B-Spline Trajectory Representation

World action models (WAMs) are large embodied policies that jointly predict future video and the actions to execute, emitting a fixed-length action chunk per inference call. Such a policy allocates its computational budget uniformly in time, unable to execute for longer over free-space motion or to spend more inference on contact-rich manipulation, which limits the throughput a WAM can reach when served in the cloud. We present SplineWAM, which adaptively compresses the action trajectory into a fixed-size window of cubic B-spline parameters, fitting the knot times to the characteristics of the motion. One parameter budget then decodes into chunks of varying temporal resolution and duration, and both the executed span and the interval until the next policy call follow from the prediction itself. Aligning the video supervision to the fitted knot times of the demonstration rather than to a uniform grid concentrates the supervised frames where the action trajectory is complex. For asynchronous deployment we introduce Jacobian-Pullback Real-Time Chunking (JP-RTC), which imposes chunk continuity on the decoded raw actions the robot executes rather than on the spline parameters, and corrects the parameters through the decoder so that the executed prefix agrees with the actions already committed. On LIBERO-Plus and RoboCasa, SplineWAM improves success rate over an action chunking WAM by $8.2$ and $4.4$ points while cutting policy calls per episode by 22% and 26%. On three bimanual real-robot tasks under asynchronous execution, it leads or matches the baseline while decoding 1.2 to 1.6 times as much executed motion per call.

Diffusion Policy (DP) is a powerful Learning from Demonstration (LfD) method for robotic manipulation, yet it suffers from discontinuous and non-smooth trajectories. Spline-based action representations promote smooth motion within individual action chunks, but existing spline-based methods neither guarantee cross-chunk $C^2$ continuity nor account for the group structure of $\mathrm{SE}(3)$. We therefore propose LieSpline-DP, a Lie-group B-spline diffusion policy that generates end-effector trajectories directly on $\mathrm{SE}(3)$ and couples consecutive plans by sharing their boundary control poses, ensuring $C^2$ continuity throughout the entire planned trajectory. Across three real-robot tasks, LieSpline-DP produces lower trajectory jerk and higher task success rates than the DP baseline. The gains are particularly pronounced in real-world tasks involving liquids and flexible objects: in our real-robot experiments, LieSpline-DP achieved a 100% success rate on both pouring and bucket hooking, whereas the DP baseline achieved only 10% and 30%, respectively.

Vision-Language-Action (VLA) models reduce inference cost by executing multiple actions per call, but longer horizons often degrade accuracy because raw chunks contain jitter and outliers. We introduce SMILE, an architecture-preserving interface that predicts B-spline coefficients and decodes them into smooth action sequences. SMILE changes only the action representation, enabling longer fixed horizons while retaining each baseline's backbone and model scale. We apply SMILE to SmolVLA, Evo1, VPP, and DAWN, improving accuracy and amortized inference efficiency across LIBERO, CALVIN, and real-world experiments. SMILE-Evo1 reaches 98.0% with a 1.1x speedup on LIBERO, while SMILE-VPP reaches an average length of 4.42 with a 1.5x speedup on CALVIN. At a matched execution horizon of 10, SMILE-SmolVLA reduces non-boundary acceleration by 78.6% and velocity sign-change rate by 42.3%. Real-world xArm tests show higher success, fewer drops, and fewer contacts. These results establish smooth coefficient-space generation as a route to accurate, efficient long-horizon VLA execution. Project page: jongwoopark7978.github.io/smilevla

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.

In this work, we present B-spline Policy (BSP), an action representation designed for accelerating robot manipulation policies. Rather than predicting discrete-time action chunks, BSP parameterizes actions as continuous B-spline curves defined by a set of knots and control points. This representation yields smooth, time-continuous trajectories that can be temporally scaled and executed by low-level controllers at higher frequencies and speeds. We show that B-spline-parameterized actions can be seamlessly integrated into standard policy learning pipelines by directly predicting B-spline parameters. Experiments on simulated and real-world tasks demonstrate that BSP significantly reduces task completion time, achieving substantial improvements over baseline methods while maintaining strong success rates. More results: https://b-spline-policy.github.io

Dynamic scene reconstruction from monocular videos remains highly challenging, as existing methods often struggle to balance global structural coherence and local fine-grained details under limited multi-view cues. To address this challenge, we propose WebSpline, a novel dynamic 3D Gaussian framework that enables structurally coherent and high-fidelity reconstruction from monocular videos with fast rendering. The core of WebSpline is the Structure-Informed Spline (SIS) representation, which models each dynamic Gaussian trajectory using a learnable cubic Hermite spline whose motion is structurally organized with an auxiliary Structural Proxy Graph (SPG). The proposed framework is optimized in two stages: (i) in the first stage, the SPG is initialized from 2D point tracks and refined with temporal rigidity regularization to establish structural coherence for moving objects across the sequence; and (ii) in the second stage, the SIS representation is initialized from the refined SPG and optimized under both spatial and structural neighborhood constraints. At inference, Gaussian motion is obtained solely by evaluating the learned SIS, enabling fast rendering. Extensive experiments on the challenging monocular dynamic scene benchmarks, iPhone and NVIDIA, demonstrate that our WebSpline achieves state-of-the-art rendering quality while rendering over 10 times faster than WorldTree, the second-best method on the iPhone dataset.

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.

Robotic manipulation requires policies that are smooth and responsive to evolving observations. However, synchronous inference in the raw action space introduces several challenges, including intra-chunk jitter, inter-chunk discontinuities, and stop-and-go execution. These issues undermine a policy's smoothness and its responsiveness to environmental changes. We propose ABPolicy, an asynchronous flow-matching policy that operates in a B-spline control-point action space. First, the B-spline representation ensures intra-chunk smoothness. Second, we introduce bidirectional action prediction coupled with refitting optimization to enforce inter-chunk continuity. Finally, by leveraging asynchronous inference, ABPolicy delivers real-time, continuous updates. We evaluate ABPolicy across seven tasks encompassing both static settings and dynamic settings with moving objects. Empirical results indicate that ABPolicy reduces trajectory jerk, leading to smoother motion and improved performance. Project website: https://teee000.github.io/ABPolicy/.

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.

Text-guided dynamic 3D character generation has advanced rapidly, yet producing high-quality motion that faithfully reflects rich textual descriptions remains challenging. Existing methods tend to generate limited sub-actions or incoherent motion due to fixed-length temporal inputs and discrete frame-wise representations that fail to capture rich motion semantics. We address these limitations by representing motion with continuous differentiable B-spline curves, enabling more effective motion generation without modifying the capabilities of the underlying generative model. Specifically, our closed-form, Laplacian-regularized B-spline solver efficiently compresses variable-length motion sequences into compact representations with a fixed number of control points. Further, we introduce a normal-fusion strategy for input shape adherence along with correspondence-aware and local-rigidity losses for motion-restoration quality. To train our model, we collate BIMO, a new dataset containing diverse variable-length 3D motion sequences with rich, high-quality text annotations. Extensive evaluations show that our feed-forward framework BiMotion generates more expressive, higher-quality, and better prompt-aligned motions than existing state-of-the-art methods, while also achieving faster generation. Our project page is at: https://wangmiaowei.github.io/BiMotion.github.io/.