SICE Journal of Control, Measurement, and System Integration, Vol. 4, No. 1, pp. 001–005, January 2011

Skill Level Evaluation for Taijiquan Based on Curve Fitting
and Logarithmic Distribution Diagram of Curvature
Toshiyuki MURAO ∗ , Yasuyuki HIRAO ∗ , and Hiroshi HASHIMOTO ∗Abstract : In this paper, we propose an objective skill level evaluation method for Taijiquan. The proposed skill levelevaluation method utilizes both curve fitting to spiral motion and classification based on a logarithmic distribution diagram of curvature for a human body part. In order to demonstrate a possibility of the proposed method, we present
experimental results with a Kinect sensor for three subjects, i.e., a beginner, intermediate and expert.
Key Words : skill level evaluation, Taijiquan, logarithmic distribution diagram of curvature, Kinect.

1.

Introduction

This paper proposes a method to give an objective evaluation of the skill level in Taijiquan based on curve fitting and alogarithmic distribution diagram of curvature.
The number of elders who start learning Taijiquan to improvetheir health is increasing with the aging of society. In order tocontinue learning Taijiquan, it is necessary to evaluate objectively how much his skill level goes up. It is known that his
passion for learning will be maintained when one can have amethod to evaluate the skill level by oneself without depending
upon the evaluation from an instructor [1],[2]. Furthermore,obtaining an objective index of the skill level is conductive tofind motion to be corrected, and leads to an improvement oflearning eﬃciency.
Learning Taijiquan on textbooks or DVD contents onlyshows shapes or time-varying movements of motion, but does
not point out how a learner should correct his bad motion.
Therefore, the instructor’s evaluation can do nothing but be received in a Taijiquan classroom. However, the fact is that mostof the evaluation is based on the subjective judgement [3],[4].
To the best of our knowledge, an objective evaluation based ona trajectory of the motion is hardly shown. Also, most previous
works for Taijiquan have focused on the movement of the center of gravity of a human body, not on the trajectory of a humanbody part, e.g., see [5].
Here, a method that objectively evaluates an aesthetic curve
based on a logarithm distribution diagram of curvature, developed in the field of the industrial design, has been proposed [6],[7]. In the field of the industrial design, objective
evaluation of curves is performed by classifying them into fivetypical patterns. Accordingly, the proposed method is able toclassify curves into the five patterns. Our idea lies in that a
trajectory of body part motion in Taijiquan is regarded as anaesthetic curve and then objective evaluation of the skill levelis obtained through the classification of the curve into the five∗

Master Program of Innovation for Design and Engineering,
Advanced Institute of Industrial Technology, Tokyo 140-0011,
Japan
E-mail: murao-toshiyuki@aiit.ac.jp, b1033yh@aiit.ac.jp,
hashimoto@aiit.ac.jp(Received June 7, 2012)(Revised January 15, 2013)

patterns.
In this paper, we apply the evaluation method of an aesthetic
curve to the evaluation of the skill level in learning Taijiquan.
We regard the proposed method as an alternative to standard instructor judgement, but not professional jury one in an oﬃcial
game. The proposed method performs curve fitting to spiralmotion in order to enhance the feature for an expert of Taijiquan before the classification using the logarithmic distribution
diagram of curvature. In the experiment, the motion captureusing a Kinect sensor makes it possible to obtain the trajectoryof a human body part in Taijiquan with simple cost. It is foundthat the diﬀerence between an expert and a non-expert can be
explained from the curve fitting and the classification throughthe experimental results.

2.

