Golf swing measurement and analysis system

[0001] The present invention relates to a method for determining the effectiveness of a golfer's swing requiring no golf club contact with the golf ball. The measurement and analysis system comprises an attachable and detachable module, that when attached to a golf club head measures three dimensional acceleration data, that is further transmitted to a computer or other smart device or computational engine where a software algorithm interprets measured data within the constraints of a multi-lever variable radius swing model using both rigid and non-rigid levers, and further processes the data to define accurate golf swing metrics. In addition, if the club head module is not aligned ideally on the club head a computational algorithm detects the misalignment and further calibrates and corrects the data.

[0002] There are numerous prior art external systems disclosures using video and or laser systems to analyze the golf swing. There are also numerous golf club attached systems using shaft mounted strain gauges and or single to multiple accelerometers and gyros to calculate golf swing metrics. However, none of these prior art approaches

[0003] U.S. Pat. No. 3,945,646 to Hammond integrates three-dimensional orthogonal axes accelerometers in the club head, and describes a means for wirelessly transmitting and receiving the resulting sensor signals. However, he does not contemplate the computational algorithms involving the multi-lever mechanics of a golf club swing required to solve for all the angles of motion of the club head during the swing with a varying swing radius. His premise of being able to obtain face angle only with data from his sensors 13, and 12 (x and y directions respectively described below) is erroneous, as for one example, the toe down angle feeds a large component of the radial centrifugal acceleration onto sensor 12 which he does not account for. He simply does not contemplate the effects of the dynamically changing orientation relationship between the inertial acceleration forces and the associated coordinate system acting on the club head constrained by the multi-lever golf swing mechanics and the fixed measurement coordinate system of the three orthogonal club head sensors.

[0004] The prior art disclosures all fail to offer a golf free swing analysis system that measures only acceleration forces on three orthogonal axes at the club head and interprets that limited data within the constraints of a multi-lever golf swing model using rigid and non rigid levers describing the mechanics of a swing, to determine the dynamically changing orientation relationship of inertial forces experienced at the club head and the orthogonal measurement axes fixed to the club head, resulting in the ability to accurately calculate numerous golf swing metrics.

[0005] The present invention is a golf swing measurement and analysis system that measures directly and stores time varying acceleration forces during the entire golf club swing. The measurement and analysis system comprises three major components; a golf club, a club head module that is attachable to and removable from the club head, and a computer program. The golf club comprises a shaft and a club head with the club head comprising a face and a top surface where the module is attached. The module comprise a means to measure acceleration separately on three orthogonal axes, and a means to transmit the measured data to a computer or other smart device where the computer program resides. The computer program comprises computational algorithms for calibration of data and calculation of golf metrics and support code for user interface commands and inputs and visual display of the metrics.

[0006] During operation the module is attached on the head of the golf club, and during the entire golf swing it captures data from the three acceleration sensors axes. The acquired swing measurement data is either stored in the module for later analysis or transmitted immediately from the module to a receiver with connectivity to a computation engine. A computational algorithm that utilizes the computational engine is based on a custom multi-lever golf swing model utilizing both rigid and non-rigid levers. This algorithm interprets the measured sensor data to determine the dynamically changing relationship between an inertial coordinates system defined by the multi-lever model for calculation of inertial acceleration forces and the module measurement axes coordinate system attached to the club head. Defining the dynamically changing orientation relationship between the two coordinate systems allows the interpretation of the measured sensor data with respect to a non-linear travel path allowing the centrifugal and linear acceleration components to be separated for each of the module's three measured axes. Now with each of the module axes measurements defined with a centrifugal component (also called the radial component), and a linear spatial transition component the swing analysis system accurately calculates a variety of golf swing metrics which can be used by the golfer to improve their swing. These swing quality metrics include:

• • [0007] 1. Golf club head time varying velocity for a significant time span before and after maximum velocity of the swing.
  • [0008] 2. Time varying swing radius for a significant time span before and after maximum velocity of the swing.
  • [0009] 3. Golf club head face approach angle of the golf club head, whether the club face is “open”, “square”, or “closed”, and by how much measured in degrees, for a significant time span before and after maximum velocity of the swing.
  • [0010] 4. Wrist cock angle during the swing, for a significant time span before and after maximum velocity of the swing.
  • [0011] 5. Club shaft lag/lead flexing during the swing, for a significant time span before and after maximum velocity of the swing.
  • [0012] 6. Club head toe down angle during the swing, for a significant time span before and after maximum velocity of the swing.
  • [0013] 7. Club head acceleration force profile for the backswing that include time varying vector components and total time duration.
  • [0014] 8. Club head acceleration force profile for the pause and reversal segment of the swing after backswing that includes time varying vector components and total time duration.
  • [0015] 9. Club head acceleration force profile for the power-stroke after pause and reversal that includes time varying vector components and total time duration.
  • [0016] 10. Club head acceleration force profile for the follow through after power-stroke that includes time varying vector components and total time duration.
  • [0017] 11. Club head swing tempo profile which includes total time duration of tempo for the backswing, pause and reversal, and power-stroke and provides a percentage break down of each segment duration compared to total tempo segment duration.

[0018] The module acceleration measurement process comprises sensors that are connected to electrical analog and digital circuitry and an energy storage unit such as a battery to supply power to the circuits. The circuitry conditions the signals from the sensors, samples the signals from all sensors simultaneously, converts them to a digital format, attaches a time stamp to each group of simultaneous sensor measurements, and then stores the data in memory. The process of sampling sensors simultaneously is sequentially repeated at a fast rate so that all acceleration forces profile points from each sensor are relatively smooth with respect to time. The minimum sampling rate is the “Nyquist rate” of the highest significant and pertinent frequency domain component of any of the sensors' time domain signal.

[0019] The sensor module also contains circuitry for storing measured digital data and a method for communicating the measured data out of the module to a computational engine integrated with interface peripherals that include a visual display and or audio capabilities. In the preferred embodiment the club head module also contains RF circuitry for instant wireless transmission of sensor data immediately after sampling to a RF receiver plugged into a USB or any other communications port of a laptop computer. The receiver comprises analog and digital circuitry for receiving RF signals carrying sensor data, demodulating those signals, storing the sensor data in a queue, formatting data into standard USB or other communication formats for transfer of the data to the computation algorithm operating on the computation engine.

[0020] An alternate embedment of this invention contemplates a similar module without the RF communication circuitry and the addition of significantly more memory and USB connectivity. This alternate embodiment can store many swings of data and then at a later time, the module can be plugged directly into to a USB laptop port for analysis of each swing.

[0021] Another alternate embodiment of this invention contemplates a similar club head module without the RF circuitry and with a wired connection to a second module mounted on the shaft of the club near the grip comprising a computational engine to run computational algorithm and a display for conveying golf metrics.

[0037] The present invention comprises accelerometers attached to the club head that allow the motion of the club head during the swing to be determined. In the preferred embodiment as shown in FIG. 1 sensors are incorporated in a club head attachable module 101. The module 101 has a front surface 102 and a top surface 103 and an inwardly domed attachment surface 107. The sensors in module 101 measure acceleration in three orthogonal axes which include: the x_f-axis 104 that is perpendicular to the front surface 102, the z_f- axis 105 that is perpendicular to x_f-axis 104 and perpendicular to the top surface 103 and the y_f-axis 106 that is perpendicular to both the x_f-axis 104 and the z_f-axis 105.

