Twist within a somersault

Human Movement Science 45 (2016) 23–39

Contents lists available at ScienceDirect

Human Movement Science journal homepage: www.elsevier.com/locate/humov

Twist within a somersault Joanne Mikl ⇑, David C. Rye Australian Centre for Field Robotics, The University of Sydney, 2006 NSW, Australia

a r t i c l e

i n f o

Article history: Received 10 September 2015 Revised 2 November 2015 Accepted 2 November 2015

Keywords: Diving Gymnastics Angular momentum Rotation axis

a b s t r a c t The twisting somersault is a key skill in diving and gymnastics. The components of twist and somersault are defined with respect to anatomical axes, and combinations of multiples of half rotations of twist and somersault define specific twisting somersault skills. To achieve a twisting somersault skill twist must be continuous; otherwise oscillations in twist while somersaulting may be observed. The posture-dependent inertial properties of the athlete and the initial conditions determine if continuous or oscillating twist is observed. The paper derives equations for the amount of somersault required per half twist, or per twist oscillation, without making assumptions about the relative magnitudes of the moments of inertia. From these equations the skills achievable may be determined. The error associated with the common assumption that the medial and transverse principal moments of inertia are equal is explored. It is concluded that the error grows as the number of twists per somersault decreases, when the medial and transverse moments of inertia diverge, and when the longitudinal moment of inertia approaches either the medial or transverse moment of inertia. Inertial property data for an example athlete are used to illustrate the various rotational states that can occur. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction The twisting somersault is an important skill in gymnastics and diving, involving three-dimensional rotation while the athlete is airborne. The skills that may be performed in competition are defined by the Fédération Internationale de Gymnastique in the Code of Points (CoP MAG, 2013; CoP TRA, 2013; CoP WAG, 2013) and by the Fédération Internationale de Natation (FINA) in the Diving Rules (DR FINA, 2015). All skills are defined in relation to the athlete’s anatomy and the sporting environment; a somersault is a rotation about a horizontal axis perpendicular to the direction of travel, and twist is a rotation about the longitudinal (head-to-toe) axis of the athlete’s body. To be awarded the difficulty points associated with a twisting somersault skill, a diver must simultaneously complete a multiple of a half twist and a multiple of a half somersault, while a gymnast must complete a multiple of a half twist and a multiple of a full somersault. Further, unintended twist in a pure somersault will result in a deduction from the athlete’s score. Thus to specify a skill, or to determine if there will be deductions related to unintentional twist, it is necessary to know the amount of twist completed within a somersault. Previous authors (Batterman, 1968; Frohlich, 1979; Rackham, 1970; Yeadon, 1993) have used rigid body mechanics to describe the phase of the twisting somersault where the majority of the twist occurs. Yeadon states ‘‘Once divers and gymnasts have started to twist in a somersault, they often appear to maintain a fixed body configuration. A rigid body may be expected to give a reasonable representation of such phases.” (Yeadon, 1993). Better performances are expected ⇑ Corresponding author at: ACFR, The Rose Street Building J04, University of Sydney, 2006 NSW, Australia. E-mail address: [email protected] (J. Mikl). http://dx.doi.org/10.1016/j.humov.2015.11.002 0167-9457/Ó 2015 Elsevier B.V. All rights reserved.

24

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

to show distinct phases, since this allows the concept of ‘‘peaking” (George, 2010) to be applied and avoids deductions for ‘‘insufficient exactness” (CoP WAG, 2013); it is therefore reasonable to expect athletes to seek to display a quasi-rigid phase where the majority of the twist rotation is completed. Since this phase is entirely airborne there can be no change to the athlete’s angular momentum and their centre of gravity will follow a parabolic trajectory. This paper focuses on the quasi-rigid aerial phase, mathematically describing the motion resulting from the constant angular momentum. The rotational aspect of the motion is addressed since gymnastic and diving skills are defined in terms of the rotation achieved rather than the translation that occurred. Equations giving twist with respect to time (Synge & Griffith, 1959; Yeadon, 1993) and an average somersault rate during a continuously-twisting somersault (Yeadon, 1993) are available in the literature. These equations are written for the special case when the athlete’s medial, transverse and longitudinal principal anatomical axes—or their equivalents in a general rigid body—correspond to the axes of the maximum, intermediate, and minimum principal moments of inertia. Gymnastic and diving skills are defined in terms of the twist completed with respect to the somersault rotation, rather than the twist completed in a period of time; consequently, the time-parameterised equations given by Synge & Griffith and by Yeadon do not directly describe specific gymnastic or diving skills. In general, the principal moments of inertia of the human body will not be equal, although two moments of inertia may be similar in some postures. Furthermore, since the body can adopt many postures it cannot be assumed that the principal axis closest to each anatomical axis is always the maximum, always the minimum or always the intermediate moment of inertia. This paper removes previous assumptions of equality of the medial and transverse moments of inertia (Batterman, 1968; Frohlich, 1979; Rackham, 1970) and assumptions regarding the order of the magnitudes of the principal moments of inertia (Yeadon, 1993). It is known (Yeadon, 1993) that when the moment of inertia about the principal axis closest to the medial anatomical axis is the intermediate principal moment of inertia then either continuous twist or twist that oscillates about the zero-twist position may be observed. Which occurs depends on the initial orientation of the athlete’s longitudinal axis with respect to the angular momentum vector. The case when the transverse moment of inertia is the intermediate-valued principal moment of inertia can be described (Yeadon, 1993) by adding a quarter-twist to the description of the twist when the medial principal moment of inertia is the intermediate moment of inertia. The situation when the moment of inertia about the principal axis closest to the longitudinal anatomical axis is the intermediate moment of inertia has not been described in the literature. The effect of a small amount of initial twist has not been previously discussed. Initial twist could be intentional—the athlete ‘cheats’—or caused by unintentional movement preceding the take-off. Good performances will display only small initial twist. This paper describes twelve cases that represent distinct rotational states governed by three factors: which moment of inertia is the intermediate-valued; the initial angle between the longitudinal axis and the angular momentum vector; and the initial twist angle. The remainder of the paper is organised as follows. First, the equations of motion are derived using rigid body mechanics. The different rotational states possible are then identified, and the equations of motion solved to determine the number of somersaults required per half twist rotation, or per one oscillation of twist. Since an athlete can remove twist (Batterman, 1968; Frohlich, 1979; Rackham, 1970; Yeadon, 1993) the skills that are achievable are those for which the number of half somersaults to be completed is less than the number of somersaults required per half twist multiplied by the number of half twists to be completed. Since assuming that the transverse and medial moments of inertia are equal greatly simplifies the equation for the number of somersaults required per half twist, when such an assumption is reasonable is discussed. One inertial property data set obtained from the literature (Huston, 2009) is used to illustrate the rotational behaviours and aid discussion. 2. Equations of motion 2.1. Frames of reference and orientation angles Two right-handed frames of reference are defined: the global frame G: {t; x, y, z} and the body frame P: {t; x, y, z}. A prepended superscript is used to denote the frame of reference for a vector. For example, the angular momentum vector measured in P is PH. The axes of the frames P or G are also distinguished by the prepended superscript. For example, the x-axis of frame P is denoted by Px and the y-axis of frame G is Gy. The global frame G has its origin at the athlete’s centre of gravity. The frame G translates with the athlete but does not rotate. Since the origin of G is the centre of gravity and it does not rotate it may be treated mathematically as an inertial frame (Smith & Kane, 1967; Synge & Griffith, 1959). The Gz-axis is vertical. The Gy-axis is the horizontal axis about which a somersault occurs, regardless of whether a pure or a twisting somersault is performed. The Gy-axis will be to the athlete’s right in a backward somersault and to the athlete’s left in a forward somersault. The Gx-axis is in the direction of travel for both forward and backward somersaults. The body frame P rotates with the athlete’s body. The origin of P is always located at the athlete’s centre of gravity, irrespective of the posture that the athlete adopts: its origin is coincident with the origin of G. In the quasi-rigid phase, since the athlete holds a single posture the anatomical landmarks will be fixed with respect to each other and the origin of P. The axes of P are parallel to the principal axes of the body as a whole and named by following anatomy. When the athlete is standing

