中 華 大 學

中 華 大 學 碩 士 論 文

題目：Time Series Analysis of Aiming Trajectory

瞄準軌跡的時序分析

系 所 別：電機工程學系碩士班

學號姓名：M09501035 黃崑書

指導教授：黃 啟 光 博士

瞄準軌跡的時序分析

Time Series Analysis of Aiming Trajectory

研 究 生：黃崑書 Student：Kun-Shu Huang

指導教授：黃啟光 博士 Advisor : 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

瞄準軌跡的時序分析

研 究 生：黃崑書 指導教授：黃啟光 博士

中華大學

電機工程學系碩士班

中文摘要

本論文主要是分析射箭選手在瞄準過程和射箭成績表現的關聯性，其瞄準過 程是利用高速攝影機以每秒 1200 個取樣頻率，並紀錄為瞄準軌跡的時序。瞄準 軌跡的時序將分為兩部份，第一部份為主要調整的瞄準過程的軌跡，第二部份為 鬆弦及放箭過程的軌跡。第一部份的軌跡的時序利用自動迴歸與外部輸入數學模 型來描述，再根據相同的數學模型來預測出選手理想完美的鬆弦及放箭過程軌 跡。然而，此預測的軌跡會與實際第二部份有所差別，此差別將是實際的鬆弦及 放箭動作所造成的。如此，就能準確分析出每位射箭選手實際鬆弦及放箭動作的 特性與箭著點的相關性。本論文針對第一部份的主要瞄準調整軌跡數學模型，以 及第二部份的軌跡差別，定義一組相關的參數，再探討組參數與箭著點的關聯性。

關鍵字:瞄準軌跡、放箭、ARMAX、時序分析、相關性、射箭

Time Series Analysis of Aiming Trajectory

Student : Kun-Shu Huang Advisor : Chi-Kuang Hwang

Institute of Electrical Engineering Chung Hua University

Abstract

In this thesis, the relationship between the aiming procedure and the arrow on the target are analyzed, and the aiming procedure is represented as the time series which are recorded by a high speed camera at 1200 frames/sec. Then, the time series of aiming procedure is partitioned into two portions, and the first portion is designated for the main aiming trajectory as well as the second potion is related to the releasing string and arrow action. Based on the time series of the first portion, the linear time invariant auto-regressive exogenous moving average (ARMAX) process is adopted to model as the intended aiming trajectory before the releasing arrow stage. We then use the ARMAX model to predict the ideal trajectory during the releasing string and arrow stage. However, for the second potion, there exits differences between the ideally predicted trajectory and real one, and these differences are caused by the releasing string and arrow action. Thus, the relationship between this typical action and the arrow on the target can be precisely analyzed. Furthermore, parameters related to ARMAX model of the first portion and the differences existed in the second potion are defined for studying their effects on the deviation of target.

TABLE OF CONTENTS

ABSTRACT (IN CHINESE)…...i

ABSTRACT (IN ENGLISH)...ii

TABLE OF CONTENTS...iii

LIST OF FIGURES...iv

LIST OF TABLES...v

CHAPTER 1 INTRODUCTION...8

CHAPTER 2PURPOSES AND METHODS OF RESEARCH………...13

2.1 Purposes and Methods...13

2.2 Hamming window and ARX Model...23

2.3 Prediction of Aiming Trajectory during the Releasing Arrow Stage………29

CHAPTER 3 RESULTS AND DISCUSSIONS………...32

3.1 Intended Horizontal Adjustment…………...32

3.2 Intended Vertical Adjustment………34

3.3 Releasing String and Arrow Stage………...…………..38

3.4 Linear Trend Analysis of the Predicted Releasing String and Arrow Stage…..…40

3.5 Trajectory Deflection Induced by the Releasing String and Arrow Motion…...43

CHAPTER 4 CONCLUSION...46

LIST OF FIGURES

Figure2.1.1: Laser light pointer installed on the bow sight………...…15 Figure2.1.2: Erosion to remove the background noise………16 Figure2.1.3: Dilation to enhance the laser pointer after removing background

noises...17 Figure2.1.4: Image processing sequence: original-binary-erosion-dilation……….18 Figure2.1.5: Next laser light point searching range based on the known coordinate

of previous one……….………19 Figure2.1.6: The selected 1800 frames (green and bold) within the original data (blue) along the X-axis……….20 Figure2.1.7: The selected 1800 frames (green and bold) within the original data

(blue) along the X-axis……….20 Figure2.1.8: The laser light point on the target becomes a long and thin

rectangle ………..22 Figure2.1.9: The laser light point transiently disappears and back again………...22 Figure2.2.1: Comparison between the original trajectory and the smoothing one by the Hamming windows with N=121………...24 Figure2.2.2: Comparison among the approximated trajectory by the ARMAX model,

the smoothing one by Hamming window, and the original one…..….26 Figure2.2.3: Time series of x_ARX, y_ARX, and r_ARX = x_ARX²+y_ARX²……...…29

LIST OF TABLES

Table 3-1-1: Correlations related to the ARMAX model along the horizontal direction………..………34

Table 3-2-1: Correlations related to the ARMAX model along the horizontal direction………...37

Table 3-3-1: Correlations based on parameters of the linear trend approximation…..40

Table 3-4-1: Correlations based on parameters of the linear trend approximation of the predicted aiming trajectory without the holding and releasing string

motion………..42 Table 3-5-1: Correlations based on parameters of the deflection caused by the holding and releasing string motion………...44

