The aim of many projectile-based sports, such as hockey, soccer, baseball, or golf, is to direct a projectile, such as a ball or puck to a certain target, such as a net or a hole. For many projectile-based sports, the projectile is struck (with a club, bat or even a player's foot) and directed to the target. The more that a player can consistently direct the projectile to the target, the better that player will perform. For this reason, many players would like to practice their game in the comfort of their own home. For the purposes of illustration of the inventions disclosed herein, the game of golf will serve as the focus of discussions. However, it will become clear to the reader that other projectile-based sports are applicable to the present invention.
A system for accurately predicting the flight and roll of a golf ball is of immense value to the golfer, whether beginner or advanced. A number of configurations of golf practice mats have been produced and demonstrated, each with accuracy limitations in predicting the flight of a golf ball, for reasons that will be elaborated below.
The collision between club and ball is a violent event, of duration less than half a millisecond, in which forces of up to 10 kN are imparted by the club-face to the ball. During the impact, the ball deforms significantly, as much as a centimeter, and slides and rolls along the club face, beginning to spin, and finally recoils to depart the club, ending the impact event.
Each individual golf ball's physical properties and the time history of the forces acting upon the ball during the impact completely determine the flight of the ball through the air and its bounce and roll upon landing (assuming a standard atmosphere and ignoring the effects of wind and small-scale random phenomena). Yet, the physics of the impact event, and what happens to the golf ball during it, remain largely intractable due to the large number of variables that must be modeled and their numerical spreads. Empirical collision studies of golf balls against fixed barriers have shown that even the coefficient of restitution of golf balls is a complex, non-linear function of impact speed, angle and force. Ball composition adds another significant variable to the modeling task: while cover softness (the thin dimpled layer surrounding the core) affects the spin-rate acquired by the ball during the collision, the ball's core affects the linear momentum acquired by the ball. When all these variables are put together, small variations in the impact event produce large variations in its outcome. This is what makes the game of golf so hard to master. The ball's velocity along with spin rate, spin axis and even dimple pattern creates the lifting force, which determines the ball's trajectory.
Upon landing, the ball's forward momentum, which acts to keep the ball rolling, is opposed by the ball's backspin and the coefficient of friction with the course surface (again determined by the cover's stickiness and the dimple pattern). The bounce and roll distance, therefore, is again a function of the ball's construction.
A diagram of the forces at work through impact is shown in FIG. 6. The line S represents the shaft of club 100, and the angle it makes with the face of the club, F X F′, is called loft. If the axes, X, Y, and Z, were centered on ball 110 at the first instance of impact, with the Z axis representing the club-head velocity vector at that instant, the ball would be struck with a “lofted, slightly open” club face. The term “lofted” refers to the effective loft angle being greater than the loft of the club, due to the angle between F and Z being obtuse, and the term “slightly open” refers to the fact that the club face is aimed slightly to the right of Z (note that the normal to the face, A, is pointing slightly to the right of Z).
At the moment of contact, with the force vector being parallel to Z, the component of the force normal to the club face, A, begins to compress the ball, while the component of the force tangential to the club face causes the ball to begin sliding and rolling over the club face. The effective loft of the club influences the energy distribution in the departing ball between the linear kinetic energy and the rotational energy it acquires. The lie of the ball also plays a large roll in determining the energy distribution: a ball that is squeezed between the club and the ground at impact will tend to acquire a greater amount of spin, at the cost of linear velocity, while the opposite will be true for the teed ball.
Although FIG. 6 depicts the moment of contact, the collision between a golf club and ball lasts up to about 0.5 seconds, during which time the club and ball stay in contact, traveling together for an inch or less. Through impact, the forces on the ball change in magnitude and direction, thus no single moment in time can be predictive of the ball's flight and roll.
The departing ball's linear and spin kinetic energy components are a complex function of the swinging club's parameters—mass, velocity, deceleration through impact, effective loft angle, etc.—and the ball's parameters—the coefficient of restitution of the ball, which, along with the ball's mass, determines the amount of linear energy transfer, and the effective coefficient of friction which determines the amount of spin energy of the departing ball.
If the spin axis is tilted away from the horizontal, the lift experienced by the ball will also cause it to veer to the left or to the right, causing the weekend golfer's much-dreaded hook and slice, but also the coveted fade and draw that the professional golfer is able to command.
At the moment of landing, the golf ball will have a downward velocity component, a forward velocity component and residual spin. These dynamical parameters of the landing ball, together with its coefficient of restitution, rolling resistance and friction determine its bounce and roll, subject to the topography of the terrain and the type of surface it lands on.
Several means have been proposed in the art for estimating a golf ball's flight based on the acquisition of post-impact ball measurements. In a tethered ball golf practice device, the ball is attached to the mat by some secure means and yet is able to move freely within a constrained volume so that the feel of hitting the captive ball is similar to that of a free ball.
Various arrangements of tethered golf balls for golf practice mats have been proposed in the literature, such as the hung ball, the cantilevered ball (side attachment), and the teed ball. Designers have in some cases attempted to reduce the effect of the tether on the measurements, but have otherwise been unable to compensate for it. In the case of a free ball golf practice device, measurement or estimation of the ball's spin rate and spin axis, a task that is essential to the simulation's accuracy, is prone to large measurement errors because of the difficulty involved.
High-speed stroboscopic photography of specially marked balls is also limited to measuring spin about a single axis. Yet, three-dimensional spin is highly determinative of the golf ball's flight and roll, since spin can generate a lifting force that is greater than the force of gravity; hence errors in estimating the spin rate and axis cause large discrepancies between the predicted and actual ball behavior.
Accordingly, there is a need for an improved system of golf simulation practice devices which compensates for the difficulties encountered with the prior art devices.