continuous trajectory modeling

This work addresses the limited adoption of Gaussian processes (GPs) in continuous-time state estimation—primarily hindered by their high theoretical barrier—by introducing a GP modeling approach formulated within a factor graph framework. By re-expressing the GP motion prior using factor graph semantics, the proposed method naturally supports asynchronous multi-sensor fusion and trajectory interpolation while yielding smooth, continuous trajectories. The authors provide three open-source implementations built on GTSAM, significantly lowering the practical entry barrier for employing GP-based continuous-time estimation and thereby facilitating its real-world deployment and application in robotic systems.

Existing continuous-time motion estimation (CTME) methods face challenges in unifying rotational (SO(3)) and translational (SE(3)) state modeling and lack analytically differentiable, closed-form trajectory representations. To address this, we propose the Gaussian Process Trajectory Representation (GPTR) framework. Its core innovation is a novel closed-form Gaussian process model based on third-order stochastic jerk (jerk derivative), enabling unified, smooth, and fully analytic differentiation of both rotational and translational state derivatives. GPTR supports joint optimization over heterogeneous sensors—including LiDAR, cameras, IMUs, and UWB—and provides an open-source, lightweight, header-only C++ library with exemplars of fully analytic Jacobian computation. Evaluated across multiple CTME benchmarks, GPTR achieves high accuracy and computational efficiency. Its open implementation significantly lowers the barrier to continuous-time trajectory modeling, facilitating downstream applications such as batch optimization, extrinsic calibration, and motion planning.

To address insufficient utilization of control inputs in continuous-time state estimation for resource-constrained robotic systems, this paper introduces, for the first time, a control-driven Gaussian process (GP) prior that systematically incorporates external control commands (e.g., velocity, acceleration) into the covariance function. The method fuses physics-informed priors with sparse ranging and high-frequency odometric measurements. It employs batch Bayesian inference to enable cross-domain generalization across heterogeneous tasks—specifically, mobile robot trajectory estimation and continuum robot shape deformation estimation. Experiments demonstrate that, under extreme sensing sparsity (5-second node spacing), the approach achieves trajectory estimation RMSEs of only 3–4 cm and 4–5°, while significantly reducing computational overhead. The proposed framework substantially improves estimation accuracy, robustness, and resource efficiency compared to conventional methods.

Existing asynchronous event-camera–IMU tightly coupled odometry methods suffer from limited accuracy and latency under high-speed motion and high-dynamic-range (HDR) conditions, primarily due to reliance on conventional discrete-time preintegration models. This paper proposes the Gaussian Process Pre-optimization (GPO) framework—a continuous-time formulation enabling analytically tractable state and Jacobian propagation at arbitrary timestamps. GPO introduces temporal Gaussian processes (TGPs) for continuous preintegration, achieving linear optimization complexity and constant-time query capability. Leveraging a lightweight two-stage optimization and asynchronous event–inertial tight coupling within a filtering paradigm, GPO natively supports fully asynchronous sensor fusion. Evaluated on both public and in-house datasets, GPO consistently improves localization accuracy and computational efficiency over state-of-the-art asynchronous fusion approaches, demonstrating superior overall performance.

To address the challenge of real-time, robust shape estimation for continuum robots operating in complex environments, this paper proposes a dynamic estimation framework based on a stochastic observer. The shape state is modeled via polynomial curvature mode coefficients, and recursive estimation is performed by fusing sparse pose sensor measurements. A noise-weighted observability matrix is innovatively introduced, and an IMM-EKF-driven adaptive switching mechanism among multi-order curvature models is designed to achieve online co-optimization of dynamical complexity and estimation accuracy. The method integrates Polynomial Curvature Kinematics (PCK), Extended Kalman Filtering (EKF), and the Interacting Multiple Model (IMM) algorithm. Both simulation and physical experiments demonstrate significant improvements in estimation accuracy and robustness, with strong adaptability to sensor noise and configuration changes. The approach effectively enables real-time motion planning and control under physical interaction.

This work addresses the instability and low accuracy of mobile robot localization under high-speed motion or on rough terrain by proposing a continuous-time, tightly coupled LiDAR-inertial odometry approach. The method employs B-spline trajectory parameterization over Lie groups to enable compact representation and simplified Jacobian computation. IMU preintegration is leveraged for online estimation of spline fitting errors, while a probabilistic adaptive voxel map and a feature re-estimation mechanism are introduced to balance computational efficiency and robustness. Comprehensive experiments on multiple challenging public datasets demonstrate that the proposed system consistently outperforms state-of-the-art methods, and ablation studies confirm the effectiveness and individual contributions of each module.

This work addresses the challenges of fusing and synchronizing heterogeneous asynchronous sensors—such as rolling-shutter cameras, LiDAR, and event cameras—in continuous-time SLAM. The authors propose G-solver, a novel framework that uniquely integrates Gaussian belief propagation with a data-driven Gaussian process motion prior to enable accurate trajectory estimation without requiring hardware synchronization. At its core, G-solver models the trajectory continuously via a Gaussian process and leverages distributed Gaussian belief propagation for efficient inference, supporting temporal interpolation across sensor modalities and automatic hyperparameter learning. Experiments demonstrate that G-solver achieves accuracy and efficiency on par with state-of-the-art continuous-time SLAM methods on both synthetic and real-world datasets, while inherently supporting distributed optimization. The implementation is publicly available.

This work addresses motion planning for continuous-time stochastic systems under both process and observation uncertainties by proposing a sampling-based planning framework that enables continuous-time probabilistic safety verification over entire trajectories. The approach constructs an offline hybrid belief propagation model that integrates continuous-time ordinary differential equation (ODE) dynamics with discrete Kalman updates, and introduces a belief barrier function as a safety checker capable of detecting potential constraint violations between sampling instants—marking the first method to achieve such intra-interval safety guarantees. Integrated with RRT/SST planners, the framework demonstrates superior performance over conventional discrete-time methods across multiple benchmark scenarios, including narrow passages, achieving higher success rates, enhanced robustness, and stronger formal safety assurances.

This work addresses the critical challenge of jointly designing sensor query rates and noise covariance under resource and cost constraints to meet prescribed trajectory estimation accuracy requirements. It presents the first formalization of this problem as a unified optimization model, leveraging semidefinite programming (SDP) within the Kalman filter error covariance framework to simultaneously optimize measurement scheduling and noise parameters. The proposed approach efficiently determines whether a given accuracy target is achievable and, when feasible, synthesizes a corresponding implementation strategy. Experimental validation demonstrates that the computed sensor configurations consistently attain the desired accuracy in both simulated and real-world scenarios, while also reliably identifying infeasible accuracy demands.