THE DESIGN AND APPLICATION OF WIRELESS MEMS INERTIAL MEASUREMENT UNITS FOR THE MEASUREMENT AND ANALYSIS OF GOLF SWINGS

Transcription

1 THE DESIGN AND APPLICATION OF WIRELESS MEMS INERTIAL MEASUREMENT UNITS FOR THE MEASUREMENT AND ANALYSIS OF GOLF SWINGS by Kevin W. King A diertation ubmitted in partial fulfillment of the requirement for the degree of Doctor of Philoophy (Mechanical Engineering) in The Univerity of Michigan 2008 Doctoral Committee: Profeor Noel C. Perkin, Chair Profeor Richard B. Gillepie Profeor Karl Groh Reearch Profeor Jame A. Ahton-Miller

2 Kevin King 2008

3 ACKNOWLEDGEMENTS I would like to extend my inceret thank you to Prof. Noel Perkin for hi time, effort and friendhip. Your contribution have greatly enhanced thi work and you have taught me a greatly deal that I will take with me to all my future endeavor. I would alo like to greatly acknowledge the upport provided by the Univerity of Michigan Office of Technology Tranfer and Commercialization, PING Golf, and Buck80 o. Without thi upport thi reearch would not have been poible. Thank you to my committee member, Prof. Karl Groh, Prof. Brent Gillepie, and Prof. Jame Ahton-Miller, for your input, time and careful evaluation of thi work. I would alo like to thank the following individual and group for their contribution to thi work; Dan Brodrick and the Michigan Office of Technology Tranfer and Commercialization for their diligent patent work and effort to publicize thi reearch, Sangwon Yoon and Prof. Khalil Najafi for their effort toward the development of initial prototype, Jutin Doyle for hi collaboration on the development of the MEMS putting analyi ytem, Joonkeol Song for hi effort in developing the calibration ytem, Andrew Buch and Bill Detriac for the deign and contruction of the putting robot, Roe Caard, Kathty Teichert and The Univerity of Michigan Women Golf Team for their participation in the tudy of putting technique, David May for algorithm development and illutration, Simon Koter for photograph and illutration, Greg Batin for hi work on validation of the golf haft flexible model, Brian Schielke for hi work to invetigate putting technique and fault, and finally to all thoe who participated in meaurement trial and who lent their upport to thi project, Thank You. ii

4 Table of Content Acknowledgement... ii Lit of Figure... v Lit of Table... xii Chapter 1 Introduction 1.1 Motivation and Background Reearch Objective and Organization of Diertation Summary of Major Reearch Contribution... 9 Chapter 2 Wirele MEMS Inertial Senor Sytem for Meauring Golf Putting Dynamic 2.1 Meaurement Theory: Kinematic of a Golf Swing Miniature Wirele IMU Hardware Performance and Accuracy Cloure Chapter 3 Undertanding Putting Stroke Kinematic via the MEMS Senor 3.1 Literature Specific to Putting Technique Two Parameter Kinematic Model for Putting Evaluation of Robot Putting Stroke Uing Kinematic Model Evaluation of Human Putting Stroke Uing Kinematic Model Identification of Common Putting Fault Cloure Chapter 4 Extenion of MEMS Inertial Senor Sytem to Highly Dynamic Golf Swing 4.1 Senor Requirement for Full Swing Meaurement of Full Swing Cloure Chapter 5 Planar Flexible Model of a Golf Club and Experimental Validation 5.1 Derivation of Planar Model (Lagrange Formulation) Experimental Validation of Planar Model Cloure Chapter 6 Three-Dimenional Flexible Model of a Golf Club 6.1 Derivation of 3D Model (Lagrange Formulation) Experimental Validation Cloure iii

5 Chapter 7 Contribution, Concluion and Future Direction 7.1 Contribution and Concluion Future Direction Appendix A Three-Dimenional Model Equation Reference iv

6 Lit of Figure Figure 2.1 (a) Club illutrated at addre (t=0) and at the end of the backwing during a putting troke. The frame ( Iˆ, Jˆ, Kˆ ) denote a lab-fixed (inertial frame) at O which coincide with the location of the centroid f of the clubface at addre. At addre, Ĵ define the target line. ( iˆ, ˆj, kˆ ) denote a body-fixed frame at f f f f (face frame), wherea ( iˆ, ˆ, ˆ j k ) denote a body-fixed frame at (haft frame) within the haft near the end of the grip. (b) Exploded club head view howing how the face frame and the haft frame are related by the manufactured lie and loft angle and Definition of dynamic loft () t, lie () t and face angle () t of the club head uing projection of the face frame ( iˆ, ˆj, k ˆ ) f f f 2.3 Partially exploded CAD drawing of a portion of the miniature IMU howing poition and orientation of MEMS accelerometer and angular rate gyro. The location and orientation of the haft reference frame ( iˆ, ˆ, ˆ j k ) i defined by the ene axe of a tri-axi accelerometer. Three angular rate gyro on mutually-orthogonal face have ene axe aligned with the axe of the accelerometer. The enor circuit board are highlighted in green Photo illutrate the major component of the fully aembled wirele IMU that fit within the mall confine of the haft of a golf club. An analog circuit include the MEMS inertial enor (ee Fig. 2.3) and a digital circuit include a microproceor, wirele tranceiver, and an antenna. A battery i houed within the hollow haft-enor coupling at the far left Calibration fixture ued to characterize the 24 calibration contant for the aembled IMU. The principal component include a leveling plate, a bulleye level, a preciion enor mounting block, and a haft with an optical encoder v

7 2.6 Single degree-of-freedom (pendulum) putting robot ued to ae the accuracy of the enor ytem during the dynamic phae of putting and at impact. Photo illutrate the principal component of the robot including a houlder joint that allow rotation about one axi a meaured by an optical encoder, adjutable (then fixed) elbow forearm, and writ joint, and a clamp that grip the club Dynamic loft, lie and face angle computed from MEMS enor ytem for a typical wing uing the putting robot Enlarged view of dynamic tracking error during a typical wing with putting robot Impact tape record of eight wing made with the putting robot. Left (putt no. 1-3) and right (putt no. 5-8) record demontrate that the clubface return at impact to ame poition at addre to within the (viual) accuracy of the tape (approximately 2 mm). The two impact tape trip are for identical etup of the robot, but allow a clear view of the impact pattern. The general dimenion of the tape are noted in the figure and the concentric circle increae in diameter by 1cm for each ucceive circle Computed Carteian coordinate for path of point f on the face of the putter baed on meaured accelerometer and gyro data for a typical wing with the putting robot (Top) Overhead view of poition data from one trial putt. (Bottom) Face angle a a function of poition. Note the highly magnified cale for the X poition coordinate (ordinate) Illutration of pendulum and gate tyle putting troke a viewed from overhead [41]. (a) Pendulum troke i alo referred to a the Pure Inline and Square (PILS) troke by Pelz. (b) Gate troke, howing the club head moving on an arc and the clubface opening and cloing during the troke Definition of two-parameter kinematic model for putting. (a) The parameter define the angle the rotation axi make with the horizontal and the parameter Larm define the perpendicular ditance from the rotation axi to the ' ' top of the grip (point ). (b) View in the J ˆ, Kˆ -plane howing the parameter L arm, Y grip, and the angle of rotation about the rotation axi (a) Definition of R p, R f, and R f, which are ued to decribe the predicted p putter path baed on the model propoed in thi ection. (b) View looking ' down the Î direction howing the contant length L tot vi

8 3.4 Image of the putting robot introduced in the previou chapter Meaured and predicted orientation angle for a robot-generated PILS putting troke. The blue, red and green curve are the meaured loft, lie, and face angle, repectively. The cyan, magenta and black curve are the correponding angle predicted by the imple, two-parameter kinematic model Meaured and predicted path for the center of the putter head during a robotcontrolled PILS putting troke. (a) Uncorrected for light mialignment of enor axe to the robot rotation axi. (b) Corrected for thi light mialignment Meaured and predicted orientation angle for a robot-generated gate troke o with 10. The blue, red and green curve are the meaured loft, lie, and face angle, repectively. The cyan, magenta and black curve are the correponding angle predicted by the imple, two-parameter kinematic model Meaured and predicted path for the center of the putter head during a robotcontrolled gate troke with 10. (a) Uncorrected for light mialignment o of enor axe to the robot rotation axi. (b) Corrected for thi light mialignment Example 15 ft putt by ubject B howing excellent agreement between meaured and modeled putter kinematic. (a) how the meaured and predicted club head lie, loft and face angle a function of time (blue, red, green = meaured; cyan, magenta, black = model prediction). (b) how the path traced by the center of the clubface in the horizontal (X-Y) plane Example 15 ft putt by ubject B howing excellent agreement between meaured and predicted putting kinematic. (a) how the meaured relationhip between the angular velocity component x and z together with that predicted by the imple model. (b) how the meaured relationhip between the rotation angle and the grip poition coordinatey grip. The meaured point are plotted for the backwing and forward wing. The bet fit model predict the green and black line for the backwing and forward wing, repectively Error metric defined by Eqn. (3.4.2) and (3.4.4) for example 15 ft putt by ubject B. (a) The three angular velocity error metric. (b) Dynamic loft error metric for the backwing and forward wing vii

9 3.12 Example 8 ft putt by ubject L howing more dicrepancy between meaured and modeled putt kinematic. (a) how the meaured and predicted club head lie, loft and face angle a function of time (blue, red, green = meaured, cyan, magenta, black = model prediction). (b) how the path traced by the center of the face of the club head in the horizontal (X-Y) plane Example 8 ft putt by ubject L exhibiting le agreement with the imple kinematic model. (a) how the meaured relationhip between the angular velocity component x and z together with that predicted by the imple model. (b) how the meaured relationhip between the rotation angle and the grip poition coordinatey grip. The meaured point are plotted for the backwing and forward wing. The bet fit model predict the green and black line for the backwing and forward wing, repectively Error metric defined by Eqn. (3.4.2) and (3.4.4) for an 8 ft putt by ubject L. (a) Angular velocity error metric. (b) Dynamic loft error metric for the backwing and forward wing uing ditinct value for L arm Two example putt illutrating different fault leading to exceive face angle at impact. Top figure illutrate the meaured path of the club head (and ball impact location) a viewed in the horizontal (X-Y plane). Bottom figure illutrate the meaured face angle a a function of poition (Y coordinate of club head) in the troke. The example to the left illutrate ignificantly different cloure rate (deg/in) on the forward veru backwing. The example to the right illutrate a period of near-zero rotation at the tart of the forward wing (a) example howing writ-break at the end of the backwing. (b) example i an exaggerated putt uing only writ and leading to very mall value for L a a reult arm 3.17 Loft angle ( in ) veru grip poition along target line ( Y grip ). Example (a) demontrate how the grip can lag the club head at impact leading to exceive loft at impact. Example (b) demontrate how grip can lead the club head at impact leading to inufficient loft at impact. Both fault develop early on in the backwing Poition of club head veru grip. Example (a) demontrate how the grip can lag the club head at impact. Example (b) demontrate how the grip can lead the club head at impact. Both fault develop early on in the backwing Strobocopic photograph of Bobby Jone golf wing which illutrate a full wing (downwing phae and follow through). [47] viii

10 4.2 Meaured and predicted orientation angle for an exaggerated robot putt. The blue, red and green curve are the meaured loft, lie, and face angle, repectively. The cyan, magenta and black curve are the correponding angle predicted by the imple, two-parameter kinematic model Meaured and predicted path for the center of the club head during a robotcontrolled putt. (a) Uncorrected for light mialignment of enor axe to the robot rotation axi. (b) Corrected for thi light mialignment (1.4 o mialignment) (a) Location and orientation of body-fixed i ˆ, ˆ, ˆ f j f k f and ˆ ˆ ˆ,, i j k frame. (b) Chapter 2 definition for lie ( ), loft ( ), and face ( ) orientation angle. 75 o o 4.5 Definition of the lie angle and it domain (a) Lie angle i poitive when i ˆf i above the horizontal planeiˆ, J ˆ. (b) Lie angle achieve it maximum of 90 o when iˆf i vertically up. (c) Lie angle i negative when i ˆf i below the horizontal planeiˆ, J ˆ o o 4.6 Definition of the face angle and it domain (a) Face angle i poitive due to counter-clockwie rotation of j from the Ĵ axi. The ˆ f _ proj face angle a hown in (b) i poitive, while that in (c) i negative Lie, loft and face angle a function of time for a full wing with 6-iron (ubject KK). The annotation decribe the approximate location of the golf haft referred to a poition on a clock. Exploded view zoom in on the angle around impact Swing path of full wing with 6-iron for example of Fig. 4.7 (ubject KK). The centroid of the clubface i repreented by the blue dot during the backwing and by the cyan dot during the downwing. The grip (enor location) i repreented by the green dot in the backwing and by the (lighter) ea-green dot in the downwing. The golf haft i repreented a a red line in the backwing and a magenta line in the downwing. The gray plane repreent the bet fit wing plane during the downwing Velocity of centroid of clubface for full wing (ubject KK). Component reported for the lab-fixed frame. The final data point are the value of the velocity component at impact Swing path of full wing 6-iron from Fig. 4.8 with an aumed impact delayed by 0.01 econd ix

11 4.11 Swing path of full wing with driver (ubject G). The centroid of the clubface i repreented by the blue dot during the backwing and by the cyan dot during the downwing. The grip (enor location) i repreented by the green dot in the backwing and by the (lighter) ea-green dot in the downwing. The golf haft i repreented a a red line in the backwing and a magenta line in the downwing. The gray plane repreent the bet fit wing plane during the downwing Orientation angle a function of time for a full wing with driver with final time 0.01 econd after impact time etimated from the enor meaurement (ubject G ) Diagram of planar model. Point, h and c denote the origin of the haft frame, the hoel (point of interection of haft and club head), and the ma center of the club head, repectively. Two generalized coordinate ( v and v ) are introduced to decribe the deflection and rotation of the end of the haft at the hoel Cantilevered Euler-Bernoulli beam ubject to terminal force F and bending moment M Kinematic relationhip between radiu of curvature and train in a beam due to bending Definition of the gravitational potential energy of the club head. The initial poition with Z=0 and a general poition Z An example illutration of how the MEMS data and the video file are ynchronized in time. Frame 300 how club head jut prior to impacting the wooden back top. Frame 301 how club head during impact Right image how the flexible club experimental etup calling out key point. Left image i a zoomed-in view of the club head that how an example calculation of the horizontal and vertical pixel-to-inch cale factor calibration Example of tatic deflection tet. The pointer haft i defined by point P1 and P4. The flexible club haft pae though point P4, P5 and P1. Point P1 and P5 define the tangent at the end of the haft and the angle formed between thi tangent and the pointer haft i denoted by v. The deflection of P1 from the pointer haft i denoted by v x

12 6.1 Schematic of 3D model. Point, h and c denote the origin of the haft frame, the hoel (point of interection of haft and club head), and the ma center of the club head, repectively. Dahed line repreent the deflected haft. The poition vector R and R c are meaured in the lab-fixed frame I ˆ, Jˆ, K ˆ wherea the poition vector r h frame ˆ ˆ ˆ,, and r c h are meaured in the body-fixed i j k Cantilevered Euler-Bernoulli beam ubject to torque T applied at the end Simulation reult of three-dimenional flexible haft model uing meaured input from MEMS enor ytem. Thi tet i decribed in Chapter 5. (a) Computed time hitorie of the two diplacement coordinate, v and w. (b) Computed time hitorie of the three rotation coordinate, v, w, and h. The reult from the 2D model are uperimpoed in black xi

13 Lit of Table Table 1.1 Performance target for a miniature, MEMS-baed IMU for reolving putting and full wing in golf and target limit on ize and battery life Statitic of initial lie and loft angle teting. Table report difference (error) between angle meaured uing the MEMS enor ytem and an optical encoder for numerou tet over a large ( 50 deg) range Mean and tandard deviation of cloure error for clubface lie, loft and face angle for ten wing uing putting robot. Reult are rounded to the nearet 0.1 degree becaue that i the reolution of the optical encoder ued for independent meaure Achieved accuracy and performance characteritic of the miniature, wirele MEMS IMU for putting dynamic Average error between predicted and meaured putt kinematic for the example 15 ft putt for ubject B Average error between predicted and meaured putt kinematic for the example 8 ft putt for ubject L Average error metric for all (120) trial conducted on a total of 10 ubject Initial and final orientation angle for trial reported in Fig Final angle are reported for both the etimated impact time a well a 10 milliecond later Initial and final orientation angle for trial reported in Fig Final angle are reported for both the etimated impact time a well a 10 milliecond later Parameter for club ued in the validation of the planar model Reult of EI calculated from tatic video tet Comparion of video meaurement to planar model prediction for tatic tet Comparion of rigid body dynamic from video and MEMS enor xii

14 5.5 Comparion of flexible body dynamic from video and planar model xiii

15 CHAPTER 1 INTRODUCTION 1.1 Motivation and Background Golf Swing Meaurement Sytem Golf wing training aid addre one of the major obtacle toward advancement in the port, namely developing a controlled and conitent wing. The mot common wing meaurement ytem utilize photographic technologie including high-peed video, motion capture camera, or trobocopic photography. High peed video, which i extenively ued to capture both club and human kinematic [1-3], require the careful alignment and calibration of two (and preferably three) high-peed camera. Once the video i captured, the rather large video data file mut be analyzed which can become computationally burdenome. In addition, the finite preciion of poitional and angular coordinate derived from digital image of a club mut be carefully conidered particularly when trying to reolve the three-dimenional motion of a complete golf wing. Alternatively, one may ue optical tracking ytem that employ tranmitting marker (often infrared LED ) placed on the golfer and the club. The ue of optical marker may increae poition accuracy and peed data analyi [4-6]. All camera-baed ytem require expenive equipment, careful et-up (alignment), and often controlled lighting. Thee limitation frequently retrict their ue to pecialized indoor facilitie, although ytem for outdoor ue alo exit. Motion capture i alo poible uing other media for triangulation including magnetic [7-8], radio frequency [9], and ultraonic marker [10]. Thee alternative may overcome ome of the limitation of optical triangulation, uch a the need for a direct line of ight, but they alo have limitation ariing from electro-magnetic and acoutic interference, portability, and cot. In addition, 1

16 all marker-baed ytem require the attachment of marker on, or in cloe proximity to, the club head which render thee method invaive to the club. Meaured wing data ha alo guided the development of mathematical model of golf wing. For intance, the multi-body model of Teu [11] and Tunoda [12] are extenively compared with experimental wing meaurement and reveal the limitation of the commonly ued double pendulum model of a golf wing. For example, Iwatubo [13] explore the validity of auming a rigid link connecting the writ and neck. The planar dynamic of the golf wing predicted by the double pendulum model wa addreed by Simon [14]. The intereted reader i referred to [15] which review the cientific literature on wing dynamic including the many mathematical model of golf wing. Pertaining to golf intruction, meaurement of club acceleration, angular velocity, and club head peed reveal ditinct difference between profeional and amateur golfer [16-18]. Undertanding and quantifying thee difference i clearly eential for golf intruction. Reearch in golf wing intruction ha largely focued on the human kinetic of the wing motion. For intance, high peed video wa ued to tudy the influence of hip and trunk motion [19-20]. Subequently, houlder [21-22] and knee [23] rotation were evaluated uing optical motion tracking. The eential timing of all of thee motion i emphaized in [15]. The golf wing meaurement technologie reviewed above all generate quantitative meaurement of wing dynamic. They alo poe one or more major limitation including 1) extenive et-up and alignment, 2) retriction to indoor ue, 3) high cot, 4) lack of portability, 5) placement of intrument (marker) on/near the head of the club, and 6) time conuming data proceing and analyi. Thi tudy introduce an alternative mean to meaure golf wing dynamic that largely overcome thee limitation. In particular, we introduce a miniature and wirele ix degree-of-freedom MEMS inertial meaurement unit (IMU) that fit entirely within the haft of a golf club at the grip end. Upon the development of a companion meaurement theory, the reulting golf wing 2