25

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

in the anatomical position the directions of the longitudinal (head-toe), transverse (right-left) and medial (back-front) axes are clear. In other postures the directions of the anatomical axes are less clear. In such cases the pelvis is selected as the reference segment of the body; the longitudinal, transverse, and medial axes of the pelvis are fixed to it and named such that they are parallel to the correspondingly-named anatomical axes when the athlete is in anatomical position. The x-axis of P is parallel to the principal axis closest to the medial axis of the pelvis, the y-axis is parallel to the principal axis closest to the transverse axis of the pelvis, and the z-axis is parallel to the principal axis closest to the longitudinal axis of the pelvis. The positive direction of the Px-axis is back-to-front in a forward somersault, and front-to-back in a backward somersault. The positive direction of the Py-axis is right-to-left in a forward somersault, and left-to-right in a backward somersault. The positive direction of the Pz-axis is from the feet to the head. Fig. 1 shows the approximate directions of the x- and z-axes of frames G and P, with an athlete performing a forward somersault. A pure somersault is illustrated and so Gy and Py are coincident and directed into the page. Should the athlete twist, G y and Py will no longer be coincident; Gy will remain into the page but Py will rotate as the athlete twists. The angle h represents the somersault rotation of frame P about the Gy axis. The definition of the Gy axis means that h and its time derivative will always be positive. The angle w denotes twist rotation of frame P about the Pz axis. It is necessary to choose a third angle / to specify the athlete’s orientation fully. It is convenient to define / as the angle between the Gy-axis and the Pz axis; from this definition / is orthogonal to h and w and it is also the complement of Yeadon’s ‘‘tilt angle” (Yeadon, 1993). When the medial and transverse principal moments of inertia are equal, a steady twisting somersault will be produced (Yeadon, 1993), and / will be the semi-vertex angle of the right circular cone traced by the longitudinal axis Pz about the angular momentum vector. Any orientation of the body frame P relative to G may be achieved by successive rotations through YXZ Euler angles (Paul, 1981; Shah, Saha, & Dutt, 2012) of h about Py, (/
p/2) about Px, and w about Pz. Euler angles are used rather than any other representation for the orientation of the body, such as quaternions, since they map directly to the definitions of somersault and twist from the sporting codes. The rotation matrix describing P relative to G is

2

G

ch

0

6 RP ¼ 4 0

1


sh 0 2 6 G RP ¼ 4

sh

32

1

0

0

32

cw
sw 0

76 76 0 54 0 s/ c/ 54 sw 0 0
c/ s/ ch

3

cw

7 05

0

1

ch  cw
sh  c/  sw s/  sw


ch  sw
sh  c/  cw s/  cw


sh  cw
ch  c/  sw

sh  sw
ch  c/  cw

3 sh  s/ 7 c/ 5 ch  s/

where s and c denote the sine and cosine operators. 2.2. Initial conditions Before solving the equations of motion it is necessary to specify appropriate initial conditions: in this case the athlete’s angular momentum vector and orientation of frame P at the start of the quasi-rigid phase of the somersault. The angular momentum will depend only on actions performed prior to the instant of take-off. The initial orientation will result from

G

z

P G

z

P

G

x

z x

P

P

x

x

z

z

P

x

G

x

P G

G

G

z G

x

z

P

P

z

G

z

G

G

x

x P

x

P

z G

x

z P

x

P

z

Fig. 1. Frames of reference G (black) and P (red), illustrated with an athlete performing a forward somersault. After O’Brien (2003). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

26

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

the orientation on take-off and any subsequent reorientation that occurs as the athlete transitions from the take-off posture to the posture to be held while twisting. To assist solving the equations of motion, while still allowing all skills in diving and gymnastics to be described mathematically, the initial conditions will be restricted. Initial values /o, ho and wo of all three orientation angles are restricted to the first quadrant, and the initial twist angle wo is assumed to be small. As a consequence of how the reference frames are defined, h will be positive regardless of whether the athlete performs a forward or backward somersault. If /o was in the second quadrant the twist would simply be in the opposite direction to when /o is in the first quadrant. It is unnecessary to consider /o in the third or fourth quadrant, since this would have the athlete starting upside-down, and results may be easily interpreted from this point since skills are defined on the basis of half somersault rotations. When /o is 0 or p, the rotation is simply a rotation about the angular momentum vector; twist and somersault are indistinguishable. Initial twist wo should be small, so that the athlete has not ‘cheated’ by already being partially twisted; it should not be set to zero, since this would ignore the effect of any unintended initial twist. As Yeadon (1993) identified, when the medial moment of inertia is the intermediate-valued moment of inertia the motion will be the same as if the transverse moment of inertia was the intermediate-valued moment of inertia and a phase shift of a quarter-twist was applied. Keeping these cases separate, however, simplifies analytical comparisons, specification of initial conditions, and interpretation in a sporting environment. The angular momentum vector H is constant in frame G. Since GH must be generated by an athlete on take-off as they enter their somersault, the possible directions that GH may take are limited. Due to the nature of potential movements preceding the take-off, and since the athlete is intending to perform a somersault about the Gy-axis, any component of H in the G x direction is clearly erroneous and can be disregarded. If the athlete leaves the ground with only somersault angular velocity, then H will be parallel to the Gy-axis. If the athlete leaves the ground with both somersault and twist angular velocity, as will occur when the athlete uses the technique known as ‘‘contact twist” (Yeadon, 1984), then H will have a component in the Gz direction. Provided that the angle of inclination of H with respect to Gy is less that p/4 the athlete’s head will still drop below the Gy-axis (Yeadon, 1984), and the motion will therefore still be perceived as a somersault. Each half-somersault rotation will be completed in the same time as if H was parallel to Gy, even though the rotations will differ. When analysing skills which are defined in half-somersault increments, it is therefore sufficient to determine h for the case where H is parallel to G y. In summary, the equations derived in this paper apply when (i) h > 0; (ii) 0 < /o < p/2; (iii) / – {0, p}, since somersault and twist would then be indistinguishable; (iv) wo P 0 and small; and v) GH = [0, H, 0]T where H is a positive scalar. 2.3. Equations of motion _ h_ and w_ gives x physical meaning. By definition, Resolving the athlete’s rigid-body angular velocity x into components /, twist w is about the z-axis of frame P, somersault h is about the y-axis of frame G, and / is a rotation about an axis parallel to the x-axis of frame P if the athlete was not twisting. Thus the angular velocity with respect to frame P is

