Quantifying Ballet Technique through Turn Kinematics for Injury Assessment ARCHNES
MASSACHUSETTS INSTITUTE
by
OF TECHNOLOLGY
Hannah Barrett
JUN 2 4 2015
LIBRARIES
Submitted to the
Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
Bachelor of Science in Mechanical Engineering
at the
Massachusetts Institute of Technology
June 2015
@ 2015 Barrett
All rights reserved
The author hereby grants to MIT permission to reproduce and to distribute
publicly paper and electronic copies of this thesis document in whole or
in part in any medium now known or hereafter created.
r
Signature of Author:
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\ 1Hannah Barrett
Department of Mechanical Engineering
May 11, 2015
Certified by:
_Signature
redacted
Di. Barb"ara Hugfty
Engineeri g
Mechanical
Instructor of
Thesis Supervi or
Signature redacted
Accepted by:
Anette Hosoi
Professor of Mechanical Engineering
Undergraduate Officer
Quantifying Ballet Technique through Turn Kinematics for Injury Assessment
by
Hannah Barrett
Submitted to the Department of Mechanical Engineering
on May 11, 2015 in Partial Fulfillment of the
Requirements for the Degree of
Bachelor of Science in Mechanical Engineering
ABSTRACT
The most common turns in ballet, pirouettesandfouett6s require precise movement to
match proper technique and prevent injury. Variation in knee angle of a dancer's
supporting leg during pirouettes of 1-5 rotations was measured using a goniometer for 12
professional and pre-professional ballet dancers. P1d angle saw no significant difference for
different numbers of rotations, but the effect of gender was significant: males plied at least
20.3 degrees deeper than females. Average knee angle while turning was less than zero for
all pirouettes, highlighting dancers' tendency to bend their supporting knee to correct for
instability, which may contribute to knee injuries. Knee angle trajectory closely matched a
minimum jerk profile, indicating that dancers encode movements in angle coordinates.
Knee angle was compared between pirouettes and fouettes using 2 goniometers and aerial
video for 3 advanced amateurs. No difference was found in preparation plid or turn angle
between turn types. Axis of rotation remained within 0.221 0.014 meters of starting
position duringfouett6s. Maximum angular velocity of the head was nearly one and a half
times that of the body duringfouett6s.
This kinematic definition of ballet technique creates a framework for movement control of
a dancer's lower extremities to prevent injury.
Thesis Supervisor: Dr. Barbara Hughey
Title: Instructor of Mechanical Engineering
3
ACKNOWLEDGEMENTS
The author would like to thank Dr. Barbara Hughey for her immense support, guidance,
and encouragement throughout the research. Further thanks to Professor Sangbae Kim,
Matt Haberland, Paul Ragaller, and Professor John Leonard for their input on experimental
design, to Angie Locknar for research assistance, Dr. Weihua Huang for statistical analysis
advice, and to Camille Henrot, Akshai Baskaran, Daniel Bramlet, Joao Luiz Almeida Souza
Ramos, Grace Young, Wyatt Ubellacker, all the professional dancers who participated in the
study, and the staff at the ballet company that made the research possible.
4
Table of Contents
1. Introduction
7
2. Background
2.1 BriefHistory of Ballet
2.2 Physics of Turning in Ballet
2.3 Definition of Knee Angle
2.4 Ballet Turn Technique
2.5 Minimum Jerk Model of Movements
2.6 Knee Misalignment and Injury
2.7 Applications of Sensors to Ballet Technique
2.8 Early Research and Motivation
7
7
8
10
10
13
14
15
15
3. Experimental Design
3.1 Overview of Subjects and Experiments
3.2 Goniometer to DetermineKnee Angle Trajectory over Time during Pirouettes
3.3 Assessment of Relevant Knee Anglesfor Pirouettes
3.4 ComparativeMeasurementsfor Fouett' Turns
3.5 Assessment of Relevant Knee Anglesfor Fouett's
3.6 High Speed Video Analysis
16
16
18
19
22
22
24
4. Pirouette Technique Assessment: Results and Discussion
4.1 PreparationPlie
4.2 Turn Angle
4.3 Turn Duration
4.4 Minimum Jerk Models
27
27
30
35
37
S. Pirouette and Fouette Comparative Analysis: Results and Discussion
5.1 PreparationPlij
5.2 Turn Angle
5.3 Evolution of Plij and Turn Angle with IncreasedRotations
5.4 Dynamic Analysis of High Speed Video
39
39
41
43
45
6. Conclusions
47
7. Further Work
48
References
49
5
List of Figures
Figure
Figure
Figure
Figure
Figure
1:
2:
3:
4:
5:
Side view of a preparation pli.
Diagram of torque applied at the start of a turn, produced by the dancer's turned-out feet.
Definition of supporting knee angle.
Steps of a right-sided ballet pirouette.
Steps of a right-sided balletfouette.
8
9
10
11
12
Figure 6: Popliteus tendon.
14
Figure 7: Average preparation plid angle as reported in the author's past research paper.
Figure 8: Average turn angle as reported in the author's past research paper.
Figure 9: Device attachment for data acquisition.
Figure 10: Sample raw data for a single pirouette completed by a professional-level subject.
Figure 11: Trimmed data from single, double, and quintuple pirouettesby a professional-level subject.
Figure 12: Normalized trials for a single, double, and triple pirouette by one subject.
Figure 13: Experimental setup for measuringfouette turns.
Figure 14: Data for afouette, showing the course of movement of both the standing and rotating legs.
Figure 15: Images from high speed video trials, including tracking dots.
Figure 16: Raw video tracking data, as X and Y coordinates.
Figure 17: Measured angle of each point relation of video tracking data.
Figure 18: Minimum plid angle for single, double, triple, quadruple, and quintuple pirouettes.
Figure 19: Magnitude of plid angles for each pirouette type, separated out by gender.
Figure 20: Magnitude of plie angle for males versus females.
Figure 21: Probability density of magnitude of plid angles for single, double, and triple pirouettes.
Figure 22: Average preparation plid angle across all dancers for pirouettes.
Figure 23: Angle during the course of each type of pirouette turn is shown by subject.
Figure 24: Single pirouette probability density of angle during rotation.
Figure 25: Double pirouetteprobability density of angle during rotation.
Figure 26: Triple pirouette probability density of angle during rotation.
Figure 27: Quadruple and quintuple pirouetteprobability density of angle during rotation.
Figure 28: Characteristic standard deviations while rotating for pirouettes.
Figure 29: Turn duration and turn fraction for each of the 5 types of pirouettes.
Figure 30: Minimum jerk profiles in joint coordinates matching plid movements.
Figure 31: Minimum jerk profiles in Cartesian coordinates matching plid movements.
Figure 32: Parametric displacement of the center of mass of a dancer during one trial.
Figure 33: Magnitude of pli6 angle for single, double, and triple pirouettes andfouett6s for each advanced
amateur subject.
rigure 34: Magnitude of p15 angle in aggregate for single, double, and triple pirouettesand fouettes
completed by the advanced amateur subjects.
Figure 35: Fouette probability density of angle during rotation.
Figure 36: Magnitude of p1id angle versus rotation number of the turn within thatfouette trial.
Figure 37: Average turn angle versus rotation number of the turn within thatfouettd trial.
Figure 38: Standard deviation during turn versus rotation number of the turn for allfouette trials.
Figure 39: Position of the center point of the head of a dancer through the course of 11 fouette turns.
Figure 40: Position of the arm of a dancer in time, position relative to the dancer's moving center.
Figure 41: Angular velocity of the head and body in time.
15
16
18
20
20
21
22
23
24
25
26
27
28
28
29
29
30
31
32
33
34
35
36
37
38
39
6
40
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46
1. Introduction
Classical dance involves precise technique that requires remarkable talent and years of
practice to perfect. At the intersection of art and science, movements in ballet depend upon
careful execution to maintain stability, proper force input and body placement to optimize
dynamics, and skillful rhythm and artistic talent for graceful performance.
Turns are one of the most common and technically complex movements in ballet. The
pirouette turn, a fundamental move in ballet, involves nimbly pivoting one or more times
about one's supporting leg. The fouettd, a more advanced turn of multiple rotations in
succession, involves similar preparation, but requires more complex leg dynamics: a bend
of the supporting leg after each rotation provides an increase in angular momentum. Thus,
experienced dancers can complete a much greater number of rotations during fouettd
turns-16 to 32 for many professional dancers-than for pirouettes-usually3 or 4.
Dancers must follow a specific pattern of knee flexures to remain balanced and ensure
proper technical execution of a turn.1 Consistency in technique is important for advanced
dancers not only to maintain proper aesthetics of the art form but also to prevent injury.2
To achieve proper technique, dancers must stably balance on their supporting leg, push off
their opposite leg with the correct amount of force for the number of turns they hope to
complete, and engage their core to ensure correct upper body alignment-making knee,
arm, and torso angle and alignment vital to successful completion of a turn. With knee, foot,
and ankle injuries accounting for approximately 60% of ballet injuries, proper turn
technique has important implications for dancers. 3
Past research on the kinematics of ballet dance has focused on corrective methods for
the alignment of the body during pirouettes.4 However, few studies have focused
specifically on the alignment of the dancer's body as it relates to technique or analysis of
the supporting leg, the portion of the body most central to maintaining proper balance and
thus preventing injury throughout a turn. Further, little work has compared stability and
control during the two most fundamental turns in ballet, pirouettes and fouettes. This study
determines a mathematical definition of proper technique as defined by optimal knee
angles for both pirouettes and fouettes and assesses the implications of deviations from
optimal angles on dancer stability during turns.
2. Background
Ballet turns involve complex movements to preserve the art form, as dictated by
tradition, and to execute the turn stably, as dictated by physics. A brief history of ballet
technique, summary of positions and terms, and a detailed breakdown of the proper steps
to complete pirouette and fouette turns are detailed below. The importance of knee angle
throughout the course of each turn is emphasized to highlight the significance of the
measurements taken.