[0038] FIG. 2 shows the preferred embodiment of the invention, which is the module 101 with three orthogonal measurement axes 104, 105 and 106 that is attached to the top surface 204 of the club head 201. The club head module 101 attachment surface 107 is attached to club head 201 top surface 204 with a conventional double sided tape with adhesive on top and bottom surfaces (not shown).

[0039] For the club head module 101 mounted perfectly on the club head 201 top surface 204 the following relations are achieved: The z_f-axis 105 is aligned so that it is parallel to the club shaft 202. The x_f-axis 104 is aligned so that is orthogonal to the z_f-axis 105 and perpendicular to the plane 203 that would exist if the club face has a zero loft angle. The y_f-axis 106 is aligned orthogonally to both the x_f-axis 104 and z_f-axis 105.

[0040] With these criteria met, the plane created by the x_f-axis 104 and the y_f-axis 106 is perpendicular to the non-flexed shaft 202. In addition the plane created by the y_f-axis 106 and the z_f-axis 105 is parallel to the plane 203 that would exist if the club face has a zero loft angle.

[0041] The mathematical label a_sx represents the acceleration force measured by a sensor along the club head module 101 x_f-axis 104. The mathematical label a_sy represents the acceleration force measured by a sensor along the club head module 101 y_f-axis 106. The mathematical label a_sz represents the acceleration force measured by a sensor along the club head module 101 z_f-axis 105.

[0042] If the club head module of the preferred embodiment is not aligned exactly with the references of the golf club there is an algorithm that is used to detect and calculated the angle offset from the intended references of the club system and a method to calibrate and correct the measured data. This algorithm is covered in detail after the analysis is shown for proper club head module attachment with no mounting angle variations.

[0043] Club head motion is much more complicated than just pure linear accelerations during the swing. It experiences angular rotations of the fixed sensor orthogonal measurement axes, x_f-axis 104, y_f-axis 106 and z_f-axis 105 of module 101 around all the center of mass inertial acceleration force axes during the swing, as shown in FIG. 3. As the golfer 301 swings the golf club 302 and the club head 201 travels on an arc there are inertial center of mass axes along which inertia forces act on the center of mass of the club head 201. These are the x_cm-axis 303, y_cm-axis 305 and z_cm-axis 304.

[0044] The three orthogonal measurement axes x_f-axis 104, y_f-axis 106 and z_f-axis 105 of module 101, along with a physics-based model of the multi-lever action of the swing of the golfer 301, are sufficient to determine the motion relative to the club head three-dimensional center of mass axes with the x_cm-axis 303, y_cm-axis 305 and z_cm-axis 304.

[0045] The mathematical label a_z is defined as the acceleration along the z_cm-axis 304, the radial direction of the swing, and is the axis of the centrifugal force acting on the club head 201 during the swing from the shoulder 306 of the golfer 301. It is defined as positive in the direction away from the golfer 301. The mathematical label a_x is the defined club head acceleration along the x_cm-axis 303 that is perpendicular to the a_z-axis and points in the direction of instantaneous club head inertia on the swing arc travel path 307. The club head acceleration is defined as positive when the club head is accelerating in the direction of club head motion and negative when the club head is decelerating in the direction of club head motion. The mathematical label a_y is defined as the club head acceleration along the y_cm-axis 305 and is perpendicular to the swing plane 308.

[0046] During the golfer's 301 entire swing path 308, the dynamically changing relationship between the two coordinate systems, defined by the module 101 measurements coordinate system axes x_f-axis 104, y_f-axis 106 and z_f-axis 105 and the inertial motion acceleration force coordinate system axes x_cm-axis 303, y_cm-axis 305 and z_cm-axis 304, must be defined. This is done through the constraints of the multi-lever model partially consisting of the arm lever 309 and the club shaft lever 310.

[0047] The multi lever system as shown in FIG. 4 shows two interdependent angles defined as angle η 401 which is the angle between the club head module 101 z_f-axis 105 and the inertial z_cm-axis 304 and the angle α 403 which is the sum of wrist cock angle and shaft flex lag/lead angle (shown later in FIGS. 7 and 7A). The angle η 401 is also the club head rotation around the y_cm-axis 106 (not shown in FIG. 4 but is perpendicular to the page at the club head center of mass) and is caused largely by the angle of wrist cock, and to a lesser extent club shaft flexing during the swing. The length of the variable swing radius R 402 is a function of the fixed length arm lever 309, the fixed length club shaft lever 310 and the angle η 401. The angle η 401 can vary greatly, starting at about 40 degrees or larger at the start of the downswing and approaches zero at club head maximum velocity. The inertial x_cm-axis 303 is as previously stated perpendicular to the inertial z_cm-axis 304 and variable radius R 402.

[0048] FIG. 5 shows the angle Φ 501 which is the club face angle and is defined as the angle between the plane 502 that is perpendicular to the club head travel path 307 and the plane that is defined for zero club face loft 203. The angle Φ 501 also represents the club head rotation around the z_f-axis 105. The angle Φ 501 varies greatly throughout the swing starting at about 90 degrees or larger at the beginning of the downswing and becomes less positive and perhaps even negative by the end of the down stroke. When the angle Φ 501 is positive the club face angle is said to be “OPEN” as shown in club head orientation 503. During an ideal swing the angle Φ 501 will be zero or said to be “SQUARE” at the point of maximum club head velocity as shown in club head orientation 504. If the angle Φ 501 is negative the club face angle is said to be “CLOSED” as shown in club head orientation 505.

[0049] FIG. 6 shows angle Ω 601 which is referred to as the toe down angle and is defined as the angle between the top of a club head 201 of a golf club with a non bowed shaft state 602 and a golf club head 201 of a golf club with bowed shaft state 603 due to the centrifugal force pulling the club head toe downward during the swing. The angle Ω is a characteristic of the multi-lever model representing the non rigid club lever. The angle Ω 601 also represents the club head 201 rotation around the x_f-axis 104 (not shown in FIG. 6, but which is perpendicular to the y_f-axis 106 and z_f-axis 105 intersection). The angle Ω 601 starts off at zero at the beginning of the swing, and approaches a maximum value of a few degrees at the maximum club head velocity.

[0050] FIGS. 7 and 7A show the angle α 403 which is the sum of angles α_wc 701, defined as the wrist cock angle, and α_sf 702, defined as the shaft flex lag/lead angle. The angle α_sf 702 is the angle between a non-flexed shaft 703 and the flexed shaft state 704, both in the swing plane 308 defined in FIG. 3, and is one characteristic of the non rigid lever in the multi-lever model. The shaft leg/lead flex angle α_sf 702 is caused by a combination of the inertial forces acting on the club and the wrist torque provided by the golfer's 301 wrists 705 and hands 706 on the shaft grip 707.

