Abstract:
An apparatus for indicating the proper pendulum putting stroke to a golfer with An elongated display device that is positioned on a floor parallel to the stroke path of the practicing golfer, A display device containing multiple microcontroller module sections, each containing visual indicators spaced uniformly along the length of the sections, Each section being responsive to commands from a peripheral computing device to turn on a specific visual indicator, a peripheral computing device such as a personal computer running a computer program which calculates the putting stroke based on desired distance, green speed, and the golfer&#39;s personal rhythm rate in beats per minute, A peripheral computing device that after calculating the putting stroke time-position data, commands the display device to turn on and off consecutive visual indicators at a predefined time in such a way as to display a putting stroke path, and show the golfer the proper time-space dynamics to strike a golf ball a desired distance.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of provisional application Ser. No. 60/750,943 filed on Dec. 16, 2005. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not Applicable 
     DESCRIPTION OF ATTACHED APPENDIX 
     Not Applicable 
     BACKGROUND OF THE INVENTION 
     This invention relates generally to the field of golf training aids and more specifically to an apparatus for indicating the proper pendulum putting stroke to a golfer. It has been known for quite some time since the advent of modern era golf greens, that the pendulum stroke is the preferred putting stroke. Before greens were fast and smooth, a wristy, hands dominated putting stroke was used by most professional golfers. With the development of smooth golf greens through agricultural grass research that has produced grass strains with fine blades that allow shorter mower cuts, a putting stroke which locks the wrists and uses the large muscles in the arms and shoulders has been adopted by the vast majority of golf professionals. This arms oriented stroke has been named the pendulum putting stroke. Almost every expert in golf instruction recommends the pendulum stroke yet, no one can actually show a golfer exactly what it is. Metronomes are usually suggested as a means of teaching a golfer how to keep a consistent back and forth rhythm. A pendulum putting stroke takes the same time to travel from the top of the backswing though impact to the top of the followthrough regardless of the putting stroke length, so golfers practice short strokes, medium strokes, and long strokes while listening to a metronome&#39;s tick tock sound. The top of the backswing is reached at the tick and the downswing is completed at the tock. Looking to the pendulum as an analog to the putting stroke, leading golf instructors see the pendulum&#39;s smooth acceleration and constant timing as an ideal device to emulate in putting. 
     Prior tempo training aides consist of devices that provide audio, visual, or vibration signals that indicate various points in the golf swing. Most indicate the beginning, backswing, and downswing phases of a full golf swing or putting stroke. Metronome devices indicate the timing between the two endpoints of the golf stroke, the top of the backswing to the end of the downswing. For a putting stroke, the metronome tick sound corresponds to the top of the backswing while the tock sound corresponds to the end of the downswing. Five U.S. patents disclose audible sound producing metronomes either worn by the golfer, placed near the golfer or placed inside the golf club handle including U.S. Pat. No. 5,743,807 to Bendo, U.S. Pat. No. 5,082,281 to Berghofer, U.S. Pat. No. 5,040,790 to Anthes, U.S. Pat. No. 3,808,707 to Fink, and U.S. Pat. No. 6,517,352 to Smith. One approach, U.S. Pat. No. 5,558,519 to Sabowitz, provides an audio or visual indication for the start of the backswing, the half swing point, the top of the backswing, and the impact point on the downswing. U.S. Pat. No. 5,423,538 describes a putting trainer that has a microswitch onto which the toe of a putter contacts in order to detect the start of the backswing when the putter is moved and the microswitch opens. Adjustment knobs set the time duration of the backswing and the downswing. Control circuitry produces a sensible signal, which is described by the inventor as a visual or audible signal detected by the golfer, at the top of the backswing and at the end of the downswing. Still another approach, U.S. Pat. Application Publication No. US 2004/0214651 A1 to Park provides a tone or vibration to indicate the start and duration of the backswing, followed by a tone or vibration to indicate the start and duration of the downswing. A series of Leds arranged in a circular fashion on the front of the display indicate for the purposes of adjustment setup, the timing of the backswing and downswing that will be executed when placed in the audio or vibration mode. 