2.1 Brief History of Ballet
Ballet is the most emblematic of all dance styles: it is the foundation for nearly every
other form of dance, from jazz to lyrical to modern, and its elegance and specific, traditional
style are well-recognized and admired by many.
7
While the beginnings of ballet can be traced back to the Renaissance courts of the 15th
century, the precise technique of ballet became well recognized in the 17th century, when
Louis XVI created the first ballet school, l'Academie Royale de Danse, in 1661, and Pierre
Beauchamps defined the five fundamental ballet positions in 1670. Over time, it
transitioned from a component of opera to a distinct art form of its own. 5
Ballet has continued to evolve through the present; many countries now have at least
one ballet school and famous company. While some modern ballet productions contain a
storyline, others use movement as a means of expression-translating human emotion
through traditional movements. In spite of the innovations in contemporary choreography,
ballet is an art form that will never lose its tradition: both training and choreography are
still centered around the fundamental positions and movements.5
As Francia Russell, mistress (teacher) of the New York City Ballet writes, "Good ballet
training produces... strong muscles, awareness and control of the entire body, and the
ability to move in many different ways at the request of a choreographer."1
2.2 Physics of Turning in Ballet
Turning in ballet involves considerations of rotational dynamics of the dancer.
Concepts of torque and rotational inertia as they relate to pirouettes and fouettes are
detailed here.
To start turning, a dancer plies-bends her knees with feet turned out-to prepare. A
side view of a plid preceding a pirouette orfouettd turn is shown in Figure 1.
Figure 1: A side view of a preparation plie, the bend of the knee that helps a dancer
prepare for a pirouette or fouettj turn. Dance instructors and professionals suggest
that a deeper pli, or greater bend, allows a dancer to push off the floor with more
control, gaining more angular momentum at the start of the turn.
8
Turning at a comfortable pace to stay balanced is vital to a dancer's stability during
rotation. To begin rotating, a dancer applies sideways, opposite forces with each of her two
turned-out feet, as shown in Figure 2. The torque at the start of the turn is thus
proportional to the frictional force resulting from her push off the floor, which allows the
dancer to begin rotating clockwise. A greater push will provide more torque and thus
produce a faster turn. The distance between the two feet also affects the torque applied,
with a greater spread of the feet making it easier to apply more torque. 2
F
d
=*F
Figure 2: Torque applied at the start of a turn is produced by the dancer's two
turned-out feet. Each applies an approximately sideways force in the direction it
faces. Frictional forces from the floor provide a torque that will allow the dancer to
rotate clockwise.
Sugano and Laws suggest that a deeper, wider plid-the bend of the knee that helps
dancers prepare for a turn-produces a more stable, technically correct ballet pirouette.6
Dance instructors encourage dancers to plid deeper for pirouettesof more than one rotation
to allow dancers to push up from the ground more easily, thus obtaining a greater torque
and angular momentum. This practice allows the dancer to turn faster and makes it more
manageable to complete multiple rotations. 2 A comparison was obtained between executed
plid angles for single, double, and multiple pirouettes to determine if professional ballet
dancers follow this suggestion and if it leads to more technically correct pirouettes, as
described in Section 2.2.
Due to conservation of angular momentum, a dancer's rotational inertia affects the
speed at which he or she rotates. During a pirouette, a dancer maintains the same position,
with her bent leg and arms very close to the body, allowing a fairly rapid spin. However,
during a fouette, a dancer whips her leg out to the front and extends her arms, increasing
her rotational inertia and thus decreasing her rotational speed. As the dancer pulls her
arms and legs back in, she speeds up at the end of afouettd rotation.
In general, a straighter body also decreases the dancer's rotational inertia, allowing her
to turn at a faster rate. Thus, proper posture of not only the dancer's arms, but also her
upper body and leg are important. Maintaining a straight leg and avoiding knee bend
allows the dancer to maintain proper posture, contributing to better balance and greater
turn speed. Understanding proper turn alignment allows dancers to properly exploit the
angular momentum of their bodies, and appear more graceful during turns.
9
2.3 Definition of Knee Angle
A dancer's supporting knee moves through a pattern of angles as she follows the steps
to complete the turn. Angle of the supporting leg is measured as deviation from initial
angle, corresponding to a straight leg, as illustrated in Figure 3. A bend of the knee is
defined as negative directional change in angle.
Figure 3: Supporting knee angle is defined as deviation from initial angle-that is,
how far the leg has bent from it's original straight position. Bent knee, the most
typical change during a turn, is defined as the negative directional change in theta,
shown above.
Based on the positions of standard ballet pirouettes and fouettes, a dancer's supporting
knee should transition through specific knee angles as he or she performs a technically
correct turn. These angles are detailed in the next section, which outlines proper ballet turn
technique.
2.4 Ballet Turn Technique
In ballet, technique refers to the proper method of executing a movement, such as a
turn or leap. The steps that a dancer should move through to complete a pirouette and a
fouett6 turn as defined by proper technique are illustrated below.
The most common and important turn in ballet, the pirouette involves moving through
a series of positions to achieve turning stability and traditional dance form. The steps of a
right-sided pirouette-the most popular direction of turning-are described here,
emphasizing the alignment of the supporting leg.
Dancers begin a pirouette with two steps of preparation. First, both arms are held
horizontal and straight while the right leg is pointed out to the side, as illustrated in Figure
4, position 1. The dancer then plies, or bends, her left leg-the supporting leg-and places
the right leg behind it, turning her knees out (position 2). The left arm remains straight out
to the side, while the right arm is placed horizontally in front of the dancer. Next, the
dancer applies a force to both feet (as shown in Figure 2) and moves into relevd, standing
on the ball of her left foot, and places her right foot at her left knee in passe, forming a
triangle with the bent leg against the thigh of the supporting leg, retire (position 3). As she
moves into this turning position, the dancer brings her arms together and pivots three
10
hundred sixty degrees clockwise. To end, the dancer places both feet down and pliesbends her knees-to stabilize (position 4).7
-
10
0(3
-10
-
B
4-20C"
-
S
V -30
CU
E -400
'-50-
0
-60 0
(2)
0.1
0.2
(1)
0.3
0.6
0.4
0.5
Fraction of Time
(2)
0.7
(3)
0.8
1
0.9
(4)
Figure 4: To perform a right-sided ballet pirouette, a dancer begins with her right
leg to the side and arms out horizontally (position 1). She then places her right leg
behind her, curls her right arm inward, and plies her supporting (left) leg (position
2). The dancer next pushes off with both feet, transfers her weight to her supporting
leg, and enters releve on the ball of her left foot with her right leg in passe as she
begins to turn (position 3); for demonstration, this position is shown here facing
forward. The dancer uses the torque generated from pushing upwards to turn
clockwise and lands facing forward in a finishing plid for balance (position 4). The
graph above the images shows a typical profile of the knee angle of the supporting
leg throughout the turn. Position 4 is not shown in the angle profile because it
occurs after the period that was analyzed.
Thefouett6 is another famous turn in ballet: professional dancers are admired for their
ability to turn for great spans of time, completing manyfouettes in a row, with as many as
thirty-two required for the most famous scene in Swan Lake. The steps of a right-sided
fouette are shown below, emphasizing the differences in leg alignment from a pirouette.
Dancers begin fouette turns with the same three initial steps as those used for a
pirouette, as shown in Figure 5, positions 1-3. The distinction begins upon the completion
of three hundred sixty degrees of rotation. At the end of each full rotation after which the
dancer desires to complete another rotation, she opens her arms, plies her left, or standing,
11
13I115
Rotating
leg, extends her right leg from retire out to the front (position 4), and whips it around to
her right (in the direction of rotation), a rond dejambe, to second position-horizontal to
the side (position 5). Finally, the dancer curves her right leg inward, returning it to the
original retirdposition (position 6) to complete the turn (position 7). Further rotations are
completed as such until the final rotation, during which the dancer turns as in a pirouette,
with the right leg in retire, and lands as in a standard pirouette.
20(6)
(5)
(3)
S-20-
repeat 4-6
-40FS-60 -
4)
(2)
-80 -1-tStanditnerg
0.15
0.2
0.15
0.2
0.25
n
-
0
0.1
0.05
-0
(1 - ---repeat 4-6
A -50 -
(2)
-100 --
-150 0
0.05
0.1
Fraction
()(2)
(3)
of Time
(4)
(5)
0.25
(6)
Figure 5: To perform a right-sided balletfouettd, a dancer begins with her right leg
to the side and arms out horizontally (position 1). She then places her right leg
behind her, curls her right arm inward, and plie's her supporting leg (position 2). She
next pushes off with both feet, transfers her weight to her supporting leg, and enters
relevd on the ball of her left foot with her right leg in pass6 as she begins to turn
(position 3), completing a pirouette. The dancer uses torque generated from pushing
off to turn clockwise and as she finishes the rotation, whips her leg out front to
begin the fouettd (position 4). She brings her leg out to the side (position 5) and
rotates as she pulls her arms and leg in (position 6). To complete additionalfouettds,
the dancer repeats steps 4, 5, and 6 as many times as desired. Finally, the dancer
does a pirouette to end, and lands in the standard position with legs together
(position 7). Position 7 is not shown in the angle profile because it occurs after the
lastfouett6 rotation.
12
(7)
Based on the four positions of a standard ballet pirouette, a dancer's supporting knee
should begin at 0 degrees (if she does not hyperextend), move to approximately negative
60-70 degrees for the preparation plie and return to 0 degrees during the extent of the
turn, as shown in the sample graph in Figure 4.
During afouette, a dancer's supporting knee should follow the same pattern, repeating
for each additional rotation. The dancer's turning leg should have an opposite pattern of
knee flexures to her supporting leg after the preparation for a turn. While the dancer is
rotating and the supporting leg is oriented at approximately 0 degrees from initial angle,
the rotating leg will be bent, at approximately 120 degrees less than initial angle.
