Geometrical Optimal Navigation and Path Planning-Bridging Theory, Algorithms, and Applications

Autonomous systems, such as self-driving cars, surgical robots, and space rovers, require efficient and collision-free navigation in dynamic environments. Geometric optimal navigation and path planning have become critical research areas, combining geometry, optimization, and machine learning to address these challenges. This paper systematically reviews state-of-the-art methodologies in geometric navigation and path planning, with a focus on integrating advanced geometric principles, optimization techniques, and machine learning algorithms. It examines recent advancements in continuous optimization, real-time adaptability, and learning-based strategies, which enable robots to navigate dynamic environments, avoid moving obstacles, and optimize trajectories under complex constraints. The study identifies several unresolved challenges in the field, including scalability in high-dimensional spaces, real-time computation for dynamic environments, and the integration of perception systems for accurate environment modeling. Additionally, ethical and safety concerns in human-robot interactions are highlighted as critical issues for real-world deployment. The paper provides a comprehensive framework for addressing these challenges, bridging the gap between classical algorithms and modern techniques. By emphasizing recent advancements and unresolved challenges, this work contributes to the broader understanding of geometric optimal navigation and path planning. The insights presented here aim to inspire future research and foster the development of more robust, efficient, and intelligent navigation systems. This survey not only highlights the novelty of integrating geometry, optimization, and machine learning but also provides a roadmap for addressing critical issues in the field, paving the way for the next generation of autonomous systems.