2.1. 3D LiDAR odometry

In 3D LiDAR SLAM, point cloud registration is performed to estimate pose changes, thereby achieving accurate localization. In the early days, the primary methods for point cloud registration were based on the Iterative Closest Point algorithm [Reference Besl and McKay17], Normal Iterative Closest Point algorithm [Reference Serafin and Grisetti18], and Normal Distributions Transform algorithm [Reference Takeuchi and Takashi Tsubouchi19]. In recent years, research has increasingly shifted to feature-based matching algorithms. Feature-based LiDAR odometry has been widely adopted in mainstream SLAM frameworks, significantly reducing matching time and thereby ensuring real-time system performance.

In 2014, Zhang et al. [Reference Zhang and Singh20] proposed the LOAM algorithm, which significantly advanced the development of 3D LiDAR SLAM. The algorithm innovatively identifies edge and planar points based on curvature-magnitude thresholds, rather than matching the entire LiDAR point cloud. Jiao et al. [Reference Jiao, Ye, Zhu and Liu21] proposed the M-LOAM algorithm for a multi-LiDAR sensor system to improve the robot’s environmental perception. In this approach, each LiDAR sensor processes the data independently before combining the features for further mapping. In addition, the experiments were conducted using multi-line LiDAR, which provides richer information compared to solid-state LiDAR.

Degeneracy remains a critical challenge in LiDAR SLAM, stemming from insufficient constraints on the robot’s state. This issue is particularly pronounced for solid-state LiDARs with a small horizontal FoV, such as when a robot navigates long, narrow indoor corridors.

Li et al. [Reference Li, Tian, Shen and Lu22] proposed a novel feature extraction method that leverages both geometry and intensity to address the degradation problem. To fully utilize the extracted features, dual-weighting functions are developed for planar and edge points, respectively, during the attitude optimization process. However, the method still exhibits susceptibility to failure when dealing with areas where features cannot be extracted based on intensity alone. Ma et al. [Reference Ma, Xu, Yuan, Zhi, Yu, Zhou and Xie23] proposed a map-building algorithm, MM-LINS, for addressing LiDAR degradation problems, but if there is minimal overlap in point clouds, the accuracy of map fusion will be significantly reduced. As concluded in [Reference Jiao, Zhu, Ye, Huang, Yun, Jiang, Wang and Liu24], actively selecting a subset of features greatly enhances the accuracy and efficiency of a LiDAR SLAM system and proves effective in addressing degeneracy. Burnett et al. [Reference Burnett, Schoellig and Barfoot25] demonstrated that incorporating an IMU into LiDAR odometry improves localization in geometrically degenerate environments. This also contributes to the advancement of semantic SLAM [Reference Chen, Milioto, Palazzolo, Giguere, Behley and Stachniss26–Reference Chen, Wang, Wang, Deng, Wang and Zhang28], where semantic information is extracted to enhance the understanding of the environment, helping to address challenges in degraded conditions. Liu et al. [Reference Liu and Zhang29] proposed a novel adaptive voxelization method to support LiDAR bundle adjustment (BA). These BA and optimization methods were further incorporated into a LOAM framework, termed BALM (Bundle Adjustment for LiDAR Mapping), to serve as a back-end for map refinement. However, the algorithm requires a good initial pose, and its performance is sensitive to any degeneracy during mapping.

2.2. Loop closure detection

The loop closure detection module is critical for reducing accumulated errors and improving mapping accuracy, especially in long-term or large-scale SLAM systems with multiple loops. By matching geometric similarities between LiDAR frames, it identifies loop closures via point cloud scan matching. Given the high dimensionality and structural sparsity of raw point clouds, feature extraction techniques compress dense data into sparse discriminative representations, enhancing detection efficiency and robustness.

