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Conclusions

A golf stroke by an accomplished golfer is usually a thing of beauty. There are a large number of books on golf, most of them are written by golfers who have owned the perfect skills of golf swing. These works are mainly concerned with the swing skill instructions that leave the reader with little understanding of what actually happens during a golf swing. Therefore, the mathematical modelling of the golf swing has been studied by many golf researchers over a period of more than 30 years in order to search for the “truth” that occurs during the golf swing and attempt to help golfers accomplish the perfect swings according to that “truth”.

The first systematic research into the mathematical and biomechanical aspects of golf swing techniques was carried out by the Royal Society Golf Group. In their work, the golf swing was modelled as a planar two-link system named the “double pendulum”. The validity of this model has been confirmed by many researchers through actual swing experiments. That is, the known inputs for the model (input torques of shoulder and wrist) would result in the outcome that is very much the same as that from the real golfers’ swings. For example, Cochran & Stobbs reported that elite golfers showed marked similarities to the “double pendulum” and that this model was an excellent mathematical analogue of the golf swing. Jorgensen used the assumed constant shoulder torque to drive the model, and the club head speed was found to be consistent with that from a professional’s swing. Some researchers such as Campbell & Reid [8] and Sprigings & Neal proposed a three-segment planar model in which the role of the torso rotation was considered, although the validity of this model has not been confirmed by the actual golf swings.

We should note that the above studies have neglected the elasticity of the golf club shaft. Some researchers considered that the vibration of the club shaft during the golf swing was closely related to the golfer’s motion, and the displacement of shaft vibration at impact greatly affected the trajectory of a golf ball. The shaft thus should be matched to the golfer’s swing speed and hand action. On the contrary, some researchers such as Milne & Davis and Brylawski believed that the shaft flexibility was not dynamically significant during the golf swing.

It has been noted that the club head speed is a function of the sequential segment velocities of the chain link that makes up the golf swing. The wrist joint would thus be expected to play an important role in the golf swing. Over the years, golf researchers have debated what kind of wrist torque can increase the final clubhead speed. In most of their studies, the golf ball was kept constant and the way the wrist action alters the clubhead speed at impact was not given explicitly. In our study, two dynamic models of golf downswing have been established to examine the combined role of the wrist action and the ball position in the improvement of the final horizontal club head speed. One model is the “double pendulum”; and the other is a new model in consideration of the bending flexibility and centrifugal stiffening of the golf shaft, which has been verified by the actual swing experiments. The results from both models clearly show that the positive wrist torque applied at the latter stage of the golf downswing would provide an advantage in the club head speed at impact. From an energy based analysis, it is also found that the application of the positive wrist torque would increase two factors that facilitate the club head speed:

1. the workproduced by a golfer; and
2. the energy transference ratio, and thus the increased club head speed at impact is achieved.

It is noted that a large amount of research has been devoted to improve golfers’ swing skills and golf club performance for decades. Among these studies, golf swing robots have formed a large body of literature. In these works, professional golfers’ swing motions were expected to be emulated by robots and the evaluation of golf club performance to be replaced by robots instead of golfers.

Most of the golf swing robots on the market have two or three joints with completely interrelated motion. This correlation only allows the users of these robots to specify the initial posture and swing velocity. The subtle adjustments during swing motion due to the various features of individual golf clubs and golfers’ arms are not possible. The significance of the dynamic interactions between a golfer’s arm and a golf club thus has been noted by some researchers. The influence of the dynamic interference force due to the different golf club characteristics on the swing motion has been investigated. However, the effect of humans’ arm masses upon the swing motion has not been studied. Therefore, we used a two-dimensional double pendulum model of the golf swing with the normalized parameters to investigate this problem, and found that the arm mass of humans is an important parameter to affect the swing performance.

Considering the vital of the dynamic interactions between a golfer’s arm and a golf club, an impedance control method based on velocity instruction is proposed for a golf swing robot to emulate different-arm-mass golfers. A model of a golfer’s swing considering the bending flexibility and centrifugal stiffening of the golf shaft is given by using the Euler-Lagrange principle and assumed modes method. A prototype of golf swing robot with one actuated joint and one passive joint is developed using the impedance control method. The comparison of swing motion is carried out between golfers and the golf swing robot. The results demonstrate that our golf swing robot can simulate the swing motions of different-arm-mass golfers.

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