Skill Scoring System for Ski’s Parallel Turns
Shinichi Yamagiwa
1
, Hiroyuki Ohshima
1
and Kazuki Shirakawa
2
1
Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Japan
2
Exercise Physiology, Health Education, Graduate School of Education, Hokkaido University, Sapporo, Japan
Keywords:
Ski, Skill Scoring, Gamification, Trainig System.
Abstract:
Dynamic posture of sports activity is one of the most important aspects to evaluate the player’s skill. Such
sports that need evaluation from the objective observation like figure skating and skiing have difficulty in
evaluation of skill. The conventional training method for such sports was the feedback of subjective comments
from the experts regarding the performance. To overcome this problem, this paper focuses on developing a
new training system to give a clear guide for body balance control to the athlete. The system gives scores and
messages for raising up the performance skills. It causes improvement of the dynamic posture. This paper
introduces a scoring system focusing on the ski’s parallel turn. The system automatically scores skill for body
balance control regarding three aspects: the tempo at turns of body balance changes between the right and the
left, the distribution of body balance, and the angle between the snow slope and the body of the skier. The
system has been implemented in an Android smartphone and evaluated the effects of the scoring functions
from the three aspects applying to a middle level skier.
1 INTRODUCTION
The skill level of any sports activity relates to the ap-
pearance of performance. If a player performs well,
the dynamic posture of his performance should be el-
egant. However, the player is not able to observe him-
self from multiple aspects, only observation by the
third person can bring the judge of the skill level. In
some sports, the judge of the skill level is evaluated
by the observation. For example, in figure skating, the
player is evaluated by judges according to the appear-
ance of the performance. Moreover, skiing is one of
the major sports that evaluates appearance during the
performance. The qualification tests mainly in Japan,
Canada and New Zealand perform examinations to
qualify the level of the ski skill by observing the ap-
pearance of the performance ((NONSTOP, )). One of
the important performance skills is a smooth parallel
turn in a gentle snow slope. A major training method
for the parallel turn is to hear comments from high
skill trainers who have qualified to the trainer license,
or to check movies during gliding in the snow slope
and to get the comments from the high skill skiers.
Thus, it is impossible to train the parallel turn by the
skier himself without any objective comments.
Regarding the ski’s parallel turn, there exist sev-
eral engineering analysis of the dynamic posture. For
example, a ski robot (Yoneyama et al., 2006) and
a simulation-based quantitative approach (Federolf
et al., 2010) were performed. These investigated
modeling methods of the dynamic posture of skiing.
These are not targeted to use it for training. Moreover,
another modeling method for the dynamic posture ap-
plying accelerometer, magnetic sensor and GPS was
also proposed by (Kondo et al., 2013). Including this
method, all those advanced researches did not dis-
cuss by what the parallel turn becomes elegance. The
skiers aim to acquire the skill that maintains elegant
and dynamic posture during the parallel turns.
In this paper, we propose a system called ski
trainer that provides scoring methods for the perfor-
mance of ski’s parallel turn and also provides how el-
egance the performance is. How elegant is evaluated
by the dynamic posture of skier regarding the tempo,
the symmetry and the distribution of body balance.
The score is calculated based on the actual accelerom-
eter data and feedbacks advices for directing the next
step of the training.
In the next section, this paper describes the back-
ground and definitions regarding the skill acquisition
process of skiing. Section 3 will propose the ski
trainer system discussing the methods to score the dy-
namic posture. Section 4 will show the evaluation of
the system regarding the effects and the validity of the
121
Yamagiwa S., Ohshima H. and Shirakawa K..
Skill Scoring System for Ski’s Parallel Turns.
DOI: 10.5220/0005070001210128
In Proceedings of the 2nd International Congress on Sports Sciences Research and Technology Support (icSPORTS-2014), pages 121-128
ISBN: 978-989-758-057-4
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

system. Finally we will conclude the paper and men-
tion the future plans.
2 BACKGROUNDS AND
DEFINITIONS
2.1 Training Methods Using Advanced
Devices in Sports
It is a very hard process for an athlete to evaluate self-
performance of sports by giving scores if there is no
guide for the ideal target performance. It is manda-
tory for a coach or a trainer or an expert player to
give comments to the player from the appearance of
the dynamic posture. However, the target direction
to change the form or posture should be presented by
some absolute standard. The best way is to apply stan-
dardized values measured by sensors as the guide to
reach the ideal dynamic posture.