     Prior putting tempo training aids suffer several disadvantages. Most of the training aids relate to the full golf swing and not specifically to the putting stroke. If applied to the putting stroke, the simpler training aids that output an audible or visual indicator at the top of the backswing and the end of the downswing provide only two data points in the overall putting stroke dynamic. These aids fail to indicate to the golfer where the putter should be between the two points. U.S. Patent Application Publication No. US 2004/0214651 A1 provides an audio beep or buzzer for the backswing start, top of the swing pause, and start of the downswing. A series of LEDs, in LED mode, placed in a small circle on the outside of the housing depicts  17  positions in a full swing pattern that flash on in sequence to indicate the backswing and downswing timing selected by the golfer prior to actual use. However, the LEDs are only viewed for device setup and are not watched while making a full swing. The Led sequence&#39;s purpose is not to show the golfer a detailed time position guide to strike a ball a desired distance. Furthermore, a full swing depiction does not provide a golfer with a relevant putting stroke guide. Although these aids provide tempo signals that aid a golfer in building a repeatable stroke, they do not show the golfer how to execute a putting stroke that will roll a golf ball a desired distance. It is the goal of the present invention to indicate to the golfer through a series of visual indicators, the proper pendulum stroke dynamics to produce a desired distance putt on a green of known green speed and with a personal putting tempo in beats per minute. Using the present invention, a golfer enters the green speed, beats per minute, and the desired distance and by following the resulting LED sequence pattern with his/her putter, will learn the backswing and downswing dynamics to stroke a golf ball a desired distance. 
     BRIEF SUMMARY OF THE INVENTION 
     The primary object of the invention is to visually indicate to a golfer, the proper putting stroke time-position relationship for a given green speed, desired distance, and golfer&#39;s personal putting stroke frequency in beats per minute. 
     Another object of the invention is to show a golfer how to produce a “perfect pendulum” putting stroke. 
     Another object of the invention is to indicate the length of the backswing and downswing to produce a putt that rolls a desired distance. 
     Still another object of the invention is to allow a golfer to practice short, medium, and long putts in a small indoor space such as an office, hotel room, or den. 
     Other objects and advantages of the present invention will become apparent from the following descriptions, taken in connection with the accompanying drawings, wherein, by way of illustration and example, an embodiment of the present invention is disclosed. 
     In accordance with a preferred embodiment of the invention, there is disclosed an apparatus for indicating the proper pendulum putting stroke to a golfer comprising: An elongated display device that is positioned on a floor parallel to the stroke path of the practicing golfer, A display device containing multiple microcontroller modules, each containing multiple light emitting diodes spaced uniformly along the length of the sections, Each section being responsive to commands from a peripheral computing device to turn on a specific LED, a peripheral computing device such as a personal computer running a computer program which calculates the putting stroke based on desired distance, green speed, and the golfer&#39;s personal putting tempo rate in beats per minute, A peripheral computing device that after calculating the putting stroke time-position data, commands the display device to turn on and off consecutive LEDs at a predefined time in such a way as to display a putting stroke path that will yield a putt length equal to the target distance that the golfer entered on the graphical user interface of the peripheral computing device. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings constitute a part of this specification and include exemplary embodiments to the invention, which may be embodied in various forms. It is to be understood that in some instances various aspects of the invention may be shown exaggerated or enlarged to facilitate an understanding of the invention. 
         FIGS. 1-3  are perspective views of the invention illustrating how a golfer would use the Pendulum Putting Stroke Training Aid. 
         FIG. 4  is a block diagram schematic of a single microcontroller module and its connection to a peripheral computing device which in the present embodiment is a personal computer. The connection in the present embodiment is a universal serial bus link USB. 
