A polynomial regression model for predicting knuckleball movements in soccer free-kick

This paper develops a polynomial regression model with physical constraints to comprehensively analyze the complex motion of knuckleballs in soccer free-kick. The predictions match accurately with the ball trajectories extracted from recorded soccer matches. The model uses a polynomial function to account for the ball’s projectile-like motion. It further incorporates a critical physical constraint that is related to the aerodynamic drag force on the ball. The regression model is implemented using a relaxed iterative method. The lsqlin solver of MATLAB is used in each iteration. The model is verified through a prominent case study – Cristiano Ronaldo’s representative knuckleball free-kick executed in April 2012. The ball trajectory is meticulously derived from image processing of video footage. The paper examines the ball position, velocity, and drag coefficient over the entire flight of the ball. Notably, a significant increase in the drag coefficient is observed during the flight, which is a critical feature of the knuckling effect. The study extensively explores the influence of physical constraints and various model parameters to ensure the regression model’s credibility, shedding light on the intricate dynamics of knuckleball free-kicks. With its simplicity, efficiency, and effectiveness, this polynomial regression model can be a powerful tool for studying knuckleball free-kicks in real time and thus improving players’ techniques in training and in the match. Besides, it can be used to create authentic flight paths of the ball in computer games.