# Topological Map-Based Autonomous Exploration in Large-Scale Scenes for Unmanned Vehicles
## 3 Methods
### Update of Movement Cost of Frontier Clusters  
As mentioned earlier, to ensure that the robot prioritizes exploring scenes outside prior information, the system needs to recognize frontier clusters that would guide the robot towards a global target and set a higher cost between them and the robot. To achieve this, we classify frontier clusters into three categories:  
The first class includes frontier clusters with agents and other frontier clusters that are adjacent to these frontier clusters. They will guide the robot towards global targets.  
The second class includes frontier clusters adjacent to the first class frontier clusters. These frontier clusters may guide the robot towards global targets. We utilize density-based spatial clustering of applications with noise (DBSCAN) algorithm to recognize these frontier clusters and rely on the following methods to identify whether they could guide the robot to global targets [28]:  
\[e^{-r_{s_{j},f^{k}}/r_{s_{j},i}}e^{\cos a^{k}-1}\cos\beta^{k}\geq\varepsilon, \tag{7}\] \[r_{s_{j},f^{k}}<r_{s_{j},R}\leq r_{s_{j},s_{i}}+\gamma,\]  
as shown in Figure 5a, \(\alpha^{k}\) is the angle between vector \(\overrightarrow{f^{k}_{ave}s_{j}}\) and \(\overrightarrow{s_{i}s_{j}}\); \(\beta^{k}\) is the angle between vector \(\overrightarrow{f^{k}_{ave}s_{j}}\) and \(\overrightarrow{Rf^{k}_{ave}}\); and \(r\) is the Euclidean distance between two points.  
However, as the robot explores, the relative position between frontier clusters and robot changes, and frontier clusters cannot always satisfy Equation (7), as shown in Figure 5b,c. Thus, to keep the consistency of determination for these frontier clusters during exploration, we employ the Dynamic Time Warping (DTW) algorithm to evaluate similarities between path \(P_{R,\mu^{i}}\) and all paths of \(P_{R,F}=\{P_{R,f^{1}},P_{R,f^{2}},\ldots,P_{R,f^{n}}\}\), as shown in Figure 5d [29]. If the similarity ranking of \(f^{k}\) satisfies the following condition, \(f^{k}\) is still believed to guide the robot towards global target \(s_{j}\):  
\[rank\Big{(}dtw\Big{(}P_{R,\mu^{i}},P_{R,f^{k}}\Big{)}\Big{)}\leq\varphi. \tag{8}\]  
The third class consists of the remaining frontier clusters. They will guide the robot to explore scenes outside prior information.  
For the frontier clusters \(f^{k^{*}}\) that guide the robot towards global targets, the movement cost from the robot to them is set as follows:  
\[c\Big{(}R,vp_{1}^{k^{*}}\Big{)}=C. \tag{9}\]  
The movement cost from robot to other frontier clusters \(f^{k}\) is computed as follows:  
\[c\Big{(}R,vp_{1}^{k}\Big{)}=\max\left\{\frac{\lambda\left(\frac{p}{P_{R,vp_{1 }^{k}}}\right)}{v_{\max}},\frac{\min\left(\left|\xi_{1}^{k}-\xi_{R}\right|,2 \pi-\left|\xi_{1}^{k}-\xi_{R}\right|\right)}{\omega_{\max}}\right\}+\lambda \cos^{-1}\frac{\left(vp_{1}^{k}-R\right)\cdot v_{R}}{\left\|vp_{1}^{k}-R\right\| \left\|v_{R}\right\|}\, \tag{10}\]  
which considers the path length, yaw change, and motion consistency, where \(p_{1}^{k}\) and \(\xi_{1}^{k}\) are coordinates, and the yaw angle of viewpoint \(vp_{1}^{k}\), \(\omega_{max}\) is the maximum angular change rate; \(\xi_{R}\) is the yaw angle of the robot; and \(v_{R}\) is the current velocity.  
As \(c(vp_{1}^{k^{*}},R)\) and \(c(vp_{1}^{k},R)\) do not impact the solution results of open-loop path planning, we set them to 0. The movement cost between all frontier clusters are calculated as follows:  
\[c\big{(}vp_{1}^{i},vp_{1}^{j}\big{)}=c\big{(}vp_{1}^{j},vp_{1}^{i}\big{)}=\max \left\{\frac{L\Big{(}p_{vp_{1}^{i},vp_{1}^{j}\big{)}}}{v_{\max}},\frac{\min \left(\left|\xi_{1}^{i}-\xi_{1}^{j}\right|\middle|2\pi-\left|\xi_{1}^{i}-\xi_{1 }^{i}\right|\right)}{\omega_{\max}}\right\}. \tag{11}\]