[0051] FIG. 8 shows the force balance for the multi-lever swing system. The term a_v 805 is the vector sum of a_x 804 and a_z 803. The resulting force is given by F_v=m_sa_v where m_s is the mass of the club head system. The term F_v 806 is also, from the force balance, the vector sum of the tensile force, F_t 807, in the shaft due to the shoulder torque 801, and F_wt 808, due to wrist torque 802. The angle between force vector F_v 806 and the swing radius, R 402, is the sum of the angles η 401 and η_wt 809.

[0052] There are several ways to treat the rotation of one axes frame relative to another, such as the use of rotation matrices. The approach described below is chosen because it is intuitive and easily understandable, but other approaches with those familiar with the art would fall under the scope of this invention.

[0053] Using the multi-lever model using levers, rigid and non-rigid, the rotation angles describing the orientation relationship between the module measured axis coordinate system and the inertial acceleration force axes coordinate system can be determined from the sensors in the club head module 101 through the following relationships:
1. a_sx=a_xcos(Φ)cos(η)−a_ysin(Φ)−a_zcos(Φ)sin(η)
2. a_sy=a_xsin(Φ)cos(η)+a_ycos(Φ)+a_z(sinΩ)−sin(Φ)sin(η)),
3. a_sz=a_xsin(η)−a_ysin(Ω)cos(Φ)+a_zcos(η)
The following is a reiteration of the mathematical labels for the above equations.

• • [0054] a_x is the club head acceleration in the x_cm-axis 303 direction.
  • [0055] a_y is the club head acceleration in the y_cm-axis 305 direction.
  • [0056] a_z is the club head acceleration in the z_cm-axis 304 direction.
  • [0057] a_sx is the acceleration value returned by the club head module 101 sensor along the x_f-axis 104.
  • [0058] a_sy is the acceleration value returned by the club head module 101 sensor along the y_f-axis 106.
  • [0059] a_sz is the acceleration value returned by the club head module 101 sensor along the z_f-axis 105.
    During a normal golf swing with a flat swing plane 308, a_y will be zero, allowing the equations to be simplified:
    4. a_sx=a_x cos(Φ)cos(η)−a_z cos(Φ)sin(η)
    5. a_sy=a_x sin(Φ)cos(η)+a_z(sin(Ω)−sin(Φ)sin(η)
    6. a_sz=a_x sin(η)+a_z cos(η)
    These equations are valid for a “free swing” where there is no contact with the golf ball.

[0060] The only known values in the above are a_sx, a_sy , and a_sz from the three sensors. The three angles are all unknown. It will be shown below that a_x and a_z are related, leaving only one unknown acceleration. However, that still leaves four unknowns to solve for with only three equations. The only way to achieve a solution is through an understanding the physics of the multi-lever variable radius swing system dynamics and choosing precise points in the swing where physics governed relationships between specific variables can be used.

[0061] The angle Φ 501, also known as the club face approach angle, varies at least by 180 degrees throughout the backswing, downswing, and follow through. Ideally it is zero at maximum velocity, but a positive value will result in an “open” clubface and negative values will result in a “closed” face. The angle Φ 501 is at the control of the golfer and the resulting swing mechanics, and is not dependent on either a_x or a_z. However, it can not be known a-priori, as it depends entirely on the initial angle of rotation around the shaft when the golfer grips the shaft handle and the angular rotational velocity of angle Φ 501 during the golfer's swing.

[0062] The angle Ω 601, on the other hand, is dependent on a_z, where the radial acceleration causes a centrifugal force acting on the center of mass of the club head, rotating the club head down around the x_f-axis into a “toe” down position of several degrees. Therefore, angle Ω 601 is a function of a_z. This function can be derived from a physics analysis to eliminate another unknown from the equations.

[0063] The angle η 401 results from both club shaft angle 702 lag/lead during the downswing and wrist cock angle 701. Wrist cock angle is due both to the mechanics and geometry relationships of the multi lever swing model as shown in FIG. 4 and the amount of torque exerted by the wrists and hands on the shaft.

[0064] Before examining the specifics of these angles, it is worth looking at the general behavior of equations (4) through (6). If both angle Ω 601 and angle η 401 were always zero, which is equivalent to the model used by Hammond in U.S. Pat. No. 3,945,646, the swing mechanics reduces to a single lever constant radius model. For this case:
7. a_sx=a_xcos(Φ)
8. a_sy=a_xsin(Φ)
9. a_sz=a_z

[0065] This has the simple solution for club face angle Φ of:

[0066] $10.\tan \left(\Phi \right)=\frac{a_{\mathrm{sy}}}{a_{\mathrm{sx}}}$

[0067] In Hammond's patent U.S. Pat. No. 3,945,646 he states in column 4 starting in line 10 “By computing the vector angle from the acceleration measured by accelerometers 12 and 13, the position of the club face 11 at any instant in time during the swing can be determined.” As a result of Hammond using a single lever constant radius model which results in equation 10 above, it is obvious he failed to contemplate effects of the centrifugal force components on sensor 12 and sensor 13 of his patent. The large error effects of this can be understood by the fact that the a_z centrifugal acceleration force is typically 50 times or more greater than the measured acceleration forces of a_sx and a_sy for the last third of the down swing and first third of the follow through. Therefore, even a small angle Ω 601 causing an a_z component to be rotated onto the measured a_sy creates enormous errors in the single lever golf swing model.

[0068] In addition, the effect of the angle η 401 in the multi lever variable radius swing model is to introduce a_z components into a_sx and a_sy, and an a_x component into a_sz. The angle η 401 can vary from a large value at the start and midpoint of the down stroke when a_z is growing from zero. In later portion of the down stroke a_z becomes very large as angle η 401 tends towards zero at maximum velocity. Also, as mentioned above, the angle η 401 introduces an a_x component into a_sz. This component will be negligible at the point of maximum club head velocity where angle η 401 approaches zero, but will be significant in the earlier part of the swing where angle η 401 is large and the value of a_x is larger than that for a_z.

[0069] The cos(η) term in equations (4) and (5) is the projection of a_x onto the x_f-y_f plane, which is then projected onto the x_f axis 104 and the y_f axis 106. These projections result in the a_xcos(Φ)cos(η) and a_xsin(Φ)cos(η) terms respectively in equations (4) and (5). The projection of a_x onto the z_f-axis 105 is given by the a_xsin(η) term in equation (6).

[0070] The sin(η) terms in equations (4) and (5) are the projection of a_z onto the plane defined by x_f axis 104 and the y_f axis 106, which is then projected onto the x_f axis 104 and y_f axis 106 through the a_z cos(Φ)sin(η) and a_zsin(Φ)sin(η) terms respectively in equations (4) and (5). The projection of a_z onto the z_f-axis 105 is given by the a_zcos(η) term in equation (6).