Belongie et al. [Reference Belongie, Malik and Puzicha30] proposed the Shape Context algorithm, which encodes the shape of the point cloud around local feature points into an image, specifically targeting the characteristics of LiDAR data. However, this algorithm requires a large amount of point cloud data. Therefore, the point cloud to be matched must be larger than the reference point cloud; otherwise, mismatches may occur. A vehicle-mounted LiDAR system, which typically operates at a fixed frequency, frequently demonstrates an inability to supply sufficient data volume to satisfy the requirements of this algorithm. Behley et al. [Reference Behley and Stachniss31] proposed the SuMa method, which was the first to employ Surfel maps to efficiently generate projection data associations and achieve loop closure detection using only LiDAR data. He et al. [Reference He, Wang and Zhang32] have proposed M2DP, which projects raw point clouds onto several 2D planes and forms signatures based on descriptors from distinct planes. Kim et al. [Reference Kim and Kim33] introduced the scan context method, which is a non-histogram-based global descriptor that divides each point cloud frame into several regions, selects representative points, and stores them in a two-dimensional array. The advantage of the scan context algorithm lies in its low computational complexity and high robustness, making it suitable for handling real-world challenges such as varying viewpoints, noise, and partial occlusions, thereby enabling efficient loop closure detection in large-scale environments. However, during the encoding process, it records only the highest point in the height direction of the point cloud, leaving room for further improvement. Scan Context++ [Reference Kim, Choi and Kim34] is an improved version of scan context that introduces translation invariance in addition to the existing rotational invariance, thereby enhancing its speed. In 2020, Wang et al. [Reference Wang, Sun, Xu, Sarma, Yang and Kong35] proposed LiDAR Iris, which uses a globally invariant descriptor with rotational invariance. By extracting binary features using the LoG-Gabor filter, a binary-encoded Iris image is generated, making full use of most of the point cloud information. This global descriptor also possesses rotational invariance, avoiding brute-force search and thereby conserving computational resources. This method enables the establishment of globally consistent maps in large-scale environments based solely on LiDAR point clouds. Zhang et al. [Reference Zhang, Wang and Wu36] proposed a novel neural network architecture called VPD-Map, which utilizes visual point clouds to generate global 3D visual descriptors for fast loop closure detection and also provides prior information for feature maps in front-end pose estimation.

Figure 1.

Using LiDAR odometry, we register scans to build submaps and extract artificial landmark ROIs. A CostMap-multilateration system enhances loop closure detection accuracy while providing coarse inter-submap matches. Keyframes are selected from submaps based on their contribution to artificial landmark ROIs, establishing submap constraints via scan-to-map registration for fine matching. Back-end optimization then minimizes submap errors, generating a globally consistent map.

Figure 1 Long description

The diagram illustrates the process of state estimation and localization in unknown environments using Simultaneous Localization and Mapping (SLAM) with LiDAR odometry. It begins with a mobile robot equipped with wheel encoders and a 3D LiDAR sensor. The process involves three main steps: submap construction, inter-submap constraints, and multi-submap constraint graph. In submap construction, scans are registered to build submaps and extract artificial landmark regions of interest (ROIs). A CostMap-multilateration system enhances loop closure detection accuracy and provides coarse inter-submap matches. Keyframes are selected from submaps based on their contribution to artificial landmark ROIs, establishing submap constraints via scan-to-map registration for fine matching. Back-end optimization then minimizes submap errors, generating a globally consistent map. The diagram includes visual representations of the CostMap, Octree, and point cloud map, as well as the flow of data through the different stages of the process.

3.1. Extrinsic LiDAR calibration

Inspired by [Reference Censi, Franchi, Marchionni and Oriolo37], we extend the research to 3D LiDAR calibration in feature-rich areas. By obtaining the wheel diameter and track width from CAD drawings, we simplify the process, reducing the calibration to only the LiDAR’s external parameters $\ell = (x, y, z, rx, ry, rz)^T$ . Since the LiDAR is horizontally mounted, we focus on calibrating $x, y, rz$ . Let $SE(3)$ be the special Euclidean group, and $q = (q_x, q_y, q_z, q_{rx}, q_{ry}, q_{rz}) \in SE(3)$ represent the robot pose. As shown in Figure 2, the transformation between the wheel odometry and the LiDAR in the global coordinate system is:

(1) \begin{equation} s^k = \ell ^{-1} \cdot r^k \cdot \ell \end{equation}

where $s^k$ and $r^k$ are the pose changes of the laser frame and odometry from $t$ to $t+1$ , respectively.

Figure 2.

The robot pose is $q_k \in \text{SE}(3)$ with respect to the world frame; the sensor pose is $\ell \in \text{SE}(3)$ with respect to the robot frame; $r_k \in \text{SE}(3)$ is the robot displacement between poses; and $s_k \in \text{SE}(3)$ is the displacement seen by the sensor in its own reference frame.

Figure 2 Long description

The diagram illustrates the relationship between a robot pose, sensor pose, and their respective displacements in a world frame. The robot pose is denoted as q_k in SE(3) with respect to the world frame, while the sensor pose is represented as ℓ in SE(3) with respect to the robot frame. The robot displacement between poses is indicated as r_k in SE(3), and the displacement seen by the sensor in its own reference frame is shown as s_k in SE(3). The diagram includes axes labeled x, y, and z for both the robot frame and the sensor frame, demonstrating the spatial relationships and transformations between these elements.

As the equation is unsolvable during straight-line motion, we calibrate by rotating along an arc. We design the loss function in Eq. (2) and use gradient descent to solve for $\ell$ :

(2) \begin{equation} \ell ^{*} = \arg \min _{\ell } \sum _{i \in C} \hat {s}^i - (\ell ^{-1} \cdot \hat {r}^i \cdot \ell ) \end{equation}

Figure 3.

