Analysis of the 'Sonic Golf System'

The 'Sonic Golf System' invented by Dr Grober is an original and rather intriguing aid not just for improving one's golf swing but also as a potential research tool for the serious golfer/instructor.  A salient feature of the system is that it is based on learning using audio biofeedback. Some 30/40 years ago biofeedback was heralded as a very promising learning tool but it never got really much outside the medical type of applications.

It is a deceptively simple instrument and yet it is quite a sophisticated tool. I will be mainly doing an analysis of the science behind the instrument and will explain in detail how some characteristics of the system, if made available commercially, could make the system even more attractive both as a learning tool for any golfer of any skill level but also as a most valuable research tool/aid for any serious golf instructor.

The instrument uses two miniature accelerometers introduced inside the shaft, a fixed distance apart. One inside the shaft at the but end, the other some fixed distance down the shaft. This is shown schematically in Fig1. The sensitive measuring axis of each accelerometer is aligned with the axis of the shaft, hence only the longitudinal acceleration along the shaft is measured.

The instrument uses a procedure often employed in precision transducers in a wide field of applications, i.e., employing a differential measuring scheme. The essential feature of such approach is to generate and subsequently subtract two signals, but with only one of them containing the information of interest. It allows, when done properly, to eliminate undesired influences due to a variety of other variables not being of interest.  

Let's have a closer look how this differential approach operates in this particular case. An accelerometer can't distinguish between gravity force and inertial accelerations. Hence having only one accelerometer would give an output signal of ±1 g when its sensitive measuring axis is aligned vertically and reduces to 0 g when positioned horizontally. However the use of two identical accelerometers and subtracting the two signals allows to completely eliminate this unwanted output signal due to gravity. See Fig2.

The aim of the instrument is to measure uniquely the acceleration due to angular motion of the club and hence not to measure any caused by linear motion of club as indicated in Fig3. Linear motion of the club in any direction results in identical acceleration signals as measured by the two accelerometers. Hence, just as for the gravity, by taking the difference of the two, the linear acceleration of the club is eliminated from the output signal.

The signals produced by the accelerometers, when there is angular motion, have different amplitudes and hence, when taking the difference, there remains an useful output signal proportional to the acceleration along the shaft. Moreover, it doesn't matter where exactly the axis of rotation of the angular motion is located along the shaft, the output signal remains the same, once more showing the advantage of a differential measuring scheme. See Fig4.

Since the golfer uses the 'sonic golf system' during a golf swing we will use for the remaining of the analysis the pendulum model for the golfswing. The pendulum model is shown in Fig5.

As shown in Fig5 the pendulum swing is characterized by the two angles φ and θ. One can derive for the acceleration measured by the two accelerometers :

The output signal is proportional to the difference between (1) and (2):

Notice from (1) and (2) that these two signals also contain the longitudinal and transverse acceleration component of the inner segment acceleration, as projected onto the club shaft at the joint. Again due to the clever differential arrangement their contributions completely disappear form the output signal (3). The pitch and volume of the digital audio signal transferred to wireless headphones that the golfer wears is hence solely related to the angular motion of the club shaft. Everything else, as we have shown above, is neatly eliminated.

Inverse Dynamics

The 'Sonic Golf System' is commercially available only with an audio output signal. However with some additions it could find another interesting niche of applications such as performing inverse dynamics.

Definition of Inverse Dynamics (Wikipedia):


'Inverse dynamics uses link-segment models to represent the mechanical behavior of connected pendulums, or more concretely, the limbs of humans, animals or robots, where given the kinematic representation of movement, inverse dynamics derives the kinetics responsible for that movement. In practice, from observations of the motion (of limbs), inverse dynamics is used to compute the associated moments (joint torques) that lead to that movement, under a special set of assumptions.'

'Equations mathematically model the behavior of a limb in terms of a knowledge domain-independent, link-segment model, such as an idealized skeleton with fixed-length limbs and perfect pivot joints. From these equations, inverse dynamics derives the torque (moment) level at each joint based on the movement of the attached limb or limbs affected by the joint. This process used to derive the joint moments is known as inverse dynamics because it reverses the forward dynamics equations of motion, the set of differential equations which yield the position and angle trajectories of the idealized skeleton's limb from the accelerations and forces applied.'


Let's see what has to done to be able to do 'inverse dynamics' with the 'sonic golf system'.

The differential equations relating the longitudinal accelerations at A and B to the motion of the pendulum are:

Whereas the output signal is proportional to the difference and given by:

The set of differential equations given by (1) and (2) can be solved for θ[t] and φ[t], if the longitudinal accelerations l_A''[t] and l_B''[t] measured at A and B are known. However these are the output signals of the two accelerometers which are measured with the 'Sonic Golf System', and hence could be made available. (Note: (1) and (3) or (2) and (3) would equally do).

Let's assume that these two longitudinal accelerations l_A''[t] and l_B''[t] at A and B have been measured and have a time history as shown above in Figs 6a/b. The solution obtained for the differential equations (1) and (2), using these two acceleration signals, is shown graphically in Figs 7a,b,c.

Knowing the time history of the angles θ[t] and φ[t],as shown in Fig7a, one can henceforth activate the motion of the two segments of the double pendulum model. It could be made a nice real time dynamic display on a lap top of which Fig8 gives a more 'static' presentation.

We are left with the task to determine the torques Q1 ad Q2 acting at the two joints in the pendulum model. To do so we have to derive the differential equations governing the two segments of the double pendulum golf model, which are given by (4) and (5).

Since we have derived θ[t] and φ[t] from solving (1) and (2), we can substitute these together with the derivatives into (4) and (5) and obtain the torques Q1 and Q2, shown in Figs 9a and Fig9b.

Conclusions

The 'Sonic Golf System' appears to be an interesting aid for improving one's golfswing. Based on audio biofeedback it constitutes an holistic approach operating on the totality of the swing motion using an audio feedback signal.

However with some additional hardware and software it could be used as a an useful research and development tool, being able to generate a real time presentation of both motion and torques, presented as a double pendulum.


mandrin