CHAPTER 1

INTRODUCTION

In competitive diving, all dives are allocated a degree of difficulty (DD) depending on dive complexity. Dives are assessed on a 10-point scale by a panel of judges. Final awards are determined by eliminating the highest and the lowest scores and summing the remaining scores. This sum is multiplied by the DD for the particular dive (Federation Internationale de Natation Amateur [FINA], 1998). Thus, a diver receives the fullest benefit from attempting dives with higher DD (Sarsfield, 1960) provided the dives are of reasonable quality.

The number of dives listed in the FINA Table of DD has increased rapidly during the past decades. For example, the forward 3½ somersault tuck from 3-metre springboard was awarded the highest DD of 2.7 in 1960 (Sarsfield, 1960). Shortly afterwards, the forward 3½ somersault pike was awarded a DD of 2.9 (The Amateur Swimming Association [ASA], 1963). Since 1990, the forward 4½ somersault tuck has been commonly adopted in competition. With the introduction of new dives, the DD of existing dives have been continuously re-evaluated and adjusted in order to allocate DD for the new dives. In 1995, FINA established a formula (Equation 1.1) to calculate DD according to factors including the number of somersaults and/or twists, flight position, approach/group, height of takeoff surface, and nature of entry (FINA, 1998). The present DD of dives in the forward group from 3-metre springboard are shown in Table 1.1.

DD = A + B + C + D + E (1.1)

where A = somersaults

B = flight position

C = twists

D = approach/group

E = unnatural entry

Table 1.1 Degree of Difficulty in the Forward Group from 3-metre Springboard

Dive number	Dive	Flight Position	DD
101C	Forward dive	tuck	1.4
101B	Forward dive	pike	1.5
103C	Forward 1½ somersault	tuck	1.5
103B	Forward 1½ somersault	pike	1.6
105C	Forward 2½ somersault	tuck	2.2
105B	Forward 2½ somersault	pike	2.4
107C	Forward 3½ somersault	tuck	2.8
107B	Forward 3½ somersault	pike	3.1
109C	Forward 4½ somersault	tuck	3.5

There are some common “lead-ups” that most coaches would use to teach multiple somersaulting dives. Performing similar dives from different heights also helps in progression of skills. For instance, the forward double somersault pike from 1-metre springboard is considered a good lead-up for the forward 2½ somersault pike from 3-metre springboard (Smith & Bender, 1973). Since feet-first entry dives are seldom adopted in competition, they are used mainly as lead-ups for head-first entry competitive dives. The usual progression of somersaulting dives within each group is: ½ tuck, ½ pike, 1 tuck, 1 pike, 1½ tuck, 1½ pike, 2 tuck, 2 pike, 2½ tuck, 2½ pike, 3 tuck, 3 pike, 3½ tuck, 3½ pike, 4½ tuck. Each successive change from tuck to pike position for the same somersault rotation and each additional somersault is accompanied by an increase in DD. For example, the increased DD of subsequent dives in the forward group from 1-metre springboard ranges from 0.1 to 0.5 (Table 1.2).

Table 1.2. Degree of Difficulty in the Forward Group from 1-metre Springboard

Dive number	Dive	Flight Position	DD	Increased DD
101C	Forward dive	tuck	1.2	-
101B	Forward dive	pike	1.3	0.1
102C	Forward somersault	tuck	1.4	0.1
102B	Forward somersault	pike	1.5	0.1
103C	Forward 1½ somersault	tuck	1.6	0.1
103B	Forward 1½ somersault	pike	1.7	0.1
104C	Forward double somersault	tuck	2.2	0.5
104B	Forward double somersault	pike	2.3	0.1
105C	Forward 2½ somersault	tuck	2.4	0.1
105B	Forward 2½ somersault	pike	2.6	0.2
106C	Forward triple somersault	tuck	2.9	0.3
107C	Forward 3½ somersault	tuck	3.0	0.1

It is evident that angular momentum at takeoff increases as the number of rotations increases (e.g. Miller, 1970, 1981; Miller & Munro, 1985b). In the forward group, the total body angular momentum increased by a factor of 3.61 times from a forward dive to a forward 2½ somersault in pike position (Hamill, Ricard & Golden, 1986). Understanding the relationship between angular momentum and somersault requirement in different body positions for an individual diver will be useful for skill progression.

Smith and Bender (1975) state that it is more difficult to hold a tight tuck position in the forward somersault than the back and reverse dives. During somersaulting dives, the body is rotating about its mass centre. There is a centripetal force causing each point mass of the body to accelerate towards the mass centre. To hold the same body configuration, the diver needs to produce greater force acting on the hip joint when the angular velocity increases. As the number of somersault increases, the corresponding angular velocity also increases. This means that in addition to increased angular momentum requirement, the diver also needs to generate greater hip torque to perform multiple somersaults.

However, the hip torque associated with different somersault rotations and body positions have not been investigated. Quantification of the centripetal force and the hip torque requirement in different dives could advance the understanding of the demands of the dives. Such knowledge would also provide practical implications for coaches and divers in preparation for learning new dives.

1.1 Statement of Purpose

The purpose of this study was:

(a) to establish the relationship between angular momentum and somersault rotations requirement;

(b) to quantify the centripetal forces acting towards the mass centre and thus the hip torque required in different somersault rotations and/or different body positions; and

(c) to determine the factors contributing to skill progression within an individual diver.

1.2 Research Question

(a) How much extra angular momentum is required to perform a set number of somersaults from a tuck to a pike position?

(b) How much extra angular momentum is required to perform an additional somersault in the same body position?

(c) What is the difference in angular momentum between a set number of somersaults in the pike position and an additional somersault in the tuck position?

(d) What is the hip torque requirement to perform a set number of somersaults in tuck or pike position?

(e) What is the hip torque requirement to perform an additional somersault in the same body position?

(f) What is the difference in hip torque requirement between a set number of somersaults in pike position and an additional somersault in tuck position?

1.3 Hypotheses

To perform an additional somersault in the same body position, the moment of inertia remains unchanged whilst the angular velocity must increase so that more rotation is completed for the same flight time. It was hypothesized that both angular momentum and hip torque requirement would increase in order to perform an additional somersault in the same body position.

For the same amount of rotation, the angular velocity of the body in tuck or pike position should be similar. Since the body is more compact in a tight tuck position, a slightly greater hip torque will be required to perform a set number of somersaults in a tight pike position than in the tuck position. On the other hand, the moment of inertia of the pike position is much larger than that in tuck position. Therefore, it was hypothesized that the contributing factor for successive change of body shape from tuck to pike position would be the increase in angular momentum.

To progress from a somersaulting dive in pike position to an additional somersault in tuck position, the angular velocity will largely increase while the angular momentum may not differ much because the moment of inertia is reduced. Therefore, it was hypothesized that a large increase in hip torque would be required to progress from a somersaulting dive in pike position to an additional somersault in tuck position.

1.4 Delimitations

The following delimitations were considered:

(a) the study only examined seven dives in the forward group from 1-metre springboard;

(b) dives with successive one somersault increment (½, 1½, 2½) in tuck position were selected;

(c) dives with successive half somersault increment (½, 1, 1½, 2) in pike position were selected; and

(d) data were obtained from one adult elite female diver.

1.5 Limitations

The limitations of this study were that:

(a) the anthropometric measurements were not directly taken from the subject to estimate body segmental inertial parameters;

(b) slight panning and tilting of the camera during recording were not adjusted; and

(c) results could only be generalized to adult elite female divers but not other population like male divers or junior divers; and

(d) results could only be generalized to dives in the forward group from 1-metre springboard.

1.6 Assumptions

It had been assumed that:

(a) a single subject design would be a good starting place to test the hypotheses;

(b) the performance of the subject was reasonably representative of others;

(c) the performance occurred in a vertical plane and that symmetrical configuration was maintained about that plane;

(d) the segmental inertial parameters were a good estimation for the subject; and

(e) the method employed was valid to test the hypotheses.

1.7 Definition of terms

The tuck position is a position that the whole body is pulled up as compactly as possible with bent at hips and knees (ASA, 1963). A good tuck position is characterized by maximum hip flexion, knee flexion, plantar flexion and shoulder protraction (Figure 1.1).

