Simultaneous Localization and Mapping (SLAM) techniques have become a fundamental research area in autonomous robotics. They enable robots and vehicles to move autonomously in unknown environments, while also building a detailed map of their surroundings in real time. Its field of application spans beyond robotics. It is common to employ SLAM techniques in areas ranging from autonomous driving [1–3] and augmented reality systems [4] to remote hostile environment exploration [5]. For these purposes, the system must be equipped with appropriate sensors. The most common choices are GNSS (Global Navigation Satellite System), inertial navigation systems, cameras, RADAR (Radio Detection and Ranging), and LiDAR (Light Detection and Ranging). Sensors are selected according to the environmental characteristics to be mapped. In GNSS-denied environments where the satellite signal is lost, LiDAR sensors are a suitable alternative. They generate a two-dimensional representation of the environment from which to accurately estimate the position and orientation of the robotic element in real time. Inertial sensors, which accumulate drift, and cameras, conditioned by visibility, are typically less efficient options [6].

The ability of these sensors to provide accurate and detailed data in real time has led to their use in numerous robotic platforms [6–8]. The rangefinder that equips the LiDAR sensor emits infrared laser signals that are reflected by the surroundings. Its receiver picks up these reflected waves, and its processor estimates the distance of the target from the sensor [3] by measuring the time-of-flight (TOF) of the signal. Additionally, an encoder provides angle measurement for each observed target. The sensor delivers a raw modulus-phase polar representation of the environment profile, which is processed into a map. LiDAR sensors capture information at high resolution. This contributes significantly to the quality of the generated maps, which is essential in applications that demand a detailed representation of the explored spaces. Recent research, such as [9], Zou et al. [10], and Kim et al. [11], has addressed issues related to the effective integration of LiDAR sensors into SLAM algorithms in order to improve the robustness and accuracy of positioning.

Despite its advantages, processing and storing large volumes of LiDAR data present challenges, especially in terms of computational resources. There are two main approaches regarding the transformation of raw data into usable maps: scan matching or feature extraction [12]. The first option revolves around the comparison of point clouds to complete the profile of the environment, with ICP (Iterative Closest Point) the most commonly used algorithm. These methods are typically overwhelmed by the computational strain as point clouds become larger [13]. In contrast, feature extraction methods extract the geometric features defining the environment, thus reducing the volume of data to be processed during subsequent stages. Such approaches significantly mitigate processing burdens within structured environments, where feature variability is reduced to straight segments, and their intersections [14].

Feature extraction techniques eventually yield a limited number of representative points on the map, which then serve as references for positioning. These points are necessarily obtained with associated estimation uncertainty, which is typically represented through covariance matrices. These covariance matrices are subsequently propagated during the estimation of successive robot poses along a trajectory and to new references generated in successive remapping steps. The estimation of covariance matrices associated with geometric features derived from LiDAR measurement standard deviations has been addressed by several authors [15–18], and their propagation to segment intersections has been addressed in [17–19]. Methodologies employed significantly condition the final results.

In this article, we propose a framework for estimating lines and corners as well as their uncertainties, which we have called Weighted Conformal Total Least Squares (WCTLS), inspired by the formulation in [20], but extended to explicitly incorporate measurement weighting and geometric normalization. The proposed approach removes scale ambiguity and improves the conditioning of the estimation problem. A comparative evaluation against the classic formulations of Unweighted Total Least Squares (UTLS) and Weighted Total Least Squares (WTLS) demonstrates that WCTLS yields lower corner uncertainty. These results show that the choice of geometric parameterization impacts the propagation of uncertainty and the reliability of landmarks in LiDAR-based SLAM.

Feature extraction methods perform two separate steps. In a first iteration, raw data is segmented into separate geometric features. Afterwards, the parameters that define each of them are characterized.

This section describes the methodology adopted for extracting straight-line segments and corner features from 2D LiDAR data, together with their associated covariance matrices. The proposed framework operates on the implicit representation of lines, allowing for a compact geometric description and direct propagation of covariance matrices. Starting from raw LiDAR points, straight segments are first estimated using Total Least Squares-based formulations. These segments are represented in implicit form and subsequently intersected to obtain corner features. The covariance matrix associated with line parameters is propagated to the corner estimates, allowing the reliability of the extracted landmarks to be quantified.

Within this framework, three TLS-based estimation strategies are considered. The first corresponds to the classical formulation by Siadat et al. [20]. The second extends this approach by incorporating range-dependent weighting, as justified in [17]. The third method is the Weighted Conformal Total Least Squares formulation proposed in this work, which introduces a geometric constraint to improve the conditioning of the estimation problem.

For clarity, the overall processing pipeline of the proposed method is summarized in Algorithm 1:

Analysis and comparison of the Total Least Squares methods discussed in the previous section was performed on data captured by SLAMTEC’s RPLIDAR S1 sensor. It is a two-dimensional sensor with a range of 40 m and a Field-of-View of 360°. According to the manufacturer, its depth accuracy is 5 cm while its angular resolution is 0.391° [37]. This information has been used in the uncertainty as input for the uncertainty propagation algorithms.

