The golf swing is one of the most difficult and complex sport motions (Dillman & Lange, 1994). Much research has been applied to the biomechanical analysis of the golf swing in an effort to understand the complex mechanics of the motion to provide a basis for improving performance (Hume et al, 2005).
A natural extension to the basic biomechanical analysis of golf swing mechanics are efforts to identify modifications that could potentially improve the swing beyond its current capabilities. Previous studies have shown that only a small percentage (20.2-26.8%) of the energy developed by the body during the downswing is transferred to the club (Nesbit and Serrano, 2005). This finding suggests that there may be an opportunity to increase this energy transfer and the resulting club head velocity, through some modification of the swing. Lampsa (1975) and later Sharp (2009) applied optimal control theory to double-pendulum models of the swing to identify joint torque profiles that maximized club head velocity for the drive shot. Both found that it was theoretically possible to increase club head velocity through modification of the arm and wrist torque profiles without exceeding the maximum torque capabilities of the subjects. In the case of Lampsa, it was revealed that the required power output of the subject was exceeded suggesting that power limiting should be considered during the search for optimal swing torque profiles (Kaneko and Sato, 1994). Sharp (2009) also presented the development of a triple-pendulum model, where shoulder, elbow, and wrist torque profiles were identified which minimized the difference between model-predicted wrist and club head positions and those obtained from subject data. The torques were then manipulated to maximize club head speed at impact. White (2006) utilized a torque driven double-pendulum model to determine means to improve the energy transfer efficiency from the arms to the club through modifications of the wrist-cock angle, release delay, and wrist torque magnitude. Ultimately, these approaches are limited by the accuracy of the model employed, and logically, cannot account for un-modeled affects. Moreover, it is difficult to provide an assessment of joint torques outside of a motion capture laboratory, thus limiting the ability to translate these results directly to the golfer and the coach. An optimization methodology that focuses primarily on the manipulation of swing trajectories (as opposed to joint torques) may provide a more visual and thus practical means towards helping the golfer improve their swing velocity.
When discussing the kinematics of the golf swing, it is natural to focus on the club head, as club head speed, direction and orientation at impact ultimate dictate the success of a shot (Jorgensen, 1999). However, the only control the golfer is able to influence over the club head is derived from the linear and angular trajectories imposed at the grip. Specifically, the path of the hands, also referred to as the hub path, is the point where the summation of the forces, torques, energy, and momentum developed by the golfer through the various joint and body movements are ultimately transferred to the club. The subtle non-circular nature of the hub path (Figure 1) has been recognized since the early days of golf biomechanical study (Cochran and Stobbs, 1969; Williams, 1966) however, its specific role in the golf swing has been ignored due to the popularity of the double-pendulum models for analyzing swing mechanics which removed this swing characteristic. Recent studies have determined that the non-circular nature of the hub path is a fundamental, yet subject dependent characteristic of the golf swing (Miura 2001; Nesbit and McGinnis, 2009), and that a reduction in radius of curvature nearing impact is indicative of skill (Nesbit 2005, Miura 2001). In addition to the trajectory of the hub path, the golfer is also responsible for controlling the three distinct angular motions of the club throughout the swing (Nesbit, 2005). Specifically, the manner in which the golfer controls the swing plane component (alpha component) of the angular motions has an important effect on the golfer/club energy transfers, and ultimately the club head velocity at impact (Jorgensen, 1970; Nesbit, 2005; Pickering and Vickers, 1999; Sprigings and Mackenzie, 2002; White, 2006).
As evidenced from the above discussions, the hub path and swing angular trajectories are important contributors to generating and maximizing club head velocity at impact. Therefore, the primary objective of this optimization study is to identify golfer-specific hub path trajectories (linear position and derivatives) and swing angular trajectories (angular position and derivatives) which maximize the club head velocity at impact while constraining the golfer kinetic inputs (force, torque, work, and power) within the empirical limits for each golfer. An important aspect of this study is to avoid the same model-based simplifications that limited previous optimization studies. A secondary objective of the optimization is to determine the most efficient hub paths and swing trajectories that minimize a specific kinetic input while maintaining the original club head velocity at impact. A possible outcome of the secondary objective is to identify particular kinematic actions that suggest specific kinetic weaknesses. Such information may prove useful for visually identifying limiting factors in a golfer's ability to generate club head velocity, and could provide insight into possible methods of improvement.
