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

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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
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• 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|SLAM Posterior

The pose of the robot at time $t$ will be denoted $s_{t}$. For a robot operating in a planar environment, this pose consists of the robot’s $x-y$ position in the plane and its heading direction. All experimental results presented in this book were generated in planar environments, however the algorithms apply equally well to three-dimensional worlds. The complete trajectory of the robot, consisting of the robot’s pose at every time step, will be written as $s^{t}$.
$$s^{t}=\left{s_{1}, s_{2}, \ldots, s_{t}\right}$$
We shall further assume that the robot’s environment can be modeled as a set of $N$ immobile, point landmarks. Point landmarks are commonly used to represent the locations of features extracted from sensor data, such as geometric features in a laser scan or distinctive visual features in a camera image. The set of $N$ landmark locations will be written $\left{\theta_{1}, \ldots, \theta_{N}\right}$. For notational simplicity, the entire map will be written as $\Theta$.

As the robot moves through the environment, it collects relative information about its own motion. This information can be generated using odometers attached to the wheels of the robot, inertial navigation units, or simply by observing the control commands executed by the robot. Regardless of origin, any measurement of the robot’s motion will be referred to generically as a control. The control at time $t$ will be written $u_{t}$. The set of all controls executed by the robot will be written $u^{t}$.
$$u^{t}=\left{u_{1}, u_{2}, \ldots, u_{t}\right}$$
As the robot moves through its environment, it observes nearby landmarks. In the most common formulation of the planar SLAM problem, the robot observes both the range and bearing to nearby obstacles. The observation at time $t$ will be written $z_{t}$. The set of all observations collected by the robot will be written $z^{t}$.
$$z^{t}=\left{z_{1}, z_{2}, \ldots, z_{t}\right}$$

It is commonly assumed in the SLAM literature that sensor measurements can be decomposed into information about individual landmarks, such that each landmark observation can be incorporated independently from the other measurements. This is a realistic assumption in virtually all successful SLAM implementations, where landmark features are extracted one-by-one from raw sensor data. Thus, we will assume that each observation provides information about the location of exactly one landmark $\theta_{n}$ relative to the robot’s current pose $s_{t}$. The variable $n$ represents the identity of the landmark being observed. In practice, the identities of landmarks usually can not be observed, as many landmarks may look alike. The identity of the landmark corresponding to the observation $z_{t}$ will be written as $n_{t}$, where $n_{t} \in{1, \ldots, N}$. For example, $n_{8}=3$ means that at time $t=8$ the robot observed the third landmark. Landmark identities are commonly referred to as “data associations” or “correspondences.” The set of all data associations will be written $n^{t}$.
$$n^{t}=\left{n_{1}, n_{2}, \ldots, n_{t}\right}$$
Again for simplicity, we will assume that the robot receives exactly one measurement $z_{t}$ and executes exactly one control $u_{t}$ per time step. Multiple observations per time step can be processed sequentially, but this leads to a more cumbersome notation.

Using the notation defined above, the primary goal of SLAM is to recover the best estimate of the robot pose $s_{t}$ and the map $\Theta$, given the set of noisy observations $z^{t}$ and controls $u^{t}$. In probabilistic terms, this is expressed by the following posterior:
$$p\left(s_{t}, \Theta \mid z^{t}, u^{t}\right)$$
If the set of data associations $n^{t}$ is also given, the posterior can be rewritten as:
$$p\left(s_{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)$$

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|SLAM as a Markov Chain

The SLAM problem can be described best as a probabilistic Markov chain. A graphical depiction of this Markov chain is shown in Figure 2.2. The current pose of the robot $s_{t}$ can be written as a probabilistic function of the pose at the previous time step $s_{t-1}$ and the control $u_{t}$ executed by the robot. This function is referred to as the motion model because it describes how controls drive the motion of the robot. Additionally, the motion model describes how noise in the controls injects uncertainty into the robot’s pose estimate. The motion model is written as:
$$p\left(s_{t} \mid s_{t-1}, u_{t}\right)$$
Sensor observations gathered by the robot are also governed by a probabilistic function, commonly referred to as the measurement model. The observation $z_{t}$

is a function of the observed landmark $\theta_{n_{t}}$ and the pose of the robot $s_{t}$. The measurement model describes the physics and the error model of the robot’s sensor. The measurement model is written as:
$$p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right)$$
Using the motion model and the measurement model, the SLAM posterior at time $t$ can be computed recursively as function of the posterior at time $t-1$. This recursive update rule, known as the Bayes filter for SLAM, is the basis for the majority of online SLAM algorithms.