2 3 32 3 2 32 3 ch  cw
sh  c/  sw s/  sw
sh  cw
ch  c/  sw 0 1 0 6 7 76 7 6 76 7 P sh  sw
ch  c/  cw 54 1 5 þ w_ 4 0 5 x ¼ u_ 6 4
sw cw 0 54 0 5 þ h_ 4
ch  sw
sh  c/  cw s/cw sh  s/ 1 0 0 0 0 1 c/ ch  s/ 2

cw

sw

0

2

3 h_  s/  sw þ /_  cw 7 P x¼6 4 h_  s/  cw
/_  sw 5: w_ þ h_  c/ Since the frame P moves with the body and is aligned with the principal axes, the inertia tensor with respect to frame P is a constant diagonal matrix; that is, the products of inertia are zero. Thus

2

3 Ixx ðh_  s/  sw þ /_  cwÞ 6 7 P H ¼ P I  P x ¼ 4 Iyy ðh_  s/  cw
/_  swÞ 5: Izz ðw_ þ h_  c/Þ The angular momentum vector is not constant in P, since it is a rotating frame, but is constant in G. In addition, since it is assumed that the angular momentum vector H is parallel to Gy,

½0; H; 0T ¼ G H ¼ G RP P H: Expanding and solving the equations for each component simultaneously, and observing common factors leads to Eqs. (1)–(3). By observing common factors in the expansion it becomes unnecessary to use the conservation of energy principle as previous authors have done (Synge & Griffith, 1959; Yeadon, 1993). This simplifies matters by removing the necessity to add an additional constant, the rotational kinetic energy, which must then be determined so that it is compatible with the angular momentum and inertial properties specified. The remainder of the paper follows from Eqs. (1)–(3).

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

27

/_ ðsin w cos wÞðIxx
Iyy Þ ¼
sin / 2 h_ ðIxx cos2 w þ Iyy sin wÞ

ð1Þ

" # w_ Ixx Iyy ¼ cos /
1 2 h_ Izz ðIxx cos2 w þ Iyy sin wÞ

ð2Þ

! 2 _h ¼ H Ixx
ðIxx
Iyy Þ sin w Ixx Iyy

ð3Þ

From Eq. (2) it is clear that an athlete wishing to increase the rate of twist within a somersault should decrease the angle /. This result is expected, since reducing / increases the component of the angular momentum vector in the direction of the athlete’s longitudinal axis. It is also clear from Eq. (2) that reducing the value of Izz relative to Iyy and Ixx will increase the ratio _ It is important to appreciate that both ratios Izz/Iyy and of the twist angular velocity w_ to the somersault angular velocity h. Izz/Ixx affect the ratio of the angular velocities of the twist to the somersault and so the values of all three moments of inertia must be known when considering athlete postures. Since H and all moments of inertia are positive, Eq. (3) shows that the somersault angular velocity h_ is always positive, although it will experience oscillation of magnitude related to the difference between Ixx and Iyy. The somersault angle h will continually increase in value. It is interesting to note that adding twist only alters the somersault angular velocity as a consequence of Iyy and Ixx being unequal, not because there is any ‘extra’ rotation; the bounds of the somersault angular velocity are H/Iyy and H/Ixx. If Ixx and Iyy are equal then h_ is constant and the athlete somersaults at a steady rate. For an athlete this is a comforting result: if they twist in the same posture for the same portion of a somersault, regardless of the number of halftwists they perform they can be confident that no ‘extra’ angular momentum is required. Each twisting somersault can be performed from the same somersault entry and the only thing that will differ with increasing numbers of twists will be the actions required for twist initiation. 3. Twist within the somersault 3.1. Categorising motion by the twist rotation observed The relationship between w and / will determine whether continuous twist, oscillating twist, or non-oscillating twist that stops at p/2 is observed. Which of the three possible motions occurs depends on if and when the angle / reaches p/2. By inspection of Eqs. (1) and (2), if w reaches p/2 first, then / reverses direction and so w continues to increase: continuous twist is produced. If, however, / reaches p/2 before w does then the twist reverses direction: the twist is oscillatory. In the special case where w and / reach p/2 simultaneously, the twist stops at p/2. Dividing Eq. (1) by Eq. (2) gives

/_
sin /  Izz ðIxx
Iyy Þ  sin w  cos w ¼ ; 2 w_ cos /½Ixx ðIyy
Izz Þ þ Izz ðIxx
Iyy Þ sin w and separating variables yields

cos / _
Izz ðIxx
Iyy Þ  sin w  cos w /¼ w_ 2 sin / ½Ixx ðIyy
Izz Þ þ Izz ðIxx
Iyy Þ sin w Both sides of this equation are in the form of the derivative of the natural logarithm. Since 0 < / < p, the denominator on the left side of the equation is positive. The denominator on the right side may take either sign, and so both situations need to be considered. After integrating and applying the initial conditions /o and wo Eq. (4) conveniently describes both:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uIxx ðIyy
Izz Þ þ Izz ðIxx
Iyy Þ sin2 w o sin / ¼ sin /o t 2 Ixx ðIyy
Izz Þ þ Izz ðIxx
Iyy Þ sin w

ð4Þ

In Eq. (4), w appears only within the sin2w term and so its initial direction is not clear. Considering Eq. (2) when 0 < / < p/2 and w is small, it can be seen that when Iyy > Izz the twist direction is to the left (w_ > 0).

Whether continuous or oscillating twist will occur can be determined from Eq. (4) by finding if / will reach p/2 for a value of w less than p/2 (in which case oscillating twist occurs) or if / can never reach p/2 (continuous twist occurs). The two critical values of /o that separate the three cases are
1

/crit1 ¼ sin

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ixx ðIyy
Izz Þ 2

Ixx ðIyy
Izz Þ þ Izz ðIxx
Iyy Þ  sin wo

ð5aÞ

28

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39


1

/crit2 ¼ sin

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Iyy ðIxx
Izz Þ

ð5bÞ

2

Ixx ðIyy
Izz Þ þ Izz ðIxx
Iyy Þ  sin wo

If /o is less than both critical values then continuous twist will occur. If /o equals the smaller of the two critical values then the twist will stop at p/2, otherwise the twist will be oscillating. To aid discussion of the rotational behaviour, the derived equations were evaluated using the inertial properties of the fiftieth percentile male from Huston (2009). Fig. 2 shows sketches of three postures of sporting interest—layout, pike and tuck—orientated so that their approximate principal axes are in the same directions. The three postures chosen reflect cases where each of the moments of inertia is the intermediate-valued moment of inertia. The calculated values are given in Table 1. Fig. 3 shows the variation in moments of inertia as the posture of Huston’s fiftieth percentile male transitions from layout, to pike, to tuck. From this plot it is clear that Iyy is the intermediate moment of inertia only for approximately the first 13° of hip flexion from layout, whereupon Iyy becomes the maximum moment of inertia, and Ixx the intermediate. At approximately 77° of hip flexion, Izz becomes the intermediate moment of inertia and then, as the athlete moves into tuck, Ixx again becomes the intermediate when the hips are held at 95° of flexion and the knees are bent at approximately 65°. The discontinuities in Ixx and Izz are due to the movement of the principal axis directions, and how the x, y, and z directions of frame P are defined. The axis names were matched with the principal direction that was closest to each anatomical axis of the pelvis; thus the principal axes will be inclined relative to the anatomical axes of the pelvis. As the athlete flexes forward, the x and z axes of frame P tip forwards, until when this angle of inclination reaches 45° the names of the x- and z-axes are swapped. This re-naming produces the discontinuities in the Ixx and Izz curves in Fig. 3. From this example, it is clear that athletes can adopt postures where each of the moments of inertia is the intermediatevalued one. This justifies the decision to allow the medial Ixx, transverse Iyy, or longitudinal Izz moment of inertia to be the intermediate-valued, so that the derived equations may be applied to any sporting posture. Eq. (4) describes the relationship between the twist angle w and the angle /. Thirteen distinct rotational states/cases may be described, depending on the rotational motion observed. Twelve of the cases are discussed below. The redundant case where Ixx = Iyy = Izz is ignored, since rotation about the angular momentum vector is the obvious result. Not all twelve cases will be observed in the sporting environment but are described here for completeness. 3.1.1. Case 1: pure somersault A pure somersault was illustrated in Fig. 1. There is no twist and the Py-axis remains coincident with the Gy-axis. A pure somersault will occur when /o = p/2 and wo = 0, regardless of the athlete’s inertial properties. 3.1.2. Case 2: no twist but not a pure somersault When Izz is the intermediate-valued moment of inertia, the upper bound of when wo may be considered small, woSmall, occurs when the denominator in Eq. (4) is zero: that is, when
1

