Optimization of the racing line of a car with PSO

This is my TIPE project for the 2023/2024 academic year, based on the theme "Sport and Games".

You can find the code of this project on my GitHub.

Introduction

I'm really fond of motorsport, which is why I decided to optimize the trajectory of a car on a track. To achieve this, I used a Particle Swarm Optimization (PSO) algorithm, a metaheuristic inspired by the behavior of bird flocks.

Soon after, I realized that the algorithm relies on many parameters that weren't detailed in the papers. Therefore, I had to conduct extensive research to find the best parameters for my problem, which became the focus of my TIPE.

The parameters of the algorithm I'm going to estimate are :
- The number of Particle
- The number of iterations
- The inertia weight w
- The cognitive coefficients c1
- The social coefficients c2

I'm going to use the French Grand Prix track, named Paul Ricard :

The papers provide the formula to update the velocity of the particles, which is:

Where w, c1 and c2 are the parameters we want to estimate.

The inertia weight parameter

The first parameter we are going to estimate is w. To do this, I will fix the other parameters and vary w.

Here are the results :

We can see that the best value for w is 0.8.

The number of iterations and particles

The second and third parameters we are going to estimate are the number of iterations and particles. Since these are both variables of nested loops, they will have the same effect on the computation time. So we're going to study there effect on the laptime.

Here are the results :

We can clearly see that after 50 iterations, the lap time stabilizes. Therefore, we can set the number of iterations to 50. As for the number of particles, we can see that it tends to reach a limit. This means that if you want better results, you can use a higher number of particles, but if you want faster results, you might opt for a lower number of particles

The social and cognitive coefficients (gbest and pbest)

The final parameters we are going to estimate are the social and cognitive coefficients. To do this, we're going to search a minimum on a plane with the simulated annealing algorithm. I ran this on a computing farm on 64 processes each time.

Here are the results :

We can see that the values for c1 only needs to be strictly positive.
As for c2, I will analyze the distribution of its values.

Here are the results :

The papers provide the formula to update the velocity of the particles, which is:

The final result of the analysis

After all these analyses, we've determined that the best parameters for the PSO algorithm are:

- The number of particles : 80-400+ (can be chosen based on the desired computation time or the precision of the results)
- The number of iterations : 50
- The inertia weight w : 0.8
- The cognitive coefficient c1 : >0
- The social coefficient c2 : 0.225

The racing line

After all these analyses, we've found the optimal parameters for the PSO algorithm. We can now proceed to compute the racing lines.

Here is one of the computed racing lines:

This concludes my TIPE project. I hope you enjoyed reading it as much as I enjoyed working on it.

If you have any questions or would like to know more about my project, feel free to contact me via email or social media.

References

• YING XIONG : Racing Line Optimization : Massachusetts Institute of Technology (2010)
• H. BUTLER, M. DALY, A. DOYLE, S. GILLIES, S. HAGEN, T. SCHAUB : The GeoJSON Format
• INTERNATIONAL CIVIL AVIATION ORGANIZATION : World Geodetic System 1984 (WGS-84) Manual
• J. KENNEDY, R.C. EBERHART : Particle swarm optimization : Proceedings of ICNN'95 - International Conference on Neural Networks, pp. 1942-1948 vol.4, (1995), Perth, Australia, doi: 10.1109/ICNN.1995.4889
• A. NICKABADI, M. MEHDI EBADZADEH, R. SAFABAKHSH : A novel particle swarm optimization algorithm with adaptive inertia weight : Applied Soft Computing, Volume 11, Issue 4, 2011, Pages 3658-3670, ISSN 1568-4946
• A.C. BANSAL, P.K. SINGH, N.R. PA : Evolutionary and Swarm Intelligence Algorithms : Springer-Nature, 2018, ISBN: 978-3-319-91341-4
• A. GRAY, E. ABBENA ,S. SALAMON : Modern Differential Geometry of Curves and Surfaces with Mathematica Third Edition : Chapman & Hall/CRC, 2006, ISBN: 1584884487