中華大學

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中華大學碩士論文

題目：The ARX-Part Effects Of Aiming Trajectory Analysis Based On ARMAX Model

系所別：電機工程學系碩士班學號姓名：M09501010 吳建鋒指導教授：黃啟光博士

中華民國九十七年八月

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ARX 部分對於射箭瞄準軌跡 ARMAX 模型的影響

The ARX-Part Effects Of Aiming Trajectory Analysis Based On ARMAX Model

研究生：吳建鋒 Student：Chien-Fung Wu 指導教授：黃啓光博士 Advisor Dr. Chi-Kuang Hwang

中華大學

電機工程學系碩士班

碩士論文

A Thesis

Submitted to Institute of Electrical Engineering Chung Hua University

In Partial Fulfillment of the Requirements For the Degree of

Master of Science In

Electrical Engineering July 2008

Hsin-Chu, Taiwan, Republic of China

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The ARX-Part Effects Of Aiming Trajectory Analysis Based On ARMAX Model

研究生：吳建鋒指導教授：黃啓光博士

中華大學

電機工程學系碩士班

中文摘要

本論文討論射箭選手在將箭射出 1.5 秒之前的瞄射軌跡，將其分成水平與垂直兩軸，分別以 ARMAX 作為其軌跡近似模型做進一步探討。經由將 ARMAX 模型上的一些參數及統計學上的相關性做處理後，可得知某些群體上及個人上的瞄準特性。在射出之前的最後 1.5 秒內，本論文欲知射箭選手原先渴望的調整方式，

換而言之就是不考慮肌肉抖動的影響。在複數平面之左半平面之中有著極點存在，表示有劇烈抖動趨勢，故利用漢彌頓窗(Hamming window)將其肌肉造成之抖動濾除。最初預期水平軸與垂直軸的調整會互相影響，也經由進一步的統計分析證實其存在。分析 ARMAX 模型參數直接影響射箭成績的相關性，並探討其相對應射箭行為之物理意義。該模型參應該具有低相關性，才會造成好的成績，

亦於本文中探討。最後對各別選手提出可改良其射箭表現的相關建議。

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The ARX-Part Effects of Aiming Trajectory Analysis Based on ARMAX Model

Student : Chien-Fung Wu Advisor : Dr. Chi-Kuang Hwang

Institute of Electrical Engineering Chung Hua University

Abstract

In this thesis, the aggressive moving average with an exogenous input (ARMAX) is adopted to model the aiming trajectory recorded during the last 1.5 second before releasing the arrow. Some useful variables related to the model are defined for analyzing the performance of archers. Through the statistic correlation approach, the individual and grouping characteristics are anticipated to be obtained from it. During that 1.5 period, the desired adjustments of archers without consideration their muscle strength are the main interest in this thesis. For expertise archers, their desired aiming style should not contain high frequency which is corresponding negative pole of the model. Therefore, a Hamming window is implemented to remove the high frequency effect. The expected coupling effect between both axes is confirmed. The direct effect on the performance related to these variables has been discussed, and the low correlation requirements among these variables for better performance are also outlined for further use. Some conceivable suggestions for individual are presented in the thesis.

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Acknowledgement

I would like to express my sincere gratitude for my advisor Dr. Chi-Kuang Hwang, and Dr. Bore-Kuen Lee for their helpful advice, patient guidance,

encouragement, and valuable support during the course of the research. I am obliged to my classmates Chia-Mao Weng, Kun-Sue Hwang, for their helpful discussions and all my friends for their listening to my mood. Finally I want to express my sincere gratitude for my parents and my grandmother for their encouragement and support.

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List of Figures

2.1 The graph of the original, smoothing by the Hamming window, and estimated

trajectories. ……….8

2.2 Three components of the intended adjustment Ux along the horizontal direction. ……….10

2.3 Three components of the intended adjustment Uy along the vertical direction. ………..11

4.1 Performance Inference procedure 1 ………..17

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List of Tables

Table 4.1.Couple effect among the defined variables

………21

TableA.1: Correlation coefficient R and its mean and variance for archer 1

………29

………30

TableA.3 Correlation coefficient R and its mean and variance for archer 3

………31 TableA.4: Correlation coefficient R and its mean and variance for archer 4

………32

………33

………35

TableA.8: Correlation coefficient R and its mean and variance for archer 8 ………36 TableA.9: Correlation coefficient R and its mean and variance for archer 9

………40 Table A13: Correlation coefficient C for archer 1

………42

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Table A15: Correlation coefficient C for archer 3

………52 Table A25: The correlation CG=C(C(TR, Variable), TR) for group or global

sense.

………53 Table A26: The correlation CGA=C(|C(TR, Variable)|, TR) for group or global

sense.

………54

Table A27: Correlations CG and CGA base on |CG|≧0.45 and |CGA|≧0.45.

………55 Table A28: The deductions based on the proposed inference and Table A.27

………56

Table A29: The summarized suggestions for each archer to improve their performance based on statistical one sigma bound

………58

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Abstract

In this thesis, the aggressive moving average with an exogenous input (ARMAX) is adopted to model the aiming trajectory recorded during the last 1.5 second before releasing the arrow. Some useful variables related to the model are defined for analyzing the performance of archers. Through the statistic correlation approach, the individual and grouping characteristics are anticipated to be obtained from it. During that 1.5 period, the desired adjustments of archers without consideration their muscle strength are the main interest in this thesis. For expertise archers, their desired aiming style should not contain high frequency which is corresponding negative pole of the model. Therefore, a Hamming window is implemented to remove the high frequency effect. The expected coupling effect between both axes is confirmed. The direct effect on the performance related to these variables has been discussed, and the low correlation requirements among these variables for better performance are also outlined for further use. Some conceivable suggestions for individual are presented in the thesis.

