EL-AGHF: Extended Lagrangian Affine Geometric Heat Flow

2025-05-30

AI Summary

Traditional affine geometric heat flow (AGHF) suffers from numerical instability and high computational cost in motion planning for both nonholonomic and holonomic systems, due to its imposition of infinite penalty on infeasible control directions. Method: This paper proposes an extended AGHF framework that couples state and dual trajectories, embedding the augmented Lagrangian principle into geometric heat flow. It explicitly models dynamical feasibility gaps via a system of coupled parabolic partial differential equations (PDEs), thereby avoiding infinite penalties. The method integrates constrained variational principles with efficient PDE solvers. Results: The approach generates fully dynamically feasible trajectories across diverse nonholonomic systems in simulation, significantly reducing computational overhead and numerical oscillations. Convergence is rigorously proved, and real-time planning applicability is empirically validated.

Technology Category

• Multiagent Systems: Multiagent Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Agentic search

Abstract

We propose a constrained Affine Geometric Heat Flow (AGHF) method that evolves so as to suppress the dynamics gaps associated with inadmissible control directions. AGHF provides a unified framework applicable to a wide range of motion planning problems, including both holonomic and non-holonomic systems. However, to generate admissible trajectories, it requires assigning infinite penalties to inadmissible control directions. This design choice, while theoretically valid, often leads to high computational cost or numerical instability when the penalty becomes excessively large. To overcome this limitation, we extend AGHF in an Augmented Lagrangian method approach by introducing a dual trajectory related to dynamics gaps in inadmissible control directions. This method solves the constrained variational problem as an extended parabolic partial differential equation defined over both the state and dual trajectorys, ensuring the admissibility of the resulting trajectory. We demonstrate the effectiveness of our algorithm through simulation examples.

Problem

• Suppress dynamics gaps in inadmissible control directions
• Unify motion planning for holonomic and non-holonomic systems
• Overcome computational cost and instability in constrained AGHF

Innovation

• Extended Lagrangian Affine Geometric Heat Flow
• Augmented Lagrangian method for constraints
• Dual trajectory for dynamics gaps