Kinematic and Kinetic Analysis of the Fouetté Turn in Classical Ballet

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1 Journal of Applied Biomechanics, 2010, 26, Human Kinetics, Inc. Kinematic and Kinetic Analysis of the Fouetté Turn in Classical Ballet Akiko Imura, Yoichi Iino, and Takeji Kojima The fouetté turn in classical ballet dancing is a continuous turn with the whipping of the gesture leg and the arms and the bending and stretching of the supporting leg. The knowledge of the movement intensities of both legs for the turn would be favorable for the conditioning of the dancer s body. The purpose of this study was to estimate the intensities. The hypothesis of this study was that the intensities were higher in the supporting leg than in the gesture leg. The joint torques of both legs were determined in the turns performed by seven experienced female classical ballet dancers with inverse dynamics using three high-speed cine cameras and a force platform. The hip abductor torque, knee extensor and plantar flexor torques of the supporting leg were estimated to be exerted up to their maximum levels and the peaks of the torques were larger than the peaks of their matching torques of the gesture leg. Thus, the hypothesis was partly supported. Training of the supporting leg rather than the gesture leg would help ballet dancers perform many revolutions of the fouetté turn continuously. Keywords: turn, dance, kinetics, balance Expert ballet dancers can continuously perform over 30 revolutions of the fouetté turn used in classical ballet dancing, whipping the gesture leg and the arms and bending and stretching the supporting leg (Figure 1 A to J). The arm ipsilateral to the gesture leg follows the motion of the leg (Laws, 1984). The other arm is kept at the side of the dancer after the leg is extended in front of the dancer until the leg moves to the side (Figure 1 C to G) (Warren, 1990). During these limb movements, the sole of the supporting foot is in contact with the floor (Laws, 1984). Both arms are kept in front of the chest during turning on tiptoe (Figure 1 J to A) (Warren, 1990). In the clockwise fouetté turn, the right leg is swung as the gesture leg and the left leg is the supporting leg. The whipping of the gesture leg and ipsilateral arm to the leg is important for the continuous turns. A frictional torque from the floor generated by the whipping supplies the angular momentum for the next revolution (Laws, 1984; Imura et al., 2008). The hip abductors and adductors of the leg are assumed to be activated for the whipping and stop of the leg, respectively (Laws, 1998). In addition, bending and stretching of the supporting leg is used during the turn to supply the frictional torque for the turn and lessen it against the turn (Imura et al., 2008). Classical ballet dancers exert a larger hip abductor torque in the supporting leg than in the rotating gesture leg in grand rond de jambe en l air in ballet (GRJL; the straightened gesture leg rotates horizontally around the supporting leg without the bending and stretching of the latter leg; in their Figure 1 of Kwon et al., 2007). The peak of the hip abduction torque of the supporting leg is estimated to be about 50% of the maximum isokinetic hip abductor torque measured at 60 deg s 1 for female ballet dancers by Hamilton et al. (1992). A lateral translation Akiko Imura (Corresponding Author), Yoichi Iino, and Takeji Kojima are with the Department of Sports Sciences, Graduate School of Arts and Sciences, University of Tokyo, Tokyo, Japan. Figure 1 Traces of the fouetté turn for a dancer from 20 to 70% of normalized time for one complete revolution of the turn. 484

2 Kinematic and Kinetic Analysis of the Fouetté Turn 485 of the pelvis for the maintenance of body balance against the gesture leg swing caused such a magnitude of torque (Kwon et al., 2007). The peak hip abductor torque of the supporting leg would be larger in the fouetté turn than in GRJL from the following reasons: The pelvis would also be translated laterally for the maintenance in the turn, and the ground reaction force (GRF) acting on the supporting foot would be larger in the turn than in GRJL because of an upward acceleration of the whole body due to the leg bending and stretching in the turn (Imura et al., 2008; Wilson et al., 2004). The knee and ankle joints as well as the hip joint of the supporting leg would be required not only to bear body weight from gravity force but also to accelerate the whole body upward though the hip, knee and ankle joints of the gesture leg would be required to accelerate the leg segments distal to their joints. Thus, it seems that the joint torques of the supporting leg are larger than