【Keywords】ARMAX, aiming trajectory, correlation, Hamming windows.

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Chapter 1 Introduction

Lots of archery researches have been conducted from different approaches in order to find the key point for improving the performance of this fine and highly skilled sport. The most important focus will be the stability of aiming style, so how to use systematic methods to evaluate it falls in the direction. A biomechanical study on the final push-pull archery has been conducted by Leroyer et al. [1]. The purpose of their study is to analyze archery performance among eight archers of different abilities by means of displacement pull-hand measurements during the final push-pull of the shoot. The archers showed an irregular displacement negatively related to their technical levels. Displacement signal analysis showed high power levels in both 0-5 Hz and 8-12Hz ranges. The latter peak corresponds to electromyographic tremor observed during a prolonged push-pull effort. The results are discussed in relation to some potentially helpful training procedures such as biofeedback and strength conditioning. Landers etal. [2] had examined novice archers to determine whether (a) hemispheric asymmetry and heart rate deceleration occur as a result of learning, and (b) these heart rates and electroencephalograph (EEG) patterns are related to archery performance. The electromyography (EMG) technology which measures the activation patterns in forearm muscles related to contraction and relaxation strategy during archery shooting, has been applied by Ertan et al., [3] to analyze for archers

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found that elite archers’ release started about 100 ms after the fall of the clicker, whereas for beginners and non-archers, their release started after about 200 and 300 ms, respectively. How the novice archers apply the taught training information under different conditions and guided them to promote their motor skills required for better archery performance have investigated by Lavisse et al. [4].

The aiming stability is the key factor affects the archery performance has been indicated by Shiang et al. [5], and it can be determined by the size of aiming locus.

They further pointed out that the aiming locus pattern is also a useful index to determine the performance. The effects of heart variable rate (HVR) related to the stability of archer has been measured by C.-T. Lo. [6]. By the frequency-domain analysis of the HVR and three main frequency domains, such as very low frequency (VLF), low frequency (LF), and high frequency (HF). The VLF component is much less defined and the HF generally represents parasympathetic activity. The LF is influenced by both sympathetic and parasympathetic activity, and the ratio of HF to LF represents the balance of parasympathetic and sympathetic activity.The results showed that the HF was higher, the LF was lower, and the LF/HF ratio was lower for the best performance.

Analysis of correlation between the aiming adjustment trajectory and the shot points has been studied [7] by Lin et al. In this thesis, ARMAX is adopted to model the aiming trajectory. We then define related variables for identifying their role upon the performance of archers. The direct effect on the performance related to these variables will be analyzed. The Hamming window is implemented to recovery the desired and anticipated adjustments of archers. Correlation method is designed to obtain the individual and grouping characteristics. Some conceivable suggestions for individual will be presented in the thesis.

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Chapter 2 System Model

2.1 Hamming Window

In this thesis, the aggressive moving average with an exogenous input (ARMAX) is adopted to model the aiming trajectory recorded during the last 1.5 second before releasing the arrow. Our goal is to measure the archer desired aiming trajectory without muscle strength. The AR part is designated for the desired adjustment to the target; the exogenous part is corresponding the offset between the center of target and the aiming point wherein occurred at the last 1.5 second instance; the moving average part is associated with the strength and stability of muscle related to the individual archer. Accordingly, a negative pole existing at the AR part is corresponding to the high frequency oscillation, so it is unsuitable for modeling as the desired and intended adjustment. This high frequency oscillation is reasonable to model as the stability of muscle strength. The original recorded data processed by the proposed ARMAX model do obtain the undesired negative pole in the AR part, even though the pole is very close to -1. Therefore, the most common Hamming window is applied to separate the high frequency oscillation from the AR part which is designed to represent the desired adjustments, as the effect of muscle strength stability.

The exogenous part of the ARMAX can be used to describe the adjustment to

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compensate the offset between the target and the current of the aiming point. For example, the current aiming point is located at the left of the center of the target, and archers usually will exert a steady force to move the bow right forward the center of the target. This steady constant force is then modeled as a constant bx . Because of the setting of this model, the right forward constant force is represented by a positive constant. Similarly, the vertical direction case is represented by the constant by.

The MA part of the ARMAX is utilized to model the muscle strength of archers.

These three coefficients are related to two zeros of the transfer function. Likewise the stability analysis based on them may have connection with the poles of AR part. The symmetric Hamming window with N=8 is implemented to smooth the recorded information, and the formulation in the time and frequency are written as follows.

Hamming(n)=0.54 0.46 cos(2 n) , 0

n N

N

− π ≤ ≤ (2-1)

Hamming weighting vector =[0.08 0.21 0.54 0.86 1 0.86 0.54 0.21 0.08]

2 2

Hamming( ) 0.54D_N( ) 0.23[D_N( ) D_N( )]

N N

π π

ω = ω + ω− + ω+ , (2-2)

where

2 2 sin( ( 1)) ( ) 2

j N

N

e N D

ω ω

− +

= .

Note that the normalization is conducted, and the difference of aiming trajectory between the original data and the one smoothing by the above Hamming window is shown in Fig 2.1.

2.2 ARMAX Model

We propose a linear time-invariant ARMAX model [8] to represent the aiming trajectory of each shot during the last 1.5 second before releasing the arrow. There are

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only two dimensional information are recorded, so two ARMAX models based on individual archer along the horizontal and vertical directions are formulated as below.

