Measurement of Three-dimensional Orientation of Golf Club Head with One Camera

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ScienceDirect Procedia Engineering 112 (2015) 455 – 460

7th Asia-Pacific Congress on Sports Technology, APCST 2015

Measurement of Three-Dimensional Orientation of Golf Club Head with One Camera Wataru Kimizuka, Masahide Onuki * Dunlop Sports Ltd., Fundamental Technology Research Dept., 6-9 3-chome Wakinohama-cho Chuo-ku Kobe 651-0072, Japan

Abstract We developed a method to measure the three-dimensional orientation of a golf club head with one camera using the NewtonRaphson method which requires less calculation time. In this method, the orientation is calculated from the relation between the coordinates of markers on the head in the coordinate system of the club head and the coordinates of markers on the photo. Golfers hit balls with Irons whose club lie angles were 62° and 60° or 64°, and we took photos of the club head hitting balls. We calculated impact lie angles and face angles of the club head using this method and also measured the ball flight with a flight measuring machine. The results are as follows: (1) When the golfers used 64° (60°) clubs, all of them hit balls at larger (smaller) impact lie angles than when using 62° clubs. (2) Most of the landing points of the ball when the golfers used 64° (60°) clubs were more to the left (right) than 62° clubs. However, the reverse was the case with some golfers, because they changed face angles. In fitting, we need to find advanced methods to change lateral deviation adding on changing club lie angle. © 2015The The Authors. Published by Ltd. Elsevier © 2015 Authors. Published by Elsevier This isLtd. an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University. Peer-review under responsibility of the the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University

Keywords: Measurement, Posture, Golf Club, Image Processing, Accuracy, Newton-Raphson Method ;

1. Introduction Golfers’ swings are different from each other, and there is a method to select the appropriate golf club for each golfer’s swing, called “Fitting Service”. Traditionally, the method of selecting the appropriate club has been based on observation of the flight of the ball hit by the golfer. Recently, though, it has been found that in order to select the appropriate club, it is important to measure the three-dimensional orientation of the club head hitting the ball, which

* Wataru Kimizuka. Tel.:+81-78-265-3118 ; fax: +81-78-265-3199. E-mail address: [email protected].

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University

doi:10.1016/j.proeng.2015.07.224

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determines the flight of the ball. The established method to measure the three-dimensional position of a point is called the direct linear transformation method (DLT method). In this method, the position in space of the point to be observed is captured by two or more cameras, and the three-dimensional coordinates of the point are calculated from the two-dimensional coordinates of the point on those pictures (1) (2). The three-dimensional orientation of a subject can be calculated from the three-dimensional coordinates of points on the subject calculated by the DLT method. This method is widely used in the field of motion analysis (3). There is a technique to measure the three-dimensional orientation of a golf club head hitting the golf ball using the above method (4). However, this method has problems such as the large space needed for the measurement, the time needed to set up the system, and the costs of building the system because of the requirement of two or more cameras. A method that measures the three-dimensional orientation of a golf club head hitting the golf ball with only one camera would resolve these problems. In fitting service, a calculation method which needs little calculation time is desirable. Thus, we developed a method to measure the three-dimensional orientation of a golf club head using the Newton-Raphson method which needs little calculation time. We tested this system by measuring golfers’ swings, specifically measuring the three-dimensional orientation of heads of irons which have different club lie angles. 2. Theory 2.1. DLT method Generally, the three-dimensional coordinates of a point are calculated by the DLT method as follows (1) (2). Fig. 2 shows the relation between the coordinates of a point P in Object space (X ,Y ,Z ) and a point Q which is the point P captured on the Digitizing plane by a camera (U ,V ). The point O is the center of the lens of the camera. A coordinate system X’Y’Z’ has this O as its origin point, with its X’ and Y’ axes parallel to the U and V axes, respectively, on the Digitizing plane UV. L is the distance between the point O and the point P along the Z’ axis. F is the distance between the point O and the point Q along the Z’ axis. The point (U 0 ,V0 ) is the intersection point of the Z’ axis with the Digitizing plane. The (U ,V ) are expressed using (X ,Y ,Z )and A1
A4B1
B4C1
C3 (camera constants) by the following equation (1).