Table 3-5-2: Comparison among the original trajectory, the smoothing one by Hamming window, and the approximated one by the ARMAX

model.………...………...………...45 Table A1.1: Information of archers………..50 Table A1.2: Archery equipment………51

Chapter 1

Introduction

Lots of researches focused on archery equipments have tried to establish the mathematical model which can more precisely match the experimental and theoretical results. The research topics of equipments can be classified as pure bow, bow-arrow, archer-bow-arrow, and arrow-grip-button. A discrete mathematical model of the simple bow has been proposed to discuss its relation to measured static as well as dynamic results [1]. Analysis of the elastic forces of a bow and a limb at various degrees of pull and the vibration of a bow after shooting has been studied to investigate the mechanical characteristics of a bow and a limb. They observed the nonlinear phenomenon between the elastic force and pulling distance. They also found the discrepancy between the patterns of the elastic forces with respect to the strength of pull between the upper and the lower limb of the bows. The discrepancy usually occurs when the limbs were more deeply pulled to cause their shapes between middle and end part of limbs were slightly different.

The analysis of the elastic forces of a bow and a limb at various degrees of pull and the vibration of a bow after shooting to investigate the mechanical characteristics of a bow and a limb has been conducted in [2]. The increasing rate of elastic force

time interval of 0.002 seconds for a while.

Pekalski divided the ballistics of the arrow into two phases: Phase 1, internal ballistics: the interaction between the arrow and the archer-bow system until the arrow leaves the string. Phase 2, external ballistics: this phase lasts from the end of phase 1 until the arrow hits the target. The governing equation of an arrow’s movement during its interaction with a bow is a linear fourth-order parabolic partial differential equation with boundary and initial conditions [3].

A whole mechanical and mathematical model of an arrow-bow motion system for arrow defection in the lateral plane has been proposed in [4]. The model has considered the mechanical properties of a string, bow limbs and a grip as an oscillator of concentrated elastic and inertial elements connected with the feathered end of the arrow. The natural modes and frequencies of bow and arrow vibration have also been solved as a form of polynomial series.

Mechanical and mathematical model of bow and arrow geometry in vertical plane in braced and drawn situations has been studied by adopting an asymmetrical scheme, rigid beams, concentrated elastic elements and elastic string in the model [5].

Theoretical and experimental results of research on the problem of archer, bow and arrow behavior have been presented and discussed.

The mathematical model of the flight of the arrow during its discharge from a bow can more completely match with reality, and this model can predict the arrow leaves the pressure button before it leaves the string [6]. In the paper, the pressure

performance of this fine and highly skilled sport. Therefore, for archery motion the United States Olympic Committee has defined archery fundamentals as: stringing the bow, stance, nock, set, pre-draw, draw, anchor, aim, release, and follow through [7].

These archery fundamentals become the important guidelines for archery researchers to follow and to find the physical, physiological and mental effect.

Most of physiological researches are based on electromyography (EMG) or electrocardiography (ECG). Activation patterns in forearm muscles, related to contraction and relaxation strategy during archery shooting, have been analyzed by (EMG) for archers with different levels of expertise; elite, beginner, and non - archers, respectively [8]. They 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.

Surface electromyography (EMG) signals of musculus flexor digitorum superficialis (MFDS) and extensor digitorum (MED) were recorded during archery shooting. The results showed that FITA scores were significantly correlated to the variance ratio of MFDS and MED. Therefore, variance ratio of MFDS and MED might be important variables for assessing shooting techniques, evaluation of archers’

progress, and selection of talented archers [9].

Both flexor and extensor digitorum activity decreased markedly immediately prior to or at arrow release. The extensor digitorum displayed a marked increase in activity just prior to release, indicating that string release was facilitated by an active extension of the fingers. It was found that highly skilled archers do not predominantly

been measured by C.-T. Lo. [11]. It is high frequency (HF) generally represents parasympathetic activity. The low frequency (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 higher HF and lower LF (lower LF/HF ratio) was for the best performance.

Differences were noted in motor skill acquisition according to the quality of the cognitive constructions made by the subjects. Better performances can be obtained at the end of the training phase by the group of "experimental" subjects. Because of the learning conditions they received, they appeared to construct better mental representations of archery motions. This effect was most significant in older subjects, at a more advanced stage of cognitive development, and clearly more efficient in mental operation. These results are interpreted with regard to cognitive theories of motor skill acquisition [12].

Shiang and Tseng (1997) indicated that aiming stability is the key factor affecting archery performance. In addition to the size of the aiming locus, the aiming locus pattern is also a useful index to determine performance [13]. Archery is a kind of sport that requires precise self-control. An archer needs strong and steady arms to maintain stability and equilibrium in every movement. Thus, the key to win a game in a contest relies on the stability when releasing the arrows, which is considered as the decisive moment. It becomes the goal for every archer to keep stable at that instant moment. Archery is described as a static sport requiring strength and endurance of the

and the aiming procedure which is the time series recorded by a high speed camera at 1200 frames/sec. The aiming procedure is further partitioned into two portions; the first portion is the major aiming phase and the second potion is related to the releasing string and arrow action. The sighting phase is the middle portion between previous two portions, and it contains the final pull-push stage under high tension for allowing the arrow to pass the clicker. This final pull-push stage usually results in some inconsistent motions to affect the archery performance. The linear time invariant auto-regressive exogenous moving average (ARMAX) process is used to model the intended aiming trajectory before the releasing arrow stage and the sighting phase.