woSmall ¼ sin

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ixx ðIyy
Izz Þ Izz ðIxx
Iyy Þ

ð6Þ

With this initial condition Eq. (4) cannot be used. It is thus necessary to consider Eqs. (1) and (2): no twist will be observed and, as the athlete somersaults, / will increase to p/2 if Ixx < Iyy, or decrease to zero if Ixx > Iyy. This case marks a transition between the athlete oscillating about the zero-twist position and oscillating about the quarter-twist position. Although an athlete is unlikely to be able to reliably achieve the exact initial orientation required for this case to be observed, it shows what occurs when wo is no longer ‘small’. What constitutes small can be seen from Eq. (6) to depend on the athlete’s inertial properties in the assumed posture: as Iyy and Izz approach each other, the value of wo must be progressively smaller

P P

z P

(a)

z

P

x

P

(b)

z P

x

x

(c)

Fig. 2. The layout (a), pike (b) and tuck (c) postures. After (CoP TRA, 2013). Frame P is aligned with the principal axis directions, and the naming of the axes is by the closest anatomical axis of the pelvis. Accordingly, there may be an angle between the anatomical axis of the pelvis and frame P; this angle will be less than 45°. For layout, the offset angle is
0.7°, for pike it is
8.1° and for tuck it is
9.6°.

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

29

Table 1 Principal moments of inertia in kg.cm2 for the fiftieth percentile male from Huston (2009). Moment of inertia

Layout

Pike

Tuck

Ixx Iyy Izz

149,334 147,552 10,406

28,679 69,427 52,957

49,489 64,550 27,271

The intermediate-valued moment of inertia is set in bold.

Fig. 3. Variation in the moments of inertia if Huston’s 17-segment representation of the fiftieth percentile male smoothly transitioned from a layout to a pike to a tuck. The x-axis shows hip flexion from layout (at 0°) to pike (at 95°, marked by the vertical line), followed by 110° of knee flexion to tuck (at 205°) with hip flexion constant at 95°. All segments move at a steady coordinated rate so that when the hip and knee angles reach those required for the pike or tuck all segments are in the correct positions.

Fig. 4. Case 3: relationship between / and twist angle w for Huston’s fiftieth percentile male in layout, with initial conditions wo = 0 and /o = 80°.

so that the oscillations in twist remain about the zero-twist position rather than about the quarter-twist position. In contrast, as Izz and Iyy diverge from each other, wo may be made progressively larger while the oscillations in twist remain about the zero-twist position. 3.1.3. Case 3: continuous twist, / maximum when twist w = 0 If Iyy is the intermediate moment if inertia, /o < /crit1 and /o – p/2 then continuous twist will be observed, with / reaching a maximum when w = 0 and a minimum when w = p/2. If wo = 0 then continuous twist is observed whenever /o < p/2. The magnitude of the oscillation of / is greater for postures where any of the following are true individually or in combination: Ixx diverges from Iyy; Izz approaches Iyy; and /o approaches 90°. Fig. 4 illustrates the relationship between w and / for Huston’s fiftieth percentile male, in layout, with initial conditions of wo = 0 and /o = 80°. In this situation the amplitude of the oscillation in / is unnoticeably small—approximately 0.15° peak to peak. 3.1.4. Case 4: oscillating twist about the quarter-twist position If Iyy is the intermediate moment of inertia, /o > /crit1 and wo > 0, then oscillation in twist about the quarter-twist position will occur. The amplitude of twist oscillation is greatest for the smallest initial value of twist. For example, if wo = 1° and

30

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

/o = 90°, then the twist angle will oscillate between w = 1° and w = 179°. Continuous twist will not be produced, and so an athlete performing a twisting somersault may only use this case if they are performing a half twist and ‘cheat’ by possessing some small initial twist and not quite finishing the half twist. This case is of concern if an athlete is intending to perform a pure somersault, but erroneously starts with some initial twist; the result will be an undesired large oscillation in twist. Case 4 may be thought of as describing the consequences of a disturbance to a body rotating about the intermediate principal axis. The amount of somersault required per oscillation, H, which is derived in Section 3.3, will determine if the consequences are of practical significance. 3.1.5. Case 5: transition between case 3 and case 4 If Iyy is the intermediate moment of inertia, /o = /crit1 and wo > 0, then twist will occur until w = p/2. It will then stop because / and w will have reached p/2 simultaneously. It is interesting to note that if an athlete started with /o = /crit1, with /o in the first quadrant and wo < 0, then from Eq. (2) the twist would still be in the positive direction and so the twist would reach zero and stop. It is unlikely that an athlete could achieve the exact initial orientation required for this case to be observed. 3.1.6. Case 6: oscillating twist about the zero-twist position, which can never have continuous twist When Izz is the intermediate moment of inertia and wo < woSmall, the twist angle w will oscillate about zero and / will oscillate between its initial value and its supplementary value. Fig. 5 illustrates the relationship between w and / for Huston’s fiftieth percentile male in pike, with initial conditions wo = 0 and /o = 80°. In this example the twist oscillation is small; approximately 9° peak to peak. The magnitude of the twist oscillations will be greater when /o is smaller, and for postures where Ixx and Izz are closer in magnitude and/or Iyy is further from the other two. Since this pike position causes Izz to be the intermediate moment of inertia, it is impossible for this example athlete to perform a twisting somersault in pike; if they use a pike at the start of their somersault or in somersaults prior to the one containing the twist, they must extend out of this posture to allow continuous twist to occur. Since typical twist oscillations in pike are small (e.g. Fig. 5)—and probably not noticeable to the unaided eye—pike is a useful posture to enter following a continuously-twisting somersault to ‘stop’ the twist. This may be one reason why divers commonly enter a pike position after completing the number of twists required in their dive, and before extending the hips and adopting a posture suitable for entry to the water. Equally, it could be a fortunate secondary effect of using a pike to increase somersault rotation at the end of the dive. 3.1.7. Case 7: continuous twist, / maximum when w = p/2 When Ixx is the intermediate moment of inertia and /o < /crit2 continuous twist will be observed. This case differs from Case 3 in that / is a maximum at w = p/2 rather than at w = 0. The value of /o must consequently be smaller than in case 3 so that continuous twist may be produced. Although this case may produce continuous twist, an athlete seeking to maximise the twist produced is advised to seek a posture where Iyy is the intermediate moment of inertia, such as Case 3. As with Case 3, the magnitude of the oscillation in / is greater for postures where any of the following are true individually or in combination: Ixx diverges from Iyy; Izz approaches Iyy; and /o approaches p/2. Fig. 6 illustrates the relationship between w and / for Huston’s fiftieth percentile male in tuck, with initial conditions of wo = 0 and /o = 60°.