Conversely, each time the supporting leg plids, the rotating leg is extended, so it should be
at approximately 0 degrees from initial angle. These angle trajectories are shown over their
corresponding leg positions in Figure 5.
During rotations of any type of turn, dancers also complete a motion called "spotting,"
which refers to the whip of a dancer's head to face the front at a faster rate than the rest of
her body. This action prevents dancers from becoming dizzy when completing many
rotations. It also may contribute to the angular momentum of a turn and preserve balance,
reducing the risk of injury.
Considering overall angular velocity as seen aerially, during a fouett&, a dancer is
constantly rotating, though he or she will slow down upon returning to the forward-facing
position and whipping his or her leg out front and speed up as he or she brings her leg back
in to the bent position. Thus, overall angular velocity of the dancer's body is expected to
oscillate between two points of nonzero magnitude and the same sign. If a dancer spots
correctly, the angular velocity of her head should oscillate between approximately zero and
some velocity of greater magnitude than the maximum magnitude of body angular velocity.
2.5 Minimum Jerk Model of Movements
Maximum smoothness describes the kinematics of unhurried movements very well.
Flash and Hogan quantify maximal smoothness as minimizing the average magnitude of
higher time derivatives of position-specifically, jerk, the time derivative of acceleration. 8
Written as an optimization problem, the position coordinate can be expressed as the
kinematic objective function:
x(t)
=
argmin 1 D 1 dx 2
J - -J- dt
x(t) D fo 2 d t3)
(1)
where x represents the coordinate of position, t is time, and D is the duration of the
movement. Choosing boundary conditions that represent simple discrete movementsthat is, setting the first and second derivative of position at the beginning and end of the
movement to zero and indicating initial and final positions-the resulting polynomial
equation is a fifth-order polynomial in time:
x(t) = xo + A [10
-
()
+6((2)
where xO is the initial position and A is the amplitude of the movement.
13
This model can be extended to multiple coordinates by optimizing in each coordinate
using the same method. Thus, this model predicts straight-line movements in x-y
coordinates when optimized in the Cartesian coordinate frame because x and y positions
both have the same movement profile.
The minimum jerk model predicts simple reaching movements expressed in Cartesian
coordinates to within 4% of experimental data, as shown in Flash and Hogan's original
tests to assess the model. However, the minimum jerk model is only a representative model
when computed in the coordinate frame in which the Central Nervous System is thought to
encode the movement. For reaching movements, the model is very representative in hand
coordinates-Cartesian coordinates-but not in joint coordinates-angles. Namely,
straight paths in Cartesian coordinates will not yield straight line trajectories from a
minimum jerk model optimized in joint coordinates. The nonlinearity of the Jacobian that
converts between coordinate frames accounts for this difference. 8
The minimum jerk model has been applied successfully to other simple point-to-point
trajectories and biomechanical tasks. 9 This study applies the minimum jerk model to the
preparation plid of the knee prior to a pirouette.
2.6 Knee Misalignment and Injury
Knee misalignment while turning can propagate through other parts of the body and
cause difficultly in turning, improper technique, and risk of injury. Hyperextending or
underextending the knee during a turn puts unnecessary stress on the knee joint and the
connecting tendon, the popliteus, as shown in Figure 6.10
Popliteus tendon
Figure 6: The tendon at the back of the knee joint, the popliteus (in red), is subject
to undesired stress during a turn if a dancer hyper- or underextends her leg. 11
Simon et. al note that knee injuries account for 20% of all dance injuries, while foot and
ankle injuries, often caused by imbalance during turns, account for approximately 40%.3
Between 67 and 95% of professional dancers sustain a foot or ankle injury during their
career, and precise technique and careful training are important to prevent such injuries.
While continuous strain on the joints has a strong influence on such injuries, monitoring
proper alignment can reduce unnecessary strain and help to prevent painful injuries or
prematurely end a dancer's career.
14
2.7 Applications of Sensors to Ballet Technique
Sensors are already used to assist dancers in improving technique. Grosshauser and
Blasing have analyzed joint angle and foot pressure of dancers performing leaps as
teachers give them feedback.1 2 With the instructor's voice as a control system, dancers
corrected these parameters as they were verbally coached on potential improvements in
technique. Berardi has similarly discussed the use of a goniometer to assess technique.' 3
The application of goniometers within dance training can be extended to help dancers
analyze their deviation from the desired knee angle during dance movements to assist
them in correcting their mistakes, improving overall technique and helping to prevent
injuries. When dancers participated in the research, an explanation of the measurements
was provided, and dancers successfully acknowledged the deviation from proper technique
present in the turns they completed upon looking at their knee angle graphs. It is evident
that sensors can have a strong impact on dancers' training, particularly for younger
dancers who are preparing to perform with competitive ballet companies.
2.8 Early Research and Motivation
Early research was conducted by the author using similar methods on experienced
collegiate dancers to assess consistency of technique across double and single pirouettes.
Variation in knee angle of a dancer's supporting leg during single and double pirouetteswas
measured using a goniometer to assess the difference in angle at the three turn positions
for nine intermediate dancers. 14
Minimum preparation plid angle was found to be -61.5
3.0 degrees for single
pirouettes and -62.4
3.4 degrees for double pirouettes, as shown in Figure 7, indicating
either that deeper plies do not necessarily allow a dancer to complete more rotations, or
that intermediate dancers do not employ this recommendation. (Note: in the remainderof
this paper, the uncertaintieslisted indicate the 95% confidence intervals on the mean values.)
70
Double
Single
60
.040
30
20
10
0
Figure 7: Average preparation plii angle was reported in the author's past research
paper as 61.5 3.0 degrees less than initial angle for single pirouettesand 62.4 3.4
degrees less than initial angle for double pirouettes.14
15
Average angle while turning (referred to throughout this paper as turn angle) was
reported as 4.1 1.3 degrees less than initial standing knee angle for single pirouettes and
5.8
1.7 degrees less for double pirouettes, as shown in Figure 8, highlighting dancers'
tendency to bend their supporting knee to correct for instability. Overlapping confidence
intervals for single and double pirouettes indicate that further testing is needed to verify
differences between the two types of pirouettes.
Double
0Single,
-2
-3
C
h-
-6-7-8
Figure 8: Average turn angle was reported in the author's past research paper as
4.1
1.3 degrees less than initial standing knee angle for single pirouettes and
5.8 1.7 degrees less for double pirouettes.'4
While this study revealed valuable information about the tendencies of recreational
dancers, determining a mathematical representation of proper technique could highlight
how well these subjects and other dancers aptly matched the strategies mastered by
professional dancers. This study provided foundation and motivation for further research
in obtaining a quantitative definition of turns in ballet dance.
3. Experimental Design
Two experiments were performed to assess the consistent joint angles that
characterize advanced technique, as exhibited by professional and pre-professional ballet
dancers. Angle of the standing leg was measured during two key steps of a pirouettepreparation plid and turn-using a goniometer, and deviation from expected angle was
analyzed to determine the consistency of each subject and mathematically define qualities
of professional level ballet technique. Additionally, two goniometers and an aerial highspeed camera were used to define the proper course of movement during fouette turns.
Stability of the standing leg was compared for pirouettes andfouettds.
3.1 Overview of Subjects and Experiments
Pirouette experiments were conducted on ten professional and two pre-professional
dancers training with a professional ballet company. The pre-professional dancers were in
their final months of training with the company at the time of measurement and have since
begun dancing professionally. Extensive ballet experience within a professional company
makes these subjects a strong representation of advanced and technically correct turns.
Eleven of these twelve subjects were reported in aggregate data. Seven of these subjects
16
elected to record data for more than two pirouettes. For these trials, dancers were
instructed to complete as many rotations as they felt comfortable with; these subjects
completed three to five rotations. Due to limited availability and company restrictions on
the professional dancers, three advanced amateurs were used for the comparative analysis
of pirouettes andfouettss.
Table 1 provides details-including professional experience, company position, turn
direction preference, and tests completed-on the subjects measured. Subjects are ordered
in terms of skill level; the table order does not correspond to the order in which subject
data is reported in Section 4. For professional dancers, the Experience column denotes the
number of years of professional experience, while for pre-professional dancers, it details
years spent dancing pre-professionally with a professional company. For advanced
amateurs, it represents years of competitive dance experience.
Table 1: List of subjects, including experience and turn types measured. Fouette experiments
completed with advanced amateurs are listed below the bold line. For second company dancers*,
the Experience column refers to the number of years spent dancing pre-professionally in a
professional company. For advanced amateurs**, it refers to years of competitive dance experience.
Pirouettes
Gender Experience
(Years)
Highest
Company
Position
Direction
of Turn
1
2
3
4
5
Fouettds
M
6
Principal
L
X
X
X
F
6
Principal
R
X
X
F
16
Soloist
R
X
X
M
6
Soloist
R
X
X
X
M
4
Soloist
R
X
X
X
M
15
Second Soloist
L
X
X
F
4
Corps de Ballet
R
X
X
F
2
Corps de Ballet
R
X
X
F
1
Corps de Ballet
R
X
X
M
1
Corps de Ballet
R
X
X
X
M
2*
Second Company
L
X
X
X
F
2*
Second Company
R
X
X
F
10**
Adv. Amateur
R
F
6**
Adv. Amateur
R
X
X
X
X
F
4**
Adv. Amateur
R
X
X
X
X
X
X
X
X
17
X
3.2 Goniometer to Determine Knee Angle Trajectory over Time during Pirouettes
A Vernier Goniometer GNM-BTA, a sensor that allows for precise measurement of joint
angle, was used to measure knee angle during each turn.15
The goniometer was attached to each subject's leg using athletic tape at four pointstwo on the thigh and two on the calf (as illustrated in Figure 9) to eliminate any undesired
degrees of freedom. Careful and rigid attachment-wrapping the tape around the leg twice
at each point of attachment-ensured accuracy of results by preventing the goniometer
from moving with respect to the subject's leg during the procedure. The goniometer was
connected to a LabQuest 2, a data acquisition module, to collect measurements of knee
angle during each pirouette.16 Each subject held the LabQuest 2 in his or her hands during
the experiment so the wires connecting the devices would not cause a safety hazard while
the dancer turned. Because dancers usually hold their hands together in a rounded out
position in front of the torso during the course of a turn, holding the device was not
uncomfortable or unnatural for the subjects and did not impact the quality of their turns.