The ARMAX model can predict the ideal trajectory during the releasing string and arrow stage. The off-track from the predicted ideal trajectory is caused by the releasing string and arrow action. Parameters related to ARMAX model of the first portion and the off-track are defined for studying their effects on the deviation of target.

This thesis is organized as follows. We begin with an introduction in Chapter 1 follows by purposes and methods of research in Chapter 2 which the Hamming window and ARX Model are used to obtain the desired aiming trajectory of each shot during the last 1.5 second before the arrow leaves the string. Experimental results and discussions are presented in Chapter 3 which we focus on difference between the desired aiming trajectory and the predicted one to study and discuss their relation with the performance. We conclude our research in Chapter 4.

Chapter 2

Purposes and Methods of Research

2.1 Purposes and Methods

In this study, six male archers from the national Hsinchu commercial vocational high school male archery team are invited to attend this experiment in which the indoor thirty-meter range is conducted at the Yuanpei university gymnasium, and the individual information is listed in Appendix I.

We begin with installing the lamps for better lighting condition of the target and the bow, and start the function of the laser light pointer, the accelerometer, four video cameras, and the wireless healthcare system for recording electrocardiogram (ECG).

Therefore, the research topic contains the aiming style, releasing arrow motion, ECG, and vibration of the bow. 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, each archer can shoot three arrows to adjust his bow sight. Then the test begins. Each archer shoots thirty-six arrows, that is, 3 arrows will be shot for a round and totally twelve rounds are performed, and the aiming trajectory of each shot has been fully

target with their bow sight. Thereafter, with the aid of one CASIO high speed video camera to capture the laser light point containing the vertical and horizontal coordinates of the aiming trajectories during the aiming stage with the sampling rate 1200 frame/second. For the lateral view perpendicular to the target plane, the other same high speed video camera is used to record the vibration of bow with two LED pointers, and another middle speed video camera is employed to capture the releasing string action with the sampling rate 440 frames/second. In order to increase the resolution at the target, we use the other regular video camera with the sampling rate 60/sec to record the arrow located on the target.

These recorded data are then processed by the K-Multimedia motion analysis system and our design motion analysis system based on the MATLAB for studying the relationship between the aiming procedure and the shot points on the target along both of the vertical and the horizontal directions. The erosion and dilation techniques in morphology are used to enhance the laser point by dilation after the erosion removes the background noise from a binary image, as shown in Figures 2.1.2 and 2.1.3. Figure 2.1.4 shows the image processing sequence step by step. Due to the laser light pointer on the target is almost a circle, its coordinate is calculated as

( , )x y =mean x y( ,_{i i}),i=1 ~ 8, where x and _i y are four corners and four edges _i coordinates of the dilation image. The difference between two consecutive images to indicate the movement of laser point, and the previous known location of laser point is utilized in the thesis to increase the efficiency of the image processing by reducing the searching size of the next image, as shown in Figure 2.1.5.

Figure 2.1.1: Laser light pointer installed on the bow sight.

Laser light pointer

Figure 2.1.2: Erosion to remove the background noise.

image matrix A

mask matrix B

Erosion (AND operation): apply mask B into image A to remove the noise

Figure 2.1.3: Dilation to enhance the laser pointer after removing background noises.

mask matrix B image matrix A

Dilation (OR operation): apply mask B into image A to enhance the laser point

Original image

Binary image with threshold 0.3

Erosion with mask matrix size 2 to remove the background noise

Dilation to enlarge the laser light point with mask size 2

Figure 2.1.4: Image processing sequence: original-binary-erosion-dilation.

Figure 2.1.5: Next laser light point searching range based on the known coordinate of previous one.

The last 1.5 second before the arrow leaving the string and the rest site of bow of original recorded data is fetched as the research object, so the corresponding 1800 frames are utilized to characterize the aiming stage for each archer. Figures 2.1.6 and 2.1.7 show the corresponding 1800 frames with bold and green line within the original recorded data. Whether the releasing string and arrow action is smooth or not can greatly affect the archery performance, so this typical releasing action is one of our research focuses. Each archer also has his own aiming style, so how to model his aiming style is also our research focus.

After carefully checking the other recorded data frame by frame, we find that the number of frames counting from the releasing string to the arrow leaving the rest site of the bow is around 20 frames, that is, 1/60 second. Therefore, we partition the 1.5 second data into two parts; the first part can represent the intended aiming trajectory of the archer, and the second part represents the releasing string and

calculated as 1/80=2.5/200 second and the upper bound can be evaluated by the average speed as 1/40=2.5/100 second. It is noted that the acceleration is nonlinear, so the value should be between 1/80 and 1/40 second.

Figure 2.1.6: The selected 1800 frames (green and bold) within the original data (blue) along the X-axis.

After align the pee sight and the center of target, archers will spend a short period of time to pull the arrow to pass the clicker. This action is no longer for aiming but to ensure the constant force exerted on the arrow to result in consist performance. This pulling action is under high tension, so it usually disturbs the intended aiming trajectory a little bit. Normally, this period is various not fixed, so we define three different periods such as 1/12, 1/15, and 1/30 seconds for studying its effect on the performance.