Advanced technologies in these days implement
internet-based communities using small sensor tech-
nologies (so called, MEMS) such as a smartphone
that is now including accelerometer, gyro sensors and
GPS. For example, Otsuka et. al. implemented a so-
cial network system (SNS) that provides a commu-
nity to encourage runners of jogging by the partici-
pants in the system (Otsuka et al., 2011). Using po-
sition information from GPS of a runner, the system
shows an ideal speed and the actual one of the runner.
Nike+ (http://nikeplus.nike.com/), which is available
in iPhone in these days, is also the similar system that
returns the information of the positions during jog-
ging. Adidas’ micoach (http://micoach.adidas.com/)
is also another system to guide physical threshold of
runner’s body. It uses heart beat information to guide
physical availability of the runner. However, these
system shows only how much wrong or how much
different from the ideal target values. Therefore the
users of the systems can not know how to improve the
skill using the information from the systems.
2.2 Training Methods and Qualification
for Skiing
In this paper we focus on skiing. There is a skill
qualification in Japan, so called badge test organized
by SAJ (Ski Association of Japan) (http://www.ski-
japan.or.jp/). To pass the qualification examination,
skiers need to join in a training course and receive
comments from the expert trainers. The qualification
is mainly decided by the appearance of the perfor-
mance. The decision of the qualification depends on
Y axis
Z axis
3D
Accelerometer
Y axis
X axis
Figure 1: The measurement axis of accelerometer piggy-
backed by skier.
observations by a few expert trainers during the exam-
ination. Therefore, it is hard to standardize the level
of the qualification among different resorts or differ-
ent snow slopes.
In Canada and New Zealand, there are the simi-
lar qualification licensing examinations organized by
Nonstop (http://www.nonstopsnow.com/). This also
causes the same problem as the one of Japan men-
tioned above.
The training to get good skill for the qualification
is mainly to watch the video or to receive comments
from the experts. However, it is hard for all skiers
to get such guides from experts. Therefore, it is im-
portant to implement a system that provides training
directions for skiers using absolute values measured
from some sensors.
2.3 Discussion
As mentioned above, it is hard for skier by himself
to acquire the higher level skill without observing ap-
pearance of his performance. Therefore, sensor data
brings possibility to train him without any comments
from the experts. Such as the systems for jogging dis-
cussed above, the similar system is needed for skiing.
However, not only sensor grabs the dynamic posture
to the skier, but the next step for getting higher skill
should be shown by the system. If not, the skier needs
the experts’ comments again to understand the data
from sensors.
In this paper, we will develop a scoring system
called Ski Trainer based on analyzing data from the
skills of experts. Additionally the system will give
skier advices to acquire higher skill. The system ap-
plies the sensors to acquire the motions of skier and
calculates the skill level. Then the system will give
the comments to the skier.
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We expect the gamification effect according to the
feedback information from the ski trainer. The skier
will try to get higher scores from the system. It pro-
motes the encouragement for the skier to reach higher
skill level by showing the score of the current skill
and also by giving advices to jump up to the next skill
level.
3 SKI TRAINER SYSTEM
3.1 Methods for Scoring
We use an accelerometer to measure performance of
skier. We defined the axes of the sensor as shown in
Figure 1. We have used data from the accelerome-
ter with three dimensional axis focusing on the three
aspects as described below. Here we use 200Hz sam-
pling data for the analysis.
1. Tempo
When a skier glides in a static tempo, the skier
has ability to adapt himself to any snow surface.
This means his skill is high if he can follow the
tempo. According to this technical aspect, we ap-
ply tempo to evaluate the skier’s skill.
When we plot the X axis of the accelerometer
values to the vertical axis of a graph as shown
in Figure 2, the tempo is calculated by measur-
ing the time distances of crossing points with the
horizontal axis of the origin (i.e. X value of ac-
celerometer becomes zero). Let us consider the
i-th crossing point t(i) sec. The tempo T(i) is cal-
culated by (t(i + 1) − t(i)) × 60 bpm. Here, we
apply a threshold condition T(i) = T + T
H
when
|T(i) − T| > T
H
and T
H
≤ T. This threshold is
needed for limiting to the domain of T(i) less than
T + T
H
because the skier would mistake to turn in
a much earlier timing than the target timing. The
distribution of the tempo is defined by;
ρ =
s
∑
N
i=0
{(T(i) − T)
2
}
N
(1)
M = (
ρ
T
× 100) (2)
where 0 < i ≤ N and T is the target tempo decided
by the skier. The score is calculated by 100 − M.