Duringfouettes, an additional goniometer was attached to the second leg using the same
method.
LabQuest 2
Vernier Goniometer
GNM-BTA
Figure 9: The goniometer was attached to the subject's leg at four points to
eliminate the degrees of freedom between the sensor and the dancer's body and
ensure measurement accuracy throughout the dancer's entire turn (left). The
goniometer is shown recording the dancer's knee bend, or pli6, as she prepares for
the pirouette while holding the LabQuest 2 data acquisition module (right).15
18
Subjects were given the opportunity to practice with the device on to ensure it was
comfortably attached and that they could turn with the desired timing. Each subject was
then asked to perform three single, three double, and three multiple (any desired number
of rotations exceeding two) pirouettes in separate trials, with pauses in between each
measurement. Data were collected at a rate of one hundred samples per second for the
duration of the turn. Dancers started and stopped the recording, and excess measurement
time-typically time spent standing before beginning a turn or walking upon completion of
a turn-was removed from each trial before analysis.
Observations were recorded for each turn regarding the quality of turns and technique
and noting if the subject fell or appeared to slightly lose balance. These observations are
quantifiable in the results as higher standard deviations from the expected angle during the
turn.
Knee angle at each relevant pirouette position was assessed to determine subjects'
accuracy and consistency in execution. Care was taken to ensure the goniometer was
oriented as close to 1800 at the start, but this is difficult to accomplish. Thus, each relevant
angle was subtracted from initial angle to account for variation in initial angle when
analyzing data.
3.3 Assessment of Relevant Knee Angles for Pirouettes
Knee angles of interest during a pirouette represent each of the stages shown above in
Figure 4, positions 2 and 3. These include minimum preparation plid angle-expected to be
about sixty degrees less than initial angle, angle during turn-expected to be
approximately the same as initial angle, and total time / fraction of run time spent turning.
Standard deviation of angle during rotation is also important: it serves as a measure of a
dancer's stability and highlights the need to correct for balance during the course of a turn.
Data were imported into Matlab and recorded data that did not represent a component
of a turn were removed, as shown in Figure 10. Because some subjects did not stand
perfectly still before or after the turn during data collection, data were trimmed using
Matlab's data cursor tools on plots of each trial. The portion of the trial that corresponded
to rotation was also selected using the data cursor tool. After a turn, some dancers
complete a finishing plid comparable to the preparation plid. However, not all dancers
exhibited this movement during tests, instead walking or standing still to stop the data.
Thus, this portion of the turn was also trimmed from each data set. The quantities of
interest were extracted from graphs of each trial and analyzed in Matlab. All angles were
recorded as the difference between initial angle and angle during the respective stage of
the turn to correct for variation in alignment of the goniometer for different subjects.
Figure 10 shows one trial of raw data, highlighting the segments that correspond to data
analyzed. Red points indicate points selected using the data cursor tool, including the start
of a trial (to eliminate excess noise if present), the start of rotation, and the end of rotation.
Blue points represent quantities determined from data: minimum plid angle and average
turn angle.
19
190
180
Turning
-tart
(Removed)
Setup
(Removed)
-170
0160
CM
* 150
E
0
0>
140
-II
-
0S130
4120
1 10
Minimum Pie
100
0
2
1
3
Finishing Pild
4
Time [s]
5
7
6
8
Figure 10: Sample raw data for a single pirouette completed by a professional-level
subject. The three red points were selected in Matlab to trim the trial, representing,
trial start and start / end of rotation. Blue points represent quantities later
determined from data, minimum plie angle and average turn angle. Points before the
start of the trial and after the end of rotation were eliminated for this analysis.
Trimmed data from three turns-a single, double, and quintuple pirouette-by the
same subject are shown in Figure 11. As expected, in general, time spent turning increases
with added rotations.
10
0
0
C)
0-50
0
-
-70
0.5
1
1.5
2
Time [s]
2.5
RA"**I**i
3
-
-Single
-Doube
60
1r.
3.5
Figure 11: Trimmed data from single, double, and quintuple pirouettesby the same
professional-level subject. The first peak shows the plid, and the perturbations near
zero show variations in the stability of the standing leg while a dancer balances and
rotates for the turn.
20
The duration of each trial was normalized in time to assess fractions of time during
which dancers were rotating. Angle change versus fraction of run time for a single, double,
and triple pirouette by the same subject is shown in Figure 12.
10
0
-20r -30
Cl)
-
E -40
0
0-50
Single
-Double
Multi e (5)
S-60 --700
0.2
0.6
0.4
Fraction of Time
0.8
1
Figure 12: Normalized trials for a single, double, and triple pirouette by one subject.
Angles are reported as deviation from initial angle. Each turn type clearly results in
a similar trajectory, where more turns generally produce more variability in angle.
Multiple trials for a single dancer are not independent measurements. Thus, to obtain
statistically accurate parameters for each type of turn (i.e. confidence intervals on plid
angle, turn angle, turn standard deviation, and turn duration), average values for each
parameter were determined for each dancer, and statistical calculations were performed
on those averages. This procedure eliminates the issue of dependence between trials
completed by the same dancer and allows for accurate inference calculations, which can
only be performed on independent measurements.
Data from the selected starting point of the run to the start of the turn (the first and
second red points in Figure 10) were compared to a minimum jerk model in joint
coordinates. The model was created for each trial as two motions: from the starting point
to the minimum plie angle and back, from the plie point to the start of turning. The root
mean square error of the model as compared to experimental data was determined.
Approximating the length of the upper thigh as 0.35 m, angle measurements were
converted to change in height of the dancer in time. Minimum jerk models were also
determined for these measurements in Cartesian coordinates, and the fit was compared to
determine the coordinate system in which dancers plan movements.
21
3.4 Comparative Measurements for Fouett Turns
Similar techniques were employed when measuring fouette turns. A Vernier
Goniometer GNM-BTA was attached to each leg of the subject and connected to a LabQuest
2, as for pirouette measurements.' 5 Figure 13 shows the experimental setup for fouettd
turns.
Figure 13: The experimental setup for measuring fouett turns. Two goniometers
were used, with one attached to each leg with athletic tape at four points, as for
pirouette data collection. The dancer also held the LabQuest 2 as in previous tests.
Additionally, a Samsung TL350 high-speed camera (set to measure at 240 frames per
second) was mounted to the ceiling, parallel to the floor, to take aerial video to allow
7
tracking the movement of subjects during the course of each turn.'
3.5 Assessment of Relevant Knee Angles for Fouettes
Similar parameters as those measured for pirouette turns were recorded for fouettd
turns. Preparation plid angle was measured as the minimum initial deviation from initial
angle, or the maximum bend of the supporting leg that the dancer completed before
pushing off to begin a turn. Angles during each rotation (defined for fouettds as all of the
segments throughout the course of a trial during which the subject is rotating with straight
supporting leg) and characteristic standard deviation of rotation were also assessed. These
values were directly compared with the parameters for pirouette turns from the group of
subjects that completed both types of turns to determine differences in the behavior of the
supporting leg between pirouettesandfouettes.
During afouettd, each valley seen in the angle data of the standing leg represents a plie,
or the start of another turn. Perturbations near zero angle for the standing leg show
variations in the stability of the standing leg while a dancer balances and rotates for the
turn, similar to those seen in the pirouette data detailed in Section 3.3. Valleys in the
22
rotating leg data occur during the turn (while the standing leg is near zero angle), and
peaks-points during which the rotating leg is straight-occur during the preparation plies
of the standing leg before each subsequent turn. Figure 14 shows the sample motion of the
supporting and rotating legs of one subject during afouettd.
K
I
20
r
0
-20
I
lilt
-40
0
E
*0
-60
CD)
0
-80
0
-100
0
-120
E
T
-140
-
-160
-
0
Standing Leg
Rotating Leg
2
4
6
Time [s]
8
10
12
Figure 14: Data for a fouett by one subject, showing the course of movement of
both the standing (left, in black) and rotating (right, in pink) legs. Angles are
reported as deviation from initial angle. Each valley in the standing leg data
represents a plid, or the start of another turn. Perturbations near zero angle for the
standing leg show variations in the stability of the standing leg while a dancer
balances and rotates for the turn. Valleys in the rotating leg data occur during the
turn, while peaks-points during which the rotating leg is straight-occur during
preparation plies of each subsequent turn.
Collected fouettd run data were also normalized in time, as shown for pirouettes in
Figure 12, to determine the fraction of run time spent completing certain actions.
Comparisons were drawn between pirouettes and fouetts by relating the data for each
turn type measured for the group of advanced amateur dancers.
23
3.6 High Speed Video Analysis
Aerial video was used to supplement goniometer data for fouette turns. Three colored
dots on the top of the dancer's head were used as a center point and means to track the
angular velocity of the head while turning. Similarly, a colored dot on each shoulder and on
the wrist of the right arm (the arm that remains bent and stationary during a turn) were
used as markers to track the angular velocity of the dancer's body throughout the course of
thefouette, as shown in the clips from the high speed video in Figure 15.
Figure 15: Images from high speed video trials, including the first frame, with the
tracking dots that marked positions throughout the entire turn shown (left), and a
later frame with the dancer's leg extended (right), with tracking dots hidden so the
markers tracked are visible.
