Vector Fields for Path Following on Lie Groups with Application in Robot Control

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.

This paper develops a singularity-free guiding vector field (SF-GVF) for path following on the special orthogonal group SO(3). First, we lift the Euclidean SF-GVF construction to SO(3), integrating the augmented-state approach with the intrinsic Lie-group geometry and obtaining a closed-form geometric guidance law whose integral curves converge to a designer-specified attitude path. The field is defined on a dense open subset of SO(3), excluding only the measure-zero antipodal set - a manifestation of the topological obstruction to continuous global stabilization on SO(3). The construction requires no per-step optimization and produces a control input intrinsically in so(3) as body angular rates. Second, we formalize the progression behavior along the path as a designer-supplied function ν(ξ), promoting the parametric speed from an implicitly resolved degree of freedom to a first-class design specification. In contrast to the Euclidean condition v = 0, which excludes vehicles with minimum-speed constraints, the corresponding condition ω= 0 on SO(3) is physically admissible for most platforms with active attitude control, making the progression behavior a design freedom structurally available on SO(3) but absent in the Euclidean setting. The framework's structural results are established under a bi-invariant Riemannian metric and hold uniformly across choices of path, progression, and Lyapunov gain. The framework is illustrated in simulation on self-intersecting paths under both constant and point-convergence progression behaviors.

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).