1 2 3

( ) ( 1) ( 2) ( 3)

( ) ( 1) ( 2) ( 3) ( 1)

x x x

x x x x x x x x x

x k a x k a x k a x k

e k c e k c e k c e k b uc k

= − + − + −

+ + − + − + − + − (2-3)

1 2 3

( ) ( 1) ( 2) ( 3)

( ) ( 1) ( 2) ( 3) ( 1)

y

y y y

y y y y y y y y

y k a y k a y k a y k

e k c e k c e k c e k b uc k

= − + − + −

+ + − + − + − + − (2-4)

where x(k) and y(k) are the time series of the aiming trajectory along the horizontal and vertical deviations; bx and by are the corresponding coefficients of exogenous inputs; coefficients ax₁~3 and ay₁~3 are the corresponding coefficients of the AR part of ARMAX mode related to the intended and desired adjustment to compensate the existing offset (the exogenous part); cx₁~3 and cy₁~3 are the associated coefficients of the MA part of ARMAX model; ex(k) and ey(k) are denoted as the driving noises.

For the intended adjustment during the final 1.5 second period, it is further classified into two parts by adopting AR3; the first part is the slowest and probably stable adjustment type and the second part is the oscillated adjustment with stable decay. It is possible that the adjustment is stable decay without oscillation, and sometimes the dominated adjustment is a little bit divergent. These cases can be categorized by the poles of the AR3 system, so the time series presentations are discussed in the next section.

Unknown coefficients along each direction are assembled as two vectors first for identification θ_x =[a a_x₁ _x₂ a_x₃c c_x₁ _x₂ c b_x₃ _x]^Tandθ_y =[a a_y₁ _y₂ a_y₃ c c_y₁ _y₂ c_y₃b_y]^T. It is noted that by observing equations (2-3) and (2-4) all of the driving noises ex(k) and ey(k) in the MA part are generated based on these two vectors, that is,

x( )

e k ,e k_x( −1) e k_x( −2), e k_x( −3), e k ,_y( ) e k_y( −1) ,e k_y( −2),and e k_y( −3),so they

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the current estimation is needed, and the associated recursive forms are written as follows.

ˆ( , _x) _x^T( , _x) _x

x k θ =φ k θ θ (2- 5)

ˆ( , _y) _y^T( , _y) _y

y k θ =φ k θ θ (2- 6) where

[ ]

( , ) ( 1) ( 2) ( 3) ( 1) ( 2) ( 3) ( 1)

x k x x k x k x k e kx e kx e kx uc kx

φ θ = − − − − − − − (2- 7)

( , ) ( 1) ( 2) ( 3) ( 1) ( 2) ( 3) ( 1)

y k y y k y k y k e ky e ky e ky uc ky

φ θ = − − − − − − −  (2-8) ˆ

( ) ( ) ( )

e kx =x k −x k (2- 9)

( ) ( ) ˆ( )

e ky = y k −y k (2-10)

The initial settings of ˆθ_x^and ˆ

θy are given from the following equations:

3 2

1 2 3 ( 0.75)( 0.5)( 0.25) 0

x ax x ax x ax x x x

λ − λ − λ − = λ − λ − λ − =

3 2

1 2 3 ( 0.75)( 0.5)( 0.25) 0

x cx x cx x cx x x x

λ − λ − λ − = λ − λ − λ − =

3 2

1 2 3 ( 0.75)( 0.5)( 0.25) 0

y ay y ay y ay y y y

λ − λ − λ − = λ − λ − λ − =

3 2

1 2 3 ( 0.75)( 0.5)( 0.25) 0

y cy y cy y cy y y y

λ − λ − λ − = λ − λ − λ − =

x y 0 b =b =

as well as initial settings ( ) ( ) 0,

x y

e k =e k = x k( )=x(1),^and y k( )= y(1)^for^k = −^{2 ~ 0}

( ) ( ) 1,

x y

u k =u k = for all k = −2 ~ 89.

Therefore, the pseudo-linear approach is needed to be adopted to estimate these two vectors, and the estimation of θ_x^andθy, denoted asθ^∧_x^and θ^∧y, can be obtained by the minimum mean square error (MMSE) criterion. We begin with defining the

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following two matricesH_xmand H_ym,

( 7) 7

(0) ( 1) ( 2) (0) ( 1) ( 2) 1

(1) (0) ( 1) (1) (0) ( 1) 1

( 6) ( 5) ( 4) ( 6) ( 5) ( 4) 1

x x x

xm

x x x m

x x x e e e

H

x m x m x m e m e m e m

+ ×

− − − −

− −

=

+ + + + + +

 

 

 

M M M M M M M

(2-11)

( 7) 7

(0) ( 1) ( 2) (0) ( 1) ( 2) 1

(1) (0) ( 1) (1) (0) ( 1) 1

( 6) ( 5) ( 4) ( 6) ( 5) ( 4) 1

y y y

ym

y y y m

y y y e e e

H

y m y m y m e m e m e m

+ ×

− − − −

− −

=

+ + + + + +

 

 

 

M M M M M M M

(2-12)

Then the estimations can be obtained by repeating the following procedure6m:1 ~ 83 ( ) ( ^T ) ¹ ^T ˆ

x m Hxm Hxm Hxm Xm

θ^∧ = ⁻ (2-13)

( ) ( ^T ) ¹ ^T ˆ

y m Hym Hym Hym Ym

θ^∧ = ⁻ (2-14) where

[ ]

ˆ ( , )_m _x ˆ(1, _x) ˆ( 7, _x) ^T

X k _θ ₌ x _θ Lx m+ _θ _and Yˆ_m_{( ,}k _θ_y₎₌yˆ(1,_θ_y)Ly mˆ( +7,_θ_y)^T.

In the end, the case similar to m=90 is repeated with the fixed dimension H_x83 and Hy83 , but associated with updated data until the difference between the current and previous estimations is below a prescribed bound. The estimated trajectory is depicted in Fig. 2.1 for comparison with the original one and the one smoothed by the Hamming window. The results show the proposed ARMAX model with Hamming window can model the original trajectory well.