         FIG. 5  is a block diagram showing four interconnected microcontroller modules in which the first microcontroller module is equipped with a USB interface that connects to a personal computer. 
         FIG. 6  is a perspective view of the top side of the first of four microcontroller modules showing the LEDs and module to module interconnecting pins. 
         FIG. 7  is a perspective view of the bottom side of the first of four microcontroller modules showing the microcontroller, and USB integrated circuits. 
         FIG. 8  is a perspective of the first and second microcontroller modules showing the three wire power and communication bus connections. 
         FIG. 9  is a drawing of the invention illustrating each of the 48 LEDs comprising a four microcontroller module Pendulum Putting Stroke Trainer. 
         FIG. 10  is a drawing of a putting stroke arc of a backswing and downswing shown above the Pendulum Putting Stroke Training Aid housing and its associated LEDs.  FIG. 10  is used to support the derivation of equations of the desired putterhead motion for the LEDs that guide the golfer&#39;s putting stroke. 
         FIG. 11  is a flow chart of the microcontroller software that executes in each of the four microcontroller modules. 
         FIGS. 12-14  are flow charts that describe the program flow of the personal computer software that controls the microcontroller modules. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Detailed descriptions of the preferred embodiment are provided herein. It is to be understood, however, that the present invention may be embodied in various forms. Therefore, specific details disclosed herein are not to be interpreted as limiting, but rather as a basis for the claims and as a representative basis for teaching one skilled in the art to employ the present invention in virtually any appropriately detailed system, structure or manner. 
     Turning first to  FIG. 1  there is shown a golfer  12  preparing to strike golf ball  14  with putterhead  17  of putter  16 . There is also shown in  FIG. 1 , a Pendulum Putting Stroke Training Aid housing  10  for indicating to golfer  12 , the proper time-position motion that the putterhead  17  should follow to strike golf ball  14  with sufficient energy to roll the ball  14  a desired length. Pendulum Putting Stroke Training Aid housing  10  is placed on the ground parallel and just outside the path of the golfer&#39;s stroke such that starting LED  30   x  is directly across from golf ball  14  at the golfer&#39;s address position. Peripheral control cable  28  connects to the USB port of personal computer  18 . In the Putting Stroke Mode, LED  30   x  turns on momentarily to indicate to golfer  12  that the putting stroke cycle will begin shortly. Golfer  12  begins to take putter  16  back away from golf ball  14  which is placed near LED  30   x  while watching Pendulum Putting Stroke Training Aid housing  10  in the vicinity of starting LED  30   x  for the indication of the beginning of the putting stroke. In this embodiment, there are a total of 48 LEDs ranging from LED  30   a  to  30   av  as shown in  FIG. 9 . Returning back to  FIG. 1 , each of the four microcomputer modules  20 ,  22 ,  24 ,  26  contain a total of twelve LEDs each. This provides one LED per inch. It should be apparent to those skilled in the art that the number of LEDs per inch as well as the total number of microcontroller modules could be increased or decreased. Prior to addressing golf ball  14  for the purpose of putting training, golfer  12  enters the desired target distance of the roll of golf ball  14 , the green speed, and the personal putting stroke tempo in beats per minute (bpm) into the graphical user interface software program running on personal computer  18 . Before the putting stroke begins, personal computer  18  calculates the time between the activation of LEDs. Referring to  FIG. 9 , during the putting stroke, personal computer  18  communicates LED activation commands to Pendulum Putting Stroke Training Aid housing  10  over the USB port through peripheral control cable  28  indicating to golfer  12 , the proper pendulum putting stroke by illuminating specific LEDs  30   a - 30   av  one at a time in precise sequence to match the time-position dynamics of a pendulum in motion as adjusted by the golfer&#39;s personal tempo. By following the activated LED  30   a - 30   av  position with putterhead  17  of putter  16 , the golfer  12  can execute a pendulum putting stroke that will yield a golf ball  14  roll distance equal to the target distance set by golfer  12  on personal computer  18 . Pendulum fulcrum  15  is the virtual base of the pendulum formed by golfer  12  and putter  16 .  