Advanced Frenet-Based Inverse Motion Planning for Autonomous Vehicles: Unified Mathematical Framework

Source abstract

This paper presents a novel advanced mathematical framework for motion planning of autonomous vehicles (AVs), centered on the simultaneous optimization of spatial trajectories and kinematic profiles within an augmented Frenet coordinate system. Addressing the inherent limitations of traditional two-stage planning approaches - namely, geometric discontinuity and the control-chattering trap - we propose a «virtual rail» concept that decouples the spatial reference curve from temporal velocity distribution while maintaining high-order differentiability (up to the 3rd derivative of curvature). By employing a finite element approach based on 3rd-order Hermite polynomials with two degrees of freedom per node for constraint satisfaction, the proposed framework ensures maximum trajectory smoothness, which is critical for actuation stability. Furthermore, we derive the complex kinematic relationships (velocity, acceleration, and jerk) directly in the curvilinear reference frame and formulate an optimization objective that balances trajectory fidelity, ride comfort and dynamic feasibility. The proposed mathematical framework demonstrates high consistency and robust convergence, validating the underlying analytical approach as a sound and rigorous basis for complex motion planning. This work establishes the foundational architecture for a series of studies focusing on higher-order vehicle dynamics and multi-segment transient maneuver planning.