17 training ytem achieve one example of a broad cla of poible port training aid baed on thi concept [24-25]. The Opportunity for a MEMS-Baed Meaurement Sytem The novel golf wing meaurement ytem decribed in thi tudy ret on the capabilitie of MEMS inertial enor including accelerometer and angular rate gyro. Ultimately, the enor data are integrated in computing the poition and orientation of the club which may execute both low and fat dynamic depending on the wing (e.g., a putting troke veru a full wing). The general ytem-level performance target for an integrated IMU that can effectively reolve golf wing are ummarized in the Table 1.1. Reolution Senor Range Sene Axe Sytem Level Performance Target Angular < 1 deg Poition < 6mm 150 deg/ & 1.5g (putting) 2500 deg/ & 20g (full wing) Three-axi acceleration Three-axi angular velocity Current Draw Battery Life Cro Section Diameter < 40mA > 2 hour cont. < 13mm Table 1.1: Performance target for a miniature, MEMS-baed IMU for reolving putting and full wing in golf and target limit on ize and battery life. MEMS accelerometer technology i relatively mature with MEMS device nearing inertial-grade performance. The tringent tability requirement of inertial-grade device etablih performance goal for MEMS accelerometer a detailed in [26] and [27]. Specific device characteritic for accelerometer that enable the golf wing IMU decribed herein include low power (e.g., 600 A/axi ), programmable range (e.g., 1.5 g to 15 g for three-axi device), and mall footprint (e.g., 5mm x 5mm for three-axi device); ee, for example, [28]. Many commercially available device may be readily employed a part of the golf wing enor ytem. MEMS angular rate gyro are ignificantly more complicated than accelerometer and their deign remain an active reearch topic; ee, for example [29-32]. A major goal i again to attain inertial grade performance which demand uperior enitivity, reolution, cale factor tability, and bia drift tability [29]. Key device characteritic for gyro that enable the golf wing 3

18 enor IMU include low power (e.g., 6 ma/axi), adjutable range (e.g., 75 deg/ to 3000 deg/), and mall footprint (e.g., 7mm x 7mm); ee, for example [33]. Several commercially available MEMS rate gyro meet thee baic requirement. The ultimate performance of the integrated IMU ret on careful calibration of all (ix) enor a detailed herein. A major conideration i enor drift but thi concern i omewhat leened in the golf application due to the rather hort integration time interval aociated with golf wing (2-3 econd typical). Inertial Meaurement Unit and Error Source A reviewed above, currently available MEMS inertial enor enable the development of a miniature and wirele ix degree-of-freedom inertial meaurement unit for meauring the dynamic of a golf wing. Within the context of port training device, the MEMS inertial enor are deployed on the port equipment and therefore meaure both acceleration and angular velocity in a body-fixed frame of reference. Thi i the configuration ued in the ditinct field of inertial navigation where body-fixed inertial enor form what i often called a trap down inertial navigation ytem [27, 34-35]. Sytem capable of accurately meauring the rigid body dynamic of an object uing a plurality of accelerometer and rate gyrocope are commonly called inertial meaurement unit (IMU). It i important to recognize and undertand the work in thi different field a many challenge in accurately meauring the dynamic of porting equipment relate to method in uing IMU to navigate hip, aircraft, pacecraft and miile. Particularly ueful are enor error model and ytem calibration method. At the device level, the dominant error ource are fixed bia, cale factor, cro-axi coupling, vibro-pendulou error and random error [27]. Vibro-pendulou error occur when the enor i ubjected to vibration along the enitivity and pendulum axe imultaneouly. Certain enor deign are either imperviou to or greatly reduce thi error mechanim. There are four categorie of bia error [27]. Fixed or repeatable bia error i a component that i predictable and preent each time the enor i witched on. Temperature-induced bia error arie from the unwanted effect of device heating and 4

19 can be corrected with uitable calibration. Switch-on to witch-on bia i a random error which varie from run to run but i otherwie contant for any one run. In-run variation are a random bia which varie throughout a run and the precie form of thi error varie from one enor type to another. The firt two bia error type (fixed and temperatureinduced) can be corrected with proper calibration. The performance of an IMU will be limited by the latter two form of bia error (witch-on and in-run). Thee two type of error can alo be compenated for if there i apriori knowledge about the motion of the body that can indicate the magnitude of thee error. Static and quai-tatic calibration method are often ued to identify enor bia, cale factor error and ytem-level mialignment of the ene axe. Dynamic calibration method are alo ued to quantify additional contribution to cale factor error a well a cro-coupling and vibro-pendulou error. Titterton [27] ugget the following error model for accelerometer and gyro to cover the majority of enor error. Accelerometer Error Model f f f actual meaured f aa x y z a x a x f B B a a S a M a w f z axa y a z a z y A v z y A y A y A (1.1.1) f a g, where f i the pecific force meaured by the accelerometer B A : A three element vector repreenting the fixed biae. B v : A 3x3 matrix repreenting the vibro-pendulou error coefficient. S A : A diagonal 3x3 matrix repreenting the cale-factor error. M A : A 3x3 kew ymmetric matrix repreenting the mounting mialignment and cro-coupling term. w A : A three element vector repreenting in-run random bia error. 5

20 Rate Gyro Error Model actual meaured x a aa x y z y z x x B B a B a a B S M w aa z a z x y x y z z y G g y ae z x ai z x G y G y G (1.1.2) B G : A three element vector repreenting the fixed biae. B g : A 3x3 matrix repreenting the g-dependent bia coefficient. B ae : A 3x3 matrix repreenting the anioelatic coefficient. B ai : A 3x3 matrix repreenting the anioinertia coefficient. S G : A diagonal 3x3 matrix repreenting the cale-factor error. M G : A 3x3 kew ymmetric matrix repreenting the mounting mialignment and cro-coupling term. w G : A three element vector repreenting the in-run random bia error. In our reearch we will not need to conider all of the error ource incorporated into the two model above, however thee error model erve a the foundation for our error correction trategie a will be decribed in Chapter 2. The Role of Shaft Flexibility The MEMS enor technology dicued above i capable of meauring the rigid body motion of a golf club. In golf, the haft of a driver and an iron i flexible and it i known to deflect during highly dynamic wing. Therefore, quantifying the contribution of flexible body dynamic to the golf wing i an important addition to a golf wing meaurement ytem for driver and iron. By contrat, there i little to no flexible body dynamic for a putter. In thi reearch, we ultimately predict the flexible body dynamic of the haft via a mathematical model for which the input i the rigid body acceleration and angular acceleration meaured by the MEMS enor. An alternative approach would be to embed enor in the haft (e.g. train gauge) to directly meaure the flexible body component a in [36]. However, thi direct approach ha major diadvantage when conidering future commercialization that include a) the invaive attachment of enor 6

21 on or in the haft (e.g., via epoxy), b) the added cot for ignificant preparation and intallation of enor during manufacturing, and c) the added ytem complexity (creation of many additional enor channel, added ignal conditioner, and ignificant temperature-dependence, etc.). The mechanic of a golf haft during a wing i well-approximated uing claical linear theorie for the bending and torion of beam. A golf haft ha length to cro-ection dimenion (i.e. a lenderne ratio) conitent with the ue of Euler-Bernoulli beam theory [37]. A mentioned above, the MEMS enor directly meaure the kinematic of the rigid motion of the haft and pecifically the rigid motion of the grip end of the club. Thi meaured motion ultimately become the input to a flexible body model for the remainder of the club. One derivation of the general theory for a flexible beam undergoing large precribed rigid body motion i developed by Kane et al. [38]. Thi theory aume the flexible motion remain mall compared to the rigid body motion. Butler and Winfield [39] meaured the deflection of the golf haft and their reult clearly demontrate that the haft deflection remain very mall correction to what i otherwie rigid body motion. The theory developed by Kane et al. [38] implifie ignificantly when applied to a golf club ince the haft ha uniform ma denity and i cone haped (tapered) which add ymmetry (e.g., eliminate product of inertia). For rotating circular beam, the model developed by Yigit et al. [40] i cloer to that needed for a flexible golf club and thu it erve a a ueful reference for thi reearch. Suitable approximation to thi model are alo likely to yield accurate prediction of the flexible motion of a golf club a decribed in Chapter 5 and Reearch Objective and Organization of Diertation Objective The objective of thi reearch i to develop an accurate golf wing meaurement and analyi ytem that i highly portable and potentially quite inexpenive. The meaurement and analyi ytem relie on deducing the rigid body dynamic of the golf 7

22 club uing embedded MEMS inertial enor. Flexible body dynamic, if and when needed, are etimated uing dynamical model of flexible golf haft. A major related objective i to identify promiing metric of wing performance by ytematically analyzing player putting troke and full wing. Thee objective are achieved through the theoretical and experimental tudie outlined in the following five chapter. Organization of Diertation Chapter 2 decribe the theory, deign and performance of a high-reolution, miniature, ix degree-of-freedom, wirele IMU for meauring putting troke. Section 2.1 open with a careful development of the meaurement theory for obtaining the kinematic of the putting troke. The theory detail how the raw enor data (body-fixed meaurement of acceleration and angular velocity) i reduced to yield the orientation, poition, and velocity of the putter relative to a lab-fixed frame of reference. Section 2.2 follow with a detailed decription of the wirele IMU deign that include enor, micro-proceor, wirele tranceiver, and battery power upply. Section 2.3 preent tet reult that ytematically explore the accuracy of the wirele IMU following precie calibration. To thi end, a putting robot i alo introduced which independently meaure the kinematic of the putting troke uing a high-reolution optical encoder. Chapter 3 employ the meaurement theory and enor hardware etablihed in Chapter 2 to identify promiing metric of putting performance. Thi chapter open with a review of the major literature on putting technique and Section 3.2 follow by introducing a imple but very ueful kinematic model for all putting troke. Thi model i then applied to the highly controlled putting troke uing the putting robot (Section 3.3) and then to the uncontrained putting troke of player (Section 3.4). The reult of Section 3.5 clearly demontrate how thi technology can uccefully deduce common fault in the putting troke of player with varying kill level. Chapter 4 extend the meaurement theory, hardware and experiment for putting troke (Chapter 2 and 3) to the meaurement and analyi of full golf wing with driver and 8

23 iron. The chapter open by ummarizing the neceary change in the hardware for reolving highly dynamic wing with driver and iron. The reulting high-range enor ytem i then ued in Section 4.2 to explore the major feature of the full golf wing. Reult for the three-dimenional trajectory of the club and the clubface lie, loft and face angle reveal the major phae of the wing including the addre, the backwing, the downwing, and the aociated writ releae jut prior to impact. The reult of Chapter 4 report the rigid body dynamic of golf club during highly dynamic full wing. The influence of haft flexibility i now examined in Chapter 5 in the context of a two-dimenional model for the flexible haft/club head ytem. Thi model include the planar bending of the haft a induced by the acceleration and angular acceleration of the grip during a wing. The derivation of the reulting two-dimenional model for a flexible club i detailed in Section 5.1 while Section 5.2 ummarize companion experimental reult. Meaurement of the bending natural frequency and tranient dynamic for a flexible club demontrate the accuracy of the two-dimenional model. The final chapter, Chapter 6, extend the theory of Chapter 5 to a full three-dimenional model of a flexible golf club. Section 6.1 ummarize the derivation of the equation of motion for the haft conidering the imultaneou bending about two orthogonal axe and torion about the long axi of the haft. Section 6.2 ummarize initial experimental reult for validating the three-dimenional model. 1.3 Summary of Major Reearch Contribution Overall thi reearch contribute a highly portable and inexpenive meaurement and analyi ytem for analyzing golf wing dynamic. Thi contribution follow from the theory, deign, and experimental evaluation of a wirele, ix degree-of-freedom, MEMS inertial meaurement unit that fit wholly within the haft of a golf club. Additional contribution follow from applying thi novel technology to undertand both putting troke and full wing. 9

24 For analyzing putting troke, the meaurement and analyi ytem reolve the threedimenional poition of the clubface to within 3 millimeter and the three-dimenional orientation of the clubface (i.e., the lie, loft and face angle) to within 0.5degree, both with milliecond update. Thi meaurement peed and ignificant preciion enable the detailed characterization of putting troke. In particular, the meaurement upport a imple kinematic model for all putting troke that unifie the pendulum tyle and gate tyle troke previouly ditinguihed in the literature. In addition, a uite of putting metric i identified that pin-point common putting fault from the meaured club head trajectory and orientation. Thee finding confirm the ignificant opportunity for uing thi technology for putting intruction and training. The ame opportunity arie for full wing upon incorporating high-range angular rate gyro and accelerometer to reolve wing made with driver and iron. By extending the meaurement theory for putting, one can reolve the rigid-body dynamic of full wing from enor data. Doing o immediately reveal two well-characterized wing plane, one for the backwing and one for the downwing, that are largely tangent to the otherwie three-dimenional trajectory of the club. In addition, one oberve the ignificant acceleration of the club head that reult from the releae of the writ in the downwing jut prior to impact. Furthermore, the predicted clubface lie, loft and face angle at addre and at impact provide immediate and valuable metric of wing performance. In addition to reolving the rigid body dynamic of a golf club, we propoe and then demontrate how the meaurement theory can be extended to predict the flexible body dynamic a well, and with no additional enor technology. Doing o relie on formulating dynamic model for the bending and twiting of the club haft that incorporate, a input, the meaured rigid body motion of the grip. Thi reearch contribute both two- and three-dimenional form of the flexible body model of a golf club and experimental data that upport their accuracy. 10

25 Beyond golf, the meaurement theory and hardware deign decribed in thi diertation hold ignificant promie for training ytem for numerou other port including, for example, baeball, hockey, tenni, and bowling. In addition, thi technology ha relevance to far broader field of application including, for example, port injury mechanic, rehabilitation cience, gait analyi, urgeon training, etc. Thi reearch ha contributed the foundation for thee upcoming application, many of which have already been initiated a a direct reult of thi tudy. 11

26 CHAPTER 2 WIRELESS MEMS INERTIAL SENSOR FOR MEASURING GOLF PUTTING DYNAMICS The objective of thi chapter i to decribe the deign and performance of a novel enor ytem. More pecifically, we focu herein on a MEMS baed IMU (inertial meaurement unit) that meaure the dynamic of the putting troke, with tringent accuracy requirement placed on the computed orientation and poition of the club head in threedimenion. We open by reviewing the meaurement theory that enable one to employ a club-fixed MEMS inertial enor to compute the kinematic of the club including dynamic lie, loft and face angle of the club head, the velocity of the club head, and the path decribed by the club head in pace. Next, we ummarize the enor hardware that incorporate MEMS inertial enor, a microproceor, and a wirele tranceiver into a miniature, battery-powered IMU that can fit within the mall confine of the golf club haft. We then preent experimental data that demontrate the accuracy of the enor ytem in meauring putting dynamic. We cloe with concluion and a ummary of the performance characteritic of the aembled enor ytem. 2.1 Meaurement Theory: Kinematic of a Golf Swing The overall objective of thi meaurement ytem i to predict the kinematic of a golf wing and pecifically the orientation, velocity, and path of the club head; ee for intance, [18-20], [41-42]. Thi paper aume that the golf club remain rigid throughout the wing, which i an appropriate aumption for putting a the club undergoe a low, mooth motion that will not lead to appreciable flexing of the haft. More preciely, we wih to predict three angle that define the orientation of the club head and the poition and velocity of the geometric center of the face of the club head by exploiting 12

27 meaurement from MEMS inertial enor mounted remotely and within the grip end of the haft of the club. Definition of Reference Frame and Orientation Angle Conider Fig. 2.1(a) which illutrate a golf club at addre, where the club i firt lined up with the ball, and at the end of the backwing for a typical putting troke. The geometric center of the face of the club i denoted by point f and the location of the enor module within the haft at the grip end i denoted by point. At addre, point f coincide with point O which i the origin of an inertial or lab frame ( I ˆ, J ˆ, K ˆ ). In the lab frame, the unit vector ˆK point vertically up (gravity g gkˆ ) and the unit vector ( I, Jˆ ) define the horizontal plane with Ĵ aligned with the initial target line of the clubface which i etablihed at addre. 13

28 iˆ kˆ ˆ j kˆ iˆf top edge kˆ f f ˆj f kˆ f (a) (b) Figure 2.1: (a) Club illutrated at addre (t=0) and at the end of the backwing during a putting troke. The frame ( Iˆ, Jˆ, Kˆ ) denote a lab-fixed (inertial frame) at O which coincide with the location of the centroid f of the clubface at addre. At addre, Ĵ define the target line. ( iˆ, ˆj, kˆ ) denote a body-fixed frame at f (face frame), wherea f f f ( iˆ, ˆ, ˆ j k ) denote a body-fixed frame at (haft frame) within the haft near the end of the grip. (b) Exploded club head view howing how the face frame and the haft frame are related by the manufactured lie and loft angle and. A body-fixed frame ( iˆ, ˆj, kˆ ) i introduced at f, on the plane of the clubface, with f f f ˆj f being the outward normal to thi plane. The unit vector iˆf lie in the plane of the face where it i choen to be parallel to the flat indexing urface that define the top of the clubface a hown in Fig. 2.1(b). The unit vector k ˆ f then follow from k ˆ ˆ ˆ f i f j f. A econd body-fixed frame ( iˆ, ˆj, kˆ ) i introduced at point within the haft at the grip. Thi frame define the three mutually-orthogonal ene axe of the accelerometer and the angular rate gyro located within the haft a detailed in the following ection. The two body-fixed frame differ by a rigid-body tranlation r / f(poition of relative to f) 14

29 and a rotation that i determined apriori by the known club geometry. The rotation matrix F S R define thi rigid body rotation per iˆ ˆ f i ˆ F S j ˆ f R j kˆ ˆ f k (2.1.1) which, in turn, depend upon two angle determined by the deign of the club. Thee two angle, referred to a the manufactured club lie and loft angle in Fig. 2.1(b), yield the following form for thi rotation matrix F R S co in in co in 0 co in in in co co co The angle and are Euler angle and the rotation matrix of equation (2.1.2) i (2.1.2) derived by firt rotating an amount about the ˆj f direction and then an amount about the i ˆf direction. Notice here that the third Euler rotation about the k ˆ f axi i 0. S The enor module within the haft meaure the acceleration of point, denoted a () t, and the angular velocity of the club, denoted S () t. Here the upercript S i ued to emphaize that the component of thee vector are meaured with repect to the bodyfixed frame ( iˆ, ˆj, k ˆ ). The overall objective i to utilize thee meaurement, from MEMS inertial enor mounted within the grip end of the haft, to deduce the orientation of the club head a well a the velocity and poition of point f dynamically during the putting troke. 15