Figure 1.1. Tuck position. (adapted from Xu & Zhang [1996]. Basic Diving Coaching Manual (p. 60). FINA. Lausanne, Switzerland)

The pike position is a position that the body bends at the hips but the legs are kept straight at the knees (ASA, 1963). The pike position can be further divided into the pike dive position (Figure 1.2a), the open pike position (Figure 1.2b) and the closed pike position (Figure 1.2c), each with variation in the degree of hip flexion and positioning of the arms (Xu & Zhang, 1996).

A somersault is a 360° rotation about a transverse axis passing through the mass centre of the body.

The trunk segment represents the head, trunk and both arms of the body.

The legs segment represents the lower body from the hip joint to the toes of both legs.

1.8 Chapter Organization

Chapter 2 reviews the related literature including the mechanics of diving and the techniques of investigation. Particular attention is given to the takeoff and somersaulting dives.

Chapter 3 includes details of methodology and calculation of angular momentum, centripetal force and hip torque. Methods of normalization and evaluation of results are also presented.

Chapter 4 displays the results obtained in this study. The relationship between somersault rotation and angular momentum, centripetal force and hip torque are highlighted. Flight time characteristics and hip angles in different period are also compared.

Chapter 5 discusses the theoretical and practical implications based on the results obtained, including a summary of the main findings of this study. The limitations are discussed along with suggestions for future research.

chapter 2

Review of Literature

This chapter contains a review of related literature. The first part deals with the mechanics of diving. Particular attention is given to the takeoff and somersaulting dives. The second part is concerned with the technique of investigation including segmental inertial parameters, film analysis and data processing.

2.1 Mechanics of Diving

The three main objectives in springboard diving are to: 1) generate sufficient angular momentum to execute required somersaults and twists; 2) obtain adequate height and thus have enough time in the air to complete the dive; and 3) travel safely away from the springboard (Miller & Munro, 1985). The angular momentum required for rotation, the height obtained during flight, and the horizontal distance travelled are all determined during the takeoff (O’Brien, 1992). Therefore, biomechanical studies of diving tend to focus on the characteristics of takeoff.

2.1.1 Takeoff

Descriptive research on the takeoff of elite divers has provided some information on the technique. Miller (1981) calculated the segmental local and remote contributions to the total body angular momentum, the vertical and horizontal velocity, and the springboard reaction force during takeoff. The author later compared the characteristics of the final approach step, the hurdle and the takeoff between male and female divers (1984). She found that men had longer flight time in the hurdle and higher vertical velocity at takeoff than their female counterparts. A study by Hamill, Ricard and Golden (1986) reported that the angular momentum generated at takeoff increased as a function of increasing number of rotation in non-twisting dives. It was found that the remote angular momentum due to the arms contributed from 17% to 74% of the total angular momentum. Hamill et al. (1986) concluded that the arms play a significant role in generating angular momentum. However, such interpretation only compares the angular momentum at takeoff with or without arms. There is no measure of how the relative arm movement to the body contributes to generating angular momentum.

Several studies have discussed the takeoff technique of the Olympic champion Greg Louganis because of his acknowledged superiority. Miller and Munro (1985a, b) analyzed Louganis’s springboard takeoff in terms of duration, shoulder, hip, knee and ankle flexion, linear and angular momentum during the depression and the recoil phase. Data obtained were then compared with eight elite female divers. The authors believed that such descriptive data could be used as a standard reference for coaches and divers. Jiang, Shu and Li (2000) developed a mathematical model based on Louganis’s takeoff. They recommended that the calculated standard function could be applied in training by comparing individual diver’s film data with Louganis’s. Although data from elite performers can provide invaluable information and insight into an individual’s optimum technique (Greig & Yeadon, 2000), it does not seem appropriate to use them as a standard reference. Valliere (1976) warned that “we copy the techniques of champions without worrying what are the essential and the nonessential components during the execution; we interpret incorrectly certain data and we even go as far as the invention of movements that could never be performed.” Therefore, caution should be taken into account individual differences when analysing a technique.

The timing and pattern of the armswing during the touchdown in the forward approach have also received attention. Results of a kinematic study indicate that the upward acceleration of the diver relative to the springboard contributes to downward deflection of the board (Miller & Munro, 1984). It was found that this upward acceleration is attributed largely to the action of the lower extremities whilst the force contributed by the upper extremities assists in pushing the board down. The authors suggested an optimum timing for the armswing at which the start of positive relative arm acceleration should coincide with the touchdown. Based on these findings, several studies followed.

Paquette (1984) observed the angle between the upper arm and a line projected vertically downward from the shoulder joint in five club divers. He found that for maximum height and thus maximum time in the air, the shoulder angle should be within the range of 73.5º and 89.6º when the diver first contacts the board. Some practical implications for coaches were given in a later study by Sprigings, Paquette and Watson (1987). There were also simple mathematical models developed for optimizing the timing of the armswing (Sprigings & Watson, 1985) and the relative force patterns of the arms, torso, and legs during the takeoff (Sprigings, Watson, Mabeganu & Derby, 1986).

Sanders and Wilson (1988) quantified the factors that affect the height achieved in forward and reverse dives and described the influence of these factors in terms of the storage and utilization of springboard strain energy. Instead of observing the upper extremities, Sanders and Wilson were interested in the flexion and extension of the lower extremities. They concluded that divers who achieved good height were characterized by a large vertical velocity at touchdown from the hurdle, and a minimization of hip flexion (forward dives) or knee flexion (reverse dives) at takeoff.

Jiang, Li, Shu and Zhang (2000) developed a mathematical model of the human body and the springboard to investigate factors that contribute to maximum height at takeoff. They suggested that a diver should jump as high as possible in the hurdle and press the board as quickly as possible during the takeoff, choosing the stiffest board setting within his ability. Their simulation results, however, were based on hypothetical data which might not represent actual performance. While obtaining good height from the hurdle and applying a strong push in the depression phase are in agreement with real performance in other studies (Miller, 1984; Sanders & Wilson, 1988), choosing the stiffest board setting does not seem plausible. Sanders and Wilson (1998) state that ‘theoretically it is desirable to maximize the magnitude and duration of the positive relative acceleration during the depression of the board.’ Using a stiffer springboard (or adjusting the fulcrum position) would in fact decrease the duration of depression.

Since Miller and Munro’s (1984) finding, it was widely accepted that armswing at optimal timing assists downward push despite the fact that the mechanisms involved are not totally clear. In vertical jump, Dapena (1999) pointed out that upward acceleration of the arms would simply produce a downward acceleration of the trunk and legs, but no change in the downward force on the ground if the legs were already executing maximal voluntary contraction. This is also true when applied to the springboard takeoff. An explanation of the arm effect by Dapena is that the downward reaction force exerted by the arms on the trunk slows down the vertical linear velocity of the hip joint. This results in a slower concentric condition which enables the leg extensor muscles to produce a greater force.

There were also a few studies concerning the takeoff angle in relation to the amount of rotation. Stroup and Bushnell (1969) measured the takeoff angle from vertical to a line drawn from the toes through the mass centre. They investigated this takeoff angle in terms of partition of total energy between translation and rotation. It was found that the partitioning and the resulting amount of rotation are dependent on the magnitude and direction of the takeoff angle. Golden (1981) found that at maximum springboard depression, body lean characterized by forward displacement of mass centre increased according to the number of rotations performed. In addition, approximately an increment of 20º in hip flexion and between 20º and 25º in shoulder extension at takeoff would be required for each additional pike somersault.

While all of the above studies used film data for kinetic and kinematic analysis, there were a few studies employing the force plate together with cinematography. Bergmaier, Wettstein and Wartenweiler (1971) pioneered the direct measurement of force exerted on the springboard by attaching a force plate onto the springboard. Miller, Hennig, Pizzimenti, Jones and Nelson (1989, 1990) installed a force plate on a 10-m platform to measure the takeoff reaction force in the backward and reverse group in a competition setting. This kind of data collection is not common because of the difficulties in mounting the force plate onto the takeoff surface. Advance in equipment design and calibration techniques may overcome these problems.

