# A Sequential Two-Step Algorithm for Fast Generation of Vehicle Racing Trajectories

arXiv: https://arxiv.org/abs/1902.00606 · HTML: https://mlsanity.com/paper/1902.00606/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2019.

## Built on by (most cited first, 5 of 5)

- [Autonomous Vehicles on the Edge: A Survey on Autonomous Vehicle Racing](https://mlsanity.com/paper/2202.07008/) (2022)
- [Real-Time Trajectory Planning for Autonomous Driving with Gaussian Process and Incremental Refinement](https://mlsanity.com/paper/2205.11853/) (2022)
- [Real-Time Optimal Trajectory Planning for Autonomous Vehicles and Lap Time Simulation Using Machine Learning](https://mlsanity.com/paper/2102.02315/) (2021)
- [Expert Knowledge-driven Reinforcement Learning for Autonomous Racing via Trajectory Guidance and Dynamics Constraints](https://mlsanity.com/paper/2603.05842/) (2026)
- [Dynamics Distillation for Efficient and Transferable Control Learning](https://mlsanity.com/paper/2605.01516/) (2026)

## Abstract (arXiv)

The problem of maneuvering a vehicle through a race course in minimum time requires computation of both longitudinal (brake and throttle) and lateral (steering wheel) control inputs. Unfortunately, solving the resulting nonlinear optimal control problem is typically computationally expensive and infeasible for real-time trajectory planning. This paper presents an iterative algorithm that divides the path generation task into two sequential subproblems that are significantly easier to solve. Given an initial path through the race track, the algorithm runs a forward-backward integration scheme to determine the minimum-time longitudinal speed profile, subject to tire friction constraints. With this fixed speed profile, the algorithm updates the vehicle's path by solving a convex optimization problem that minimizes the resulting path curvature while staying within track boundaries and obeying affine, time-varying vehicle dynamics constraints. This two-step process is repeated iteratively until the predicted lap time no longer improves. While providing no guarantees of convergence or a globally optimal solution, the approach performs very well when validated on the Thunderhill Raceway course in Willows, CA. The predicted lap time converges after four to five iterations, with each iteration over the full 4.5 km race course requiring only thirty seconds of computation time on a laptop computer. The resulting trajectory is experimentally driven at the race circuit with an autonomous Audi TTS test vehicle, and the resulting lap time and racing line is comparable to both a nonlinear gradient descent solution and a trajectory recorded from a professional racecar driver. The experimental results indicate that the proposed method is a viable option for online trajectory planning in the near future.