U=

A1 X + A2Y + A3 Z + A4 C1 X + C 2Y + C3 Z + 1

V=

B1 X + B2Y + B3 Z + B4 C1 X + C 2Y + C3 Z + 1

(1)

The coordinates (X ,Y ,Z ) of a point whose (U ,V ) are known can be calculated by the following steps. 1. Photograph the point with two or more cameras. 2. Derive four or more equations by substituting (U1 ,V1 ) (U 2 ,V2 ) , , … of the point in the above equations. 3. Calculate the (X ,Y ,Z ) by applying the least-square method to the four or more equations. F

L

Q(U, V) U

O(X0, Y0, Z0) (U0, V0)

X’ V

P(X, Y, Z)

Z’ Z

Y’

Object Plane

Digitizing Plane (Film) X

Y Object Space

Fig. 1 Direct Linear Transformation method

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2.2. Method to derive the three-dimensional orientation of an object with one camera The three-dimensional coordinates of one independent point cannot be determined with one camera; two or more cameras are needed, as explained above. However, if the positional relations of plural fixed and separated points on an object to each other (i.e. the coordinates of the points in the coordinate system of the object) are known, the expressions of these relations can be added to the equations used to derive (X ,Y ,Z ) , the three-dimensional coordinates of these points in the Object Space. In the following, the unknowns to be determined are the threedimensional coordinates (X 1 ,Y1 , Z1 ) of a representative point on the object and the three-dimensional orientation (α , β ,γ ) of the object. In this study, the orientation of the club head (α , β ,γ ) is expressed in terms of rotation angles for transforming the Object Space coordinate system XYZ to the object coordinate system X’Y’Z’ as follows. First, the coordinate system XYZ is rotated α degrees around the Y axis, resulting in Coordinate System 2. Then, Coordinate System 2 is rotated β degrees around the Z axis, resulting in Coordinate System 3. Finally, Coordinate System 3 is rotated γ degrees around the X axis, resulting in coordinate system X’Y’Z’. Equations (1) can be rewritten as

( A1 − C1U ) X + ( A2 − C2U )Y + ( A3 − C3U ) Z + ( A4 − U ) = 0

(2)

( B1 − C1V ) X + ( B2 − C2V )Y + ( B3 − C3V ) Z + ( B4 − V ) = 0 The three-dimensional coordinates of the points on the object in Object Space are expressed by the equations

R i = ( X 1,Y1,Z1 ) + T ⋅ ri


3

Here, ri (i = 2,
N) are the vectors from the representative point on the object to the other N-1 points in the coordinate system of the object, and T is the rotation matrix consisting of the functions of α, β, γ for transforming from the coordinate system of Object Space to the coordinate system of the object. Thus, R i is a function of ri consisting of (X 1 ,Y1 , Z1 ) and (α , β ,γ ). By substituting (X 1 ,Y1 , Z1 ) and (U1 ,V1 ) of the reference point and R i and (U i ,Vi ) of the other points in the equations (2), the following 2N non-linear simultaneous equations are obtained.

( A1 − C1U 1 ) X 1 + ( A2 − C 2U 1 )Y1 + ( A3 − C3U 1 ) Z1 + ( A4 − U 1 ) = 0 ( B1 − C1V1 ) X + ( B2 − C 2V1 )Y1 + ( B3 − C3V1 ) Z1 + ( B4 − V1 ) = 0 ( A1 − C1U 2 ) R2 ,X + ( A2 − C 2U 2 ) R2 ,Y + ( A3 − C3U 2 ) R2 ,Z + ( A4 − U 2 ) = 0 ( B1 − C1V2 ) R2 ,X + ( B2 − C 2V2 ) R2 ,Y + ( B3 − C3V2 ) R2 ,Z + ( B4 − V2 ) = 0 ( A1 − C1U 3 ) R3 ,X + ( A2 − C 2U 3 ) R3 ,Y + ( A3 − C3U 3 ) R3 ,Z + ( A4 − U 3 ) = 0 ( B1 − C1V3 ) R3 ,X + ( B2 − C 2V2 ) R3 ,Y + ( B3 − C3V3 ) R3 ,Z + ( B4 − V3 ) = 0
( A1 − C1U N ) RN ,X + ( A2 − C 2U N ) RN ,Y + ( A3 − C3U N ) RN ,Z + ( A4 − U N ) = 0 ( B1 − C1VN ) RN ,X + ( B2 − C 2VN ) RN ,Y + ( B3 − C3VN ) RN ,Z + ( B4 − VN ) = 0 Here, Ri, X , Ri, Y , Ri, Z are the X,Y,Z components of R i . Finally, the six unknowns (X 1 ,Y1 , Z1 ) and (α , β ,γ ) can be derived by applying the Newton-Raphson method to the 2N equations (4) if ri are known and N ≥ 3 . 3. Experiment 3.1. Measurement system Fig. 2 is a picture of a measurement system. A camera takes a photo of the club head hitting the ball. Then, the three-dimensional orientation of a club head is calculated from the coordinates of markers in the picture as shown in Fig. 3. To investigate the measurement accuracy of this system, we developed a jig with two rotation mechanisms which can set a club head at an intended face angle and lie angle with 0.1 degree accuracy. Fig. 4 shows the jig.

