### robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Structure and Sparsity in SLAM

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## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Structure and Sparsity in SLAM

At any given time, the observations and controls accumulated by the robot constrain only a small subset of the state variables. This sparsity in the dependencies between the data and the state variables can be exploited to compute the SLAM posterior in a more efficient manner. For example, two landmarks separated by a large distance are often weakly correlated. Moreover, nearby pairs of distantly separated landmarks will have very similar correlations. A number of approximate EKF SLAM algorithms exploit these properties by breaking the complete map into a set of smaller submaps. Thus, the large EKF can be decomposed into a number of loosely coupled, smaller EKFs. This approach has resulted in a number of efficient, approximate EKF algorithms that require linear time [36], or even constant time $[1,5,7,49]$ to incorporate sensor observations (given known data association).

While spatially factoring the SLAM problem does lead to efficient EKFbased algorithms, the new algorithms face the same difficulties with data association as the original EKF algorithm. This book presents an alternative solution to the SLAM problem which exploits sparsity in the dependencies between state variables over time. In addition to enabling efficient computation of the SLAM posterior, this approach can maintain multiple data association hypotheses. The result is a SLAM algorithm that can be employed in large environments with significant data association ambiguity.

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

As shown in Section 1.2, correlations between elements of the map only arise through robot pose uncertainty. Thus, if the robot’s true path were known, the landmark positions could be estimated independently. Stated probabilistically, knowledge of the robot’s true path renders estimates of landmark positions to be conditionally independent.

Proof of this statement can be seen by drawing the SLAM problem as a Dynamic Bayes Network, as shown in Figure 1.5. The robot’s pose at time $t$ is denoted $s_{t}$. This pose is a probabilistic function of the previous pose of the robot $s_{t-1}$ and the control $u_{t}$ executed by the robot. The observation at time $t$, written $z_{t}$, is likewise determined by the pose $s_{t}$ and the landmark being observed $\theta_{n_{t}}$. In the scenario depicted in Figure $1.5$, the robot observes landmark 1 at $t=1$ and $t=3$, and observes landmark 2 at $t=2$. The gray region highlights the complete path of the robot $s_{1} \ldots s_{t}$. It is apparent from this network that this path “d-separates” [80] the nodes representing the two landmarks. In other words, if the true path of the robot is known, no information about the location of the first landmark can tell us anything about the location of the second landmark.

As a result of this relationship, the SLAM posterior given known data association (1.2) can be rewritten as the following product:
$$p\left(s^{t}, \theta \mid z^{t}, u^{t}, n^{t}\right)=\underbrace{p\left(s^{t} \mid z^{t}, u^{t}, n^{t}\right)}{\text {path posterior }} \underbrace{\prod{n=1}^{N} p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^{t}\right)}_{\text {landmark estimators }}$$
This factorization states that the full SLAM posterior can be decomposed into a product of $N+1$ recursive estimators: one estimator over robot paths, and $N$ independent estimators over landmark positions, each conditioned on the path estimate. This factorization was first presented by Murphy [66]. It is important to note that this factorization is exact, not approximate. It is a result of fundamental structure in the SLAM problem. A complete proof of this factorization will be given in Chapter 3 .

The factored posterior (1.4) can be approximated efficiently using a particle filter $[20,51,75]$, with each particle representing a sample path of the robot. Attached to each particle are $N$ independent landmark estimators (implemented as EKFs), one for each landmark in the map. Since the landmark filters estimate the positions of individual landmarks, each filter is low dimensional. In total there are $N, M$ Kalman filters. The resulting algorithm for updating this particle filter will be called FastSLAM. Readers familiar with the statistical literature should note that FastSLAM is an instance of the Rao-Blackwellized Particle Filter [23], by virtue of the fact that it combines a sampled representation with closed form calculations of certain marginals.
There are four steps to recursively updating the particle filter given a new control and observation, as shown in Table 1.1. The first step is to propose a new robot pose for each particle that is consistent with the previous pose and the new control. Next, the landmark filter in each particle that corresponds with the latest observation is updated using to the standard EKF update equations. Each particle is given an importance weight, and a new set of samples is drawn according to these weights. This importance resampling step corrects for the fact that the proposal distribution and the posterior distribution are not the same. This update procedure converges asymptotically to the true posterior distribution as the number of samples goes to infinity. In practice, FastSLAM generates a good reconstruction of the posterior with a relatively small number of particles (i.e. $M=100$ ).

Initially, factoring the SLAM posterior using the robot’s path may seem like a poor choice because the length of the path grows over time. Thus, one might expect the dimensionality of a filter estimating the posterior over robot paths to also grow over time. However, this is not the case for FastSLAM. As will be shown in Chapter 3, the landmark update equations and the importance weights only depend on the latest pose of the robot $s_{t}$, allowing us to silently forget the rest of the robot’s path. As a result, each FastSLAM particle only needs to maintain an estimate of the current pose of the robot. Thus the dimensionality of the particle filter stays fixed over time.

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

FastSLAM has two main advantages over the EKF. First, by factoring the estimation of the map into in separate landmark estimators conditioned on the robot path posterior, FastSLAM is able to compute the full SLAM posterior in an efficient manner. The motion update, the landmark updates, and the computation of the importance weights can all be accomplished in constant time per particle. The resampling step, if implemented naively, can be performed in linear time. However, this step can be implemented in logarithmic time by organizing each particle as a binary tree of landmark estimators, instead of an array. The $\log (N)$ FastSLAM algorithm can be used to build a map with over a million landmarks using a standard desktop computer.

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

$$p\left(s^{t}, \theta \mid z^{t}, u^{ t}, n^{t}\right)=\underbrace{p\left(s^{t} \mid z^{t}, u^{t}, n^{t}\right)} {\text {路径后 }} \underbrace{\prod {n=1}^{N} p\left(\theta_{n} \mid s^{t}, z^{t}, u^{t}, n^ {t}\right)}_{\text {landmark estimators }}$$

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