平均时间。此外，它是一种最佳的释放算法，因为所有不需要的内存在不再被引用时都会立即释放。

## 机器人代写|SLAM代写机器人导航代考|Victoria Park

FastSLAM 算法在悉尼大学的基准 SLAM 数据集上进行了测试 [37]。如图 3.13 所示，配备激光测距仪的仪表车辆反复驶过澳大利亚悉尼的维多利亚公园。维多利亚公园是测试 SLAM 算法的理想场所，因为公园的树木是机器人激光扫描的独特特征。编码器测量车辆的速度和转向角。使用局部最小值检测器从激光数据中提取到附近树木的距离和方位测量值。车辆行驶了大约 30 分钟，行驶距离超过4 ķ米. 该车辆还配备了 GPS 以捕获地面实况数据。由于树叶和建筑物的遮挡，地面实况数据仅可用于整个遍历的一部分。虽然地面实况可用于机器人的路径，但没有地面实况数据可用于地标的位置。

FastSLAM的输出如图3.14( b)(c)。GPS路径显示为虚线，FastSLAM的输出显示为实线。所得路径的 RMS 误差仅超过 4 米。4 ķ米遍历。该实验使用 100 个粒子进行。

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