# Curve-Localizability-SVM Active Localization Research for Mobile Robots in Outdoor Environments
## 2 Curve Localizability Estimation  
[PERSON] and other researchers [16] proposed a static localizability estimation method based on 2D grid map and 2D laser sensor model by introducing the concepts of Fisher information and CRB inequality of Statistics. Concerning about the states of mobile robots and the corresponding sensor model in outdoor environments with uneven ground, we extended the 2D indoor localizability index to 3D outdoor environments, and proposed a curve localizability index based on 3D OctoMap [23] and 3D Lidar sensor, which can better adapt to the observation state of robots. In order to make full use of the omni-direction observation characteristics of the 3D Lidar, the lidar was placed horizontally above the mobile robot, and ensured that the laser beam would not be affected by robot hardware mechanisms. On the meanwhile, for the  
Figure 1: Framework of Curve-Localizability-Based Active Localization Algorithm. purpose of simplifying the calculation, we assume that the mobile robot platform can be approximated as a particle, which means, the orientation of 3D Lidar remains constant if the platform is in the same position and different orientations. With this prerequisite, we can define the coordinate system of the lidar as follows. As shown in Figure 2a, \((x,y,z)\) is the position of the 3D Lidar, and \((\varphi,\psi,\theta)\) represents its rotation angles about the \(x\) axis, \(y\) axis, and \(z\) axis. Therefore the pose of the lidar can be expressed as \(p=(x,y,z,\varphi,\psi,\theta)\). Considering that we are using an omni-direction 3D lidar with a horizontal field of 360 degrees, the yaw angle \(\theta\) has no effect on the lidar observation data, so the pose model can be simplified as \((x,y,z,\varphi,\psi)\) when calculating the curve localizability index. We have made a strong assumption that the robot platform could be approximated as a particle before, but the effects on \(\varphi\) and \(\psi\) of different orientations of robots cannot be ignored in real environments. According to the kinematics and dynamics characteristics of the robot platform, the maximum angle changes caused by robot hardware structures during movement were defined as \(\Phi\) and \(\Psi\). Therefore, we can use the 3D Lidar observation model shown as Figure 2b to traverse the interval of \([\varphi-\Phi,\varphi+\Phi]\) and \([\psi-\Psi,\psi+\Psi]\) for index calculation. We define that a frame of laser observation data \(Z\) contains a set of discrete lidar points information. Suppose that the expected distance of point \(i\) in \(Z\) is \(r_{iE}\), where \(i=1,2,...,N\), the discrete robot localizability matrix can be defined as:  
\[\hat{J}(p)=\sum_{i=1}^{N}\frac{1}{\sigma^{2}}(\frac{\Delta r_{iE}}{\Delta p})^ {T}(\frac{\Delta r_{iE}}{\Delta p}) \tag{1}\]  
Substituting the pose \(p\) of the 3D Lidar into Formula (1), we can obtain the general form of the curve localizability matrix:  
\[\hat{J}_{curve}(p)=\sum_{i=1}^{N_{1}}\frac{1}{\sigma_{i}^{2}} \tag{2}\]  
Figure 2: 3D Omni-Direction Lidar Model. According to the simplifying method of 3D Lidar pose mentioned above, and given different weights to the lidar observation data from different rotation and elevation angle, the matrix can be simplified as:  
\[\hat{f}_{curve}(p)= \tag{3}\] \[\sum_{i=1}^{N_{1}}\frac{\omega_{i}}{\sigma_{i}^{2}} \begin{bmatrix} \frac{\Delta r_{i}^{2}}{\Delta x^{2}}&\frac{\Delta r_{i}^{2}}{\Delta x\Delta y}& \frac{\Delta r_{i}^{2}}{\Delta x\Delta z}&\frac{\Delta r_{i}^{2}}{\Delta x \Delta z\Delta y}&\frac{\Delta r_{i}^{2}}{\Delta x\Delta y}\\ \frac{\Delta r_{i}^{2}}{\Delta x\Delta y}&\frac{\Delta r_{i}^{2}}{\Delta y\Delta y }&\frac{\Delta r_{i}^{2}}{\Delta x\Delta y}&\frac{\Delta r_{i}^{2}}{\Delta y \Delta y}&\frac{\Delta r_{i}^{2}}{\Delta y\Delta y}\\ \end{bmatrix} \] \[\sum_{i=1}^{N_{1}}\frac{\omega_{i}}{\sigma_{i}^{2}}