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0 10 20 30 40 50 60 70 80 90 6.8

7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8

time

function value

smooth y estimation smooth y y orignial

Fig. 2.1 The graph of the original, smoothing by the Hamming window, and estimated trajectories.

2.3 Time Series

The time series of the ARMAX of Equations(2-3) and (2-4) can be written as follows:

A z x t_x( ) ( )=B z uc t_x( ) _x( )+C z e t_x( ) ( )_x (2-15)

A z y t_y( ) ( )=B z uc t_y( ) _y( )+C z e t_y( ) ( )_y (2-16)

or

( ) ( ) ( )

x x x x

x x

B uc t C z e z

x t = A z + A z (2-17)

( ) ( ) ( )

y y y y

y y

B uc t C z e t

y t = A z + A z (2-18)

whereA z_x( )= −1 a z_x₁ ⁻¹−a z_x₂ ⁻² −a z_x₃ ⁻³, A z_y( )= −1 a z_y₁ ⁻¹−a z_y₂ ⁻²−a z_y₃ ⁻³

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1 2 3

x( ) x x x

C z =c z⁻ +c z⁻ +c z⁻ ,C z_y( )=c z_y₁ ⁻¹+c z_y₂ ⁻² +c z_y₃ ⁻³,B_x =b z_x ⁻³,

andB_y =b z_y ⁻³.

Owing to uc_x =uc_y =1, we have ( ) ( )

x y 1

Uc z Uc z z

= = z

− ^{and the} corresponding ARX parts for both axes are

4

3 2

1 2 3

( )

( ) ( 1)( )

_1 1 2 3

( 1) ( 1) ( 2) ( 3)

x x x

x x x x

B Uc z z b

A z z z a z a z a

Rx Rx Rx Rx

z z z Px z Px z Px

= − − − −

 

=  + + + 

− − − −

 

(2-19)

and

4

3 2

1 2 3

( )

( ) ( 1)( )

_1 1 2 3

( 1) ( 1) ( 2) ( 3)

y y y

y y y y

B Uc z z b

A z z z a z a z a

Ry Ry Ry Ry

z z z Py z Py z Py

= − − − −

 

=  + + + 

− − − −

 

(2-20)

where Px1, Px2, Px3, Py1, Py2, and Py3 are the roots of z³−a z_x₁ ²−a z_x₂ −a_x₃ =0

andz³−a z_y₁ ²−a z_y₂ −a_y₃ =0, respectively. These roots are called as the poles of the system, and their associated residue can be calculated as fallows:

3

1 3

2 3

3

_1 ( 1)( 2)( 3)

1 ( 1)( 2)( 3)

2 ( 1)( 1)( 3)

3 ( 1)( 1)( 2)

x

z

x

z Px

x

z Px

x

z Px

Rx b z

z Px z Px z Px

Rx b z

z z Px z Px

Rx b z

z z Px z Px

Rx b z

z z Px z Px

=

= − − −

3

1 3

2 3

3

_1 ( 1)( 2)( 3)

1 ( 1)( 2)( 3)

2 ( 1)( 1)( 3)

3 ( 1)( 1)( 2)

y

z

y

z Py

y

z Py

y

z Py

Ry b z

z Py z Py z Py Ry b z

z z Py z Py Ry b z

z z Py z Py

=

= − − −

(2-21)

The intended horizontal and vertical adjustments of the exerting force Ux(k) and Uy(k), relating to the offset bx, by , and the AR part of the ARMAX are written as

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1 ( )

( ) ( )

x x

x

B Uc z

Ux k A z

−  

= Ζ  

  (2-22)

( ) _1 1( 1) 2( 2) 3( 3)

_1( ) 1( ) 2( ) 3( )

k k k

Ux k Rx Rx Px Rx Px Rx Px

Ux k Ux k Ux k Ux k

= + + +

= + + + (2-23)

1 ( )

( ) ( )

y y

y

B Uc z

Uy k A z

−  

= Ζ  

 

  (2-24) ( ) _1 1( 1) 2( 2) 3( 3)

_1( ) 1( ) 2( ) 3( )

k k k

Uy k Ry Ry Py Ry Py Ry Py

Uy k Uy k Uy k Uy k

= + + +

= + + + (2-25)

where

_1 _1

Ux =Rx ,Ux k1( )=Rx Px1( 1)^k,Ux2( )k =Rx2(Px2)^k, Ux k3( )=Rx Px3( 3)^k,

_1 _1

Uy =Ry , Uy k1( )=Ry Py1( 1)^k, Uy2( )k =Ry2(Py2)^k, andUy k3( )=Ry Py3( 3) .^k Their initial values (k=0) are as the same as their associated residue, and their final values (k=90) are defined as Ux_1=Rx_1, Ux1=Ux1(90), Ux2=Ux2(90),

3 3(90)

Ux =Ux , Uy_1=Ry_1, Uy1=Uy1(90), Uy2=Uy2(90), andUy3=Uy3(90) for comparing their role playing in the aiming trajectory time series.

2.4 Notations And Physical Meanings

The intended horizontal and vertical adjustments of the exerting force along both directions Ux(k) and Uy(k) have been defined, and their initial values and the final values are also evaluated for comparison. Now the physical meanings of them are illustrated by the graph. Three components of the Ux(k), in which the dominated exponential type of adjustment Ux1(k), the oscillation type with exponential decay envelope adjustment Ux2(k)+Ux3(k) and the constant type Ux_1 are depicted in Fig.