Logarithmic Distribution Diagram of Curvature

A classification method through a logarithmic distribution diagram of curvature which shows characteristics of an aesthetic
curve, i.e., curvature changes and a volume of a curve has been
proposed in [6],[7]. The logarithmic distribution diagram ofcurvature is represented on a double logarithmic coordinates
system whose horizontal and vertical axes represent an intervalof a radius of a curvature ρ̄i and a ratio of a length of a segmental curve to a total length of a curve s̄i , respectively. Theinterval ρ̄i and the ratio s̄i are calculated as follows:

ρ̄i =

s 
ρi
i
, s̄i = log10
,
S
S

(1)

where ρi , si and S are a radius of a curvature at a constitutional
point ai , a length of a segmental curve and a total length of acurve, respectively.
In the field of the industrial design, it is reported that the logarithmic distribution diagram of curvature for an aesthetic curve
can be classified into five typical types [6],[7]. The five typicaltypes of the logarithmic distribution diagram of curvature areshown in Fig. 1. The eﬀectiveness of the classification method
based on a logarithmic distribution diagram of curvature has
been verified through a few thousand real products including
industrial products, shaped forms by nature and artifacts [8].

c 2012 SICEJCMSI 0001/11/0401–0001 

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SICE JCMSI, Vol. 4, No. 1, January 2011

Type 1

Type 2

Table 1 Years of experience and frequency of training of Taijiquan.

Type 3horizontal

0

minus
Type 4plus

minus+plus

Five typical types of logarithmic distribution diagram of curvature.

x

x

Curve
FittingttFig. 2 Curve fitting using the spiral equations.

3.

Frequency of Training(times per a week)

4
8
32

2
2 or 3
5–7

Type 5plus+minus

Fig. 1A (Beginner)
B (Intermediate)
C (Expert)

Years of Experience(years)Skill Level Evaluation Method for Taijiquan

In previous works [6],[7], the classification method throughthe logarithmic distribution diagram of curvature is only applied to a shape of a product in the field of the industrial design.
In this paper, we expand it to a skill level evaluation of humanmotion in Taijiquan.
3.1 Curve Fitting Using Spiral Equation
We interviewed an expert of Taijiquan to understand characteristics for motion of an expert in Taijiquan qualitatively. In
the interview result, it is found that the motion of a Taijiquan
expert has the feature of spiral motion. Then, it is indicated thatthe motion of an expert has a good curve fitting to an oscillation
which is amplified or decayed.
In order to enhance the feature of an expert in Taijiquan,
we perform curve fitting with the least-square method using a
trigonometric function with the damping term which represents
the feature of a spiral. The equations of the curve fitting for the
position data p = [x, y, z]T are shown below.
⎫
⎧ n
⎪
⎪

⎪
ax t ⎪⎬⎨,(2)bix sin(cix t + dix )+eix cos(cix t + dix ) ⎪x = f x +e 2 ⎪
⎪
⎪
⎭
⎩
i=1
⎫
⎧ n
⎪

ay t ⎪⎪⎪⎬⎨
y = fy +e 2 ⎪, (3)biy sin(ciy t + diy )+eiy cos(ciy t + diy ) ⎪
⎪
⎪
⎭
⎩
i=1
⎫
⎧ n
⎪
⎪

⎪
az t ⎪⎬⎨, (4)z = fz +e 2 ⎪biz sin(ciz t + diz )+eiz cos(ciz t + diz ) ⎪
⎪
⎪
⎭
⎩
i=1

where fk , ak , bik , cik , dik and eik (k = x, y, z) are unknown parameters. The spiral equations (2)–(4) are proposed based on
the equation representing the damped oscillation. In order to
fit multiple peaks of the trajectory, polynomials of trigonometric functions are utilized. Figure 2 shows a result of the curve
fitting using the spiral equations for a curve.
3.2 Skill Level Evaluation Method Based on Curve Fitting
and Logarithmic Distribution Diagram of Curvature
In this subsection, we propose the skill level evaluation
method for Taijiquan based on the curve fitting and logarithmic

distribution diagram of curvature. Our idea is that the trajectoryof the expert’s motion in Taijiquan is regarded as an aesthetic
curve.
Firstly, we measure a trajectory of a human body part using a motion capture system, e.g., a Kinect system. It should
be noted that our proposed evaluation method does not need
the movement of the center of gravity of a human body which
is more diﬃcult to obtain accurately. Secondly, we utilize thesmoothing diﬀerential method in order to remove noise of human motion and sensor noise from the measured position data.
The equation of the smoothing diﬀerential method is shown as
follows (See [9]):ṗ(tn ) =1(−p(tn−3) − p(tn−2 ) − p(tn−1 )12+p(tn+1 ) + p(tn+2 ) + p(tn+3 )) ,(5)