Here, we use T
H
= T.
Figure 3 shows an example of calculating T(i)
used in the equation above. The upper waves
than the horizontal axis of origin show the left
side turns. Besides, the lower waves show the
right side turns. Figure 3 illustrates the compar-
ison of the tempo graph between a high and a
-10
-5
0
5
10
t(s)
X axis(m/s
2
)
(
)
(
)
(
)
60}1{
×
−
+
=
ititiT
(
)
it
(
)
1
+
it
Figure 2: A tempo graph example.
low skill skiers. Figure 3(a) shows the one of a
high skill skier with the badge test qualification.
Figure 3(b) shows the tempo graph of a low skill
skier. Here, each skier defines the target tempo
to a suitable one at which he performs parallel
turns easily. The high skill skier follows the tar-
get tempo, which targets to 60, stably. However,
another skier does not follow the target tempo,
which targets to 70, at all. Thus, the tempo is
one of the important aspects to decide the skill of
skier. Using the distribution calculated above, we
can represent the score to inform the skier’s skill.
This means that the skier can knowthe levelwith a
numerical and absolute value and they can under-
stand what the main target to accomplish is, that
is, the skill to glide any slope at a concrete static
tempo.
2. Symmetry
A good balance between the right and the left side
during gliding with parallel turn is mandatory for
maintaining a good form of skiing. The balance is
evaluated by two aspects. One is the ratio of body
balance changes between the right and the left
sides. We call this the balance ratio. Another is
the angles of body against the snow slope during
turns. We call this the balance angle. These are
analyzed by the accelerometer’sX-Y data mapped
to a scatter diagram. Figure 4(a) shows the scat-
ter diagram. The raw data from the accelerome-
ter during parallel turn shapes like a reversed V
character. The top of the diagram of Y axis is 9.8
m/s
2
, which is 1G. The bottom of it becomes 2G
because there does not exist any force from the
head to the snow slope more than the gravity as
illustrated in Figure 4(a). When we move data to
Y direction for 1G (Step 1 in Figure 4(a)) and re-
place the sign of each Y value of the point (Step2
in Figure 4(a)), it becomes like a V character such
SkillScoringSystemforSki'sParallelTurns
123

(a) High skill skier
(b) Low skill skier
20
40
60
80
100
120
-10
-5
0
5
10
Tempo(bpm)
X axis(m/s
2
)
Acc X Tempo
0
20
40
60
80
100
120
140
-5
0
5
10
X axis(m/s
2
)
Tempo(bpm)
0-10
Figure 3: Comparison of tempo between (a) a high skill
skier (the upper) and (b) a low skill one (the lower).
as Figure 4 (b). Using this diagram, we decide the
scores for the balance ratio and the balance angle.
We call this diagram the symmetry diagram in this
paper.
Regarding the balance ratio, we count the number
of points in the diagram separating the right and
the left side of the Y axis. If a side of Y axis has a
larger number of points, the body balance causes
an inclination to the side. If the both sides have
the same number of points, it keeps ideal symme-
try for the parallel turn that obtains well balance
between the right and the left sides of the body.
Thus, we score the balance ratio using the equa-
tion below;
100−
|L
y
− R
y
|
L
y
+ R
y
× 100 (3)
where L
y
is the number of points in the left side
and R
y
is the one of the right side of Y axis in the
diagram.