2.1.2 Airborne movement

Basically, diving movements can be classified into somersaulting dives and twisting dives (O’Brien, 1992). Somersaulting dives involve rotation of the body about the transverse axis passing through the mass centre. Twisting dives require rotations about the transverse axis and the longitudinal axis at the same time. The angular momentum which determines the amount of somersault and twist in the air is determined at the instant of takeoff (Eaves, 1969; Miller, 1970; O’Brien, 1992; Rackham, 1969, 1975; ASA, 1963). Throughout the airborne phase, air resistance is negligible such that the only force acting on the body is the weight which passes through the mass centre. Therefore, by the rotational equivalent of Newton’s First Law of Motion, angular momentum of the body remains constant once the diver takes off from the springboard (Bartlett, 1997b; Hay, 1993; Sarsfield, 1960; ASA, 1963).

Twisting Dives

Considerable studies had focused on the twisting dives from qualitative observation (Barrow, 1959; Narcy, 1966; Hennessy, 1993; Reeves & Petersen, 1992; Ball, 1993) to quantitative description (Van Gheluwe & Duquet, 1977; Sanders, 1995, 1999). Some researchers conducted experimental studies (Bartee & Dowell, 1982) while others tried to explain the motion with application of laws of mechanics (Batterman, 1974; Frohlich, 1979, 1980). The mechanics of twisting somersaults are not well understood until recent advancement in knowledge gained through mathematical modeling and simulation (Van Gheluwe, 1981; Yeadon, 1984, 1987, 1990a,b,c, 1993; Yeadon, Atha & Hales, 1990). A detailed discussion of twisting dives is not considered since it is beyond the scope of the present study.

Somersaulting Dives

In this section, the angular momentum of different dives and the hip torque generated by muscle contraction are discussed. Special emphasis is placed on the angular momentum for different rotation requirements in the forward group.

Angular Momentum. In somersaulting dives, the body is rotating about the transverse axis passing through the mass centre. The total angular momentum of the body is conserved during the flight time, that is, from the instant when the body loses contact with the springboard until the hands break the water.

Rackham (1975) states that if a diver jumps off the board with no angular momentum, ‘no action on the part of the diver during a jump can cause him to rotate and enter the water head-first’ (p. 178). His misinterpretation of conservation of angular momentum was corrected by Frohlich (1979) who demonstrated a 90º rotation after the body was in the air on a trampoline. It is in fact possible to generate rotation without any change in total body angular momentum (Frohlich, 1979, 1980). The principle of transfer of angular momentum within body segments was later explained by Hay (1993) and Bartlett (1997b) in a clearer way.

Angular momentum is the product of the body’s moment of inertia about the axis of rotation and its angular velocity. During the somersault, the angular momentum of the body can be calculated by Equation (2.1):

L = Iw (2.1)

where L = angular momentum about the mass centre,

I = moment of inertia about the transverse axis,

w = angular velocity about the transverse axis.

Since L remains constant during the flight, any increase in I must be compensated by a proportionate decrease in w, and vice versa.

Rackham (1975) estimated that the moment of inertia of the body in tuck position (I_t) and closed pike position (I_p) are approximately one quarter and one half of that in straight position (I_s) respectively, that is:

I_{t :} I_{p :} I_s = 1 : 2 : 4

This results in a reverse relationship of the angular velocities:

w_{t :} w_{p :} w_s = 4 : 2 : 1

where t = tuck, p = pike, and s = straight position.

A diver can control the amount of rotation by altering the body shape (Frohlich, 1980). Therefore, the timing and technique of coming out from tuck and pike somersaults is critical for a good entry (O’Brien, 1992; Rackham, 1969; Sarsfield, 1960; Xu & Zhang, 1996). For somersaults in the straight position, this can only be done by back and/or arm movement (ASA, 1963; Rackham, 1975).

Some studies had quantified the angular momentum of the body during somersaulting dives. Miller (1970) developed a four-segment simulation model to investigate the mechanics of airborne phase in diving. She calculated the angular momentum when the diver was in a quasi-rigid state with all his segments moving at the same angular velocity. Values ranged from 24.4 - 28.2 kgm²s^-1 in a forward dive pike, 27.1 – 30.5 kgm²s^-1 in a forward dive straight, 94.1 kgm²s^-1 in a forward 1½ somersault pike and 106.2 kgm²s^-1 in forward 2½ somersault pike.

Miller (1981) later used a linked segment method to calculate the angular momentum at last contact with the springboard of a male diver performing a forward 3½ somersault in the tuck position. The angular momentum for this diver was found to be 39.3 kgm²s^-1. Using the same method, the average angular momentum of Greg Louganis’s forward dive straight and the forward 3½ somersault pike were calculated as 17.5 kgm²s^-1 and 70.2 kgm²s^-1 respectively (Miller & Munro, 1985b).

Hamill, Ricard and Golden (1986) reported that total body angular momentum increased as a function of rotational requirement. In the forward group, the total body angular momentum increased by a factor of 3.61 times from ½ to 2½ rotations in pike position. The average angular momentum of three divers in the forward ½, 1½ and 2½ somersaults were 20.1 kgm²s^-1, 50.9 kgm²s^-1 and 72.4 kgm²s^-1 respectively.

Sanders and Wilson (1987) quantified the angular momentum requirements of the forward 1½ somersault dive with and without twist by the quasi-rigid method. The angular momentum of the forward 1½ somersault in the pike position reported in their study were 42 kgm²s^-1, 45 kgm²s^-1 and 58 kgm²s^-1 for three divers of body weight 662N, 600N and 756N respectively.

In summary, it is evident that the angular momentum increases as the number of rotation increases. However, since the moment of inertia depends on the body weight and mass distribution, data comparison across subjects and/or studies is not meaningful unless there is proper normalization of data. Webb (1997) divided the absolute angular momentum by the moment of inertia of the body in a straight position and 2p (Equation 2.2). The resulting value gave an indication of how many rotations per second the body would have completed in the standardized straight position.

H _S/S = (2.2)

where H _S/S = number of straight somersaults per second

H = absolute angular momentum

Is = moment of inertia in straight position

This method of normalization takes into account the differences in moment of inertia between subjects.

Hip Torque. A diver can change body position by muscular contraction forcing the body into a shorter or longer radius resulting in faster or slower rate of rotation (Harper, 1966). In somersaulting dives, the diver pikes or tucks up after takeoff for somersault rotation. In the forward and the inward group, this is often known as the ‘head chasing toes’ effect in diving terminology (Moriarty, 1961; Smith & Bender, 1973). The work done by the muscles about the hip joint to change the body shape from straight to tuck or pike position increases the rotational kinetic energy of the body (Equation 2.3).

KE = Iw² (2.3)

Rackham (1975) discussed the muscle work in the pike jump and the inverted pike. He states that when the pike position is assumed in the air, little strength is required to raise the legs and none at all to maintain the pike position. This is because there is only the force of gravity acting on the body whilst it is in free fall. He demonstrated this theory by dropping two loosely pivoted sticks to the ground. The sticks were held in a position similar to the pike position before being dropped with a heavy weight attached to the free end of one stick. It was found that there was no change in the relative position of the two sticks throughout the flight.

While Rackham’s theory is clearly correct when the body is free falling from a specific body position, this does not apply to the spinning body during somersault. Harper (1966) defined the force related to the radius and speed of rotation as “centrifugal force”, which was the outward force from the center of rotation caused by a rotating body. In-depth discussion on this force was not followed however.

During somersaulting dives, the body is rotating about its mass centre. There is a centripetal force causing each point mass of the body to accelerate towards the mass centre. In an open pike position in which the arms are abducted laterally, this force is produced by the hip flexors and the abdominal muscles; whilst in the closed pike and the tuck position, this force is also provided by the pulling action of arms. The faster the rotation, the greater the centripetal force. This hypothesis is consistent with Smith and Bender (1975) who state that it is more difficult to hold a tight tuck position in the forward somersault than the back and reverse dives. To hold the same body configuration, the diver needs to produce greater force acting on the hip joint when the angular velocity increases.

The centripetal force, thus the hip torque, produced in different somersault rotations and body positions have not been investigated however. Quantification of this force and hip torque requirement can provide knowledge and understanding of the demands of the dives.

