In the field of robot technology, accurate and reliable sensor extrinsic calibration is crucial for achieving high-precision localization and navigation in LiDAR-inertial fusion systems. However, many existing calibration methods rely on obtaining triaxial excitation from inertial sensors, which becomes problematic when LiDAR and inertial sensors are mounted on ground robots with restricted movement. This limitation often leads to performance degradation or even failure in calibration. To address this issue, we propose a novel calibration method for ground robot LiDAR-inertial odometry based on planar features in LiDAR point clouds. This approach leverages the inherent planar structures in environments commonly encountered in robot technology applications, such as indoor floors and outdoor terrains, to achieve robust and accurate extrinsic parameters without requiring full-axis motion.
The proposed method consists of two main phases: LiDAR point cloud preprocessing and optimization. In the preprocessing phase, we perform ground segmentation to extract ground points and estimate the LiDAR’s installation height relative to the ground. Additionally, we extract planar features from the point clouds by employing an adaptive voxelization grid, which helps in fitting plane normal vectors. The optimization phase utilizes these features to calibrate the extrinsic parameters through three types of constraints: preliminary optimization based on planar features, optimization based on octree voxel grid spatial occupancy, and Z-axis displacement correction using ground constraints. By integrating these elements, our method ensures complete six-degree-of-freedom (6-DOF) extrinsic parameters, enhancing the overall performance of robot technology systems in various scenarios.
The core of our method lies in the efficient use of planar features to establish residuals. Specifically, we minimize the distance from LiDAR points to their corresponding planes to rapidly converge the extrinsic parameters to a small error range. This is achieved by maintaining a sliding window of multiple LiDAR frames and jointly solving the residuals between point clouds and normal vectors across frames. The transformation from the LiDAR frame to the inertial sensor frame, denoted as $T^I_L$, is represented as a rotation matrix $R^I_L$ and a translation vector $t^I_L$. The global point cloud map $M$ in the world frame is constructed as follows:
$$ M = \sum T^W_I (R^I_L P_L + t^I_L) $$
where $P_L$ represents the point cloud in the LiDAR frame, and $T^W_I$ is the pose matrix of the inertial sensor. The optimization goal is to find the optimal $T^I_L$ that minimizes the residuals, leading to a high-quality global point cloud map. This process is integral to advancing robot technology by ensuring precise sensor alignment.
In the ground segmentation step, we use the Patchwork++ algorithm to divide the LiDAR point cloud $P_L$ into ground points $P_G$ and non-ground points $P_{\neg G}$ based on a concentric sector model. The segmentation is defined as:
$$ P_L = P_G \cup P_{\neg G} $$
$$ \text{Sec} = \bigcup_{m \in \text{num}_z} Z_m $$
$$ Z_m = \{ P_m \mid L^m_{\min} \leq \rho_m \leq L^m_{\max} \} $$
where $\text{Sec}$ denotes the concentric sectors, $Z_m$ is the $m$-th zone, $\rho_m = \sqrt{x_m^2 + y_m^2}$ is the radius of points in $P_m$, and $L^m_{\min}$ and $L^m_{\max}$ are the minimum and maximum radius boundaries. The ground points $P_G$ are derived by applying thresholds on height and other geometric properties, which is essential for reliable robot technology operations in varied environments.
For Z-axis correction, we incorporate ground constraints to address the unobservable degrees of freedom in planar motion. By directly measuring the inertial sensor’s height vector $d_I$ from the ground and using it as a prior, we establish the following constraint equations:
$$ (R^G_I R^I_L R^L_G) = 1 $$
$$ e^T_3 (R^L_I R^I_G d_I – d_L) = e^T_3 t^I_L $$
where $e_3 = (0,0,1)^T$, $R^L_G$ is the rotation matrix from the ground frame to the LiDAR frame obtained from ground segmentation, and $d_L$ is the LiDAR height vector. This correction ensures accurate extrinsic parameters in all six dimensions, which is critical for the stability and accuracy of robot technology applications.
Planar feature extraction involves processing each LiDAR scan ring to identify points with small distance variations, classifying them into plane points $P_{\text{plane}}$ and corner points $P_{\text{cornor}}$ based on a threshold $\mu$:
$$ P_{\text{plane}} = \sum (P_L \mid \text{dis}_{P_L} \leq \mu) $$
$$ P_{\text{cornor}} = \sum (P_L \mid \text{dis}_{P_L} > \mu) $$
We then voxelize the point cloud to associate features across frames, computing the covariance matrix $A$ for each voxel to derive the plane normal vector. The centroid $\bar{p}$ and covariance are given by:
$$ \bar{p} = \frac{1}{N} \sum_{i=1}^N p_i $$
$$ A = \frac{1}{N} \sum_{i=1}^N (p_i – \bar{p})(p_i – \bar{p})^T $$
The smallest eigenvalue of $A$ corresponds to the plane’s normal vector, which is used in the optimization process. This step enhances the robustness of robot technology systems by ensuring consistent feature matching.
The sliding window optimization jointly solves residuals over $n$ frames. The point cloud in the world frame $P^W_i$ for the $i$-th frame is projected as:
$$ P^W_i = (T^W_I)_i T^I_L P^L_i $$
The extrinsic parameter $T^I_L$ is optimized by minimizing the sum of residuals $\xi_k$, where each residual represents the distance from points in the $k$-th frame to the corresponding plane in the first frame:
$$ T^I_L = \arg \min \sum_{i=k}^n \| \xi_k \|^2 $$
$$ \xi_k = \sum_{n=1}^N n^T (P^n_{\text{plane}} – P^1_{\text{plane}}) $$
Here, $n^T$ is the normal vector of the first frame’s plane, and $N$ is the number of planes extracted. This optimization rapidly converges the extrinsic parameters, demonstrating the efficiency of our method in robot technology contexts.
