Simulation and Modeling of Free Kicks in Football Games and Analysis on Assisted Training

The aim of this study was to develop a new method for the determination of lift on spinning baseballs. Inertial trajectories of (a) ball surface markers during the first metre of flight and (b) the centre of mass trajectory near home-plate were measured in a pitch using high-speed video. A theoretical model was developed, incorporating aerodynamic Magnus-Robins lift, drag and cross forces, which predicts the centre of mass and marker trajectories. Parameters including initial conditions and aerodynamic coefficients were estimated iteratively by minimizing the error between predicted and measured trajectories. We compare the resulting lift coefficients and spin parameter values with those of previous studies. Lift on four-seam pitches can be as much as three times that of two-seam pitches, although this disparity is reduced for spin parameters greater than 0.4.

This study involved a theoretical and an experimental investigation of the direct free kick in soccer. Our aim was to develop a mathematical model of the ball's flight incorporating aerodynamic lift and drag forces to explore this important 'set-play'. Trajectories derived from the model have been compared with those obtained from detailed video analysis of experimental kicks. Representative values for the drag and lift coefficients have been obtained, together with the implied orientation of the ball's spin axis in flight. The drag coefficient varied from 0.25 to 0.30 and the lift coefficient from 0.23 to 0.29. These values, used with a simple model of a defensive wall, have enabled free kicks to be simulated under realistic conditions, typical of match-play. The results reveal how carefully attackers must engineer the dynamics of a successful kick. For a central free kick some 18.3 m (20 yards) from goal with a conventional wall, and initial speed of 25 m x s(-1), the ball's initial elevation must be constrained between 16.5 degrees and 17.5 degrees and the ball kicked with almost perfect sidespin.