Artificial landmark point cloud obtained by light intensity thresholding.

Figure 3 Long description

A heat map visualizes light intensity distribution in a 3D space. The axes are labeled X, Y, and Z. The color scale ranges from blue, indicating low-light intensity, to red, indicating high-light intensity. The heat map shows a cylindrical object with dimensions 0.075 meters in diameter and 0.7 meters in height. The light intensity varies across the space, with higher intensity near the center and lower intensity towards the edges. The heat map provides a detailed view of how light intensity changes in different regions of the 3D space.

3.2. EKF-submap construction

The detection of artificial landmarks $z_t^r$ at time $t$ is implemented through the following components: first, the artificial landmarks are created by applying high-reflection film on the objects (in this paper, the reflector film is applied on the cylinders), which can be easily distinguished from external objects by the level of light intensity, as shown in Figure 3.

We denote the laser points that hit a reflector as $S = \{ S_i (x_i, y_i, z_i, f_i) \mid i \in \{ 1, 2, \ldots , n \} \}$ , where the artificial landmark is a cylindrical object with a known radius $R$ . Given that the mobile platform operates in real time on the same horizontal plane, the position of the artificial landmark along the Z-axis is disregarded, and only its planar coordinates are calculated within the search window $W = \{ (x_{\text{min}}, y_{\text{min}}), (x_{\text{max}}, y_{\text{max}}) \}$ . The search is conducted in step increments of $\delta _{\text{step}}$ , ultimately determining the optimal center position of the landmark $z_t^r(z_x, z_y)$ . The calculation is provided in Eq. (3).

(3) \begin{equation} \begin{cases} (z_x, z_y) = \arg \min \sum _{w \in W} Dis_w \\ Dis_w = \left(n*R^2 -\sum _{i=0}^{n}(w_x-x_i)^2+(w_y-y_i)^2\right) \end{cases} \end{equation}

The submap construction framework is derived from LiDAR odometry. We propose leveraging artificial landmarks to construct precise submaps. In LOAM, high-reflectivity objects are typically filtered out because LiDAR measurements exhibit reduced accuracy when encountering high-reflection surfaces [Reference Lin and Zhang14]. However, this study posits that even in degenerate environments, these high-intensity objects with marginally degraded precision can still provide coarse pose estimation, thereby potentially alleviating degradation scenarios. To address this, we propose integrating artificial landmark feature points into the pose optimization process. Specifically, we establish data associations between odometry predictions and retro-reflector pillars in the map. Let $S_k$ denote the point cloud cluster of an artificial landmark observed in the current frame, and let $L_m$ represent the corresponding retro-reflector pillar center in the map. For each point in $S_k$ , its distance to the associated pillar center $L_m$ should theoretically equal the pillar radius $R$ .

(4) \begin{equation} p_w = R_kp_l+t_k \end{equation}

where $(R_k, t_k)$ is the LiDAR pose when the last point in the current frame is sampled and needs to be determined by the pose optimization. The corresponding point-to-point residual is then computed according to Eq. (5) and incorporated into the pose optimization problem.

(5) \begin{equation} r_{p2p}(p_w) = ||p_w - L_m|| - R \end{equation}

After obtaining the pose estimation from LOAM, we construct a traditional EKF framework. The detailed algorithmic procedure is outlined in Figure 4, which consists of three main steps: prediction, observation, and correction. First, state prediction is performed using the left and right wheel speeds $v_l(t)$ and $v_r(t)$ at time $t$ . Subsequently, process noise $Q_r$ and $Q_l$ are introduced, and the Kalman gain $K_t$ is updated to process sensor observations collected from artificial landmarks and the LOAM system to optimize the state estimation.

Figure 4.

Workflow for EKF-based submap creation.

Figure 4 Long description

The diagram illustrates the workflow for EKF-based submap creation. It is divided into three main sections: Prediction Model, Observation Models, and Correction. The Prediction Model section includes Wheel Odometry and the estimation of the robot's state. The Observation Models section details the adjustment noise and updates through reflector observation and LOAM. The Correction section involves updating the current LiDAR states and the covariance matrix. Arrows indicate the flow of information between these sections, showing how the robot's state is estimated and updated through various models and observations.

First, the robot’s initial pose for the new submap in the global coordinate system is given by $M_i^G = M_{i-1}^G \oplus T^i_{i-1}$ , where $M_{i-1}^G$ is the previous submap’s pose in the global coordinate system, and $T^i_{i-1}$ is the alignment transformation between the new submap $M_i$ and the previous submap $M_{i-1}$ .

The new submap is then initialized with $M_0^G = (0, 0, 0, 0, 0, 0)^T$ . To balance the time required for submap updates and the accuracy of submaps, a new submap is initialized when either the number of observations processed in the submap reaches a certain threshold or a large error occurs between the observations and predictions.

