## 2 The Proposed Method
### Closed-Loop Detection and Back-End Optimization  
Though a scan-to-map matching strategy is used which can limit the accumulated error, with the continuous movement of the robot, the error will still continue to accumulate. Moreover, in order to reduce the error of closed-loop detection in the low-texture environment, the local map is applied to eliminate the accumulated error. The sparse pose adjustment (SPA) [30] method for back-end optimization is used. When a new frame is added to the local map, a 3D search window along the direction of \(x\), \(y\), and \(\theta\) around the estimated LiDAR pose is defined first. Then, searching the window step by step is carried out in the local map by calculating the total probability of all points in the frame. If the total probability exceeds the threshold, then it is regarded as a successful closed-loop detection. The relative pose between the LiDAR and the corresponding local  
To verify the performance of the proposed method, we conducted some quantitative and qualitative experiments. For the quantitative comparison, because of the limitation of ground truth data, we conducted the experiments in the constrained environment, where we could measure the length with an accurate handheld laser range finder. The comparison was made among the Cartographer (LiDAR-only) [40] and Cartographer with IMU fusion-based method, and the proposed fusion-based method, which is discussed in Section 3.2.  
There are some state-of-the-art 2D LiDAR-based SLAM methods, such as Hector SLAM [7], GMapping [10], KartoSLAM [41], CoreSLAM [42], LagoSLAM [43], and Cartographer [40], which have been proposed, and some papers on the performance evaluation are published [44; 45; 46; 47]. As mentioned in reference [44], among the former five methods (Cartographer was proposed later), KartoSLAM, GMapping, and HectorSLAM show better performance than the other two methods. Both KartoSLAM and LagoSLAM are graph-based optimizations, but the computation load of LagoSLAM is higher than that of KartoSLAM. In particular, KartoSLAM showed the best performance in the real world because the SPA solver is employed and it is a full SLAM approach. In this paper, we focus on the accuracy of the map and real-time application, so the methods including KartoSLAM and Cartographer are used for the evaluation. Since Cartographer is the most state-of-the-art method, we take it as the baseline method in this paper. The experiment results will be introduced in Section 3.3.  
The dataset used in Section 3.3 are from reference [48; 49; 50]. The performance of the algorithms can be evaluated by the accuracy of the map, which is usually defined by the distance between the obtained map and ground truth map or the accuracy of the robot trajectory, but it is usually difficult to generate the ground truth. Additionally, many open standard datasets were recorded with only one sensor. All these factors make the comparison a little hard. In this paper, we apply the proportion of occupied and free cells to the quantitative comparison, which has been proposed in reference [44] for quality evaluation. As shown in Figure 6, the perceptible feature of the picture of a map is the accuracy of the walls or edges. That is, for two maps which are similar in most of the areas, the more blurs in a map means the lower its quality. It is obviously straightforward and no ground truth is needed.  
Figure 5: The mobile robot used for the experiment and its architecture. (**a**) The mobile robot used for the experiments; (**b**) The architecture of the mobile robot.