2019 IEEE 8th Global Conference on Consumer Electronics (GCCE)

Generalized Principal Motion Analysis:
Classiﬁcation of Sit-to-Stand Motions
Tomoyuki Iwasaki, Shogo Okamoto, Yasuhiro Akiyama, and Yoji Yamada
Department of Mechanical Systems Engineering, Nagoya University, Nagoya, Japan

Abstract—We propose a generalized principal motion analysis(GPMA) method for analyzing temporally evolving motions of
redundant systems, such as human motions. GPMA ﬁnds basefunctions, which are called principal motions, that maximallyseparate distinctive types of motions and that weaken the effectsof repeated errors within each type of motion. As an exampleof human motions, we measured 15 types (3 participants × 5conditions) of sit-to-stand motions by a camera-based motion
capture system. Each type of motion was repeated 10 times. We
then compared GPMA and PMA in terms of their ability toclassify the type of motions. GPMA correctly classiﬁed all typesof motions, whereas PMA correctly classiﬁed only 81% of them,which shows that GPMA has a better ability to classify motions.
Index Terms—principal motion analysis, time-series datathe effects of motion repetition errors. We analyzed the sit-tostand motions of knee-pain holders as an example of humanmotions and then compared the capabilities of GPMA andPMA. Especially, we evaluated their ability to classify sitto-stand motions recorded by a camera-based motion capturesystem.
II. A NALYSIS METHODS
A. Notion of Motion Variables
Suppose that a human motion at a certain moment isrepresented by p joint angles. For joint i (i = 1, ..., p) at k-thtrial (k = 1, ..., k ′ ), the time-series data vector θik , which isaligned to the data length u, is given asθik = (θik1 , ..., θikl , ..., θiku )T .

I. I NTRODUCTION
Techniques to distinguish human motions are important fora variety of applications including security and sports science.
A human motion is composed of the time-series data ofmultiple variables such as joint angles. We exemplify sit-tostand motions from a chair. It is a dynamic motion of thetotal joint motions of the lower limbs and the trunk, which
requires the load on the lower-limb joints during the posturalchange from sitting to standing position. In human motion
detection problems, typically, statistics computed from themotion data are discussed. For the analysis of sit-to-standmotions, Anan et al. [1] and Turcot et al. [2] divided eachmotion into a few phases and compared several characteristic
values of joint variables for each phase. Christiansen et al.
[3] and Samman et al. [4] discussed the time-series data foreach joint variable. Thus, many previous studies did not aim toanalyze multivariate motion data continuously and statistically.
The maximum joint moments during the sit-to-stand motions
have been also typically discussed [5], [6].
Human motions are sometimes analyzed by principal motion analysis (PMA) [7], [8]. This method can reduce dimension without losing the information of interlocked multiple
degrees of freedom of redundant systems. Hence, PMA has
been leveraged for solving the problems of redundant motion
systems including motion generation of animated robots [7]and classiﬁcation of human motions [9]. However, PMA
is affected by the variation in inter-individual motion and,
therefore, it may not correctly classify human motions ofmultiple participants.
In this work, we improved PMA and propose a generalized
principal motion analysis (GPMA) method that is robust to(1)Using this, we create an extended column vector xk as
xk = (θ1k T , ..., θik T , ..., θpk T )T .

(2)

(s)

Here, xk represents a time-series column vector of the experiment participant s (s = 1, ..., s′ ) at trial k and x̄ represents themean value of the motions of all the experiment participants.
If the dataset contains s′ k ′ trials, the time-series data of all′ ′the motions are represented by a matrix X (∈ Rs k ×pu ), as
follows:
(1)

(s)

(s′ )

X = (x1 − x̄, ..., xk − x̄, ..., xk′ − x̄)T .

(3)

B. Principal Motion Analysis (PMA)
Park et al. [7] extended the principal component analysis(PCA) method to make it applicable to time-series data andachieved a representation of human motions with a reduced
number of variables. PCA is a multivariate analysis methodthat identiﬁes the combinations of correlated variables, which
are called principal components. These components, which arecalculated from the same data pool, are linearly independentof each other.
PMA interlocks the variables along the temporal dimension
as well as among multiple variables. The combination ofinterlocked variables in the temporal direction is called theprincipal motion. PMA is a linear analysis method, and it ispossible to interpret the meanings of the obtained principalmotions.
From the eigenvector expansion of covariance matrix
X T X,X ∼ Y V T,
(4)

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(1)

(s′ )

where Y = (y1 , ..., yk′ )T is a score matrix that represents
how much each observed motion includes the informationof each principal motion and V = (v1 , ..., vq , ..., vr ) is aprincipal motion matrix composed of r eigenvectors corre′ ′sponding to the r largest eigenvalues. vq (∈ Rs k ×1 ) is theq-th principal motion vector. Any motion is represented by alinear combination of the principal motion vectors weighted
by the corresponding scores:(s)(s)xk ∼ V yk .