The first part is defined as the first 1680 frames of recorded data, and it is used to represent the predetermined aiming trajectory of archers. The intended or predetermined aiming trajectory before the releasing arrow stage is modeled by an ARMAX process. Usually, the initial pee site has offset with the target, so the archer will adjust his bow to compensate this offset till the finish of the alignment between the pee site and the target. The adjustment is assumed to be consisted of three components. The first component is the constant adjustment related to the exogenous input; the second component is exponentially decay adjustment corresponding to the stable and convergent pole of the AR part. Most of time, archers will adjust back and forth to modify the small over-adjustment, this modification is related to two complex poles of the AR part.

The data processing and definition are illustrated as follows: the defined positive horizontal axis is pointed to right hand side of archers, and the positive vertical one is pointed to the ground. When the released string impacts on the

and 2.1.7 by checking the last frame of the selected 1800 frames before the high frequency oscillation.

Figure 2.1.8: The laser light point on the target becomes a long and thin rectangle.

2.2 Hamming window and ARX Model

One of our purposes is to acquire the archer desired aiming trajectory without affecting by the muscle strength, so we use Hamming window to smooth out the effect of high frequency variation of muscle strength. The symmetric Hamming window with N=121 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 ≤ ≤ (2-2-1)

2 2

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

N N

π π

ω = ω + ω− + ω+ , (2-2-2)

where

2 2 sin( ( 1)) ( ) 2

j N

N

e N D

ω ω

ω ω

−

= + .

The temporary staying vision of human is about 24 frames/second, so we assume that the adjusting rate should be around twice of 24 frames/second, that is, 1/10=120/1200 second. Accordingly, the symmetric Hamming window with N=121 is reasonable to smooth the recorded information. The normalization of the Hamming window is also conducted, and the smoothing result as compared with the original data is shown in Figure 2.2.1.

Unit: pixels Figure 2.2.1: Comparison between the original trajectory and the smoothing one by

the Hamming windows with N=121.

In this thesis, we propose a linear time-invariant ARMAX model [17] to represent the aiming trajectory of each shot during the last 1.5 second before leave the arrow. In other words, this model is applied to obtain the desired or intended aiming trajectory from the original data. The associated ARX part can be used to describe the adjustment to compensate the offset between the target and the current of the aiming point. The MA part can model the average muscle strength of archers. The 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.

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

1 2 3

1 2 3

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

( ) ( 1) ( 2) ( 3)

y

y y y y y

y y y y y y

y k a y k a y k a y k b u k e k c e k c e k c e k

= − + − + − + −

+ + − + − + − (2-2-4)

where x k( ) and y k( ) are the time series of the aiming trajectory along the horizontal and the vertical directions, respectively; b_x and b_y are the corresponding coefficients of exogenous inputs; a_x1~3 and a_y1~3 are the corresponding coefficients of the AR part; c_x1~3 and c_y1~3 are the associated coefficients of the MA part; e k _x( )

and e k are denoted as the driving noises. The unknown coefficients are _y( ) assembled as two vectors for identification.

1 2 3 1 2 3

[ ]^T

x a ax x a c cx x x c bx x

θ = (2-2-5)

1 2 3 1 2 3

[ ]^T

y a ay y ay c cy y cy by

θ = (2-2-6)

The current driving noises e k_x( )= x k( )−x kˆ( ) and e k_y( )= y k( )−y kˆ( ) are substituted into Equations (2-2-3) and (2-2-4) to form one-step-ahead predictor as

1 2 3

1 2 3

ˆ( ) ( 1) ( 2) ( 3) ( 1)

( 1) ( 2) ( 3)

x x x x x

x x x x x x

x k a x k a x k a x k b u k c e k c e k c e k

= − + − + − + −

+ − + − + − (2-2-7)

1 2 3

1 2 3

ˆ( ) ( 1) ( 2) ( 3) ( 1)

( 1) ( 2) ( 3)

y y y y y

y y y y y y

y k a y k a y k a y k b u k c e k c e k c e k

= − + − + − + −

+ − + − + − (2-2-8)

written as follows.

( )

ˆ( | _x) ^T_x , _{x x}

x k θ =ϕ k θ θ (2-2-9)

( )

ˆ ( | _y) ^T_y , _{y y}

y k θ =ϕ k θ θ (2-2-10) where

( )

( , ) [ 1 ( 2) ( 3) ( 1) ( 2) ( 3) 1]

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

ϕ θ = − − − − − −

( )

( , ) [ 1 ( 2) ( 3) ( 1) ( 2) ( 3) 1]

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

ϕ θ = − − − − − −

Comparison among the original trajectory, the smoothing one by the Hamming window, and the approximated one by the ARMAX model are shown in Figure 2.2.2.

Unit: pixels.

Figure 2.2.2: Comparison among the original trajectory, the smoothing one by Hamming window, and the approximated one by the ARMAX

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

( ) ( ) ( ) ( ) ( ) ( )

x x x x x

A z x t =B z u t +C z e t (2-2-11) ( ) ( ) ( ) ( ) ( ) ( )

y y y y y

A z y t =B z u t +C z e t (2-2-12)

whereA z_x( ) 1= −a z_x1 ⁻¹−a z_x2 ⁻²−a z_x3 ⁻³,B z_x( )=b z_x ⁻³, C z_x( ) 1= −c z_x1 ⁻¹−c z_x2 ⁻²−c z_x3 ⁻³,