Tracker1 8 video analysis software was used to analyze the high-speed video, using
color thresholds to track the movement of the colored markers on the dancer's body as
Cartesian coordinates in time, as shown in Figure 16. Tracking of these points in time was
used to measure change in angular velocities of the body and head to assess the
contribution of spotting tofouettd turns.
24
I-A
-D
-B-C
nr
0
0
0
0
5
5
10
10
-0.3
15
-
0.5
0
0
5
10
1
15
5
10
5
10
-5
5
10
ii
0
5
-0
10
10
n
-
-5
0
0
-
0
n -r%
0
15
0
5
10
0
15
1.-
1.-
0
-I
0
-I
0
5
10
15
15
0
5
10
15
Time [s]
Figure 16: Raw video tracking data, as X and Y coordinates, of points A (center red
marker), B (green head marker), C (blue head marker), and D (green arm marker) in
time.
Occlusion of several markers as the dancer turned made it challenging to track certain
parts of the dancer's movement; thus, to preserve accuracy of the measurements, the four
colored dots shown in the left image of Figure 15 were the only points tracked in the video
data. The green and blue dots on the right and left sides of the dancer's head were also
hidden from video for small portions of the trial, so manual selection was used to track the
markers during those sections.
The orientation of the lines connecting (1) the red center point (point A) and the green
head marker (point B), (2) point A and the blue head marker (point C), (3) points B and C,
and (4) points A and the green arm marker (point D) were determined from the
measurements as an inverse tangent in Matlab, and data were then manipulated such that
the magnitude of the bearing continually increased, as shown in Figure 17.
25
90
-
80
70-60
s0
_
40-
10
20
10
2
4
6
Time [s]
10
12
14
-
Figure 17: Measured angle of each point relation (1 - between points A and B, 2
between points A and C, and 3 - between points B and C). While measurements are
slightly different-as limited by the precision of the software-the patterns of angle
changes are comparable, such that all can be reasonably used to estimate angular
velocity.
The derivative of each relation in time was estimated using a centered difference
approximation to determine angular velocity. Angular velocity for the body was estimated
based on relation 4, while angular velocity for the head was estimated as the average
between relations 1, 2, and 3, removing outliers that occurred with spikes in the estimated
angular velocity vectors due to measurement noise.
26
4. Pirouette Technique Assessment: Results and Discussion
To quantify a kinematic representation of proper pirouette technique, preparation plid
angle, average and standard deviation of turn angle, and fraction of time rotating were
determined.
4.1 Preparation Plie
100
--
-
-
-
-
-
Preparation plie angle was measured for each subject. 95% confidence intervals on
magnitude of preparation plid angle by subject are reported in Figure 18.
Ingle
Do
MTrIple
909-
MQuedruple
80 - =Wuntupe
70
~60
-.
-
5040m302010
1
2
3
4
5
6
Subject Number
7
8
9
10
11
F
F
F
F
F
M
F
M
M
M
M
Figure 18: Minimum plid angle for each type of pirouette-single, double, triple,
quadruple, and quintuple-is shown with 95% confidence intervals for eleven of the
subjects measured. Only seven subjects opted to record data for more than two
pirouette turns. Only one subject chose to complete a quadruple pirouette and two
completed quintuples.
Results indicate that professional dancers tend to plid to a similar level prior to
completing pirouettes. As predicted, these results approach approximately sixty degrees.
Some variation occurred, which may have been due to height and weight of each
dancer: dancers bend their knees to the level that feels comfortable or necessary prior to
turning. While height and weight of each subject is not known (due to restrictions by the
professional company), gender was recorded. Subjects 6, 8, 9, 10, and 11, the dancers that
exhibited the largest plid angles generally, are males. In fact, the correlation coefficient
relating gender (when expressed as a numerical indicator, males as 1 and females as 0) and
plid angle is 0.69. This implies a strong correlation between gender and plid angle-males
appear more likely to plid more deeply. The discrepancy between male and female plid
angles is shown in Figure 19, with values separated out by number of pirouette rotationssingles, doubles, triples, quadruples, and quintuples.
27
6
I
e m * e Seece Sos 0 0 5 US 0 @SS
0
2
30
S
5
Os
@0
s@
50
40
eFemale
o
80
70
60
Magnitude of Pile Angle, A6
010
Figure 19: Magnitude of plkd angles is shown for each pirouette type, separated out
by gender. Clearly, regardless of turn type, males tend to plid more deeply than
females, implying that plids angle depends on body geometry (e.g. height).
2C
Averaging across all turn types, males and females show statistically significant
differences in p116 angle at any reasonable significance level. The average p116 angle of males
is at least 20.3 degrees greater than that of females with 95% confidence, as shown in
0
o9
07
5
~~~~ 40
30~
Figure 20.
Female
80Male
70o60-
10
. 30-
E
20100
Figure 20: Magnitude of pPii angle for males versus females. With 95% confidence,
the average p it of males is at least 20.3 degrees greater than that of females.
Clearly gender is a strong predictor of ped angle. The height and weight disparity
between males and females implies that body geometry-height and/or weight-may
affect preparation p11e angle; however, additional tests that include height and weight
measurements are needed to verify if a statistically significant difference exists.
Turn type was also assessed as a predictor of pc angle. Probability density histograms
of ptle angles for single, double, and triple pirouettes (those with the greatest amount of
data collected) are shown in Figure 21. The data do not appear to follow a normal
distribution; other factors may affect dancers' p11e angles.
28
0.3
0.3 r
0.3
0.25
0.25-
0.25
0.2-
0.2
0.15-
0.15
0.2 I
>-
0.15
.0
0.1
I
0.1
0.1 j
0.05
0.05 [
-
'2
a.
0.05
IL
0
0
0
Magnitude of PlHe Angle, A0
40
L
80-A
Figure 21: Probability density of magnitude of plid angles for single, double, and
triple pirouettes. Plots do not appear to follow a specific distribution. The black line
along the x-axis represents the 95% confidence interval.
Pli angles aggregated across subjects by turn type are reported with 95% confidence
intervals in Figure 22.
110
Single
Double
Triple
Quadruple
Quintuple
100
90
CD 80
70
a. 60
0
50
40
C1
30
20
10
0
Figure 22: Average preparation plie angle across all dancers is reported with 95%
confidence intervals. Single pirouetteshave a plid angle of 59.2 4.0 degrees below
10 for triple
4.6 below for double pirouettes, 64
starting angle, with 61.9
19 for quadruple pirouettes, and 74 11 for quintuple pirouettes.
pirouettes, 88
Overlapping confidence intervals indicate that a significant difference does not exist
between plid angles for each type of turn as exhibited by professional ballet dancers.
Note that only one subject completed quadruple and two subjects quintuple
pirouettes.
Observed plid angle magnitude was 59.2 4.0 degrees for single pirouettes, 61.9 4.6
degrees for double pirouettes, 64 10 for triple pirouettes, 88 19 for quadruple pirouettes,
and 74 11 for quintuple pirouettes. A smaller standard deviation for single and double
29
turns indicates that dancers know how much they must plid to prepare for a lower number
of rotations. Standard deviation increased with number of rotations, indicating that more
variability existed in plid angle across dancers for greater number of rotations. It should be
noted that only one subject completed quadruple pirouettes and two subjects quintuple
pirouettes. Thus, additional data would make the assessment of differences in angle across
turn types more robust. The confidence intervals overlap: while the observed mean
showed deeper pli angles for greater turns, the differences are not statistically significant.
Paired t-tests comparing the differences of means across turn types yielded similar
results. The differences between pli6 angles for singles and doubles and for singles and
triples were not statistically significant at the 5% level. However, the difference for doubles
versus triples was significant; with 95% confidence, the average plie angle for double
pirouettes is at least 2.7 degrees less than that for triple pirouettes. Additional data (only
four subjects completed triple pirouettes) would better verify this result. Comparisons for
quadruple and quintuple pirouettes were not possible due to limited data.
The results of these experiments do not validate the hypothesis that dancers tend to
plid more to complete a greater number of turns as instructors often suggest. Rather, it
appears that body geometry-namely height and/or weight-could be a stronger predictor
based on the high correlation between gender and plid angle.
4.2 Turn Angle
Angle during the portion of the trial for which the dancer was rotating was measured
for each turn by each subject and aggregated to assess its difference from initial angle. 95%
confidence intervals on turn angle for each subject by turn type are reported in Figure 23.
10
5-
T
0
4
E -5-
~-10-
I
-15Trip e
MQuadruie
-20
1
II
2
3
1
1
4
5
Sulbject Number
0
11
Figure 23: 95% confidence intervals for angle during the course of each type of
pirouette turn is shown by subject. Typical averages were within 10 degrees of
initial angle, with the majority of subjects having an average angle below zero (bent
leg), as expected to maintain balance.
30
Across all dancers and trials, angles were aggregated to examine the probability
distribution of values observed. Figure 24 shows a probability distribution of all angles
during the turning portion of single pirouettes, the confidence interval on average angle
while turning (shown as a black dot and line on the x-axis), and a Gaussian fit. The results
are approximately normally distributed, as evident by the Gaussian fit to the data, a normal
distribution of mean -3.2 and standard deviation 4.4. Average angle during rotation for
single pirouettes was found to be -2.0
2.1 degrees. The upper limit of the confidence
interval is only slightly above zero, indicating statistically significant evidence that average
turn angle during single pirouettes is less than initial angle.
0.12
0.1
I
0.08
'0.06O.04
20.04-
0.02
0
-15
-10
-5
5
Deviation from starting angle,
10
is
)
-20
Figure 24: Single pirouette probability density of angle during rotation, measured
as deviations from initial angle, as observed during all trials. The dashed line
denotes the zero degree mark. The black line along the x-axis represents the 95%
2.1
confidence interval for angle during rotation for single pirouettes, -2.0
degrees. The observed results are approximately normally distributed, as indicated
by the Gaussian fit of mean -3.2 and standard deviation 4.4. The upper limit of the
confidence interval is only slightly above zero, indicating statistically significant
evidence that average turn angle during single pirouettes is less than initial angle.