30 Defining the Orientation of the Club Head We begin by defining the orientation of the club head which i equivalent to decribing the orientation of the body-fixed frame ( iˆ, ˆj, kˆ ) relative to the lab frame ( Iˆ, Jˆ, K ˆ ). To f f f thi end, we introduce three orientation angle for the club head referred to a the dynamic loft, lie, and face angle. Thee three angle, which are commonly mentioned in the literature on golf, are critical in undertanding the kinematic of a golf wing. Depite their common ue however, there doe not appear to be precie definition for thee angle. We anwer thi hortcoming by introducing the following definition in reference to Fig loft lie face target line Figure 2.2: Definition of dynamic loft () t, lie () t and face angle () t of the club head uing projection of the face frame ( iˆ, ˆj, k ˆ ). f f f Begin by defining the loft angle a the angle formed between the normal to the face, ˆ f j, and it projection, ˆproj j, onto the horizontal plane. Next, define the lie angle a the angle formed between the top edge of the face, i ˆf, and it projection, i ˆproj, onto the horizontal plane. Finally, introduce the face angle a the angle formed between j and the original target line Ĵ. When 0, the face angle i aid to be cloed ; ˆproj wherea when 0, the face angle i aid to be open. Ideally one deire that 0 at impact o that the ball travel olely along the original target line. For convenience below, 16

31 we alo introduce the angle formed between iˆproj and Î which i determined by the prior three angle. To etablih the relationhip between and the other three angle one could ue any of the contraint propertie of direction coine matrice. For example uing the contraint that the column of the matrix mut be orthogonal, one could take the dot product of column 1 and 2, then olve for in term of the other three angle. Recognize that the dynamic loft,, and lie,, of the club head vary ignificantly with time during the wing and that they are alo ditinct from the previouly introduced manufactured loft,, and lie,, of the club which remain contant. The dynamic orientation of the club head can now be defined by the rotation matrix L F R between the body-fixed frame at f and the lab frame at O per Iˆ iˆ f ˆ L F J R ˆj f Kˆ kˆ f Uing the above definition for (,,, ), thi rotation matrix become (2.1.3) L R F co in in co in co in in co co in co co co in in co co co in in in co co co co in co in co (2.1.4) Deducing the Intantaneou Orientation of the Club Head from Senor Data We hall now demontrate how thi rotation matrix can be contructed upon integrating the angular velocity of the club, S () t, to obtain the intantaneou or dynamic clubface loft, lie and face angle. The integration procedure begin by etablihing the initial condition of the clubface orientation angle a follow. At addre (t=0), the club i at ret, and thu S (0) 0 (2.1.5) and 17

32 a (0) gkˆ S (2.1.6) In (2.1.6), we recognize that the MEMS accelerometer, which meaure dynamic ignal down to DC, detect an apparent upward acceleration equivalent to gravity during addre. Tranforming thi meaured initial acceleration of point from the haft frame to the face frame via (2.1.2) yield a F (0) F R S a S (0) a F (0) iˆ a F (0) ˆj a F (0) kˆ gkˆ x f y f z f (2.1.7) F F F where ( a (0), a (0), a (0)) are the three component of gravity with repect to the face frame. x y z By definition at addre, (0) 0 which then implifie L R F (2.1.4). Uing thi implified form, (2.1.3) yield Kˆ in (0) iˆ in (0) ˆj co (0) co (0) co (0) kˆ (2.1.8) f f f Subtituting (2.1.8) into (2.1.7) enable the olution of the initial clubface loft angle, and the initial lie angle, F a (0) 1 y (0) in g (2.1.9) F 1 a (0) (0) in x g (2.1.10) from the meaured initial acceleration data. Computation of (0) follow readily from the ame implified form of (2.1.4) by exploiting the propertie of thi orthogonal tranformation. For intance, uing the fact that the firt two column are orthogonal yield 1 in (0) in (0) (0) in co (0) co (0) (2.1.11) With the above initial condition on the orientation angle, we now proceed to integrate the meaured angular velocity vector forward in time. To thi end, we rewrite the rotation matrix L R F (2.1.4) in term of tandard Euler parameter [43] 18

33 L e1 e2 e3 e4 e1e 2 e3e4 e1e 3 e2e4 F R 2 e e e e e e e e 2 e e e e 2 e e e e 2 e e e e e e e e (2.1.12) Evaluating both (2.1.4) and (2.1.12) at addre (t=0) and equating component lead to the following initial condition for the four Euler parameter in (0) co (0) co (0)in (0) e1 (0) (2.1.13) 4 e (0) 4 in (0) co (0) e2 (0) (2.1.14) 4 e (0) 4 in (0) co (0)in (0) in (0) co (0) in (0) e3 (0) (2.1.15) 4 e (0) 4 1 e4 (0) 1 co (0) co (0) co (0) co (0) co (0) co (0) (2.1.16) 2 Next, we tranform the meaured angular velocity with component in the haft frame to it equivalent repreentation with component in the face frame per R F F S S The Euler parameter tate equation for the face frame are [43] e e e F x e3 e4 e 1 F y e2 e1 e 4 F z e1 e2 e 3 (2.1.17) F 1 e 2 (2.1.18) which may now be integrated forward in time uing the initial condition (2.1.13)- (2.1.16) above and the meaured angular velocity. Thi computation mut alo atify the contraint e e e e 1 which can be achieved in a variety of way (e.g., via a penalty formulation following [44]). The reulting Euler parameter at each time tep can then be ued to recontruct the dynamic orientation angle of the face frame. Equating the component of (2.1.4) and (2.1.12) and olving for the dynamic clubface loft, lie and face angle yield 19

34 1 ( t) in 2( e2e3 e1e 4) (2.1.19) 1 ( t) in 2( e1e 3 e2e4 ) (2.1.20) 1 2( e1e 2 e3e4 ) ( t) in co (2.1.21) Deducing the Velocity and Poition of the Club Head from Senor Data Upon determining the dynamic orientation of the club head relative to the lab frame, one may now turn attention to computing the abolute velocity and poition of the clubface in the lab frame by uitable integration of the accelerometer data. We begin by firt determining the abolute velocity and poition of point. The acceleration of point in the haft i originally meaured with repect to the haft frame. It equivalent repreentation with repect to the lab frame can be contructed by two ucceive rotation L L F F S S a R R a gkˆ Here, we alo ubtract gravity o that the remaining vector repreent the abolute acceleration of point. Integration of thi reult t L L v ( t) a ( t) dt 0 (2.1.22) (2.1.23) L yield the abolute velocity of point in the lab frame recognizing that v (0) 0. One further integration yield the abolution poition of point in the lab frame where t L L L 0 r ( t) v ( t) dt r (0) r (0) r (0) r (0) L L L f / f (2.1.24) (2.1.25) 20

35 L At addre, r (0) 0 by definition. Furthermore, f r R R r L L F F S S f f (2.1.26) S where r f i a contant vector known apriori from the club geometry. Puruing our objective, one may now compute the abolute velocity and poition of point f in the lab frame via v v R R r L L L F F S S L f f (2.1.27) XIˆ L L L r ˆ f r r f YJ (2.1.28) ZKˆ where X,Y, and Z are Carteian coordinate that decribe the poition of point f in a lab fixed frame. 2.2 Miniature Wirele IMU Hardware Having etablihed the meaurement theory above, we now decribe the inertial enor S unit that provide the requiite acceleration a () t of point and the angular velocity of the club, S () t, both meaured with repect to the haft frame. Begin with Fig. 2.3, which illutrate an exploded view of the portion of the IMU that contain the MEMS inertial enor. 21

36 enor-haft coupling gyro fixture k-axi rate gyro j-axi rate gyro 3-axi accelerometer at point kˆ ˆ j iˆ i-axi rate gyro main board Figure 2.3: Partially exploded CAD drawing of a portion of the miniature IMU howing poition and orientation of MEMS accelerometer and angular rate gyro. The location and orientation of the haft reference frame ( iˆ, ˆ, ˆ j k ) i defined by the ene axe of a tri-axi accelerometer. Three angular rate gyro on mutually-orthogonal face have ene axe aligned with the axe of the accelerometer. The enor circuit board are highlighted in green. Thi portion of the deign conit of three enor circuit board (green) arranged to form three-mutually orthogonal plane. Mounted on the main board i a tri-axi accelerometer which define point. The orientation of the ene axe of thi accelerometer define the haft fixed frame ( iˆ, ˆj, k ˆ ). Three ingle-axi angular rate gyro are illutrated having ene axe aligned with ( iˆ, ˆj, k ˆ ). The i-axi rate gyro i mounted on the main board wherea the j- and k-axi gyro are mounted on eparate board that define mutuallyorthogonal plane with the main board. The orientation of all three plane i achieved by a mall, machined gyro fixture that alo provide mechanical upport. The entire enor aembly i captured at the upper end by a cylindrical coupling (portion hown) that aure the proper poitioning and alignment of the IMU within the haft of the club. The coupling form a cloe lip fit with the inner diameter of the haft and poitive orientation of the IMU i provided via a mall key way machined on the end of the haft. A photo of 22

37 the entire aembled IMU i hown in Fig. 2.4 which alo how the phyical cale of the deign relative to the grip end of a golf club. Shaftenor coupling and 6V battery 2 orthogonal rate gyro 3-axi accelerometer Analog Circuit Rate gyro Microproceor and RF tranceiver Digital Circuit Figure 2.4: Photo illutrate the major component of the fully aembled wirele IMU that fit within the mall confine of the haft of a golf club. An analog circuit include the MEMS inertial enor (ee Fig. 2.3) and a digital circuit include a microproceor, wirele tranceiver, and an antenna. A battery i houed within the hollow haft-enor coupling at the far left. The complete deign yield a very compact and lightweight IMU that add minimal ma to a golf club a detailed below. An analog circuit, powered by a regulated 5V ource, include the MEMS inertial enor decribed above a well a a battery houed within the haft-enor coupling at the far left. Alo at the far left i a mall cap that finihe the end of the deign and it i the only portion viible when the IMU i intalled. A miniature power witch, which protrude from thi cap, i alo viible in the photograph. A digital circuit finihe the oppoite end of the deign and it i powered by a regulated 3V ource. The major digital component include a microproceor (for performing 10- bit A/D, controlling RF tranmiion, and other circuit function), a low-power RF tranceiver (2.4GHz), and a urface-mount antenna. The total continuou current draw of the aembled wirele IMU i 37mA including a 28mA draw from the analog component and a 9mA draw from the digital component. The total aembled ma i 25.1gm with 7.1gm (28%) of that total attributed the 6V 23

38 battery which provide continuou operation for two hour. The total ma i extremely mall relative to that of a golf club (e.g., 490gm) rendering it virtually undetectable. Note that the total ma of the IMU i alo a mall fraction of the ma of a grip alone (60gm). The performance and accuracy limit of the deign are carefully evaluated next. 2.3 Performance and Accuracy The putting troke i an excellent candidate for thi enor ytem ince 1) the dynamic of the troke are well within the deign range of commercial MEMS accelerometer and angular rate gyro, and 2) the outcome of the putt i highly dependent on the orientation and poition of the club head [41]. In addition, the entire troke i completed within a very hort time frame (typically 1-2 econd), which minimize the advere influence of enor drift, and within a modet bandwidth (e.g., 25 Hz). The putting troke, however, doe place tringent accuracy limit on the computed orientation angle of the clubface. For example, to make a 20 foot putt, the face angle mut remain within 1 degree of the intended target line; refer to Fig The accuracy of the computed clubface orientation angle, club head velocity, and club head poition ret on: 1) the performance limit of the inertial enor, 2) the calibration of the fully aembled IMU, and 3) the data reduction (integration) method employed. Fortunately, the low noie and bia drift tability characteritic of mot commercial MEMS inertial enor lead to acceptable reolution limit for acceleration and angular velocity over the hort duration, low bandwidth dynamic of the putting troke. For intance, under thee condition, the noie limit of commercially available accelerometer and angular rate gyro remain le than 2 mg and 0.3 deg/ec, repectively. 24

39 Sytem-Level Calibration The ytem-level calibration of the wirele IMU employ algorithm adapted from [34-35]. In particular, the IMU i ubject to three precie 180 degree rotation about each of the enor axe ( iˆ, ˆj, kˆ ) in ucceion; refer to Fig Prior to each rotation, the elected active axi i aligned with gravity (and to within 0.03 deg via a bulleye level) o that the aociated accelerometer undergoe a 2g change following a 180 degree rotation about a horizontal axi. Thi procedure i achieved uing the cutom calibration fixture pictured in Fig. 2.5 that include a leveling plate (with bulleye level), a preciion mounting block with a bored hole for fixturing the IMU, and a ground haft equipped with a high-reolution optical encoder. The encoder i ued to inure a 180 degree rotation to within the encoder reolution limit ( 0.07 deg). Shaft and Optical Encoder Senor Block Leveling Plate Bulleye Level Figure 2.5: Calibration fixture ued to characterize the 24 calibration contant for the aembled IMU. The principal component include a leveling plate, a bulleye level, a preciion enor mounting block, and a haft with an optical encoder. Calibration of the accelerometer i achieved by noting the um and difference of the acceleration ignal before and after each 180 degree rotation. The angular rate gyro are calibrated by requiring their integrated ignal ultimately yield the controlled 180 degree rotation. Thi procedure characterize 24 calibration contant for the aembled IMU. 25

40 Thee contant include the cale factor, the two cro-axi enitivity cale factor, and the dc offet for each axi of the ix degree-of-freedom enor ytem. The cro-axi enitivitie originate from minute but unavoidable mialignment of the individual axe of each enor relative to the axe ( iˆ, ˆj, k ˆ ) defined by the haft of the club. Once calibrated, the enor ytem can be effectively ued to meaure putting troke. Prior to doing o, however, we report next on the expected accuracy of thee meaurement. Accuracy We hall evaluate the accuracy of the enor ytem by eparately evaluating the accuracy achieved 1) at addre of the ball, 2) during the putt, and 3) at impact of the ball. In each of thee phae, the kinematical quantitie obtained uing the enor ytem are benchmarked againt thoe meaured uing independent intrument (e.g., optical encoder). We begin by conidering time t 0 when the club i at addre with the ball. At addre, the club i momentarily till and the accelerometer ignal yield the initial lie and loft of the club head a dicued in Section 2.2; refer to Eq. (2.1.9) and (2.1.10). To independently meaure the initial orientation of the club, we again employ the calibration fixture which enable one to meaure the lie and loft to within the limit of preciion of the bulleye level ( 0.03 deg) and the optical encoder ( 0.07 deg). To thi end, the platform i firt leveled which etablihe the origin (zero angle) for the optical encoder. The platform i then rotated about a horizontal axi to a new orientation a meaured by the encoder and then held tationary while data are collected from the enor. The initial orientation angle calculated by the MEMS enor ytem are compared to thoe meaured by the encoder over a wide range of initial lie and loft a reported in Table Thee reult clearly demontrated that the MEMS enor ytem reolve the initial lie and loft angle to within 1 4 deg. 26

41 Accuracy at Addre Range of Lie Angle 50 deg Range of Loft Angle 50 deg Number of Tet 42 Number of Tet 40 Average Angular Error 0. 01deg Average Angular Error deg Table 2.1: Statitic of initial lie and loft angle teting. Table report difference (error) between angle meaured uing the MEMS enor ytem and an optical encoder for numerou tet over a large ( 50 deg) range. The initial lie and loft angle form the initial condition for the ubequent computation of the dynamic lie, loft and face angle during a putting troke; refer to Section 2.2. To evaluate the enor accuracy during thi dynamic phae and at impact with the ball, we deigned an intrumented putting robot a illutrated in Fig Thi i a paive ingle degree-of-freedom robot that execute a perfect pendulum-tyle putt [41] by winging freely under gravity. Even though thi ytem i completely paive it will be referred to a a robot, ince previou golf literature refer to imilar fixture a robot. The robot wing i driven by gripping the upper arm and pulling the arm back to make the deired backwing and then releaing it, thereby allowing the forward wing to occur freely under the effect of gravity. The club i clamped at the grip below an adjutable writ joint which i at the dital end of an adjutable forearm. An upper arm connect an adjutable elbow joint to a houlder joint which i upported by a horizontal haft captured by a pair of bearing. The haft incorporate an optical encoder for precie meaurement of the rotation of the club ( 0.1deg) about the haft at the houlder joint. The pendulum robot yield a highly repeatable putting troke wherein the forward wing repeat (in revere) the backwing. Thu, at impact, the club head mut return to the ame poition and orientation it had at addre. Thi required cloure provide a powerful mean to evaluate the accuracy of the enor ytem during the dynamic phae of the putting troke and with realitic putting peed and range of motion. 27

42 Optical Encoder Rotation Axi Shoulder Elbow Writ Forearm Figure 2.6: Single degree-of-freedom (pendulum) putting robot ued to ae the accuracy of the enor ytem during the dynamic phae of putting and at impact. Photo illutrate the principal component of the robot including a houlder joint that allow rotation about one axi a meaured by an optical encoder, adjutable (then fixed) elbow forearm, and writ joint, and a clamp that grip the club. We next report reult obtained from ten trial meaured with the enor ytem embedded in a club wung by the putting robot. The reult include the cloure error at impact a well a the dynamic tracking error during the backwing and forward wing. By cloure error, we refer to the difference in the computed club head lie, loft and face angle between addre and impact. Ideally, thee difference hould be zero a the putting robot enforce a kinematically cloed path from addre to impact. In addition, the angle meaured by the encoder i equal to the clubface loft angle which provide a quantitative mean to ae the accuracy of the enor ytem all throughout the putting troke. We refer to difference between thee latter two meaure a the dynamic tracking error. 28

43 Angle (deg) Dynamic Orientation Throughout Putt Loft -12 Lie Face Time (ec) Figure 2.7: Dynamic loft, lie and face angle computed from MEMS enor ytem for a typical wing uing the putting robot. Figure 2.7 illutrate typical reult for the dynamic loft, lie and face angle for a wing uing the putting robot from addre (t=0 ec) to impact (t 1.5 ec.). The initial (nonzero) loft and lie angle at addre are ubtracted from thee reult to better illutrate the cloure error at impact. Thu, all three angle tart at zero and hould ultimately end at zero at impact for perfect cloure. In the example hown however, there i a minute cloure error for the loft angle which finihe approximately at degree a hown in the enlarged view. The tatitic of ten uch cloure tet are reported in Table 2.2 which include the mean and tandard deviation for the cloure error for the lie, loft and face angle, rounded to the nearet 0.1 degree, which i the reolution of the optical encoder ued to benchmark the IMU accuracy. Thee reult confirm that the wirele, miniature IMU and the aociated integration algorithm (Section 2.2) yield cloure error on the order of 0.1 deg (tandard deviation) yielding highly accurate meaurement of putting dynamic. 29