3.3. Artificial landmark extraction in submap

This subsection details the principles for extracting artificial landmarks and selecting keyframes from high-reflection ROIs in submaps. Specific steps are outlined in Algorithm 1. Lines 1–8 collect LiDAR frame identifiers and point cloud data for the ROI regions, incorporating them into the corresponding ROI set using prior information from the EKF to generate data-associated observations. Lines 9-11 determine the artificial landmark location based on the point cloud data within the ROIs, in conjunction with Eq. (3).

Algorithm 1

Artificial Landmark Extraction in Submap

Algorithm 1 Long description

The image displays an algorithm for extracting artificial landmarks in submaps. It involves collecting LiDAR frame identifiers and point cloud data for regions of interest (ROIs) and incorporating them into the corresponding ROI set. The algorithm uses prior information from the Extended Kalman Filter (EKF) to generate data-associated observations. It also determines the artificial landmark location based on the point cloud data within the ROIs, in conjunction with a specific equation.

Figure 5.

(a) CostMap builds on reflectors. (b) Optimal data association in similar environments.

Figure 5 Long description

The diagram consists of two parts labeled (a) and (b). Part (a) shows a grid with artificial landmark centers marked in red, expanded areas around them, and a highest scoring triangle formed by connecting these landmarks. The maximum truncation distance is indicated by a line segment. Part (b) depicts submap landmarks and observation landmarks, with scores assigned to matches between them. The diagram illustrates the transformation between the wheel odometry and the LiDAR in the global coordinate system, focusing on calibrating the parameters x, y, and rz.

3.4. CostMap-multilateration for loop closure detection

Loop closure detection identifies similar regions within submaps. To accelerate the search process, we directly project artificial landmarks onto the horizontal plane. This detection is implemented via CostMap-Multilateration, as detailed in Algorithm 2.

On completion of the new submap $M_{i+1}$ , an octree map $O_{i+1}$ is built using reflector-configured triangles, serving as loop closure detection input. Algorithm 2 lines 7–10 identify similar triangles via octree traversal.

To find corresponding reflectors (data association) while avoiding false loop closures caused by overly similar triangles, a cost map is introduced, as shown in Figure 5. The maximum expansion distance $d_{\text{max}}$ is defined based on LiDAR ranging measurements and artificial landmark recognition errors. A high-resolution grid map with resolution $res$ is then constructed, where grids surrounding artificial landmarks are proportionally expanded according to their distance from the landmark center. The cost function is defined in Eq. (6). Figure 5(b) depicts six reflectors in a highly similar arrangement, a common occurrence in structured industrial environments. When the reflector set {O0, O1, O3} is observed, five similar triangles can be matched within the submap. Relying solely on an octree-based method would fail to achieve correct data association under such ambiguity. However, by employing the CostMap’s scoring mechanism, the triangle {L1, L2, L4} is identified as the correct match due to its highest score. This process effectively excludes other geometrically similar candidate triangles, such as {L0, L1, L5}, thereby resolving the data association problem in environments with repetitive geometries and ensuring robust performance. Concurrently, this approach suppresses false positives in loop closure detection, thereby preventing incorrect loop closures, which could lead to eventual mapping failure.

(6) \begin{equation} \text{cost} = 0.1 + 0.8 \cdot \frac {d}{d_{\text{max}}} \end{equation}

where $d$ represents the Euclidean distance from the center of a grid to the center of the artificial landmark. Grids that lie beyond the maximum truncation distance are assigned a value of $C_{\text{max}}$ .

By integrating the cost map in lines 10–25 of Algorithm 2, the data association set $A_{\text{min}{\underline{\ }}{\text{cost}}} = \left \{ \{r_i\}, \{R_i\} \mid 1 \leq i \leq n \right \}$ is obtained, yielding the optimal pose transformation $T_n^m$ from submap $M_n$ to $M_m$ – the result of coarse matching between the two submaps. Herein, $T_n^m$ can be obtained through multilateration based on the data association results.

Algorithm 2

Finding the Optimal Data Association

Algorithm 2 Long description

The image displays an algorithm designed to find the optimal data association between submaps. The algorithm involves initializing minimum cost and iterating through various combinations of landmarks. It calculates distances and checks for triangle formations within the submaps. If triangles are found, it computes the cost of associating data and updates the minimum cost accordingly. The process involves multilateral calculations and comparisons to determine the best data association.