2.2, respectively; similarly, those components of the Uy(k), Uy2(k)+Uy3(k) and Uy_1 are also shown in Fig. 2.3. Moreover, the initial values of Ux2(k)+Ux3(k) is equal to Rx2+Rx3, and in case of the complex pair of Px2 and Px3, we have Rx2+Rx3 =2Rx2r,

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which is most common in this experiment. The final values of Ux1=Ux1(90), Ux2+Ux3= Ux2(90)+Ux3(90) can be checked at the last points of the graph Fig. 2.2.

The settling time T-x=m is defined that the five consecutive time instances in which

|Ux(k)- mean(Ux(86~90)) |<0.05* mean(Ux(86~90)), for k=m, m+1, …, m+4.

Accordingly, the longest settling time m=85. The same definition is also applied for T-y, so the absolute value of difference between these two settling times is then formulated as | T-x - T-y|. The settling time T-x of Fig. 2.2 is m=84 because the dominated pole Px1= 0.999 is too close to 1. The settling time T-y of Fig. 2.3 is m=69 which is also dominated by the Py1=0.965. Usually, the settling time can be dominated by the complex pole pair with a slow exponential decay envelope, that is, the combined effect of the real part of Px2 (Px2r), its initial value Rx2+Rx3 =2Rx2r and Rx_1. Sometimes, the oscillation frequency also plays an important role in the settling time T-x, so the oscillation frequency ( 2 / Tπ ) is the same as the phase angle of the complex pole pair, so they are defined as follows:

-1

=tan (imagnary( 2) / 2 )

Ax Px Px r

Ay Py Py r

(2-26)

The oscillation frequency Ax=0.608=2 /10.334π with the period T=10.33 and

=0.422 2 /14.889

Ay = π with the period T= 14.889 can be observed from Fig 2.2 and Fig. 2.3, respectively.

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0 10 20 30 40 50 60 70 80 90 -3

-2.5 -2 -1.5 -1 -0.5

UX1

0 10 20 30 40 50 60 70 80 90

-0.1 -0.05 0 0.05 0.1

UX2+UX3

0 10 20 30 40 50 60 70 80 90

1.5 2 2.5 3 3.5 4

UX₁

Fig 2.2 Three components of the intended adjustment Ux along the horizontal direction.

0 10 20 30 40 50 60 70 80 90

0 0.5 1 1.5 2 2.5

UY1

0 10 20 30 40 50 60 70 80 90

-0.1 -0.05 0 0.05 0.1 0.15

UY2+UY3

0 10 20 30 40 50 60 70 80 90

-3.5 -3 -2.5 -2 -1.5 -1

UY1

Fig 2.3 Three components of the intended adjustment Uy along the vertical direction.

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2.4.1 Notations:

TR: the radius distance of the arrow from target center.

X: the horizontal direction.

Y: the vertical direction.

bx, by: the exogenous inputs along the horizontal and vertical directions.

|bx|, |bx|: the absolute value of the exogenous inputs.

Px1, Py1: the first dominated real pole of ARMAX model.

Px2r, Py2r: the real part of complex pair pole or the second dominated real pole.

Px_1, Py_1: the pole is 1.

Rx1, Ry1: the corresponding residue of Px1 and Py1, respectively.

Rx2, Ry2: the corresponding residue of Px2 and Py2, respectively.

Rx2r, Ry2r: the real parts of Rx2 and Ry2, respectively.

Rx2a, Ry2a: the absolute values of Rx2 and Ry2, respectively.

Rx_1, Ry_1: the corresponding residue of Px_1 and Py_1, respectively.

Ux1: the combined effect of Px1 and Rx1.

Uy1: the combined effect of Py1 and Ry1.

Ux2: the real part of combined effect of Px2 and Rx2.

Uy2: the real part of combined effect of Py2 and Ry2.

Ux_1: the combined effect of Px_1 and Rx_1 (the same as Rx_1).

Uy_1: the combined effect of Py_1 and Ry_1 (the same as Ry_1).

Ux, Uy: the total effect of exogenous inputs bx and by, respectively.

UR: UR= Ux²+Uy²

T-x, T-y: the settling time of Ux and Uy, respectively.

T-d: | T-x - T-y| the absolute value of difference between settling times of T-x and T-y.

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C(v1,v2): the correlation between variables v1 and v2 based on individual archer.

CG(v1,v2): the correlation between TR and C(v1,v2) based on twelve archers (global sense).

CGA(v1,v2): the correlation between TR and |C(v1,v2)| based on twelve archers (global sense).

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Chapter 3 Experiment Setup

And Analysis Method

3.1 Experiment Setup

In our experimental setting, a laser pen is mounted at the bow handle for capturing the aiming trajectory by using a digital video camera. Using the laser pen and the digital camera, both the vertical and horizontal aiming trajectory coordinates during a suitable period before releasing the arrow can be accurately recorded. On the other hand, the vertical and horizontal shot coordinates are captured by another camera placed in front of the target. These recorded data are then processed by APAS (Ariel Performance Analysis System) motion analysis system for studying the aiming procedure and the shot points along the vertical and the horizontal directions.

A total of twelve elite archers with stable archery skill attended this experiment, and the aiming trajectory of each archer during the one and half second period before releasing the arrow is recorded. The distance between start line and arrow target is 70 meters. The arrows with almost equal weights are carefully selected by an 18-meters shooting pretest to ensure uniform targeting performance. The experiment proceeds according to the usual competition procedure. Before the test, they can shoot three

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arrows, that is, 3 arrows will be shot for a round and totally 4 rounds are performed.