where p(tn ) = [x(tn ), y(tn ), z(tn )] is the measured position dataat time tn .
Next, the curve fitting is performed using the spiral equations (2)–(4). It should be remarked that the motion of an expert has to be fit to the spiral well. In other word, the result of
curve fitting by using the spiral equations (2)–(4) could catchthe turning points in the motion. In particular, a key point for
curve fitting in this research is that the shape of the fitting curve
matches that of the measured data at the inflection points.
Finally, we draw the logarithmic distribution diagram of curvature in Section 2 using the radius of the curvature. If theresult is classified as one of the five typical patterns in Fig. 1, it
should be evaluated that the curve is aesthetic, i.e., the motion
approaches an expert’s motion. Considering the real instructor’s judgement criteria, the skill level of a Taijiquan player
could be high if his motion is similar to an expert’s one. Therefore, the proposed method can evaluate the skill level throughthe classification between experts and others.
Figure 3 shows the flow diagram of the skill level evaluation
for Taijiquan based on the curve fitting and logarithmic distribution diagram of curvature. It is important that the curve is
fit to the spiral well and the logarithmic distribution diagram ofcurvature is classified into the five typical types. Our proposed
objective evaluation of the skill level for Taijiquan will be obtained using both the curve fitting and the classification basedon the logarithmic distribution diagram of curvature.
T

4. Experiment
In this section, we present experimental results in order toverify the eﬀectiveness of the proposed method.
4.1 Experimental Setup
In this experiment, we compare motion of three human subjects, i.e., a beginner, an intermediate and an expert. Table 1
shows years of experience and frequency of training of Taijiquan for the three subjects. An experimental environment isshown in Fig. 4. The human motion capture system used in theexperiment, consists of an oﬀ-the-shelf Microsoft Kinect [10]3SICE JCMSI, Vol. 4, No. 1, January 2011

MotionCaptureSystem

1st Check Pointx

Numerical
IntegralCurve
Fitting

z

y
y

z

z

z

SmoothingDifferential

x

y

x

(a) Measured position dataof a human body part(b) Velocity dataxy(c) Noise reduction position data(d) Curve fitting position data

2nd Check Point
Type 1

Type 2Type 3horizontal

Hi

siClassification
minus

0Type 4

Fig. 3plus+minus

Hi(f) Logarithmic distribution diagram of curvaturet(e) Radius of curvature

plus
Type 5minus+plus(g) Classification into five typical types

Flow diagram of the skill level evaluation for Taijiquan based on the curve fitting and logarithmic
distribution diagram of curvature.

KinectyxzFig. 5

Fig. 4 Experimental environment.

and a laptop. The Microsoft Kinect, which utilizes the laptop,
has a depth sensor, a RGB camera and a special microchip totrack movement with a bone model. A player does not need any
marker in order that the Kinect sensor system provides full bodythree dimensional motion capture. Hence, an advantage of thesystem is that both cost and a burden for the player are small.
The Kinect sensor system creates a digital skeleton based on thebone model and depth data obtained from an infrared projector
and a camera. The coordinate on the human body is automatically determined by the Kinect sensor system. Matching thedigital skeleton to image data received from the RGB camera,
the Kinect sensor system is capable of automatically calibrating
the human motion based on their physical attribute.
After calibration, the Kinect sensor system can provide accuracy of depth reconstruction in the order of 1–4 cm at therange of 1–4 m [11],[12]. Typical errors for the pose estimation accuracy can be kept within about 5 cm [13]. The Kinect
sensor motion capture algorithm works best when players are
standing, the view is unobstructed and players body limbs are
not interacting with objects [13]. In order to keep accuracy of
the Kinect sensor motion capture, we set the Kinect sensor in
the front of the Taijiquan players. Here, the position data canbe derived automatically from the Kinect motion capture system, after a certain motion which an evaluator focused on is
extracted manually from whole motion.
Three human subjects performed Yunshou, which is one of
the 24 formulas of Taijiquan. Figure 5 shows movement of a