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Bayes Filter Derivation

The Bayes Filter can be derived from the SLAM posterior as follows. First, the posterior (2.6) is rewritten using Bayes Rule.
$$p\left(s_{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)=\eta p\left(z_{t} \mid s_{t}, \Theta, z^{t-1}, u^{t}, n^{t}\right) p\left(s_{t}, \Theta \mid z^{t-1}, u^{t}, n^{t}\right)$$
The denominator from Bayes rule is a normalizing constant and is written as $\eta$. Next, we exploit the fact that $z_{t}$ is solely a function of the pose of the robot $s_{t}$, the map $\Theta$, and the latest data association $n_{t}$, previously described as the measurement model. Hence the posterior becomes:
$$=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) p\left(s_{t}, \Theta \mid z^{t-1}, u^{t}, n^{t}\right)$$
Now we use the Theorem of Total Probability to condition the rightmost term of $(2.10)$ on the pose of the robot at time $t-1$.
$$=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \int p\left(s_{t}, \Theta \mid s_{t-1}, z^{t-1}, u^{t}, n^{t}\right) p\left(s_{t-1} \mid z^{t-1}, u^{t}, n^{t}\right) d s_{t-1}$$

The leftmost term inside the integral can be expanded using the definition of conditional probability.
\begin{aligned} &=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \ &\int p\left(s_{t} \mid \Theta, s_{t-1}, z^{t-1}, u^{t}, n^{t}\right) p\left(\Theta \mid s_{t-1}, z^{t-1}, u^{t}, n^{t}\right) p\left(s_{t-1} \mid z^{t-1}, u^{t}, n^{t}\right) d s_{t-1} \end{aligned}
The first term inside the integral can now be simplified by noting that $s_{t}$ is only a function of $s_{t-1}$ and $u_{t}$, previously described as the motion model.
\begin{aligned} &=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \ &\qquad \int p\left(s_{t} \mid s_{t-1}, u_{t}\right) p\left(\Theta \mid s_{t-1}, z^{t-1}, u^{t}, n^{t}\right) p\left(s_{t-1} \mid z^{t-1}, u^{t}, n^{t}\right) d s_{t-1} \end{aligned}
At this point, the two rightmost terms in the integral can be combined.
$$=\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \int p\left(s_{t} \mid s_{t-1}, u_{t}\right) p\left(s_{t-1}, \Theta \mid z^{t-1}, u^{t}, n^{t}\right) d s_{t-1}$$
Since the current pose $u_{t}$ and data association $n_{t}$ provide no new information about $s_{t-1}$ or $\Theta$ without the latest observation $z_{t}$, they can be dropped from the rightmost term of the integral. The result is a recursive formula for computing the SLAM posterior at time $t$ given the SLAM posterior at time $t-1$, the motion model $p\left(s_{t} \mid s_{t-1}, u_{t}\right)$, and the measurement model $p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right)$.
\begin{aligned} &p\left(s_{t}, \Theta \mid z^{t}, u^{t}, n^{t}\right)= \ &\eta p\left(z_{t} \mid s_{t}, \Theta, n_{t}\right) \int p\left(s_{t} \mid s_{t-1}, u_{t}\right) p\left(s_{t-1}, \Theta \mid z^{t-1}, u^{t-1}, n^{t-1}\right) d s_{t-1} \end{aligned}

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

s^{t}=\left{s_{1}, s_{2}, \ldots, s_{t}\right}s^{t}=\left{s_{1}, s_{2}, \ldots, s_{t}\right}

u^{t}=\left{u_{1}, u_{2}, \ldots, u_{t}\right}u^{t}=\left{u_{1}, u_{2}, \ldots, u_{t}\right}

z^{t}=\left{z_{1}, z_{2}, \ldots, z_{t}\right}z^{t}=\left{z_{1}, z_{2}, \ldots, z_{t}\right}

n^{t}=\left{n_{1}, n_{2}, \ldots, n_{t}\right}n^{t}=\left{n_{1}, n_{2}, \ldots, n_{t}\right}

p(s吨,θ∣和吨,在吨)

p(s吨,θ∣和吨,在吨,n吨)

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|SLAM as a Markov Chain

SLAM 问题可以最好地描述为概率马尔可夫链。该马尔可夫链的图形描述如图 2.2 所示。机器人当前位姿s吨可以写成前一个时间步的位姿的概率函数s吨−1和控制在吨由机器人执行。这个函数被称为运动模型，因为它描述了控制如何驱动机器人的运动。此外，运动模型描述了控制中的噪声如何将不确定性注入到机器人的姿态估计中。运动模型写为：
p(s吨∣s吨−1,在吨)

p(和吨∣s吨,θ,n吨)

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Bayes Filter Derivation

p(s吨,θ∣和吨,在吨,n吨)=这p(和吨∣s吨,θ,和吨−1,在吨,n吨)p(s吨,θ∣和吨−1,在吨,n吨)

=这p(和吨∣s吨,θ,n吨)p(s吨,θ∣和吨−1,在吨,n吨)

=这p(和吨∣s吨,θ,n吨)∫p(s吨,θ∣s吨−1,和吨−1,在吨,n吨)p(s吨−1∣和吨−1,在吨,n吨)ds吨−1

=这p(和吨∣s吨,θ,n吨) ∫p(s吨∣θ,s吨−1,和吨−1,在吨,n吨)p(θ∣s吨−1,和吨−1,在吨,n吨)p(s吨−1∣和吨−1,在吨,n吨)ds吨−1

=这p(和吨∣s吨,θ,n吨) ∫p(s吨∣s吨−1,在吨)p(θ∣s吨−1,和吨−1,在吨,n吨)p(s吨−1∣和吨−1,在吨,n吨)ds吨−1

=这p(和吨∣s吨,θ,n吨)∫p(s吨∣s吨−1,在吨)p(s吨−1,θ∣和吨−1,在吨,n吨)ds吨−1

p(s吨,θ∣和吨,在吨,n吨)= 这p(和吨∣s吨,θ,n吨)∫p(s吨∣s吨−1,在吨)p(s吨−1,θ∣和吨−1,在吨−1,n吨−1)ds吨−1

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