3.5. Building constraints and optimization

Upon completing loop closure detection, coarse inter-submap matches are obtained. To establish higher-precision constraints, keyframes derived from submaps undergo fine matching via scan-to-map registration, ultimately constructing submap constraints. This workflow essentially highlights two core components:

1. Selection of Keyframes: The selection of keyframes is based on their contribution to the matched reflector pillar ROI. To avoid selecting an excessive number of adjacent keyframes, only the keyframe with the highest contribution weight among adjacent ones is extracted. If multiple keyframes exhibit equal contribution weights, the centrally located keyframe is selected. Following this criterion, a predefined number of keyframes are chosen to participate in submap matching and constraint relationship construction.

2. Scan-to-Map Matching: From the previous section, it has been established that the optimal transformation from submap $M_i$ to submap $M_j$ is denoted as $T_i^j$ . The set of keyframes selected from submap $M_i$ is represented as $K = \{ k_1, k_2, \ldots , k_n \}$ , with the pose of each keyframe in submap $M_i$ given by $T_{i_n}$ . Accordingly, the initial predicted pose of the keyframes within submap $M_j$ can be expressed as $T_{j_n} = T_i^j \times T_{i_n}$ . During the construction of the submap, the LOAM front-end module records information pertaining to key corner features and planar features within the map. By introducing the keyframes and their respective initial poses, the final matched pose information is obtained according to the method described in [Reference Lin and Zhang14], and is denoted by $\hat {T}_{j_n}$ . This approach ensures accurate alignment of keyframes, facilitating an optimal estimation of the relative transformation between submaps.

As the robot moves, optimization is performed by incorporating both odometry constraints and loop closure constraints. Specifically, an optimization problem is formulated by adding adjacent submaps and establishing constraints for loop closure detection. In this paper, the set $ M^G = \{M^G_0, M^G_1, \ldots , M^G_t\},{} M^G \in SE(3)$ represents the positional information of a sequence of submaps in the global context, while the loop closure constraints between submaps are denoted by pairs $(i, j) \in C$ . To quantify the discrepancy between the observed and predicted transformations, we define the error $ e_{ij}$ as shown in Eq. (7):

(7) \begin{equation} e_{ij}(p_i, p_j, O_{ij}) = O_{ij} \boxminus \hat {O}_{ij}(M^G_i, M^G_j) \end{equation}

The error term $e_{ij}$ represents the difference between the observed transformation $O_{ij} \in SE(3)$ , which corresponds to the loop closure constraint between nodes $i$ and $j$ , and the predicted transformation $\hat {O}_{ij} \in SE(3)$ , derived from odometry and the estimated positions of the submaps. The operator $\boxminus$ denotes a mapping from the local neighborhood of $SE(3)$ to its tangent space, enabling effective comparison between transformations. The predicted transformation is given by $\hat {O}_{ij} (M^{G}_i, M^{G}_j) = {M^{G}_i}^{-1} M^{G}_j = T_{jn} \hat {T}_{in}^{-1}$ , which represents the relative transformation between submaps $M^{G}_i$ and $M^{G}_j$ in the global reference frame.

This optimization problem can be formulated as a nonlinear least squares problem, as shown in Eq. (8). The system aims to optimize the robot’s pose by minimizing the error.

(8) \begin{equation} \left \{ \begin{aligned} P^* &= \arg \min _M \sum _{(i,\,j) \,\in\, C} R_{ij} \\ R_{ij} &= e_{ij}^T e_{ij} \end{aligned} \right . \end{equation}

where $R_{ij}$ signifies the negative log-likelihood function of a constraint between $M^G_i$ and $M^G_j$ .

Figure 6.

Logistics robot and sensor platform with LIVOX Lidar.

Figure 6 Long description

The image consists of two separate photos. The first photo shows a logistics robot equipped with various components. The robot features a Livox Mid 70 Lidar sensor mounted on top, a power supply unit, and wheel encoders for tracking movement. The second photo depicts a room with artificial landmarks placed on the walls and door frame, likely used for navigation and localization purposes. The combination of these images illustrates the setup and environment for a logistics robot utilizing Lidar technology and artificial landmarks to navigate and map unknown environments.

4.1. Implementation and evaluation setup

To verify the feasibility of the algorithm, we collected real-world environmental data using the AGV, as shown in Figure 6(a). The AGV is equipped with a differential drive chassis and utilizes a LIVOX Mid-70 3D LiDAR. The Mid-70 LiDAR has a circular FoV of 70.4 degrees both horizontally and vertically, with a significantly reduced near blind zone of 5 centimeters. During the experiments, the LiDAR operated at 10 Hz, capturing 10,000 points per frame. To provide an initial pose estimate for the algorithm, a KINCO servo motor drives the robot, with an encoder resolution of 65,536 pulses per revolution and a data acquisition frequency of 20 Hz. The data processing system is equipped with an 11th Gen Intel(R) Core(TM) i7-11370H processor, featuring an x86_64 architecture with 4 cores and 8 threads.

Figure 7.

Submap construction qualitative experiment. $\times$ represents mapping failure and $\checkmark$ represents mapping success.