On the other hand, the balance angle is calculated
by processing an approximated line from the ori-
gin of the symmetry diagram of Figure 4(b). How
to calculate the line is shown in the APPENDIX
section. For example, it shows the lines from the
origin of the diagram to the left and the right sides
of Y axis as depicted in Figure 4 (b). When the
angles of those lines are respectively a
L
and a
R
,
(a) Mapping original data from accelerometer
(b) V character mapping to show simple view
of symmetry diagram
-5
0
5
10
15
20
-20 -10 10 20
X axis(m/s
2
)
Y axis (m/s
2
)
-25
-20
-15
-10
-5
0
-20 -10 10 20
Y axis (m/s
2
)
Step 2. Reversing the sign
X axis(m/s
2
)
Step1. Shifting 1G to the upper side
Figure 4: Scatter diagram of a V character mapping X-Y
axis of accelerometer data. When we move the data for 1G
from the original and reverse the sign of the data regarding
Y direction, the shape becomes like a V character.
we calculate the score using the equations
below;
100− 100 × (
|a
L
|
|a
R
|
− 1), |a
L
| ≥ |a
R
| (4)
100− 100 × (
|a
R
|
|a
L
|
− 1), |a
L
| < |a
R
| (5)
If this ratio is more than 100 or less than 50, the
score is 0 because it means that the body balance
between the right and the left sides varies widely.
3. Dynamicity
We have experimented with six participants to
measure the maximum acceleration to the left and
the right side of the body balance. Figure 5(a)
shows the symmetry diagram with four time trials
of parallel turns performed by a high skill skier.
It does not become more than 2G (9.8 m/s
2
×
2 =19.6m/s
2
) to the right and the left side. On
the other hand, as a comparison, the case of the
low skill skier shows that the domain of data be-
tween the right and the left sides becomes small
as depicted in Figure 5(b). In this case, the ap-
pearance of the parallel turns becomes compact
and is not elegant. When the acceleration to both
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(c) Tempo graph (d) Symmetry diagram (e) Score(a) Data recording mode (b) Data selecting
Figure 6: Screenshots of the ski trainer system implemented on an Android smartphone.
-20 -15
-10 -5
5 10 15 20
X axis(m/s
2
)
Y axis (m/s
2
)
-20
-15
-10
-5
0
5
10
15
20
(a) A high skill skier
(b) A low skill skier
-20 -15 -10 -5 5 10 15 20
X axis(m/s
2
)
Y axis (m/s
2
)
-20
-15
-10
-5
0
5
10
15
20
Figure 5: Symmetry diagrams of (a) a high skill and (b) a
low skill skiers. The force to the right and the left sides
never become more than 2G. The data domain in the case
of low skill skier becomes small.
sides is maintained largely, the appearance of the
parallel turn seems elegant because the angle of
the body becomes keen against snow slope and
the dynamic body balance is controlled widely.
Therefore, we decide the score for the dynamicity
to provide a higher score when a larger accelera-
tion is observed from the accelerometer using the
equation below;
100− (
L
max
+ R
max
2
)/19.6 (6)
where the maximum acceleration to the left and
the right are L
max
and R
max
respectively. This
equation calculates an average among L
max
and
R
max
, and the ratio is calculated by dividing 2G.
Applying the tempo, the symmetry and the dy-
namicity to the scoring methods, we can evaluate the
skier’s skill of his parallel turn technique as men-
tioned above. The tempo shows the ability of the skier
if he can adapt himself to the snow surface. The sym-
metry gives advice to correct skier’s turn to obtain it
beautiful. The dynamicity brings a factor if he can
perform elegant parallel turn.
The scoring methods bring a new feature for the
training system, which extends the system to output
messages used as advices to the skier. We define the
messages regarding the scores for the tempo, the sym-
metry and the dynamicity. When the score of tempo
is more than 80, it outputs a message ”gliding in a
good tempo”. If it is less than 40, it outputs ”follow
the target tempo”. When the score of balance angle is
0, it shows a message ”inclination is too significant”
to the right or the left according to the approximation
line in the symmetry diagram. If it is 50 or more, the
message ”the balance is good” is outputted. When the
score of dynamicity is more than 70, it outputs ”ele-
gant gliding”. If not, it shows ”use more side edges to
line curves”.
Using these three scoring methods and messaging
output rules, we implement a system for ski’s parallel
turn called Ski Trainer.
3.2 Implementation
We have implemented the scoring methods discussed
above in a smartphone application. The screen-
shots of the application in an Android smartphone
are shown in Figure 6. The application includes the
functions below. There are two modes in the applica-
tion: the recording mode and the analysis mode. The
SkillScoringSystemforSki'sParallelTurns
125

Figure 7: Wearing the ski trainer piggybacked by a holster
small backpack.