FIG. 2  shows golfer  12  at the top of the backswing portion of an example putting stroke whereby the last LED in the stroke is LED  30   aj . In this example pendulum stroke, golfer  12  stops putter  16  backswing progress at LED  30   aj . Personal computer  18  then sends LED commands over the USB port through peripheral control cable  28  to Pendulum Putting Stroke Training Aid housing  10  to start the downswing portion of the putting stroke. Each successive LED is activated one at a time until the LED lined up with the initial starting position of the ball is reached at which time golfer strikes ball  14  with putterhead  17 . In  FIG. 3  the end of the example putting stroke is shown. Putter  16  and putterhead  17  is lined up with LED  30   m  which in this example, activates at the end of the putting stroke mode as commanded by personal computer  18  over the USB port through peripheral control cable  28  to indicate the end of the putting stroke arc and ball  14  is rolling past the end of the Pendulum Putting Stroke Training Aid housing  10 . 
     As shown in  FIG. 9 , in the preferred embodiment, Pendulum Putting Stroke Training Aid housing  10  comprises four microcontroller module sections  20 ,  22 ,  24 , and  26  each of which has 12 LEDs uniformly spaced 1 inch apart for a total of 48 LEDs spanning a 4 foot length. Each microcontroller module is connected to a common 3-wire bus and placed inside Pendulum Putting Stroke Training Aid housing  10  which consists of an elongated clear acrylic tube. In the preferred embodiment, the elongated clear tube is straight. However, it would not be inconceivable to those skilled in the art to shape the elongated housing into a curve that resembles the shape of a pendulum arc. Referring now to  FIG. 4 , a block diagram of the microcontroller module and its connection to personal computer  18  through USB cable  28  is shown. Microcontroller  32  I/O circuitry interfaces and controls 12 LEDs  30   a - 30   l  through LED Driver  40 . Microcontroller Serial Data In input  34  receives serial commands from personal computer  18  USB port over USB Cable  28  to USB Interface integrated circuit  42  located on the first microcontroller module. Power +V  33  and ground  35  are derived from the USB bus USB Interface  42 . As shown in  FIG. 5 , four microcontroller modules are interconnected to the three-wire bus which comprises V+ power  33 , ground  35 , and serial data  34 . Serial data out from the USB Interface  42  from personal computer  18  through USB cable  28  is distributed to all four microcontroller Serial In inputs. Only the first microcontroller module  20  includes the USB Interface  42 . The USB integrated circuit is not installed on microcontroller modules  22 ,  24 , and  26 . Serial data sent from personal computer  18  is received by all four microcontroller modules  20 .  22 ,  24 ,  26  on every transmission. Each microcontroller module microcontroller  32  is programmed with its local address. The first microcontroller module  20  is programmed with address  0 , microcontroller module  22  is programmed with address  1 , microcontroller module  24  is programmed with address  2 , and microcontroller module  26  is programmed with address  3 . During reception of a command byte, these address bits are compared with the address specified within a command byte for a match. If the address matches the local address, the LED number specified within the command is turned on, otherwise all LEDs are turned off in the microcontroller module. LED drivers  40  activate LEDs  30 . Refer now to  FIG. 9 . The first microcontroller module  20  controls LEDs  30   a - 30   l , the second microcontroller module  22  controls LEDs  30   m - 30   x , the third microcontroller module  24  controls LEDs  30   y - 30   aj , and the fourth microcontroller module  26  controls LEDs  30   ak - 30   ay .  FIG. 6  shows a perspective drawing of the first microcontroller module  20  showing the top LED side, while  FIG. 7  shows the back side of the first microcontroller module  20  revealing the microcontroller  32 , LED Driver  40 , and USB Interface  42 . As shown in  FIG. 6  twelve equally spaced LEDs  30   a - 30   l  are placed 1 inch apart. The three-wire bus connections appear at the rightmost portion of the circuit board at  33 ,  34 , and  35  and appear also at the leftmost end of the microcontroller module. As shown in  FIG. 8 , two boards are interconnected by inserting jumper wires  36  between identical circuit board pads of the first microcontroller module  20  and the second microcontroller module  22 . The same process is used to connect module  22  to the next module (not shown) and so on. In this scheme, power, ground and serial data transmitted from personal computer  18  of  FIG. 5  are passed to the next microcontroller module in line. 