3.2 Analysis Method

We note that the variable TR is the most important variable which is directly related to the performance of archers. The sorting method according to the value TR based on twelve shots of each archer is conducted, and the first shot is corresponding to the best shot for individuals. Then the mean of TR of twelve shots of individual archer is calculated as the criterion for categorized these twelve archers, so the archer 1 is the archer with the best performance with the smallest mean of TR. Because the ARMAX model is adopted to model the aiming trajectory associated with each shot and even different archers, the essential variables and their physical meanings related to this model are defined and illustrated in Section 2.4. The detail information, such as TR, bx, by, |bx|, |bx|, Px1, Py1,Px2r, Py2r, Rx1, Ry1, Rx2, Ry2, Rx2r, Ry2r, Rx2a, Ry2a, Rx_1, Ry_1, Ux1, Uy1,Ux2, Uy2, Rx_1, Ry_1, Ux, Uy, UR, T-x, T-y, T-d, Ax for each archer is listed in Appendix I. Therefore, the most popular correlation approach is utilized to identify the relationship among these essential variables. For the individual case, the correlations as C(v1,v2) among the above variables are computed, and the results of correlation are shown in Appendix II.

In this thesis, the main objective is to find the crucial effects which are suitable for most archers, and to propose some conceivable suggestions for each archer based on group sense or global sense. According to the individual affiliated correlation and its mean of TR, the correlation approach is managed to repeat again to obtain the group or global senses and defined as CG(v1,v2). The results are listed in Appendix III. Since the range of C(v1,v2) is form -1 to 1. If both positive and negative C(v1,v2)

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have the same effect relative to the key variable TR, then the effect will be migrated by the CG(v1,v2) approach. In order to recover this missing effect, the absolute value of C(v1,v2) relative to the variable TR is defined as CGA(v1,v2). The results of CGA(v1,v2) are also shown in Appendix III. For convenience the threshold 0.45 for correlation coefficients CG and CGA is applied to screen out the strong relationship for further analysis. The screened outcomes of CG and CGA with this threshold are also listed in Appendix III, respectively. Owing to the symmetric of CG and CGA, we further arrange the CG outcome into the low triangle and the CGA into the upper triangular, and the results will be utilize for analysis in the next chapter.

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Chapter 4 Analysis Result and Discussion

4.1 Analysis Result

Base on the associated correlations CG and CGA as well as their means in Chapter 3 and Appendix III, we design an inference algorithm to classify them into several groups. The inference basically is derived from the sufficient condition and necessary condition. Due to the interval of the correlation [-1,1], if the mean of correlation( C ) is greater than zero, then the positive correlation (PC) becomes the sufficient condition and the negative correlation (NC) is the necessary condition.

Similarly, for the CGA case the absolute value of C is needed to implement to result in new interval [0,1], in which the zero is corresponding to low correlation (LC) and the value 1 is associated with high correlation (HC). Since CG and CGA are conducted with TR, the larger and positive CG indicates that the good performance (GP) is related to the more negative correlation, in other words the bad performance (BP) is relative to the more positive correlation. Combining with the previous discussed case, that is the mean of correlation is greater than zero, we can say that if the correlation is positive (PC) then the corresponding performance is good (GP). Thus we use the abbreviation PCGP to represent the above inference that is also equivalent to the abbreviation BPNC.

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In case of |CG| = |CGA|, they are four possible cases as outline in the Fig.4.1.

The first case is CGA>0 and CG>0 indicates that the original correlation C is distributed inside the interval [0 1], the mean of C is greater than zero, by fallowing the previous inference we can obtain the positive correlation as the domination sufficient condition which infers PCBP and the equivalent GPLC. The second case (CGA>0, CG<0) and third case (CGA<0, GA>0) have the same interval [-1 0] which is different form the first case and the forth case (CGA<0, GA<0). Thereafter, the mean of correlation associated with the second and third cases is less than zero, so we can deduce NCBP (GPLC) for the second case and the deduction NCGP (BPLC) for the third case.

Fig. 4.1. Performance Inference Procedure 1

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4.2 Couple Effects

To obtain the minimum radial deviation TR is the main concern of archers, and the radius is a nonlinear coupling with the horizontal and the vertical axis. The variables generated from the proposed ARMAX model for each axis can be expected to have couple effect among each other, and their effect are outlined as follows.

Table 4.1.Couple effect among the defined variables

Variable Coupled Variables Variable Coupled Variables bx Ry2r , Ay, Ry2, Uy1 by Ux1, Ux2

|bx| Ay, Uy1, Py1, Uy2 |by| Ay, Uy2, Py2r, Ry2r

Px1 Ay, T-y, Uy1 Py1 Rx1, Rx_1, Ux1, |bx|, Ax, Ux2

Px2r Ry1 Py2r Rx1, Rx_1, Ux1, Ax, Rx2r, Ux2

Ax Ry1, Ry2r, Ry_1, Uy1, Uy, Py1, Py2r, Ay, T-y

Ay |bx|, Px1, Rx2a, T-x, bx, Ax, Rx2r

Rx1 |by|, Py1, Py2r, Ry2a, Ry_1

Ry1 Ax, Ux2, Px2r , Rx2r

Rx2r |by|,Ry2r, Py2r, Ay, Ry1, Ry_1, Uy2

Ry2r Bx, Ax, Rx2r, Ux1 ,Ux2

Rx2a Ay, Uy1, |by| Ry2a Rx1, Rx_1, Ux1, |bx|,Ux Rx_1 |by|, Py1, Py2r, Ry2a

Ry_1

Rx_1 Ax, Ux2, Ux, Rx1, Rx2r, Rx_1,Ux1

T-x Ay, T-y T-y Ux1, Px1, Ax, T-x

Ux1 by, |by|, Py1, Py2r, Ry2r,Ry2a,T-y, y_1

Uy1 |bx|, Ax, Rx2a, Ux, bx, Px1

Ux2 by , |by|, Ay, Ry1, Ry_1, Uy2 ,Uy,Py1,Py2r, y2r

Uy2 Ux2, bx|, Rx2r

Ux Ry_1,Py1 ,Py2a Uy Ux1,Px1,Ux

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4.3 Direct Key Effects

The performance is directly related to the variable TR, so in this section we focus on this particular one. We start with the horizontal axis in which three significant correlations CG or CGA associated with variable Ux1, Px1, and Ux exist. The last two variables with positive correlations CGA suggest that for the better archer these two variables should have little effect on the performance. For the first variable Ux, related to the Px1 and Rx1, we have a negative CG=C(C(TR,Ux1),TR) value indicates that the better performance have a positive C(TR,Ux1). The positive C(TR,Ux1) can be further explained that the better performance is, the smaller value Ux1 is. The previous inference that Px1 should have little effect on performance, so this sounds a little confliction may be explained that both correlation are not highly significant and the other factors Rx1or the coupling effect.