Acc X Tempo
20
40
60
80
100
120
140
-5
0
5
10
X axis(m/s
2
)
Tempo(bpm)
0-10
(a) Before applying the ski trainer to the skier
20
40
60
80
100
120
140
-5
0
5
10
X axis(m/s
2
)
Tempo(bpm)
0-10
(b) After applying the ski trainer to the skier
Figure 8: Performance comparison regarding the tempo be-
fore and after applying the ski trainer.
recording mode saves the accelerometer data to the
smartphone. It includes a sound guide function that
outputs a target tempo by a metronome sound. The
skier performs parallel turn following the sound. Us-
ing it as the target tempo, the score for the tempo is
calculated.
A screenshot of the recording mode is shown in
Figure 6 (a). It includes the accelerometer data and
the gyroscope data measured by the sensors equipped
in the smartphone. The STBY button starts to record
the motion data and saves it in a CSV file stored in the
smartphone.
The analysis mode has three steps to show the
scores. The first step is to select the interval of the
parallel turns as shown in Figure 6(b). This is done
by a touch and a drag operation to select the duration
of the parallel turns. The ski trainer application has
a comparing function among several performances at
a time. Therefore, the second step selects data sets
Table 1: Tempo data T(i) before and after applying the ski
trainer. The i in the first column corresponds to the sequence
number of turns.
i T
before
(i) T
after
(i)
1 35 42
2 140 49
3 20 69
... ... ...
23 140 75
... ... N/A
33 49 N/A
ρ 47.0 9.8
Score 54 90
to evaluate and compare it simultaneously. For ex-
ample, if a skier wants to compare his performance
with the previous ones, this function provides visual
comparison methods effectively and he can consider
the difference among performances. The final step re-
sults the scoring. This has three screens illustrated in
Figure 6(c)-(e); tempo, dynamic/symmetry and score,
respectively. The tempo screen shows the differences
among the target and the actual tempos acquired by
performances. The example shows two tempo graphs
at a time. The dynamic/symmetry screen shows the
symmetry diagram. It represents visually the differ-
ence of dynamicity among performances. The score
screen shows a triangle where the corners represent
the scores of tempo, the symmetry (the average of
balance ratio and the balance angle) and the dynam-
icity respectively. When the score becomes high, the
triangle shapes large. The shape of the triangle intu-
itively informs the skier the lack of skills. The exam-
ple screen shows the messages in Japanese language
mode.
4 EVALUATION
We have asked a middle level skier to use the ski
trainer who has 10 year experience of skiing, but who
does not have any qualification for any ski licenses.
We compare his performance before and after we ap-
ply him the ski trainer by comparing twice on two
different days. He wear the smartphone in his back
as shown in Figure 7 and he heard the target tempo
from a noise canceling earphone. Figure 8(a) and Fig-
ure 9(a) show the results from the ski trainer on the
first trial day of the skier. After considering the mes-
sages and graphs outputted from the ski trainer on the
second day, his skill level has become higher such as
Figure 8(b) and Figure 9(b). It becomes better per-
formance when the score and messages from the ski
trainer are applied as advice to him. Let us compare
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(a) Before applying the ski trainer to the skier
(b) After applying the ski trainer to the skier
-20 -15 -10 -5 5 10 15 20
X axis(m/sec
2
)
Y axis (m/s
2
)
-20
-15
-10
-5
0
5
10
15
20
25
-20 -15 -10 -5 5 10 15 20
X axis(m/sec
2
)
Y axis (m/s
2
)
-20
-15
-10
-5
0
5
10
15
20
25
Figure 9: Performance comparison regarding the symme-
try/dynamicity before and after applying the ski trainer.
the effect of the ski trainer quantitatively among the
tempo, the symmetry and the dynamicity.
• Tempo
As we can see in Figure 8, the tempo has become
stable due to the effort of the skier on the 2nd
day. His target tempo is 70. He tried to make
the score high following the message from the
ski trainer system. Table 1 shows a part of the
tempo data calculated from the tempo graph. The
score before using the ski trainer was 54. How-
ever, he improved his skill to 90. Therefore, we
have confirmed that this method provides an ef-
fective training to guide the timing related matters
of ski’s parallel turn. The skier tries to glide any
slope by following the target tempo. As the re-
sult, the trials raise the skill to turn in any snow
environment.