Fig. 5. Case 6: relationship between / and twist angle w for Huston’s fiftieth percentile male in pike, with initial conditions wo = 0 and /o = 80°. Over time / and w will change to trace anticlockwise around the loop.

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

31

The value of /o required to achieve continuous twist in tuck is much smaller than that in layout. As a consequence, although it is possible to twist in tuck, it not a practically useful twisting posture. Further, even when continuous twist was produced the large oscillations in / would detract from the performance. 3.1.8. Case 8: oscillating twist about the zero-twist position When Ixx is the intermediate moment of inertia and /o > /crit2 twist oscillating about the zero-twist position will be observed. The oscillations in w increase in magnitude as /o approaches /crit2, as Ixx approaches Iyy, and as Izz diverges from Ixx. Fig. 7 illustrates the relationship between w and / for Huston’s fiftieth percentile male, in tuck with initial conditions wo = 0 and /o = 80°. Comparing Figs. 5 and 7, which show the oscillations in the w-/ phase plane for pike and tuck respectively given the initial conditions wo = 0 and /o = 80°, it is clear that tuck shows greater twist oscillations than pike. The twist oscillations in tuck would more readily be observable by the naked-eye than the oscillations that occur in pike, and so for the purpose of appearing to cease twist the pike posture is more suitable than tuck. Nevertheless, tuck could be used to prevent continuous twist. The smaller value of Iyy in tuck compared to pike would increase the somersault rate in general; this would allow the remaining somersault to be completed in less time. Further the fact that the knees are already flexed would make for a safer landing on the feet, if the flight time is such that the athlete does not have sufficient time to ‘open-out’ prior to landing. Thus, tuck may still be the posture of choice over pike for skills were the athlete must complete multiple somersaults with only some containing twist. Tuck is commonly used in full twisting double back somersaults in gymnastics where the twist is completed in the first somersault and tuck is used for the second somersault. 3.1.9. Case 9: transition between Case 7 and Case 8 When Ixx is the intermediate moment of inertia and /o = /crit2, w and / both increase, simultaneously reaching p/2; the twist therefore stops at a quarter twist. This is the ‘‘singular solution” presented by Yeadon (1993). It is unlikely that an athlete could achieve the exact initial orientation required for this case to be observed in practice. 3.1.10. Case 10: steady continuously twisting somersault When Ixx = Iyy, a steady twisting somersault is produced, with / = /o constant and twist w increasing at a steady rate. This case is equivalent to Yeadon’s ‘‘rod mode” (1993). Letting Ixx = Iyy = Ia, from Eq. (2) the twist-to-somersault rate is

w_ ðIA
Izz Þ cos / ¼ _h Izz

ð7Þ

This case represents the transition between Case 3 and Case 7. For Huston’s fiftieth percentile male, one posture where Ixx equals Iyy occurs at approximately 13° of hip flexion and zero knee flexion (Fig. 3). This is only one of many postures that will show continuous twist, and thus it should not be generally assumed that Ixx = Iyy when describing twisting somersaults. The error associated with this convenient simplifying assumption will be analysed in Section 3.4. 3.1.11. Case 11: Ixx = Izz, oscillating twist about the zero-twist position When Ixx = Izz, Eq. (4) reduces to sin / ¼ sin /o ðcos wo = cos wÞ and the twist oscillates about the zero twist position. The oscillations increase in magnitude only as a result of a change in the initial conditions; changing the relative magnitude of the moments of inertia while maintaining Ixx = Izz will not change the magnitude of the oscillations in twist. When wo = 0, this case is equivalent to Yeadon’s ‘‘disc mode” (1993a). This case is unlikely to be observed; as was seen in Fig. 3, Huston’s fiftieth percentile male did not assume a posture where Ixx = Izz. 3.1.12. Case 12: Iyy = Izz, oscillating twist about the quarter-twist position When Iyy = Izz, Eq. (4) reduces to sin / ¼ sin /0 ðsin wo = sin wÞ and the twist oscillates about the quarter-twist position. The oscillations increase only as a result of a change to the initial conditions, not the value of any of the moments of inertia. This

Fig. 6. Case 7: relationship between / and twist angle w for Huston’s fiftieth percentile male in tuck, with initial conditions wo = 0 and /o = 60°.

32

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

Fig. 7. Case 8: relationship between / and twist angle w for Huston’s fiftieth percentile male in tuck, with initial conditions of wo = 0 and /o = 80°. Over time / and w will trace the loop in an anticlockwise direction.

case is unlikely to be observed; as was seen in Fig. 3, when transitioning from a layout to a pike to a tuck, Huston’s fiftieth percentile male did not assume a posture where Iyy equalled Izz. 3.2. Observations across the cases It is theoretically possible to transition through all the cases by altering the moments of inertia or the initial conditions. Fig. 8 diagrammatically illustrates the relationships between cases. Altering the initial conditions will shift the curves in Figs. 4–7 up or down. By shifting the curves up, / reaches 90° sooner and causes continuous twist to become oscillating twist: Case 3 becomes Case 4 via Case 5, and Case 7 becomes Case 8 via Case 9. It is interesting to note that due to the opposite direction of the movement of / from its initial value in Cases 3 and 7, when initial twist is present the curve effectively shifts up for Case 3 but down for Case 7. Thus, when seeking to reduce / so as to increase the twist rate, any initial twist should be avoided when Iyy is the intermediate moment of inertia. When Ixx is the intermediate moment of inertia, increasing the initial twist—within the limit where a deduction will be incurred—will increase the twist rate. Altering the moments of inertia, such as when changing to a new body posture, will change the magnitude and direction of the oscillation in /, and hence which cases are observed. As Iyy approaches Ixx, such as when starting to flex at the hips from a layout position, the oscillations of / in Case 3 reduce and the rotational state becomes Case 10; then, as Iyy passes Ixx so that Ixx becomes the intermediate moment of inertia, the rotational state transitions to Case 7. If the initial conditions are unchanged then, as Ixx moves away from Iyy, the oscillations in / increase and may be sufficient for / to reach 90° before w reaches 90°; we have thus transitioned to Case 8. If from Case 8, Izz approaches Ixx then the oscillations in twist would decrease, continuing to decrease as Izz becomes the intermediate moment of inertia. Allowing Iyy to approach Izz means that the value of wo which may be considered small decreases. Then, as Iyy moves to become the intermediate moment of inertia, the value of wo that may be considered small is so small that the oscillation in twist moves through Case 2 to Case 12, Case 4 and then finally to Case 3 with continuous twist when Iyy is the intermediate moment of inertia. The moment of inertia ratios that are desirable when seeking to perform a twisting somersault, or to prevent twist, will be considered in more detail in Section 3.3. From the relationship between w and / (Eq. (4)), however, it is expected that postures where Iyy is the intermediate-valued moment of inertia will be the most useful for twisting somersaults since they will display continuous twist with larger values of /. In addition, postures where Izz is the intermediate moment of inertia will be useful for preventing twist, since they will show smaller twist oscillations for the same initial conditions. In the range of postures of interest here, the human body will generally have non-equal principal moments of inertia, and it is unreasonable to expect that an athlete will be sufficiently well-controlled as to achieve the precise initial conditions required for the transition cases. Transition cases should therefore be regarded as limiting situations that occur as /o approaches the smaller of /crit1 or /crit2. The cases of sporting interest are therefore identified as Cases 3, 4, 6, 7 and 8. Case 10 will also be revisited when examining the consequences of assuming that the medial Ixx and transverse Iyy principal moments of inertia are equal. 3.3. Somersault required per half twist Using the relationship between / and w (Eq. (4)), the twist angle w may be eliminated from Eq. (2) and the resulting equation integrated to give the somersault angle h as a function of /. The nature of the relationship between w and / implies that,