The sensor was operated in a stationary configuration and placed on existing furniture within the environment. In environment A, the sensor was positioned at an approximate height of 1 m above the floor, while in environment B, the height was approximately 1.5 m. In both cases, the sensor remained fixed during the acquisition process. For each environment, 100 consecutive LiDAR scans were acquired from the same position using the standard configuration of the RPLIDAR S1 sensor, operating at a scan frequency of 10 Hz.

The experimental evaluation of the UTLS, WTLS, and WCTLS methods has been conducted in two structured environments, which we have identified as environment A and environment B, as shown in Fig. 1. Environment A, characterized by small dimensions, is approximately 4.5 m long and 2.5 m wide. On the contrary, environment B presents larger dimensions, about 18 m long by 9 m wide. Environment A corresponds to a structured indoor domestic space (room-like scenario) where the dominant geometric elements are walls, columns, and a wardrobe. These elements generate several planar surfaces and orthogonal intersections that produce the line segments and corner features used in the experiments. Environment B corresponds to a laboratory mainly composed of cabinets, columns, and partition panels. This configuration creates multiple parallel walls and partial occlusions typical of indoor workspaces, leading to the set of detectable line segments and corner intersections.

The segmentation of raw data has been performed using the Iterative End Point Fit methodology. A threshold distance of 40 mm has been set for the IEPF algorithm. This value corresponds to an intermediate operating point that yields a sufficient number of geometric features to represent the environment while avoiding over-segmentation of the LiDAR scan. A sensitivity analysis over multiple LiDAR scans showed that increasing the threshold reduces the number of detected corners as expected, while the propagated corner uncertainty and the relative performance of UTLS, WTLS, and WCTLS remain consistent across the tested range.

To facilitate the geometric interpretation of the results, a representative scan from each environment is first analyzed in detail. This representative-scan analysis is used to illustrate the extracted line segments, corner locations, and associated propagated covariance ellipses. Subsequently, the statistical consistency of the three methods is assessed over the full set of 100 repeated scans acquired in each environment.

The scans yielded by the sensor show a section of both environments in polar coordinates. The origin of coordinates is defined by the sensor position. Environment A is defined by 935 points, which are condensed to 21 segments and 18 corners. Environment B is defined by 36 straight segments and 23 corners, extracted from the 887 points provided by the laser rangefinder. In Fig. 2, both locations are shown with the corners numbered clockwise, since this is the rotational direction of the laser rangefinder.

Tables 1 and 2 show the standard deviations of the estimated corner coordinates obtained by UTLS, WTLS, and the proposed WCTLS formulation for a representative scan in environments A and B, respectively. For both environments, UTLS and WTLS yield nearly identical standard deviation values for all corners. Differences between both methods remain below a few percent and do not alter the overall magnitude of the uncertainty, confirming that the introduction of measurement weighting alone does not significantly improve the reliability of the extracted corner features.

In contrast, the proposed WCTLS formulation reduces the standard deviations of the corner coordinates. In environment A (Table 1), the standard deviations σxc and σyc obtained with WCTLS are typically reduced by factors ranging from approximately 3 to 6 with respect to UTLS and WTLS. For instance, corner 1 exhibits a reduction from (42.02,35.08)mm with UTLS to (13.43,10.43)mm with WCTLS, while corner 18 decreases from (156.12,174.29)mm to (59.12,66.36)mm. Similar reductions are observed across the remaining corners, including those associated with comparatively large coordinate standard deviations.

A comparable trend is observed in environment B (Table 2), which presents a more challenging geometric configuration with larger distances and several partial occlusions. While UTLS and WTLS again yield virtually identical results, WCTLS achieves a systematic reduction in the standard deviations of the corner coordinates. Typical reductions range between factors of 2 and 4, depending on the local corner geometry. These results indicate that the improvement introduced by WCTLS is consistent across different indoor environments and not restricted to a particular spatial configuration.

To provide an intuitive visualization of the uncertainty associated with the corner estimates, Fig. 3 shows the covariance ellipses obtained for a scan in both experimental environments. The ellipses represent the propagated covariance of the estimated corner coordinates and are displayed with an expansion factor of 10 to improve visual interpretation. As observed in the figure, the uncertainty ellipses obtained with UTLS and WTLS are nearly identical in most cases, confirming that both formulations lead to similar uncertainty estimates when applied to the considered datasets. A small discrepancy can be observed in corner 1 of Environment A, where the two methods produce slightly different ellipse sizes. However, these differences remain minor compared to the overall uncertainty scale. In contrast, the proposed WCTLS formulation consistently produces smaller uncertainty ellipses for all detected corners in both environments. This reduction indicates that the proposed normalization strategy improves the conditioning of the estimation problem and leads to more stable covariance propagation in the corner estimation process. Consequently, although all methods provide equivalent geometric corner locations, WCTLS achieves a consistent reduction in the estimated positional uncertainty of the extracted features. This visual result is fully consistent with the numerical values reported in Tables 1 and 2.

