Singularity-free Guiding Vector Field for Robot Navigation (2012.01826v3)

Abstract: Most of the existing path-following navigation algorithms cannot guarantee global convergence to desired paths or enable following self-intersected desired paths due to the existence of singular points where navigation algorithms return unreliable or even no solutions. One typical example arises in vector-field guided path-following (VF-PF) navigation algorithms. These algorithms are based on a vector field, and the singular points are exactly where the vector field diminishes. In this paper, we show that it is mathematically impossible for conventional VF-PF algorithms to achieve global convergence to desired paths that are self-intersected or even just simple closed (precisely, homeomorphic to the unit circle). Motivated by this new impossibility result, we propose a novel method to transform self-intersected or simple closed desired paths to non-self-intersected and unbounded (precisely, homeomorphic to the real line) counterparts in a higher-dimensional space. Corresponding to this new desired path, we construct a singularity-free guiding vector field on a higher-dimensional space. The integral curves of this new guiding vector field is thus exploited to enable global convergence to the higher-dimensional desired path, and therefore the projection of the integral curves on a lower-dimensional subspace converge to the physical (lower-dimensional) desired path. Rigorous theoretical analysis is carried out for the theoretical results using dynamical systems theory. In addition, we show both by theoretical analysis and numerical simulations that our proposed method is an extension combining conventional VF-PF algorithms and trajectory tracking algorithms. Finally, to show the practical value of our proposed approach for complex engineering systems, we conduct outdoor experiments with a fixed-wing airplane in windy environment to follow both 2D and 3D desired paths.

• The paper presents a novel method using higher-dimensional unbounded guiding vector fields to eliminate singular points that limit global convergence in traditional Vector Field-guided Path Following (VF-PF) algorithms for complex paths.
• The methodology involves transforming problematic paths into higher-dimensional spaces homeomorphic to the real line, allowing for singularity-free vector field construction and ensuring global convergence through extended dynamics.
• Theoretical proofs and outdoor experiments with fixed-wing aircraft demonstrate the approach's practical applicability, verifying robust path-following accuracy for paths previously impossible with conventional methods.

Overview of the Singularity-free Guiding Vector Field for Robot Navigation

The paper presented explores the critical challenge of robot path-following navigation, specifically addressing issues related to singular points that considerably limit the global convergence capabilities of conventional Vector Field-guided Path Following (VF-PF) algorithms. The novel contribution lies in providing a methodology to circumvent these limitations, allowing more robust navigation of mobile robots along desired paths, including those that are self-intersected or simple closed paths.

Key Contributions

The paper establishes that conventional VF-PF algorithms struggle with singular points—locations where the vector field becomes zero—hindering reliable navigation particularly when paths are self-intersected or simple closed. The authors methodologically demonstrate the mathematical impossibility of achieving global convergence using these traditional approaches.

To counter the singularity problem, the research introduces higher-dimensional unbounded guiding vector fields. This involves transforming the problematic paths into higher-dimensional spaces where these singularities can be effectively eliminated, and global convergence assured.

Methodology

1. Transformation to Higher Dimensions: The authors propose mapping self-intersected or simple closed paths to higher-dimensional unbounded counterparts which are homeomorphic to the real line. This transformation facilitates a singularity-free guiding vector field construction in a higher-dimensional ambient space.
2. Extended Dynamics: The paper introduces extended dynamics that ensure the integral curves of the newly constructed vector fields converge globally to these higher-dimensional paths and consequently, when projected, align with the physical desired paths in lower dimensions.
3. Mathematical Assurance: Rigorous theoretical analysis rooted in dynamical systems theory affirms that the transformation and vector field construction guarantee robust path-following behavior. This is achieved through the systematic elimination of singular points.

Results and Implications

The authors provide theoretical proof paired with practical experiments, including outdoor trials with fixed-wing aircraft under variable windy conditions, verifying the global convergence and practical applicability of the proposed approach. The method enhances path-following accuracy while enabling the navigation of complex paths, which are not possible with conventional VF-PF algorithms.

From a theoretical perspective, the research advances the understanding of vector-field-guided path-following controls by overcoming topological limitations through dimensional transformation. Practically, it improves the robustness and versatility of robotic systems navigating complex environments, setting the stage for future innovations in autonomous vehicle guidance and control.

Future Prospects

The implications of this research potentially revolutionize AI-based robotics navigation, opening avenues for further exploration into path-following algorithms with extended application scopes, such as in unmapped terrains or dynamic environments. Subsequent research could explore optimizing these higher-dimensional transformations or integrating collision avoidance mechanisms into the robot navigation framework.

Conclusively, this paper represents a significant step forward in navigating singularity-free paths, enabling the seamless operation of autonomous robots across more complex, unstructured environments.