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

33

Fig. 8. The cases and the means by which to move from one case to another. The arrows show the changes to the initial conditions and the moments of inertia required to move from one case to another. The symbol ‘‘?” represents ‘‘approaches”.

if oscillating twist is produced, w will require the same amount of somersault as an oscillation of / and, if continuous twist is produced, w will increase by p/2 with every oscillation of /. Somersault skills are named in increments of a half twist; the amount of somersault ‘available’ is 2p divided by the number of half twists, and the number of somersaults per half-twist, or oscillation of w gives the somersault ‘required’. For a skill to be achievable, the somersault required must be less than the somersault available; equality between the somersault available and the somersault required is not necessary provided that the athlete is able to perform a suitable ‘‘tilt-twist” technique (Yeadon, 1984), by which they can return / to p/2. Cases 3, 4, 6, 7 and 8 have been identified as those of sporting interest and so this section will derive equations for the amount of somersault required for an of oscillation in / for these five cases. To aid expressing the equations in a standard form of Elliptic integrals, it is convenient to define the following constants: 2

2

a ¼ Ixx ðIyy
Izz Þ  cos2 /o
Izz ðIxx
Iyy Þ  sin wo sin /o b ¼ Ixx ðIyy
Izz Þ 2

2

c ¼ Iyy ðIxx
Izz Þ
sin /o ðIxx ðIyy
Izz Þ þ Izz ðIxx
Iyy Þ  sin wo Þ 2

2

¼ Ixx ðIyy
Izz Þ  cos2 /o þ Izz ðIxx
Iyy Þð1
sin /o sin wo Þ f ¼ Izz ðIxx
Iyy Þ þ Ixx ðIyy
Izz Þ ¼ Iyy ðIxx
Izz Þ 2

2

g ¼ sin /o ðIxx ðIyy
Izz Þ þ Izz ðIxx
Iyy Þ  sin wo Þ ¼ f
c ¼ b
a Rewriting Eq. (4) using these constants gives

g

2

sin / ¼

2

b þ ðf
bÞ sin w

;

which may be rearranged to give 2

2

sin w ¼

g
b sin / 2

ðf
bÞ sin /

:

Substituting this into Eq. (2) gives

/_ ¼
sin / h_

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðg
b sin /Þðf sin /
gÞ 2

Ixx Iyy sin /
g

:

ð8Þ

34

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

Inverting and integrating gives

Z h¼

/

/o

2

g
Ixx Iyy sin / rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    d/ 2 2 sin / g
b sin / f sin /
g

ð9Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let u = cos/; thus du/d/ = –sin/ and d/ ¼
du= 1
u2 . This is the same substitution used by Yeadon (1993) when seeking an equation for / with respect to time. It allows the equation to be written in a form that can be more easily identified with Elliptic integrals of the first and third kind. Following this substitution, Eq. (9) becomes

h¼g

Rx

Rx
du=d/ ffi d/
Ixx Iyy xo pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðg
bð1
u2 ÞÞðf ð1
u2 Þ
gÞ R cos / du du pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ Ixx Iyy cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / 2 2 2 2

1
du pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1
u2 ðg
bð1
u2 ÞÞðf ð1
u2 Þ
gÞ 1
u2

xo

¼
g

R cos /

cos /o ð1
u2 Þ

R cos /

¼
g

ðbu þg
bÞðf
g
du Þ

cos /o ð1
u2 Þ

du pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

ðbu2
aÞðc
fu2 Þ

ðbu þg
bÞðf
g
fu Þ

o

þ Ixx Iyy

R cos /

cos /o

ð10Þ

du pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

ðbu2
aÞðc
fu2 Þ

The first integral in Eq. (10) may be written as an Elliptic integral of the third kind and the second integral as an Elliptic integral of the first kind. The notation used here follows that of Mathematica (Weisstein, 2014a,b) as Mathematica provides the most current and accessible means by which to evaluate standard Elliptic integrals. The notation used for incomplete Elliptical integrals of the first and third kinds is
1

Fðsin

Z

2

y

y; k Þ ¼ 0

Pðn; sin
1 y; k2 Þ ¼

dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1
y 1
k y2

Z

y

ð1


0

ny2 Þ

dy ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1
y2 1
k y2

where k is known at the elliptical modulus and n is the elliptic characteristic; it is essential that 0 < k2 < 1. Each case must be evaluated individually since the signs of the constants in Eq. (8) are different for each case. Eq. (10) must therefore be rearranged to ensure that 0 < k2 < 1 and achieve the form of the standard Elliptic integrals. To determine the period of / as a function of the somersault angle h it is necessary to perform a definite integration. Due to symmetry in the motion it is only necessary to evaluate the integral between the bounds of / at its minimum and the lesser of its maximum and p/2. To obtain the full integral, the reduced integral must be multiplied by two in the case of continuous twist, since / oscillates within two bounds in the first quadrant. For oscillating twist, the reduced integral must be multiplied by four to obtain the full integral since / oscillates from its minimum to p/2 then to its supplement, back to p/2 then finally returning to its minimum. The amount of somersault required per oscillation in /, expressed in units of radians of somersault rotation, for Cases 3, 4, 6, 7 and 8 are




2f

   bc
af p af 2Ixx Iyy p af þ pffiffiffiffiffi F ; ;1
;1
bðc
f Þ 2 bc bc 2 bc

H3 ¼ pffiffiffiffiffi P bc

H4 ¼

!  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   
4f c p bc 4Ixx Iyy bc
af p bc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P F ; ; þ ; bc c
f 2 bc
af 2 bc
af bc
af 


4g

   a p af 4Ixx Iyy p af þ pffiffiffiffiffiffiffiffiffi F ; ; ; b 2 bc 2 bc
bc

H6 ¼ pffiffiffiffiffiffiffiffiffi P
bc


2b



H7 ¼ pffiffiffiffiffi P af

   af
bc p bc 2Ixx Iyy p bc ; ;1
þ pffiffiffiffiffi F ;1
f ða
bÞ 2 af af 2 af


4b

! 

