# Sensor Model (Measurement Model)

## 1 Sensor Model简介

$bel(x_t) = \eta \, \underbrace{p(z_t \mid x_t)}_{\text{Sensor Model}} \int \underbrace{p(x_t \mid x_{t-1}, u_t)}_{\text{Motion Model}} bel(x_{t-1}) \, \mathrm{d} x_{t-1}$

$z_t = \{ z_t^1, z_t^2, \cdots, z_t^K\}$

$p(z_t \mid x_t, m) = \prod_{k=1}^K p(z_t^k \mid x_t, m)$

## 2 Sensor

We classify sensors using two important functional axes: proprioceptive/exteroceptive and passive/active.
Proprioceptive sensors measure values internal to the system (robot); for example, motor speed, wheel load, robot arm joint angles, battery voltage.
Exteroceptive sensors acquire information from the robot’s environment; for example, distance measurements, light intensity, sound amplitude.
Passive sensors measure ambient environmental energy entering the sensor. Examples of passive sensors include temperature probes, microphones, and CCD or CMOS cameras.
Active sensors emit energy into the environment, then measure the environmental reaction.

1 列出了一些常用的传感器。

### 2.1 Range Finders

• Sensors based on measuring the time-of-flight (TOF) of a pulse of emitted energy traveling to a reflecting object, then echoing back to a receiver.
• The phase-shift measurement (or phase-detection) ranging technique involves continuous wave transmission as opposed to the short pulsed outputs used in TOF systems.
• Sensors based on frequency-modulated (FM) radar. This technique is somewhat related to the (amplitude-modulated) phase-shift measurement technique.

## 3 Sensor Model

### 3.1 Beam Models of Range Finders

Beam Model对下面四类错误进行了建模：
(1) small measurement noise,
(2) errors due to unexpected objects,
(3) errors due to failures to detect objects,
(4) random unexplained noise.

$p(z_t^k \mid x_t, m) = \begin{pmatrix} z_{\text{hit}} \\ z_{\text{short}} \\ z_{\text{max}} \\ z_{\text{rand}} \end{pmatrix}^{\mathsf{T}} \cdot \begin{pmatrix} p_{\text{hit}}(z_t^k \mid x_t, m) \\ p_{\text{short}}(z_t^k \mid x_t, m) \\ p_{\text{max}}(z_t^k \mid x_t, m) \\ p_{\text{rand}}(z_t^k \mid x_t, m) \end{pmatrix}$

#### 3.1.1 Beam Model中四类错误相关的概率模型

Beam Model中四类错误对应的测量概率分布如图 3 所示。

##### 3.1.1.1 第一类错误(Small Measurement Noise)

$p_{\text{hit}}(z_t^k \mid x_t, m) = \begin{cases} \eta \frac{1}{\sqrt{2\pi \sigma_{\text{hit}}^2}} e^{- \frac{(z_t^{k} - z_t^{k*})^2}{2\sigma_{\text{hit}}^2}} & \text{if} \; 1 \le z_t^k \le z_{\text{max}} \\ 0 & \text{otherwise} \\ \end{cases}$

$\eta = \left( \int_0^{z_{\text{max}}} \frac{1}{\sqrt{2\pi \sigma_{\text{hit}}^2}} e^{- \frac{(z_t^{k} - z_t^{k*})^2}{2\sigma_{\text{hit}}^2}} \, \mathrm{d} z_t^{k} \right)^{-1}$

$$p_{\text{hit}}(z_t^k \mid x_t, m)$$ 的示意图如图 3 中图(a)所示。

##### 3.1.1.2 第二类错误(Unexpected Objects)

One way to deal with such objects is to treat them as part of the state vector and estimate their location; another, much simpler approach, is to treat them as sensor noise.

$p_{\text{short}}(z_t^k \mid x_t, m) = \begin{cases} \eta \lambda_{\text{short}} e^{-\lambda_{\text{short}} z_t^k} & \text{if} \; 1 \le z_t^k \le z_t^{k*} \\ 0 & \text{otherwise} \\ \end{cases}$

$\eta = \left( \int_0^{z_t^{k*}} \lambda_{\text{short}} e^{-\lambda_{\text{short}} z_t^k} \, \mathrm{d} z_t^{k} \right)^{-1} = \frac{1}{1 - e^{-\lambda_{\text{short}} z_t^{k*}}}$

$$p_{\text{short}}(z_t^k \mid x_t, m)$$ 的示意图如图 3 中图(b)所示。

##### 3.1.1.3 第三类错误(Failures to Detect Objects)

$p_{\text{max}}(z_t^k \mid x_t, m) = \begin{cases} 1 & \text{if} \; z_t^k = z_{\text{max}} \\ 0 & \text{otherwise} \\ \end{cases}$

$$p_{\text{max}}(z_t^k \mid x_t, m)$$ 的示意图如图 3 中图(c)所示。

##### 3.1.1.4 第四类错误(Random Unexplained Noise)

$p_{\text{rand}}(z_t^k \mid x_t, m) = \begin{cases} \frac{1}{z_{\text{max}}} & \text{if} \; 0 \le z_t^k \le z_{\text{max}} \\ 0 & \text{otherwise} \\ \end{cases}$

$$p_{\text{rand}}(z_t^k \mid x_t, m)$$ 的示意图如图 3 中图(d)所示。

#### 3.1.4 Beam Model的缺点

Beam Model和测距仪的几何结构和物理性质紧密相关，它有下面的缺点：
(1) not smooth for small obstacles and at edges.
(2) not very efficient.

http://ais.informatik.uni-freiburg.de/teaching/ss15/robotics/slides/07-sensor-models.pdf
Probabilistic Robotics, by Sebastian Thrun, 6.3.5 Limitations of the Beam Model

### 3.2 Likelihood Field Models for Range Finders

Beam Model测距仪的几何结构和物理性质紧密相关，这导致用它计算的概率不是平稳的(not smooth)，机器人位姿的细微变化可能导致计算的结果的差异非常大，Likelihood Field Models可以克服Beam Model的这个缺点。

### 3.3 Feature Based Sensor Models

A key advantage of this approach is the enormous reduction of computational complexity: While inference in the high-dimensional measurement space can be costly, inference in the low-dimensional feature space can be orders of magnitude more efficient.

In many robotics applications, features correspond to distinct objects in the physical world. In robotics, it is common to call those physical objects landmarks, to indicate that they are being used for robot navigation.
The most common model for processing landmarks assumes that the sensor can measure the range and the bearing of the landmark relative to the robot’s local coordinate frame.

Created: <2016-06-18 Sat 00:00>

Last updated: <2018-01-02 Tue 15:52>

Creator: Emacs 25.3.1 (Org mode 9.1.4)