Subjects and testing protocol
Four amateur golfers, three males and the one female had their golf swings analyzed for this study. All subjects are right-handed and their relevant data are given in Table 1. A diversity of skill levels and swing styles was the criteria for selecting these subjects in an attempt to yield a range of results (Nesbit and Serrano, 2005; Nesbit, 2005). Stylistically, the male scratch and male 5H subjects had aggressive, powerful, and quick swings, whereas the male 13H and female 18H subjects had smoother, longer, and slower swings. All subjects used the same club (driver of length = 1.092 m; mass = 0.382 kg; cg location from top of club = 0.661 m; ICG = 0.07104 kg.[m.sup.2]). All subjects were informed of the purposes of the study, and gave written consent for the following testing procedures, and the use of their data for research purposes, in accordance with local IRB requirements. A rigid triad of passive reflective markers was attached to the club near the bottom of the grip. The three-dimensional paths of these markers were tracked at 200 Hz using an 8-camera motion capture system (Motion Analysis Corporation, Santa Rosa, CA, USA). The system was calibrated until the combined 3D residual for all cameras was under 1.00 mm (Test/retest of static marker locations varied by less than 0.20 mm for a given calibration) prior to testing. Subjects were asked to execute a series of swings that included hitting a ball into a net after being advised to swing the club in a manner similar to hitting a driver in a competitive situation where distance and accuracy were both important. The subjects were instructed to practice swinging the club as many times as necessary until they became comfortable with the testing situation and felt they could swing "normally" and consistently. Subsequently, a minimum of eight swings from each subject were recorded and tracked then presented to the subjects for their review. It was found that the club head velocities were consistent among the acceptable trials within a maximum range of 5% for all subjects. The subjects each selected what they considered to be representative swings in terms of club head velocity, impact feel, partial flight of the ball, and overall visual assessment of the motion capture data. One of the self-selected swings from each subject was then analyzed for this study. This manner of conducting trials and selecting swings for subsequent analyses is consistent with previous studies (Nesbit and Serrano, 2005; Nesbit, 2005; Nesbit and McGinnis, 2009).
Forward kinematics model
The free-body-diagram of the golf club model is shown in Figure 2. The club model with representative mass and inertia properties, constrained the swing to one nonmoving vertical plane (Coleman and Anderson, 2007), ignored rotations about the club shaft, and treated the shaft as rigid. These simplifications are consistent with many biomechanical models of the golf swing (Budney and Bellow, 1979; 1982; Cochran and Stobbs, 1969; Jorgensen, 1970; Lampsa, 1975; Neal and Wilson, 1985; Vaughn, 1981; Williams, 1966) with the exception that this model did not constrain the hub path to follow a constant radius circular arc. This model included the primary kinematic and kinetic parameters responsible for affecting club head velocity, and the golfer/club interacting forces, torques, and energy transfers (Nesbit, 2005).
The X-Y coordinate system illustrated in Figure 2 is in the plane of the swing and fixed to the ground (global coordinate system). The position of the grip along the hub path, relative to the global coordinate system, is described by the radial coordinate R and the transverse coordinate [theta].
The following scalar equations of motion were developed from Figure 2:
[F.sub.X] = [MA.sub.GX] (1)
[F.sub.Y] = Mg = [MA.sub.GY] (2)
T + [F.sub.X][L.sub.G] sin [gamma] - [F.sub.Y][L.sub.G] cos [gamma] = [I.sub.G] [??] (3)
where [F.sub.X] and [F.sub.Y] are the X and Y components of the applied linear force, M is the mass of the club, [A.sub.GX] and [A.sub.GY] are the X and Y components of the acceleration of the club mass center (G), g is the acceleration of gravity, T is the applied swing torque, [L.sub.G] is the distance from the grip point to the club mass center, [I.sub.G] is moment of inertia of the club about the mass center, and [gamma], [??]...