Redundant inertial sensor setup and constrained optimization for accurate golf putting orientation and trajectory estimation

The putting stroke in golf is a crucial component of the game, accounting for approximately 45% of the total strokes in a match. While professional players show 96% accuracy from 1 meter, performance drops dramatically to 65% from 2 meters. To maintain high accuracy, training is essential with precise trajectory feedback provided by specialized tools. Thus, the ability to maintain this level of precision lies in the accurate reconstruction of the putter's trajectory. Although different analysis systems exist, their main limitations are the impossibility of performing analysis in the open field and the high cost of equipment. In this context, inertial sensors (IMUs) which integrate an accelerometer and a gyroscope prove to be a feasible solution thanks to their relatively low cost and availability in real scenarios. The single IMU-based trajectory is typically estimated by double integrating the accelerometer signals after having removed the gravity vector projection obtained through a sensor-fusion filter for orientation computation. However, trajectories obtained using this non-optimized pipeline are not accurate enough due to the non-optimal setting of the filter parameter(s) and the rapid accumulation of measurement errors over time. In fact, position drifts can easily reach errors up to 0.5 m after a very few seconds. The goal of this thesis was the accurate reconstruction of the trajectory of the putter equipped with a redundant multi-IMU configuration through a constrained optimization framework. The accelerometer and gyroscope measurement errors were modeled as additive terms which are considered constant within the single trial duration. The objective function was designed to find the optimal values of the latter and the parameter values of the sensor fusion filter by minimizing the orientation of the IMUs rigidly attached to the putter throughout the trial. The following constraints were also based on the specific knowledge of the task to avoid drifts in the velocity and position components of each IMU. During the static phase before and after the stroke in which the club was stationary: the norm of the accelerometer signal was set to be close to the gravity acceleration (condition 1), the mean of the acceleration and velocity was set to be null (conditions 2 and 3), and the maximum velocity value was set to be close to zero (condition 4). During the swing phase, the position of the putter head must remain non-negative, with a maximum height of 0.1 m. Due to the irregular shape of the putter, a rigid model was developed using the Denavit-Hartenberg convention to set the IMUs in a common and meaningful reference system. Model parameters were fitted through a static acquisition using the stereophotogrammetric system (SP). The putter was equipped with two IMUs (Xsens-MTw) placed on the shaft and head. Then, 22 stroke trials were recorded in a controlled environment. The putter reference orientation and trajectory were acquired by the SP. The accuracy during a putting stroke was evaluated in terms of root mean square error (RMSE) values for both position and orientation during the swing phase only. The optimal parameters allowed to achieve an average RMSE of 1.8 deg in orientation and 0.08 m in position, in comparison with the “non-optimized” where the errors were 1.7 deg in orientation and 0.15 m in position. These results seem to suggest the potentiality of a redundant IMU sensor configuration, marking a significant step towards integrating IMU technology into putting analysis.