1 2 3

1 2 3

( ) 1

y y y y

A z = −a z⁻ −a z⁻ −a z⁻ ,B z_y( )=b z_y ⁻³, andC z_y( ) 1= −c z_y1 ⁻¹−c z_y2 ⁻²−c z_y3 ⁻³. Owing tou_x =u_y = , we have 1 ( ) ( )

x y 1

U z U z z

= = z

− and the corresponding ARX parts for both axes are

4

3 2

1 2 3

( ) 1 2 3

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

x x x

x x x x

B U z z b Rxu Rx Rx Rx

A z z z a z a z a z z z Px z Px z Px

⎧ ⎫

= − − − − = ⎨⎩ − + − + − + − ⎬⎭

4

3 2

1 2 3

( ) 1 2 3

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

y y y

y y y y

B U z z b Ryu Ry Ry Ry

A z z z a z a z a z z z Py z Py z Py

⎧ ⎫

= − − − − = ⎨⎩ − + − + − + − ⎬⎭

where Px1 ~ 3 and Py1 ~ 3 are the roots of the z³−a z_x1 ²−a z_x2 −a_x3 = and 0

3 2

1 2 3 0

y y y

z −a z −a z−a = , respectively.

The intended horizontal and vertical adjustments of the exerting force x_ARX( )k and ( )y_ARX k related to the offsets bx and by are written as:

_ _1 _ 2 _ 3

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

( ) ( ) ( )

k k k

ARX

ARX u ARX ARX ARX

x k Rxu Rx Px Rx Px Rx Px

x x k x k x k

= + + +

= + + + (2-2-13)

_ 3( ) 3( 3)^k

xARX k =Rx Px , y_{ARX_u} =Ryu y, _{ARX_ 1}( )k =Ry Py1( 1) ,^k

_ 2( ) 2( 2) ,^k

yARX k =Ry Py and y_{ARX_ 3}( )k =Ry Py3( 3)^k.

The intended horizontal and vertical adjusting trajectories in terms of time series related to the exerting force along both directions x_ARX( )k and y_ARX( )k have been defined and their initial values and the final values are also evaluated for comparison as shown in Figure 2.2.3. The exponential typex_{ARX_1}( )k andy_{ARX_1}( )k for the stable adjustment, the constant type x_{ARX u_} and y_{ARX_u} for compensation of the initial offset, and the possible oscillation type with exponential decay x_{ARX_ 2}( )k and

_ 2( )

yARX k for the modification of over adjustment are also defined. We outline

notions used in this thesis as follows.

,

X Y : the positive horizontal direction is pointed to the right hand of archer and the

positive vertical directions is pointed to the ground.

, ,

T T T

x y r : horizontal, vertical, and radial deviations of arrows on the target,

respectively.

r : the radial deviation of arrows on the target relative to the mean of 36 shots for R

each archer.

, , ,

bx by bx by : exogenous inputs and their absolute value.

1, 1

Px Py : the smallest real pole of the ARMAX model

2, 2

Px Py : the second smallest real pole or the complex pair pole.

_1, _1

Px Py : the pole is one.

1, 1, 2, 2, 3, and 3

Px Py Px Py Px Py , respectively.

,

Sx Sy : the settling time.

,

Sx Sy Sx− −Sy : difference or its absolute value of settling time between two axes.

Unit: pixels.

Figure 2.2.3: Time series of x_ARX, y_ARX, and r_ARX = x_ARX²+y_ARX² .

2.3 Prediction of Aiming Trajectory

during the Releasing Arrow Stage

As mentioned above the time series of aiming procedure has been partitioned into two portions, and the first portion is designated for the main aiming trajectory as well as the second potion is related to the releasing string and arrow action. Based on the time series of the first portion, the linear time

The releasing string and arrow stage can further be divided into two sections.

The first section is corresponding to the duration for pulling the arrow to pass the clicker and button. This action is no longer for the intended aiming of archer but to exert a consistent force on the arrow to result in better performance. Usually, this pulling action is under very high tension to hold the string, so it always induce inconsistent factor on the performance. Moreover, this period is varied not fixed, so three possible periods such as 1/12, 1/15, and 1/30 seconds are selected for studying its effect. Additionally, the second section belongs to the releasing string and arrow action which spends around 1/60 second, i.e., 20 frames.

Based on the above illustrations, we use the estimated coefficients of ARMAX model which represents the intended aiming trajectory during the first 1680 frames to predict the next 120 frames associated with the releasing string and arrow stage, as formulated as Equations 2-3-1 and 2-3-2. The releasing string and arrow action results in differences between the ideally predicted trajectory and real one. Thus, the relationship between this typical action and the arrow on the target can be analyzed.

( )

| 1680 | 1680 | 1680

ˆ ˆ ˆ

ˆ( | _{x k} ) ^T_x , _{x k x k}

x k θ ₌ =ϕ k θ ₌ θ ₌ (2-3-1)

( )

| 1680 | 1680 | 1680

ˆ ˆ ˆ

ˆ( | _{y k} ) ^T_y , _{y k y k} , 1681 ~ 1800

y k θ ₌ =ϕ k θ ₌ θ ₌ k = (2-3-2) For three different durations, the differences between the ideally predicted trajectory and real one are defined as follows.

( )k = − e ( )k = −

average of the first ten frames and the last ten frames are also defined based on three durations.