H8 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P af
bc

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    a p af 4Ixx Iyy af
bc p af ; ; þ : ; F a
b 2 af
bc 2 af
bc af

Now that equations describing the somersault rotation required per oscillation of / have been obtained, Figs. 9–12 show the number of somersaults Nh = H/2p as a function of /o or wo, when wo is held at zero or /o is held at 90°, for the three postures of the fiftieth percentile male given in Table 1. The plot scales are chosen to cover regions of interest in a sporting application. It is clear from Fig. 9 that on decreasing the initial value of /o—that is, reducing the initial angle between the athlete’s longitudinal axis and their angular momentum vector—less somersault rotation is required to achieve a half twist. The curve has a vertical asymptote at /o = 90° since /o cannot equal p/2 for twist to occur. As /o approaches zero, so does the somersault required to achieve a half-twist, yet /o cannot equal zero since twist and somersault would then be indistinguishable.

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

35

Fig. 9. Case 3: dependence of number of somersaults required per half twist, Nh, on initial condition /o with wo = 0 for Huston’s fiftieth percentile male in layout.

Fig. 10. Case 4: dependence of number of somersaults required per half twist, Nh, on initial condition wo with /o = 90° for Huston’s fiftieth percentile male in layout.

Fig. 11. Case 6: dependence of number of somersaults required per half twist, Nh, on initial condition /o with wo = 0 for Huston’s fiftieth percentile male in pike.

Which twisting somersault skills can be achieved by this athlete depend on the value of /o he is able to obtain. From Fig. 9, a value of /o less than 88° will allow a half twist to be achieved, less than 86° will allow a full twist and less than 83° will allow a double twist.

36

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

Fig. 12. Case 7 (continuous twist: blue) and case 8 (oscillating twist: red): dependence of number of somersaults required per half twist, Nh, on initial condition /o with wo = 0 for Huston’s fiftieth percentile male in tuck. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

When /o is p/2, Case 4 has vertical asymptotes at both wo = 0 and p/2 (Fig. 10) since these values correspond respectively to athlete rotation about the transverse and medial principal axes, and so no twist occurs. Fig. 10 shows the behaviour for small wo, such as could occur because of a disturbance or an error. The figure shows what may be considered the bounds of acceptable unintended initial twist, since 10° is commonly specified as the angle at which to apply deductions. The minimum number of somersaults required is 6.8 for the oscillation, meaning close to a half twist may be observed in 3.4 somersaults. If the athlete is not performing this many somersaults, a slight unintended twist at take-off will not be of concern for producing undesired twist. In gymnastics, double layout somersaults are the hardest multiple-somersault layout skill currently performed, and the layout is not held for the entirety of the two somersaults since some flexion at the hips is a part of the follow through from take-off and the preparation for landing. In diving, multiple somersaults in the layout posture are not common. Thus, for this particular athlete, unintended twist in the layout posture is not a significant practical concern. Since different athletes will have different inertial properties, this conclusion is not broadly applicable. The pike only produces oscillating twist, with the magnitude of the oscillations increasing as /o decreases. The shape of Fig. 11 indicates that as the magnitude of the oscillations increases, so does the number of somersaults required per oscillation in /. This makes pike a good posture for preventing twist: the twist that will occur is small compared to tuck and layout and, since for larger oscillations the number of somersaults required increases, the athlete may have landed before the maximum twist has been reached. For this example athlete, in Fig. 12, the transition between continuous and oscillating twist for the tuck occurs at /o just above 60°. Since this would require the athlete to perform actions which tilt their longitudinal axis by 30° towards the angular momentum vector before continuous twist is observed, tuck is not a good posture for seeking to perform twisting somersaults. It would be more useful for preventing twist. Unlike pike, the number of somersaults per oscillation decreases as /o decreases, and so the number of somersaults decreases as the oscillation size increases, meaning that the oscillation will be more readily be seen. Tuck may be used to prevent twist, but pike will be more effective. 3.4. Assuming equality of the transverse and medial moments of inertia In prior analyses of the twisting somersault it has been assumed (Batterman, 1968; Frohlich, 1979; Rackham, 1970) that the transverse and medial moments of inertia are equal. This assumption considerably simplifies the equations of motion but may not be sufficiently accurate to estimate the capacity of an athlete to perform a particular twisting somersault skill. This section provides an assessment of the error introduced by the assumption, in terms of the difference in the number of somersaults required to achieve a particular amount of twist. If it is assumed that the transverse Iyy and medial Ixx moments of inertia are equal then the twist-to-somersault angular velocity ratio is described by Case 10. Since Ixx and Iyy will in practice be close rather than equal, their average value Ia may be used. Integrating Eq. (7) remembering that / is constant in this situation, and dividing by 2p gives the number of somersaults required for any amount of twist:

N¼

Izz ðw
wo Þ 2pðIA
Izz Þ cos /

ð11Þ

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

37

Assuming that Ixx and Iyy are equal will always predict continuous twist. Thus, when making this assumption it is essential that the true situation does in fact display continuous twist. Continuous twist is observed only in Cases 3 and 7 when the medial and transverse moments of inertia are unequal. Accordingly, /o must be less than the value that marks the transition between continuous and oscillating twist; that is /oCrit1 or /oCrit2 depending on whether Iyy or Ixx is the intermediate moment of inertia. The focus here is the difference in the number of somersaults required per half twist that is predicted with and without assuming that the medial and transverse moments of inertia are equal, Nh(error), since the number of somersaults required predicts if a particular skill is achievable by an athlete. The difference is the number of somersaults predicted under the assumption of equality of Ixx and Iyy given by Eq. (11), and the actual number of somersaults required, H3/2p or H7/2p, as appropriate. The difference will decrease as the ratio Ixx/Iyy approaches unity, as the ratio Iyy/Izz becomes large, and as /o decreases. One may think that quoting a percentage difference between Ixx and Iyy would be sufficient to decide when the assumption of equality is reasonable; however, the value of Izz compared to Ixx and Iyy, as well as the initial conditions will also affect the difference in the number of somersaults predicted, and so must also be considered. Fig. 13 shows the difference in the number of somersaults predicted with and without making the assumption of equality when /o is set to 80° for a reasonable range of inertia ratios Ixx/Iyy and Iyy/Izz. Huston’s fiftieth percentile male in the layout posture is located at coordinates (14.15, 1.012), marked by the black dot. The inertial ratios of this example person in the pike and tuck postures fall outside the domain of the plot and, in any event, both postures cause oscillatory twist for /o = 80° so the assumption of equality is clearly wrong. To see how the difference in the predictions of the required number of somersaults depends on /o, consider Huston’s fiftieth percentile male in the layout posture. Layout is chosen since it will predominantly produce continuous twist. Fig. 14 shows the difference in the predictions. It can clearly be seen that the error in the number of somersaults, Nh(error) decreases as /o decreases. The assumption of equality has caused the number of somersaults required to be overestimated; the athlete will finish the twist ‘early’, leaving a longer time to prepare for landing. In contrast, if the assumption underestimated the somersault required the athlete would not complete the skill. Thus a larger error is acceptable when it is an overestimate. Based on the error values in Fig. 14 it would be reasonable to assume equality of Ixx and Iyy for values of /o less than 86°, or if the athlete is performing at least a full twist. The error would then be less than 0.03 of a somersault, or 10°. Ten degrees is a typical point at which deductions will start to be made by judges. It must be remembered that this conclusion is athleteand posture-specific. If, rather than specifying the initial conditions, the assumption of equality of the medial and transverse moments of inertia is used to predict the initial value of /o required to achieve a particular skill, and the error is still measured as the difference in the number of somersaults required, the error pattern will differ from Fig. 13. This is because the various inertial property ratios will require different values of /o to achieve the same skills under of the assumption of equality of the medial and transverse moments of inertia. The value of /o alters the error. It is thus essential that the quantity being predicted using the equations of motion with Ixx = Iyy is known before deciding for what inertial property combinations the assumption of equality is reasonable.

