A Performance Analysis System for the Sport of Bowling

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$Course Syllabus - Penn State Harrisburg Math/Computer Sciences ...$

$A Simulator for the MARIE Architecture - Penn State Harrisburg Math ...$

3.4.1 Finding the Linear Velocity of the Ball The following values have already either been measured or calculated: 1) The total distance D T the ball traveled from release to pin impact (nominally 60 feet from foul line to head pin). 2) The time T it took for the ball to travel D T feet. 3) The average linear velocity of the ball v ave = D T / T. 4) The number of revolutions R T that occurred in time T. 5) The period t i and angular velocity ω i = 1 / t i of each revolution i in R T . Using these values, an equation can be developed that gives the linear velocity v i for each revolution i of the ball. Recalling the previous calculations for R R and R P (the fractional revolutions at release and pin impact), D T can be represented as D T = R R · (v 1 t 1 ) + v 1 t 1 + v 2 t 2 +....+ v n t n + R P · (v n t n ) = (1+ R R ) · v 1 t 1 + v 1 t 1 + v 2 t 2 +....+ v n t n + (1 + R P ) · v n t n . By letting t 1 ' = (1 + R R ) · t 1 , t n ' = (1 + R P ) · t n , and t i ' = t i for 1 the above equation can be reduced to the summation R T D T = Σ (v i t i '). i=1 Recall the assumption that the kinetic energy K of the ball remains constant from release to pin impact, and let K i be the kinetic energy of the ball during revolution i. Then K i = m / 2 (v i + kω 2 i ), where ω i = 1/t i ' and K 1 = K i = K n , for 1 ≤ i ≤ n. From the above, m / 2 (v 2 1 + kω 2 1 ) = m / 2 (v 2 i + kω 2 i ), and solving for v i in terms of v 1 , v i = (v 2 1 + k(ω 2 1 - ω 2 i )) ½ . For constant energy, no energy is lost to friction, and the above equation reveals that friction acts solely to transfer energy from linear momentum to angular momentum. An expression can now be obtained for each velocity v i in terms of v 1 . Substituting that expression into the previous summation yields R T D T = Σ (v 2 1 + k(ω 2 1 - ω 2 i )) ½ · t i '. i=1 The value of D T is nominally 60 feet, and this summation can be used in developing a converging iterative solution for v 1 . The remaining values for v i can be generated from the value found for v 1 . 52
In order to find the value for v 1 , an initial "seed" value v is needed to start the iteration. A good first guess for v is the average linear velocity v ave of the ball, v ave = D T / T, The actual value for v 1 (the initial velocity) is greater than v ave , since the ball is constantly slowing down after release. Plugging v ave into the above summation will result in a value D' that is less than D T . D' can be used to arrive at the next guess, as follows: v' = v + (D T - D') / R T . Iteration continues in this fashion (guess, calculate, adjust), until the difference between D T and D' does not exceed a predefined threshold, at which point a value for v 1 has been found. The velocities for the remaining revolutions are obtained by plugging v 1 and the values for ω 1 and ω i into v i = (v 2 1 + k(ω 2 1 - ω 2 i )) ½ . 3.4.2 Finding the Distance from the Foul Line Using the values for v i from this equation, a method for converting time (sample index) to distance for each sample in the waveform can be derived. With such a conversion, it is possible to find the distance the ball has traveled at any time in the waveform, and also to graph the linear and angular velocities of the ball with respect to distance. The location of each revolution on the lane relative to the foul line, and the ball loft distance (the distance the ball traveled before impacting the lane) can also be determined. Recall that R T D T = Σ (v i