1 2 3 1 2

( ( 1681 ~ 1690)) ( ( 1791 ~ 1800)) ( ( 1701 ~ 1710)) ( ( 1791 ~ 1800)) ( ( 1761 ~ 1770)) ( ( 1791 ~ 1800)) ( ( 1681 ~ 1690)) ( ( 1791 ~ 1800))

( ( 1701 ~ 17

x mean x k mean x k

x mean x k mean x k

x mean x k mean x k

y mean y k mean y k

y mean y k

Δ = = − =

Δ = = − =

Δ = = − =

Δ = = − =

Δ = =

3

10)) ( ( 1791 ~ 1800)) ( ( 1761 ~ 1770)) ( ( 1791 ~ 1800))

mean y k

y mean y k mean y k

− =

Δ = = − =

1 2 3 1 2

ˆ ( (ˆ 1681 ~ 1690)) ( (ˆ 1791 ~ 1800)) ˆ ( (ˆ 1701 ~ 1710)) ( (ˆ 1791 ~ 1800)) ˆ ( (ˆ 1761 ~ 1770)) ( (ˆ 1791 ~ 1800)) ˆ ( (ˆ 1681 ~ 1690)) ( (ˆ 1791 ~ 1800)) ˆ

x mean x k mean x k

x mean x k mean x k

x mean x k mean x k

y mean y k mean y k

y mea

Δ = = − =

Δ = = − =

Δ = = − =

Δ = = − =

Δ =

3

ˆ ˆ

( ( 1701 ~ 1710)) ( ( 1791 ~ 1800)) ˆ ( (ˆ 1761 ~ 1770)) ( (ˆ 1791 ~ 1800))

n y k mean y k

y mean y k mean y k

= − =

Δ = = − =

The least square to approximate these two aiming trajectories as a linear trend can result in two parameters for each duration, that is, the slopes

1, 2, 3, ˆ1, ˆ2, ˆ3, 1, 2, 3, ˆ1, ˆ2, ˆ3

mx mx mx mx mx mx my my my my my my and the associated offsets

1, 2, 3, ˆ1, ˆ2, ˆ3, 1, 2, 3, ˆ1, ˆ2, ˆ3

bmx bmx bmx bmx bmx bmx bmy bmy bmy bmy bmy bmy .

Chapter 3

Results and Discussions

The aiming trajectory can somehow indicate the stability of archers, so in this thesis we focus on this topic to study its relation with the performance. Based on the previous defined parameters, the correlation coefficient approach is carried out through this chapter. We start with the parameters of the first portion related to ARMAX model follows by the investigation of the second portion regarding the release of string and arrow stage.

3.1 Intended Horizontal Adjustment

In this section, the relationship between the arrow deviations on the target and the parameters of the ARX part of the ARMAX model is investigated. For all six archers, the magnitude of poles of the ARMAX are less than one, so it indicates that their intended or desired aiming trajectories all belong to the exponentially stable type which confirms their expert on the archery sport. We assign the number to each archer from 1 to 6 according their performance in this experiment, so the best one is archer 1.

There are two interesting findings regarding the complex pole. The first one is that along the vertical direction archer 4 also has all complex pole pairs and archer 2 has almost all complex pole pairs. The second one is that for archer 1 the complex pole

that the slower the pole is, the more negative horizontal deviation is (left hand). The residue is corresponding to how and how much the associated pole affects the adjusting trajectory. From correlationsC R x( 2 ,x_T ) = C R x( 3,x_T) = 0 .4 7 1 6 , we know that the more positive Rx2 or Rx3 is, the more positive x is (right hand). _T The same value of these two correlations can confirm that all the second and third poles are the complex pole pair. The most important parameter related to the performance is the radial deviation, and for archer 1, archer 3 and archer 5, their intended horizontal aiming trajectories do not affect their performance.

For archer 2, negative correlations C R x( 1,r_T) C R x( 3,r_R) imply that the more positive Rx1 and Rx3 are, the better performance is. The indication of three positive correlations ( C R x( _ 1,r_T) , C R x( 2 ,r_T ) ,significant C R x( 2 ,r_R) ) is reverse. Besides, the additional negative correlationsC R x( 2 ,x_T ),C R x( 2 , y_T ) and the positive C R x( 2 ,r_T) suggest that the better performance (small r , the right _T hand x , and downward _T y ) is strongly connected to the negative _T Rx2. Similarly, the implication of all positive C R x( _ 1,r_T) C R x( _ 1,x_T) C R x( _ 1, y_T)is that the better performance (small r , the left hand _T x , and upward _T y ) is related to the _T negative value of Rx_1.

For archer 4, significant negative C R x( 1,r_R) C R x( 1,r_T) C R x( _ 1,x_T)and positive C R x( 1,x_T) C R x( _ 1,r_R) C R x( _ 1,r_T) suggest that better performance is associated with positive Rx1, negative Rx_1, and right hand x . Positive _T correlations (C R x( 3,y_T) C( R x3 , y_T)) show that most of Rx3 are positive, and

deviation r from the target center is. This may be caused by his pee sight mismatch _T with the target center to result in a good concentration of the arrows but poor performance.

Table 3-1-1: Correlations related to the ARMAX model along the horizontal direction.