[0071] The angle Ω 601 introduces yet another component of a_z into a _sy. The angle Ω 601 reaches a maximum value of only a few degrees at the point of maximum club head velocity, so its main contribution will be at this point in the swing. Since angle Ω 601 is around the x_f-axis 104, it makes no contribution to a_sx, so its main effect is the a_zsin(Φ) projection onto the y_f-axis 106 of equation (5). Equations (4) and (5) can be simplified by re-writing as:
11. a_sx=(a_xcos(η)−a_zsin(η))cos(Φ)=f(η)cos(Φ) and
12. a_sy=(a_xcos(η)−a_zsin(η))sin(Φ)+a_zsin(Ω)=f(η)sin(Φ)+a_z sin(Ω) where
13. f(η)=a_xcos(η)−a_zsin(η). From (11):

[0072] $14.f\left(\eta \right)=\frac{a_{\mathrm{sx}}}{\cos \left(\Phi \right)}$
which when inserted into (12) obtains:
15. β_sy=α_sx tan(Φ)+a_zsin(Ω)

[0073] From equation (15) it is seen that the simple relationship between a_sx and a_sy of equation (10) is modified by the addition of the a_z term above. Equations (4) and (6) are re-written as:

[0074] $16.a_x=\frac{a_x}{\cos \left(\eta \right)\cos \left(\Phi \right)}+\frac{a_z\sin \left(\eta \right)}{\cos \left(\eta \right)}$ $17.a_z=\frac{a_{\mathrm{sz}}}{\cos \left(\eta \right)}-\frac{a_x\sin \left(\eta \right)}{\cos \left(\eta \right)}.$
These equations are simply solved by substitution to yield:

[0075] $18.a_z=a_{\mathrm{sz}}\cos \left(\eta \right)-a_{\mathrm{sx}}\frac{\sin \left(\eta \right)}{\cos \left(\Phi \right)}.19.a_x=a_{\mathrm{sz}}\sin \left(\eta \right)+a_{\mathrm{sx}}\frac{\cos \left(\eta \right)}{\cos \left(\Phi \right)}.$

[0076] Equation (19) can be used to find an equation for sin(η) by re-arranging, squaring both sides, and using the identity, cos²(η)=1-sin²(η), to yield a quadratic equation for sin(η), with the solution:

[0077] $20.\sin \left(\eta \right)=\frac{a_x{a}_{\mathrm{sz}}+\frac{a_{\mathrm{sx}}^2}{{\cos }^2\left(\Phi \right)}\sqrt{1-{\cos }^2\left(\Phi \right)\left(\frac{a_{\mathrm{sz}}^2-a_x^2}{a_{\mathrm{sx}}^2}\right)}}{a_{\mathrm{sz}}^2+\frac{a_{\mathrm{sx}}^2}{{\cos }^2\left(\Phi \right)}}.$

[0078] To get any further for a solution of the three angles, it is necessary to examine the physical cause of each. As discussed above the angle η 401 can be found from an analysis of the angle α 403 , which is the sum of the angles α_wc 701, due to wrist cock and α_sf 702 due to shaft flex lag or lead.

[0079] Angle α 403, and angle η 401 are shown in FIG. 4 in relationship to variable swing radius R 402, fixed length arm lever A 309, and fixed length club shaft lever C 310. The mathematical equations relating these geometric components are:
21. R²=A²+C²+2AC cos(α)
22. A²=R²+C²−2RC cos(η)
Using R² from equation (21) in (22) yields a simple relationship between α and η:
23. a=cos⁻¹((R cos(η)−C)−C)/A)
The swing radius, R 402, can be expressed either in terms of cos(α) or cos(η). Equation (21) provides R directly to be:
24. R={square root over (C²+A²+2ACcos(α))}.
Equation (22) is a quadratic for R which is solved to be:
25. R=C cos(η)+{square root over (C²(cos(η)−1)+A²)}.

[0080] Both α 403 and η 401 tend to zero at maximum velocity, for which R_m=A+C.

[0081] The solutions for the accelerations experienced by the club head as it travels with increasing velocity on this swing arc defined by equation (25) are:

[0082] $26.a_z=\frac{V_{\Gamma }^2}{R}-\frac{d{V}_R}{dt}$ $27.a_x=\frac{2}{R}{V}_R{V}_{\Gamma }+R\frac{d}{dt}\left(\frac{V_{\Gamma }}{R}\right)$
The acceleration a_z is parallel with the direction of R 402, and a_x is perpendicular to it in the swing plane 308. The term V_Γ is the velocity perpendicular to R 402 in the swing plane 308, where Γ is the swing angle measured with respect to the value zero at maximum velocity. The term V_R is the velocity along the direction of R 402 and is given by dR/dt. The swing geometry makes it reasonably straightforward to solve for both V_R and its time derivative, and it will be shown that a_z can also be solved for which then allows a solution for V_Γ:

[0083] $28.V_{\Gamma }=\sqrt{{\mathrm{Ra}}_z+R\frac{d{V}_r}{dt}}$
Now define:

[0084] $29.a_{z-\mathrm{radial}}=\frac{V_{\Gamma }^2}{R}$
so that:
30. V_Γ={square root over (Ra_Z-radial)},
Next define:

[0085] $31.a_{\mathrm{ch}}=\frac{d{V}_{\Gamma }\left(t\right)}{dt}=\frac{\Delta V_{\Gamma }\left(t\right)}{\Delta t},$
Because (31) has the variable R 402 included as part of the time derivative equation (27) can be written:

[0086] $32.a_x=a_{\mathrm{ch}}+\frac{2}{R}{V}_R{V}_{\Gamma }$

[0087] Also equation (26) can be written:

[0088] $33.a_z=a_{z-\mathrm{radial}}-\frac{d{V}_R}{dt}$
The acceleration a_v 805 is the vector sum of a_x 804 and a_z 803 with magnitude:

[0089] $34.a_v=\sqrt{a_x^2+a_z^2}=\frac{a_x}{\sin \left(\beta \right)}=\frac{a_z}{\cos \left(\beta \right)}$ $\mathrm{where}$ $35.\beta ={\tan }^{-1}\left(\frac{a_x}{a_z}\right)$
The resulting magnitude of the force acting on the club head is then:
36. F_v=m_sa_v
FIG. 8 shows this force balance for F_v 806. If there is no force F_wt 808 acting on the golf club head due to torque 802 provided by the wrists, then F_v 806 is just F_t 807 along the direction of the shaft, and is due entirely by the arms pulling on the shaft due to shoulder torque 801. For this case it is seen that:
37. β=η for no wrist torque.

[0090] On the other hand, when force F_wt 808 is applied due to wrist torque 802:
38. β=η+η_wt where:
39. F_wt=F_vsin(η_wt).
The angle η_wt 809 is due to wrist torque 802. From (38):

[0091] $40.\eta =\left(1-\frac{{\eta }_{\mathrm{wt}}}{\beta }\right)\beta =C_{\eta }\beta$
where C_η<1 is a curve fitting parameter to match the data, and is nominally around the range of 0.75 to 0.85. From the fitted value:
41. η_wt=(1−C_η)β

[0092] Using (41) in (39) determines the force F_wt 808 due to wrist torque 802.