Further refinement is achieved through octree voxel grid spatial occupancy optimization. After obtaining a refined extrinsic parameter $T^I_{L(\text{refine})}$ from the initial optimization, we generate a global point cloud map $M_{\text{refine}}$ and minimize its spatial occupancy:
$$ T^I_L = \arg \min \{ \text{occupancy}(M_{\text{refine}}) \} $$
$$ M_{\text{refine}} = \sum T^W_I T^I_{L(\text{refine})} P_L $$
This step ensures that the point cloud map occupies the fewest voxels, indicating high alignment quality and accurate extrinsic parameters. Such optimizations are vital for advancing robot technology in dense mapping scenarios.
To validate our method, we conducted experiments on two open-source datasets: AI-Lab and M2DGR. These datasets represent diverse environments, including open plains and dynamic scenes with obstacles, which are common in robot technology applications. We compared our method with state-of-the-art algorithms like Open-calib, Li-Init, and LI-Calib. The evaluation metrics included mean absolute error (MAE) for translation and rotation parameters. The translation error $E_T$ is defined as:
$$ E_T = \| t_{\text{pred}} – t_{\text{rel}} \| $$
where $t_{\text{pred}}$ is the predicted translation vector and $t_{\text{rel}}$ is the ground truth. The MAE for rotation and translation at time $t$ is calculated as:
$$ \text{MAE} = \frac{1}{n} \sum_{i=1}^n \| \hat{p}_i – p_{\text{rel}} \| $$
where $\hat{p}_i$ is the estimated value and $p_{\text{rel}}$ is the true value. Our method consistently outperformed others, as shown in the following tables, underscoring its superiority in robot technology calibration tasks.
| Algorithm | Roll MAE (°) | Pitch MAE (°) | Yaw MAE (°) | tx MAE (m) | ty MAE (m) | tz MAE (m) |
|---|---|---|---|---|---|---|
| Open-calib | 0.425 | 0.303 | 0.611 | 0.040 | 0.031 | 0.200 |
| Li-Init | 0.724 | 0.713 | 0.904 | 0.088 | 0.072 | 0.112 |
| LI-Calib | 0.558 | 0.629 | 0.958 | 0.095 | 0.062 | 0.146 |
| Our Method | 0.271 | 0.293 | 0.404 | 0.028 | 0.021 | 0.017 |
On the AI-Lab dataset, our method reduced the average rotation error by 43.73% and the average translation error by 76.33% compared to other algorithms. This significant improvement highlights the effectiveness of our approach in robot technology applications where precise calibration is essential.
| Algorithm | Roll MAE (°) | Pitch MAE (°) | Yaw MAE (°) | tx MAE (m) | ty MAE (m) | tz MAE (m) |
|---|---|---|---|---|---|---|
| Open-calib | 0.513 | 0.445 | 0.914 | 0.108 | 0.099 | 0.200 |
| Li-Init | 0.851 | 0.520 | 1.005 | 0.098 | 0.112 | 0.092 |
| LI-Calib | 0.692 | 0.274 | 0.673 | 0.041 | 0.123 | 0.164 |
| Our Method | 0.368 | 0.152 | 0.697 | 0.069 | 0.076 | 0.054 |
For the M2DGR dataset, our method achieved a 36.47% reduction in average rotation error and a 41.52% reduction in average translation error. Although the errors were slightly higher due to dynamic obstacles and fewer planar features, our method maintained robust performance, demonstrating its adaptability in complex robot technology environments.
We also conducted real-world experiments using a ground robot platform equipped with a RS-Helios-16P LiDAR and a CH110 IMU. The robot traversed various terrains, including flat ground, corridors, and uneven surfaces, following a figure-eight path to collect data. The initial extrinsic parameters were set with deliberate biases: 10° offsets in Roll, Pitch, and Yaw angles, and 0.2 m offsets in translation components. After calibration, the global point cloud map showed improved consistency, with clearer object contours and accurate height proportions, which is crucial for reliable robot technology operations.
To quantitatively assess the calibration accuracy, we used the calibrated extrinsic parameters as initial inputs for the FAST-LIO2 LiDAR-inertial odometry algorithm and evaluated the absolute trajectory error (ATE). The ATE at the $i$-th frame is defined as:
$$ F_i = Q^{-1}_i S P_i $$
where $P_i \in SE(3)$ is the estimated pose, $Q_i \in SE(3)$ is the ground truth pose, and $S \in Sim(3)$ is the similarity transformation matrix. The root mean square error (RMSE) of the ATE is calculated as:
$$ \text{RMSE}(F_{1:n}, \Delta t) = \sqrt{ \frac{1}{m} \sum_{i=1}^m \| \text{trans}(F_i) \|^2 } $$
In our experiments, the robot traveled a total distance of 62.393 m. The trajectory obtained using our calibration method had an ATE RMSE of 0.173718 m, compared to 0.185077 m with Li-Init, representing a 6.54% improvement. This enhancement in localization accuracy underscores the practical benefits of our method for robot technology systems.
In conclusion, our proposed calibration method effectively addresses the challenges of extrinsic parameter estimation for ground robots by leveraging planar features and ground constraints. It achieves high accuracy in both rotation and translation parameters without requiring full-axis motion, making it highly suitable for robot technology applications. The experimental results on datasets and real-world scenarios confirm its robustness, accuracy, and practicality, contributing to the advancement of autonomous navigation in robot technology.