(4)

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Using this jig, we conducted experiments to investigate the measurement accuracy of this system. The results are as follows: (1) The measurement error can be decreased by applying markers over a wider area on the surface of a head. (2) In the case where the orientation of a head significantly deviates from the standard orientation, the measurement accuracy is low compared with the case of a small deviation. (3) The measurement error can be reduced by calculating the coordinates of the center of a marker from the positions and the luminance of each pixel of the marker instead of visually selecting the pixel at the center of the marker. (4) The average driver measurement error was 0.1° for face angle and 0.4° for lie angle. The average iron measurement error was 0.1°for face angle and 0.3° for lie angle (5).

Camera

Strobe

Fig. 2 Measurement system

Fig. 3 Example of photos used for measurement

Fig. 4 Jig for setting the angle of club heads

3.2. Measurement of golfers’ swings We measured the three-dimensional orientation of club heads when golfers hit balls. We prepared three 7 iron clubs whose club lie angles were 60°, 62°, and 64° and put five markers on each face. Fourteen golfers hit balls with the iron whose club lie angle was 62°. After that, seven of these golfers hit balls with the iron whose club lie angle was 64° and the other seven golfers hit balls with the iron whose club lie angle was 60°. In each shot, we took a photo of the club head hitting the ball and calculated the impact lie angle and face angle from the coordinates of the markers in the photo. We also measured the ball speed, launch angle, and spin ratio with our ball measurement system (6) and calculated the lateral deviation of the ball landing point. 4. Results and discussion Table 1 shows impact lie angles, face angles, lateral deviations, and the difference between these values using different clubs. The impact lie angle is defined as the angle between the face lines and the ground, and positive values show that the golfer hit the ball toe-up relative to the ground. The face angle is defined as the angle between

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the club face and the plane perpendicular to the target line; positive values show that the golfer hit the ball with an open face, the club facing to the right of the target line. The lateral deviation is the position of the ball landing point relative to the target line, and positive values show that the ball landing point is to the right of the target line. All the seven golfers using the 64° iron hit the ball with a larger impact lie angle with 64° iron than with the 62° iron, and among these, the lateral deviation of five golfers was more to the left with the 64° iron than with the 62° iron. Fig. 5 left shows the results of Golfer B as an example of this. Although the three results with each club were dispersed, Golfer B tended to hit the ball with a larger impact lie angle with the 64° iron than the 62° iron, and these lateral deviation with the 64° iron tended to be more to the left than with the 62° iron. We surmise that this was because the normal vector of the face plane tilted to the left due to toe up, and the amount of spin in the hook direction tended to increase. However, the lateral deviation of the two other golfers was more to the right with the 64° iron than with the 62° iron. Fig. 5 right shows the results of Golfer F as an example of this. Although the three results with each club were dispersed, Golfer F tended to hit the ball with a larger impact lie angle with the 64° iron than the 62° iron, but the lateral deviation tended to be more to the right with 64° iron than with the 62° iron. We surmise that this was because Golfer F changed his swing so that the face angle was open despite toe up. It is unlikely that Golfer F changed his swing upon watching the trajectory of balls hit with the 64° iron, because the first shot with the 64° iron also deviated to the right. We consider that Golfer F changed his swing because he noticed the toe up of the head when addressing the ball, and felt that his ball would deviated to the left if he didn’t change his swing. Additionally, there were similar results from the seven golfers who swung 62° and 60° irons. Fig. 6 left shows the results of Golfer L as an example of a golfer whose lateral deviation with the 60° iron was more to the right than with the 62° iron. On the other hand, Fig. 6 right shows the results of Golfer N as an example of a golfer whose lateral deviations with the 60° iron was more to the left than with the 62° iron. It is unlikely that Golfer N changed his swing upon watching the trajectory of balls hit with the 60° iron because the first shot with the 60° iron also deviated to the left. We consider that Golfer N changed his swing because he observed his club head when addressing the ball like Golfer F and noticed that it was toe down. Thus, it was found that when golfers change the lie angle of the club, their impact lie angles change in the same direction and the lateral deviation of the most of these golfers also changes, accordingly, but the lateral deviation of some of them changed in the direction opposite the change in the impact lie angle because they changed their face angles. Thus, in fitting, we need to find advanced methods to change lateral deviation adding on changing club lie angle. Table 1: Results of measuring lie angle, face angle and lateral deviation of golfers. Club Head Lie Angle 64°