their matching joint torques of the gesture leg during the swing phase of the turn. The scientific analyses of ballet techniques would increase dancers and instructors understanding of stresses placed on the dancer s body. This would be useful for designing a training program, enhancing stage performances and preventing the incidence of injuries against growing physical demands placed on today s dancers (Koutedakis, 2005a). The motion analyses of ballet techniques themselves as well as the measurements of cardiorespiratory fitness, body flexibility and body composition (Koutedakis, 2005b) would contribute to the understanding. The movement intensities of both legs in the fouetté turn have not been investigated though the vertical components of joint torques at the hip joint of the gesture leg and both shoulder joints have been reported in the turn (Imura et al., 2008). The purposes of this study were to investigate the joint kinetics of the limbs during the fouetté turn and to estimate their movement intensities for the understanding. The hypothesis of this study was that the intensity of the movement would be higher in the supporting leg than in the gesture leg during the swing phase of the turn (Figure 1 C to G). segments for each limb after Ae et al. (1992). The tape was wound around both knee and ankle joints and metatarsophalangeal articulations. The markers were attached to the skin or the clothing on the relevant anatomical landmarks; both anterior superior iliac spines, the middle of both posterior superior iliac spines, skin on the tibia in both lower legs, calcaneal process and the center of both lateral and medial malleoli. Half casts, which were molded from fiberglass casting tape with more than three markers were applied to the surfaces of the thighs and lower legs for the determination of the segment coordinates and angular velocities (Figure 2). After proper warming up, the dancer performed more than five continuous fouetté turns to a same music (third act, coda, in Don Quixote Grand Pas de deux) on a force platform (type 9281B, Kistler) several times wearing soft ballet shoes: The dancer turned not on tiptoe but on the ball of the supporting foot owing to the shoes. The turns were filmed using three 16 mm cine cameras (1PL and 1P, Photo-Sonics, Inc.) at nominal film speeds of 150 fps, 150 fps and 100 fps, respectively. The actual film speeds were determined using time markers of 100 Hz from a pulse generator recorded at the side of the films. The average speeds were fps, fps and 99.4 Participants Methods Seven healthy female ballet dancers (age 27.7 ± 1.7 years; height 1.58 ± 0.06 m; body mass 47.8 ± 3.0 kg) who had performed classical ballet for 20.6 ± 3.2 years participated in this study. All the dancers preferred the fouetté turn in the clockwise direction with the left leg being the supporting leg, which this study used. The experimental procedure was approved by the ethics committee of the University of Tokyo and written informed consent of each subject was obtained before data collection. Experimental Procedure Spherical markers and adhesive tape were used on the lower limbs of the dancer to make a model of three rigid Figure 2 Marker configuration established from relevant landmarks on the dancer s body for calculating joint kinematics and kinetics.

3 486 Imura, Iino, and Kojima fps, respectively. The GRF acting on the supporting foot was measured at a sampling rate of 1000 Hz using the force platform synchronized electronically with the three cine cameras: The synchronization was done with the generator. To determine the positions of the hip joint centers by a functional method (Gamage & Lasenby, 2002), the motions of the lower limbs of the dancer were captured with two video cameras (DXC-200A, SONY) at 60 Hz. The motions were flexion and extension, and then abduction and adduction with about 90 range of motion. The positions of the markers in the videotaped images were digitized using a motion analysis system (Frame-DIAS II, DKH). Three-dimensional (3D) coordinates of the markers were reconstructed using similar procedures to those for the cine camera data. The functional method by Gamage & Lasenby (2002) assumes that there is no error in the digitization of the markers and that the joint is a true spherical joint. The bias caused by the assumptions was corrected using the algorithms presented by Halvorsen (2003). Data Analysis We analyzed one revolution from each dancer s best trial judged by herself. The positions of the markers and the joint centers in the film were manually digitized with a film motion analyzer (NAC Image Technology, Inc.). Three-dimensional coordinates of the digitized points were reconstructed