Movement of a part of Yunshou.

part of Yunshou to which the proposed evaluation method isapplied. In this experiment, we focused on the motions of theirleft wrists, since it was found through experimental analysis
that features for movement of Yunshou prominently emergedin them.
The curve fitting was carried out by using the least-square
method with Scilab. We set the initial values of the parameters
fk , ak , bik , cik , dik and eik (k = x, y, z) in Eqs. (2)–(4) to 1.
4.2

Experimental Results

We present the experimental results for the skill level evaluation for Taijiquan in Figs. 6–8. Figures 6–8 show the results of
the left wrist motion of the beginner, intermediate and expert,
respectively. The dashed lines denote the measured position
data with the smoothing diﬀerential method, and the solid lines
denote the result of the curve fitting by using the spiral equations in Eqs. (2)–(4). The curve fitting results of the x axis andz axis are presented in Figs. 6–8 (a) and (b), respectively. Also,
Figs. 6–8 (c), (d) and (e) show the curve fitting results of the yaxis with n = 1, n = 2 and n = 3, respectively, in Eq. (3). Although the good curve fitting results of the x axis and z axis are
obtained with n = 1 in Eqs. (2) and (4), that of the y axis could
not catch the feature of the turning point. Figures 7 (f) and 8(f) depict the logarithmic distribution diagram of curvature byusing the curve fitting results of the x and z axis with n = 1 andthat of the y axis with n = 3.
From Fig. 6, it can be verified that the curve fitting in thecase of the beginner cannot pick up the feature of the turning
point from the measured data at 14 s and 14.6 s obviously, inspite of increasing the number of the trigonometric function’s
component n. Whereas the good curve fitting can be obtained,
Fig. 7 indicates that the logarithmic distribution diagram of cur-

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SICE JCMSI, Vol. 4, No. 1, January 2011
-0.1

2.45

2.7

0

-0.5

2.4

-0.5

z

x

z

x

2.6
2.5

2.35

2.4
-1.0
10

12

14

16

10

12

14

16

t
t
(a) Curve fitting result with x axis (n = 1) (b) Curve fitting result with z axis (n = 1)
0.44

-1.0

51

52

53

54

2.3

55

51

52

53

54

55

51

52

53

54

55

t
t
(a) Curve fitting result with x axis (n = 1) (b) Curve fitting result with z axis (n = 1)

0.44
0.42

0.4

y

y

y

0.42

y

0.42

0.42

0.4

0.4

0.4

0.38
10

12

14

16

t
(c) Curve fitting result with y axis (n = 1)

0.38
10

12

14

16

t
(d) Curve fitting result with y axis (n = 2)

0.38

51

52

53

54

55

t
(c) Curve fitting result with y axis (n = 1)

0.38t(d) Curve fitting result with y axis (n = 2)

0.44

0
0.42

0.42

si

y

y

-1
0.4

0.4

-2

0.38
10

12

14

0.38

16

t
(e) Curve fitting result with y axis (n = 3)

51

52

53

54

55

t
(e) Curve fitting result with y axis (n = 3)

-3
-3

-2

-1

0

1

Hi
(f) Logarithmic distribution diagram ofcurvature2

Fig. 6 Curve fitting of the beginner.
Fig. 8 Curve fitting and the logarithmic distribution diagram of curvatureof the expert.