Fig. 13. Error expressed as the number of somersaults required when assuming equality of Ixx and Iyy less the number of somersaults required when not assuming equality, when /o is set to 80°. The layout posture of Huston’s fiftieth percentile male would have coordinates (14.15, 1.012) as marked by the black dot. The white region is where the posture displays oscillating twist.

38

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

Fig. 14. Error in number of somersaults due to assuming that Ixx = Iyy for Huston’s fiftieth percentile male in layout as a function of /o when wo = 0.

4. Conclusions Equations have been derived that are applicable to the quasi-rigid phase of a twisting somersault. These equations show that the twist may be, for practical purposes, continuously-rotating or oscillating, and allow calculation of the somersault required per half rotation or full oscillation of twist. This description is directly applicable to the twisting somersault skills that are defined by the governing sporting bodies in terms of integer numbers of full- or half-twist rotations per somersault. An athlete can assume postures where each of the moments of inertia is the intermediate-valued one; this was affirmed with one example athlete. As a consequence, no assumptions were made regarding the order of the magnitudes of the principal moments of inertia, so the presented equations may be applied to any posture that an athlete may use. Three example postures—layout, pike and tuck—having different moments of inertia as the intermediate-valued one were used to illustrate the amount of somersault required per half twist, or oscillation in twist, and discuss how this will affect skill achievement. As expected, the somersault angle continuously increases, although the somersault angular velocity may oscillate about a constant value, with the magnitude the oscillations depending on the difference between the medial and transverse moments of inertia, and the magnitude of the oscillations in twist. When seeking to maximise the twist rate with respect to the somersault rate, an athlete should adopt the following general strategies.  Reduce the initial angle between their principal longitudinal axis and their angular momentum vector.  Avoid any initial twist when they are using a posture where Iyy is the intermediate-valued moment of inertia.  Use as much initial twist as can be achieved without incurring a deduction when Ixx is the intermediate-valued moment of inertia.  Choose a posture with Iyy as the intermediate moment of inertia since it will display continuous twist sooner when decreasing the angle between their principal longitudinal axis and the angular momentum vector  Choose a posture with the smallest possible Izz value relative to Iyy and Ixx, as this boosts the twist-to-somersault angular velocity ratio. Of the three example postures evaluated, layout was the most suitable for performing twisting somersaults. When seeking to prevent continuous twist a posture with Izz as the intermediate-valued moment of inertia appears to be the best choice, since it can only display oscillating twist. It is, however, possible that a posture with Ixx as the intermediate moment of inertia may lead to twist oscillation with a smaller amplitude and/or a longer period, depending on the values of the other two moments of inertia. The two specific postures should therefore be compared for the specific athlete and skill of interest. For the example athlete used here, pike is superior to tuck for preventing continuous twist, and for causing the aerial motion to appear as if the twist had ceased. The implications of assuming that the medial and transverse moments of inertia are equal were discussed. It was shown that the difference between the required number of somersaults per half twist predicted with and without making this assumption depends on the ratios Ixx/Iyy and Iyy/Izz, as well as on the initial conditions. Knowledge of the ratio of Ixx to Iyy is thus insufficient to decide whether making the assumption is justified. Less error is observed when Ixx/Iyy is closer to unity, for larger values of Iyy/Izz, and for smaller angles between the athlete’s principal longitudinal axis and their angular momentum vector. For the example case given, it would be reasonable to assume equality for the layout posture when seeking to perform at least a full twist within a somersault.

J. Mikl, D.C. Rye / Human Movement Science 45 (2016) 23–39

39

Conflict of interest The authors have no financial or personal relationships with any people or organisations that could inappropriately influence this work. Acknowledgements The authors acknowledge the support of the Australian Research Council (ARC) through LP100200245. Joanne Mikl was supported by an ARC Australian Postgraduate Award and by a University of Sydney Vice-Chancellor’s Research Scholarship. References Batterman, C. (1968). The Techniques of Springboard Diving. Cambridge, USA: MIT Press. CoP MAG, 2013. 2013-2016 Code of Points: Men’s Artistic Gymnastics. Lausanne, Switzerland: Fédération Internationale de Gymnastique. Retrieved July 1, 2015, from We Are Gymnastics!: http://www.fig-gymnastics.com/site/rules/disciplines/art/. CoP TRA, 2013. 2013-2016 Code of Points: Trampoline Gymnastics. Lausanne, Switzerland: Fédération Internationale de Gymnastique. Retrieved July 1, 2015, from We Are Gymnastics!: http://www.fig-gymnastics.com/site/rules/disciplines/tra/. CoP WAG, 2013. 2013-2016 Code of Points: Women’s Artistic Gymnastics (Nov. 2014 ed.). Lausanne, Switzerland: Fédération Internationale de Gymnastique. Retrieved July 1, 2015, from We Are Gymnastics!: http://www.fig-gymnastics.com/site/rules/disciplines/art/. DR FINA, 2015. Part V Diving Rules 2015-2017. In FINA Handbook 2015-2017. Lausanne, Switzerland: Fédération Internationale de Natation. Retrieved July 1, 2015, from http://www.fina.org/H2O/index.php?option=com_content&view=article&id=4161&Itemid=184. Frohlich, C. (1979). Do springboard divers violate angular momentum conservation? American Journal of Physics, 47(7), 583–592. George, G. S. (2010). Championship Gymnastics: Biomechanical Techniques for Shaping Winners. Carlsbad, USA: Wellness Press. Huston, R. L. (2009). Principles of Biomechanics. Boca Raton, USA: CRC Press. doi: 10.1201/9781420018400. O’Brien, R. F. (2003). Springboard and Platform Diving (second ed.). Champaign USA: Human Kinetics. Paul, R. P. (1981). Robot Manipulators: Mathematics, Programming and Control. Cambridge, USA: MIT Press. Rackham, G. W. (1970). The fascinating world of twist: Twist by somersault transfer. Swimming Times, 47(6), 263–267. Shah, S. V., Saha, S. K., & Dutt, J. K. (2012). Dynamics of Tree-Type Robotic Systems. Dordrecht: Springer. Smith, P. G., & Kane, T. R., 1967. Technical report no. 171: The reorientation of a human being in free fall. Stanford USA: Division of Engineering Mechanics, Stanford University. Synge, J. L., & Griffith, B. A. (1959). Principles of Mechanics (third ed.). New York: McGraw-Hill. Weisstein, E.W., 2014a. Elliptic integral of the third kind. Retrieved October 21, 2014, from MathWorld, A Wolfram Web Resource: http://mathworld.wolfram.com/EllipticIntegraloftheThirdKind.html. Weisstein, E.W., 2014b. Elliptic integral of the first kind. Retrieved October 21, 2014, from http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html. Yeadon, M. R. (1993). The biomechanics of twisting somersaults: Part 1: Rigid body motions. Journal of Sports Sciences, 11(3), 187–198. Yeadon, M. R., 1984. Mechanics of twisting somersaults (PhD Thesis), Loughborough University of Technology. Retrieved from https://dspace.lboro.ac.uk/.