Infinite-Dimensional Path Planning Framework for Autonomous Navigation

 🧠 Core Concept 

Rather than discretizing the environment or relying on local potential fields, I represent obstacle-laden maps as continuous higher dimension surfaces using summations of exponential bump functions. These surfaces encode obstacle regions as high-cost zones in an augmented spatial dimension, allowing the shortest collision-free path to be formulated as a geodesic on this surface. 

By applying the Euler–Lagrange equations, the planner computes the globally optimal path directly — avoiding the local minima traps common in artificial potential field (APF) and graph-based methods.

 🔍 Key Features 

• Boundary Value Problem Formulation: Guarantees collision-free solutions by solving differential equations that inherently respect obstacle boundaries.
  
  

• Real-Time Updates: Capable of incorporating sensor feedback to replan paths dynamically in unknown or changing environments.
  
  

• Dynamic Obstacle Handling: Supports time-varying obstacle positions and recomputes optimal trajectories accordingly.
  
  

• Minimal Penalty in Unknown Maps: Demonstrated less than 1.5% path length penalty compared to fully known environments.

🛠 Technical Contributions 

• Developed full analytical model for static and dynamic multi-obstacle environments.
  
  

• Implemented in MATLAB using bvp4c for solving high-order nonlinear differential equations.
  
  

• Validated performance under various scenarios, including real-time replanning, obstacle detection, and closely spaced constraints.

🧩 Method Summary

Obstacle map  →   Higher dimension surface →Euler–Lagrange → BVP solution → Optimal path

📌 Applications

• Autonomous UAV navigation

• Collaborative mobile robots

• Surgical path shaping with minimal tissue disruption

• CNC path optimization

• Obstacle avoidance in unknown terrain
  
  

📍Status & Next Steps 

    Now extending to multi-agent coordination and SLAM-based feedback.