• Symmetry
Before applying the ski trainer system, the skier
tends to use the ski edge in the right side mainly
because his dominant leg is the right side. There-
fore, the right side of the vertical axis in the sym-
metry graph of Figure 9(a) includes the larger
number of points comparing to the left side. The
numbers of the points in the right and the left sides
of the vertical axis in the graph are 3658 and 1342
respectively. This result causes the balance ratio
a bad result. The score was 54. Moreover, the
balance angle shows also inclination to the right
side. The angles of the approximation lines are
also 2.1 and 2.8 in the right and the left sides of the
vertical axis respectively. The score was 67. On
the other hand, after the participant skier trained
himself to correct his performance referring the
symmetry diagram of the ski trainer, the balance
between the right and the left side has becomes
well. The numbers of the points in the right and
the left sides of the vertical axis has become 2597
and 2135 respectively. Additionally, the angles of
the approximation lines have become 2.6 and 2.4
for the right and the left sides respectively. Thus,
the scores of the balance ratio and the balance an-
gle have become 90 and 92 respectively. Accord-
ing to the observation above, we confirmed that
the symmetry diagram provides enough informa-
tion for considering an inclination to a side, and
also it intuitively provides a correct training di-
rection regarding the technique for body balance
control.
• Dynamicity
The symmetry diagram before applying the ski
trainer shows that the dynamicity becomes be-
tween 9.7m/s
2
in the right side and 8.0m/s
2
in
the left side. Therefore, the skier’s parallel turn
has an inclination to the right side, also was very
compact (not dynamic). The score of the dynam-
icity was 45. Due to the score, the symmetry dia-
gram and the messages from the ski trainer, he has
become aware of using the both sides of ski edges
and has moved his balance largely to the right and
the left sides. Then the dynamicity has become
9.4m/s
2
and 9.0m/s
2
on the right and the left sides
respectively. The score has become 46. Thus, be-
cause the acceleration to the side edges has be-
come large according to the advices from the ski
trainer, his parallel turn has become more elegant
than before although the scores do not change sig-
nificantly.
As we have discussed in this section, the ski
trainer system actually improved a skier’s parallel
turn. Originally, the training had to be provided by
the objective advices from experts by observing ap-
pearance of the parallel turn of a trainee skier. On
the other hand, our system brought a novel training
method by the skier himself using accelerometer data
from a smartphone. We also confirmed that the nu-
merical scoring method to the performance, the vi-
sual guides and the messages from the tempo graph
SkillScoringSystemforSki'sParallelTurns
127

and the symmetry diagram promotes the self-training.
Thus, we conclude that the ski trainer system is effec-
tive to training in the ski’s parallel turns.
5 CONCLUSIONS
We have designed and implemented a training system
using accelerometer on a smartphone for ski’s parallel
turn. We focused on the tempo, the symmetry and
the dynamicity during the parallel turn. Applying the
system to a middle level skier, we have a good effect
for improving the perspective of his technique. As the
next step of this research, we are planning to develop
a matching technique to identify the skill level from
multiple symmetry diagrams. For example, we will
try to match a symmetry diagram to a group of the
same diagrams. Then we will design and implement
a new system that classifies the skill level to another
skier’s skill such as an Olympic athlete, and outputs
a message such as ”Your skill is similar to the gold
medalist of this year”.
ACKNOWLEDGEMENT
This research is partially supported by Tateishi Sci-
ence and Technology Foundation. And also this work
is partially supported by KAKENHI (24240085)
Grant-in-Aid for Scientific Research (A) and KAK-
ENHI (24300020) Grant-in-Aid for Scientific Re-
search (B). We thank to Prof. Yuji Yamamoto at
Nagoya University, Japan for his proofreading of this
paper content.
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APPENDIX
The approximation line for the balance angle is cal-
culated using the least squares method. For any ac-
celerometer data of X and Y axis represented by (x
i
,
y
i
) where i is 0 ≤ i < N, we calculate a that makes
minimal of;
S(a) =
∑
(y− ax)
2
Because S
′
(a) = 0,
∑
xy− a
∑
x
2
= 0
a =
∑
xy
∑
x
2
The scoring equation for the balance angle uses the
data in the right side to calculate a
R
and the one of the
left side to calculate the a
L
. The calculation for the
score uses the absolute values of a
R
and a
L
.
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