Statistically modelling tennis racket impacts with six degrees of freedom

The International Tennis Federation (ITF) is responsible for protecting the nature of tennis. The ITF uses computational models to predict how trends in equipment parameters could affect the games future. The current ball-racket impact model is limited to non-spinning, on-axis, normal ball impact simulations. The aim of this project was to develop a model of oblique, spinning, on- and off-axis ball-racket impacts. Large scale test data (n > 1000) was collected using an impact rig and calibrated high-speed cameras. Impacts for a range of realistic velocities, spin rates and impact locations were collected, measured using automated image processing algorithms to digitise ball centroids. An established spin measurement method was improved to correct for perspective errors associated with the proximity of the cameras to the test volume. The automated algorithms were validated with experimental data and manual methods. Multi-variate polynomial models to predict the lateral and vertical components of rebound velocities and rebound spin rate were trained and validated using a curve fitting toolbox and ‘n-fold and leave one out cross-validation’ method. Second order models best fit the training data, with the low predictive errors. Root-mean-squared errors were calculated using a test dataset, independent of the training data. These were 0.57 m·s-1 for the lateral rebound velocity model, 0.48 m·s-1 for the vertical rebound velocity model and 30.5 rad·s-1 for the rebound spin rate model. Variance was partially explained by experimentally established inherent variability of the ball and stringbed. Model output confidence was established by simulating small changes in model inputs. The simulated lateral and vertical components of rebound velocity, but not the simulated spin rate, were an order of magnitude greater than the measurement precision. The new models were combined with ball aerodynamics and ball-to-surface impact models to simulate tennis court trajectories for oblique, spinning, on- and off-axis ball-racket impacts. Increasing stringbed stiffness or the lateral offset of impact location were found to decrease rebound velocity and increase rebound angle – markedly so for a 60 mm lateral offset. Increasing lateral offset also increased the rebound spin rate.