Club Head Lie Angle 62° Golfer

Lie Angle[°]

Face Angle

Deviation [°]

Lie Angle[°]

Face Angle

Difference Deviation [°]

Lie Angle[°]

Face Angle

Deviation [°]

A

-4.7

2.6

4.0

-2.2

4.6

3.0

2.5

2.0

-1.0

B

-1.2

6.4

8.7

0.7

4.0

-0.3

1.9

-2.4

-9.0

C

-2.7

-0.1

3.0

-2.3

-0.8

-1.0

0.4

-0.7

-4.0

D

-3.2

2.1

7.0

-1.5

-0.1

1.7

1.6

-2.2

-5.3

E

-2.0

4.0

-2.7

-0.8

0.5

-10.7

1.2

-3.5

-8.0

F

-1.1

3.9

-2.0

0.9

5.1

4.7

2.1

1.1

6.7

G

-1.0

3.8

-6.0

2.0

5.2

-4.7

3.0

1.3

1.3

H

0.5

4.5

-5.7

-0.4

5.8

-4.0

-0.9

1.3

1.7

I

8.1

7.8

12.3

6.5

7.2

19.3

-1.6

-0.6

7.0

J

7.9

7.7

-5.0

4.7

4.7

-15.0

-3.2

-3.0

-10.0

K

3.4

6.4

-1.3

3.2

4.7

-7.0

-0.2

-1.6

-5.7

L

2.6

4.4

-5.0

1.5

5.0

2.0

-1.1

0.6

7.0

M

0.2

5..4

-0.7

-0.9

5.3

4.0

-1.1

0.0

4.7

N

3.1

3.5

-4.7

1.8

1.8

-8.0

-1.3

-1.7

-3.3

Wataru Kimizuka and Masahide Onuki / Procedia Engineering 112 (2015) 455 – 460 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3


62° 64°

3 2 1

2

1 3

2

Impact lie angle [degree]

Impact lie angle [degree]

460

1.5

1

1

62° 64°

3

0.5

2

0 -0.5 -1

1 2 3

-1.5 -10


     Lateral deviation of golf ball landing point [yd]

-5 0 5 10 15 Lateral deviation of golf ball landing point [yd]

Golfer B

Golfer F

Fig. 5 Examples of measurement of golfers’ swings with clubs whose lie angles were 62° and 64°

1

62°

4

60°

3.5

3

2

2.5

2

2

3

1.5

1

1 0.5

3

0

Impact lie angle [degree]

Impact lie angle [degree]

4 3.5

3

3

2

2.5 2

1

2

3

1.5 1

62°

1

0.5

60°

0 -15

-10 -5 0 5 Lateral deviation of golf ball landing point [yd]

Golfer L

-15

-10 -5 0 Lateral deviation of golf ball landing point [yd]

Golfer N

Fig. 6 Examples of measurement of golfers’ swings with clubs whose lie angles were 62° and 60°

5. Conclusion We developed a method to measure the three-dimensional orientation of a golf club head based on one photo taken by one camera. In this method, the orientation is calculated from the relations between the coordinates of markers attached to the head in the coordinate system of the club head and the coordinates of the markers in the photo. Using this method, we measured the three-dimensional orientation of club heads when golfers hit balls, and found that when the club lie angle of the golfers’ club changed, the impact lie angle always changed accordingly, and the lateral deviation of the ball landing points usually changed in accordance with the impact lie angle, but that with some golfers, lateral deviation changed in the direction opposite the change in impact lie angle. Thus, in fitting, we need to find advanced methods to change lateral deviation adding on changing club lie angle. References [1] Abdel-Aziz, Y. I. and H. M. Karara, “Direct linear transformation from comparator coordinates into object space in close-range photogrammetry”, ASP Symposium on Close-Range Photogrammetry, American Society of Photogrammetry, Falls Church (1971). [2] Ikegami,Y., Sakurai, S., “D.L.T. method”, Japanese Journal of Sports Science, 10-3 (1991), pp. 191-195. [3] D.Gordon E.Robertson, Research Methods in Biomechanics, (2008), pp. 39-48. [4] Nakasuga, M., Hashimoto, R., “The 3D-Measurement of Golf Club Head Movement”, Symposium on Sports Engineering : Symposium on Human Dynamics, Vol.1996 pp.166-169 (1996). [5] Kimiduka, W., Onuki, M., “Measurement of Three-Dimensional Postures of Golf Club Head through One Set of Camera”, Symposium on Sports Engineering : Symposium on Human Dynamics, Vol.2013 209 (2013). http://golf.dunlop.co.jp/diw/index.html