Continuous-time trajectory representation has gained significant popularity in recent years, as it offers an elegant formulation that allows the fusion of a larger number of sensors and sensing modalities, overcoming limitations of traditional discrete-time frameworks. To bolster the adoption of the continuous-time paradigm, we propose a so-called Gaussian Process Trajectory Representation (GPTR) framework for continuous-time motion estimation (CTME) tasks. Our approach stands out by employing a third-order random jerk model, featuring closed-form expressions for both rotational and translational state derivatives. This model provides smooth, continuous trajectory representations that are crucial for precise estimation of complex motion. To support the wider robotics and computer vision communities, we have made the source code for GPTR available as a light-weight header-only library. This format was chosen for its ease of integration, allowing developers to incorporate GPTR into existing systems without needing extensive code modifications. Moreover, we also provide a set of optimization examples with LiDAR, camera, IMU, UWB factors, and closed-form analytical Jacobians under the proposed GP framework. Our experiments demonstrate the efficacy and efficiency of GP-based trajectory representation in various motion estimation tasks, and the examples can serve as the prototype to help researchers quickly develop future applications such as batch optimization, calibration, sensor fusion, trajectory planning, etc., with continuous-time trajectory representation. Our project is accessible at https://github.com/brytsknguyen/gptr .

翻译：连续时间轨迹表示近年来获得了显著关注，其提供了优雅的数学表述，能够融合更多传感器与感知模态，克服了传统离散时间框架的局限性。为促进连续时间范式的应用，我们提出了一种称为高斯过程轨迹表示（GPTR）的框架，用于连续时间运动估计（CTME）任务。本方法的突出特点在于采用三阶随机加加速度模型，具备旋转与平移状态导数的闭式表达式。该模型提供了平滑、连续的轨迹表示，这对精确估计复杂运动至关重要。为支持更广泛的机器人学与计算机视觉研究社区，我们已将GPTR的源代码发布为轻量级头文件库。选择此格式旨在便于集成，使开发者能够将GPTR纳入现有系统而无需大量代码修改。此外，我们还提供了一套基于所提高斯过程框架的优化示例，包含LiDAR、相机、IMU、UWB因子及闭式解析雅可比矩阵。实验验证了基于高斯过程的轨迹表示在各种运动估计任务中的有效性与高效性，这些示例可作为原型帮助研究者快速开发未来应用，如批量优化、标定、传感器融合、轨迹规划等，均基于连续时间轨迹表示。本项目可通过 https://github.com/brytsknguyen/gptr 访问。