Singularity-Free Guiding Vector Fields on SO(3) with Designer-Specified Progression Behavior

The existing singularity-free guiding vector field (SF-GVF) with an additional virtual coordinate can eliminate singular points (i.e., points where the vector field vanishes) inherent in conventional GVFs and guarantee global convergence of robot trajectories to closed and self-intersecting desired paths. However, the desired speed given by the GVF along the desired path in the original lower-dimensional space cannot be arbitrarily specified but depends on path parameterizations. One possible workaround is to partially normalize the physical projection of the SF-GVF and assign a user-designed speed. However, we show that this workaround may introduce new singularities since the normalization denominator can become zero. To address this issue, we propose a new SF-GVF with prescribed physical speed (PPS). The integral curves of the new SF-GVF converge exponentially to the desired path from any initial condition in the higher-dimensional space (including virtual dimension); more importantly, the robot's physical speed converges to the PPS, while the path-error dynamics remain invariant under regular reparameterizations of the desired path. We further develop a saturated acceleration control law for second-order kinematic models. Finally, comparative simulations and 3D path-following experiments with a quadrotor under different PPS profiles validate the theoretical results and demonstrate the effectiveness of the proposed approach.

Guiding Vector Fields (GVFs) are a powerful tool for robotic path following. However, classical methods assume smooth, ordered curves and fail when paths are unordered, multi-branch, or generated by probabilistic models. We propose a unified framework, termed the Score-Induced Guiding Vector Field (SGVF), which leverages score-based generative modeling to construct vector fields directly from data distributions. SGVF learns tangent fields from point clouds with unit-norm, orthogonality, and directional-consistency losses, ensuring geometric fidelity and control feasibility. This approach removes the reliance on ad-hoc path segmentation and enables guidance along complex topologies such as branching and pseudo-manifolds. The study establishes a correspondence between score vanishing in diffusion models and GVF singularities and highlights representational capacity near sharp path curvatures. Experiments on robotic navigation in planar environments demonstrate that SGVF achieves reliable path following in scenarios where classical GVFs fail, underscoring its potential as a bridge between generative modeling and geometric control. Code and experiment video are available at https://github.com/czr-gif/Guiding-Vector-Field-Generation-via-Score-based-Diffusion-Model.

Accurately steering a robot to a target configuration is fundamental in engineering, yet remains challenging for nonholonomic mobile robots. Vector fields (VFs) provide a natural framework by specifying desired motion directions throughout the workspace and enabling direct integration with feedback control. However, most existing VF-based methods cannot explicitly generate trajectories satisfying curvature constraints. Actuator limits are therefore often enforced by input saturation, which may invalidate stability guarantees and degrade closed-loop performance when not considered in controller design. In addition, these methods usually ensure only asymptotic convergence without an explicit settling-time bound. To address these issues, we propose a generalized motion planning and control framework consisting of a finite-time curvature-constrained vector field (FT-C2VF) and a saturation-free control law. Depending on the motion objective, the framework drives the robot to the target configuration in finite time or through it periodically. First, the FT-C2VF is constructed using complementary gains to achieve finite-time convergence while ensuring that the curvature of its integral curves is continuous, bounded, and monotonically decreasing with the radial ratio. Second, an almost globally C1-smooth, saturation-free controller is developed to track the FT-C2VF without Jacobian information, while keeping all control inputs within prescribed actuator limits. Third, dynamical-systems analysis establishes almost-global finite-time stability of the target equilibrium. Numerical simulations show improved performance over representative VF-based methods, and outdoor experiments on an Ackermann-steered vehicle confirm the effectiveness and robustness of the proposed approach.