using a direct linear transformation (DLT) method (Adbel-Aziz & Karara, 1971). Thirty markers placed in a 1.9 m 1.6 m 1.8 m space were used to determine DLT coefficients for each camera. The root mean square (RMS) errors in the reconstructed 3D coordinates of the markers were less than 3.3 mm. The films from the three cameras were synchronized using the algorithms of Pourcelot et al. (2000) after the correction of the film speeds by the time markers of 100 Hz. This synchronization method is based on that the two-dimensional (2D) coordinate of a marker calculated from the reconstructed 3D coordinates of the marker using the DLT coefficients becomes the same with the original 2D coordinate for the reconstruction of the 3D coordinate if the original 2D coordinates of the marker from the different cameras are synchronized well. The 3D coordinates were filtered with a fourth order, zerophase lag, Butterworth digital low pass filter with a cut off frequency of 10 Hz. GRF data were also filtered with a fourth order, zero-phase lag, Butterworth digital low pass filter with a cut off frequency of 10 Hz. This avoids a large distortion of the joint torques determined with inverse dynamics at impact phases (Bisseling & Hof, 2006): The underestimation of segmental accelerations caused by the calculation using low-pass filtered positional data and the use of raw GRF data may overestimate joint torques at the phases. The center of pressure (COP) was determined from the filtered GRF and corrected after Sommer et al. (Sommer et al., 1997) to eliminate systematical errors, probably due to minute inclinations of the sensors of the force platform (Bobbert & Schamhardt, 1990). COP was used to determine the external moment of the GRF and the free moment: The latter moment is a force couple about a vertical axis resulting from frictional forces between the foot and the force platform (Holden & Cavanagh, 1991). The position of the COP is inaccurate when the vertical component of GRF is small (Bobbert & Schamhardt, 1990) and so both moments were assumed to be zero when the component was less than 10 N. The torques exerted at the hip, knee and ankle joints were determined using the following equation described by Hof (1992): M = M ( r r ) F r r ) m g k e e [ ( i ] k e k i i= 1 k k d + [ ( ri rk ) mia i ]+ ( Ii i ) i= 1 i= 1 dt In this equation, m i is the mass of segment i, I i is the inertia tensor of segment i, r k is the position vector of the proximal joint center of segment k, r i is the position vector of the center of mass of segment i, r e is the position vector of the COP, i is the angular velocity of segment i, a i is the acceleration of the center of mass of segment i, g is the acceleration due to gravity, M k is the external moment acting at the proximal joint of segment k or joint torque, M e is the external moment, F e is the external force and t is the time. F e and M e were both zero for the arms and gesture leg and were equal to GRF and the moment due to GRF plus the free moment acting on the foot for the supporting leg, respectively. The joint reaction force acting from the thigh to the pelvis at the hip joint of the gesture leg was determined from the acceleration of the center of mass of the leg and its mass. The inertial properties of the limbs were estimated using the coefficients reported by Ae et al. (1992). The setup and global coordinate system of the experimental site were the same as those of the previous study (Imura et al., 2008). All the coordinate systems were right handed orthogonal systems. Each joint coordinate system of a dancer was defined using two vectors from two adjacent body segments and their cross product when she was standing with the limbs flexed: Each vector joined the joint centers at both ends of the segment (Table 1). The hip, knee and ankle joint centers were used to define the systems. The orientation of the local coordinate system of a segment was the same as that of the joint coordinate system at the proximal joint of the segment. The coordinates of the three markers attached on a segment during the fouetté turn were used to determine the local coordinate system of the segment during the motion with the help of the rotational matrix relating the orientation of the three markers to the local coordinate system during the standing posture with the limbs flexed. The sets of three markers of the casts on the thighs and the lower legs, the makers on the calcaneal process and both lateral and medial malleoli and the center of the ankle joint were used to define the rotation matrices for the lower limb segments. k

