RESPLE2: Inertial-Informed Recursive Non-Uniform Spline Estimation for Multi-Sensor Odometry with Global Refinement

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.

Visual-inertial navigation systems estimate six-degree-of-freedom motion by fusing visual and inertial data. Modern discrete-time methods with IMU preintegration provide strong accuracy and efficiency, but keyframe-based representations can be less flexible when residuals must be evaluated at arbitrary timestamps or when motion-dependent temporal resolution is needed. Continuous-time splines address this issue by representing the trajectory as a smooth temporal function, but uniformly spaced knots can under-represent rapid dynamics or over-parameterize static intervals. This letter proposes ASPIRE-VINS, a continuous-time VINS framework that combines adaptive knot placement (AKP), multi-resolution splines (MRS), and 3D measurement-space residuals (3D-MSR). AKP allocates knots according to local motion variation, MRS adds bounded local refinement in tangent space, and 3D-MSR provides bearing consistency by aligning transformed features with calibrated observation rays in 3D measurement space. Experiments show that ASPIRE-VINS achieves competitive or lower trajectory errors than the compared baselines, demonstrating the effectiveness of motion-adaptive continuous-time trajectory modeling under diverse motion and sensing conditions.

Accurate relative localization is critical for multi-robot cooperation. In robot groups, measurements from different robots arrive asynchronously and with clock time-offsets. Although Continuous-Time (CT) formulations have proved effective for handling asynchronous measurements in single-robot SLAM and calibration, extending CT methods to multi-robot settings faces great challenges in achieving high-accuracy, low-latency, and high-frequency performance. In particular, existing CT methods suffer from the inherent query-time delay of unclamped B-splines and high optimization latency. This paper proposes CT-RIO, a novel Continuous-Time Relative-Inertial Odometry framework. We adopt Clamped Non-Uniform B-splines (C-NUBS) to represent states, eliminating the query-time delay. We further augment C-NUBS with closed-form extension and shrinkage operations that preserve the spline shape, making it suitable for online estimation and enabling flexible knot management. This flexibility leads to the concept of a knot-keyknot strategy, which supports spline extension at high frequency while retaining sparse keyknots for adaptive relative motion modeling. We then formulate a reference-centric sliding-window relative localization problem that operates purely on relative kinematics and inter-robot constraints. To enable low-latency and high-frequency estimation, we decompose the tightly coupled optimization into robot-wise subproblems and solve them in parallel using asynchronous block coordinate descent. Extensive experiments show that CT-RIO converges from time-offsets as large as 264 ms to sub-millisecond within 3 s, and achieves RMSEs of 0.046 m and 1.8 degree. It consistently outperforms evaluated published methods, with improvements of up to 60% under high-speed motion.