44 Angular Cloure Error for MEMS IMU Lie Cloure Error (deg) Loft Cloure Error (deg) Face Cloure Error (deg) Mean Stand. Dev Table 2.2: Mean and tandard deviation of cloure error for clubface lie, loft and face angle for ten wing uing putting robot. Reult are rounded to the nearet 0.1 degree becaue that i the reolution of the optical encoder ued for independent meaure. Figure 2.8: Enlarged view of dynamic tracking error during a typical wing with putting robot. Figure 2.8 illutrate an enlarged cale view of the dynamic tracking error defined a the intantaneou difference between the loft angle computed uing the MEMS enor and that reported by the optical encoder. The loft angle change by approximately 15 degree during the wing and the maximum tracking error remain maller than 0.35 degree and ha an average of 0.15 degree. The average of the ten trial wa 0.11 degree. The dither in Fig. 2.8 i due to comparing the analog output of the enor to the digital ignal from the optical encoder. Since the reolution of the encoder i 0.1 degree the fluctuation of approximately 0.1 degree een in Fig. 2.8 i expected. Thee reult confirm that the tracking error are alo on the order of 0.1 degree. 30

45 The reult above conider the accuracy of the predicted lie, loft and face angle obtained uing the rate gyro data and the algorithm ummarized in Section 2.2. The accelerometer data are alo ued in tandem with the rate gyro data to predict the club head peed, the club head path, and ultimately the impact poition of the ball on the clubface; refer again to Section 2.2. The putting robot require that the clubface at impact align with it poition and orientation at addre. To demontrate that the robot actually achieve thi cloed path, we directly meaure the location of the impact uing impact tape for a number of wing. Impact tape leave a viible mark where ball impact occur a een in the image of Fig Thee image confirm that the club path cloe at impact to the ame poition a at addre to within the (viual) reolution of the tape (approximately 2 mm). ROBOT PUTTS 1-3 TOP 2.0 cm ROBOT PUTTS 4-8 TOP HEEL TOE 5.7 cm BOTTOM BOTTOM Figure 2.9: Impact tape record of eight wing made with the putting robot. Left (putt no. 1-3) and right (putt no. 5-8) record demontrate that the clubface return at impact to ame poition at addre to within the (viual) accuracy of the tape (approximately 2 mm). The two impact tape trip are for identical etup of the robot, but allow a clear view of the impact pattern. The general dimenion of the tape are noted in the figure and the concentric circle increae in diameter by 1cm for each ucceive circle. The predicted path of the club head for a typical putt i illutrated in Fig which illutrate the computed Carteian coordinate of point f on the clubface relative to the L lab-fixed frame; refer to r f in Fig The path largely lie in the (vertical, o Jˆ Kˆ plane) a expected. The maximum backwing poition occur at Y 32cm where the clubface ha alo been lifted Z 3cm above it height at addre. The X motion 31

46 Po (cm) reache a maximum 12mm which hould be zero baed on robot contruction, but i the um total effect of finite preciion on aligning the enor in the golf club and aligning the club in the robot. At impact, the cloure on poition i excellent a obervable in the exploded view. Here the maximum poition cloure error i le than 3mm and it occur in the vertical (Z) direction. The average and tandard deviation of the poition cloure error over ten wing are mm for the X-direction and mm for the Z- direction. Thee tight error on poition cloure again confirm the accuracy of the deign and the data reduction algorithm for reolving the dynamic of a putting troke. 1 5 Poition of Clubface in Lab Fixed Reference Frame X -25 Y Z Time (ec) Figure 2.10: Computed Carteian coordinate for path of point f on the face of the putter baed on meaured accelerometer and gyro data for a typical wing with the putting robot. The accuracy meaure above focu largely on the cloure of the club head poition and orientation at impact when uing the putting robot. Given that the backwing and the forward wing of the robot trace the ame path, one can alo evaluate the accuracy of the MEMS enor ytem at intermediate time during the wing. For example, Fig. 2.11(top) illutrate the computed path of the club head during the backwing (blue curve) and the forward wing (red curve) for a typical putt uing the putting robot. The companion face angle data are preented in Fig. 2.11(bottom). The forward path trace nearly perfectly the path of the backwing; note the highly magnified cale for motion in the lab-baed X- 32

47 Face Ang. (deg) X-po (mm) direction (direction tranvere to target line). Likewie, the face angle remain cloe to zero and retrace the ame curve to a high degree. The mall but ytematic departure from the target line ( Ĵ -axi) in each cae arie from a very mall initial mialignment between the iˆf axi of the enor frame and the fixed axi of rotation of the putting robot. For thi trial, the maximum deviation of the club head path during the backwing and the forward wing i le than 1.5mm. Similarly, the maximum deviation between the face angle along thee path remain le than 0.03 degree. Over ten uch trail, the average and tandard deviation for thee path and angular error were mm and degree, repectively. Thee reult again confirm that the MEMS enor ytem and the aociated numerical algorithm yield outtanding poitional and angular accuracie for the objective of reolving a putting troke. Overhead View of Club Head Path Backwing Forwardwing Face Angle a Function of Y-Po Y-po(cm) Figure 2.11: (Top) Overhead view of poition data from one trial putt. (Bottom) Face angle a a function of poition. Note the highly magnified cale for the X poition coordinate (ordinate). 33

48 2.4 Cloure Thi chapter preent the theory, deign, and experimental evaluation of a miniature, wirele ix degree-of-freedom inertial meaurement unit (IMU) for computing the dynamic of a golf wing. The deign integrate a three-axi MEMS accelerometer, three ingle-axi MEMS angular rate gyro, a microproceor, and a wirele (RF) tranceiver in forming a battery-powered IMU that net within the mall confine of the haft of a golf club. The inertial enor, located at the grip end of the haft, are ued to compute the kinematic (e.g., velocity, poition and orientation) of the club head located at the oppoite end of the haft. The requiite meaurement theory i carefully developed and the accuracy of the aembled enor ytem i ytematically evaluated by experiment. The experiment utilize a putting robot that execute a highly repeatable pendulum troke for putting. The robot i equipped with an optical encoder for independent aement of wing kinematic. The achieved accuracy and performance of the MEMS enor ytem for the target application (putting) are ummarized in Table 2.3. Accuracy and Performance Angular Accuracy Performance Initial Orientation 02deg. Total Ma 25 g Abolute Orientation 05deg. Total Current Draw 37 ma Poition Accuracy X-Section Diameter 13 mm Abolute Poition 3 mm Continuou Run-Time 2 hr Table 2.3: Achieved accuracy and performance characteritic of the miniature, wirele MEMS IMU for putting dynamic. By employing the meaurement theory, the enor ytem yield the poition of the club head with an abolute accuracy of 3 mm and the orientation of the club head with an abolute angular accuracy of 0.5 degree. The reulting deign yield a highly portable, inexpenive, and highly accurate enor ytem for meauring golf wing dynamic. Poible ue include golf wing training, cutom club fitting, and club deign. Thi enor ytem alo addree many hortcoming of exiting camera-baed wing training 34

49 ytem. The example enor ytem for putter i readily adapted to other golf club (iron and driver) by appropriately changing the dynamic range of the inertial enor. 35

50 CHAPTER 3 UNDERSTANDING PUTTING STROKE KINEMATICS VIA THE MEMS SENSOR The objective of thi chapter i to ue the MEMS enor detailed in the previou chapter to quantify the putting troke and thereby advance the fundamental knowledge of putting technique. In realizing thi objective, we alo propoe a generalization of all putting troke by introducing a two-parameter kinematic model of putting. Thi model unifie the previou claification of putting troke which were categorized a either pendulum tyle troke or gate tyle troke [41]. The MEMS enor technology allow u to quantitatively explore thee two putting tyle and to demontrate their trong relationhip through a imple kinematic model. In addition, thi chapter introduce a novel diagnotic tool for analyzing putting troke and quantitative metric for common putting fault. We begin by reviewing the literature pecific to putting technique. 3.1 Literature Specific to Putting Technique A ucceful putt require kill in reading the green, aligning the putt, and then triking the ball with the club. Thi technology, which focue on mechanic of the golf wing, i a tool to evaluate the lat kill. Neverthele, we recognize the importance of reading the green and properly aligning the putt at addre. The putting troke ultimately determine the initial velocity and angular velocity of the ball following impact. Thee initial condition for the ball following impact reult from three prior condition at the time of impact, namely: 1) the velocity of the club head, 2) the orientation of the club head, and 3) the impact location of the ball on the clubface. A the mean to explain how player develop thee impact condition, coache and teacher often tudy related apect of the putting troke that are eaier to decribe uch a the 36

51 putter path (e.g., the path traced by the centroid of the club head during putting) and the power ource (e.g., the ource of the kinetic energy of the club). After urveying the writing of three well-repected teacher, David Pelz [41], Stan Utley [45], and Edward Craig [46], we ummarize a contemporary undertanding of the putting troke. David Pelz, who i a coach of PGA player and author of Dave Pelz Putting Bible [41], advocate a kinematically imple and thu repeatable troke often referred to a the pendulum tyle troke. In thi troke, the putter head never deviate from the intended target line and the putter face remain quare to thi target line. Thu, following addre, the putter head i brought back along the target line and raied lightly above the green in proportion to the length of the backwing. The forward wing i the revere motion which i ideally powered olely by the gravitational potential energy of the club a if it were a imple pendulum; refer to Fig. 3.1(a). (a) (b) Figure 3.1: Illutration of pendulum and gate tyle putting troke a viewed from overhead [41]. (a) Pendulum troke i alo referred to the Pure Inline and Square (PILS) troke by Pelz. (b) Gate troke, howing the club head moving on an arc and the clubface opening and cloing during the troke. Pelz conclude that the pendulum tyle troke, or the Pure Inline and Square PILS troke, i the troke player hould adopt ince the ue of gravity alone to power the club lead to a highly repeatable troke. He alo mention that the gate or creen door troke taught by Harvey Penick, a highly repected golf teacher, i baed on a pendulum motion where the axi of rotation i inclined from the horizontal. Thi i a keen 37

52 obervation that we later return to in developing a imple kinematic model for all putting tyle. Pelz aert that if the axi of rotation i inclined from the horizontal, the player mut upply a reaction force (actually a reaction torque) to maintain thi angle againt gravity. Thi reaction force i a perturbation to the PILS troke. Stan Utley, who i a current profeional player, a current coach of PGA player, and author of The Art of Putting [45], teache putting uing a natural, free-flowing wing. He advocate for the putting troke being a miniature verion of a full wing and thi alo lead to a gate tyle putting troke. The gate tyle wing derive it name from the analogy of a winging gate whoe hinge line i not plumb. A the gate open, the center of ma of the gate raie lightly and then fall a the gate cloe. Kinematically, the club head during a gate wing follow a lightly curved path, when viewed from overhead, the clubface open relative to the target line during the backtroke, and then cloe during the forward troke. Ideally, the clubface remain perpendicular to the putter path throughout the troke o that at impact the face i again quare to the target line; refer to Fig. 3.1(b). Edward Craig, who i the editor of Britian top golf magazine Golf Monthly and author of Putt Perfect [46], dicue the potential benefit of both the gate and the pendulum troke. Craig focue on the motion of the houlder, arm and writ and conclude that the motion of thee body part hould be independent of the putter path. He ay that the houlder mucle hould be the power ource for the backwing. The arm and writ hould provide a rigid connection from the rotation axi to the putter. The lower body hould remain till. 3.2 Two-Parameter Kinematic Model for Putting A keen reader may note that the decription of the gate troke above lack a precie definition. By contrat, the pendulum (or PILS) troke definition preciely decribe the path and orientation of the putter head. Thu, while there are definitive mean to tet whether a troke come cloe to an ideal pendulum troke, no analogou mean have been 38

53 propoed to tet whether a troke come cloe to an ideal gate troke. We hall now introduce a two-parameter kinematic model of the putting troke that quantitatively predict the path and orientation or the putter head for all troke, including ideal pendulum and gate troke. Later, we alo how that thi model correlate well with the meaured putt from player of varying kill level. ' Î ˆK ' p ˆK ' ' Ĵ L arm Rotation Axi R p L arm Rotation Axi ˆK Y grip f Î (a) (b) Figure 3.2: Definition of two-parameter kinematic model for putting. (a) The parameter define the angle the rotation axi make with the horizontal and the parameter Larm define the perpendicular ditance from the rotation axi to the top of the grip (point ' ' ). (b) View in the J ˆ, Kˆ -plane howing the parameter L arm, Y grip, and the angle of rotation about the rotation axi. Figure 3.2 illutrate the propoed model which ha a ingle rotational degree of freedom and i defined by the two parameter and L arm. i the angle that the rotation axi make with the horizontal and L arm i the perpendicular ditance from the rotation axi to the top of the grip. The top of the grip i eentially point, the origin of the body fixed enor frame i ˆ, ˆ, ˆ j k defined in the previou chapter. 39

54 Etablihing the Model Parameter from Experimental Data We now illutrate how to contruct the model parameter and Larm from the meaured data from a player putt. Doing o alo allow u to predict the ideal kinematic of the player putt a predicted by a model that mot cloely fit the player natural troke. Thi i a novel approach in that the player troke i ultimately compared to a tandard that cloely fit the natural troke of the player rather than to ome pre-determined tandard (ay a perfect pendulum troke). Figure 3.2(a) illutrate a lab-fixed frame I ˆ, Jˆ, K ˆ ' ' ', where ' Î i aligned with the rotation axi. Thi frame i ditinguihed from the econd lab-fixed frame I ˆ, Jˆ, K ˆ by the rotation about the Jˆ ' Jˆ axi. The angular velocity vector of the club i along ' I and thu orthogonal to Ĵ. The Î and ˆK component of, are related to through The quantitie x and x and z repectively, z tan (3.2.1) x z can be derived from the enor meaurement of. The poition vector R lie wholly within the ˆ' ', ˆ p J K plane and the length of thi vector i the parameter L arm. One may compute L arm from the Ĵ component of the grip poition Y grip, upon computing the rotation angle a uggeted in Fig. 3.2(b). i computed by integrating () t uing (0) 0 at addre. Thee quantitie are related through L arm Ygrip in (3.2.2) The quantity Y grip i derived from the enor meaurement a detailed in Chapter 2. 40

55 Predicting the Ideal Putt Baed on the Model Having etablihed a method for determining the model parameter, we now predict the kinematic of the ideal putt a predicted by the model. The key kinematical quantitie to predict include the three club head orientation angle (lie, loft, and face angle) and the path of the putter head. The model aume that the club rotate about a contant rotation axi, I '. Thu, the Euler parameter for thi rotation are eaily derived [43]. Explicit expreion for the rotation axi and Euler parameter a function of and are co Iˆ 0Jˆ in Kˆ co in e1 2 e 2 in 0 2 e 3 in in 2 e4 co 2 (3.2.3) In the previou chapter the relationhip between the Euler parameter and the clubface orientation angle (lie, loft, and face angle) were derived and are reproduced here a 1 ( t) in 2( e2e3 e1e 4) 1 ( t) in 2( e1e 3 e2e4 ) 1 2( e1e 2 e3e4 ) ( t) in co By ubtituting the reult of Eq. (3.2.3) into Eq. (3.2.4), the imple kinematic model yield the lie, loft and face angle of the club head throughout the putting troke. (3.2.4) To predict the putter path, we now develop an expreion for the vector R f p illutrated in Fig. 3.3(a). 41

56 At Addre (t=0) ' Î R p ' ˆK p ' Î ˆK ' ' Ĵ View along axi f R f R f p Î ˆK R f p L tot (a) (b) Figure 3.3: (a) Definition of R p, R f, and R f, which are ued to decribe the predicted p putter path baed on the model propoed in thi ection. (b) View looking down the ' Î direction howing the contant length L tot. Figure 3.3(b) illutrate the ditance L tot, which i the ' ˆK component of the vector when the club i at addre 0. Thi ditance i contant a it i determined by the (contant) putter geometry and the parameter L arm. To derive an expreion for R f, firt p we etablih R f at addre and then R p for all time t a follow. R 0 (0) ˆ (0) ˆ (0) ˆ f X I Y J Z K (3.2.5) ˆ' ˆ ' R p Larm in J Larm co K (3.2.6) Iˆ, Jˆ, K ˆ frame while R i conveniently R i mot conveniently written uing the f written uing the I ˆ ', Jˆ ', K ˆ ' frame. The component of R f the initial poition of the golf club a detailed in Chapter 2 and p R are the reult of calculating R p f p can be derived by L inpection of Fig The tranformation between thee two frame, ' L C, which i required prior to adding thee two vector, i given by 42

57 ˆ' ˆ ˆ I I co 0 in I ' ˆ ' L L J C Jˆ Jˆ ˆ ' K Kˆ in 0 co Kˆ (3.2.7) Evaluating equation (3.2.6) at t 0 and 0, and applying equation (3.2.7) yield the following expreion for the vector R at addre f p R X Z Iˆ YJˆ X Z L Kˆ f p 0 co in in co The ditance L tot i the club i rotating about the ' ˆK component. The follow where the upercript L and ' ' ' arm ' Î axi. The final expreion for ' Î component remain contant ince the R f p (3.2.8) at any time t i given a ' L are ued to denote which lab-fixed frame i ued to decribe each vector quantity. ' L R co in ˆ co in ˆ in co ˆ f X Z I Y L p tot J Y Ltot K ' ' L L L L R C R f p C ' L L L L ' C f p T ' ' ' (3.2.9) Equation (3.2.9) can now be ued to predict the path of the center of the putter face, point f, throughout the putt. Shortly we hall demontrate that thi imple model correlate well with the meaured putting troke from ubject player. Before analyzing the player troke however, we firt preent meaurement of putt made uing the putting robot decribed in the previou chapter. Doing o allow u to confirm the accuracy of the kinematic model in a mechanim that truly poee jut a ingle degree-of-freedom. 3.3 Evaluation of Robot Putting Stroke Uing Kinematic Model The putting robot introduced in the previou chapter i now ued to demontrate the utility of the two-parameter kinematic model under highly controlled condition. Figure 3.4 reintroduce the putting robot which i eentially a ingle degree-of-freedom 43

58 pendulum with adjutable linkage extending from the axi of rotation to the grip on the club. Optical Encoder Rotation Axi Shoulder Elbow Writ Forearm Figure 3.4: Image of the putting robot introduced in the previou chapter. The putting robot wa ued to perform both PILS troke and gate troke by imply controlling the orientation of the axi of rotation. For the PILS troke, the angle 0. o For example gate troke, the robot bae wa inclined to achieve 10. Meaured putting data for both type of troke i ummarized below. Starting with the PILS troke, Eq. (3.2.1) wa ued to etimate an average inclination angle o 1.1 from computed reult of x and z. Thi rather mall angle i quite reaonable conidering that the robot bae i eentially level. Direct meaurement of the parameter L arm provide arm L in. Thi agree well with the value Larm 21.51in (and tandard deviation 0.37 in) predicted from Eq. (3.2.2) baed on the average of 10 trial. Having etablihed that parameter of the model and the phyical et up agree, we next preent reult for the predicted and meaured clubface orientation and path. The model predict the orientation of the putter face throughout the putting troke uing Eqn. (3.2.3) - (3.2.4). The enor meaurement provide direct meaurement of thee 44