Figure 7 Long description

The image consists of one real-world photo and three submap construction results displayed side by side. The real-world photo shows a hallway with doors and a floor. The three submap construction methods are labeled as LOAM-livox, BALM, and Ours. Each method shows a 3D map of the hallway with different levels of accuracy. The LOAM-livox method has a red circle indicating a mapping failure. The BALM method has a red circle indicating another mapping failure. The Ours method has a red checkmark indicating a successful mapping. The image demonstrates the effectiveness of different submap construction techniques in accurately mapping a hallway environment.

The experiments were conducted in an indoor setting simulating dynamic scenarios within a generic factory environment. Artificial landmarks were non-uniformly positioned at varying heights across multiple indoor regions. At selected positions, the landmarks were wall-mounted, as shown in Figure 6(b). Experimental data were collected from the onboard robotic platform. This dataset was subsequently used to systematically evaluate our proposed algorithm and perform comparisons against other methods.

The proposed method is deployed to compare with competing state-of-the-art methods. These include works on LOAM-livox, BALM, EKF, Scan Context, and M2DP (in loop closure detection experiments) and the proposed method. Since accurate ground truth cannot be obtained indoors, the end-to-end error is used for comparison in quantitative evaluation.

4.2. Qualitative evaluation

4.2.2. Focused study on the number and placement of artificial landmarks

To validate the impact of the number and arrangement of reflectors on our algorithm, we collected six sets of data in the environment shown in Figure 8.

Figure 8.

Real environment. (a) Start environment (b) End environment (c) Path and reflectors distribution map.

Figure 8 Long description

The image consists of three parts labeled (a), (b), and (c). Part (a) shows a room with a red start point, green reflectors labeled 1 to 4, and a blue arrow indicating the path direction. Part (b) depicts a hallway with a red start point, green reflectors labeled 1 to 5, a red end point, and a blue arrow showing the path direction. Part (c) is a map illustrating the layout of the environment with walls, a red start point, a red end point, green reflectors, and a blue arrow indicating the path direction.

For each data collection, we removed one reflector according to the serial numbers of the green icons in the figure, allowing us to observe the algorithm’s performance under different numbers and arrangements of reflectors. We compared our algorithm with two other methods, and the results are presented in Figure 9.

Figure 9.

Experiment on the number and placement of artificial landmarks.

Figure 9 Long description

The image presents a comparison of three different SLAM methods: LOAM-livox, BALM, and an unnamed method referred to as 'Ours'. The comparison is based on the number of reflectors used, ranging from 1 to 6. Each row corresponds to a different number of reflectors, and each column represents a different SLAM method. The images show 3D reconstructions of an environment, with the quality and accuracy of the reconstructions varying based on the number of reflectors and the method used. The 'Ours' method consistently shows more accurate and detailed reconstructions compared to LOAM-livox and BALM, especially as the number of reflectors decreases.

The results indicate that when LiDAR degeneracy occurs, both the BALM and LOAM-Livox algorithms exhibit significant performance degradation. Removing the first four reflectors (corresponding to the serial numbers) has little impact on the results, as reflectors No. 5 and No. 6 play a crucial role in compensating for information loss during vehicle rotation. However, when only one reflector was present, our algorithm inevitably produced mapping errors – specifically, two reflectors were visibly reconstructed in the map instead of one. This is because the LiDAR failed to continuously scan the column during degeneracy. The experiment clearly demonstrates that reflectors should be deployed as supplementary information in areas prone to LiDAR degeneracy.

Combining the experimental results and algorithmic principles, we propose several key considerations for deploying reflectors: in scenarios prone to LiDAR degeneracy, at least two reflectors should be deployed to provide necessary geometric constraints; to avoid point cloud overlap or feature confusion, the horizontal spacing between adjacent reflectors should be no less than 1 m; to support reliable loop closure detection, multiple reflectors should be arranged in an asymmetric configuration with randomized orientations to eliminate ambiguity caused by symmetry; and in spatially constrained narrow areas, reflectors should be placed outside the sensor’s minimum blind zone (typically at a distance greater than 0.5 m) – a requirement that can be effectively met by mounting them on vertical surfaces such as walls, thereby increasing their spatial separation and enhancing observability and system robustness.

4.2.3. Circular indoor scene mapping experiments

To qualitatively analyze the loop closure effects, we selected a circular area indoors and manually placed artificial landmarks within it. The mapping results of the three algorithms are depicted in Figure 10.

Figure 10.

Mapping performance comparison diagram: LOAM-LIVOX shows increased mapping errors in circular environments, resulting in mapping failure.