Angles were also aggregated for double pirouettes, as seen in Figure 25. The observed
results are approximately normally distributed, with a similar mean and a greater standard
deviation than those observed for single pirouettes; the Gaussian fit to the data is a normal
distribution of mean -2.7 and standard deviation 6.2. The distribution for double
pirouettes, however, has larger spread than that for single pirouettes, indicating more
variation over the course of rotation, perhaps due to the need to adjust for stability.
Average angle during rotation was found to be -2.0
3.2 degrees. The upper limit of the
confidence interval is above zero; there is not statistically significant evidence that average
turn angle during double pirouettesis less than initial angle.
31
0.1
0.09
0.08:0.07
C 0.06
20.05
2 0.04
0- 0.03
0.02
0.01
0
-30
-20
-10
0
Deviation from starting angle, i
10
20
Figure 25: Double pirouette probability density of angle during rotation, measured
as deviations from initial angle, as observed during all trials. The dashed line
denotes the zero degree mark. The black line along the x-axis represents the 95%
3.2
confidence interval for angle during rotation for double pirouettes, -2.0
degrees. The observed results are approximately normally distributed, with a
Gaussian fit of mean -2.3 and standard deviation 4.3, a similar mean and a greater
standard deviation to those observed for single pirouettes. The upper limit of the
confidence interval is above zero; there is not statistically significant evidence that
average turn angle during double pirouettes is less than initial angle.
Angles for triple pirouettes are shown in Figure 26. The observed results are slightly
left skewed, but a Gaussian fit to the data showed that it could be considered an
approximately normal distribution, of mean -2.3 and standard deviation 4.3. Average angle
during rotation was found to be -1.9
5.1 degrees. The upper limit of the confidence
interval is above zero; there is not statistically significant evidence that average turn angle
during double pirouettes is less than initial angle. Note the increase again in angle spread,
indicating increased need for dancers to bend their knees to correct for instability during
triple pirouettes.
32
0.12
0.1
. 0.08
0.06
0.04
0.02
0
-20
-15
-10
-5
0
5
Deviation from starting angle,
10
15
(
-25
Figure 26: Triple pirouette probability density of angle during rotation, measured
as deviations from initial angle, as observed during all trials of triple pirouettes. The
dashed line denotes the zero degree mark. The observed results appear slightly left
skewed, but they can be considered approximately normally distributed as seen by
the applied Gaussian fit in black. The black line along the x-axis represents the 95%
5.1
confidence interval for angle during rotation for multiple pirouettes, -1.9
degrees. The upper limit of the confidence interval is above zero; there is not
statistically significant evidence that average turn angle during double pirouettes is
less than initial angle.
Distributions were also produced for quadruple and quintuple pirouettes. Adjacent
histograms for these turn types are shown in Figure 27. Both distributions are left skewed
and have a very large spread, indicating that dancers tend to bend their legs much more
often than they tend to hyperextend and this bending has a great deal of variation. Since
only one and two subjects completed these turn types, respectively, it is difficult to express
reasonable confidence intervals on turn angle. For quadruple pirouettes, average turn angle
was -12
12; this does not appear to be a representative result for quadruple pirouettes,
however. Similarly, for quintuple pirouettes, average turn angle measured was -5
32. The
high uncertainty resulted from the small number of measurements (two dancers). While
quadruple and quintuple pirouettesdo appear to have greater deviations from starting
angle, further tests are required to produce generalized results that represent typical
performance of professional dancers.
33
0.12
0.12,
0.12,
0.1
0.1
0.08-
0.08
0.06
0.06-
0.04
0.04
-30
-25
-20
-15
-10
-5
-20
-15
-10
0
(
Deviation from starting angle,
Figure 27: Quadruple (left) and quintuple (right) pirouette probability density of
angle during rotation, measured as deviations from initial angle, as observed during
all trials. Dashed lines denotes the zero degree mark. Results are aggregated from
the single subject that completed quadruple pirouettes and the two subjects that
completed quintuple pirouettes during testing and are thus not representative of
how professional dancers complete these turns generally. The observed results are
left skewed with a large spread. Averages were found to be -11.9 and -4.7,
respectively, as indicated by the black dots on the x-axis.
The large deviations from initial angle observed for quadruple and quintuple pirouettes
may have been influenced by the instructions of the experiment: subjects were told to
complete as many pirouettes as they were comfortable with for these trials. Some dancers
may have chosen to test their limits, performing a number of rotations they knew they
could complete without falling but not to a level of technical expertise the same as their
other turns. Because insufficient data were collected for these turn types, the remaining
observed parameters are only reported for single, double, and triple pirouettes.
While the amount of variation differs slightly by number of rotations, distributions of
angles while rotating for all types of pirouettes measured demonstrate that even
professional dancers cannot maintain a completely straight leg during turns. Thus, it can be
hypothesized that dancers use their supporting leg as a control system for balance while
rotating during turns. As a dancer feels an imbalance, represented by a slight lean off her
axis of rotation, she appears to bend her supporting leg to lower her center of gravity and
regain stability as she continues to rotate. The supporting leg of a dancer appears to change
based on neural feedback, likely involving position of the center of gravity, tilt of the axis of
rotation, and comfort with the turn. Use of more advanced motion capture technology
could help to determine a model of the supporting leg's reaction to disturbances.
Characteristic standard deviation for each turn type was determined as the average of
the standard deviations of each dancer. Figure 28 shows the observed values for single,
double, and triple pirouettes, 2.00
0.36, 3.17
0.77, and 3.2
34
2.0, respectively.
5
6
Single
Double
Triple
-
C
0
C
2M
01
Figure 28: Characteristic standard deviation while rotating was found to be
2.00 0.36 for single pirouettes, 3.17 0.77 for double pirouettes, and 3.2 2.0 for
triple pirouettes.
While standard deviation while rotating appeared to increase with the number of
rotations of the pirouette, no significant difference was identified in standard deviation
between double and triple pirouettes.The difference between standard deviation of turn
angle for single and double pirouettesis at least 1.15 degrees with 95% confidence. Thus,
completing multiple turns (regardless of number of rotations, it appears) results in more
instability of the dancer's standing leg.
4.3 Turn Duration
Both turn duration in seconds and percentage of the trial during which the dancer was
rotating-with the supporting leg approximately straight and the opposite leg bent-were
measured. Figure 29 shows difference in turn duration and turn fraction for each type of
pirouette.
No significant difference was found in duration between turn types. Single turns lasted
0.97 0.43 seconds, doubles 0.93 0.22 seconds, triples 1.58 0.83 seconds, quadruples
1.61 0.49 seconds, and quintuples 1.52 0.53 seconds.
However, fraction of time turning yielded statistically significant differences between
some pirouettes of different numbers of rotations. Single turns lasted 22.5 2.1 percent of
total run time, doubles 34.3 2.7 percent, triples 36.0 4.1 percent, quadruples 52.8 6.6
6.7 percent. Single turns were notably different than the
percent and quintuples 58.9
others; doubles and up saw very narrowly overlapping confidence intervals.
+
Graphing turn fraction against number of rotations of the pirouette, a statistically
0.011 and intercept 0.135
significant linear relationship was found, of slope 0.091
0.037, shown in the right panel of Figure 29.
35
0.65 - -
*
1
data
slope = 0.091
1
1
1
,
0.7
2.5
0.011
0.62-
0.55-
.2 0.5
C
-
0
0.45
-
1.5 -
0.40.350.3-
0.25
0.501
0
1
4
3
2
Number of Rotations
5
0.2
6
1
4
3
2
Number of Rotations
5
Figure 29: Turn duration (left) and turn fraction (right) are shown for each of the 5
types of pirouettes measured versus number of rotations. It is clear that a
correlation exists between number of rotations and fraction of time turning: a linear
0.011 fits the data well. Turn duration, however, does not
model of slope 0.091
show such a correlation since dancers were allowed to turn at their desired pace.
This difference occurs because during the tests, dancers were not mandated to go at a
particular speed-there was no music or count in the background. Thus, dancers were able
to complete the turns measured at their own preferred pace, whether fast or slow, and this
preference varied greatly across subjects as the high standard deviations and thus
overlapping intervals on turn duration indicate. Regardless, dancers are very skilled at
maintaining pace, like an internal beat, when completing moves without music. Thus, the
dancers completed the preparation plids prior to each turn at a comparable pace to the turn
itself. This tendency explains why 95% confidence intervals on turn fractions for each turn
type were very narrow and some did not overlap.
A statistically significant difference in fraction of run time turning exists across
pirouettes of different rotations. Turn fraction is not directly proportional to number of
turns, i.e. a double pirouette does not take twice as long as a single pirouette, but rather it
has an affine relationship, with a "setup time" of approximately 13.5% of the run length-a
mechanical constraint on how fast dancers can turn-and an additional 9.1% of run length
for each additional rotation. Dancers naturally complete a greater number of rotations at a
faster pace. The additional angular momentum needed to complete extra rotations results
in a greater rotational speed. Completing turns too slowly makes it much more difficult to
maintain stability because it forces the dancer to balance on his or her supporting leg in
relevd for a longer period of time.
36
4.4 Minimum Jerk Models
Minimum jerk models were compared to observed trajectories of the angle of the
standing leg during the plid phase of pirouettes. Plots of two sample fits are shown in Figure
30.
10
10
0
0
CD 10
5
L-20
E
1-30
E
2-40
-30
1-40
C
0
%-50
.- 50
-60
-60
-RMSE=1.
1.0
RME3.8
-70C
0.5
I
Time [s]
:4.7
-701
0
1.5
0.5
1
2
1.5
Time [s]
2.5
Figure 30: Minimum jerk profiles in joint coordinates (dotted lines) are shown for
the first (red) and second (pink) half of plie movements as compared to data (blue)
for two archetypal profiles of plies. Root mean square error of the two profiles is
reported in the legend in degrees. The left profile represented a smooth movement
into pli, exhibited by the majority of dancers measured. The right profile represents
a step into a plid, so the first half of the trajectory does not follow a single minimum
jerk movement.
