Motion Planning for Control-Affine Systems by Geometric Heat Flow with Fine-Tuning

In this work, we address the problem of motion planning for control-affine systems. We first employ the Geometric Heat Flow (GHF) method to generate an admissible trajectory that adheres to kinematic constraints. However, due to inherent limitations of GHF, the resultant trajectory's final state may deviate from the goal state. To address this issue, we introduce a novel gradient descent algorithm for fine-tuning, which iteratively adjusts the trajectory to minimize the final state error while maintaining compliance with kinematic constraints, and ensuring that the control effort remains reasonably small. Simulation examples confirm the efficacy and efficiency of our algorithm, including parallel parking and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$180^\circ$</tex> turn of a simple car.