Maintaining stable and accurate localization during fast motion or on rough terrain remains highly challenging for mobile robots with onboard resources. Currently, multi-sensor fusion methods based on continuous-time representation offer a potential and effective solution to this challenge. Among these, spline-based methods provide an efficient and intuitive approach for continuous-time representation. Previous continuous-time odometry works based on B-splines either treat control points as variables to be estimated or perform estimation in quaternion space, which introduces complexity in deriving analytical Jacobians and often overlooks the fitting error between the spline and the true trajectory over time. To address these issues, we first propose representing the increments of control points on matrix Lie groups as variables to be estimated. Leveraging the feature of the cumulative form of B-splines, we derive a more compact formulation that yields simpler analytical Jacobians without requiring additional boundary condition considerations. Second, we utilize forward propagation information from IMU measurements to estimate fitting errors online and further introduce a hybrid feature-based voxel map management strategy, enhancing system accuracy and robustness. Finally, we propose a re-estimation policy that significantly improves system computational efficiency and robustness. The proposed method is evaluated on multiple challenging public datasets, demonstrating superior performance on most sequences. Detailed ablation studies are conducted to analyze the impact of each module on the overall pose estimation system.
Figures & tables
Fig. 1: Dataset platforms and mapping performance. Despite the varying motion patterns and sensor configurations across different dataset platforms, the proposed method achieves robust and accurate pose estimation in all cases.
Fig. 2: Schematic diagram of system overview.
Fig. 3: The black line denotes a continuous trajectory, represented by a 4th-order spline. The solid green circle denotes the IMU measurement, while the hollow green and red circles represent the affected and unaffected historical estimated states, respectively.
Fig. 4: Schematic diagram of representation error of continuous time spline curve.
Fig. 5: Voxel representation and decomposition diagram.
Fig. 6: The schematic diagram illustrates three prediction and estimation strategies, with the rightmost strategy being the prediction and re-estimation strategy used in this paper.
Fig. 7: The experiments are conducted on four public datasets with diverse platforms and environmental conditions. M2UD is collected using an all-terrain vehicle operating in staircases, forests, and playgrounds, featuring rugged terrain and low-speed motion. MARS-LVIG is captured by a quadrotor UAV flying in a downward-facing view at high speed. Diter++ is collected using a quadruped robot navigating complex terrain with significant vibrations and irregularities. MCD is acquired from a ground mobile robot operating in campus environments, characterized by relatively high-speed motion and a LiDAR sensor with a limited field of view.
Visual-inertial odometry (VIO) is widely used for mobile robot localization, but its long-term accuracy degrades without global constraints. Incorporating ranging sensors such as ultra-wideband (UWB) can mitigate drift; however, high-accuracy ranging usually requires well-deployed anchors, which is difficult to ensure in narrow or low-power environments. Moreover, most existing visual-inertial-ranging (VIR) fusion methods rely on discrete time-based filtering or optimization, making it difficult to balance positioning accuracy, trajectory consistency, and fusion efficiency under asynchronous multi-sensor sampling. To address these issues, we propose a spline-based continuous-time state estimation method for VIR fusion localization. In the preprocessing stage, VIO motion priors and UWB ranging measurements are used to construct virtual anchors and reject outliers, thereby alleviating geometric degeneration and improving range reliability. In the estimation stage, the pose trajectory is parameterized in continuous time using a B-spline, while inertial, visual, and ranging constraints are formulated as factors in a sliding-window graph. The spline control points, together with a small set of auxiliary parameters, are then jointly optimized to obtain a continuous-time trajectory estimate. Evaluations on public datasets and real-world experiments demonstrate the effectiveness and practical potential of the proposed approach.
Yu-An Liu, Li Zhang
School of Mathematics, Hefei University of Technology, Hefei 230031, China
Reliable odometry is essential for mobile robots as they increasingly enter more challenging environments, which often contain little information to constrain point cloud registration, resulting in degraded LiDAR-Inertial Odometry (LIO) accuracy or even divergence. To address this, we present BIEVR-LIO, a novel approach designed specifically to exploit subtle variations in the available geometry for improved robustness. We propose a high-resolution map representation that stores surfaces as voxel-wise oriented height images. This representation can directly be used for registration without the calculation of intermediate geometric primitives while still supporting efficient updates. Since informative geometry is often sparsely distributed in the environment, we further propose a map-informed point sampling strategy to focus registration on geometrically informative regions, improving robustness in uninformative environments while reducing computational cost compared to global high-resolution sampling. Experiments across multiple sensors, platforms, and environments demonstrate state-of-the-art performance in well-constrained scenes and substantial improvements in challenging scenarios where baseline methods diverge. Additionally, we demonstrate that the fine-grained geometry captured by BIEVR-LIO can be used for downstream tasks such as elevation mapping for robot locomotion.
Patrick Pfreundschuh, Turcan Tuna, Cedric Le Gentil +3
Autonomous Systems Lab, ETH Zürich, Switzerland · Robotic Systems Lab, ETH Zürich, Switzerland · Mobile Robotics Lab, ETH Zürich, Switzerland
This note presents a concise mathematical formulation of tightly-coupled LiDAR-Inertial Odometry within an iterated error-state Kalman filter framework using a VoxelMap representation. Rather than proposing a new algorithm, it provides a clear and self-contained derivation that unifies the geometric modeling and probabilistic state estimation through consistent notation and explicit formulations. The document is intended to serve both as a technical reference and as an accessible entry point for a foundational understanding of the system architecture and estimation principles.