Guiding vector fields enable smooth path following on 3D rotation space
Singularity-Free Guiding Vector Fields on SO(3) with Designer-Specified Progression Behavior
Robotics
Summary
Controlling how objects rotate in 3D space, like drones or satellites, often faces tricky points where the math breaks down or becomes unstable. The authors propose a new way to guide rotations smoothly along a path without hitting these problematic points, except in very rare cases. They also let users specify how fast they want to move along the path, which previous methods couldn’t do in this space. Their approach creates control inputs that match physical constraints and works broadly for different paths and speeds.
What this means in practice
- •For drone control engineers: Generate smooth rotational commands for UAVs to follow desired orientation paths without instability.
- •For satellite attitude teams: Design attitude guidance laws for satellites that avoid singularities and specify precise rotation speeds along orbits.
Authors
Jesus Bautista, Hector Garcia de Marina
Abstract
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.