Correlations Archer 1 Archer 2 Archer 3 Archer 4 Archer 5 Archer 6 ( 1, _T)

C Px x -0.4523 0.1951 -0.0027 -0.1349 -0.0784 0.0426

( 1, _R)

C R x r -0.2566 -0.0832 -0.1746 -0.7571 -0.1766 -0.1273

( 1, _T)

C R x r -0.108 -0.3084 -0.06 -0.727 -0.1776 0.0039

( 1, _T)

C R x x 0.1398 -0.2483 -0.05 0.7407 0.0924 -0.1374

( 2 , _R)

C R x r -0.0848 0.5739 -0.1163 -0.093 0.1613 0.2169

( 2 , _T)

C R x r 0.0929 0.3879 0.0197 -0.0721 0.1648 -0.0802

( 2 , _T)

C R x x 0.4716 -0.3191 0.038 0.158 -0.082 0.2117

( 2 , _T)

C R x y 0.0801 -0.4639 -0.1953 0.1344 -0.0334 -0.0815

( 3, _R)

C R x r -0.0848 -0.3595 -0.0581 0.0111 -0.0498 -0.3297

( 3, _T)

C R x r 0.0929 -0.2094 0.0211 0.017 -0.0324 0.3118

( 3, _T)

C R x x 0.4716 0.0509 0.1188 0.1362 -0.1424 -0.3287

( 3, _T)

C R x y 0.0801 0.1382 0.0435 0.4375 -0.0754 0.3114

( 3 , _T)

C R x y 0.1582 -0.1128 -0.1352 0.549 -0.0961 -0.0406

( _ 1, _R)

C R x r 0.2681 0.0762 0.179 0.7571 0.0979 0.1299

( _ 1, _T)

C Rx r 0.1039 0.3048 0.0591 0.7269 0.0963 -0.0088

( _ 1, _T)

C Rx x -0.1753 0.2606 0.0468 -0.7407 -0.0492 0.1403

( _ 1, _T)

C Rx y -0.0659 0.406 0.1274 -0.1074 0.0641 -0.0057

3.2 Intended Vertical Adjustment

For archer 1 with the best performance, there is only one little significant

complex pole pairs (related toPy2,Py3,Ry2,Ry not to 3 Ry_1,Ry ), the vertical 1 deviationy is highly related to_T Ry_1 and Ry by checking the significant positive 1

( _ 1, _T)

C R y y and negativeC R y( 1,y_T) . For the performance, r has the same but _R weak connection withRy_1 and Ry , as compared with the1 y case. Conversely, _T

x is weakly related to T Ry_1 andRy (1 C R y( 1,x_T) < 0and C R y( _ 1,x_T)> 0 ) but with reverse way as referred with the y case. _T

Archer 4 with all complex pole pairs has the same situation as archer 2 (significant C R y( _ 1,y_T)> 0andC R y( 1,y_T) < 0 ) that y is highly related to _T

_1

Ry and 1Ry instead of Ry and 2 Ry . Unlike archer 2, his phase of the 3 complex pair AgY will weakly affect the radial deviations on the target, as indicated by positive C A gY r( , _R) andC A gY r( , _T) . Even though archer 3 has only four correlations worth investigation, there are three correlations related to performance, as shown in Table 3-2-1.

It is noted that for archer 5 all significant correlations are related to the complex pole pair, and there are four positive performance-related correlations C Py( 3,r_R)

( 3, _T)

C Py r C A gY r( , _R) C A gY r( , _T) as well as the other three y related _T

correlations C Py( 2,y_T) C AgY y( , _T) C R y( 3, y_T ) . The meaning of positive ( , _T)

C A gY r is that for the complex pair pole the higher ratio of between its real part and imagery part AgY will deteriorate the performance.

As observed the bold frame from Tables 3-1-1 and 3-2-1, total seven

symptom of the same number of pairs with opposite signs is also true for the x and _T

y case, and his most significant pair is the pair (T C R y( 1,x_T ) , C R y( 1, y_T ) ). The finding can double verify his poor performance in this experiment.

We now look into the settling time effect which dose not directly affect the performance for all six archers but affect the vertical deviation y . For archer 3, _T archer 4, and archer 5, their vertical setting time of aiming trajectories will influence the vertical deviation on the target. There is a remarkable negative correlation

(

^, T

)

C Sx y for archer 4, that is, if his horizontal adjustment becomes steady faster, then the arrow goes lower on the target. Besides, combination of the significant

(

^, T

)

C Sx−Sy y and insignificant ^{C Sx}

(

^{−Sy y, T}

)

suggests that if his horizontal adjustment becomes steady earlier than his vertical adjustment (Sx−Sy< ), then the 0 arrow also goes lower on the target. However, the case for archer 3 is caused by absolute difference of both axial settling times, and the situation can be explained that if he aligns the pee sight along one of directions following by the other direction will result in lower deviation on the target.

Table 3-2-1: Correlations related to the ARMAX model along the horizontal direction.

Correlations Archer 1 Archer 2 Archer 3 Archer 4 Archer 5 Archer 6

( 1, _R)

C P y r 0.2112 -0.0244 -0.1299 -0.0492 0.0604 -0.3764

( 1, _T)

C Py r 0.1665 0.0438 -0.1431 -0.0256 0.0728 0.4643

( 1, _T)

C Py x -0.0821 0.2562 0.3406 0.1253 -0.0178 -0.3738

( 1, _T)

C Py y 0.0899 0.1198 0.0163 0.1944 0.2131 0.4653

( 2, _T)

C Py x -0.3792 -0.2846 -0.0525 0.1047 -0.1828 0.0491 ( 2, _T)

C Py y -0.0213 -0.3332 -0.0986 -0.0384 -0.5027 -0.0476 ( 3, _R)