     Referring to  FIG. 11  microcontroller software flow is shown. Step S 1  indicates the power up start of the microcontroller program. In step S 2 , the programmed local bus address ( 0 - 3 ) is read from program memory and placed into a ram variable named MYADDR. This will be used later in the serial receive interrupt service routine step S 6 . In S 3 , all LEDs are turned OFF. In step S 4 , the serial receive interrupt is enabled. In step S 5 , the microcontroller program stays in a forever background idle loop. When personal computer  18  of  FIG. 5  sends an LED command, it consists of an LED number ( 1 - 12 ) and two destination address bits (bit  5  and bit 6 ). When bits  5  and  6  are 0, the LED command is addressed to microcontroller  20 . When bit  5  is a 1 and bit  6  is a 0, the second microcontroller  22  is addressed. When bit  5  is a 0 and bit  6  is a 1, the third microcontroller  24  is addressed. And, when bits  5  and  6  are both 1, the fourth microcontroller  26  is addressed. As shown in step S 6  of  FIG. 11 , the microcontroller serial receive interrupt service routine is called when a serial command byte is received from personal computer  18  of  FIG. 5 . The received byte is stored in RXCHAR in step S 7 . The address bits are extracted from RXCHAR and shifted into the lower two bits and placed into RXADDR in step S 8 . In step S 9 , the received byte&#39;s destination address, RXADDR is compared with the microcontroller&#39;s local address MYADDR. If the address does not match, step S 11 , all LEDS are turned OFF in step S 12 . If the address matches, the command is for this microcontroller and the LED specified in RXCHAR ( 1 - 12 ) is turned ON in step S 10 . The microcontroller software then, in either case returns back to the background forever loop in step S 13 . 
     As indicated in  FIG. 12 , personal computer PC Control software flowchart is shown. The PC Control software starts in step S 20 . In step S 21 , the green speed GREENSPEED, beats per minute of the golfer&#39;s tempo BPM, and the desired target distance to hit the golf ball DISTANCE is selected by golfer  12  of  FIG. 1  prior to stroking golf ball  14  of  FIG. 1 . There are two main modes of operation of the Pendulum Putting Trainer Aid: Pendulum Swing Mode and Putting Stroke Mode that is selected in step S 22 . In addition to selecting GREENSPEED, BPM, and DISTANCE, golfer  12  of  FIG. 1  selects Pendulum Swing Mode or Putting Stroke Mode on personal computer  18  of  FIG. 1 . 
     In Pendulum Swing Mode, LEDs within Pendulum Putting Training Aid enclosure  10  of  FIG. 1 , “swing” back and forth as a pendulum would do starting at the top of the backswing. Golfer  12  of  FIG. 9  swings his/her putter  16  of  FIG. 9  in synchronization with the moving LEDs  30   a - 30   av  to get a feel for the pendulum swing. The golfer may increase or decrease the Beats Per Minute selection on the Personal computer  18  to find the tempo that best suits his/her personal timing. The Putting Stroke Mode is used to strike a golf ball a predicted distance guided by the Pendulum Putting Training Aid. It guides the golfer from address in the middle of the Pendulum Putting Training Aid housing  10  of  FIG. 9  LED  30   x  to the backswing in which speed gradually builds up to a peak mid way to the top of the backswing, slowing down to a slight stop at the top of the backswing followed by the downswing that builds up speed and reaches maximum speed at impact with the golf ball  14  at the starting position and continuing through to the followthrough shown in  FIG. 3 . 
     Pendulum Swing mode begins in S 23  of  FIG. 12  where, based on GREENSPEED, BPM, and DISTANCE, the length of the pendulum swing is calculated. Each LED ( 30   a - 30   av ) of  FIG. 9  is spaced 1 inch apart. The time for the pendulum to move from one LED position to the next LED in sequence in milliseconds is calculated in step S 24  of  FIG. 12  for each of the LEDs that comprise the pendulum swing. LEDs that fall outside of the particular pendulum swing resulting from golfer&#39;s selections, are not activated. Each LED ON time is stored into LED_PENDULUM array is step S 25 . This array has 48 elements, one for each of the 48 LEDs used in this embodiment. To begin the pendulum swing, the LED associated with the top of the backswing is turned ON in step S 26 . State variable, STATE is set to DOWNSWING to indicate to the timer interrupt, that occurs later in the process, which way the pendulum is moving so that the correct next LED will be activated. The timer is loaded with the value from LED_PENDULUM array representing the top of the backswing calculated in step S 27  and the timer is started in step S 33 . 
