EL-AGHF: Extended Lagrangian Affine Geometric Heat Flow

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.