This paper presents a novel planning and control strategy for competing with multiple vehicles in a car racing scenario. The proposed racing strategy switches between two modes. When there are no surrounding vehicles, a learning-based model predictive control (MPC) trajectory planner is used to guarantee that the ego vehicle achieves better lap timing performance. When the ego vehicle is competing with other surrounding vehicles to overtake, an optimization-based planner generates multiple dynamically-feasible trajectories through parallel computation. Each trajectory is optimized under a MPC formulation with different homotopic Bezier-curve reference paths lying laterally between surrounding vehicles. The time-optimal trajectory among these different homotopic trajectories is selected and a low-level MPC controller with control barrier function constraints for obstacle avoidance is used to guarantee system's safety-critical performance. The proposed algorithm has the capability to generate collision-free trajectories and track them while enhancing the lap timing performance with steady low computational complexity, outperforming existing approaches in both timing and performance for a autonomous racing environment. To demonstrate the performance of our racing strategy, we simulate with multiple randomly generated moving vehicles on the track and test the ego vehicle's overtake maneuvers.
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