Two profiles were observed for preparation plies amongst dancers measured. The first,
shown in the left panel of Figure 30, represents a smooth entry into a plik, while the right
panel, which shows a small jump up in the first half of the trajectory, represents a step into
a plid. Observed root mean square error of minimum jerk profiles is shown in Table 2.
Results are reported for those dancers that exhibited no-step plies only-all but three-and
are separated out into values for the first half of the plie, the second half of the plid, and for
the trial overall.
Table 2: Root mean square error in degrees for minimum jerk profiles for the no-step preparation
plies.
First Half
RMSE
6.18
Second Half
3.87
0.85
37
0.49
Overall
7.40
0.65
In general, minimum jerk was a better model for the second half of the pli trajectory,
with a root mean square error of 3.87 0.49 degrees, versus 6.18 0.85 degrees for the
first half of the trajectory. Overall, the root mean square error between the minimum jerk
0.65 degrees, within about 10% of the
model and the whole pli trajectory was 7.40
maximum model deflection.
Across all trials, coefficient of determination R 2 for a minimum jerk model in joint
0.038. The minimum jerk model is thus a
coordinates to observed plie data was 0.873
very good fit for this motion in joint coordinates; 87% of the observed movements are well
explained by the model.
By converting joint coordinates-the angle of the standing leg-into change in height,
minimum jerk models in Cartesian coordinates were also determined. Coefficient of
determination R 2 for a minimum jerk model in Cartesian coordinates to observed plid data
was 0.710 0.090-a much worse fit than that of the model in joint coordinates. Figure 31
shows the analogous plots to those in Figure 30 for the model computed in Cartesian
coordinates.
0
0
-0.02-0.04
-E
-0.05-
Ec
-0.06
0.1
--
F-0.08
-0.1
E)
E
0-0.12
-0.15
.2-0.14>-0.16
-
-0.18
-
---
-
Data
-RMSE
-0.2
-Data
- -
-0.026
=0.019 m
--
1
0
0.5
1
Time [s]
1.5
.2
RMSE
= 0.062 m
= 0.0069 m
-0.251
2
0
0.5
1
1.5
Time [s]
2
2.5
3
Figure 31: Minimum jerk profiles in Cartesian coordinates (dotted lines) are shown
for the first (red) and second (pink) half of pli movements as compared to data
(blue) for two archetypal profiles of plids. Root mean square error of the two
profiles is reported in the legend in meters. Plots are analogous to those in Figure
31, indicating that the minimum jerk model is a better fit to the data in joint
coordinates.
This poor fit results from the nonlinearity of the trajectory of a dancer's leg in
Cartesian coordinates. Plotting the x and y position of a dancer's center of mass for one trial
yields, as expected, a curved path, shown in Figure 32.
38
0
-0.02-0.04-
-0.06-0.08-0.1
-0.12-0.14-0.16-0.18-0.21
-0.35
-0.3
-0.25
-0.2
-0.15
X [in]
-0.1
-0.05
0
0.05
Figure 32: Parametric displacement of the center of mass in Cartesian coordinates
of a dancer during one trial.
The nonlinearity of the path of the leg in Cartesian coordinates and the goodness of fit
in joint coordinate space implies that for dancing movements, the Central Nervous System
encodes movements in joint space rather than Cartesian coordinates.
5. Pirouette and Fouette Comparative Analysis: Results and Discussion
To compare dancers' ability to maintain standing leg stability between pirouettes and
fouett6s, preparationplid angle and angle while rotating were evaluated for each type of
turn by advanced amateur dancers.
In addition, the dynamics of fouettds were studied by comparing turn angle and plie
angle across rotations within the same trial. Aerial video was used to compare the
movement of the head with that of the body and assess the quality of spotting during the
turns.
5.1 Preparation Plie
Magnitude of preparation plid angle was averaged for each turn type for all advanced
amateurs that completed both pirouettes andfouettes. Results are shown in Figure 33.
39
n
I
90 IDuble
ITripleI
S80
FoueM
70
C
.50
0
*
40-
C 30i2010
01
2
Subject Number
3
Figure 33: Magnitude of plie angle for single, double, and triple pirouettes and
fouettes completed by the advanced amateur subjects.
These subjects showed very comparable plid angles across all turn types, as
demonstrated by the aggregated data in Figure 34. Average plid angle magnitude was
68 22 for single pirouettes, 71 17 for double pirouetes, 70 20 for triple pirouettes, and
65
18 for fouettes. A small number of samples resulted in wide confidence intervals. No
significant difference can be observed between plid angle before pirouettes as compared to
before fouettes. When averaged across subjects, only subject 3 showed a significant
difference between plie angles for single pirouettes and fouettes at the 10% level. While it
appears that some dancers plid less prior to completing their first fouettd-perhaps
because they know they will soon be able to plie again and regain angular momentumthese tests were not conclusive in identifying a difference between single pirouette and
fouettd preparationplid angles. Further tests with additional subjects would help to identify
if a significant difference exists.
40
Single
Double
Triple
Foueft
90.
80<70
60
50
I-so
040
0
~30
1 20
10
0
Figure 34: Magnitude of plid angle for single, double, and triple pirouettes and
fouettes completed by the advanced amateur subjects. When averaged over the
three subjects, none of the differences are significant.
5.2 Turn Angle
Turn angle for fouettds was assessed on several levels-in aggregate for all rotations
and as it evolved throughout the trial as number of completed rotations increased.
When aggregated across all rotations, average angle while rotating for fouett turns
was similar to that observed for pirouette turns: it falls just below zero, or at a slight bend
of the leg. However, the observed standard deviation far exceeded that observed for any
number of pirouette rotations, leading to a 95% confidence interval that extends from 15
degrees below initial angle to 10 degrees above initial angle. Statistically significant
evidence that average turn angle during fouettds is less than initial angle does not exist
because the confidence interval contains zero. Figure 35 shows a histogram of angles
observed during all rotations for recorded fouette turns. The observed results are
approximately normally distributed; a Gaussian of mean -0.8 and standard deviation 6.5 is
a good fit.
41
0.07
0.06-
-
0.05
C
-
A 0.04
-
2 0.03
0.02-
-
0.01
0
-25
-20
-15
-10
-5
0
Deviation from starting angle, 0
5
10
15
Figure 35: Fouette probability density of angles during periods of rotation,
measured as deviations from initial angle. The black line along the x-axis represents
the 95% confidence interval for angle during rotation for single pirouettes, -2.4
12.6 degrees. The observed results are approximately normally distributed, as
indicated by the Gaussian fit of mean -0.8 and standard deviation 6.5. Statistically
significant evidence that average turn angle duringfouettes is less than initial angle
does not exist because the confidence interval contains zero.
Comparing the observed interval on average angle while rotating for fouettes with
those observed for different numbers of pirouettes by the same subjects, it is clear average
angle while rotating cannot be distinguished between turn types. However, the large
standard deviation and spread in the distribution for fouette turns is evident in the wider
confidence interval on average fouettd turn angle. Table 3 shows results for turn angle on
single, double, and triple pirouettes and fouettds.
Table 3: Average angle while rotating for single, double, and triple pirouettes andfouettes, reported
with 95% confidence intervals.
Single
Pirouettes
Average Angle
while Rotating
0.1
0
5.9
Double
Pirouettes
Triple
Pirouettes
Fouettis
-3.0
-3.4
-2
9.2
42
4.4
13
5.3 Evolution of Plie and Turn Angle with Increased Rotations
Plid angle, turn angle, and turn standard deviation were measured as rotations
increased throughout the course offouettd trials.
Plid angle prior to each subsequent turn was found to be relatively consistent by
subject as number of rotations increased, and deviation from average plie angle was found
to be the same across subjects. The left panel of Figure 36 shows the relation between plhd
angle and rotation number for the three advanced amateurs measured, with each subject's
data in a different color.
Difference in plie angle and subject's average phid angle for all turns after the first (not
including preparation plie) and rotation number were compared. The correlation was not
statistically significant at the 5% level, as shown in the linear model in the right panel of
Figure 36.
80
o
75-
4
0
0
00 X x
PlieAngle vs. RotationNumber
1
x
6
-
0
70
X
X
0
0
-X
6
5-
5
0
00-0
0
0)6
0.....X
0
..........
2
X....
......
o-,-2 -X
50 -
X
.2X
0
40 -
35
0
X
-
45 -
-
2
-6
S0
0
4
8
6
Rotation Number
'
10
.6
12
X
X
x
X@--------.Confidence
,IIII
8
6
4
2
Rotation Number
x
ta
Bounds
10
Figure 36: (Left) Magnitude of plid angle versus the rotation number of the turn
within that fouettd trial. Each subject is indicated with a different color. (Right)
attempted linear model relating the difference between p11d angle and average plie
angle to rotation number. The red line indicates the fit, which falls very close to the
gray zero degree reference line. Correlation was not statistically significant at the
5% level.
Therefore, rotation number does not have a significant effect on the plie angle prior to
that rotation; namely, dancers maintain a relatively standard plie angle, and their plid
angles, regardless of rotation number, remain relatively constant.
Average angle of the supporting leg during each rotation was also measured for fouettd
turns. Like with plid angles, dancers maintain a relatively constant average turn angle, even
as rotations increase. Rotation number was not a significant predictor of average turn angle
at the 5% level. Figure 37 shows the relation between average turn angle and rotation
number, with each subject's data in a different color.
43
15
0
10
0
0
00
0
0
5
4)
@O0
I-B)
0
00
0
8o~
o
0
0
cc
-5
00
0
-10
-15
0
0
2
4
8
6
Rotation Number
10
12
Figure 37: Average turn angle versus the rotation number of the turn within that
fouette trial. Each subject is indicated with a different color.