4 Kinematic and Kinetic Analysis of the Fouetté Turn 487 Table 1 Definitions of joint coordinate systems Joint Vector Joint Coordinate Systems Hip x : from HJC to KJC y : x the vector joining both HJCs External (+) / internal ( ) rotation axis Abduction (+) / adduction ( ) axis z : x y Extension (+) / flexion ( ) axis Knee x : from KJC to AJC External (+) / internal ( ) rotation axis y : x the vector from KJC to HJC Extension (+) / flexion ( ) axis z : x y Valgus (+) / varus ( ) axis Ankle x : from AJC to MPJC of the foot Eversion (+) / inversion ( ) axis y : x the vector from AJC to KJC Plantar flexion (+) / dorsiflexion ( ) axis z : x y Abduction (+) / adduction ( ) axis Note. HJC: hip joint center, KJC: knee joint center. AJC: ankle joint center. The directions of the axes in the systems of both lower limbs are adjusted to these notations of the axes. The hip joint angles were determined using Euler angles by the rotation of the local coordinate system of the thigh against the system of the pelvis with the following sequences: Mediolateral-anteroposterior-longitudinal (Table 1 ) (Yeadon, 1990). The angles of the knee and ankle joints were described by the flexion-extension motion using the angular differences of the longitudinal axes of the proximal and distal segments. The angular velocities of the limb segments were determined using Poisson s equation (Wittenburg, 1977) after the determination of the rotation matrices from three markers on the segments using Veldpaus s algorithm (Veldpaus et al., 1988). The joint angular velocities of the limbs were determined through subtraction of the angular velocity of a proximal segment from that of the adjacent distal segment. Normalization of the Kinetic Variables and Estimation of the Relative Magnitudes of Joint Torques The joint torque of each dancer was normalized using the product of the whole body weight and the height of the dancer. The joint angular velocities and the normalized torques were projected onto the axes of relevant joint coordinate systems. Duration of the turn was divided by the time taken for one revolution and was shown in percent. The 0% (Figure 1C) was the time when the heel of the supporting foot touched the force platform from a standing on the ball of the foot: The time was judged from a spike of GRF. The range of the time when the averages of all the dancers could be determined was from 20 to 40% of time for one revolution (0.85 ± 0.04 s). This time duration was narrower than the range of time when the digitization of the markers for each dancer was possible because the relative orientations of the dancers to the cameras at 0% of time were not the same. The relative magnitudes of the peak joint torques of the limbs except the peak hip internal and external rotator torques in the turn were estimated using the maximum isokinetic torques at 60 deg s 1 by ballet dancers (Hamilton et al., 1992) or the maximum isometric torques by sedentary females (Dean et al., 2004). The relative magnitude of the internal rotator torque was estimated from the relationship between maximum isokinetic torque at 30 deg s 1 by dancers and the joint angle (Gupta et al., 2004): The maximum isokinetic torque at the joint angle where the peak torque appeared in the turn was determined through the linear interpolation of the torques measured at 9 and 23 degrees of internal rotation. The relative magnitude of the external rotator torque was estimated from the relationship between the maximum isokinetic torque by sedentary females and joint angular velocity (Cahalan et al., 1989): The maximum isokinetic torque at the joint angular velocity where the peak torque appeared in the turn was determined through the linear interpolation of the torques measured at 86 and 111 deg s 1 of external rotation. Statistics The differences in mean peak joint torques between both hip, knee and ankle joints were tested using two-tailed paired t tests. The significance level was set as p <.05, considering the multipleness of the tests (Holm, 1979). The tests were done 10 times. Results Movement Intensities of Both Legs The peaks of the hip abductor, knee extensor and plantar flexor torques of the supporting leg were higher than those of their matching torques of the gesture leg (p <.05, Table 2). The peak of the hip extensor torque of the supporting leg tended to be higher than the peak of the gesture leg (p =.024 without taking account of multiple tests, Table 2). The peak hip internal rotator torque of the gesture leg tended to be higher than that of the supporting leg (p =.0085 without taking account of multiple tests, Table 2). The hip abductor, hip extensor, knee extensor and plantar flexor torques of the supporting leg tended to be larger than their matching torques of the gesture leg during the swing phase (Figure 3).

