Summary
The paper addresses how to smoothly guide the rotation of objects in three dimensions without running into problematic points called singularities. The authors extend existing methods from flat spaces to the more complex space of 3D rotations, represented mathematically by SO(3). They create a way to control the rotation that avoids points where control normally breaks down and allow the designer to specify exactly how fast the object moves along a desired rotational path. This approach is mathematically guaranteed to work almost everywhere and produces commands that are directly usable for controlling physical devices like robots or drones. Simulations show the method working on complex rotation paths, confirming its effectiveness.
SO(3)rotation groupvector fieldsingularitypath followingLie groupattitude controlRiemannian metricangular velocitygeometric control
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.