     Referring to  FIG. 9 , in Putting Stroke Mode, LEDs  30   a - 30   av  within Pendulum Putting Training Aid enclosure  10 , indicate a real-time putting stroke path from the starting address position as shown in  FIG. 9 , back towards the top of the backswing as shown in  FIG. 2  and down through the impact with golf ball  14  and on to the top of the followthrough as shown in  FIG. 3 . Golfer  12  of  FIG. 9  swings his/her putter  16  of  FIG. 9  in synchronization with the moving LEDs  30   a - 30   av  to generate a putting stroke that upon impact with golf ball  14 , rolls golf ball  14  the distance selected by golfer  12  on personal computer  18 . 
     Referring back to  FIG. 12 , Putting Stroke Mode begins in step S 28  where, based on GREENSPEED, BPM, and DISTANCE, the length of the stroke backswing is calculated as well as the stroke downswing. In step S 29 , the time between each LED activation on the backswing is calculated and stored into LED_PENDULUM array in step S 30 . In step S 30 , the time between each LED activation on the downswing is stored into LED_PENDULUM array. To begin the putting stroke mode, the LED associated with the middle of the Pendulum Putting Training Aid, LED  30   x  of  FIG. 1  is turned On briefly in step S 31  to indicate the start of the backswing. State variable, STATE is set to BACKSWING to indicate to the timer interrupt, that occurs later in the process, which way the pendulum is moving so that the correct next LED will be activated. The timer is loaded with the value from LED_BACKSWING array representing the middle of the backswing in step S 32  and the timer is started in step S 33 . 
     As indicated in  FIG. 13 , PC Control Timer Interrupt is shown in step S 35 . Upon timer expiration, step S 35  is executed. In step S 36 , the mode is checked. If MODE is Putting Stroke Mode, program execution branches to Putting Stroke Timer Handler in step S 37 . If MODE is Pendulum Swing Mode, the next LED to turn ON is transmitted in step  38  to Pendulum Putting Training Aid enclosure  10  of  FIG. 1 . If the end of the swing path is reached in step S 39 , STATE is flipped to its opposite state in step S 40 . Program returns from the timer interrupt service routine in step S 41 . 
     Referring now to  FIG. 14 , step S 37  is the start of Putting Stroke Timer Handler. In step S 42 , the next LED to activate is determined and transmitted to the Pendulum Putting Training Aid enclosure  10  of  FIG. 1 . If STATE is BACKSWING in step S 43 , a check for the end of the backswing is performed in step S 45 . If the end of the backswing is reached, STATE is flipped to DOWNSWING in step S 47  and control is returned back to the background idle loop in step S 51 . If STATE is not BACKSWING in step S 43 , step S 44  checks if STATE is DOWNSWING. If STATE=DOWNSWING, a check for the end of the downswing is checked in step S 46 . If the end of the downswing has been reached in step  46 , STATE is set to WAIT and the timer is set for ½ seconds in step S 48 . If STATE is not DOWNSWING in step S 44 , STATE is checked for WAIT in S 49 . If STATE is WAIT, STATE is set to BACKSWING, the timer is set to the first entry of LED_BACKSWING array, and the middle LED  30   x  of  FIG. 1  is turned on to start another cycle of Putting Stroke Mode in step S 50 . Control then returns to the background idle loop in step S 51 . 
     Refer now to  FIG. 10  for the description of the equations used to calculate the backswing length  72  and downswing dynamics based on GREENSPEED, BPM, and DISTANCE. The length of the virtual pendulum putter is represented by L  60 . The top pivot point of the pendulum diagram is fulcrum  15  as shown in  FIGS. 9 and 10 . The starting position of the putter head is represented by point  74 . The top of the backswing position of the putter head is  72 . The angle formed by lines  60  and  62  is angle THETA  66 . The length of the pendulum stroke swing would correspond to  72  for the top of the backswing and  76  for the top of the followthrough. Line  64  represents the pendulum putter at the end of the followthrough position. Line x  70  of  FIG. 10 , is the projection of  72 , the top of the backswing, onto x-axis floor plane  73 . Height h  75  is the height of the putter head  75  above floor plane x  73 . The arc representing the path of the putter head from address position to the top of the backswing is  71 . Pendulum Putting Training aid housing  10  is placed on the putting surface and is controlled by Personal Computer  18  through USB cable  28 . 
     Given a desired target golf ball roll distance, selected green speed to simulate, pendulum putter length, putter head mass, and personal putting tempo, the object of the invention is to show a practicing golfer through equally spaced visual indicators the putting stroke backswing and downswing dynamics which, if followed, will result in a putting stroke that will roll a golf ball the desired distance. 
     Given a desired target distance, Distance, and a selected green speed, Greenspeed, the initial required golf ball speed, Vball can be calculated from:
 