For the vertical case, there is only one significant CGA(0.52) with the variable by, so it suggests that the lower correlation C(by, TR) is good for better performance. The variable by is related to the vertical offset at the 1.5 second instance, so this deduction is reasonable. The good performance of archers should have very consistent adjustments along the vertical direction regardless of the vertical offset. Another three variables Py1(CG=-0.89), Ay(CG=-0.69), and T-y(CG=-0.56) all have negative correlations, so they all indicate that the better performance is connected with positive correlations C(TR, Py1), C(TR, Ay) and C(TR, T-y). The most significant one is related to Py1 which is the dominated pole along the vertical direction, and its positive correlation implies that the better performance is associated with the smaller value of pole, in which the exponential decay is fast. This implication is conceivable. Variable Ay is defined as the phase angle of the complex pair Py2 and Py3, and it is also

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corresponding to the oscillation frequency of the desired adjustment. So the positive C(TR, Ay) have the physical meaning that the slower oscillation adjustment frequency along the vertical direction can result in better performance. The fast decay of the dominated pole and the slower oscillation adjustment frequency can always result in a fast settling time, so it is confirmed with a positive C(TR, T-y). Because the smaller radius (better performance) and the fast settling time have a positive relationship. The large T-d (CG=0.78) time difference can be easily resulted from the fast settling time T-y along the vertical direction, so this deduction can reconfirmed by positive C(TR, Py1), C(TR, Ay) and C(TR, T-y). The last significant variable Ry2a (CG=0.76) indicates that the negative correlation C(TR, Ry2a) in which the better performance can be achieved by larger Ry2a. This variable is a function of many variables, so it is not easy to explain its effect straightforward. In this section, the direct effect on the performance has been elaborately discussed their physical meanings and their relationship connected to each other.

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Chapter 5 Conclusion

The most popular aggressive moving average with an exogenous input (ARMAX) has been adopted and tried to model the aiming trajectories of the twelve archers. It is noted that the recorded trajectories have been processed to represent the last 1.5 second before releasing the arrow. Variables defined from the model are utilized to identify and analyze their roles affecting the performance in direct or indirect way through the individual and global statistic correlation approach. Based on the significant results, some conceivable checking points are suggested to archers for improving their performance. To obtain the desired adjustments of archers without affecting by the stability of muscle strength is the main target in this thesis. The desired aiming style of expertise should not be a high frequency adjustment one. The Hamming window can smooth out the muscle strength effect to obtain their desired one. A simple inference based on the sufficient and necessary conditions principle has been proposed to link these variables with the performance. Coupling between both axes has been identified for related variables. Variables have the most important direct effect on the performance have been analyzed as much as possible based on physical sense. Variable Ux, related to the Px1 and Rx1, has a negative CG value indicates that the better performance have a positive C(TR,Ux1). The positive C(TR,Ux1) further suggests that the better performance is relative to the smaller value Ux1. For the vertical case, there is only one significant CGA(0.52) with the variable by, so it

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suggests that the lower correlation C(by, TR) is good for better performance. Archers with good performance are supposed to have very consistent adjustments along the vertical direction regardless of the vertical offset. The dominated pole Py1 has positive correlation with TR implies that the better performance is associated with the smaller value of pole or the fast exponential decay. We have deduced that the slower oscillation adjustment frequency Ay is directly related to better performance. The fast decay of the dominated pole and the slower oscillation adjustment frequency are also linked to a fast settling time T-y. Moreover, the large T-d (CG=0.78) time difference can be easily resulted form the fast settling time T-y, so this deduction can double confirmed by positive C(TR, Py1), C(TR, Ay) and C(TR, T-y).

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REFERENCE

[1] P. Leroyer, J. V. Hoecke, and J. N. Helal,“Biomechanical study of the final push-pull archery,” Journal of Sports Sciences, London, 11(1), pp. 63-69, 1993.

[2] D. M. Landers, M. Han, W. Salazar, S. J. Petruzzello, K.A. Kubitz, and T. L.

Gannon, “Effects of learning on electroencephalographic and electrocardiographic patterns in novice archers,” International Journal of Sport Psychology, 25(3), pp. 313-330, 1994.

[3] H. Ertan, B. Kentel, S.T. Tumemer, and F. Korkusuz,“Activation patterns in forearm muscles during archery shooting,” Human Movement Science, Amstererdam, 22(1), pp. 34-45. 2003.

[4] D. Lavisse, D. Deviterne, and P. Perrin, “Mental processing in motor skill acquisition by young subjects,” International Journal of Sport Psychology, Rome, 31(2), pp. 364-375, 2000.

[5] T.-Y. Shiang and C.-J. Tseng, “A new quantitative approach for archery stability analysis,” International Society of Biomechanics Congress, Tokyo, pp. 142, 1997.