C Py r -0.1584 0.1946 -0.2428 -0.233 0.464 0.0213

( 3, _T)

C Py r 0.0475 0.0828 -0.3146 -0.2582 0.433 0.0366

( , _R)

C AgY r 0.0458 0.019 -0.3028 0.3466 0.4299 0.0366

( , _T)

C AgY r 0.1098 0.0273 -0.2438 0.3445 0.4106 0.0221

( , _T)

C AgY y -0.1416 -0.0601 0.152 -0.0216 0.4704 0.0235

( 1, _R)

C R y r -0.1387 0.3319 0.1666 -0.1443 -0.0345 0.6226

( 1, _T)

C R y r -0.1105 0.1189 0.1335 -0.1612 -0.0638 -0.4184

( 1, _T)

C R y x 0.2435 -0.3982 0.2734 -0.0649 -0.025 0.6317

( 1, _T)

C R y y -0.2143 -0.6129 -0.1634 -0.4973 -0.0832 -0.415

( 2 , _R)

C R y r -0.0788 -0.0243 -0.0745 0.0397 0.2416 0.3603

( 2 , _T)

C R y r -0.1499 -0.0478 -0.175 0.0174 0.2275 -0.4065

( 2 , _T)

C R y x -0.0727 -0.0704 -0.2905 -0.0929 0.0335 0.3612

( 2 , _T)

C R y y -0.1851 -0.176 0.0889 -0.1664 0.133 -0.4065

( 3, _R)

C R y r -0.33 0.2025 0.2394 0.0397 -0.2629 0.3053

( 3, _T)

C R y r -0.1578 0.0965 0.3359 0.0174 -0.2697 -0.3644

( 3, _T)

C R y x 0.1975 -0.2677 -0.2043 -0.0929 -0.0343 0.3015

( 3, _T)

C R y y 0.0259 -0.0266 0.022 -0.1664 -0.55 -0.3659

( 3 , _R)

C R y r -0.1207 0.0492 0.1609 0.081 -0.1229 0.2765

( 3 , _T)

C R y r -0.0861 -0.0356 0.204 0.057 -0.1316 -0.3597

( 3 , _T)

C R y x 0.1127 -0.26 -0.2441 -0.1392 0.0479 0.2726

( 3 , _T)

C R y y -0.1198 -0.1091 0.0612 -0.1669 -0.2542 -0.3612

( _ 1, _R)

C R y r 0.1785 -0.3766 -0.1901 0.1354 0.0361 -0.6339

3.3 Releasing String and Arrow Stage

In this section, the relationship between the arrow deviations on the target and the parameters related to releasing stage is investigated. For all six archers, the vertical slope of holding and releasing motion mx is mostly steeper than that along ₁ the horizontal direction my by the same sign of ₁ C mx

(

1−my y1, _T

)

and

(

1 1 , _T

)

C mx −my y , as shown in Table 3-3-1. This observation seems justifiable because the holding and releasing string motion mostly belongs to the horizontal movement. All positive C b y

(

_m 1, y_T

)

for all six archers exhibit that the vertical offset

1

b y can result in the oriented deviationm y . _T

For the best accomplishment in this experiment archer 1, the offset b y _{m 1} generated from the linear trend approximation of firmly pulling and releasing string stage can effect the performance (radial deviation on the targetr ) and majorly the _T vertical deviation y_T , as referred by two significant and positive

(

_m 1, _T

)

C b y r C b y

(

_m 1, y_T

)

in Table 3-3-1. Furthermore, we can find that most of his noticeable correlations are distributed along the vertical direction, except only one correlation C b x x

(

_m 1, _T

)

is in the horizontal direction. Besides, this noticeable

(

_m 1, _T

)

C b x x is existed exclusively for archer 1. Opposite sign and nearly same significance pair (C my

(

1 ,y_T

)

and C my y

(

1, _T

)

) indicates that the majority of his

my are negative or upward. 1

3-3-1, and the significant one pair is (C b y r

(

_m 1, _R

)

,C b y r

(

_m 1, _T

)

. The domination of this significance is governed by the horizontal deviation as referred to his

(

_m 1, _T

)

C b y x and C b y

(

_m 1, y_T

)

. Majority of mx is positive or rightward, as ₁

inferred from two similar values ofC

(

m x1, r_T

)

andC

(

m x1 , r_T

)

, but it can influence its orthogonal deviation y which may be one of reasons to result in his _T bad performance. Similarly, his extraordinary and abnormal negative C b y x

(

_m 1, _T

)

suggests that the vertical movement can deeply affect the horizontal deviation.

The second best archer 2 likes the archer 6 has many noticeable correlations during this period. Similarly, he has cross directional effect between the movement and orthogonal deviation, but his correlations are less significant than that of archer 6, for example, C mx y

(

1, _T

)

,C mx

(

1 ,y_T

)

,C b y x

(

_m 1, _T

)

. The negative C mx x

(

1, _T

)

is a serious but fortunately not too significant scenario, because its implication is that a rightward motion during this stage can cause a leftward deviation on the target. Thus it is worth for him to find the causes and then to correct it.

In this stage, we can find that archers 4 and 5 have only one significant correlation C b y y

(

_m 1, _T

)

and C my y

(

1, _T

)

, respectively. Nevertheless, for archer 5 the

unusual negative correlation C my y

(

1, _T

)

indicates that a downward slope of holding and releasing action can result in an upward deviation on the target. This may be one important finding for him and his coach to improve his performance.