2.2 Techniques of Investigation

2.2.1 Determination of segmental inertial parameters

Biomechanical movement analysis requires body inertial parameters including the mass, position vector of the centre of mass, and the moment of inertia of each segment. These parameters are commonly determined by experimental methods, regression equations, and geometrical modelling.

The disadvantages of experimental methods such as the quick release method, the oscillation method and the pendulum method are that they are inaccurate, impractical, time-consuming and that only partial data sets can be obtained. Attempts had been made to obtain estimates of such segmental inertial parameters from anthropometric measurements of an individual by using linear regression equations based upon cadaver data (Barter, 1957; Clauser, McConville & Young, 1969; Hinrichs, 1985). Yeadon and Morlock (1989) found that non-linear equations were superior to linear equations and that non-linear equations could provide reasonable estimates of segmental moments of inertia even when the anthropometric measurements lie outside the sample range.

Ideally, these segmental inertial parameters should be personalized to the subject being analyzed (Bartlett, 1997a). This can be achieved by geometric modelling. An early example was Hanavan’s (1964) model which had been criticized because of its inappropriate geometric solids. Other models require more measurements but give more accurate results. For example, Jensen’s (1976, 1978) model requires the digitization of photographs of the subject. Hatze’s (1980) model involves 242 anthropometric measurements taken directly from the subject which can be taken in 80 minutes, whilst Yeadon’s (1990b) requires 95 measurements taking between 20 to 30 minutes. It should be noted that all these models make simplifying assumptions such as uniform density over a given cross-section. Systematic errors may be expected and there are no criterion values of the segmental inertia parameters with which to compare the calculated values (Yeadon, 1990b).

The most recent developments in medical imaging techniques or mass scanning techniques are still facing practical problems. Kwon (2000) summarized some recent findings on the effects of the methods of body segmental parameters on the experimental simulation of complex human airborne movements. The applicability of selected estimation methods in simulation of these movements was also discussed.

2.2.2 Image Analysis

Cine cameras have long been used for movement analysis. With the introduction of video cameras, cine cameras are still preferred because of their excellent picture quality. Until recent improvement in video recording technology, video cameras have become more widely accepted for image analysis (Bartlett, 1997b). For two-dimensional (2D) analysis, the motion of the body landmarks of interest are assumed to occur in a single plane, although sporting activities are not general planar (Bartlett, 1997a, b). Three-dimensional (3D) analysis that requires at least two cameras gives a better representation of the movements studied.

2.2.3 Determination of spatial coordinates

Several different methods have been used to transform digitized image coordinates into object spatial coordinates. The most common reconstruction method with fixed camera orientation is the Direct Linear Transformation (DLT) or modifications thereof (Abdel-Aziz & Karara, 1970; Hatze, 1988). The DLT requires at least six control points for calibration of each camera. Once the 11 DLT parameters of each camera are known, and the associated image coordinates are obtained, the 2D/3D spatial coordinates can be computed.

The DLT method has some shortcomings when used to analyze events that take place over large areas. To solve the problems, techniques of using panning and/or tilting cameras have been developed (e.g. Dapena, 1978; Yeadon, 1989b; Yu, Koh & Hay, 1993). Still, problems of relative camera positioning and computation time limitation for the existing 3D reconstruction technique for moving cameras arise (Allard, Stokes & Blanchi, 1995). Since panning techniques are essential for the analysis of many sports, Yeadon and Challis (1994) recommended that further effort should be invested to develop simple, accurate and reliable techniques for the measurement of body positions using panning cameras.

2.2.4 Synchronization of cameras

Synchronized data sets can be obtained by genlocking video or phase-locking cine cameras. Such synchronization needs a physical connection between cameras, which is not always possible, especially in competition settings. This problem can be circumvented by having a timing device in the field of view of all cameras and then obtaining synchronized data sets by using an interpolating procedure (Bartlett, 1997a). In cases where this is not possible, the digitized data can be used to determine the time of a particular field in the time scale of the other camera (Yeadon, 1989). More recently, a general method for synchronizing digitized video data using a mathematical approach based upon the DLT technique has been developed (Yeadon & King, 1999).

2.2.5 Data smoothing

The sampled signal can be considered to be the sum of the true signal, systematic noise and random noise (Yeadon & Challis, 1994). The systematic noise including lens distortion, incorrect marker placement, calibration errors, and skin marker movement should be identified and minimized. The random noise can be reduced by a number of techniques, for example, Butterworth filters, truncated Fourier series (Hatze, 1981), quintic splines (Wood & Jennings, 1979) and generalized cross-validated splines (Woltring, 1985). Each of these techniques uses a mathematical function to approximate the displacement data. When selecting a noise removal technique, consideration must be given to the nature of the technique and how the degree of smoothing is selected (Yeadon & Challis, 1994). Challis and Kerwin (1988) compared a variety of techniques and found that quintic splines produced more accurate first and second derivatives than truncated Fourier series and the Butterworth filter.

2.3 Summary

Many studies had examined the takeoff techniques and twisting and somersaulting characteristics in diving. The angular momentum and takeoff angle for different amounts of rotation in somersaulting dives have also been investigated. However, little was known about the hip torque required in somersaulting dives with different rotations and body positions.

CHAPTER 3

METHODOLOGY

3.1 Data Collection

Seven dives of one female diver from 1-metre springboard in the video ''Diving My Way'' (The Athletic Institute, 1990) were chosen for this study. The subject was an Olympic bronze medallist, World champion and Pan American champion. The seven selected dives are listed in Table 3.1.

Table 3.1. Seven Selected Dives from 1-metre Springboard

Dive number	Dive	Flight Position
101C	Forward dive	tuck
103C	Forward 1½ somersault	tuck
105C	Forward 2½ somersault	tuck
101B	Forward dive	pike
102B	Forward somersault	pike
103B	Forward 1½ somersault	pike
104B	Forward double somersault	pike

3.2 Digitization

The videotape was time-coded for use with the Target video digitizing system (Kerwin, 1995). Eight body landmarks were digitized in each field of the movement phase. These eight body landmarks were toes, ankle, knee, hip, shoulder, elbow and wrist of the left side and the centre of the head. The movement phase began with the field in which the foot first lost contact with the springboard, and ended when the hands broke the water. 2D analysis was considered adequate for the present study, though 3D analysis would be more desirable.

3.3 Body inertial parameters

Since anthropometric measurements could not be taken directly from the subject, segmental inertial parameters were estimated using the measurements from four female divers (Yeadon, 1989b). The anthropometic measurements were entered into the inertia model developed by Yeadon (1990). Body mass was adjusted such that the total body density was 1.00kgm^-3. Four different sets of segmental inertial parameters were obtained and used for further calculation. Using four sets of inertial parameters could demonstrate how body weight affects the angular momentum and hip torque requirement of the same dive.

3.4 Calculation

From the digitized coordinates and the body inertial parameters, a Fortran 77 program was written to calculate the angular momentum, centripetal force and associate hip torque in each dive. The program and a sample of results were listed in Appendix A and Appendix B.

3.4.1 Angular velocity

For each dive, the somersault angle and hip angle in each field were calculated. The somersault angle was determined by the angular rotation of a line joining the shoulder and the hip. The hip angle was defined as the shoulder-hip-knee angle. From the time history of hip angle, the time period of a tight pike or tuck position was determined. The average angular velocity during this period was calculated by Equation 3.1. Since average values rather than derivatives were taken, data smoothing was considered not necessary for this study.