[6] C.-T. Lo, S.-H. Huang, T.-M. Hung, “A study of the relationship between heart rate variability and archery performance,” International Journal of Psychophysiology , Vol.69, pp. 276-316, 2008.

[7] K.-B. Lin and C.-K. Hwang, “Analysis of correlations between aiming adjustment trajectory and target,” the 13th International Conference in Medicine and Biology, Tainan, Taiwan, pp. 132-133, 2003.

[8] L. Ljung, System identification, 2nd edition. PTR: Prentice-Hall, 1999.

[9] K.-B. Lin and C.-K. Hwang, “Relation between the low score and the slow arrow speed,” The Impact of Technology on Sports, ASTA, pp. 392-398, 2005.

[10] K.-B. Lin, Y.-H. Lin and C.-K. Hwang, “Relationship between arrow speed and shot Points based on a radial deviation sorting method,” 15th International Conference on Mechanics in Medicine and Biology, pp191-192, 2006.

[11] K.-B. Lin and C.-K. Hwang and Y.-H. Lin, “Archery releasing style of an Olympic bronze medal archer,” The Impact of Technology on Sports, ASTA, pp. 522-527, 2005.

[12] C.-K. Hwang, K.-B. Lin and Y.-H. Lin, “Analysis of bronze medal archers aiming style based on aiming time and arrow speed.” 15th International

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Conference on Mechanics in Medicine and Biology, pp189-190, 2006.

[13] K.-B. Lin, C.-K. Hwang and Y.-H. Lin, “Archery performance analysis based on the coefficients of AR2 model of aiming trajectory,” The Impact of Technology on Sport, pp. 747-752. September, 2007.

[14] C.-K. Hwang, K.-B. Lin and Y.-H. Lin, “Mathematical model of the aiming trajectory,” The Impact of Technology on Sport Ⅱ, pp. 741-746, September, 2007.

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Appendix

The associated value of variables based on ARMAX

Here, we provide the value of variable associated with ARMAX model for each archer. Correlation coefficient R between variable TR and the set of defined variables as well as their mean and variance for each archer are listed in Tables.A1~12. In order to use this information easily, the corresponding correlation coefficient C(TR, Variable) is also organized as a matrix form, as listed in Tables A13~24.

The correlation CG=C(C(TR, Variable), TR) for group or global sense is listed in Table A25, and the correlation CGA=C(|C(TR, Variable)|, TR) is also listed in Table A26. Moreover, if their absolute value is greater or equal to 0.45, then they are rearranged in Table A27 for further use. The deductions based on the proposed inference and the information in Table A.27 is also listed in TableA28. The summarized suggestions for each archer to improve their performance based on statistical one sigma bound are presented in Table A29.

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TableA.1 Correlation coefficient R and its mean and variance for archer 1

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TableA2: Correlation coefficient R and its mean and variance for archer 2

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TableA16: Correlation coefficient C for archer 4

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中 華 大 學

中 華 大 學 碩 士 論 文

題目：The ARX-Part Effects Of Aiming Trajectory Analysis Based On ARMAX Model

系 所 別：電機工程學系碩士班 學號姓名：M09501010 吳建鋒 指導教授：黃啟光 博士

中華民國 九 十 七 年 八 月

ARX 部分對於射箭瞄準軌跡 ARMAX 模型的影響

The ARX-Part Effects Of Aiming Trajectory Analysis Based On ARMAX Model

研 究 生：吳建鋒 Student：Chien-Fung Wu 指導教授：黃啓光 博士 Advisor Dr. Chi-Kuang Hwang

中華大學

電機工程學系碩士班

碩士論文

A Thesis

Submitted to Institute of Electrical Engineering Chung Hua University

In Partial Fulfillment of the Requirements For the Degree of

Master of Science In

Electrical Engineering July 2008

Hsin-Chu, Taiwan, Republic of China

The ARX-Part Effects Of Aiming Trajectory Analysis Based On ARMAX Model

研 究 生：吳建鋒 指導教授：黃啓光 博士

中華大學

電機工程學系碩士班

中文摘要

The ARX-Part Effects of Aiming Trajectory Analysis Based on ARMAX Model

Student : Chien-Fung Wu Advisor : Dr. Chi-Kuang Hwang

Institute of Electrical Engineering Chung Hua University

Abstract

Acknowledgement

Contents

1 Introduction 1

2 System Model 3

2.1 Hamming Window 3

2.2 ARMAX Model 4

2.3 Time Series 8

2.4 Notations and Physical Meanings 10

2.4.1 Notations 13

3 Experiment Setup and Analysis Method 15

3.1 Experiment Setup 15

3.2 Analysis Method 16

4 Analysis Result and Discussion 18

4.1 Analysis Result 18

4.2 Couple Effects 21

4.3 Direct Key Effects 22

5 Conclusion 24

Appendix 28

List of Figures

List of Tables

Abstract

Chapter 1

Introduction

Chapter 2

System Model

2.1 Hamming Window

2.2 ARMAX Model

[ ]

[ ]

2.3 Time Series

2.4 Notations And Physical Meanings

2.4.1 Notations:

Chapter 3

Experiment Setup

And Analysis Method

3.1 Experiment Setup

3.2 Analysis Method

Chapter 4

Analysis Result and Discussion

4.1 Analysis Result

4.2 Couple Effects

4.3 Direct Key Effects

Chapter 5

Conclusion

REFERENCE

Appendix

The associated value of variables based on ARMAX

中華大學

中華大學碩士論文

系所別：電機工程學系碩士班學號姓名：M09501010 吳建鋒指導教授：黃啟光博士

中華民國九十七年八月

研究生：吳建鋒 Student：Chien-Fung Wu 指導教授：黃啓光博士 Advisor Dr. Chi-Kuang Hwang

研究生：吳建鋒指導教授：黃啓光博士