Existing approaches to trajectory planning for autonomous racing employ sampling-based methods, generating numerous jerk-optimal trajectories and selecting the most favorable feasible trajectory based on a cost function penalizing deviations from an offline-calculated racing line. While successful on oval tracks, these methods face limitations on complex circuits due to the simplistic geometry of jerk-optimal edges failing to capture the complexity of the racing line. Additionally, they only consider two-dimensional tracks, potentially neglecting or surpassing the actual dynamic potential. In this paper, we present a sampling-based local trajectory planning approach for autonomous racing that can maintain the lap time of the racing line even on complex race tracks and consider the race track's three-dimensional effects. In simulative experiments, we demonstrate that our approach achieves lower lap times and improved utilization of dynamic limits compared to existing approaches. We also investigate the impact of online racing line generation, in which the time-optimal solution is planned from the current vehicle state for a limited spatial horizon, in contrast to a closed racing line calculated offline. We show that combining the sampling-based planner with the online racing line generation can significantly reduce lap times in multi-vehicle scenarios.
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