  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.

Robotics