t i '). i=1 By performing the summation from i = 1 to R i-1 , such that R i-1 D i = Σ (v i t i '), i=1 D i then gives the distance from the foul line where revolution i started. The value for each of the v i 's was found previously, and v i can now be assigned to each sample of revolution i. Since the distance D i from the foul line to the start of each revolution i has also been found, the distance for each sample in revolution i, relative to the foul line can be calculated. Define s i,0 to be the first sample of revolution i, so that s i,0 , s i,1 , …, s i,n-1 are the n samples of revolution i and let d i,j be the corresponding distance for sample s i,j . A method for finding the velocity v i and the distance D i for s i,0 has already been given. Assuming a constant velocity during revolution i, the distances for the remaining samples of revolution i can be generated from the equation d i,j = D i + (j / f) · v i , where 0 ≤ j < n, and f is the sampling frequency (currently 120 Hz). Thus, a conversion from time (sample index) to distance for each sample has been found, and it is now possible to graph the linear and angular velocities of the ball with respect to distance from the foul line, as well as indicate the location of each impact. 53
Page 1 and 2:
The Pennsylvania State University T
Page 3 and 4:
TABLE OF CONTENTS Section I: Introd
Page 5 and 6:
LIST OF FIGURES Figure 2-1 SMARTDOT
Page 7 and 8: SECTION I: INTRODUCTION, BACKGROUND
Page 9 and 10: eceiver that stores, processes, and
Page 11 and 12: 1.3.2 Product Development Phase The
Page 13 and 14: SECTION II: COLLECTING THE DATA - T
Page 15 and 16: 2.2 Design Constraints The feasibil
Page 17 and 18: 2.3 SENSING REQUIREMENTS The module
Page 19 and 20: Pin Deck Light Foul Line Figure 2-2
Page 21 and 22: 2.4.1 Ambient Light Sensor The ambi
Page 23 and 24: 9 28 8 RST V CC V CC 5 6 hyst 7 3 +
Page 25 and 26: 2.6 HARDWARE/SOFTWARE INTERACTION D
Page 27 and 28: 2.6.3 Baud Rate The module communic
Page 29 and 30: Configuration settings and the samp
Page 31 and 32: 2.7.5 The Ball is Rolling If the wa
Page 33 and 34: covered by the wand). When the modu
Page 35 and 36: 2.8.1 Detecting Release Detecting t
Page 37 and 38: Light Level 100 90 80 70 60 50 40 3
Page 39 and 40: By filtering the data, and then cou
Page 41 and 42: The 'F300x can write to any portion
Page 43 and 44: One possibility is to use a transfo
Page 45 and 46: 2.10 SUMMARY This portion of the re
Page 47 and 48: The ball's angular velocity increas
Page 49 and 50: Light Level 100 90 80 70 60 50 40 3
Page 51 and 52: the fundamental frequency has been
Page 53 and 54: Since the fundamental frequency and
Page 55 and 56: Figure 3-7: MASTER "Analysis" Scree
Page 57: known, the linear momentum (and thu
Page 61 and 62: 3.5 ASSUMPTIONS AND ERROR ANALYSIS
Page 63 and 64: 3.5.3 External Forces and Friction
Page 65 and 66: 3.6.1 Implementation and Performanc
Page 67 and 68: Figure 3-11a: Effects of Phase Shif
Page 69 and 70: 3.6.3 The "Perfect" Game to Analyze
Page 71 and 72: Since the change in angular velocit
Page 73 and 74: BIBLIOGRAPHY [1] United State Paten
Page 75 and 76: APPENDIX A - SMARTDOT MODULE EMBEDD
Page 77 and 78: Appendix A: SMARTDOT Module Embedde
Page 85 and 86: APPENDIX B - SMARTDOT MODULE SOURCE
Page 87 and 88: Appendix C: SMARTDOT Module Command
Page 89 and 90: Figure C-3 Appendix C: SMARTDOT Mod
Page 91 and 92: APPENDIX E - 300 GAME MASTER SCREEN
Page 93 and 94: Appendix E: 300 Game Analysis Frame
Page 95 and 96: Appendix E: 300 Game Analysis Frame
Page 97: Appendix E: 300 Game Analysis Frame
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A Performance Analysis System for the Sport of Bowling

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