(3.1)

where w = angular velocity

f = somersault angle

t = time

3.4.2 Moment of inertia

It was assumed that the body maintains a fixed position during the somersaulting phase. This rigid system can be represented by a two-segment model shown in Figure 3.1. The trunk segment includes the head, trunk, and both arms. The leg segment includes the thighs, shanks and feet of both legs. The following body inertial parameters can be obtained using the digitized data and the inertial parameters:

I_t = moment of inertia of the trunk

I_l = moment of inertia of the legs

m_t = mass of the trunk

m_l = mass of the legs

t = distance between the mass centre of the trunk and the hip joint

l = distance between the mass centre of the legs and the hip joint

where H = hip joint centre

G = mass centre of the whole body

G_t = mass centre of the trunk

G_l = mass centre of the legs

d = distance between G_t and G_l

d_t = distance between the G_t and G

d_l = distance between the G_l and G

a = angle between the trunk and the legs at the hip joint

a can be determined from digitized data and segmental inertial parameters. The distances d, d_t and d_l can be calculated as follows:

m_td_t = m_ld_l

d_t = d_l

Since d = d_t + d_l

= d_t + d_t

= d_t (1 + )

= d_t ()

By the Cosine Law, d =

Therefore d_t = () (3.2)

Similarly, d_l = () (3.3)

Knowing the mass (m), the moment of inertia of individual segment (I_G), the distance between the segmental mass centre and the whole body mass centre (r), the moment of inertia of the whole body (I) can be determined by using the Parallel Axis Theorem (Equation 3.4).

I = S (I_G + mr²)

= (I_t + m_td_t²) + (I_l + m_ld_l²)

= I_t + I_l + () +

()

= I_t + I_l + ()

I = I_t + I_l + () (3.4)

3.4.3 Angular momentum

Angular momentum is defined as the product of moment of inertia and angular velocity. Given that the moment of inertia of the body and the average angular velocity during a tight tuck or pike position are known, the angular momentum of the body can then be calculated by Equation 2.1.

3.4.4 Centripetal force

During the somersault rotation, every point mass of the body is accelerating towards the mass centre. This centripetal acceleration is provided by a centripetal force pointing towards the mass centre. Figure 3.2 shows the centripetal force at G_t and G_l where F = centripetal force

b = ÐG_tG_lH

The corresponding centripetal force towards the mass centre of the body can be calculated by

F_t = m_td_tw²

F_l = m_ld_lw²

Since m_td_t = m_ld_l, the two centripetal forces are equal in magnitude but opposite in direction. For simplicity, this force is calculated by using data from the legs segment only.

F = m_ld_lw² (3.5)

3.4.5 Hip torque

For 2D motion, the torque about a point is defined as the force multiplied by the perpendicular distance. During somersault, the torque about the hip at G_t and G_l are equal in magnitude and opposite in direction. After sinb is solved by Sine Law (Equation 3.6),

= (3.6)

the torque about the hip joint (T) can be determined (Equation 3.7).

T = F • l sinb (3.7)

3.5 Data Analysis

3.5.1 Normalization

Four sets of results were obtained by using the four different inertial data sets. The torque values for each dive were normalized by body mass and leg length (Equation 3.8). Leg length was defined as the length from hip to ankle.

(3.8)

where T₁ = hip torque in subject 1

T₁ⁿ = normalized hip torque in subject 1

m_av = average body mass of four subjects

m₁ = body mass of subject 1

z_av = average leg length of four subjects

z₁ = leg length of subject 1

For each dive, and average normalized torque value can be calculated (Equation 3.9).

(3.9)

From this average value, the torque required for any diver can be estimated once the body mass and leg length of the diver are known (Equation 3.10).

(3.10)

where T_s = hip torque in subject S

m_s = average body mass of subject S

z_s = leg length of subject S

Similarly, the values of centripetal force and angular momentum were normalized in the same way (Equation 3.11 – 3.16). The standard deviations of the average normalized angular momentum, centripetal force and torque value were also calculated.

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

where L₁ = angular momentum of subject 1

L₁ⁿ = normalized angular momentum of subject 1

L_av = average normalized angular momentum of four subjects

L_s = angular momentum of subject S

F₁ = centripetal force in subject 1

F₁ⁿ = normalized centripetal force in subject 1

F_av = average normalized centripetal force in four subjects

F_s = centripetal force in subject S

3.5.2 Evaluation on angular momentum

The angular momentum during the first 5 frames and the last 5 frames in each dive was calculated to evaluate the value obtained during a tight pike or tuck position. The average value of the five frames was used. For comparison, the angular momentum was also calculated using three other body segments to determine the somersault angle (Figure 3.3). The four segments were hip to ankle (r₁), shoulder to hip (r₂), shoulder to knee (r₃), and hip to knee (r₄).

3.5.3 Estimation of angular momentum

Two methods were used to estimate the angular momentum required for dives with further somersault rotations, namely forward 2½ somersault pike, forward 3½ somersault tuck and forward 3½ somersault pike.

Method 1: Linear Regression

Linear regression equation was used to estimate the angular momentum required for dives with further somersault rotations in tuck and pike position respectively. The data of the forward somersault pike was discarded due to its unusual short flight time and large angular momentum.

Method 2: Calculation in Standard Straight Position

The values of angular momentum were further normalized based on Webb’s (1997) method. The average normalized values can be multiplied by the flight time and then divided by the moment of inertia in standard straight position and 2p (Equation 3.17). The resulting SS value gives an indication of how many somersaults can be performed in a standard straight position. The standard straight position is defined as the body position of no hip flexion or hyperextension, and with straight legs together and straight arms down on each side. Inertial parameters of Subject 1 were used for the present study. This method of normalization allows comparison across subjects and studies, and also takes into account the different flight time in each dive.

(3.17)

where SS = number of somersaults in standard straight position

I_SS = moment of inertia in standard straight position

The moment of inertia of the pike position in a forward 1½ somersault pike and that of the tuck position in a forward 1½ somersault tuck of Subject 1 were taken as the standard pike and tuck position. These values were used together with the calculated I_SS, SS and Lⁿ values to compute the percentage flight time in tight body position. The calculating procedures for dives in tuck position are as follows:

f =

=

n_T =

n_T = SS(1-t) + SSt (3.18)

where f = angular rotation

w_SS = angular velocity in standard straight position

w_T = angular velocity in standard tuck position

n_T = number of somersaults in standard tuck position

I_T = moment of inertia in standard tuck position

t = percentage flight time in tight position

Similarly, the percentage flight time for dives in pike position can be given by:

n_P = SS(1-t) + SSt (3.19)

where n_P = number of somersaults in standard pike position

I_P = moment of inertia in standard pike position

The calculated percentage time in tight position in each dive was compared with the actual percentage time obtained from film data. To estimate for dives with further somersault rotations, it was assumed a flight time of 1.30s during which a diver hold in a tight position in no more than 70% of time. This assumption ensures enough time to come-out for a good entry. The angular momentum required for a forward 2½ somersault pike, forward 3½ somersault tuck and forward 3½ somersault pike were first estimated in number of somersaults in standard straight position, and then de-normalized into average Lⁿ values.

3.5.4 Estimation of hip torque

Linear regression equation was used to estimate the hip torque required for a forward 2½ somersault pike, forward 3½ somersault tuck and forward 3½ somersault pike. Again, data of the forward somersault pike was discarded due to its unusual short flight time and large angular momentum.

CHAPTER 4

RESULTS

4.1 Physical characteristics

Anthropometric measurements of four female divers (Yeadon, 1989b) were used to calculate body segmental inertial parameters. Table 4.1 shows the estimated body mass and leg length of Subject 1 to 4. The average body mass and the leg length are 59.76kg and 0.783m respectively.

Table 4.1 Estimated Body Mass and Leg Length

Body parameters	S1	S2	S3	S4	Mean ± SD
Mass (kg)	56.74	57.60	62.60	62.08	59.76 ± 3.01
Leg length (m)	0.813	0.801	0.742	0.776	0.783 ± 0.032

4.2 Flight time and hip angle

Flight time (t), time period in tight pike or tuck position (t-tight), and percentage time of tight period (%t) for each dives are displayed in Table 4.2. Average hip angle throughout flight (AvH), average hip angle (H) and the average angle between the trunk and the leg segment (a) during the tight body position are also compared.

4.2.1 Dives in tuck position

For dives in the tuck position, the diver jumped higher when the somersault requirement increased as reflected by the flight time. Also, the more the rotation, the longer she maintained a tight tuck position. The flight time for ½ to 2½ rotations ranges from 1.08s to 1.29s, during which 11% to 68% was hold in a tight position. The gradual decrease in AvH value from ½ to 2½ rotations implies an increasing degree of hip flexion on average.

