Thien-Minh Nguyen, Ziyu Cao, Kailai Li, William Talbot, Tongxing Jin, Shenghai Yuan, Timothy D. Barfoot, Lihua Xie
In this paper, we propose a third-order, i.e., white-noise-on-jerk, Gaussian Process (GP) Trajectory Representation (TR) framework for continuous-time (CT) motion estimation (ME) tasks. Our framework features a unified trajectory representation that encapsulates the kinematic models of both $SO(3)\times\mathbb{R}^3$ and $SE(3)$ pose representations. This encapsulation strategy allows users to use the same implementation of measurement-based factors for either choice of pose representation, which facilitates experimentation and comparison to achieve the best model for the ME task. In addition, unique to our framework, we derive the kinematic models with the closed-form temporal derivatives of the local variable of $SO(3)$ and $SE(3)$, which so far has only been approximated based on the Taylor expansion in the literature. Our experiments show that these kinematic models can improve the estimation accuracy in high-speed scenarios. All analytical Jacobians of the interpolated states with respect to the support states of the trajectory representation, as well as the motion prior factors, are also provided for accelerated Gauss-Newton (GN) optimization. Our experiments demonstrate the efficacy and efficiency of the framework in various motion estimation tasks such as localization, calibration, and odometry, facilitating fast prototyping for ME researchers. We release the source code for the benefit of the community. Our project is available at https://github.com/brytsknguyen/gptr.
Panoramic multi-object tracking is important for industrial safety monitoring, wide-area robotic perception, and infrastructure-light deployment in large workspaces. In these settings, the sensing system must provide full-surround coverage, metric geometric cues, and stable target association under wide field-of-view distortion and occlusion. Existing image-plane trackers are tightly coupled to the camera projection and become unreliable in panoramic imagery, while conventional Euclidean 3D formulations introduce redundant directional parameters and do not naturally unify angular, scale, and depth estimation. In this paper, we present $\mathbf{S^3KF}$, a panoramic 3D multi-object tracking framework built on a motorized rotating LiDAR and a quad-fisheye camera rig. The key idea is a geometry-consistent state representation on the unit sphere $\mathbb{S}^2$, where object bearing is modeled by a two-degree-of-freedom tangent-plane parameterization and jointly estimated with box scale and depth dynamics. Based on this state, we derive an extended spherical Kalman filtering pipeline that fuses panoramic camera detections with LiDAR depth observations for multimodal tracking. We further establish a map-based ground-truth generation pipeline using wearable localization devices registered to a shared global LiDAR map, enabling quantitative evaluation without motion-capture infrastructure. Experiments on self-collected real-world sequences show decimeter-level planar tracking accuracy, improved identity continuity over a 2D panoramic baseline in dynamic scenes, and real-time onboard operation on a Jetson AGX Orin platform. These results indicate that the proposed framework is a practical solution for panoramic perception and industrial-scale multi-object tracking.The project page can be found at https://kafeiyin00.github.io/S3KF/.