5 488 Imura, Iino, and Kojima Table 2 Peaks of selected joint torques and their times of appearance during the fouetté turn Body Part Joint Torque Maximum Torque Strength (% Maxima) P Values Supporting Leg Hip Abduction 0.14 ± ± 0.02 a 110% * Adduction ± ± 0.03 a 10% 0.19 Flexion ± ± 0.03 b 70% 0.15 Extension 0.13 ± ± 0.02 b 120% Internal rotation ± ± d 33% External rotation ± ± c 160% 0.75 Knee Flexion ± ± a 30% 0.54 Extension 0.14 ± ± 0.02 a 100% * Ankle Plantar flexion 0.12 ± ± a 150% Dorsiflexion ± ± a 13% Gesture Leg Hip Abduction ± ± 0.03 a 36% Adduction ± ± 0.04 a 40% Flexion ± ± 0.03 b 57% Extension ± ± 0.02 b 38% Internal rotation 0.03 ± ± d 86% External rotation ± ± c 83% Knee Flexion ± ± a 33% Extension ± ± 0.03 b 17% Ankle Plantar flexion ± ± a 4% Dorsiflexion ± ± a 22% Note. Each torque value was divided by the product of the whole body mass and the height for each dancer. The value after the plus/minus sign is 1 standard deviation (SD). Ratios of the peak torques to their maxima were estimated using maximum torque values of the previous studies as follows: a, Hamilton et al., 1992; b, Dean et al., 2004; c, Gupta et al., 2004; d, Cahalan et al., The p value is not taken account of multiple tests. *Indicates that the peak torque of the supporting leg is statistically different from that of the gesture leg (p < 0.05). Hip Joint Kinematics and Kinetics of Both Legs The hip joint of the gesture leg began to abduct with its flexed position of about 1 radian as the knee joint of the leg extended (Figure 4.1 a, from 0% to 19% of time). The abductor torque changed to the adductor torque at the hip joint when the abduction velocity of the joint started to decrease (Figure 4.1 c, 11% of time). The joint abducted and flexed about 0.8 and 0.5 radian in total, respectively (Figure 4.1 b). The hip joint of the supporting leg began to extend with its adducted and flexed position of about 0.1 radian and 0.4 radian, respectively (Figure 5.1 a, b after 7% of the time). Though the joint abducted only 0.1 radian at the largest (Figure 5.1 a), the abductor torque of the joint was greater than the extensor torque at the joint for not a short period around the heel contact (Figure 5.1 c, from 14.5 to 19% of time). Knee Joint Kinematics and Kinetics of Both Legs The knee joint of the gesture leg extended until 6% of time and then flexed (Figure 4.2 a, b). The knee flexor torque was exerted almost throughout the entire time analyzed (Figure 4.2 c). The knee joint of the supporting leg flexed and then extended, exerting a large extension torque (Figure 5.2 a, b and c, at 6% of time). Ankle Joint Kinematics and Kinetics of Both Legs The ankle joint of the gesture leg scarcely changed all through the time analyzed (Figure 4.3 a). The ankle joint of the supporting leg dorsiflexed and then plantar flexed, exerting a large plantar flexor torque except around the beginning of dorsiflexion and the end of plantar flexion. (Figure 5.3 c, around 15% and 32% of time). Discussion The hypothesis of this study was that the intensity of the movement would be higher in the supporting leg than in the gesture leg during the swing phase of the fouetté turn (Figure 1 C to G). As for only relatively high intensity joint torques, the peaks of the hip abductor, knee extensor and plantar flexor torques of the supporting leg were

6 Figure 3 Comparison of selected joint torques between both legs. Solid and dotted lines indicate the averaged joint torques of the gesture leg and supporting leg, respectively. Thin lines drawn above and under a thick line indicate one standard deviation. Figure 4 Averaged joint kinematics and kinetics of the hip joint (4.1), knee joint (4.2) and ankle joint (4.3) of the gesture leg for the seven dancers. (a) joint angles determined as Euler angles (b) joint angular velocities and (c) joint torques. Bw: body weight, ht: height. Er = external rotation; ir = internal rotation; Ab = abduction; ad = adduction; Ex = extension; fl = flexion; Val = valgus rotation; var = varus rotation; Df = dorsiflexion; Pf = plantar flexion; Ev = eversion; in = inversion. 489