 V ball= sqrt [(36.1*Distance)/GreenSpeed]
 
     Using conservation of momentum for a putter head striking a golf ball, Vball is the velocity of the golf ball after impact. Vputter is the velocity of the putter head before impact. The equation for the velocity of the ball after impact is:
 
 V ball=[(1 +e )/(1 +M ball/ M putter)] V putter
 
where e is the coefficient of restitution of the ball and the putter head, Mputter is the mass of the putter head, and Mball is the mass of a golf ball. A typical putter head mass is 325 g while a golf ball is 46 g. The coefficient of restitution between the steel putter head and a golf ball is approximately 0.78.
 
     Given the initial ball speed, Vball, and Putter Head mass, Mputter, the required putter head speed to achieve Vball speed is:
 
 V putter=[(1 +M ball/ M putter)/1.78 ]*V ball
 
     Given the putter speed at impact of Vputter, the height, h, at which the pendulum putter would need to start to reach Vputter at the bottom of the arc would be:
 
 h =( V putter^2)/(2 g ), where g in the gravitational acceleration constant of 32.2 ft/sec^2.
 
     However, instead of using a gravitational constant of 32.2 ft/sec^2, an equivalent g′ will be developed taking into consideration the golfer&#39;s personal putting tempo rate. 
     The equation for ½ period (one way swing) of a pendulum of length L, is:
 
 T=pi*sqrt[L/g] , where g is the gravitational acceleration constant of 32.2 ft/sec^2 and pi is 3.14159
 
     The golfer&#39;s personal tempo is expressed in beats per minute (BPM). The period, T expressed in seconds is:
 
 T= 60/BPM
 
     Setting the golfer&#39;s personal putting tempo period of 60/BPM equal to the period of a pendulum:
 