59 ame quantitie per the theory developed in Chapter 2. Figure 3.5 compare the orientation angle of the putter face predicted by the model and meaured from the enor data for a typical PILS troke by the putting robot. The predicted loft angle (cyan curve) and the meaured loft angle (blue curve) lie nearly on top of each other demontrating excellent correlation between the imple model and the enor meaurement. Similarly good agreement i oberved for the face angle (black curve = predicted v. green curve = meaured). A light diparity exit between the meaured lie angle (red curve) and the predicted lie angle (magenta curve). Thi modet difference derive from a light mialignment of the enor axe in the putter relative the axi of rotation of the robot a revealed next in an analyi the putter path. 45

60 Orientation Angle (deg) 2 Actual v. Predicted Orientation Angle 0-2 Figure 3.5: Meaured and predicted orientation angle for a robot-generated PILS putting troke. The blue, red and green curve are the meaured loft, lie, and face angle, repectively. The cyan, magenta and black curve are the correponding angle predicted by the imple, two-parameter kinematic model. Having demontrated good agreement between the predicted and meaured orientation of the putter face, we now examine the putter path. Figure 3.6 how the predicted putter path (cyan curve) and the calculated putter path (backwing = blue curve; forward wing = red curve). Notice that the meaured path i very nearly traight (a predicted by the model) and that diplacement along the lab-fixed X axi are greatly enlarged due to the refined vertical cale ued in thi figure (cale unit of 0.1 in). The reult hown in the Fig. 3.6(a) are uncorrected for the mall mialignment of the enor axe relative to the rotation axi of the putting robot. Thi mialignment derive from everal ource including: 1) the imprecie manner of gripping the club, 2) any light mialignment of the enor axe relative to the grip, and 3) the machining tolerance controlling the placement and orientation of the bearing upporting the robot haft. Figure 3.6(b) how the data corrected for mialignment, uing a correction of 1.6 o. Thi correction i computed from Loft Lie -14 Face Time (ec) _ Y t X 1 f _ mea tb X f model t b mi tan f _ mea b X X Y tan f _ corr f _ mea f _ mea mi (3.3.1) 46

61 X-po (in) X-po (in) where t b denote the time at the end of the backwing and X f, Y f are the Iˆ, Jˆ Carteian coordinate of clubface centroid, point f Overhead view of PILS robot putt, uncorrected Backwing Forwardwing Model Overhead view of PILS robot putt, corrected Backwing Forwardwing Model Y-po (in) (a) Y-po (in) (b) Figure 3.6: Meaured and predicted path for the center of the putter head during a robotcontrolled PILS putting troke. (a) Uncorrected for light mialignment of enor axe to the robot rotation axi. (b) Corrected for thi light mialignment. The meaured putter path, once corrected for light enor mialignment, i hown Fig. 3.6(b) and it encircle the path predicted by the model. In particular, the meaured path lie within 0.1 in of the predicted path with the greatet deviation occurring at the tart of the backwing. Thi early error likely derive from clearance and compliance of the putting robot mechanim. During the backwing, the robot arm i pulled backward from ret and any lateral clearance or compliance allow the putter head to move lightly along the lateral X axi. Overall, thee tet of a nearly perfect PILS troke by the robot demontrate that the imple kinematic model accurately predict the path and orientation of the putter throughout thi troke a expected. The putting robot wa then ued to execute a nearly perfect gate troke. To thi end, the rotation axi of the robot wa inclined approximately ten degree uing him beneath the bae of the robot. Uing Eqn. (3.1.1)-(3.1.2), we compute the average = 9.6 (tandard deviation 0.1 o ) and the average calculated L arm = in (tandard deviation o 47

62 1.14 in ) baed on ix trial. (The larger tandard deviation for thee parameter relative to thoe for the PILS trial i a likely the reult of mall wobbling of the robot bae on the added him.) Figure 3.7 report the dynamic lie, loft and face angle for a repreentative gate troke predicted by the two-parameter kinematic model and a meaured by the enor. A in the cae of the PILS troke, the dynamic loft predicted for the gate troke i in near-perfect agreement with the meaured loft. Similarly, the predicted dynamic face angle i in reaonably cloe agreement with the meaurement. Unlike the PILS troke where the face angle remained eentially zero, the gate troke require that the clubface open during the backwing (achieving a maximum of o 2 for thi modet example for which o 10 ) and then cloe back to quare on the forward wing. Thi eential opening and then cloing of the face angle during the putt i what many golf teacher and profeional identify a the key difference between the gate and PILS troke. Here again, the larget dicrepancy between the model and meaurement occur for the lie angle and it again i largely attributed to a light and unknown mialignment of the enor axe relative to the axi of rotation of the robot. Thi mall dicrepancy aide, thee reult demontrate that one way to generate a gate troke i to imply incline the axi of rotation. In the next ection we will preent further finding that ugget that mot player exhibit a gate troke developed in thi very manner. 48

63 Orientation Angle (deg) 2 Actual v. Predicted Orientation Angle Loft Lie Face Time (ec) Figure 3.7: Meaured and predicted orientation angle for a robot-generated gate troke o with 10. The blue, red and green curve are the meaured loft, lie, and face angle, repectively. The cyan, magenta and black curve are the correponding angle predicted by the imple, two-parameter kinematic model. The predicted and meaured putter path for thi example gate troke i illutrated in Fig Figure 3.8(a) i not corrected for mialignment between the enor and the robot, wherea Fig. 3.8(b) i corrected 2.92 o. The corrected meaured path again encircle and cloely follow that predicted by the imple kinematic model and with the mall deviation attributable to robot clearance/compliance. 49

64 X-po (in) X-po (in) Overhead view of 10 o inclined robot putt, uncorrected Backwing Forwardwing Model Overhead view of 10 o inclined robot putt, corrected Backwing Forwardwing Model Y-po (in) (a) Y-po (in) (b) Figure 3.8: Meaured and predicted path for the center of the putter head during a robotcontrolled gate troke with 10. (a) Uncorrected for light mialignment of enor o axe to the robot rotation axi. (b) Corrected for thi light mialignment. Comparing the path hown in Fig. 3.5 and 3.7 reveal that the gate troke induce obervable curvature to the path. The imple kinematic model clearly how why. The actually 3-D path (of the center of the face of the putter) remain a circular path but now in a plane inclined from vertical by the angle. The projection of thi 3-D path onto the horizontal (X-Y) plane lead to the illutrated curved arc whoe curvature i proportional to. Thu, for larger, a will be een in many player putt, thi curvature may be ubtantially greater than the modet example illutrated by thi gate troke by the putting robot. By contrat, the putter path remain a traight line in the horizontal (X-Y) for an ideal PILS troke. Overall, the reult preented in thi ection demontrate that the propoed two-parameter kinematic model accurately decribe the idealized PILS and gate troke performed by the putting robot. In the next ection, we turn our attention to applying thi model to the meaured putting troke from player of variou kill level. 50

65 3.4 Evaluation of Human Putting Stroke Uing Kinematic Model It wa expected that the two-parameter kinematic model of putting agree well with meaured putt uing the putting robot ince the robot i a ingle degree-of-freedom mechanim for which L arm and remain contant. The major finding within thi ection i that thi imple model remain a very good approximation for player who naturally move free of the contraint of the putting robot. We hall begin by firt introducing quantifiable metric for comparing meaured putting troke to the model. Metric for Comparing Meaured Putting Stroke to Model Return to Fig. 3.2 and recall that the two-parameter kinematic model repreent a ingle degree-of-freedom ytem allowing rotation about the inclined axi denoted by Therefore, how well thi model decribe player putting ret in part on how cloely player putting can be decribed by rotation about a fixed (inclined) axi. To quantify thi idea, we propoe the following metric which decribe the degree to which the meaured angular velocity of the club i aligned with a contant axi of rotation inclined by an etimated angle. Let the club angular velocity vector be repreented a Iˆ Jˆ Kˆ x y z where ( Iˆ, Jˆ, Kˆ ) i the lab-baed frame; refer to Fig Next, let ' I. ave repreent the average value of computed uing Eq. (3.2.1) for all time t for which i greater (3.4.1) than ome reaonably mall value, ay 5 deg/ec. For lower angular velocitie, the putter i eentially at ret. Next we introduce the following three error meaure that decribe the deviation of the meaured angular velocity vector from one aligned with the rotation axi defined by ave. 51

66 ( t) co ave x( t) deg x _ err ( t) for ( t) 5 () t y () t deg y _ err ( t) for ( t) 5 () t (3.4.2) ( t) in ave z ( t) deg z _ err ( t) for ( t) 5 () t If thee error metric remain mall, then the putter i indeed rotating largely about a fixed axi a determined by the computed angle ave. The two-parameter kinematic model alo aume that the ditance L arm remain contant thereby determining the length of an equivalent pendulum from the enor module to the axi of rotation; refer again to Fig We etimate L arm by computing it average from Eqn. (3.2.2) uing computed value of Ygrip () t and () t from the enor data. We then compute the predicted club rotation model from the imple kinematic model by again uing Eqn. (3.2.2) with the average value of L arm. The actual club rotation mea () t can be computed directly from the enor data according to ˆ t () ' mea t I dt 0 (3.4.3) uing mea (0) 0 at addre. The error between the meaured rotation and that predicted by the model model ( t) mea ( t) deg err ( t) for 5 () t mea 1 model in Ygrip Larm 1 yield a fourth criterion for comparing a player putt to the imple kinematic model. Thee four criteria will now be applied to player putt to ae how well thee putt conform to the aumption of the imple kinematic model. (3.4.4) 52

67 Orientation Angle (deg) X-po (in) Analyi of Player Putt The putt preented in thi ection are example elected from putting trial conducted over the pat three year involving more than ten ubject putting in variou condition; from putt in the laboratory, to putt on artificial putting urface, to putt on actual putting green. We begin by analyzing an example putt from a tudy conducted at the Univerity of Michigan golf coure. In thi tudy, we elected two (approximately traight and level) putt of 15 ft and 5 ft and requeted each ubject to complete three putt at both ditance. The example 15 ft putt illutrated in Fig. 3.9 wa completed a collegiate-level competitive player. Illutrated are the orientation and putter path meaured by the enor and predicted by the (bet-fit) kinematic model Actual v. Predicted Orientation Angle Subject B 15ft Putt Overhead view of clubhead path Subject B 15ft Putt 1 Backwing Forwardwing Model Loft -12 Lie Face Time (ec) (a) Y-Po(in) (b) Figure 3.9: Example 15 ft putt by ubject B howing excellent agreement between meaured and modeled putter kinematic. (a) how the meaured and predicted club head lie, loft and face angle a function of time (blue, red, green = meaured; cyan, magenta, black = model prediction). (b) how the path traced by the center of the clubface in the horizontal (X-Y) plane. The meaured feature illutrated in Fig. 3.9 are in excellent agreement with the model prediction. In Fig. 3.9(a), the meaured dynamic loft (blue curve) and lie (red curve) angle cloely follow the predicted angle (cyan and magenta, repectively). The 53

68 z (deg/ec) in() meaured face angle (green) alo follow a very imilar trend to the predicted angle (black), which how the clubface opening during the backwing and cloing during the forward wing. In Fig. 3.9(b), the predicted path (cyan) i cloely traced by the meaured clubface path (blue = backwing, red = forward wing); again, note the greatly magnified cale on the vertical axi. Thi example putt exhibit the main characteritic of a gate troke; 1) face open during the backwing and cloe during the forward wing, and 2) the projection of the putter path onto the horizontal plane i ditinctly curved. The overall cloe agreement with the model prediction i trong evidence that the model kinematic cloely approximate actual putting kinematic for thi ubject. Figure 3.10 preent another view of the ame data that allow direct etimation of the model parameter and (viual) comparion of the error metric etablihed by Eqn. (3.4.2) and (3.4.4) Lab Fixed Ang. Vel. x v. z Subject B 15ft Putt Backwing Forward Swing Sin() v. Y grip Subject B 15ft Putt Senor Data Model x (deg/ec) (a) Y grip (in) (b) Figure 3.10: Example 15 ft putt by ubject B howing excellent agreement between meaured and predicted putting kinematic. (a) how the meaured relationhip between the angular velocity component x and z together with that predicted by the imple model. (b) how the meaured relationhip between the rotation angle and the grip poition coordinatey grip. The meaured point are plotted for the backwing and forward wing. The bet fit model predict the green and black line for the backwing and forward wing, repectively. Figure 3.10(a) report the (lab-fixed) angular velocity component component z veru the x. The experimental data are plotted with the blue dahed line and the olid 54

69 red line i the bet fit line to thi data. The lope of thi trend line i ued to etimate the model parameter ave, which i equivalent to averaging Eqn. (3.2.1) for all time t. In thi example, the reult of the model cloely approximate the experimental data. Figure 3.10(b) report in mea veru grip Y where mea i calculated via Eqn. (3.4.3). Data point for the backwing are plotted in blue and thoe for the forward wing are plotted in red. Separate linear trend line are fit to the back and forward wing. After reviewing the putt from numerou ubject, it became apparent that mot putt exhibit two ditinct trend line when viewed a illutrated in Fig. 3.10(b). Per Eqn. (3.1.2), the invere of the lope of a trend line in thi figure provide an etimate for the model parameter L arm. Thu, one can compute an effective Larm for the backwing which i often ditinct from that computed for the forward wing. In the next ection, we will further dicu how a meaurable change in L arm from the backwing to the forward wing repreent a common putting fault. In thi example, however, the two trend line lie nearly on top of each other indicating that a ingle value of L 17.1in for the model cloely approximate the kinematic of both the back and forward wing. For completene, we report in Fig the error metric defined by Eqn. (3.4.2) and (3.4.4). arm 55

70 Error Metric Error Metric Angular Velocity Error Metric Subject B 15ft Putt 1 (a) x err y err z err Time (ec) Dynamic Loft Error Metric Subject B 15ft Putt Time (ec) (b) bck err fwd err Figure 3.11: Error metric defined by Eqn. (3.4.2) and (3.4.4) for example 15 ft putt by ubject B. (a) The three angular velocity error metric. (b) Dynamic loft error metric for the backwing and forward wing. The reult of Fig demontrate that the error metric remain reaonably mall throughout the putt. The jump in the data in Fig. 3.11(a) at approximately 1.45 and 1.60 and the gap in the data of Fig. 3.11(b) occur at the end of the backwing and the tart of the forward wing where the angular velocity remain le than the threhold (5 deg/). The time average of thee five metric are reported in Table 3.1. Thee average error remain le than 5% indicating a uperb overall fit to the aumption of the imple kinematic model. Subject B x_ err (%) y _ err (%) z _ err (%) bck _ err (%) fwd _ err (%) 15ft Putt 1 Average STD Table 3.1: Average error between predicted and meaured putt kinematic for the example 15 ft putt for ubject B. A a econd example, a putt from a different collegiate-level player i preented next that illutrate one of the more extreme cae where larger error develop. Thi example putt i taken from trial conducted with player from the Univerity of Michigan Women 56

71 Orientation Angle (deg) X-po (in) Golf Team. During thee trial, three ubject completed 5 putt from 4 ft, 10 putt from 8 ft, and 10 putt from 15 ft. The example choen i one 8 ft putt from ubject L. Actual v. Predicted Orientation Angle Subject L 8ft Putt 2 Overhead view of clubhead path Subject L 8ft Putt Backwing Forwardwing Model -8 Loft -10 Lie Face Time (ec) (a) Y-Po (in) (b) Figure 3.12: Example 8 ft putt by ubject L howing more dicrepancy between meaured and modeled putt kinematic. (a) how the meaured and predicted club head lie, loft and face angle a function of time (blue, red, green = meaured, cyan, magenta, black = model prediction). (b) how the path traced by the center of the face of the club head in the horizontal (X-Y) plane. From Fig. 3.12(a), note that the meaured (blue) and predicted (cyan) loft angle match well indicating that thi troke conform to a major prediction of the imple kinematic model. The face angle how a very imilar trend but the lie angle how oppoite trend. A further comparion i provided in Fig

72 z (deg/ec) in() Lab Fixed Ang. Vel. x v. z Subject L 8ft Putt Backwing Forward Swing Sin() v. Y grip Subject L 8ft Putt Senor Data Model x (deg/ec) Y grip (in) (a) (b) Figure 3.13: Example 8 ft putt by ubject L exhibiting le agreement with the imple kinematic model. (a) how the meaured relationhip between the angular velocity component and together with that predicted by the imple model. (b) how the x z meaured relationhip between the rotation angle and the grip poition coordinatey grip. The meaured point are plotted for the backwing and forward wing. The bet fit model predict the green and black line for the backwing and forward wing, repectively. In Fig. 3.13(a) the meaured angular velocity component increaingly deviate from the model trend (red) for x > 40 deg/ec, which occur toward the end of the forward wing. Fig. 3.13(b) now clearly how two ditinct trend line for the backwing and the forward wing, indicating ditinct value for Larm for thee part of the putting troke. The abrupt change in Larm occur at the tranition from the back to the forward wing, an iue that i dicued in the next ection on common putting fault. 58

73 Error Metric Error Metric Angular Velocity Error Metric Subject L 8ft Putt 2 (a) x err y err z err Time (ec) Dynamic Loft Error Metric Subject L 8ft Putt Time (ec) (b) bck err fwd err Figure 3.14: Error metric defined by Eqn. (3.4.2) and (3.4.4) for an 8 ft putt by ubject L. (a) Angular velocity error metric. (b) Dynamic loft error metric for the backwing and forward wing uing ditinct value for L arm. Figure 3.14 illutrate the time hitorie of the five error metric for thi example putt and the time average of thee error metric are reported in Table 3.2. Overall, the error (ranging from 3%-17%) are approximately twice a large a in the lat example and they are alo on the extreme ide of all meaurement taken on elite golfer. Neverthele, the model till appear to predict the kinematic of the putt with reaonably good fidelity (to within 20%). Subject L 8ft Putt 2 Ave x_ err Ave y _ err Ave z _ err Ave bck _ err (%) (%) (%) (%) (%) Average STD Table 3.2: Average error between predicted and meaured putt kinematic for the example 8 ft putt for ubject L. Ave fwd _ err 59

74 Simple Kinematic Model Correlation Error for 10 Subject Subject Ave x_ err Ave y _ err Ave z _ err Ave bck _ err Ave fwd _ (%) (%) (%) (%) (%) J B R L A KK JK K T BT Overall Table 3.3: Average error metric for all (120) trial conducted on a total of 10 ubject. Table 3.3 report the compilation of error metric for a total of 120 trial conducted on a total of 10 ubject. For each ubject, the average error metric over all meaured putt i reported to etablih the overall fit of the kinematic model to that player. The ubject are arranged by kill level, with the firt five ubject having played collegiate-level competitive golf and the lat two ubject having jut been introduced to the game. Over all of thee trial, the overall average error metric (bottom row) confirm that the imple kinematic model remain accurate to within approximately 10%. Thi new and fundamental undertanding of the putting troke a viewed uing the imple kinematic model might have ignificant diagnotic value in player aement and training. err 3.5 Identification of Common Putting Fault Having hown that human putting troke are reaonably well predicted by the imple kinematic model, we will now examine the ource of deviation from that model. The model dicued thu far focue on the kinematic of the putting troke. In addition, one may alo conider the force and moment developed by the player at the grip that drive the putting troke. Thee force and moment are often looely referred to a the power ource of a putt in popular decription of the game. To aid our dicuion of putting technique, we firt comment on the power ource. 60

