A Third-Order Gaussian Process Trajectory Representation Framework with Closed-Form Kinematics for Continuous-Time Motion Estimation

In this article, we propose a third-order, i.e., white-noise-on-jerk, Gaussian process trajectory representation (TR) framework for continuous-time motion estimation (ME) tasks. Our framework features a unified TR that encapsulates the kinematic models of both <inline-formula><tex-math notation="LaTeX">$\mathrm{SO(3)}\times \mathbb {R}^{3}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathrm{SE(3)}$</tex-math></inline-formula> 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 make a better choice for the ME task. In addition, unique to our framework, we derive the kinematic models with the <italic>closed-form temporal derivatives of the local variables of <inline-formula><tex-math notation="LaTeX">$\mathrm{SO(3)}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathrm{SE(3)}$</tex-math></inline-formula></italic>, which so far has only been approximated based on 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 TR, as well as the motion prior factors, are also provided for accelerated Gauss–Newton optimization. Our experiments demonstrate the efficacy and efficiency of the framework in various ME tasks, such as localization, calibration, and odometry, facilitating fast prototyping for ME researchers. We release the source code for the benefit of the community.