Figure 10 Long description

The image contains three diagrams side by side comparing the mapping performance of different LiDAR methods in indoor environments. The first diagram on the left, labeled LOAM-LIVOX, shows increased mapping errors in circular environments, resulting in mapping failure. The second diagram in the middle, labeled BALM, also indicates mapping failure with highlighted areas of error. The third diagram on the right, labeled Ours, demonstrates successful mapping with minimal errors, as indicated by the red circles and arrows. The diagrams include visual representations of the mapping process and highlight the effectiveness of the new method in handling narrow spaces and thin walls, as well as sharp turns, which are challenging for solid-state LiDARs with constrained horizontal fields of view.

Evidently, LOAM-livox exhibits drift at the third turn due to insufficient environmental features in degenerate scenarios, ultimately causing mapping failure. Similarly, BALM demonstrates significant mapping errors at the second turn and fails to complete the mapping process. In contrast, our proposed algorithm correctly identifies loop closure upon returning to the mapping origin and successfully constructs a globally consistent map.

4.3. Quantitative evaluation

4.3.1. Evaluation on loop closure detection

To evaluate the performance of our algorithm in 3D SLAM, we compared it with two state-of-the-art methods: Scan Context and M2DP. An ablation study was conducted by removing the CostMap component from the CostMap-Multilateration framework (denoted as “w/o CostMap” in the table) to validate its effectiveness. To assess the stability of our algorithm’s performance, we collected data in six different environments. During the experiments, data acquired from the same environment were used to form positive loop-closure pairs, whereas data from different environments formed negative pairs (i.e., non-loop-closure pairs).

The performance of loop closure detection is quantitatively evaluated using three standard metrics: accuracy $A_c$ , precision $P_r$ , and recall $R_e$ , defined as follows: $A_c = \frac {TP + TN}{TP + TN + FP + FN}$ , $P_r = \frac {TP}{TP + FP}$ , and $R_e = \frac {TP}{TP + FN}$ . Here, $TP$ , $FP$ , $FN$ , and $TN$ denote true positive, false positive, false negative, and true negative, respectively. The evaluation results of the loop closure detection algorithms on these data are presented in Table I.

Table I.

Loop closure detection results comparison in multiple indoor scenarios.

Table I Long description

The table presents a comparison of loop closure detection results across six indoor scenes using four different methods: M2DP, Scan Context, w/o CostMap, and Our. The table is divided into three sections, each representing two scenes. Each section lists the precision, recall, and accuracy for each method. The methods are evaluated using three metrics: precision, recall, and accuracy. The 'Our' method consistently shows the highest performance across all scenes, with precision, recall, and accuracy values often reaching one. The 'M2DP' method shows varying performance, with high accuracy in Scene 2 but lower values in other scenes. The 'Scan Context' method generally performs worse than the others, with notably low precision and recall in Scene 1. The 'w/o CostMap' method shows strong performance, particularly in Scenes 1, 2, and 3. The table highlights the best results in bold and the worst results in the same group of tests underlined.

Best results are boldened, and the same group of tests’ worst results are underlined.

As shown by the experimental results, both Scan Context and M2DP generally perform moderately across most scenarios. This is because they generate feature descriptors from the raw sensor data. However, the limited field of view and sparse point cloud from a single-frame solid-state LiDAR (Livox Mid-70) often provide insufficient information. Consequently, these methods struggle to correctly identify loop closures due to information scarcity. Although they achieve relatively high recall in a few specific scenarios, their precision and accuracy in indoor environments with similar layouts are generally lower than those of our method. The algorithm with CostMap removed can only achieve loop closure detection by identifying a unique similar triangle. Nevertheless, this approach has two inherent limitations. First, when multiple similar triangles are observed in a single scan relative to the environment, the inability to determine a unique correspondence results in detection failure; second, the accidental presence of a similar triangle in two entirely unrelated environments is more likely to trigger a false loop closure. These findings demonstrate that our method establishes an evaluation framework that integrates information from all reflective markers, enabling more accurate loop closure identification.

To mitigate timing fluctuations induced by OS scheduling and CPU dynamic frequency scaling, we conducted 20 identical trials measuring loop closure detection and fine matching durations (Figure 11). Experimental results demonstrate a mean total constraint construction time of 38.3584 ms, with loop closure detection consuming 7.20 ms and fine matching accounting for 31.1512 ms of the total processing time.

Figure 11.

Overall elapsed time for loopback constraint construction.

Figure 11 Long description

The bar graph compares the runtime in milliseconds for fine match time and loop closure detection time across different numbers of loop detections. The x-axis represents the number of loop detections, ranging from 0 to 20. The y-axis represents the runtime in milliseconds, ranging from 0 to 100. The graph features two sets of bars for each number of loop detections: one for fine match time in light green and one for loop closure detection time in dark gray. The fine match time consistently shows higher values compared to the loop closure detection time. Notable trends include peaks in fine match time at 5, 10, and 15 loop detections, reaching approximately 80 milliseconds. The loop closure detection time remains relatively low throughout, with values generally below 20 milliseconds. The color scheme differentiates between fine match time and loop closure detection time, with a legend indicating the corresponding colors. All values are approximated.

