Time-Optimal Trajectory Planning in Highway Scenarios Using Basis-Spline Parameterization

Dorpmüller, Philip; Schmitz, Thomas; Bejagam, Naveen; Bertram, Torsten · 2023 · OpenAlex-citations

DOI: 10.1109/itsc57777.2023.10422490

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Summary

This paper addresses the challenge of time-optimal trajectory planning for automated vehicles in highway scenarios, specifically focusing on ensuring recursive feasibility and handling independent time horizons for longitudinal and lateral motion. Existing gradient-based motion planners often rely on time scaling of B-spline breakpoints (BPs), which fails to guarantee that a feasible solution in one planning step remains feasible in the next (recursive feasibility) and cannot accommodate different arrival times for target velocity and lane changes. To resolve these limitations, the authors propose a nonlinear optimization framework that treats spline breakpoints as optimization variables rather than fixed time scales. The method utilizes B-spline parameterization to represent the vehicle’s trajectory, ensuring time-continuous feasibility by enforcing constraints on the spline’s convex hull rather than at discrete time steps. The optimization problem minimizes a cost function comprising the time to reach the terminal manifold and the integral of squared jerk, subject to dynamic feasibility, collision avoidance, and heading angle constraints. Collision avoidance is modeled using axis-aligned ellipses to overapproximate vehicle shapes, with constraints formulated via B-spline arithmetic operations. A key innovation is a breakpoint removal strategy that adapts the knot vector during closed-loop simulation, allowing the planning horizon to shrink while maintaining a sparse problem formulation and ensuring convergence into the terminal manifold. The problem is solved using the IPOPT solver within the CasADi framework. The approach was evaluated in a highway overtaking scenario using the CARLA simulator, involving an ego vehicle accelerating from 50 km/h to 70 km/h while changing lanes amidst five obstacle vehicles. The results demonstrate that the proposed method successfully generates collision-free, dynamically feasible trajectories that satisfy recursive feasibility. The evaluation highlights the trade-off between solution quality and computational effort based on the number of breakpoints; increasing the number of lateral breakpoints reduced the initial cost by allowing more flexible maneuver timing, such as delaying the lane change until an obstacle passed. The breakpoint removal strategy effectively managed the shrinking horizon, with the vehicle reaching its target velocity and lane while the optimizer progressively removed interior breakpoints. The planned trajectories showed asymptotic convergence, with the closed-loop trajectory aligning with the open-loop plan as active constraints were resolved. The significance of this work lies in its ability to provide a theoretically sound, time-continuous motion planning solution that guarantees recursive feasibility, a critical property for robust real-time control in automated driving. By optimizing breakpoints directly and employing a novel removal strategy, the method overcomes the limitations of uniform time scaling, enabling independent optimization of longitudinal and lateral dynamics. This contributes to the field of intelligent transportation systems by offering a robust framework for safe and efficient trajectory generation in complex, dynamic highway environments.

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