Efficient path–velocity coupled trajectory planning for autonomous vehicles using sparse normal plane constrained trajectories

Frenet-based trajectory planning is widely used in autonomous vehicles (AVs) because it simplifies arbitrary road geometries into straight-line representations. However, its application to path–velocity coupled trajectory optimization (PVCTO) is severely limited by the implicit Frenet–Cartesian conversion, which requires online projection computations and precludes explicit gradient evaluation. To address these limitations, we propose a sparse normal plane constrained trajectory (SNPCT) representation that leverages the inverse mapping from the Frenet frame to the Cartesian frame to explicitly parameterize trajectories over a discretized reference path. This parameterization ensures analytical expressions for lateral deviation and its first derivative relative to the reference line. Moreover, SNPCT provides a sparse representation that reduces the dimensionality of the optimization variables, thereby mitigating the curse of dimensionality. Building upon SNPCT, we exploit the differential flatness of the bicycle model to compactly formulate the PVCTO problem under road constraints, guaranteeing the dynamic feasibility of the resulting trajectories. Extensive benchmarks in realistic driving scenarios demonstrate that the proposed method significantly outperforms state-of-the-art approaches in both computational efficiency and trajectory quality. Source codes of this work are available at https://github.com/zhangxi33/snpctPlanner .