[0093] To solve for angle Ω 601 as previously defined in FIG. 6 the force balance shown in FIG. 9 is applied to accurately determine the toe down angle Ω 601. A torque 901 acting on club head 201 with mass M is generated by the acceleration vector 902 on the z_cm-axis 304 with magnitude a_z acting through the club head 201 center of mass 903. The center of mass 903 is a distance 904 from the center axis 905 of club shaft 202 with length C 310 and stiffness constant K. The mathematical label for distance 904 is d. Solving the force balance with the constraints of a flexible shaft K gives an expression for Ω 601:

[0094] $42.\Omega =\frac{{\mathrm{dC}}_{\Omega }}{C}\left(\frac{\frac{{\mathrm{Ma}}_z}{\mathrm{KC}}}{1+\frac{{\mathrm{Ma}}_z}{\mathrm{KC}}}\right)$

[0095] It is worth noting that from equation (42) for increasing values of a_z there is a maximum angle Ω 601 that can be achieved of d C_Ω/C which for a typical large head driver is around 4 degrees. The term C_Ω is a curve fit parameter to account for variable shaft stiffness profiles for a given K. In other words different shafts can have an overall stiffness constant that is equal, however, the segmented stiffness profile of the shaft can vary along the taper of the shaft.

[0096] An equation for angle Φ 501 in terms of angle Ω 601 can now be found. This is done by first using equation (17) for a_z in equation (15):

[0097] $43.a_{\mathrm{sy}}=a_{\mathrm{sx}}\frac{\sin \left(\Phi \right)}{\cos \left(\Phi \right)}+a_{\mathrm{sz}}\cos \left(\eta \right)\sin \left(\Omega \right)-a_{\mathrm{sx}}\frac{\sin \left(\eta \right)\sin \left(\Omega \right)}{\cos \left(\Phi \right)}$
Re-arranging terms:
44. (a_sy−a_sz cos(η)sin(Ω))cos(Φ)=a_sxsin(Φ)−a_sxsin(η)sin(Ω)
Squaring both sides, and using the identity cos²(Φ)=1-sin²(Φ) yields a quadratic equation for sin(Φ):

[0098] $45.{\sin }^2\left(\Phi \right)\left[a_{\mathrm{sx}}^2+{\left(a_{\mathrm{sy}}-a_{\mathrm{sy}}\cos \left(\eta \right)\sin \left(\Omega \right)\right)}^2\right]-2a_{\mathrm{sx}}^2\sin \left(\Phi \right)\sin \left(\eta \right)\sin \left(\Omega \right)+{a_{\mathrm{sx}}^2\left(\sin \left(\eta \right)\sin \left(\Omega \right)\right)}^2-{\left(a_{\mathrm{sy}}-a_{\mathrm{sz}}\cos \left(\eta \right)\sin \left(\Omega \right)\right)}^2=0$
Equation (45) has the solution:

[0099] $46.\sin \left(\Phi \right)=\frac{1}{2b_1}\left[-b_2+\sqrt{b_2^2-4b_1{b}_3}\right]$
where the terms in (46) are:
b₁=a_sx²+(a_sy−a_szcos(η)sin(Ω))²
b₂=−2a_sx²sin(η)sin(Ω)
b₃=a_sx²(sin(η)sin(Ω))²−(a_sy−a_szcos(η)sin(Ω))²

[0100] Equations (42) for Ω 601, (46) for Φ 501, and (20) for η 401 need to be solved either numerically or iteratively using equations (32) for a_x, (33) for a_z, and (25) for R 402. This task is extremely complex. However, some innovative approximations can yield excellent results with much reduced complexity. One such approach is to look at the end of the power-stroke segment of the swing where V_R and its time derivative go to zero, for which from equations (32), (33), (35) and (40):

[0101] $47.\eta =C_{\eta }{\tan }^{-1}\left(\frac{a_{\mathrm{ch}}}{a_{z-\mathrm{radial}}}\right)$
In this part of the swing the a_sx term will be much smaller than the a_sz term and equation (18) can be approximated by:
48. a_z=a_z-radial=a_szcos(η).
During the earlier part of the swing, the curve fit coefficient C_η would accommodate non-zero values of V_R and its time derivative as well as the force due to wrist torque 802.

[0102] The maximum value of η 401 is nominally around 40 degrees for which from (48) a_ch/a_z-radial=1.34 with C_η, =0.75. So equation (47) is valid for the range from a_ch=0 to a_ch=1.34 a_z-radial, which is about a third of the way into the down-stroke portion of the swing. At the maximum value of η 401 the vector a_v 805 is 13 degrees, or 0.23 radians, off alignment with the z_f axis and its projection onto the z_f axis 105 is a_sz =a_vcos(0.23)=0.97a_y. Therefore, this results in a maximum error for the expression (48) for a_z=a_z-radial of only 3%. This amount of error is the result of ignoring the a_sx term in equation (18). This physically means that for a_z in this part of the swing the a_z-radial component value dominates that of the a_sx component value. Equation (47) can not be blindly applied without first considering the implications for the function f(η) defined by equations (13) and (14), which has a functional dependence on cos(Φ) through the a_sx term, which will not be present when (47) is used in (13). Therefore, this cos(Φ) dependence must be explicitly included when using (47) to calculate (13) in equation (12) for a_sy, resulting in:
49. a_xy=(a_xcos(η)−a_zsin(η))tan(Φ)+a_zsin(Ω).

[0103] Equation (49) is applicable only when equation (47) is used for the angle η 401.

[0104] A preferred embodiment is next described that uses the simplifying equations of (47) through (49) to extract results for Φ 501 and η 401 using (42) as a model for Ω 601. It also demonstrates how the wrist cock angle α_wc, 701 and shaft flex angle α_sf 702 can be extracted, as well as the mounting angle errors of the accelerometer module. Although this is the preferred approach, other approaches fall under the scope of this invention.

[0105] The starting point is re-writing the equations in the following form using the approximations a_z-=a_z-radial and a_x=a_ch. As discussed above these are excellent approximations in the later part of the swing. Re-writing the equations (4) and (49) with these terms yields:
50. a_sx=a_chcos(Φ)cos(η)−a_z-radialcos(Φ)sin(η)
51. a_sy=a_chtan(Φ)cos(η)+a_z-radialsin(Ω)−a_z-radialtan(Φ)sin(η)
52. a_z-radial=a_szcos(η)

[0106] Simplifying equation (31):

[0107] $53.a_{\mathrm{ch}}=\frac{dV}{dt}$

[0108] In this approximation V=V_Γ is the club head velocity and dt is the time increment between sensor data points. The instantaneous velocity of the club head traveling on an arc with radius R is from equation (29):

[0109] $54.V=\sqrt{a_{z-\mathrm{radial}}R}=a_{z-\mathrm{radial}}^{1/2}{R}^{1/2}\mathrm{for}\mathrm{which}:55.a_{\mathrm{ch}}=\frac{dV}{dt}=\frac{1}{2}\left(\frac{1}{R}\frac{dR}{dt}+\frac{1}{a_{z-\mathrm{radial}}}\frac{d{a}_{z-\mathrm{radial}}}{dt}\right)\sqrt{{\mathrm{Ra}}_{z-\mathrm{radial}}}$
Using equation (52) for a_z-radial in (55):

