Problem

High-performance racing has always been an exciting and challenging field, where the goal is to achieve the fastest possible lap time around the track. In Formula One (F1) racing, drivers and engineers continuously search for ways to optimize the performance of their cars and outpace their competitors. One approach to improving lap times is to use optimal control strategies, which aim to find the most efficient path a racing car can take around the track.

Optimal control strategies in racing involve sophisticated algorithms that calculate the optimal acceleration, braking, and steering inputs at each point on the track. These strategies consider the dynamic capabilities of the car, track conditions, and driver preferences to maximize speed while maintaining control and efficiency. By applying these techniques, F1 teams can fine-tune their race strategies, improve cornering speeds, manage tire wear more effectively, and ultimately shave valuable seconds off their lap times. This relentless pursuit of optimization and innovation makes optimal control a cornerstone of modern racing engineering.

Our Approach

To achieve the fastest lap times, F1 cars need to maintain maximum velocity while cornering, which requires a path that balances the trade-off between minimizing distance traveled and maximizing speed. The optimum path for an F1 car lies somewhere in between, and finding this path is a challenging problem that requires a deep understanding of optimal control, vehicle dynamics, and track geometry.

A. Geometry Problem

The maximum speed that can be achieved by a race car while navigating a curve with a certain radius is limited by the maximum centripetal force that can be generated by the tires. Therefore, to minimize lap time, it is essential for the driver to strike a balance between minimizing the distance traveled and maximizing the speed attained while staying within the limits of the car’s centripetal force capabilities.

B. Minimum Curvature Trajectory

In Formula One racing, the minimum curvature trajectory refers to the path a car takes through a turn that minimizes the distance traveled and maximizes the speed while actively maintaining vehicle control. To follow the minimum curvature trajectory, the driver will typically approach the turn from the outside edge of the track and aim to hit the apex, or the point where the turn is at its tightest, with the inside wheels of the car. The driver will then accelerate out of the turn, using the full width of the track to maintain momentum.

C. Quadratic Formulation

Closed natural cubic splines are a type of interpolation method used in minimum curvature problems to construct the minimum curvature trajectory. The cubic spline algorithm creates a smooth curve by connecting several cubic polynomials at specified data points, such as the center of turns and straight sections on a track. In this context, the closed natural cubic spline approach ensures the spline is continuous and maintains a continuous first and second derivative at all data points. This continuity guarantees a smooth trajectory that minimizes curvature, ultimately resulting in faster lap times. The effectiveness of closed natural cubic splines in reducing lap times and enhancing race car performance has been well-demonstrated in motor sports applications.

D. Trajectory Optimization

We are now going to present the Nonlinear Program formulation of our problem to optimize the trajectory for controls, taking into account vehicle dynamics and track constraints. We utilized the IPOPT (Interior Point OPTimizer) Nonlinear Programming (NLP) solver to solve this problem efficiently. IPOPT is well-suited for handling complex NLPs, ensuring that we can derive an optimal trajectory that respects the dynamics of the vehicle and adheres to the constraints imposed by the track layout.

Results