Table 4.2 Flight time and Hip angle

Dive	t(s)	t-tight(s)	%t	AvH	H	a
101C	1.08	0.12	11%	117.36°	39.14°	48.91°
103C	1.16	0.36	31%	71.55°	39.93°	42.20°
105C (1^st)	1.29	0.88	68%	54.11°	47.88°	56.78°
(2^nd)					36.18°	44.06°
101B	1.10	0.06	5%	118.12°	46.06°	39.39°
102B	1.04	0.16	15%	108.08°	59.87°	42.70°
103B	1.32	0.55	42%	70.52°	32.36°	20.96°
104B (1^st)	1.10	0.67	61%	57.99°	48.49°	41.49°
(2^nd)					35.44°	29.99°

Note. 1^st and 2^nd refer to the first and second somersault of the dive.

Apparently, the diver demonstrated a similar degree of hip flexion in the forward dive tuck (39.14°) and the forward 1½ somersault tuck (39.93°). However, the computed a value in the forward dive (48.91°) is greater than that in the forward 1½ somersault (42.20°), indicating a lesser degree of knee flexion in the latter. This means that the diver has adopted a less compact tuck position in the forward 1½ somersault tuck. Futher, it is observed from both the H and the a value that the diver adopted a tighter position during the second somersault than the first somersault in the forward 2½ somersault tuck.

4.2.2 Dives in pike position

The flight time of dives in the pike position varies in two folds. Firstly, the head-entry dives (101B and 103B) have longer flight time than the feet-first entry dives (102B and 104B) on average. Secondly, the flight time increases when somersault rotation increases, as seen in the case of dives in tuck position. However, the forward 1½ somersault pike has the longest flight time of 1.32s, which is even longer that that of the forward 2½ somersault tuck (1.29s). On the other hand, the forward somersault pike has the shortest flight time of 1.04s.

Similar to the dives in tuck position, the percentage time in tight pike position increases whilst the AvH value decreases from the forward dive pike to the double somersaults pike. The forward somersault pike demonstrates an exceptionally large AvH (108.8°), H (59.87°) and a (42.70°) value. The large H and a value indicates that the diver did not adopt a very tight pike position. The large AvH value shows that there is not much hip flexion throughout the flight on average. This suggests that the diver came out from the lose pike position very early and that she adopted a rather straight body position during most of the flight time.

4.3 Angular momentum, centripetal force and hip torque

Table 4.3 presents the normalized values of angular momentum, centripetal force and hip torque. In the forward 2½ somersault tuck and forward double somersault pike, the subject held a tighter position during the second somersault. Therefore, the angular velocity, and thus the centripetal force and hip torque were calculated separately for the first and the second somersault.

Table 4.3 Angular Momentum, Centripetal Force and Hip Torque

Dive	Parameter	S1	S2	S3	S4	Mean ± SD
101C	L^n (kgm^2s^-1)	13.69	15.33	19.01	14.50	15.63 ± 2.35
	F^n (N)	80.28	86.52	99.42	86.26	88.12 ± 8.07
	T^n (Nm)	16.89	19.77	24.92	18.92	20.13 ± 3.42
103C	L^n (kgm^2s^-1)	31.49	35.11	43.69	33.18	35.87 ± 5.42
	F^n (N)	426.71	457.87	526.29	456.65	466.88 ± 42.15
	T^n (Nm)	94.92	110.78	139.72	105.86	112.82 ± 19.12
105C	L^n (kgm^2s^-1)	37.14	41.29	51.66	40.70	42.71 ± 5.79
	F^n (N) -1^st	556.81	598.27	688.92	595.21	609.80 ± 56.02
	- 2^nd	624.80	631.41	771.45	671.58	674.81 ± 67.66
	T^n (Nm) -1^st	121.28	140.29	177.27	133.81	143.16 ± 24.07
	- 2^nd	135.04	157.52	198.61	150.89	160.51 ± 27.09
101B	L^n (kgm^2s^-1)	20.95	23.48	29.88	22.23	24.13 ± 3.97
	F^n (N)	77.38	82.84	95.78	82.35	84.59 ± 7.86
	T^n (Nm)	26.27	30.82	38.82	29.44	31.34 ± 5.34
102B	L^n (kgm^2s^-1)	38.89	44.01	55.43	42.23	45.14 ± 7.18
	F^n (N)	283.39	312.91	360.79	316.40	318.38 ± 31.92
	T^n (Nm)	94.53	112.58	141.27	109.49	114.47 ± 19.53
103B	L^n (kgm^2s^-1)	29.18	32.57	41.09	30.94	33.45 ± 5.28
	F^n (N)	122.14	133.38	154.86	131.36	135.44 ± 13.84
	T^n (Nm)	41.86	49.00	61.85	46.56	49.82 ± 8.55
104B	L^n (kgm^2s^-1)	45.26	50.74	63.80	48.16	51.99 ± 8.19
	F^n (N) -(1^st)	435.24	472.68	547.47	467.45	480.71 ± 47.49
	-(2^nd)	372.8	406.56	470.79	402.20	413.09 ± 41.29
	T^n (Nm)-(1^st)	136.67	159.77	201.51	152.00	162.49 ± 27.73
	-(2^nd)	130.19	152.53	192.27	145.14	155.03 ± 26.51

Note. Lⁿ, Fⁿ and Tⁿ denotes normalized angular momentum, centripetal force and hip torque. 1^st and 2^nd refer to the first and second somersault of the dive.

In the forward 2½ somersault tuck, the tighter body configuration in the second somersault leads to an increased centripetal force and hip torque. In contrast, the tighter pike in the second somersault the forward double somersault pike results in decreased centripetal force and hip torque. This is due to the reduced radius from the segmental mass centre to the whole body mass centre.

For dives in tuck position, the angular momentum, centripetal force and hip torque increase gradually from the forward dive to the forward 2½ somersault. A similar trend is observed in dives in pike position with an exception of the forward somersault pike which demonstrates an extraordinary high value in all three aspects. Figure 4.1 to Figure 4.3 illustrate a linear relationship between somersault rotation and the corresponding angular momentum, centripetal force and hip torque respectively. Average angular momentum and the higher centripetal force and hip torque values are plotted for the forward 2½ somersault tuck and forward double somersault pike.

For the same somersault rotation, it is expected that dives in the pike position require more angular momentum than dives in tuck position. This is true in the case of the forward dive of which the angular momentum for the pike and tuck position is 24.13 kgm²s^-1 and 15.63 kgm²s^-1 respectively. However, the forward 1½ somersault tuck possesses more angular momentum (35.87 kgm²s^-1) than the forward 1½ somersault pike (33.45 kgm²s^-1). It is suggested that the forward 1½ somersault tuck in this particular case requires more angular momentum due to its relative shorter flight time (1.16s) compared to that of forward 1½ somersault pike (1.32s).

4.4 Angular momentum calculated at different time period

The angular momentum during the first 5 frames, tight body position and the last 5 frames was computed using four different segments (r₁ – r₄) to determine somersault angle. Results along with centripetal force and hip torque in the forward 1½ somersault pike of Subject 1 are shown in Table 4.4.

The angular momentum, centripetal force and hip toque during the tight somersaulting phase are more or less the same using four different segments to determine somersault angle. However, the angular momentum calculated during the first 5 and the last 5 frames do not agree with that calculated during the tight somersaulting phase. These computed values also vary inconsistently across the four different methods of calculation.

Table 4.4 Angular Momentum calculated at Different Times in Forward 1½ Somersault Pike

Segment	F5L(kgm^2s^-1)	L(kgm^2s^-1)	L5L(kgm^2s^-1)	F(N)	T(Nm)
r1	-1.20	29.10	52.58	123.19	42.21
r2	25.52	28.77	7.58	120.42	41.27
r3	16.88	28.72	30.25	119.99	41.12
r4	6.50	28.26	64.46	116.19	39.82

Note. F5 denotes first 5 frames, L5 denotes last 5 frames. L, F and T values are not normalized.

4.5 Estimation of angular momentum

4.5.1 Linear regression

The estimated angular momentum for dives with additional somersault rotation using linear regression equation are presented in Figure 4.4 and 4.5.

From the corresponding regression equation, the angular momentum required for forward 2½ somersault pike, forward 3½ somersault tuck and forward 3½ somersault pike were calculated (Table 4.8).