[0110] $56.a_{\mathrm{ch}}=\frac{1}{2}\left(\frac{1}{R}\frac{dR}{dt}+\frac{1}{a_{\mathrm{sz}}}\frac{d{a}_{\mathrm{sz}}}{dt}-\tan \left(\eta \right)\frac{d\eta }{dt}\right)\sqrt{{\mathrm{Ra}}_{\mathrm{sz}}\cos \left(\eta \right)}$
During the early part of the downswing, all the derivative terms will contribute to a_ch, but in the later part of the downswing when R is reaching its maximum value, R_max, and η is approaching zero, the dominant term by far is the da_sz/dt term, which allows the simplification for this part of the swing:

[0111] $57.a_{\mathrm{ch}}=\frac{1}{2}\left(\frac{1}{a_{\mathrm{sz}}}\frac{d{a}_{\mathrm{sz}}}{dt}\right)\sqrt{{\mathrm{Ra}}_{\mathrm{sz}}\cos \left(\eta \right)}$
With discreet sensor data taken at time intervals Δt, the equivalent of the above is:

[0112] $58.a_{\mathrm{ch}}=\frac{\sqrt{R\cos \left(\eta \right)}}{\Delta t}\left(\sqrt{a_{\mathrm{sz}}\left(t_n\right)-a_{\mathrm{sz}}\left(t_{n-1}\right)}\right)$
It is convenient to define the behavior for a_ch for the case where R=R_max and η=0, so that from equation (52) a_z-radial=a_sz, which defines:

[0113] $59.a_{\mathrm{chsz}}=\frac{\sqrt{R_{\max }}}{\Delta t}\left(\sqrt{a_{\mathrm{sz}}\left(t_n\right)-a_{\mathrm{sz}}\left(t_{n-1}\right)}\right)$
Then the inertial spatial translation acceleration component of the club head is:

[0114] $60.a_{\mathrm{ch}}=a_{\mathrm{chsz}}\frac{\sqrt{R\cos \left(\eta \right)}}{\sqrt{R_{\max }}}$

[0115] Substituting equation (52) and (60) back into equations (50) and (51) we have the equations containing all golf swing metric angles assuming no module mounting angle errors in terms of direct measured sensor outputs:
61. a_sx=a_chsz({square root over (R cos(η))}/{square root over (R_Max)})cos(Φ)cos(η)−a_szcos(η)cos(Φ)sin(η)
62. a_sy=a_chsz({square root over (R cos(η))}/{square root over (R_Max)})tan(Φ)cos(η)+a_szcos(η)sin(Ω)−a_szcos(η)tan(Φ)sin(η)
Using equation (62) to solve for Φ, since this is the only equation that contains both η and Ω, yields:

[0116] $63.\tan \left(\Phi \right)=\frac{a_{\mathrm{sy}}-a_{\mathrm{sz}}\cos \left(\eta \right)\sin \left(\Omega \right)}{a_{\mathrm{chsz}}\left(\sqrt{R\cos \left(\eta \right)}/\sqrt{R_{\mathrm{Max}}}\right)\cos \left(\eta \right)-a_{\mathrm{sz}}\cos \left(\eta \right)\sin \left(\eta \right)}$

[0117] Now there are two equations with three unknowns. However, one of the unknowns, η, has the curve fit parameter C_η that can be iteratively determined to give best results for continuity of the resulting time varying curves for each of the system variables. Also, there are boundary conditions from the multi-lever model of the swing that are applied, to specifics points and areas of the golf swing, such as the point of maximum club head velocity at the end of the downstroke, where:

• • [0118] 1. For a golf swing approaching max velocity the value of η approaches zero,
  • [0119] 2. Ω is at a maximum value when centrifugal force is highest, which occurs at maximum velocity.
  • [0120] 3. The club face angle, Φ, can vary greatly at maximum club head velocity. However, regardless of the angle at maximum velocity the angle is changing at a virtual constant rate just before and after the point of maximum club head velocity.
    This knowledge allows for all equations to be solved, through an interactive process using starting points for the curve fit parameters.

[0121] The angle Ω 601 is a function of a_sz through equations (42), (48) and (52). The curve fit constant, C_Ω, is required since different shafts can have an overall stiffness constant that is equal, however, the segmented stiffness profile of the shaft can vary along the taper of the shaft. The value of C_Ω will be very close to one, typically less than 1/10 of a percent variation for the condition of no module mounting angle error from the intended alignment. Values of C_Ω greater or less than 1/10 of a percent indicates a module mounting error angle along the y_cm-axis which will be discussed later. Re-writing equation (42) using (52):

[0122] $64.\Omega =\frac{C_{\Omega }{\mathrm{dm}}_s{a}_{\mathrm{sz}}\cos \left(\eta \right)}{C\left(\mathrm{KC}+m_s{a}_{\mathrm{sz}}\cos \left(\eta \right)\right)}$
The constants in equation (64) are:

• • [0123] C_Ω Multiplying curve fit factor applied for iterative solution
  • [0124] d Distance from housel to center of gravity (COG) of club head
  • [0125] m_s mass of club head system, including club head and Club Head Module
  • [0126] a_sz The measured z_f-axis 105 acceleration force value
  • [0127] K Stiffness coefficient of shaft supplied by the golfer or which can Be determined in the calibration process associated with the user profile entry section of the analysis program
  • [0128] C Club length
    The angle η 401 is found from equation (47):

[0129] $65.\eta =C_{\eta }{\tan }^{-1}\left(\frac{a_{\mathrm{ch}}}{a_{z-\mathrm{radial}}}\right)$
The curve fit parameter, C_η, has an initial value of 0.75.

[0130] An iterative solution process is used to solve equations (61), (63), and (64), using (65) for η 401, which has the following defined steps for the discreet data tables obtained by the sensors:

