SE3Kit: A Lightweight Python Library for Specialized Geometric Primitives in Robotics

Optimal trajectory generation for rigid-body motions on Lie groups can be formulated as a variational problem that minimizes energy functionals defined by Riemannian metrics. While closed-form solutions exist for special cases such as product metrics and rest-to-rest boundary conditions, solving the general problem with arbitrary boundary states and coupled rotational-translational metrics often requires computationally expensive numerical boundary value solvers. These limitations restrict the use of geometrically consistent trajectory generation in real-time robotic planning and control. This paper presents a learning-based framework for approximating higher-order smooth trajectories on SE(3) under general left-invariant Riemannian metrics. The method parameterizes body-twist trajectories using high-order polynomials and relies on a neural network to learn a subset of the polynomial coefficients and the trajectory duration. The remaining coefficients are analytically determined to enforce the boundary conditions. The training of the network is guided by losses derived from Euler-Lagrange optimality conditions, metric-weighted smoothness objectives, and feasibility constraints. The metric-conditioned framework enables generalization across diverse metric structures and motion conditions. Extensive numerical experiments demonstrate that the proposed approach generates smooth trajectories that closely approximate solutions from numerical optimization while achieving millisecond-level inference times. We demonstrate two practical applications of the proposed framework: real-time generation of diverse motion primitives with waypoint traversal, and refinement for quadrotor flight under dynamic conditions. These results suggest that learning-based motions with geometric structure can provide an efficient alternative to conventional optimization-based methods for trajectory generation on SE(3).

Dynamically feasible trajectory optimization for rigid-body systems is naturally formulated on the special Euclidean group SE(3) but is challenging when dynamics are available only as black-box computations without derivatives. This paper formulates the Trajectory Bundle Method (TBM) for motion planning implicitly on SE(3). Bundles are constructed in the Lie algebra and propagated through nonlinear rigid-body dynamics using exponential and logarithmic maps, enabling derivative-free planning of non-Euclidean trajectories. We show that Euclidean TBM interpolation error is bounded quadratically by bundle diameter and extend this result to SE(3), where the bound additionally depends on a local Lipschitz constant of the Log map. Numerical experiments corroborate these bounds. Finally, we demonstrate SE(3) TBM by optimizing an acrobatic, collision-free fixed-wing maneuver through a rotated aperture without explicit models or derivatives of the vehicle dynamics, aerodynamics, or collision model.

Vector-field-based methods are widely used for robot control and are often applied to the path-tracking problem. Some vector field approaches require repeatedly computing the distance between the robot configuration and the curve, as well as the corresponding closest point. Recently, vector fields have been extended to Lie Groups. In this case, this computation can be expensive, especially when performed at high control frequencies on embedded platforms. This paper proposes a method for efficiently computing the distance between a point and a curve represented as what is called a G-polynomial curve, which is a curve representation that generalizes polynomial curves to matrix Lie groups. The proposed approach exploits the structure of these curves to reduce the problem to a small number of polynomial root-finding computations. Simulation results show that the method significantly reduces computation time while maintaining accuracy compared to existing optimization-based approaches. Practical formulas are also provided for the case of the group SE(3), and the method is validated experimentally on a robotic manipulator. The methodology is implemented in a computational package, available online.