75 The power ued during the putting troke may derive from two principal ource; namely, keletal mucle and gravity. Notable teacher, Pelz [41], Utley [45], and Craig [46], advocate that the mucle acro the houlder joint hould be the main keletal mucle recruited in putting while the elbow and writ joint hould remain eentially locked. The houlder mucle are ued in the backwing to draw the club backward oppoite the target line and upward againt gravity. The expert concur that the forward wing hould be powered by gravity alone. Under thee circumtance the forward wing kinematic reduce to that of a gravity-driven pendulum allowing rotation about a (poibly inclined) fixed axi. Similarly, the club and the player arm form a rigid body pendulum. In cae of very long putt and/or very low green, it may become kinematically impoible for player to develop the very long backwing required to power the forward wing by gravity alone. In uch intance, player will actively recruit mucle to provide the requiite additional power. With thi baic undertand of putting dynamic we move forward to the econd goal of thi chapter, to demontrate how the MEMS enor technology yield a novel tool for the analyi of putting technique. Specifically, we will demontrate how the enor can meaure common fault in putting technique. Mot coache would concur that fault in putting technique trace to one or more of the following principal ource: Improper face angle at impact Off-center impact of the ball on the clubface (related to putter path) Improper ue of writ Improper hand poition relative to the club head at impact Thee are common fault pecifically aociated with the putting troke (back and forward wing) and they are ditinguihed from any additional fault due to improperly aligning the putt and/or improperly reading the green. We hall explore each of thee principal ource in turn and provide experimental data to upport our claim that the MEMS enor technology can identify fault in the putting troke. 61

76 Face Ang. (deg) Face Ang. (deg) X-po (in) X-po (in) Improper Face Angle at Impact The face angle at impact i the ingle mot critical parameter affecting the outcome of the putt. According to Pelz [41], an improper face angle at impact adverely affect the putt four time more than an improper path of the putter alone. For example, conider a traight 10 ft putt with the initial target line directed exactly to the center of the hole. A mere one degree open or cloed face angle at impact will caue the ball to mi the hole. It i therefore imperative to meaure and to undertand how the face angle change during the putting troke in order to detect the root caue of thi mot common putting fault. A very effective way to view how error in the face angle develop i to plot the face angle a a function of poition during the back and forward wing a illutrated next Overhead view of clubhead path Subject J Putt 1 Backwing Forwardwing Model Overhead view of clubhead path Subject L 15ft Putt 8 Backwing Forwardwing Model Face Angle a function of Y-Po 5 Face Angle a function of Y-Po Y-po (in) Y-po (in) Figure 3.15: Two example putt illutrating different fault leading to exceive face angle at impact. Top figure illutrate the meaured path of the club head (and ball impact location) a viewed in the horizontal (X-Y plane). Bottom figure illutrate the meaured face angle a a function of poition (Y coordinate of club head) in the troke. The example to the left illutrate ignificantly different cloure rate (deg/in) on the forward veru backwing. The example to the right illutrate a period of near-zero rotation at the tart of the forward wing. Figure 3.15 preent two example putt having ditinct root caue for exceive face angle at impact. The two bottom figure plot the face angle a a function of poition within the troke (a determined by the Y-coordinate of the club head) for both the back (blue) and forward (red) wing. Alo hown i the predicted relation uing the imple 62

77 kinematic model (cyan). The top portion of the figure how the meaured path of the club head a viewed in the horizontal (X-Y) plane which i dicued further below. From the bottom left figure, notice that the clubface open at a lower rate during the backwing than the model predict and then cloe at a rate approximately equal to that of the model on the forward wing. (Thee face angle rate are meaured in degree per inch of diplacement along target line.) Our meaurement of over 100 trial from 10 ubject reveal that face angle fault frequently derive from different face angle rate for the back and forward wing. Thi type of meaurement i extremely difficult to make uing exiting camera-baed ytem and thu uch diagnoe have not been readily available for player feedback/training. Second, the imple kinematic model ugget an expected or target face angle rate that further dicriminate the root caue problem of the putting fault. For the example putt on the right, the backwing ha a well-defined face angle rate that cloely matche the target rate predicted by the model. By contrat, the face angle rate for the forward wing matche that of the backwing (and model) only for the lat two-third of the forward wing extending from approximately 8Y 0in. For the firt one-third of the forward wing, extending from approximately 11.5 Y 8in, the face angle rate i coniderably le and ultimately lead to a net open face angle at impact of about 3 o. Again, thi analyi pin-point the root caue of the putting fault which in thi cae i a low face angle rate at the very tart of the forward wing. 63

78 Off-Center Impact The location of impact between the ball and the clubface i the econd mot important condition that determine ball dynamic following impact. Off-center impact produce le momentum tranfer to the ball and may alo allow the ball to diverge from the intended target line due to udden, inadvertent change to the face angle. For example, impact toward the heel (converely the toe) of the club may allow the ball to veer left (converely veer right) of the intended target line. To invetigate where ball/head impact occur, we now examine the final poition of the club head from the meaured club head path. In doing o, we tacitly aume that the player ha centered the ball on the clubface at addre (which i alo part of our teting protocol). The impact location for the two prior example putt illutrated in Fig can be deduced by inpection of the club head path in the top figure. For the example hown to the left, the impact occur toward the toe (hence the center of the clubface finihe with X 0 in thi example) by approximately X 0.3in. Notice further that the club head remain largely outide (i.e., on the far ide of) the target line X 0 and remain there until jut before impact. The proper path for both the ideal PILS and gate troke with poitive i one that remain even with or inide the target line. Only in the eldom een tyle where 0 would the putter path be expected to lie outide the target line. In thi example, 0 and therefore the putter path would be expected to be inide the target line. The club head path predicted by the imple kinematic model provide a poible target for the correct path for thi ubject. By contrat, for the example hown on the right, the club head at impact finihe nearly exactly at the ame poition where it began ( X 0 in) and thu thi impact location i eentially perfect. Although thi putt achieve proper impact location, the putter path for remain largely inide the predicted path. 64

79 Improper Ue of Writ Many player unlock or hinge their writ during the putting troke which violate the advice of expert a well a the contraint of the imple kinematic model. A mentioned above, the expert concur that the putting troke derive principally from rotation about the houlder joint keeping the elbow and writ joint locked. The enor technology provide a quick mean to detect inadvertent joint rotation. For example, the reult hown in Fig. 3.16(a) illutrate a putt having a ditinctly maller value for Larm during the forward wing ( 9.2in) compared to that (14.7 in) for the backwing. A a reult, the player hand lag behind the club head during the forward wing and create a final putter loft angle at impact that i about 3 o greater than that at addre. Thi change in L arm and the reulting lag of the hand are meaureable putting fault. In the previou ection, we noted that many player putt exhibit thi type of change in L arm. In nearly all example oberved to date (a in the example of Fig. 3.16), the change occur preciely at the tranition from the backwing to the forward wing. One poible explanation for thi abrupt change i that the player allow hi/her writ to break or hinge at thi tranition. That make ene conidering that the putter achieve it greatet angular acceleration at thi tranition and thu the reaction moment on the writ alo achieve a maximum. The example reult illutrated in Fig. 3.16(b) repreent an exaggerate putt intentionally executed uing only writ rotation. The exceedingly mall value for L arm (4.7 and 3.6 in) meaured for thi putt confirm that the axi of rotation i uncharacteritically cloe to the top of the grip due to rotation largely about the writ joint. Thi limiting cae confirm that any rotation about the writ during normal putting will indeed decreae L arm a oberved at the tart of the forward wing in the example to the left. The ability to identify if and where a player allow their writ to break i a powerful aid in diagnoing thi common putting fault. 65

80 in() in() Backwing Forward Swing Sin() v. Y grip Subject J Putt Y grip (in) (a) arm _ b arm _ f 14.7in 9.2in Figure 3.16: (a) example howing writ-break at the end of the backwing. (b) example i an exaggerated putt uing only writ and leading to very mall value for Larm a a reult. Improper Hand Poition Relative to Club Head L L Backwing Forward Swing Sin() v. Y grip All Writ arm _ b 4.7in 3.6in arm _ f Y grip (in) (b) L L It i alo well-undertood that putting fault arie from the improper poitioning of the player hand (grip) relative to the club head during the back and/or forward wing. Ideally, the grip hould be approximately even with the clubface at impact to obtain the proper loft at impact. When the clubface i forward of the hand at impact, the loft at impact i greater than the pre-et loft deigned for the putter. Too much loft keep the ball in flight too long, which contribute to poor ball tracking along the target line. Converely, when the clubface i too far behind the hand at impact, the loft at impact i maller than the pre-et loft deign for the putter. Too little loft caue the ball to impact the ground too quickly (and poibly re-bound) which again lead to poor ball tracking. Data for two putt illutrated below demontrate both of thee fault; namely 1) a cae where the hand exceively lead the club head at impact, and 2) a cae where the hand exceively lag the club head at impact. Careful analyi of the data pin-point exactly where thee fault arie in thee example. 66

81 in() in() Backwing Forward Swing Sin() v. Y grip Initial Tranlation Backwing Forward Swing Sin() v. Y grip Writ Backwing Y grip (in) (a) L L arm _ b arm _ f 36.6in 19.7in arm _ b arm _ f 5.0in 16.6in Y grip (in) (b) L L Figure 3.17: Loft angle ( in ) veru grip poition along target line ( Y grip ). Example (a) demontrate how the grip can lag the club head at impact leading to exceive loft at impact. Example (b) demontrate how grip can lead the club head at impact leading to inufficient loft at impact. Both fault develop early on in the backwing. The two example putt are illutrated by the reult of Fig Figure 3.17 report the meaured change in in( ) a a function of grip poition Y grip during the backwing (blue) and forward wing (red) and a predicted by the imple kinematic model (green and black). Fig report the poition (Y coordinate) of the hand/grip (green) and the club head (blue) for the ame trial. Starting with Fig. 3.17, notice that the forward wing (red) for both example putt largely conform to the prediction of the kinematic model (black). Thu, any fault deriving from improper hand poition will largely develop during the backwing. For the example on the left in Fig. 3.17, the backwing begin with a ubtantial change in the grip poition ( Y grip ) but with little change in the loft angle ( in ). Thu, the player begin thi backwing by tranlating the club backward without a commenurate rotation of the club. A a reult, the club head i drawn back at the tart of the backwing by an amount nearly equal to that of the grip a clearly een in the reult of Fig. 3.18(a). The unuually large value for L arm (36.6 in) computed for the backwing i conitent with the fact that backwing begin with tranlation and little rotation. (For the limiting cae of pure tranlation Larm ). The forward wing then proceed a predicted by the 67

82 Y poition (in) Y poition (in) kinematic model and doe not correct any of the error developed at the tart of the backwing. A a reult, the player hand ubtantially lag (-2 in) the club head at impact and the clubface ha exceive loft. By contrat, the example hown in Fig. 3.17(b) and 3.18(b) illutrate a putt for which the hand lead the head ubtantially (+4 in) at impact leading to inufficient loft. The root caue of thi putting fault i a writ break at the very tart of the backwing. Notice from Fig. 3.17(b), that the backwing ha an unuually mall value (5.0 in) for Larm which i conitent with a writ break; recall reult for the all-writ putt illutrated in Fig. 3.16(b). By contrat, the forward wing i more typical of a gate troke for which L 16.7 in and hence it doe not exhibit a poible counter-correcting writ break. arm Thu, the player hand remain coniderable forward of the club head throughout the backwing and throughout the forward wing where they finih at impact with a large (4 in) lead over the club head. Thi putt, while rather different from the example on the left, reinforce the ame concluion; namely, the backwing failed to follow the kinematic of the imple pendulum model where club motion develop olely through houlder rotation Y value of grip poition v clubhead poition Initial Tranlation -10 Clubhead Grip Time(ec) (a) Y value of grip poition v clubhead poition Writ Backwing -14 Clubhead Grip Time(ec) (b) Figure 3.18: Poition of club head veru grip. Example (a) demontrate how the grip can lag the club head at impact. Example (b) demontrate how the grip can lead the club head at impact. Both fault develop early on in the backwing. 68

83 3.6 Cloure Thi chapter advance our fundamental knowledge of putting troke by exploiting meaured putting kinematic provided by the MEMS enor ytem. In addition, thi chapter introduce a unified kinematic model for putting that encompae the previou pendulum tyle and gate tyle troke dicued in the literature; ee, for example, [41]. Finally, thi chapter introduce a uite of putting metric that can pin-point common putting fault. In total, thi meaurement and analyi ytem provide ignificant and new capabilitie for putting intruction. The imple kinematic model for putting reduce to a ingle-degree of freedom pendulum with two parameter and L arm determined by the player preferred putting tyle. The parameter define the (average) inclination of the axi of rotation of the pendulum and the parameter Larm define the ditance from the axi of rotation to the grip. Both and L arm are computed from meaured enor data. Doing o yield a reaonably accurate prediction of the club head orientation (lie, loft and face angle) for both the back and forward wing. Comparion of the meaured veru the predicted club head orientation yield coniderable inight into the root caue of principal putting fault including the mot common fault, namely an improper face angle at impact. In uing the kinematic model, we often oberve that the forward wing and backwing yield ditinct value of the parameter L arm. In particular, abrupt change in Larm reveal that the player ha broken otherwie locked writ during the troke which lead to a econd major putting fault. Thi improper ue of the writ can produce exceively large or exceively mall loft angle at impact. Upon adding ditinct value for Larm for the forward and backwing, the imple kinematic model remain in good agreement with meaured kinematic. The lat ection of thi chapter introduce example metric derived from the enor data that pin-point common putting fault. By tudying numerou putt from many ubject, it 69

84 became apparent that putting error mot often originate at the beginning of the backwing or at the tranition between the back and forward wing. A fundamental reaon for thi i that thee point repreent the period of greatet angular acceleration, hence greatet reaction moment at the writ. 70

85 CHAPTER 4 EXTENSION OF MEMS INERTIAL SENSOR SYSTEM TO HIGHLY DYNAMIC GOLF SWINGS The objective of thi chapter i to extend the meaurement theory and hardware deign developed in Chapter 2 for putter to the meaurement of highly dynamic golf wing that arie with driver and iron. The immediate objective i to meaure the rigid body motion of the golf club during what we refer to a a full wing. The additional flexible motion of the club, which may be induced by izable rigid body acceleration, i dicued in both Chapter 5 and 6. Repreentative reult of full wing meaurement and analye will be preented to demontrate the ucceful application of thi technology for driver and iron. We open by firt reviewing the requirement for the enor to upport thi new range of meaurement. 4.1 Senor Requirement for Full wing During a full wing a illutrated in Fig. 4.1, the golf club achieve ignificantly larger acceleration and angular velocity than in putting. Therefore the enor ytem of Chapter 2, that wa optimized to reolve the putting troke, i ill-uited to reolve a full wing. A deign capable of meauring the full wing require enor that can accurately reolve angular velocity upward of 2000 deg/ec and acceleration (at the grip) in the range of g. Thee rate follow from the obervable fact that the golf club travel from the top of the backwing (haft at the 3 o clock poition in Fig. 4.1) to the impact poition (haft at the 6 o clock poition in Fig. 4.1) within a mere 0.2 to 0.3 econd [15]. 71

86 Figure 4.1: Strobocopic photograph of Bobby Jone golf wing which illutrate a full wing (downwing phae and follow through). [47] Baed on thee high range a econd-generation of the deign decribed in Chapter 2 (refer to Fig. 2.4) wa created with rate gyro capable of meauring up to 3000 deg/ec and a tri-axi accelerometer with a 15 g dynamic range. No additional hardware change where required for thi econd-generation deign. In particular, the firtgeneration deign, which ampled the enor at 1 khz, till provide ample during highly dynamic downwing. The meaurement and analyi of full wing uing thi econd-generation high-range MEMS enor i the focu for the remainder of the chapter. 4.2 Meaurement of Full Swing Low Speed Swing Uing Putting Robot In Chapter 2, a imple putting robot wa introduced that wing a golf club on a pendulum path under the action of gravity alone. The putting troke conidered were low peed and with loft range conitent with putting. We ued thi robot to etablih that the firt-generation enor ytem accurately meaure the orientation and poition of the clubface. Chapter 3 then introduced a two-parameter kinematic model of the putting troke that wa hown to predict the putt executed with the robot to a high degree of accuracy. We now employ the robot and the two-parameter kinematic model to etablih the accuracy of the econd-generation enor for exaggerated pendulum tyle troke near the kinematic limit of the robot. Thee troke, which induce club head peed two 72

87 Orientation Angle (deg) to three time thoe of typical putting troke, are till ubtantially lower than full wing a reported later. Figure 4.2 and 4.3 preent the reult obtained for an example exaggerated pendulum robot putt meaured uing the high-range enor deign. Figure 4.2 report the meaured loft, lie and face angle for a putt that reulted in nearly a o 50 change of loft o ( 25 i ubtantial for putting). Alo hown are the predicted loft, lie and face angle from the imple two-parameter model of Chapter Actual v. Predicted Orientation Angle Robot Swing 6 Iron -40 Loft -45 Lie Face Time (ec) Figure 4.2: Meaured and predicted orientation angle for an exaggerated robot putt. The blue, red and green curve are the meaured loft, lie, and face angle, repectively. The cyan, magenta and black curve are the correponding angle predicted by the imple, two-parameter kinematic model. The agreement apparent in Fig. 4.2 between the meaured loft angle (blue curve) and the predicted loft angle (cyan curve) i excellent. There are mall dicrepancie between the meaured and predicted face angle (green and black, repectively) and lie angle (red and magenta, repectively). Overall, the meaured lie, loft and face angle at impact return to within 0.2 o of their value at addre, which i poitive tetimony of the overall accuracy of the high-range deign for an exaggerated putt. 73

88 X-po (in) X-po (in) Overhead view of clubhead path Robot Swing 6 Iron Backwing Forwardwing Model Overhead view of clubhead path corrected Robot Swing 6 Iron Backwing Forwardwing Model Y-po (in) (a) Y-po (in) (b) Figure 4.3: Meaured and predicted path for the center of the club head during a robotcontrolled putt. (a) Uncorrected for light mialignment of enor axe to the robot rotation axi. (b) Corrected for thi light mialignment ( 1.4 o mialignment) Conider next the club head path a one would oberve from overhead in the horizontal (X-Y) plane. Figure 4.3 illutrate the meaured club path and that predicted by the twoparameter model where Y denote poition along the intended target line and X denote a lateral deviation from that line. The cale for the lateral deviation i greatly exaggerated in the figure. The uncorrected path of Fig. 4.3(a) how a one-ided error due to a mall mialignment of the enor axe to the robot axi of rotation. Following the method of Ch. 3 (refer to Eq ), we correct for thi modet ( 1.4 o ) mialignment to arrive at the reult of Figure 4.3(b) which illutrate very good agreement (within 0.25 inch) between the meaured and the predicted path for thi exaggerated robot putt. Overall, thee tet demontrate that the high-range enor yield accurate reult for the low peed dynamic of putting that can be independently meaured uing the putting robot. The high-range MEMS enor incorporated in thi econd-generation deign are reported to maintain their linearity throughout their meaurement range [28, 33]. Given thi expected enor linearity, one may reaonably expect to achieve a imilar level of accuracy for the higher peed aociated with full wing a dicued next. 74