• • [0131] 1. Determine from sample points of a_sz the zero crossing position of a_chsz. This is the point where the club head acceleration is zero and therefore the maximum velocity is achieved. Because the samples are digitized quantities at discrete time increments there will be two sample points, where a_chsz has a positive value and an adjacent sample point where a_chsz has a negative value.
  • [0132] 2. Course tune of Ω 601: Use initial approximation values to solve for the numerator of tan (Φ) of equation (63) with respect to the sample point where a_ch passes through zero:
    • [0133] a. Numerator of tan (Φ)={a_sy−a_sz cos(η)sin(Ω)}
    • [0134] b. The numerator of tan (Φ) in equation 63 represents the measured value of a_sy minus a_z-radial components resulting from angle Ω with the following conditions at maximum velocity:
    • [0135] i. Toe down angle Ω, which is at its maximum value at maximum club head velocity, where maximum a_sz is achieved at η=0, for which a_sz=a_z-radial From equation (52).
    • [0136] ii. Angle η 401, which is a function of wrist cock and shaft flex lag/lead, is zero when maximum velocity is reached and a_ch is zero.
    • [0137] c. Use the multiplying constant C_Ω to adjust the Ω 601 equation so that the tan (Φ) numerator function sample point value, equivalent to the first negative sample point value of a_ch, is set to the value zero.
  • [0138] 3. Use new course tune value for the Ω 601 function to calculate Φ 501 from equation (63) for all sample points.
  • [0139] 4. Next, fine tune the multiplying constant C_Ω of the Ω 601 function by evaluating the slope of Ω 501, for the point pairs before, through, and after maximum velocity.
    • [0140] a. Examine sample point pairs of the total tan (Φ) function given by equation (63) before maximum velocity, through maximum velocity, and after maximum velocity, evaluating slope variation across sample pairs.
    • [0141] b. Evaluate sequential slope point pairs comparing slopes to determine a variation metric.
    • [0142] c. Tune multiplying constant C_Ω of Ω 601 function in very small increments until the slope of Φ 501 of all sample point pairs are equivalent.
    • [0143] d. Now the value of the Ω function is defined but the value of η is still given with the initial value of C_η=0.75. Therefore, even though the value of Φ 501 is exact for values very near max velocity where η 401 approaches zero, values of Φ 501 are only approximations away from maximum velocity since Φ 501 is a function of η 401, which at this point is limited by the initial approximation.
  • [0144] 5. Calculate all sample points for the for the following functions:
    • [0145] a. The fine tuned function Ω 601
    • [0146] b. Approximate function η 401 with C_η=0.75.
    • [0147] c. Function Φ 501 from equation (63)
    • [0148] i. Which will be exact for sample points close to maximum velocity
    • [0149] ii. Which will be an approximation for the sample points away from max velocity because the function η 401 is still an approximate function.
  • [0150] 6. Tune the multiplying curve fit constant C_η of the η 401 function using equation (61). This is done by rewriting equation (61) into a form which allows the comparison of a_sx minus the a_sz components which must be equal to a_chsz. The evaluation equation is from (61):
    a. {a_sx+a_szcos(η)cos(φ)sin(η)}/{cos(φ)cos(η)}=a_chsz({square root over (R cos(η))}/{square root over (R_Max)})
    • [0151] b. If everything were exact, the two sides of this equation would be equal. If not, they will differ by the variance:
      Variance={a_sx+a_szcos(η)cos(φ)sin(η)}/{cos(φ)cos(η)}−a_chsz({square root over (R cos(η))}/{square root over (R_Max)})
    • [0152] c. This variance metric is summed across a significant number of sample points before and after maximum velocity for each small increment that C_η is adjusted.
    • [0153] d. The minimum summed variance metric set defines the value of the constant C_η for the η 401 function.
  • [0154] 7. Compare the value of C_η obtained at the conclusion of the above sequence with the starting value of C_η, and if the difference is greater than 0.1 repeat steps 3 through 7 where the initial value for C_η in step 3 is the last iterated value from step 6.d. When the difference is less than 0.1, the final value of C_η has been obtained.
  • [0155] 8. Angle α 403 is now solved from equation (23) with η 401 across all sample points:
    α=cos⁻¹((R cos(η)−C)/A)
    • [0156] a. α 403 represents the sum of wrist cock angle and shaft flex lag/lead angle as defined by α=α_wc+α_sf.
    • [0157] b. In a standard golf swing the wrist cock angle is a decreasing angle at a constant rate during the down stroke to maximum club head velocity. Therefore, the angle can be approximated as a straight line from the point where wrist cock unwind is initiated.
    • [0158] c. The slope of the angle α_we 701 is:
    • [0159] i. [α_wc (at wrist cock unwind initiation)-α_wc (club head max Velocity)]/ΔT, where ΔT is the time duration for this occurrence.
    • [0160] d. Since α_wc 701 goes to zero at the point of maximum velocity and the time duration αT is known, the function of angle α_wc 701 is now defined.
  • [0161] 9. The shaft flex angle α_sf 702 is now defined as α_sf=α−α_wc for all sample points during down stroke. Any deviation from the straight line function of α_wc 701 is due to shaft flex.
    The iterative analysis solution described above is based on the club head module being mounted so that the x_f-axis 104, y_f-axis 106, and z_f-axis 105 associated with the club head module 101 are aligned correctly with the golf club structural alignment elements as previously described in FIG. 2.

[0162] Since the module 101 attaches to the top of the club head 201, which is a non-symmetric complex domed surface, the mounting of the club head module 101 is prone to variation in alignment of the x_f-axis 104, z_f-axis 105, and y_f-axis 106 with respect to the golf club reference structures described in FIG. 2.

[0163] During mounting of the club head module 101, as shown in FIG. 10, the front surface 102 of the club head module 101 can easily be aligned with the club face/club head top surface seam 1002. This alignment results in the y_f-axis 106 being parallel to the plane 203 which is the plane created if the club face has zero loft. Using this as the only alignment reference for attaching the club head module 101 to the club head 201, two degrees of freedom still exist that can contribute to club module 101 mounting angle errors. The module 101 mount angle errors can be described with two angles resulting from the following conditions:

• • [0164] 1. The module 101 being mounted a greater distance away or closer to the club face seam 1002 causing an angle rotation around the y_f-axis 106 causing the x_f-axis 104 and z_f-axis 105 to be misaligned with their intended club structure references. The mathematical label that describes this angle of rotation is λ 1103 (as shown in FIG. 11).
  • [0165] 2. The module 101 being mounted closer to or farther away from the club shaft 202 causing an angle rotation around the x_f-axis 104 causing the y_f-axis 106 and the z_f-axis 105 to be misaligned with the intended club structure references. The mathematical label that describes this angle of rotation is κ 1201 (as shown in FIG. 12).

[0166] The issue of mounting angle variation is most prevalent with the club head module 101 being rotated around the y_f-axis. As shown in FIG. 11, the club head module 101 is mounted with the x_f-axis 104 parallel to the plane 1101 that is defined as perpendicular to the shaft axis 1102. With this condition met the angle value λ=0 1103 indicates no rotation around the y_f-axis 106 (not shown but is perpendicular to drawing surface). As shown in FIG. 11A, the club head module 101 is mounted closer to the club face seam 1002 causing a negative value for the angle λ 1103 between the plane 1101 and the x_f-axis 104. As shown in FIG. 11B, the club head module 101 is mounted further from the seam 1002 resulting in a positive value for the angle λ 1103 between the plane 1101 and the x_f-axis 104. On a typical club head, and depending on how far back or forward on the club head dome the module 101 is mounted, the mounting error angle λ 1103 typically varies between −1 degrees and +6 degrees. This angle creates a small rotation around the y_f-axis 106 resulting in a misalignment of the x_f-axis 104 and also the z_f-axis 105. This mounting error can be experimentally determined using a standard golf swing.