2.7

z

x

0
2.5

-0.5

2.3
-1.0

20

22

24

26

20

22

24

26

t
t
(a) Curve fitting result with x axis (n = 1) (b) Curve fitting result with z axis (n = 1)
0.4

5. Conclusions

y

y

0.4

0.35

0.3

0.35

20

22

24

26

t
(c) Curve fitting result with y axis (n = 1)

0.3

20

22

24

26

t
(d) Curve fitting result with y axis (n = 2)
0

0.4

y

si

-1
0.35
-2

0.3

20

22

24

26

t
(e) Curve fitting result with y axis (n = 3)

ture histogram is classified as the plus type which is one of thefive typical types. This implies that the motion trajectory of theexpert which needs the feature of the spiral can be regarded as
an aesthetic curve in the field of the industrial design. Fromthese experimental results, it can be confirmed that the objective evaluation of the skill level for Taijiquan can be presented
by using both the curve fitting and the classification based on
the logarithmic distribution diagram of curvature.

-3
-3

-2

-1

0

1

Hi
(f) Logarithmic distribution diagram ofcurvature2

Fig. 7 Curve fitting and the logarithmic distribution diagram of curvatureof the intermediate.

vature in the case of the intermediate cannot be classified into
the five typical types in Fig. 1. On the other hand, in the caseof the expert, it is found that the curve fitting can capture theturning point of the measured data, and the logarithmic curva-

In this paper, we present the objective skill level evaluation
method for Taijiquan. This method utilizes both the curve fitting to the spiral motion and the classification based on the logarithmic distribution diagram of curvature for a human bodypart. The advantage of the proposed method does not need tomeasure the movement of the center of gravity of a human body.
Although the number of subjects are small, we present experimental results of the beginner, intermediate and expert with
the Kinect in order to demonstrate a possibility of the proposed
method.
In the field of Taijiquan, standard evaluations to skill levelare usually given by each expert instructor. It is fact that anevaluation method has not been formalized yet so that anybody
can measure skill level easily. In this background, we have tostart the verification of the proposed evaluation method basedon one evaluation of only one reliable instructor. As the result, we could provide equivalent experimental results to a realworld skill level evaluation one, i.e., an instructor visual judgement one. However, we know that the more we increase thenumber of subjects, the more objective our proposed evaluation
become. Possible future work involves an increase in the num-

SICE JCMSI, Vol. 4, No. 1, January 2011

ber of subjects and Taijiquan formulas for experimental verification and numerical evaluation for skill level of Taijiquan. We
will also improve the acquisition procedure of position data to
be able to extract a judged motion from whole motion automatically.

Acknowledgments
The authors would like to thank Mr. Kazuo Sayama in Japan
Health Taijiquan Association for his helpful insight as an expert of Taijiquan. They would also like to thank Mr. Satoshi
Kawata, Mr. Yusaku Satoh, Mr. Takashi Nakano and Mr. Hasicholu in Advanced Institute of Industrial Technology for theireﬀorts to build experimental systems, as well as the editors andthe reviewer whose many insightful comments and suggestions
greatly improved this manuscript.

Toshiyuki MURAO (Member)He received his B.E. degree in the Department of Electrical & Computer Engineering from Kanazawa University in 2003, his M.E. degree in the Division of Electronics and Computer Science from Kanazawa University
in 2005, and his Dr. of Engineering degree in the Department of Mechanical and Control Engineering from
Tokyo Institute of Technology in 2008. Since 2006, hehas been an Assistant Professor of Advanced Institute of Industrial Technology, Japan. His research focuses on passivity-based visual feedback
control of robot systems. He is a member of IEEE and IEEJ.

Yasuyuki HIRAOHe received his B.E. and M.E degrees in Engineering
Science from Osaka University in 2003 and 2005, respectively, and his Master of Technology in Innovation for
Design and Engineering degree from Advanced Instituteof Industrial Technology in 2012. He is currently with
the System & Software Technology Platform, Sony Corporation, Japan.

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5

Hiroshi HASHIMOTO (Member)He received his Ph.D. degree in science engineeringfrom Waseda University, Tokyo, Japan in 1990. He iscurrently a Professor in the Master Program of Innovation for Design and Engineering, Advanced Institute ofIndustrial Technology, where he does research on intelligent robots, cybernetic interfaces, vision systems, welfare technology, and e-learning. He is a member of IEEEand IEEJ.