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

Abstract

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.

Autonomous racing using learning Model Predictive Control

“…The goal of the above minimum time optimal control problem is to steer the dubins car from the starting state x S to the terminal point x F , while satisfying input saturation constraints and avoiding an obstacle. The obstacle is represented by an ellipse centered at (x obs , y obs ) = (27, −1) with semiaxis (a x , a y ) = (8,6). At iteration 0, we compute a first feasible trajectory using a brute force algorithms and we use the closed-loop data to initialize the LMPC (12) and (14) with N = 6.…”

From the Racetrack to the Road: Real-Time Trajectory Replanning for Autonomous Driving

“…While using nonlinear optimization to find the optimal trajectory for a full circuit is computationally expensive, Timings and Cole showed that convex optimization can be used to reduce the required computation time for these problems [8]. Kapania et al achieved further reduction in computation time by using alternating convex subproblems to find approximately optimal trajectories [9]. This approach was experimentally validated by testing on an autonomous vehicle as was the receding horizon approach of Gerdts et al [10].…”

Adaptive Trajectory Planning and optimization at Limits of Handling

“…Through the use of nonlinear programming, Perantoni et al [14] computes the time-optimal speed profile and racing line for an entire race track, although computational limitations require the trajectories to be computed offline. Kapania and Gerdes [15] presents an experimentally validated algorithm that reduces computational expense by breaking down the combined lateral/longitudinal vehicle control problem into two sequential subproblems that are solved iteratively.…”