7 490 Imura, Iino, and Kojima Figure 5 Averaged joint kinematics and kinetics of the hip joint (5.1), knee joint (5.2) and ankle joint (5.3) of the supporting leg for the seven dancers. (a) joint angles determined as Euler angles (b) joint angular velocities and (c) joint torques. Bw: body weight, ht: height. Er = external rotation; ir = internal rotation; Ab = abduction; ad = adduction; Ex = extension; fl = flexion; Val = valgus rotation; var = varus rotation; Df = dorsiflexion; pf = plantar flexion; Ev = eversion; in = inversion. about 2.8, 5.4 and 40 times, respectively, higher than the peaks of their matching torques of the gesture leg (Table 2). The peak hip extensor torque of the supporting leg tended to be higher than that of the gesture leg. However, the peak hip internal rotator torque of the gesture leg tended to be higher than that of the supporting leg. Thus, the hypothesis was partly supported in terms of the peak joint torques. The magnitudes of torque exertion of the hip abductor, hip extensor, knee extensor and plantar flexor torques of the supporting leg were generally larger than those of their matching torques of the gesture leg during the phase (Figure 3). This also supports the hypothesis. Large joint torques were required for the supporting leg in the fouetté turn (Figures 5.1 c, 5.2 c and 5.3 c). The large abduction torque would be required owing to the lateral translation of the pelvis (Kwon et al., 2007) and a large GRF due to the upward acceleration of the whole body. The normalized peak hip abductor torque of the turn was about 1.9 times higher than that in GRJL (Kwon et al., 2007) though the vertical component of the peak GRF was about 2.6 times of the body weight in the turn. The peak of the torque appeared at the time when the peak GRF appeared in the turn (Figure 6). The GRF in GRJL is presumably almost equal to the dancer s body weight because the gesture leg moves in the horizontal plane and the remainder of the body is stationary (Kwon et al., 2007). Hence, the normalized lever length i.e., the length/ height of hip adduction for the GRF would be shorter in the turn than in GRJL. The reason for the difference is unclear. The large valgus torque at the knee joint of the leg (Figure 5.2 c) would be explained from a similar reason to that for the abductor torque at the hip joint. The large knee extensor and plantar flexor torques would be brought about for the upward acceleration of the whole body with a relatively deep leg bending around the heel contact (Figures 5.2 a, 5.3 a). From the estimation of the required joint torque intensities for the fouetté turn (Table 2), it is suggested that strength training of the supporting leg improves the performance of the turn. Training of the hip abductors can be especially beneficial because ballet bar work being a typical training method in classical ballet would result in providing many bending and stretching movements for the knee and ankle joints and few training opportunities for the hip abductors if the dancer is not conscious with the muscles.

8 Kinematic and Kinetic Analysis of the Fouetté Turn 491 Figure 6 Stick pictures of both legs with GRF when the peak hip abductor torque was observed in a representative dancer. (a) A view from above of the dancer (enlarged) (b) A sagittal view of the dancer. (c) A view from the position of the cine camera placed diagonally forward left of the dancer. We estimated the relative magnitudes of some peak joint torques exerted by the dancers in the fouetté turn using the values reported in the previous studies (Table 2). However, the conditions for the measurements of the torques in the fouetté turn were not necessarily the same as those of the reports about joint angles and joint angular velocities. Though the torques measured at 60 deg s 1 are comparable with those measured in maximum isometric contraction (Wickiewicz et al., 1984), or about 70 95% of maximum isometric torques (Horstmann et al., 1999), some errors may be included in our estimation because of the possible differences in muscle length, muscle shortening velocity and co-contraction of muscles between the measurements. Further studies would be necessary for more accurate estimation of the relative magnitudes of torque exertion. The dancers turned the fouetté on the force plate with soft ballet shoes. The joint torque components vertical to the floor would have changed if they had turned the fouetté wearing shoes which had a different frictional coefficient against the floor. In summary, female ballet dancers exerted the hip abductor torque of the gesture leg for the leg whipping and stopped it with the hip adductor and flexor torques of the leg in the fouetté turn. The turn required the larger hip abductor, knee extensor and plantar flexor torques of the supporting leg. Strength training for the hip abductors, knee extensors and plantar flexors of the supporting leg would help ballet dancers perform many revolutions of the fouetté turn continuously. Acknowledgments Figures 1 and 2 were reprinted from Human Movement Science (2008; 27[6]: , Imura A, Iino Y and Kojima T, Biomechanics of the continuity and speed change during one revolution of the Fouetté turn, by permission of Elsevier Science. We thank Y. Miyagi for introducing the dancers in the current study and L. Griffin for her advice on the writing of this paper. References Adbel-Aziz, Y.I., & Karara, H.M. (1971) Direct linear transformation from computer coordinates into object coordinates in close-range photogrammetry. In: Proceedings ASP