Although the representative-scan analysis is useful to visualize the geometric behavior of the three methods, it does not by itself characterize scan-to-scan variability. For this reason, an additional repeatability analysis was conducted using 100 consecutive scans acquired from a fixed sensor position in each environment. For each scan, the median area of the uncertainty ellipses corresponding to the detected corners was computed. This per-scan median ellipse area provides a compact scalar measure of the typical propagated corner uncertainty while ensuring that each scan contributes equally to the statistical analysis, independently of the number of corners detected in that scan. Fig. 4 shows the statistical distribution of these per-scan median ellipse areas for the three estimation methods. Each box represents the interquartile range, corresponding to the interval between the first and third quartiles and therefore containing the central 50% of the scans. The horizontal line inside each box denotes the median value, while the whiskers extend to 1.5 times the interquartile range. Outliers, corresponding to scans with unusually large uncertainty values, are represented by circles. A logarithmic scale is used in the vertical axis to facilitate the visualization of relative differences in uncertainty magnitude.

In environment A, the box plots in Fig. 4a show that UTLS and WTLS yield very similar uncertainty levels across the repeated scans. The median ellipse area per scan is approximately 4.5×103mm2 for both methods, with values of 4498.5±330.3mm2 and 4594.7±355.0mm2, respectively. In contrast, the proposed WCTLS formulation produces a substantial reduction in the typical uncertainty, yielding a median ellipse area per scan of 626.2±178.7mm2. This corresponds to an approximate sevenfold reduction in the typical per-scan median ellipse area with respect to the classical TLS formulations. The box plot also reveals a noticeably smaller dispersion of the uncertainty values for WCTLS, indicating a more stable behavior across repeated scans.

A similar trend is observed in environment B, as shown in Fig. 4b. In this case, UTLS and WTLS again exhibit nearly identical behavior, with median ellipse areas per scan of 8125.3±914.1mm2 and 8123.6±911.1mm2, respectively. The WCTLS method consistently produces lower uncertainty values, with a median ellipse area per scan of 3468.5±640.9mm2. Although the improvement is less pronounced than in environment A, WCTLS still reduces the typical uncertainty by a factor of approximately 2.3. The box plot distributions further show that this reduction is systematic across the repeated scans, rather than being driven by a small number of favorable cases.

The experimental results show that this improvement in WCTLS does not come from the weighting of measurements, which is already present in WTLS, but from the elimination of the scale ambiguity inherent in the implicit line representation used in classical TLS formulations. In UTLS and WTLS, the parameters (a,b,c) are not uniquely defined, which negatively affects numerical conditioning and amplifies the propagated covariance associated with the corner estimates. By enforcing a geometrical normalization, the WCTLS formulation decouples orientation and distance parameters and yields better-conditioned covariance matrices. This results in smaller and less anisotropic error ellipses, as evidenced both by the per-corner uncertainty values and by the global ellipse area distributions.

Simplifying the 2D LiDAR SLAM process by extracting geometric primitives from the mapped environment is becoming increasingly common. In this paper we have analyzed different variants of the Total Least Squares method for obtaining geometric features in the mapping process, proposing a method that improves the conditioning of the estimation problem and the reliability of the extracted geometric features.

Experimental results obtained from two real indoor environments show that all analyzed TLS variants (UTLS, WTLS, and WCTLS) yield equivalent geometric positions for the extracted corners. This confirms that the proposed method does not alter the underlying geometric structure of the map. However, significant differences arise in the associated covariance-based uncertainty estimates. While UTLS and WTLS provide nearly identical uncertainty values, the proposed WCTLS formulation consistently achieves a reduction in the standard deviations of the estimated corner coordinates. The standard deviations of the corner coordinates obtained with WCTLS are reduced by factors ranging from approximately 3 to 6 in environment A and between 2 and 4 in environment B when compared to UTLS and WTLS. This improvement is further supported by the analysis of the propagated error ellipse areas computed over 100 repeated scans acquired from fixed sensor positions. In this repeatability analysis, WCTLS significantly reduces the typical ellipse area and exhibits lower dispersion across scans, indicating a more stable behavior of the propagated uncertainty estimates.

These results demonstrate that the improvement achieved by WCTLS is not driven by measurement weighting alone, but by the elimination of the scale ambiguity inherent to the implicit line representation used in conventional TLS approaches. By enforcing a geometric normalization, WCTLS yields better-conditioned covariance matrices and more compact uncertainty estimates for derived corner features.

Despite these promising results, the proposed formulation has been validated in structured indoor environments, where geometric primitives such as lines and corners are clearly observable. Further investigation will be necessary to assess whether the approach can be generalized to more complex or less structured scenarios. It should be noted that no external ground-truth measurements of the true corner positions were available in the considered environments. Consequently, the experimental results demonstrate a consistent reduction in the propagated uncertainty associated with the extracted corners, but they do not directly establish an improvement in absolute geometric accuracy with respect to an external reference.

Future work will focus on exploiting the improved corner uncertainty provided by WCTLS in downstream SLAM tasks, such as data association, map optimization, and robot localization, assessing its potential impact on overall navigation accuracy and robustness.