Figure 12.

Long circular corridor environmental. (a) Clockwise path (b) Counterclockwise path.

Figure 12 Long description

The diagram illustrates a long circular corridor with two different path directions. In part (a), the path direction is clockwise, starting and ending at marked points. The corridor measures 52.8 meters in length, 19.9 meters in width, and 2.2 meters in height. In part (b), the path direction is counterclockwise, also starting and ending at marked points. The diagram includes labels for start and end points and path directions.

Figure 13.

Circular corridor indoor experiments(Clockwise). (a) Results of LOAM-livox map building. (b) Results of BALM map building. (c) Results of EKF map building. (d) Results of proposed map building. Only the proposed methodology completes the building of the map.

Figure 13 Long description

The image contains four separate graphs labeled (a) through (d), each representing different map-building methodologies in circular corridor indoor experiments. Graph (a) shows the results of LOAM-livox map building, marked with a failure label and a red line indicating a distance of 0.3835 meters between the start and end points. Graph (b) displays the results of BALM map building, also labeled as a failure. Graph (c) illustrates the results of EKF map building, again labeled as a failure. Graph (d) presents the results of the proposed map-building methodology, labeled as a success with a red line indicating a distance of 0.0345 meters between the start and end points. The proposed methodology is the only one that successfully completes the map building.

4.3.2. Evaluation on end-to-end distance

To quantify algorithmic accuracy improvements, we conducted dual experiments in a 59-meter-long circular indoor corridor (width: 19.9 m), with its primary segment spanning 52.8 m × 2.2 m (Figure 12). Given the susceptibility of corridor corners to degeneracy, reflective columns were strategically deployed at all four corners of the ring-shaped region for algorithm validation. After designating an origin point, the AGV traversed clockwise and counterclockwise paths returning to the origin. End-to-end positioning error was quantified by comparing LiDAR poses between initial and final frames.

To compare the mapping performance, we conducted experiments on the same dataset. The results of the two experiments are shown in Figures 13 and 14, respectively, and their trajectory plots are shown in Figure 15. Since our AGV operates on a plane, we focused on the planar trajectory.

Figure 14.

Circular corridor indoor experiments(Counterclockwise). (a) Results of LOAM-livox map building. (b) Results of BALM map building. (c) Results of EKF map building. (d) Results of proposed map building. Only the proposed methodology completes the building of the map.

Figure 14 Long description

The image displays four maps representing indoor corridor experiments conducted in a counterclockwise direction. Each map illustrates the results of different methodologies for map building. The first map, labeled LOAM-livox, shows a distorted structure with a failure indication and a distance measurement of 17.2454 meters between the start and end points. The second map, labeled BALM, depicts a vertical structure with a failure indication. The third map, labeled EKF, also shows a distorted structure with a failure indication. The fourth map, labeled Ours, presents a successful map building with a clear rectangular corridor structure and a distance measurement of 0.1898 meters between the start and end points.

Figure 15.

Trajectory of circular corridor indoor experiments. (a) Clockwise mapping trajectory. (b) Counterclockwise mapping trajectory.

Figure 15 Long description

Two line graphs depict the trajectories of circular corridor indoor experiments. The left graph shows the clockwise mapping trajectory, while the right graph shows the counterclockwise mapping trajectory. Each graph has four lines representing different methods: ALLOAM in red, EKF in green, BLAM in blue, and LOAM-livox in cyan. The x-axis represents the x-coordinate in meters, ranging from negative to positive values, and the y-axis represents the y-coordinate in meters, also ranging from negative to positive values. The trajectories illustrate the paths taken during the experiments, with each method showing distinct patterns and deviations.

From the above experiments, it can be seen that the LOAM-Livox algorithm shows significant deviations in both the rotational and translational directions at the corners of narrow corridors due to geometric degeneracy. In the end-to-end distance tests, the differences between the start and end points were 34.2834 m and 17.2454 m, respectively. By contrast, the BALM algorithm produces obvious mapping errors during the initial mapping stage of the long corridor when mapping clockwise, ultimately failing to construct a valid map. When mapping counterclockwise, significant degeneration occurs at corridor corners, resulting in an inability to build a map. The EKF algorithm can achieve mapping in areas with artificial landmarks; however, once it moves away from such areas, significant mapping errors occur, making it unable to complete the mapping process. While our proposed algorithm does have certain LiDAR frame data errors at corners, it generally achieves normal mapping in both experiments. After identifying loop-closure submaps, the relationship between them was corrected, with the final end-to-end errors measured as 0.0345m and 0.1898m, respectively.