4.5.2 Calculation in standard straight position

Table 4.5 shows the moment of inertia in standard straight, pike and tuck position using body inertial parameters of Subject 1. The tuck to pike to straight ratio is approximately 1: 1.3: 3.1.

Table 4.5 Moment of Inertia in Standard Straight, Pike and Tuck Position

Body Position	Moment of Inertia (kgm^2)
Standard Straight	8.31
Standard Pike	3.49
Standrad Tuck	2.71

Table 4.6 Angular Momentum normalized in Number of Somersaults in Standard Straight Position

Dive No.	SS
101C	0.32
103C	0.80
105C	1.05
101B	0.51
102B	0.90
103B	0.85
104B	1.09

The normalized angular momentum expressed in number of somersaults in standard straight position (SS) are displayed in Table 4.6. A linear relationship is observed in both pike and tuck somersaulting dives (Figure 4.6). After such normaliztion in which flight time was corrected, the forward 1½ somersault pike possesses more angular momentum than the forward 1½ somersault tuck as hypothesized.

Table 4.7 compares the calculated percentage time in standard pike or tuck position with the actual values in tight body position from the film data. It should be noted that the pike position in the forward dive pike and forward somersault pike were far less compact than the standard pike position during somersault, resulting in a negative (-1%) or very small value (8%) in calculated percentage time. On the other hand, the tuck position in forward dive tuck was more compact than the standard tuck position during somersault as discussed above. As a consequence, the calculated value (27%) was much bigger than the actual value (11%). As the somersault rotation increases, the calculated values were more comparable to the actual values.

Table 4.7 Percentage Flight Time in Pike or Tuck Position

Dive No.	Actual %t in tight position	Calculated %t in standard position
101C	11%	27%
103C	31%	42%
105C	68%	67%
101B	5%	-1%
102B	15%	8%
103B	42%	55%
104B	61%	60%

Assuming a flight time of 1.30s during which 70% was in standard pike or tuck position, the angular momentum of a forward 2½ somersault pike, forward 3½ somersault tuck and forward 3½ somersault pike were estimated. Results were comparable to values obtained using a linear regression equation. Table 4.8 shows the estimated angular momentum expressed in number of somersaults in standard straight position and the absolute average values obtained from calculation and linear regression.

Table 4.8 Angular Momentum estimated from Calculation and Linear Regression

Dive No.	SS	Calculated L^n (kgm^2s^-1)	Linear Regression L^n (kgm^2s^-1)
105B	1.27	51.03	56.65
107C	1.43	57.46	57.82
107B	1.78	71.53	73.90

4.6 Estimation of Hip Torque

Figure 4.7 and Figure 4.8 display the linear regression equations used to estimate the hip torque requirement for a forward 2½ somersault pike, forward 3½ somersault tuck and forward 3½ somersault pike. The calculated values were shown in Table 4.9.

Table 4.9 Estimated Hip Torque Requirement

Dive No.	Hip Torque (Nm)
105B	171.73
107C	238.20
107B	249.31

4.7 Contributing factors of increasing somersault rotation

From the estimated values of angular momentum and hip torque, the relationships between somersault rotation, angular momentum and hip torque requirement are outlined in Figure 4.9. The percentage increase in angular momentum and hip torque for each successive increment in degree of difficulty were calculated:

(a) progression from n somersault to (n + 1) somersault in the same body position,

(b) progression from n somersault tuck to n somersault pike, and

(c) progression from n somersault pike to (n + 1) somersault tuck.

Dive		Angular Momentum (SS)			Hip Torque (Nm)
101	C	0.32		20.13
	B		0.51			31.34
103	C	0.80		112.82
	B		0.85			49.82
105	C	1.05		160.51
	B		1.27			171.73
107	C	1.43		238.20
	B		1.78			249.31

From Figure 4.9, it is obseverd that there is a positive increment for each of the progressions except for the hip torque in forward 1½ somersault tuck to forward 1½ somersault pike. As explained above, this might be due to the angular momentum of the forward 1½ somersault tuck being higher than usual. Neglecting the above dive, a general pattern of successive increment in DD can be summarized as below.

(a) Progression from n somersault to (n + 1) somersault in the same body position:

An increment in both angular momentum and hip torque is required for each somersault progression and in pike or tuck position.

(b) Progression from n somersault tuck to n somersault pike:

From a forward dive tuck to a forward dive pike, a nearly equal proportion increase in angular momentum and hip torque is requred. In the forward 2½ and 3½ somersault, the percentage increase in angular momentum (21.0% and 24.5%) is much higher than that in hip torque (7.0% and 4.7%).

(c) Progression from n somersault pike to (n + 1) somersault tuck:

For all dives, it is obvious that the percentage increase in hip toque is much greater than that in angular momentum. The increment in hip torque (angular momentum) for each successive progression from ½ pike to 1½ tuck, 1½ pike to 2½ tuck and 2½ to pike 3½ tuck are 260% (56.9%), 222% (23.5%) and 38.7% (12.6%).

In summary, the contributions of angular momentum (L) and hip torque (T) increment to each successive progression in DD are illustrated in Figure 4.10.

101C		101B
	L+T	L+T
103C		103B
	L+T	L+T
105C		105B
	L+T	L+T
107C		107B

Figure 4.10. Contributions of angular momentum and hip torque increment to successive progression in dives in the forward group.

CHAPTER 5

DISCUSSION

5.1 Successive somersault increment

The major findings of the present study support the research hypotheses. To progress from n somersault to (n+1) somersault in the same body position, the contribution of increased angular momentum and hip torque are both important. To progress from n somersault tuck to n somersault pike, a large increase in angular momentum is crucial. To progress from n somersault pike to (n+1) somersault tuck, similar angular momentum is required whilst a large increase in hip torque is essential.

If a diver is under-rotated in a multiple somersaulting dive in the tuck position, an increase in angular momentum and/or a tighter tuck position should be adopted to increase the speed of rotation. It is likely that the diver has generated sufficient angular momentum for the dive but has not adopted a very tight tuck position. This is because a large hip torque is needed to hold a tight tuck position during somersault. In this case, the under-rotation can be corrected by holding a tighter tuck position during somersault. For coaches who have been emphasizing the generation of angular momentum at takeoff, it should be noted that the tightness of body position in the airborne phase is of equal importance. Therefore, coaches should pay attention to the tightness of shape in addition to the generation of angular momentum especially when the diver is under-rotated.

Data from this study suggest that the tuck or pike position in the first somersault is less compact than that of the second somersault in forward 2½ somersault. The sooner the diver adopts a tight position, the more the rotation completed during flight. Hence, it is crucial that the diver generates enough angular momentum at takeoff and adopts a tight position as quickly as possible for fast rotation. On the other hand, the faster the rotation, the more difficult to control for good entry. The timing for come-out is another critical factor determining the quality of multiple somersaulting dives.

Besides, the results from forward 2½ somersault indicate a tighter tuck position requires larger hip torque than a less tight tuck position. In contrast, a more compact pike position requires less hip torque than a lose pike position. This implies that adopting a tighter pike position is mechanically efficient in both speeding up rotation and reducing hip torque. A compact position is also more aesthetically appealing, which is a very important factor to get high marks in competition.

It is estimated that the angular momentum required for a forward 3½ somersault tuck is similar to that for a forward 2½ somersault pike. With a 0.4 increase in DD (Table 1.2), a diver may benefit from attempting a forward 3½ somersault tuck if he/she can perform a forward 2½ somersault pike. During the progression, emphasis should be put on holding a tight tuck position and coming out at the right time.

5.2 Muscular strength

In the four sets of body inertial parameters, the standard deviation for angular momentum and hip torque are greater than 10% in general. This implies that angular momentum and hip torque are dependent on body mass distribution and segmental length. The heavier the body, the more the force required to generate angular momentum and maintain tight body position. Therefore, relative strength training is preferable and more meaningful to absolute strength training according to the nature of the sport.

O’Brien (1992) stressed that

The diver can best accomplish the pike position by moving the arms so the elbows are at the knees … at this point the arms warp around the backs of the knees ….This method of assuming the pike allows the diver to quickly achieve a very compact position, resulting in greater acceleration. The alternative technique is for the diver to grasp the backs of the knees with the hands and then begin pulling into a tight position …. It is not as effective for multiple somersault dives (p.73).