89 High Speed Swing of Human Subject The example full wing preented in the remainder of thi chapter all have the general form of the wing previouly illutrated in Fig Thee wing were meaured in the laboratory and with an impact made by triking a oft practice ball. The wing we hall dicu, while not exhautive in number, capture the major characteritic (e.g., range of motion, club head peed, and club trajectory) common to all full wing. Before doing o, however, we mut reviit the definition of the clubface lie, loft and face angle for the rather large range of motion aociated with full wing. During a full wing thee three orientation angle exhibit a coniderably greater dynamic range than during a putt. The firt definition of thee orientation angle, a given in Chapter 2, did not explicitly decribe their domain of definition which mut now be undertood to meaningfully dicu full wing. iˆ kˆ ˆ j kˆ loft iˆf kˆ f f ˆj f kˆ f lie face target line (a) (b) Figure 4.4: (a) Location and orientation of body-fixed ˆ ˆ ˆ f, f, f iˆ, ˆ, ˆ j k frame. (b) Chapter 2 definition for lie ( ), loft ( ), and face ( ) orientation angle. i j k and Figure 4.4 contain image reproduced from Ch. 2 to refreh the reader with the definition for the f and body fixed reference frame and the orientation angle. We begin by returning to the definition of the lie angle a illutrated in Fig. 4.4(b). The lie angle i the angle formed between the horizontal plane and the unit vector i ˆf. A 75

90 o o hown in Fig. 4.5 the lie angle, which i defined over the domain 90 90, i poitive when iˆf i above the horizontal plane (a), achieve it maximum when i ˆf i vertically up (b), and i negative when i ˆf i below the horizontal plane (c). ˆK ˆK ˆK Î i iˆf Ĵ ˆf _ proj Î iˆf Ĵ Î i ˆf _ proj Ĵ iˆf (a) (b) (c) o o Figure 4.5: Definition of the lie angle and it domain (a) Lie angle i poitive when i ˆf i above the horizontal planeiˆ, J ˆ. (b) Lie angle achieve it maximum of 90 o when iˆf i vertically up. (c) Lie angle i negative when i ˆf i below the horizontal planeiˆ, J ˆ. Equivalently, when the ˆK component of the vector i ˆf i greater than zero a in Fig. 4.5(a), then > 0. When the ˆK component of the vector i ˆf i le than zero a in Fig. 4.5(c), then < 0. When iˆf Kˆ then 90 o. Converely, when i Kˆ, then 90 o. ˆf In an analogou manner, the loft angle i defined by the angle formed between the unit vector ˆj f and the horizontal plane; refer to Fig. 4.4(b). Doing o etablihe the domain o o for the loft angle a

91 Î ˆK ˆj f ˆj f _ proj Ĵ Î ˆj f ˆj f _ proj ˆK Ĵ Î ˆj f ˆj f _ proj ˆK Ĵ (a) (b) (c) o o Figure 4.6: Definition of the face angle and it domain (a) Face angle i poitive due to counter-clockwie rotation of j from the Ĵ axi. The face ˆ f _ proj angle a hown in (b) i poitive, while that in (c) i negative. We now review the definition of the face angle employing the illutration of Fig The face angle i the angle formed between the Ĵ axi and the projection of ˆj f onto the horizontal plane, denoted ˆ f _ proj j in thi figure. Recall from Chapter 2 that, by definition, the face angle i zero at addre ince the target line ( Ĵ ) i etablihed at addre. The face angle may then continuouly change a the wing develop. A poitive face angle reult when change counter-clockwie from the Ĵ axi. A negative face angle reult when increae clockwie from the Ĵ axi. Thu, the domain of the face o o angle i In Fig. 4.6(a), i poitive a it reulted from the counter-clockwie rotation of ˆj f _ proj from the Ĵ axi. Figure 4.6(b) and 4.6(c) how the ame ˆj f vector, but with poitive in (b) and negative in (c). The face angle in Fig. 4.6(b) i again poitive ince it reulted from the counter-clockwie rotation of ˆ f _ proj j from the Ĵ axi. By contrat, the face angle in Fig. 4.6(c) i conidered negative ince it reulted from the clockwie rotation of ˆ f _ proj j from the Ĵ axi. 77

92 Angle (deg) We now proceed to the analyi of an example full wing uing a 6-iron. Figure 4.7 illutrate the time hitory of the club head orientation (lie, loft and face angle) for a typical full wing imilar to that illutrated in Fig o clock Dynamic Orientation Throughout Swing Subject KK 6 Iron 9 o clock approx. 3 o clock 9 o clock Impact Loft -125 Lie Face Time (ec) Addre Backwing Downwing Figure 4.7: Lie, loft and face angle a function of time for a full wing with 6-iron (ubject KK). The annotation decribe the approximate location of the golf haft referred to a poition on a clock. Exploded view zoom in on the angle around impact. Focu firt on the lie angle (red) which provide a clear indication of the phae of a full wing including 1) the addre, 2) the backwing, and 3) the downwing. The lie angle tart near o 0 a the club haft i held till at addre at approximately the 6 o clock poition. The lie angle then increae ignificantly a the backwing begin. When the club haft reache the 9 o clock poition in the backwing, the unit vector iˆf i nearly aligned with vertical and therefore the lie angle i nearly 90 o. The lie angle then decreae a the backwing continue, paing through o 0 at approximately the 12 o clock poition of the haft, and then becoming negative a the haft head toward the 3 o clock poition. The haft frequently approache or pae through the 3 o clock poition near the end of the backwing at which intant the lie angle achieve it minimum a clearly obervable in Fig The enuing downwing phae i extremely hort (approximately 0.4 econd in thi example tarting at t 1.3ec ) a the haft eentially revere it motion. A the haft pae back through the 9 o clock poition ( t 1.6ec ) the lie angle 78

93 again achieve a local maximum. From that point onward, the lie angle decreae rapidly to another local minimum at or near impact with the ball. At impact the lie angle return nearly to the value it tarted with at addre and ideally the final lie angle hould be o 0 a thi maximize the potential contact area between the clubface and the ball. The face and loft angle of the club alo reveal many key element of the full wing. For example, the face angle monotonically decreae during the backwing a the clubface open and then monotonically increae during the downwing a the clubface cloe. During the downwing, there i a very pronounced increae in the rate of thi cloing a the haft pae back through the 9 o clock poition. Starting approximately at the 9 o clock poition, player aggreively releae their writ and rotate the haft about it long axi thereby increaing the rate at which the clubface i cloing. The two traight line uperimpoed on the face angle in Fig. 4.7 highlight thi large change in the cloing deg rate. In particular, 250 ec jut before the 9 o clock poition and then change to deg 1000 ec after the 9 o clock poition for thi example wing. Thi releae alo occur very cloe to the local maximum oberved in the lie angle. Simultaneouly, the loft angle alo increae greatly at and beyond thi releae. A a reult, the entire angular velocity of the club increae very ubtantially at and beyond the releae. A the club approache impact ( t 1.720ec in thi example), the loft angle and face angle continue to change rapidly while the lie angle i nearly tationary a bet een in the exploded view in the right ide of Fig Jut a in the putting troke, the final orientation of the clubface at impact hould be the ame a that at addre. Meaurable difference in lie, loft and face angle from addre to impact are important wing metric which we alo referred to a cloure error in Chapter 2. Table 4.1 report the initial (at addre) and final (at impact) orientation angle for the example wing of Fig. 4.7, and we remind the reader that the face angle at addre i zero by definition. The table lit value for lie, loft and face angle at the time of impact ( t 1.720ec ) a bet etimated from the meaured data a well a value 10 milliecond after thi etimated impact time. The rather large change in the final loft 79

94 and face angle during thi very hort (0.010 econd) period are further tetimony to the large angular velocity of the club head near impact. There i therefore ignificant enitivity of the final loft and face angle with repect to the etimated time of impact. Subject KK Initial Lie Final Lie Initial Loft Final Loft Final Face 6 Iron (deg) (deg) (deg) (deg) (deg) t 1.722ec imp t 0.01ec imp Table 4.1: Initial and final orientation angle for trial reported in Fig Final angle are reported for both the etimated impact time a well a 10 milliecond later. The initial lie angle =10.2 o at addre indicate that the toe of the club head i initially higher than the heel. It i deirable to trike the ball with = 0 o a thi maximize the opportunity to impact the ball on the center, weet pot, of the clubface. The data of Table 4.1 confirm that the final lie angle for thi wing i nearly 0 o and hence the error at addre wa effectively corrected during the wing. The manufactured loft of the example club i 30.5 o o thi ubject tarted with le loft than deired at addre and then reduced the loft further at impact. The clubface i therefore de-lofted at impact yet the loft i alo increaing ignificantly during the impact proce a een by the 7 o change in a mere 10 milliecond. If the face angle i not zero at impact, then the initial ball trajectory will not be aligned with the target line etablihed at addre. The negative value of the face angle at impact ( 8.3 o ) indicate that the initial trajectory would be to the right of the target line. However, the poitive value ( 6.7 o ) a mere 10 milliecond later confirm that the clubface pae through the quare poition ometime time in between. Overall the time hitorie of the lie, loft and face angle provide ignificant inight into each phae of the wing and the orientation of the clubface at impact. To gain further inight into the full wing, we now preent a three-dimenional view of the poition of the club in Fig. 4.8, a computed from the meaured enor data uing the algorithm detailed in Chapter 2. Thi poition data ha been corrected for enor drift 80

95 by auming that the final club head poition return to it poition at addre. While thi eentially zero the poition error at impact, it alo yield very trong condition for eliminating drift error in the rate gyro and accelerometer leading to accurate prediction of club head orientation and wing path. In future work, we will eek to relax thi aumption o that the impact location of the ball on the clubface can alo be predicted. 81

96 v h Figure 4.8: Swing path of full wing with 6-iron for example of Fig. 4.7 (ubject KK). The centroid of the clubface i repreented by the blue dot during the backwing and by the cyan dot during the downwing. The grip (enor location) i repreented by the green dot in the backwing and by the (lighter) ea-green dot in the downwing. The golf haft i repreented a a red line in the backwing and a magenta line in the downwing. The gray plane repreent the bet fit wing plane during the downwing. The path of the club in the figure i divided into the backwing and the downwing phae. During the backwing, the clubface centroid i repreented by the blue dot, the haft i repreented by the red line, and the grip poition (enor origin) i repreented by the green dot. In the downwing, the clubface centroid i repreented by a cyan dot, the haft i repreented a a magenta line, and the grip poition by the (lighter) ea-green dot. The time interval between ucceive image for the backwing i 0.010ec and 0.005ec for the downwing. The gray plane uperimpoed on the club trajectory repreent the bet fit plane to the haft poition data during the downwing phae. From the perpective illutrated in Fig. 4.8, the poition of the haft and the club head at the end of the backwing can be readily identified. Thi poition i often ued to ae the quality of the backwing. The downwing arc i eaily obervable in Fig. 4.8, and it arc length could be calculated both for the grip and the club head. The length of the downwing arc for the club head i commonly aociated with the final club head peed 82

97 Vel (ft/) for which a longer wing arc often correlate with a greater club head peed at impact. The reader can infer the peed of the club head by noting that higher peed lead to greater pacing between ucceive image. Quantitatively, the intantaneou club head velocity i readily available from calculation following the algorithm detailed in Chapter 2 and the reult i hown in Fig. 4.9 where the three component of the club head velocity vector are plotted a function of time. For the wing conidered here, the club head peed at impact wa 52.7 mph (and a typical 5-iron (0.5 longer than a 6-iron) peed i be 77 mph, [48]). Thi wing wa intentionally performed at a reduced peed in the laboratory for afety. The direction of the club head velocity at impact i alo a key factor in determining the direction the ball will travel. From Fig. 4.9, one oberve that the lab-fixed Y component of the velocity dominate at impact a expected ince the Y axi define the target line etablihed at addre. The non-zero X velocity component at impact however confirm that the club head trajectory i not perfectly aligned with the intended target line. 80 Velocity of Club Face in Lab fixed ref. Frame Subject KK 6 Iron X: Y: X: Y: X: Y: X-Axi Y-Axi Z-Axi Time (ec) Figure 4.9: Velocity of centroid of clubface for full wing (ubject KK). Component reported for the lab-fixed frame. The final data point are the value of the velocity component at impact. The image in Fig. 4.8 how that the backwing and the downwing are largely confined to two plane which we hall refer to a the wing plane ; refer to grey plane in the figure for the downwing. A wing plane can be defined by two angle: the angle v that 83

98 the plane make with the vertical Z axi, and the angle h that the plane make with the target line Y; refer to Fig We often oberve that the wing plane that bet fit the backwing i ditinct from the one that bet fit the downwing and thi difference may later prove to be important in characterizing thee phae of the wing. In the example wing illutrated in Fig. 4.8, the vertical angle 21.3 o for the backwing and v v 25.9 o for the downwing. Similarly, the horizontal angle h =17.6 o for the backwing and h =5.8 o for the downwing indicating that the wing plane for the downwing i rather cloely aligned with the target line a expected. Recall that the reult of Table 4.1 demontrate that the loft and face angle at impact are enitive to the etimated impact time. Moreover, the impact time oberved in the enor data actually occur lightly earlier than the aociated impact with a perfectly rigid golf club, a aumed in our analyi of the data. During a highly dynamic golf wing, the golf haft flexe and twit, and at impact the club head normally lead the grip. In other word, the Y poition of point f (on the face of the club head) at impact i greater than the Y poition of point (within the grip) at impact [18, 39, 49]. The analyi of the enor data tacitly aume that the golf haft remain rigid during the wing, an aumption that will alo be re-examined in Chapter 5 and 6. Due to thee fact, the meaured wing (for a flexible club) would then take lightly longer to reach impact if the club were truly rigid. We believe that thi time delay might be approximately 0.01 econd for the wing illutrated in Fig Extending thee reult by an additional 0.01 econd lead to the reult hown in Fig Notice now that the grip end of the club at impact finihe very cloe to it initial poition at addre. In fact, the final haft orientation at impact i very imilar to it initial orientation at addre. 84

99 Figure 4.10: Swing path of full wing 6-iron from Fig. 4.8 with an aumed impact delayed by 0.01 econd. We conclude thi ection by preenting reult for an example wing with a driver wung at full peed. Doing o confirm much of what wa oberved with the lower wing with the 6-iron dicued previouly. 85

100 Figure 4.11: Swing path of full wing with driver (ubject G). The centroid of the clubface i repreented by the blue dot during the backwing and by the cyan dot during the downwing. The grip (enor location) i repreented by the green dot in the backwing and by the (lighter) ea-green dot in the downwing. The golf haft i repreented a a red line in the backwing and a magenta line in the downwing. The gray plane repreent the bet fit wing plane during the downwing. Figure 4.11 illutrate the three-dimenional trajectory of the club and Fig report the aociated time hitorie for the clubface lie, loft and face angle. For thi wing, the vertical wing plane angle, v, i 39.7 o and 43.7 o for the backwing and downward wing, repectively. Thee angle are ignificantly greater than thoe for the 6-iron wing. Thi i expected ince the driver i ignificantly longer that the 6-iron (46in for driver veru 37.25in for the 6-iron). At impact the club head velocity now attain 87.5mph which i very typical of the club head peed for driver [48]. The horizontal wing plane angle, h, i 5.2 o and 4.3 o for the backwing and downward wing, repectively. Thee mall value confirm that both of thee wing plane are well aligned with the target line. The reulting lie, loft and face angle, illutrated in Fig. 4.12, alo confirm the major feature noted previouly for the wing with a 6-iron (and a different ubject). Thi coniderable agreement lend confidence to the future ue of thee wing ignature for comparing wing of player with different kill level. Notice again that the lie angle 86

101 Angle (deg) clearly demarcate the important phae of the wing including addre, the backwing, and the downwing. The final value of the lie, loft and face angle at impact are again an important metric of wing performance. 100 Dynamic Orientation Throughout Swing Subject G Driver Loft -125 Lie Face Time (ec) Figure 4.12: Orientation angle a function of time for a full wing with driver with final time 0.01 econd after impact time etimated from the enor meaurement (ubject G ). Table 4.2 report the cloure data for thi wing with a driver. The lie, loft and face angle value reported in the firt row follow from the etimated impact time from the enor data. The value of thee angle 0.01 econd later are alo reported in the econd row for comparion. Again, notice that while the final lie angle i eentially tationary, the loft and face angle continue to change rather rapidly near impact. Note alo that the wing for thi ubject nearly matche the manufactured club lie at addre but decreae the lie at impact by approximately 7 degree. The initial loft at addre i very cloe to the manufactured club loft of 9 o and the final loft alo remain reaonably cloe to thi value. The open face angle at impact ugget that the initial ball trajectory would be right of the target line baed on the etimated impact time from the enor data; ee value in firt row. By contrat, if the impact time were delayed by a mere 10 milliecond, then the face angle i very lightly cloed. Clearly, the determination of the actual impact time (or impact time interval) i critical to an accurate determination of the orientation of the club head during impact with the ball. 87

102 Subject G Initial Lie Final Lie Initial Loft Final Loft Final Face Driver (deg) (deg) (deg) (deg) (deg) t 1.462ec imp t 0.01ec imp Table 4.2: Initial and final orientation angle for trial reported in Fig Final angle are reported for both the etimated impact time a well a 10 milliecond later. 4.3 Cloure Thi chapter decribe the ucceful extenion of the hardware and analyi method developed firt for putting in Chapter 2 to full wing uing iron and driver. For low peed wing that can be independently meaured uing the putting robot, we confirm that the econd-generation high-range enor ytem accurately predict the planar and exaggerated pendulum troke. The remainder of the chapter analyze the high peed wing by human ubject uing both iron and driver. The analyi of wing made with iron and driver reveal important feature common to all full wing. Firt, the three major phae of a full wing, including the addre, the backwing and the downwing, are readily ditinguihed uing the dynamic lie angle. Second, the loft and face angle change very rapidly during the final portion of the downwing following the releae of the writ tarting at approximately the 9 o clock poition with impact occurring at approximately the 6 o clock poition. Both the backwing and the downwing are largely confined to ditinct wing plane that are well defined from the three-dimenional club trajectorie computed from the enor data. The orientation of the wing plane with repect to vertical and with repect to the target line etablihed at addre may become ueful metric in future tudie of golfer kill. In addition, the meaured club head peed and the lie, loft and face angle at impact are of immediate ue in aeing golfer kill. However, we alo note that the meaurement of the impact time or the impact time interval require further tudy and definition, perhap by independent mean. 88

