Learning Smooth Driving Policies via Bézier-Parameterized Flow Matching

We propose a kinematically structured flow-matching planner for offline goal-conditioned behaviour cloning of driving data. Rather than directly predicting unconstrained trajectories in the ambient state-action space, the model parameterizes the action component of the learned vector field through low-order Bézier control points for heading and speed, then lifts these actions to form joint trajectories using a differentiable kinematic rollout. This induces a trajectory-level generative model whose samples remain aligned with the kinematics model used for execution, reducing lateral-velocity drift and while maintaining smoothness without relying on post-hoc filtering. The model is trained with a flow-matching objective over full trajectories together with auxiliary supervision on fitted Bézier control points, enabling the use of a compact local motion representation while preserving conditional planning performance under scene and goal contexts.