For good technique, the hip torque for pulling into shape is solely provided by hip flexors and abdominals whilst the arms only assist in maintaining the established compact shape. When a diver has problems in holding a tight shape during multiple somersaulting dives, it is advised that additional strength training for hip flexors and abdominal muscles should be included.

Attention should also be drawn to the type of muscle contraction in strength training. The principle of specificity states that training must be geared to the particular activity (Goodbody, 1986). Specific strength training to a movement results in optimization of intermuscular coordination for that particular movement. During a somersaulting dive, a diver pulls into shape from straight to pike or tuck position by concentric contraction of hip flexors and abdominal muscles. After a tight shape is formed, isometric contraction of these muscle groups provides the forces to hold this tight position during somersault. While there are many exercises involving concentric contraction of hip flexors and abdominals to improve isotonic strength, it is difficult to assimilate the isometric contraction during somersault. The hip torque required to hold a tight pike or tuck position when the body is stationary is far less than that required during somersault.

Performing a series of fast forward or backward roll in tight tuck position can give the diver some idea of holding a tight position during rotation. Performing multiple somersaults on a trampoline with a spotting belt is another good way to feel how much hip torque is needed to hold a tight position during somersault. The coach can make use of his/her body weight to add the diver additional force for rotation. The more the somersaults, the larger the additional force needed. The speed of rotation can be controlled by pulling the ropes at the right time with appropriate force (Xu & Zhang, 1996).

Krug, Reiss and Knoll (2000) investigated the training effect of a “somersault simulator” on somersault rotation in diving and gymnastics. Athletes were fastened into the somersault simulator with a seatbelt and could be rotated in tuck, pike or straight position. The angular velocity of the somersault simulator can reach up to 700 deg/s, which is similar to 2½ somersault in diving. It is believed that the speed of rotation in forward 3½ somersault from 1-metre springboard would be similar to that of forward 4½ somersault from 3-metre springboard, which was reported to be 1300deg/s (Krug et al., 2000). Increasing the speed of rotation and modifying the design such that the diver has to maintain a tight position without any support, such simulator can be an useful training device for developing specific strength in hip flexors and abdominals.

5.3 Hip Torque Requirement

The estimated hip torque requirement for a forward 2½ somersault pike, forward 3½ somersault tuck and forward 3½ somersault pike were 171.7Nm, 238.2Nm and 249.3Nm respectively. Andersson, Sward and Thorstensson (1988) measured the maximum isokinetic strength of trunk muscles in 14 female gymnasts (age = 18 ± 3 years, weight = 57 ± 6kg, height = 163 ± 4cm). The peak torques at a constant angular velocity of 15 deg/s for hip flexion and trunk flexion were found to be 177.8 ± 3.3Nm and 85.5 ± 2.0Nm. Since the physical characterisitcs of the gymnasts in Andersson et al.’s (1988) study are similar to those in the present study, absolute hip torque values can be compared across the two studies. It appears that the estimated hip torque values in this study lie within the possible range suggested by Andersson et al. Thus, the diver should be able to generate the required hip torque for additional somersault rotation.

5.4 Angular Momentum calculated at different times

Normalizing the angular momentum in number of straight somersaults being performed is a better way than using solely the absolute value. The absolute values show that forward 1½ somersault pike possess less angular momentum than forward 1½ somersault tuck. In fact, dive in pike position should possess more angular momentum than that in tuck position for the same amount of rotation. After correcting the time differences in the dives, the resulting values are more sensible. Moreover, such method of normalization takes into account body inertial parameters of the subject. This allows further comparison across subjects and studies.

The angular momentum generated at takeoff should remain constant throughout the flight. Table 4.4 shows that there is a great inconsistency in the values of angular momentum calculated during the first 5 frames, tight body position and the last 5 frames in forward 1½ somersault pike. During the tight position, the angular momentum, centripetal force and hip torque are more or less the same using four different segments to determine somersault angle. This indicates that using either segment is a good measure of somersault angle during the tight phase. However, the inconsistency in the first 5 and the last 5 frames suggests that none of the four segments is suitable for measuring the somersault angle at that time. This is because the body is not a rigid system but flexes after takeoff and extends before entry. Using a single segment only determines the relative angular rotation of that particular segment rather than the whole body rotation. Therefore, a single segment cannot be used to determine the whole body somersault angle unless the body is rotating as a rigid system.

5.5 Evaluation on calculation procedures

Using a linear regression equation to predict the angular momentum for dives with additional somersault rotation does not take into account the percentage of time the diver is in tight position. There might be a risk that the resulting values require an impossibly long time to be in tight body position. To avoid any unreasonable results, angular momentum were also estimated by a calculation method.

The flight time was chosen to be 1.30s based on the data of actual dives from the 1-meter springboard. For instance, 1.32s was recorded for forward 1½ somersault pike and 1.29s for forward 2½ somersault tuck. It was assumed that the diver could obtain a similar height for more difficult dives, though there might be a compromise in height for increased angular momentum (Golden, 1984; Miller, 1981, 1984; Sanders & Wilson, 1988). The actual and calculated percentage time in tight position for the forward 2½ somersault tuck are 68% and 67% respectively. Since the diver takes time to pull into shape and extend for a good entry, choosing 70% of flight time in tight position is considered reasonable.

The calculation also assumes that the transfer of standard straight position to pike or tuck position is instantaneous. Even though the diver is never in a standard straight position throughout the dive, it is assumed that the moment of inertia while pulling into shape and extending for entry are comparable to that of a standard straight position. The moment of inertia of the body with shoulder flexion and hip flexion should be similar to a position without any flexion (Figure 5.1).

Webb (1997) calculated the moment of inertia in standardised straight position of 14 gymnasts (height = 1.43 ± 0.02m). Values obtained ranged from 3.576 kgm² to 5.345 kgm². The moment of inertia calculated for standard straight position in the present study was 8.31 kgm². Although the absolute value is relatively bigger than those reported by Webb (1997), the tuck to pike to straight ratio is 1:1.3:3.1, which is comparable to 1:2:4 estimated by Rackham (1975). Therefore, the values used in this study appear to be reasonable.

The resulting angular momentum estimated from linear regression equation and calculation are very close (Table 4.8). Thus, all the assumptions made in the calculation procedures can be justified. It is concluded that both methods provide a good estimation of angular momentum for dives with further somersault rotation. Hence, linear regression equation can be used to further estimate the hip torque requirement.

5.7 Conclusion

This study has investigated the contribution of increased angular momentum and hip torque in the progression of multiple somersaulting dives with increasing degree of difficulty. Results suggest that a large increase in angular momentum is needed to progress from n somersault tuck to n somersault pike, whilst similar angular momentum but a large increase in hip torque is required to progress from n somersault pike to (n+1) somersault tuck. When a diver is under-rotated at entry, attention should be drawn not only to the angular momentum generation at takeoff, but also the tightness of shape during somersault. Specific strength training in hip flexors and abdominal muscles is recommended for divers who have difficulties in adopting a tight body position in multiple somersaulting dives.

5.8 Recommendation for further investigation

Only four dives in the pike position and three dives in the tuck position from a 1-metre springboard were used in this study. It should be noted that the diver might not perform all dives to a high standard, particularly in feet-first entry dives which are not commonly adopted in competition. It is recommended that dives should be taken from 3-metre springboard so that more dives can be examined, for example, forward 2½ somersault pike, forward 3½ somersault tuck, forward 3½ somersault pike and forward 4½ somersault tuck. Moreover, since many major events include only the 3-metre springboard and not the 1-metre springboard, results obtained from the 3-metre springboard will be more applicable to the competitive arena. Competitive divers might be more familiar with dives from the 3-metre springboard than the 1-metre springboard.

An additional improvement would be an increased number of subjects where anthropometric measurements are taken directly from the subjects. This will allow comparison across different styles and techniques for the same dive; for example, adopting an open pike or a closed pike in a forward 1½ somersault pike. Within each subject, results from different dive groups, namely the backward, reverse and inward group, can also be compared.

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APPENDIX A

LISTING OF FORTRAN PROGRAM DIVEPROJ