## 3 Topological Map Representation
### Topological Map Preservation  
The preservation of topological map is the process of integrating topological information and occupied information to form RGB image information. In order to describe the two kinds of map information conveniently, a conical space model is proposed, which is controlled by the parameter TM, the parameter OM, and integer one. The height of the cone is determined by the integer one, and the height is always one. The radius of the bottom surface of the cone is determined by the parameter OM, and the value of the polar coordinates of the bottom surface is determined by the parameter TM, which is inspired by the HSV color space theory [30]. An example process is shown in Figure 3 below. In the figure, the red line represents the conversion process of the topological information, the green line represents the conversion process of the occupied information, and the blue line represents the synthesis process between the two pieces of information. It should be noted that the relationship between the grid cell and topological vertex is \"many-to-one\", which means that a topological vertex corresponds to multiple grid elements. In other words, multiple grid cells form a region, and a region is represented by a topological vertex. In the process of preservation, the information of the topological vertex is extracted from the topological map firstly; then, the occupied information of the grid cell corresponding to the topological vertex is found in the metric map, and the two pieces of information are converted into the parameters TM and OM. In the conical space model, a cross-section can be obtained by these two parameters. The cross-section represents the result of the synthesis of two kinds of information. However, such information is still unusable, and needs to be converted again. The blue triangle cross-section is shown in Figure 3. Therefore, the information converter is proposed to convert the cross-section information into RGB information, which is often used in the field of vision. The RGB value is used to represent a grid cell, and then each grid cell is operated in the same way. Finally, the color map containing the topological information and occupancy information is saved. In the following, the TM and OM parameters are explained in detail:  
TM is measured by the angle, and the range of values is [0, 360]. It is calculated counterclockwise from red. This value is used to represent the topological neighborhood, which can represent at most 360 topological neighborhoods. In order to show a higher degree of discrimination, values can be taken at intervals, such as 20 color values for the identification of the topological region, and 18 kinds of topological region identifiers can be used. OM represents the degree of proximity to the spectral color, and the range of values is [0, 1]. This value is used to represent whether the obstacle is occupied at a point in the map. The larger the value, the greater the likelihood that there is an obstacle at that point; the smaller the value, the greater the likelihood that the point may be free. Then, in the beginning of topological map preservation, the occupied information and topological information need to be converted. The conversion formulas are as shown in Equation (6):  
\[\left\{ \begin{array}{l}TM=a+b\times P_{i}\\ OM=k\times p(m_{i})\end{array} \right. \tag{6}\]  
where \(a\) is the start value, \(b\) is the interval value, \(k\) is the certain coefficient, and \(a=0,b=20,k=1\) is taken in the paper. Figure 3: Topological map preservation process. After that, some intermediate variables should be computed from these two parameters, and the formulas are shown in Equation (7), where \(V=1\) is taken in the paper. Then, with the help of intermediate variables, the RGB information can be obtained by Equation (8):  
\[\left\{ \begin{array}{c}h=\left\lfloor\frac{TM}{60}\right\rfloor\quad(\text{ mod6})\\ f=\frac{TM}{60}-h\\ p=V\times(1-OM)\\ q=V\times(1-f\times OM)\\ g=V\times(1-(1-f)\times OM)\end{array} \right.