9 492 Imura, Iino, and Kojima symposium on close range photogrammetry. Falls Church, VA: American society of photogrammetry, pp Ae, M., Tang, H., & Yokoi, T. (1992) Estimation of inertia properties of the body segments in Japanese athletes (in Japanese). In the Society of Biomechanics, editors. Biomechanisms 11. Tokyo: University of Tokyo Press, pp Bisseling, R.W., & Hof, A.L. (2006). Links handling of impact forces in inverse dynamics. Journal of Biomechanics, 39, Bobbert, M.F., & Schamhardt, H.C. (1990). Accuracy of determining the point of force application with piezoelectric force plate. Journal of Biomechanics, 23, Cahalan, T.D., Johnson, M.E., Liu, S., & Chao, E.Y. (1989). Quantitative measurements of hip strength in different age groups. Clinical Orthopaedics and Related Research, 246, Dean, J.C., Kuo, A.D., & Alexander, N.B. (2004). Age-related changes in maximum hip strength and movement speed. Journal of Gerontology Biological Sciences, 59, Gamage, S.S.H.U., & Lasenby, J. (2002). New least squares solutions for estimating the average centre of rotation and the axis of rotation. Journal of Biomechanics, 35, Gupta, A., Fernihough, B., Bailey, G., Bombeck, P., Clarke, A., & Hopper, D. (2004). An evaluation of differences in hip external rotation strength and range of motion between female dancers and non-dancers. British Journal of Sports Medicine, 38, Halvorsen, K. (2003). Bias compensated least squares estimate of the center of rotation. Journal of Biomechanics, 36, Hamilton, W.G., Hamilton, L.H., Marshall, P., & Molnar, M. (1992). A profile of the musculoskeletal characteristics of elite professional ballet dancers. American Journal of Sports Medicine, 20, Hof, A.L. (1992). An explicit expression for the moment in multibody systems. Journal of Biomechanics, 25, Holden, J.P., & Cavanagh, P.R. (1991). The free moment of ground reaction in distance running and its changes with pronation. Journal of Biomechanics, 24, Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6, Imura A., Iino, Y. & Kojima T. (2008). Biomechanics of the continuity and speed change during one revolution of the Fouetté turn. Human Movement Science, 27, Koutedakis, Y. (2005a). Fitness for dance. Journal of Dance Medicine and Science, 9(1), 5 6. Koutedakis, Y. (2005b). The significance of muscular strength in dance. Journal of Dance Medicine and Science, 9(1), Kwon, Y.H., Wilson, M.A., & Ryu, J.H. (2007). Analysis of the hip joint moments in grand rond de jambe en l air: skilled versus novice ballet dancers. Journal of Dance Medicine and Science, 11, Laws, K. (1984). The physics of dance. New York: Schermer Books. Laws, K. (1998). Momentum transfer in dance movement. Medical Problems of Performing Artists, 13, Pourcelot, P., Audigié, F., Degueurce, C., Geiger, D., & Denoix, J.M. (2000). A method to synchronise cameras using the direct linear transformation technique. Journal of Biomechanics, 33, Sommer, R., Kohler, D., & Calame, C. (1997) Center of pressure (COP) enhancement in piezoelectric force plates. In: Proceedings of XVIth Congress of the International Society of Biomechanics; August 25 29: Tokyo (Japan). Tokyo Metropolitan University, p. 18. Veldpaus, F.E., Woltring, H.J., & Dortmans, L.J.M.G. (1988). A least-squares algorithm for the equiform transformation from spatial marker coordinates. Journal of Biomechanics, 21, Warren, W.G. (1990). Classical ballet technique (p. 22). Florida: University of South Florida Press. Wickiewicz, T.L., Roy, R.R., Powell, P.L., Perrine, J.J., & Edgerton, V.R. (1984). Muscle architecture and force-velocity relationships in humans. Journal of Applied Physiology, 57, Wilson, M.A., Lim, B.O., Kwon, Y.H. (2004). A three-dimensional kinematic analysis of grand rond de jambe en l air: Skilled versus novice ballet dancers. Journal of Dance Medicine and Science, 8, Wittenburg, J. (1977). Dynamics of systems of rigid bodies (p. 29). Stuttgart, Germany: B.G. Teubner. Yeadon, M.R. (1990). The simulation of aerial movement: I. The determination of orientation angles from film data. Journal of Biomechanics, 23,