[0167] For a linear acceleration path the relationship between true acceleration and that of the misaligned measured value of a_sx is given by the following equations where a_sx-true is defined as what the measured data would be along the x_f-axis 104 with λ=0 1103 degrees. A similar definition holds for a_sz-true along the z_f axis 105. Then:
66. α_sx-true=α_sx/cos(λ)
67. α_sz-true=α_sz/cos(λ)
However, the travel path 307 is not linear for a golf swing which creates a radial component due to the fixed orientation error between the offset module measurement coordinate system and the properly aligned module measurement coordinate system. As a result, any misalignment of the club head module axis by angle λ creates an a_z-radial component as measured by the misaligned x_f-axis 104. The a_z-radial component contributes to the a_sx measurement in the following manner:
68. α_sx=α_sx-true+α_szsin(λ)
The angle λ 1103 is constant in relation to the club structure, making the relationship above constant, or always true, for the entire swing. The detection and calibrating correction process of the mounting variation angle λ 1103 is determined by examining equations (50) and (53) at the point of maximum velocity where by definition:

• • [0168] η goes to zero
  • [0169] a_ch goes to zero
    Therefore, at maximum velocity a_sx-true must also go to zero. At maximum velocity:

[0170] $69.a_{\mathrm{sx}-\mathrm{true}}=a_{\mathrm{sx}}-a_{\mathrm{sz}}\sin \left(\lambda \right)=0$ $70.\lambda ={\sin }^{-1}\left(\frac{a_{\mathrm{sx}}}{a_{\mathrm{sz}}}\right)$

[0171] Now the measured data arrays for both the affected measurement axis x_f-axis 104 and z_f-axis 105 must be updated with calibrated data arrays.
71. α_sx-cal=α_sy−α_szsinλ
72. α_sz-cal=α_sz/cos λ
The new calibrated data arrays a_sx-cal and a_sz-cal are now used and replaces all a_sx and a_sz values in previous equations which completes the detection and calibration of club head module mounting errors due to a error rotation around the y_f-axis 106.

[0172] Now the final detection and calibration of the club head module 101 mounting error angle κ 1201 around the x_f-axis 104 can be done. As shown in FIG. 12, the angle κ 1201 is zero when the club head module 101 is perfectly mounted, defined as when the club head module 101 axis y_f-axis 106 is parallel with the plane 1101, that is perpendicular to the shaft axis 1102. As shown in FIG. 12A when the club head module 101 is mounted closer to the shaft the y_f-axis 106 intersects the plane 1101 creating a negative value for the angle κ 1201. As shown in FIG. 12B the angle κ 1201 is a positive value resulting from the intersection of the y_f-axis 106 and the plane 1101 when the module 101 is mounted further away from the shaft.

[0173] The detection of mounting error angle κ 1201 is achieved by evaluating C_Ω resulting from the iterative solution steps 2 though 4 described earlier. If C_Ω is not very close or equal to one, then there is an additional a_z-radial contribution to a_sy from mounting error angle κ 1201. The magnitude of mounting error angle κ 1201 is determined by evaluating Ω 601 at maximum velocity from equation (64) where for no mounting error C_Ω=1. Then the mounting angle κ 1201 is determined by:
73. κ=(C_Ω−1)(dm_sα_szcos(η))/(C(KC+m_sα_szcos(η)))
As previously described for mounting angle error λ, the mounting error angle κ 1201 affects the two measurement sensors along the y_f-axis 106 and the z_f-axis 105. Consistent with the radial component errors resulting from the λ 1201 mounting angle error, the κ 1201 mounting angle error is under the same constraints. Therefore:
74. α_sy-cal=α_sy−α_szsin(κ)
75. α_sz-cal=α_sz/cos λ
The new calibrated data arrays a_sy-cal and a_sz-cal are now used and replaces all a_sy and a_sz values in previous equations which complete the detection and calibration of club head module mounting errors due to a mounting error rotation around the x_f-axis 104 .

[0174] Thereby, the preferred embodiment described above, is able to define the dynamic relationship between the module 101 measured axes coordinate system and the inertial acceleration force axes coordinate system using the multi-lever model and to define all related angle behaviors, including module 101 mounting errors.

[0175] All of the dynamically changing golf metrics described as angle and or amplitude values change with respect to time. To visually convey these metrics to the golfer, they are graphed in the form of value versus time. The graphing function can be a separate computer program that retrieves output data from the computational algorithm or the graphing function can be integrated in to a single program that includes the computational algorithm.

[0176] The standard golf swing can be broken into four basic interrelated swing segments that include the backswing, pause and reversal, down stroke, also called the power-stroke, and follow-through. With all angles between coordinate systems defined and the ability to separate centrifugal inertial component from inertial spatial translation components for each club head module measured axis, the relationships of the data component dynamics can now be evaluated to define trigger points that can indicate start points, end points, or transition points from one swing segment to another. These trigger points are related to specific samples with specific time relationships defined with all other points, allowing precise time durations for each swing segment to be defined. The logic function that is employed to define a trigger point can vary since there are many different conditional relationships that can be employed to conclude the same trigger point. As an example, the logic to define the trigger point that defines the transition between the back swing segment and the pause and reversal segment is:

• • [0177] If a_z-radial(tn)<1.5 g
  • [0178] AND
  • [0179] a_sx-linear(tn)=0
  • [0180] AND
  • [0181] AVG(a_sx-linear(tn-5) thru a_sx-linear(tn))<-1.2 g
  • [0182] AND
  • [0183] AVG(a_sx-linear(tn) thru a_sx-linear(tn+5))>+1.2 g By defining the exact time duration for each swing segment and understanding that each swing segment is related and continuous with an adjacent segment, the golfer can focus improvement strategies more precisely by examining swing segments separately.

[0184] By incorporating a low mass object that is used as a substitute strike target for an actual golf ball the time relationship between maximum club head velocity and contact with the strike target can be achieved. The low mass object, such as a golf waffle ball, can create a small perturbation which can be detected by at least one of the sensor measurements without substantially changing the characteristics of the overall measurements. In addition, the mass of the substitute strike object is small enough that it does not substantially change the inertial acceleration forces acting on the club head or the dynamically changing relationship of the inertial axes coordinate system in relation to the module measured axes coordinate system.

[0185] The data transfer from the club head module 101 to a user interface can take place in two different ways: 1) wirelessly to a receiver module plugged into a laptop or other smart device, or 2) a wired path to a user module that is attached to the golf club near the golf club grip.

[0186] The preferred embodiment as shown in FIG. 13 demonstrates the module 101 transmitting measured data through a wireless method 1303 to a receiver module 1301 that is plugged into a computer laptop 1302. The receiver module 1301 transfers the data through a USB port to the computer laptop 1302 where the data is processed by the computational algorithm and displayed to the golfer 301.

[0187] In another embodiment, as shown in FIG. 14, the club head module 101 communicates swing data through a wired connection 1401 to a user interface module 1402 that is attached to the club shaft 202 below the grip 1403. The interface module 1402 contains the processing power to compute the metrics and display those metrics on the graphical and text display 1404.

[0188] The approach developed above can also be applied for a golf club swing when the golf club head contacts the golf ball. For this case, the above analysis returns the values of the three angles and club head velocity just before impact. Using these values along with the sensor measurements after impact describing the change in momentum and the abrupt orientation change between the module's measured sensor coordinate system and the inertial motional acceleration force coordinate system will enable the determination of where on the club head face the ball was hit, and the golf ball velocity.

[0189] Although specific embodiments of the invention have been disclosed, those having ordinary skill in the art will understand that changes can be made to the specific embodiments without departing form the spirit and scope of the invention. The scope of the invention is not to be restricted, therefore, to the specific embodiments. Furthermore, it is intended that the appended claims cover any and all such applications, modifications, and embodiments within the scope of the present invention.