(5)

C. Generalized Principal Motion Analysis (GPMA)
We extended the linear discriminant analysis (LDA) method,a generalized form of PCA, to make it applicable to time-series
data and propose a (GPMA) method. LDA ﬁnds the reduced
space of observed samples so that the variance between individuals is increased and that within individuals is decreased.
Therefore, in the motion analysis of multiple participants,
GPMA can emphasize individual differences and depress thefactors caused by repetition errors within individuals.
′A motion matrix of experimental participants A (∈ Rs ×pu )is computed as′

A = (x̄(1) − x̄, ..., x̄(s) − x̄, ..., x̄(s ) − x̄)T ,(6)

where x̄(s) is the mean of the motions of each participant.
Moreover, a matrix of repetition errors within individuals B′ ′(∈ Rs k ×pu ) is computed as
(1)

(s′ )

(s)′

B = (x1 − x̄(1) , ..., xk − x̄(s) , ..., xk′ − x̄(s ) )T .

(7)

Let the covariance matrices AT A and B T B of A and B
be C and D, respectively. Then, we regularize these matrices
and obtain Cr , Dr , respectively. A principal motion matrix
V and a score matrix Y are obtained from the eigenvector(s)
expansion of Dr−1 Cr . Motion xk is approximated by a linear
combination of the principal motions as well as by PMA:
(s)

(s)

xk ∼ V yk ,(s)where yk(8)is a row vector of the score matrix Y .

III. DATA COLLECTION OF SIT- TO - STAND MOTIONS
We recorded sit-to-stand motions of people under some
controlled conditions. In the ﬁrst condition, three healthy
male participants normally stood up from a chair. In theother conditions, they imitated the motion of those with lightor severe pain on either the right or the left knee. Before
the imitation, they received a lesson on stand-up motions ofpatients with knee impairment. The level of pain was adjusted
by the imbalance ratio of body weight support. In the light pain
condition, the participant loaded his weight on the foot of the
supposedly healthy side by 10−20 % compared to the affectedside. In the severe pain condition, the healthy side supportedmore weight than that of the affected side by 30−40 %. Eachcondition was repeated 10 times. In total, three participants
performed sit-to-stand motions under ﬁve conditions and thenthe motions of 15 people were imaginarily prepared.

Fig. 1. Scatter plots of the ﬁrst and second principal motion scores of the PMA(top) and GPMA (bottom). The markers indicate the mean scores, whereas
the ellipses indicate the standard deviations. Here, PA, PB, and PC stand forparticipant A, B, and C, respectively. Condition 1: helthy; condition 2: lightpain on the left knee; condition 3: severe pain on the left knee; condition 4:
light pain on the right knee; condition 5: severe pain on the right knee. Theellipses of GPMA are invisible because they are extremely small.

Sit-to-stand motion is a full-body motion with upper bodyand lower limb motions. Therefore, the joint angles measured
in this experiment were trunk forward angles and hip ﬂexion
angles as upper-body angles, and knee ﬂexion angles andankle dorsiﬂexion angles as lower-limb angles. We deﬁned thebeginning and the end of the sit-to-stand motion as follows.
The beginning of the sit-to-stand motion was deﬁned as a point
when the trunk forward angular velocity reached 5 deg/s in
0.4 s. The end of the sit-to-stand motion was when either theleft or the right hip joint angle reached its maximum extension.
IV. R ESULTS
Fig. 1 shows the distribution of the ﬁrst and second principal
motion scores for PMA (top) and GPMA (bottom). Bothﬁgures show the distribution of ﬁrst and second principal
motion scores on the principal motion plane. The markers
in the ﬁgures represent the mean scores among 10 trials foreach combination of the condition and participant. The ellipses
represent their standard deviations.
We tested whether arbitrary pairs of 15 motions could bestatistically distinguished. Using the PMA scores, we could
statistically classify 85 of 105 (15 C2 ) pairs (MANOVA, p <0.05). On the other hand, using the GPMA scores, we could680

correctly classify all types of motion pairs. In the result of thePMA (top of Fig. 1), the ellipses of the standard deviations
were large, whereas in the result of the GPMA (bottom ofFig. 1), they were extremely small. GPMA decreased therepetition errors within individuals and identiﬁed the principalmotions, which increased the statistical differences betweenindividuals. GPMA exhibits a good ability to classify themotions of individuals.
V. C ONCLUSIONSWe proposed a generalized principal motion analysis(GPMA) method to improve the capability of the principal motion analysis (PMA) method. In the result of the performance
test of sit-to-stand motions recorded by a camera-based motion
capture system, GPMA correctly classiﬁed all the participants’
motions, whereas PMA correctly classiﬁed only 81 % of them.
ACKNOWLEDGEMENT
This study was in part supported by NEXT Kakenhi(#19K21584).
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