103 CHAPTER 5 PLANAR FLEXIBLE MODEL OF A GOLF CLUB AND EXPERIMENTAL VALIDATION The objective of thi chapter i to derive and validate a model which predict the flexible motion of a golf club. The overall motion of the club can be decompoed into the rigid body motion of the grip and the flexible motion of the haft relative to the grip; refer to Chapter 2. Recall that the flexible motion i driven by the rigid body motion and that the enor ytem preciely meaure the latter. Thu, a model for the flexible body motion may now be developed with the meaured rigid body motion a the excitation. The overall motion, of coure, ultimately define the abolute motion of the club head and predicting thi overall motion i now the goal of thi chapter. Thi chapter etablihe thi modeling concept by conidering the impler cae of a purely planar motion of the club head. Chapter 6 extend thi planar model to a full three-dimenional model. 5.1 Derivation of Planar Model (Lagrange Formulation) kˆ iˆ x ˆK g ( xt, ) v h kˆc c v Î Figure 5.1: Diagram of planar model. Point, h and c denote the origin of the haft frame, the hoel (point of interection of haft and club head), and the ma center of the club head, repectively. Two generalized coordinate ( v and v ) are introduced to decribe the deflection and rotation of the end of the haft at the hoel. iˆc Figure 5.1 illutrate a male and flexible beam (the haft of the club) attached to the club head (rigid body) of ma m. Thee component form the dynamic model of the golf 89

104 club. The lab fixed frame I ˆ, K ˆ i introduced with the unit vector ˆK aligned with vertical g gkˆ. Point, located at the grip end of the club, repreent the location of the MEMS accelerometer a previouly introduced in Chapter 2. The body fixed reference frame i ˆ, ˆ k i located at and define the tangent and normal direction of the undeformed or rigid haft (olid line in Fig. 5.1). A econd body-fixed frame i ˆ, ˆ c k i located at the ma center of the club head c, and define the tangent and normal direction of the deformed haft at the end of the haft denoted a point h. Point h i introduced at the interection of the haft and the club head and it i commonly referred to a the hoel. The coordinate xt, repreent the deflection of the flexible haft from it undeformed configuration with x being the independent patial coordinate meaured along the undeformed haft from toward h. The deflection of the haft at point h (where x=l) i denoted a v (the deflection at the hoel) and the lope of the haft at thi ame point h i denoted a v (the rotation of the deformed haft relative to the undeformed haft at the hoel). c It i aumed that the haft bend a an inextenible and male Euler-Bernoulli beam. The jutification for uing the linear approximation inherent in Euler-Bernoulli beam theory ret on the oberved fact that a typical maximum deflection of the haft at the hoel in a highly dynamic wing remain le than 15 cm [39] relative to a typical haft length of 110 cm [50]. Thu the haft, which i truly lender, undergoe deflection that are alo reaonably mall. Likewie, the ma of a typical haft i approximately 110 gm (teel) or 75 gm (graphite) [50] relative to the ma of a typical head which i 200 gm (driver) or 265 gm (iron) [50]. Thi fact, coupled to the fact that the haft only change direction once, between the back and forward wing, during the wing (prior to impact) ugget that one can reaonably ignore the ma of the haft to a firt approximation. The jutification for ignoring the extenion of the haft ret on the fact that the haft i coniderably tiffer in extenion than bending and that any poible extenion due to the pin up ha negligible influence on the club dynamic. Of coure, any of thee 90

105 aumption could be relaxed in the future hould they omeday be deemed important for improved model accuracy. Diipation, while initially ignored in the derivation, i appended to the model at the end. We now eek to decribe the deflection of the beam, xt,, wholly in term of the two generalized coordinate, v and v. Firt, we etablih the boundary condition for the beam. At point we require that the flexible beam remain tangent to the undeformed or rigid beam. The kinematic of thi rigid beam are directly meaured via the enor ytem. Thu, the flexible beam mut atify a clamped boundary condition at the origin,, of the haft frame. Meanwhile, the lope and deflection at the oppoite end, point h, are the generalized coordinate v and v, repectively. Conider next Fig. 5.2 which how a cantilevered beam ubjected to a terminal force and moment. x ( xt, ) h F M Figure 5.2: Cantilevered Euler-Bernoulli beam ubject to terminal force F and bending moment M. By uperpoition of claical olution for thi Euler-Bernoulli beam, the diplacement xt,, and the lope of the beam centerline, ' xt, 2 2 Fx Mx ( x, t) 3L x 6EI 2EI 2 x Fx '( x, t) FL M EI 2EI, are (5.1.1) Evaluating thee function at x=l and olving for F and M in term of v and v yield 12EI 6EI F 3 v 2 v L L 6EI 4EI M 2 v v L L (5.1.2) 91

106 Subtituting (5.1.2) into (5.1.1) yield xt, 2 2 x vl 3 vl x vl 2x v (5.1.3) 3 L Thi diplacement, now expreed a a function of the two generalized coordinate, will be ued in determining the train energy of the beam. The potential energy due to elatic deformation can be generalized a: Vtrain 1 Fx (5.1.4) 2 with F being a generalized force and x being a generalized diplacement. If we conider a beam ubject to a normal tre, the potential energy per unit volume i dvtrain 1 (5.1.5) 2 Applying Hooke law (and ignoring Poion contraction), the train energy per unit volume i imply. dvtrain E (5.1.6) Figure 5.3 illutrate the kinematic relationhip between the train, the curvature 1 of the neutral axi, and the ditance y of a beam element from the neutral axi. 92

107 y L ( y) y L Neutral Axi Figure 5.3: Kinematic relationhip between radiu of curvature and train in a beam due to bending. Combining thi reult with (5.1.6) and integrating over the volume of the beam yield the train energy: 1 y Vtrain E dvol 2 vol Vtrain E dx y da 2 L A 1 1 Vtrain EI dx 2 L 2 2 (5.1.7) recalling that integral of 2 y over the cro ection area i defined the area moment of inertia I. Following the tandard ign convention introduced in Bedford [37], the relationhip between curvature and the diplacement of the neutral axi, xt,, i 2 1 ( xt, ) x which, when ubtituted into (5.1.7) yield 2 (5.1.8) (, ) 2 2 xt x (5.1.9) L Vtrain EI dx Subtituting the diplacement (5.1.3) into (5.1.9) and integrating along the length of the beam lead to the following expreion of the beam train energy wholly in term of the two generalized coordinate 93

108 V train EI 3 v 3L vv L v (5.1.10) 3 L The 2D model conider the club motion in the vertical I ˆ, Kˆ plane, therefore the potential energy due to gravity mut alo be included. Fig. 5.4 provide two illutration to aid in the development of an expreion for the gravitational potential. The gravitational potential of the club head of ma m i Vgrav mgz (5.1.11) where Z i the height of the head meaured in the inertial frame. Refer to Fig. 5.4 where Z i the ˆK component of the poition vector defined in the inertial frame. R c. Here, capital R denote a poition vector Vgrav mgz kˆ iˆ kˆ g R iˆ r c Undeformed Beam Flexed Beam Î ˆK c Î ˆK R c c h Initial Poition Z=0 General Poition Figure 5.4: Definition of the gravitational potential energy of the club head. The initial poition with Z=0 and a general poition Z 0. The vector R XIˆ ZKˆ and can be written a R R rc with r c r c r. Here, the h c c h lower cae r denote a poition vector meaured in a body-fixed frame. At thi point it i convenient to introduce the coordinate tranformation from i ˆ, ˆ k to i ˆ, ˆ c k c and from 94

109 I ˆ, K ˆ to iˆ, kˆ per the following illutration. Thee tranformation will be referred to a c C and C L repectively. Î iˆ kˆ ˆK kˆ iˆc kˆc iˆ v iˆ co in ˆ 1 ˆ c v v i v i ˆ k in co ˆ 1 ˆ c v v k v k iˆ co in ˆ I dt kˆ in co ˆ ; K (5.1.12) In doing o, we alo aume that the angle v remain mall to be conitent with the Euler-Bernoulli hypothei of mall rotation of the beam cro ection. Expanding the term of r c lead to T T ˆ ˆ c r ˆ ˆ ˆ ˆ c h x i z k C x i z k x z i z x k r ˆ ˆ h vi Lk r x z i L z x kˆ c c c c c c c c c c v c c v c ˆ v c v c c v c Tranforming the reult of (5.1.13) to the inertial reference frame and ubtituting into provide R X IˆZ Kˆ R c Derived from meaurement co L z x x z Iˆ L z x in x z Kˆ in in co c v c v c v c c v c v c v c Z Z co L z x x z c v c v c v c which yield the gravitational potential wholly in term of the two generalized (5.1.13) R c (5.1.14) coordinate. The gravitational potential and the train energy contitute the total potential energy function for the model. We now turn our attention to decribing the kinetic energy. Since the haft i conidered to be male the kinetic energy reduce to that of the club head. The total kinetic energy i thu given by 1 1 T mv v I 2 2 T 2 c c yy c (5.1.15) 95

110 where v c i the abolute velocity of point c, c i the abolute angular velocity of the club head and I yy i the centroidal ma moment of inertia about the out of plane axi ˆc j. The following equation block (5.1.16) detail the expanion ued to develop the kinetic energy. Thee expanion demontrate that the vector v c and c can be written in term of known geometrical quantitie, namely L, x c, and z c, known meaurand from the inertial enor ytem, namely v x, v z, (refer to Chapter 2), and finally the generalized coordinate and their derivative,, v v, v, v. vc vh c rc h r ˆ ˆ c h xc vzc i zc vxc k from eq. (5.1.13) ˆ c v jc ˆ y j (meaured by enor) ˆ ˆ ˆ ˆ c y v j Note: J j jc vh v vh h r v ˆ ˆ vxi vzk (meaured by enor) r ˆ ˆ h vi Lk v ˆ h vi v v L iˆ v kˆ v h x y v z v y c v ˆ x L y v xc yv z cv xc vv i v z x z kˆ z v y y v c y v c v c v v At thi point we introduce Lagrange equation: d T T Vtrain Vgrav 0 i 1,2 dt q i qi qi with all neceary quantitie developed above in term of the generalized coordinate, denoted a q i. Evaluating Lagrange equation yield the following two equation of motion (5.1.16) (5.1.17) 96

111 Mq Cq Kq F 0 v q v m mzc M 2 2 mzc I yy mxc mz c 0 2mxc y C 2mxc y 0 12EI 2 6EI m y m 3 zcy x cy 2 L L K 2 6EI EI m zcy x cy m 2 xc y zc y xcvx zcvz Lx c y L L 2 m xc y vzy g in v x L z c y 2 F ml y xc gxc co vxy xc vz xc gzc in zcvzy zcvx Lz cy I yy mzc mxc y (5.1.18) where nonlinear term have alo been dropped. The matrix equation of (5.1.18) are a coupled et of 2 nd order ordinary differential equation (ODE ) for the two unknown generalized coordinate. There are numerou way to numerically olve thee ODE with the meaured data from the MEMS enor. One method i to firt decouple the highet order derivative term to arrive at a et of econd order equation which can then be readily cat in firt order form for implementation in tandard ODE olver. We purued thi direction to integrate the equation of motion uing tandard Matlab olver. 5.2 Experimental Validation of Planar Model Having developed a planar model that decribe the flexible motion of a golf club, we now eek to utilize thi model in predicting the total motion of the club head. Before doing o however, a number of model parameter mut be determined from experiment or from known geometric and inertial propertie of the club being analyzed a reviewed below. 97

112 The parameter x c and zc decribe the location of the ma center, c, relative to the hoel, h, and are a function of the club head geometry. The parameter L i the ditance from to h a determined by the haft geometry and the placement of the enor within the haft. I yy and m define the club head inertia propertie for the planar model. The haft flexural rigidity EI can be etimated by numerou method. One method i to etablih E from the pecific haft material and then determine an average I, baed on the haft geometry. Golf haft are tapered and therefore I varie along the length of the haft. In the derivation above, we aumed an average value of I though the model could readily be modified to account for the exact taper. A pragmatic alternative i to imply meaure the average EI for the entire aembled club experimentally by fitting a model for an equivalent uniform haft to the experimental data for tatic haft deflection a decribed further in thi chapter. We ought to develop a imple experimental method to directly meaure the generalized coordinate, v and v, for a club. We elected to ue a tandard (30Hz frame rate) video camera and a pecially deigned club with dynamic both low and large enough to reolve with a video camera. The club wa alo deigned to make the calculation and identification of the model parameter relatively traight forward. A tandard golf haft wa ued with a large aluminum head for thi purpoe a hown later in Fig The aociated model parameter are reported in Table 5.1. Note that the club head ma (4 lb) i about an order of magnitude larger than that tandard club (0.46 lb) which lead to the intentional low and exaggerated haft flexing in thee tet. 98

113 Parameter Value Unit Parameter Value Unit L 3.69 ft m c 4.18 lbm 2 x c 0.15 ft I zz z c 0.00 ft EI 188 Table 5.1: Parameter for club ued in the validation of the planar model. 2 lbm ft lbf ft 2 Video Calibration Three calibration tep are needed in order to make an accurate comparion between the video data, the meaurement from the MEMS enor, and ultimately the output from the planar model. The firt tep i to ynchronize in time the data from the MEMS enor and the camera. Thi entail determining the frame rate of the camera and identifying a tarting event which i clearly viible in the camera and imultaneouly detectable in the MEMS enor data. The total time of the video clip i divided by the total number of frame in that clip to determine an average frame rate per ec Total Clip Time Frame Rate ( )= ( ) frame Total frame F To enure good ynchronization of a video clip, the club head wa firt impacted on a (wooden) backtop block a illutrated in Fig Thi impact caue high frequency (5.2.1) vibration of the haft which i eaily identified in the MEMS enor output, a alo hown in Fig From thi tet the impact frame in the video and the impact time in the enor data are determined. Thi information, along with the frame rate, enable the ynchronization of event between the video and enor data per 1 Event Frame Impact Frame ( timpact te) (5.2.2) Frame Rate The time when an event occur in the MEMS data are denoted a t e. 99

114 Frame 300 Frame 301 Impact with backtop block Raw angular velocity data from MEMS enor Figure 5.5: An example illutration of how the MEMS data and the video file are ynchronized in time. Frame 300 how club head jut prior to impacting the wooden back top. Frame 301 how club head during impact. The econd tep in calibrating the video i to covert pixel dicretization to dimenion of length. Thi calibration i accomplihed by viewing an object of known length in the video. For thee experiment, the reference length ued were the edge of the club head. The calibration i done in both the horizontal and vertical direction to account for any difference caued by the camera len. The vertical and horizontal length of the club head were meaure uing a dial caliper (accurate to ). Figure 5.6 below identifie key point on the tet club and include an exploded view of the club head ection. The exploded view of the club head how example data for calibration of the horizontal and vertical pixel-to-inch cale factor. 100

115 3.499 P2 P3 P4 video 222h 119v 262h 119v v h 79v P1 Backtop Block h Figure 5.6: Right image how the flexible club experimental etup calling out key point. Left image i a zoomed-in view of the club head that how an example calculation of the horizontal and vertical pixel-to-inch cale factor calibration. The pixel-to-inch cale factor calibration are given by in SFv pixel vert, SF h in pixel horz (5.2.3) The third tep in calibration i to meaure the mialignment of the camera vertical axi from true vertical. The MEMS enor accurately meaure the enor orientation relative to gravity. The enor i mounted in the haft and therefore calculate the orientation of the haft relative to vertical. The orientation of the haft in the video can be calculated from point P1 and P2 hown in Fig. 5.6, uing the following relationhip. h h 1 2 P1 pixel, P2 pixel v1 v2 h" ( h h ) SF 2 1 v" ( v v ) SF 2 1 v" h" 1 video tan h v (5.2.4) 101

116 The vertical mialignment of the camera axi, vert MEMS enor meaurement, MEMS, repreent the difference of the, and video, calculated from analyzing the video. The amount of mialignment wa one degree on average for the following tet. The other point noted in Fig. 5.6 include the poition of two clamp attached to the haft that can be ued to hold a econd and rigid pointer haft jut in front of the club. The clamp alo erve a viible reference point for determining location on the haft during analyi of the video. When the pointer haft i attached to the club, it track the orientation of the grip ection of the haft with no deflection and thu follow the dynamic of a perfectly rigid haft. Three different method for determining the orientation of the grip end of the haft where the enor i mounted were explored. The firt method ue a long pointer haft of nearly equal length to the flexible haft and mounted in front of the flexible haft relative to the camera. Thi long pointer haft would offer the bet opportunity to oberve the motion of the rigid haft were it not for the unavoidable vibration caued during movement. A mall damping element wa added to the top the club head to partially addre thi concern. The econd attempt wa to ue a rather hort (approximately 1 ft long) pointer haft which, being tiffer, did not uffer the ame vibration. However, the horter length lead to poorer reolution of poition and orientation from the pixilated video output. The final method wa to ue the haft clamp a a reference point for the poition of the grip end of the club and then ue the MEMS enor data to etablih the orientation of the rigid portion of the haft in the video. Thi final method proved to be inaccurate and the reult will not be preented. We preent next three type of tet that erve to validate the flexible golf club planar model. Thee tet include 1) tatic meaurement of haft deflection, 2) dynamic meaurement of the natural frequency of the haft/head ytem, and 3) the tranient meaurement of the haft/head ytem dynamic under meaured acceleration of the grip. 102

117 Tet 1: Validation of Static Deflection The implet validation tet i to compare the output of the planar model with the tatic deflection of the club under gravity a meaured by video. Thi tet conit of holding the haft at a known and contant angle relative to vertical and then meauring the reulting tatic deflection of the haft. The reult of thi tet were alo ued to determine the average EI for the club. Under tatic condition, the planar model reduce to ( 0, 0, 0, v v v v 0) v v y y v v x z x z 12EI 6EI 3 2 L L in v mg 6EI 4EI mg xc co zc in v 2 L L which can readily be olved for the tatic haft diplacement and rotation at the hoel. (5.2.5) Equation (5.2.5) can alo be olved for EI in term of the diplacement at the hoel. Thi latter calculation reult in EI (for thi club 0) 2 mgl 2L 3zc in 3xc co z c 6 The variable v and, meaured from a video image, are then ued to etimate EI. v (5.2.6) Figure 5.7 i a typical video frame analyzed during the tatic teting. Recall that the generalized coordinate of the flexible model are the deflection of the hoel v and the rotation of the haft at the hoel v, both meaured relative to the undeformed haft a hown in Fig

118 P4 v P1 v P5 P1 Figure 5.7: Example of tatic deflection tet. The pointer haft i defined by point P1 and P4. The flexible club haft pae though point P4, P5 and P1. Point P1 and P5 define the tangent at the end of the haft and the angle formed between thi tangent and the pointer haft i denoted by v. The deflection of P1 from the pointer haft i denoted by v. The tet were conducted at three nominal angle 15, 20 and 30 degree. At each angle v, v, and were computed a detailed in (5.2.7) below. v flex h v 2 2 v in in h h h SF in P1 ' P1 h v v v SF in P1 ' P1 h and v 1 rigid tan h rigid v v v SF rigid P4 ' P1 v h h h SF rigid P4 ' P1 h v 1 flex flex tan h flex v v v SF flex P1 P5 v h h h SF flex P1 P5 h (5.2.7) The calculated EI for each trial, uing the computed value of v and, are given in Table