60/BPM= pi*sqrt[L/g′] 
 
     g′ in the above equation represents the equivalent gravitational constant based on the golfer&#39;s personal tempo. 
     Solving for g′:
 
 g′=L *BPM/365
 
     A standard putter is approximately 36 inches long. However, in executing a “pendulum” stroke where the arms and shoulders move back and forth with little hand action, the effective length of the putter when considering it as a pendulum is longer than 36 inches due to the fact that a properly executed pendulum stroke points the butt of the grip just above the golfer&#39;s navel throughout the stroke. This point is approximately 44 inches or 3.7 feet on an average adult male. This point is the virtual fulcrum of a standard length putter projected towards the golfer and is shown as item  15  in  FIG. 9  and  FIG. 10 . Letting L=3.7 feet, the period of one swing of the pendulum putter is:
 
 T= 3.14159 *sqrt[  3.7/32.2]=1.06 seconds
 
     This corresponds to a swing rate of: (60 sec/min*( 1/1.06 sec)=56.3 beats per minute. 
     Most professional golfers exhibit an actual swing tempo rate in the range of 70 to 120 beats per minute whereas a true pendulum rate would be 56 beats per minute. A golfer therefore, would not perform a true pendulum stroke in most cases in that it would be too slow. The pendulum however will be used as a model for a proper stroke in regard to its smooth gravitational acceleration aspects, but not its exact timing. In order to develop equations of motion for the putting stroke while taking into account actual swing rates of expert golfers, a modified gravitational constant g′ is used. The g′ utilized is an equivalent gravitational constant based on the personal tempo rate of the golfer. For a given golfer personal tempo rate, a corresponding g′ is used to generate the equations of motion for the putting stroke. 
     For example, if the golfer&#39;s personal tempo rate is 80 beats per minute, the equivalent g′ would be: 65 ft/sec^2 approximately twice that of Earth&#39;s gravity. 
     Refer to  FIG. 10 . With the equation for g′, the peak speed of the putter at impact can be determined from Vputter=sqrt[2g′h], where h is the height  75  of the top of the backswing above the floor plane  73 . 
     In terms of h:
 
 h =( V putter^2)/2 g′ 
 
     The backswing length x  71  can be determined from the height h  75  from:
 
 x=sqrt[L^ 2−( L−h )^2].
 
     Therefore, for a given desired golf ball roll distance, x is the backswing starting position that will yield a downswing stroke following gravitational acceleration constant g′ that will result in the target golf ball roll distance. 
     The goal of the calculations is to determine the putter pendulum backswing height h  75 , that will yield a maximum pendulum putter downswing velocity at impact using the equivalent g′ based on the golfer&#39;s personal tempo rate such that after collision with putterhead  17  of  FIG. 1 , golf ball  14  of  FIG. 1  will roll a desired distance based on a given green speed. In usage, the golfer selects distance, personal tempo rate, putter head mass, pendulum putter length, and green speed on personal computer  18  of  FIG. 1  and the Pendulum Putting Stroke Training Aid shows the golfer the proper swing path dynamics through illuminating LEDs  30   a - 30   av  of  FIG. 9  to strike golf ball  14  a desired distance. 
     In another embodiment of the Pendulum Putting Stroke Training Aid, a generate audio file button feature within the graphical user interface running on personal computer  18  of  FIG. 1  generates a sound file that can be loaded onto personal music players. Pressing the generate audio file button (not shown) generates multiple cycles of a putting stroke at the tempo rate selected. The resulting personal computer sound file can then be listened to by the golfer on a green while practicing putting. Refer to  FIG. 9 . For each time increment between LEDs, n cycles of a phase continuous frequency related to the height of putterhead  17  are generated and stored in a personal computer way file. A special beep sound is placed in the file at the precise moment of impact of putterhead  17  with golf ball  14 . By listening to the resulting audio file, the golfer  12  is provided with an audio version of the Pendulum Putting Stroke Training Aid that he/she can take to the practice putting green. 
     While the invention has been described in connection with a preferred embodiment, it is not intended to limit the scope of the invention to the particular form set forth, but on the contrary, it is intended to cover such alternatives, modifications, and equivalents as may be included within the spirit and scope of the invention as defined by the appended claims.