Standard deviation during turn saw much more variation across rotations than the
other two parameters, but rotation number was again not a significant predictor of this
parameter at the 5% level. Figure 38 shows the relation between standard deviation of
turn angle and rotation number, with each subject's data in a different color.
12
0
10k
0
I-
8
0
0
00
0
0
0
cc
'U
6
0
0
0
o 8
4
09
0
0
0 0
0
0
2
0
8
01
0
2
4
6
8
Rotation Number
10
12
Figure 38: Standard deviation during turn versus the rotation number of the turn
within thatfouette trial. Each subject is indicated with a different color.
Clearly dancers are able to maintain relatively consistent plies, average turn angles, and
turn quality independent of rotation number.
44
5.4 Dynamic Analysis of High Speed Video
Movement of points A, B, C, and D, as defined in Section 3.6 and reproduced in Figure
39, was tracked in time and assessed to determine consistency offouettes.
To complete what is considered a perfect turn, a dancer theoretically must keep his or
her center point, the location of his or her axis rotation, exactly still. However, in practice,
this is nearly impossible for dancers to achieve: they must readjust their position
constantly to maintain stability. Thus, minimizing traveling during turns is considered best
practice, but small deviations are accepted. A parametric plot of the movement of the
center point of the head (red marker A) is shown in Figure 39. The direction of movement
in time is indicated by the color gradient on the right hand side.
0.5
12
0.410
0.3
8
-
0.2
>-
6
0.1
E
4
0-
-0.11
0.4
2
-0.2
0
0.2
0.4
0
X [m]
Figure 39: Position of the center point of the head of a dancer through the course of
11 fouettg turns. An image of the point tracked is shown to the left. The change of
position in time (right) is indicated by the color gradient, scaled to the duration of
the turn in seconds. A technically perfect fouette would require the center point to
remain still at its initial position; however, in practice, this is very challenging for
dancers to maintain. Minimizing movement of one's central axis during turns is the
accepted best practice.
As Figure 39 shows, this advanced amateur dancer was able to remain within half a
meter in any given direction from her starting point, a very small radius. Average deviation
from the center point was 0.221 0.014 meter (less than 9"), under 15% of the dancer's
wingspan.
Overall movement in time produced nearly perfect circles across all completed
rotations, as shown in Figure 40. Tracked motion of point D in time, representing the
overall motion of the subject's body, produced circles of different centers (left). Subtracting
out the distance from the center point (A) produced nearly overlapping circles (right). The
dancer maintained a steady radius between her axis of rotation and her hand during the
course of a turn.
45
0.8
0.6
0.4
0.2
>-I
1
0.5.
0.8
210.6
0
.90.4
0
-0.2
-0.4
0.2
-0.5
-0.5
0
X[m]
0.5
0
0
-0.5
0.5
Figure 40: Position of point D-the steady arm-of a dancer in time (left) and
position of point D relative to the dancer's moving center, point A (right). The
subject maintained a relatively constant distance between her axis of rotation and
steady arm through the course of her turns.
Due to time constraints of the subjects, only one dancer was measured with video
analysis. Analysis of additional dancers may yield further insight on how capable advanced
and/or professional dancers are of minimizing travel while turning.
Data collected was limited by the accuracy of the video-tracking software: slight noise
in positions measured produced a noisier angular velocity estimate. Higher resolution
video and more advanced tracking software may produce a clearer measurement.
Angular velocity was determined for each of the relations defined in Section 3.6 to
assess the quality of a dancer's spotting, as defined in Section 2.4, by comparing the change
in rotational speed of the head and body during the course of each trial. Overall angular
velocity of the dancer's body is expected to oscillate between two points of nonzero
magnitude and the same sign, while angular velocity of the head should oscillate between
approximately zero and a value of greater magnitude than the maximum body angular
velocity. Figure 41 shows angular velocity values for the head and body of a dancer as
estimated from measured angles.
25.
20 0
15
C
10
s
5
VI.1
0
~0
2
4
6
Time [s]
a
10
12
Figure 41: Angular velocity of the head (red) and the body (green) in time. Angular
velocity of the body varied between 4.85 0.72 and 6.97 0.12 radians per second,
while angular velocity of the head showed greater amplitude of oscillation, between
1.93 0.15 and 10.60 0.35 radians per second.
46
14
As expected, angular velocity of the body varied between 4.85 0.72 and 6.97 0.12
radians per second, while angular velocity of the head showed greater amplitude of
oscillation, between 1.93
0.15 and 10.60
0.35 radians per second. Thus, maximum
angular velocity of spotting is about one and a half times that of a dancer's body while
turning. These data represent a preliminary look at the contribution of spotting, a practice
that helps dancers maintain balance and avoid dizziness, to turns. However, because
spotting is a fundamental part of correct turns, it is difficult to assess if it contributes to a
dancer's overall speed or stability during a turn. Further analysis across a greater number
of advanced and/or professional dancers may provide a stronger overall estimate of the
difference in angular velocity of spotting and turning and the potential contribution of
spotting to injury prevention.
6. Conclusions
Minimum preparation plid angle, turn angle, and turn standard deviation did not differ
significantly across different numbers of rotations in pirouettes. The insignificant difference
in preparation plie angles shows that a deeper plid does not necessarily improve a dancer's
ability to complete more turns. Instead, plie angle is likely influenced by gender, and thus
the geometry of the leg: magnitude of plie angle was found to be at least 20.3 degrees
greater for males than for females.
Similarity in angle while rotating for single, double, and triple pirouettes shows that for
professional and pre-professional dancers, there is little difference in skill in performing
one to three pirouette rotations. Average angle during the turning portion of the pirouette
was 2.0 2.1 degrees less than initial standing knee angle for single pirouettes, 2.0 3.2
degrees less for double pirouettes, and 1.9
5.1 degrees less for triple pirouettes.
Characteristic standard deviation during rotation was 2.00 0.36 for single pirouettes, 3.17
0.77 for doubles, and 3.2 2.0 for triples. These results emphasize dancers' tendency to
bend their supporting knee to correct for instability: a combination of neural feedback on
balance and dancer comfort with the turn cause the dancer to bend and straighten the
supporting knee while rotating. However, bending one's knee while turning can lead to
fatigue injuries and greater risk of falling; as such, this is not good practice. The positive
deviations from average turn angle illustrate some dancers' tendency to hyperextend
during stable points in a turn-also a dangerous practice. These results could be used to
help dancers correct their knee angle during each stage of a pirouette to reduce risk of
injury and improve technique.
Since dancers were instructed to complete turns at their own pace, turn duration in
seconds varied greatly by subject. However, fraction of time turning saw a statistically
significant difference between single and double pirouettes, 22.5
2.1 and 34.3
2.7
percent of total run time, respectively. Turns required at least 13.5% of run time with an
added 9.1% of run time for each additional rotation.
Goodness of fit of the minimum jerk model to plid movements in joint space rather than
in Cartesian coordinates implies that the Central Nervous System encodes bending motions
in dance in joint coordinates (angles) instead of in Cartesian coordinates, the encoding
coordinates for reaching movements of the arm.
47
In comparing fouett6s to pirouettes, no difference could be distinguished between turn
types for both preparation pli6 and angle while rotating. Rotation number was not a
significant predictor of plie angle or angle while turning during each successive rotation for
fouettes. Dancers can successfully maintain consistent plid and turn angles when turning
continuously duringfouettds.
For the one advanced amateur recorded with high speed video, axis of rotation
remained within 0.221 0.014 meter of starting position during fouettes-minimal travel
but sufficient to correct for instability. Advanced dancers are thus able to minimize travel
to a small fraction of their wingspan-in this example, below 15 percent-while
completing manyfouettds in succession. Angular velocity of the body varied between 4.85
0.72 and 6.97 0.12 radians per second, while angular velocity of the head showed greater
amplitude of oscillation, between 1.93 0.15 and 10.60 0.35 radians per second. Thus,
maximum angular speed of spotting is approximately one and a half times the angular
velocity of the body while turning.
This kinematic definition of proper ballet technique, with deviations from expectation
representing potential for improvement, creates a framework for interpreting dancer
movement control that could contribute to the understanding of neural control of a
dancer's lower extremities to prevent injury.
7. Further Work
Use of more advanced movement tracking technologies could allow for further
development and testing of a dynamic model (such as the minimum jerk model) that
matches the form of proper technique as quantified during these experiments. The use of
motion capture technology for this study was investigated early on, but the issue of
occlusion and time and resource constraints prevented its use for the study. Motion
capture technology would enable tracking of further degrees of freedom and could
potentially enable computation of a more detailed mathematical model of pirouette and
fouette motions in multiple degrees of freedom.
While complicated, modeling the standing leg as a feedback mechanism could provide
greater understanding of the inputs (neural and mechanical) that a dancer uses to control
his or her movement during turns. More detailed comparison of pirouette andfouetti turns,
particularly with more advanced dancers, could yield additional insight on the standing leg
as a control system that corrects for instability during turns, allowing dancers to maintain
their balance and prevent falls.
48
References
Laws, "Physics and the Art of Dance: Understanding Movement," New York: Oxford
University Press (2002), pp. 62-82.
2 A. Watkins, "Dancing Longer and Stronger: A Dancer's Guide to Improving Technique and
Preventing Injury," New Jersey: Princeton Book Co. (1990).
3
J. Simon et. al, "Prevalence of Chronic Ankle Instability and Associated Symptoms in
University Dance Majors," Journal of Dance Medicine & Science, Vol. 18 Issue 4 (2014),
pp. 178-184.
1 K.
4 T.
Huang, "Automatic Dancing Assessment Using Kinect," Smart Innovation, Systems and
Technologies, Vol. 21 (2013), pp. 511-520.
5L. Kirstein. "Four Centuries of Ballet: Fifty Masterworks," New York: Dover Publications
(1984), pp. 49-97.
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